![2.3 Factor Analysis 7
2.3 Factor Analysis
Similar to the probabilistic PCA method, factor analysis concentrates on the latent
variables t whose distribution are Gaussian, and the original measurement variables
x are treated as linear combination of t plus small additive white noise e. The aim
of FA is to find the most probable parameter set = {P, } in the model structure,
which is described as follows (Bishop 2006)
x = Pt + e (2.13)
where P = [p1, p2, . . . , pl] ∈ Rm×l
is the loading matrix, just as that in PCA. The
variances matrix of measurement noise e is represented by = diag{λi}1,2,...,m,
in which different noise levels of measurement variables are assumed. If all the
λ are assumed to be of the same value, then FA is equivalent to PPCA. If all λi,
i = 1, 2, . . . , m are assumed to be zero, then FA becomes PCA. Therefore, FA is the
general description of Gaussian latent model structure. PPCA and PCA are just two
special cases of FA.
In the FA model, the latent variable t is assumed to be Gaussian distribution with
zero mean and unity variance, which is t ∈ N(0, I), where I is the unity matrix with
appropriate dimension. Precisely, the distribution of the latent variable t is
p(t) = (2π)−k/2
exp
−
1
2
tT
t
(2.14)
Then, the conditioned probability of the measured process variable x is given as
p(x|t) = (2π)−m/2
|e|−1/2
exp
−
1
2
(x − Pt)T
−1
e (x − Pt)
(2.15)
Based on Eqs. (2.14) and (2.15), the probability of x can be calculated as
p(x) =
p(x|t)p(t)dt = (2π)−m/2
|C|−1/2
exp
−
1
2
xT
C−1
x
(2.16)
where C = e +PPT
is the variance matrix of the process data, thus the distribution
of x can be represented as x ∈ N(0, e + PPT
), |C| is to calculate the discriminant
value for the C.
The parameter set = {P, e} can be estimated by the expectation and maxi-
mization (EM) algorithm, which is an iterative likelihood maximization algorithm.
EM is a widely used algorithm for parameter estimation due to its simplicity and ef-
ficiency. Besides, it can also handle incomplete datasets, which is also very attractive
to the application in process industry. The EM algorithm can be partitioned into two
steps: the E-step and the M-step. The maximum likelihood function of the process
data can be given as
L =
n
i=1
ln{p(xi, ti)} =
n
i=1
ln{p(ti|xi)p(xi)} (2.17)](https://image.slidesharecdn.com/2115521-250518162830-7756bfa1/85/Multivariate-Statistical-Process-Control-Process-Monitoring-Methods-And-Applications-1st-Edition-Zhiqiang-Ge-27-320.jpg)
![8 2 An Overview of Conventional MSPC Methods
where the posterior distribution of each latent variable can be calculated as
p(ti|xi) = (2π)−k/2
|M|−1/2
exp
−
1
2
(ti − Qxi)T
M−1
(ti − Qxi)
(2.18)
where Q = PT
(PPT
+ e)
−1
, M−1
= I − QP. In the E-step of the EM algorithm,
the expectation value of the likelihood function L is calculated as follows, which is
based on the first and second statistics of the latent variable
E(L) = −
n
i=1
1
2
xT
i −1
e xi − xT
i −1
e PE(ti|xi) +
1
2
tr[P−1
e PE(titT
i |xi)]
− (n/2) ln e + cont (2.19)
where E(·) and tr(·) are expectation and trace calculators, cont represents a con-
stant value, and the first and second statistics of the latent variables are given as
follows
t̂i = E{ti|xi} = Qxi (2.20)
E{titT
i |xi} = I − QP + QxixT
i QT
(2.21)
In the M-step of the EM algorithm, by maximizing the expectation value of the
likelihood function, taking the partial derivatives of L with respect to the parameter
set = {P, e}, and setting them to zero, the optimal values of the two parameters
can be determined, which are given as
P =
n
i=1
xit̂T
i
n
i=1
E{titT
i |xi}
−1
(2.22)
e
=
n
i=1
diag xixT
i − Pt̂ixT
i
n
(2.23)
By calculating the E-step and M-step iteratively, the final optimal parameters for the
FA model can be obtained.
For process monitoring, the T2
and SPE monitoring statistics based on the FA
model can be constructed as follows
T 2
i = t̂i
2
= xT
i QT
Qxi ≤ χ2
α,k (2.24)
êi = E{ei|xi} = (I − PQ)xi (2.25)
SPEi = −1/2
e ei
2
= xT
i (I − PQ)T
−1
e (I − PQ)xi ≤ χ2
α,m (2.26)](https://image.slidesharecdn.com/2115521-250518162830-7756bfa1/85/Multivariate-Statistical-Process-Control-Process-Monitoring-Methods-And-Applications-1st-Edition-Zhiqiang-Ge-28-320.jpg)
![2.4 Independent Component Analysis 9
2.4 Independent Component Analysis
Independent component analysis (ICA) is a statistical and computational technique
for revealing hidden factors that underlie sets of random variables, measurements, or
signals. ICA was originally proposed to solve the blind source separation prob-
lem. To introduce the ICA algorithm, it is assumed that m measured variables,
x(k) = [x1(k), x2(k), . . . , xm(k)] at sample k can be expressed as linear combina-
tions of r( ≤ m)unknown independent components [s1, s2, . . . , sr ]T
, the relationship
between them is given by Hyvarinen and Oja (2000)
XT
= AS + E (2.27)
where n is the number of measurements, X ∈ Rn×m
is the data matrix, A =
[a1, a2, . . . , ar ] ∈ Rm×r
is the mixing matrix, S = [s1, s2, . . . , sr ] ∈ Rr×n
is the
independent component matrix, and E ∈ Rl×n
is the residual matrix. The basic
problem of ICA is to estimate the original component S and the mixing matrix A
with X, therefore, the objective of ICA is to calculate a separating matrix W so that
the components of the reconstructed data matrix Ŝ become as independent of each
other as possible, given as
Ŝ = WXT
(2.28)
Before applying the ICA algorithm, the data matrix X should be whitened, in order
to eliminate the cross-correlations among random variables. One popular method
for whitening is to use the eigenvalue decomposition, considering x(k) with its
covariance Rx = E{x(k)x(k)T
}, the eigenvalue decomposition of Rx is given by
Rx = U UT
(2.29)
The whitening transformation is expressed as
z(k) = Qx(k) (2.30)
where Q = −1/2
UT
. One can easily verify that Rz = E{z(k)z(k)T
} is the identity
matrix under this transformation. After the whitening transformation, we have
z(k) = Qx(k) = QAs(k) = Bs(k) (2.31)
where B is an orthogonal matrix, giving the verification as follows
E{z(k)z(k)T
} = BE{s(k)s(k)T
}BT
= BBT
= I (2.32)
Therefore, we have reduced the problem of finding an arbitrary full-rank matrix A
to the simple problem of finding an orthogonal matrix B. Then, we can estimate as
follows
ŝ(k) = BT
z(k) = BT
Qx(k) (2.33)](https://image.slidesharecdn.com/2115521-250518162830-7756bfa1/85/Multivariate-Statistical-Process-Control-Process-Monitoring-Methods-And-Applications-1st-Edition-Zhiqiang-Ge-29-320.jpg)
![10 2 An Overview of Conventional MSPC Methods
From Eqs. (2.28) and (2.33), we can get the relation between W and B
W = BT
Q (2.34)
There are two classic measures of non-Gaussianity: kurtosis and negentropy (Hy-
varinen and Oja 2000). Kurtosis is the fourth-order cumulant of a random variable,
unfortunately, it is sensitive to outliers. On the other hand, negentropy is based on the
information-theoretic quantity of differential entropy. When we have obtained the
approximate forms for negentropy, Hyvarinen (1997) introduced a very simple and
efficient fixed-point algorithm for ICA. This algorithm calculates the independent
components one by one through iterative steps. After all vectors bi (i = 1, . . . , m)
have been calculated and put together to form the orthogonal mixing matrix B, then
we can obtain ŝ(k) and demixing matrix W form Eqs. (2.33) and (2.34), respectively.
Dimension reduction of ICA is based on the idea that these measured variables
are the mixture of some independent component variables. The performance and
interpretation of ICA monitoring depend on the correct choice of the ordering and
dimension of the ICA model. Because negentropy is used for measurement of non-
Gaussianity, we can select suitable number of ICs by checking their negentropy
values. If the negentropy value of current IC is zero or approximately zero, it indicates
that the non-Gaussian information of the process has already been extracted, and
the rest of the process is Gaussian, which can be analyzed by conventional MSPC
methods such as PCA.
Based on the identified ICA model, two monitoring statistics can be constructed
for monitoring, which are defined as I2
and SPE, given as (Lee et al. 2004; Ge and
Song 2007)
I2
= sT
s (2.35)
SPE = eT
e (2.36)
where s and e are independent components and residuals, respectively.
2.5 Kernal Principal Component Analysis
As a simple linear transformation technique, PCA compresses high-dimensional
data into low dimension with minimum loss of data information. When the algo-
rithm is carried out in the feature space, KPCA is obtained. This algorithm was
originally proposed by Schölkopf et al. (1998). Suppose the original training dataset
is x1, x2, . . . , xn ∈ Rm
, where n is the sample number and m is the number of
process variables. The feature space is constructed by using a nonlinear mapping:
Rm (·)
−
−
→ Fh
, where ( · ) is a nonlinear mapping function and h is the dimension in
feature space, which is assumed to be a large value. Similar to PCA, the covariance
matrix in the feature space F can be calculated as
F
=
1
n
n
i=1
[(xi) − c][(xi) − c]T
(2.37)](https://image.slidesharecdn.com/2115521-250518162830-7756bfa1/85/Multivariate-Statistical-Process-Control-Process-Monitoring-Methods-And-Applications-1st-Edition-Zhiqiang-Ge-30-320.jpg)
![2.6 Conclusion 11
where c is the sample mean in the feature space. Conveniently, denote ¯
(xi) =
(xi) − c as the centered feature space sample. The kernel principal component can
be obtained by the eigenvalue problem below
λv =
F
v =
1
n
n
i=1
[ ¯
(xi)T
v] ¯
(xi) (2.38)
v =
n
i=1
αi ¯
(xi) (2.39)
where λ and v denote eigenvalue and eigenvector of the covariance matrix F ,
respectively. The αi is the coefficient which indicated that the kernel principal com-
ponent is spanned by feature space training samples. Multiplying ¯
(xj ) with both
sides of Eq. (2.38), the following equation is obtained
λ[ ¯
(xj )v]= ¯
(xj )Fv (2.40)
The problem can be transformed to Eq. (2.41) by introducing a kernel matrix K̄ij =
¯
(xi) · ¯
T
(xj ), where the kernel matrix is centered, α should be scaled as α2
=
1/nλ to ensure the normality of v (Choi et al. 2005)
λα =
1
n
¯
Kα (2.41)
The score vector of qth observation is calculated as
tq,k = [vk · ¯
(xq)] =
n
i=1
αk
i [ ¯
(xq) ¯
T
(xi)] =
n
i=1
αk
i K̄qi (2.42)
Then, T2
and SPE statistics can be developed for process monitoring.
2.6 Conclusion
In this chapter, several typical multivariate statistical process control methods have
been introduced, including PCA, PLS, ICA, FA, and KPCA. Based on these basic
MSPC methods, more specific and complex process monioring methods can be
further developed, some of which will be introduced in the following chapters.](https://image.slidesharecdn.com/2115521-250518162830-7756bfa1/85/Multivariate-Statistical-Process-Control-Process-Monitoring-Methods-And-Applications-1st-Edition-Zhiqiang-Ge-31-320.jpg)
![14 3 Non-Gaussian Process Monitoring
It is also important to note that all the reported ICA-based monitoring methods
are based on the assumption that the essential variables which drive the process are
non-Gaussian and discarded Gaussian parts are utilized to set up a single squared
prediction error (SPE) monitoring chart. Actually, complex multivariate processes
are always driven by non-Gaussian and Gaussian essential variables simultaneously.
Separating the Gaussian information from the un-modelled Gaussian uncertainty
will facilitate the diagnosis of faults which occur in different sources. Therefore, a
two-step ICA-PCA information extraction strategy has been proposed and used for
process monitoring.
Due to complicated confidence limit determination of the traditional ICA-based
monitoring method, which is based on the kernel density estimation (KDE), support
vector data description (SVDD) has been introduced to improve the determination
of confidence limits for the non-Gaussian components (Tax and Duin 1999, 2004).
Compared to KDE (Lee et al. 2004), the SVDD-based method is more computation-
ally efficient, which relies on a quadratic programming cost function. To monitor
discarded Gaussian parts of the process information, PCA might be the straight-
forward but not the optimal approach in which the projected directions have the
maximum variations irrespective of the underlying information generating structure.
More precisely, the process recordings always consist of inevitable measurement
noises, while PCA is not capable of characterizing the different influences from
common process Gaussian impelling factors and random variations due to measure-
ment equipments. A probabilistic generative model (probabilistic PCA, PPCA) was
employed to address this issue via the expectation and maximization (EM) algorithm
(Kim and Lee 2003). However, PPCA assumes the same noise level of all measure-
ment variables, which cannot be easily satisfied in practice. To this end, a general
extension of PPCA model, factor analysis (FA) can model different noise levels of
measurement variables. To determine the number of essential variables, lots of use-
ful approaches have been reported, including Akaike Information Criterion (AIC),
Bayesian Information Criterion (BIC), SCREE test, PRESS cross validation test
(Chiang et al. 2001) and more recent fault detection criteria (Valle et al. 1999; Wang
et al. 2002, 2004).
3.2 Two-step ICA-PCA Information Extraction Strategy
for Process Monitoring
3.2.1 Process Monitoring Method
Giving the data matrix X, assume r independent components (ICs) are extracted,
S = [s1, s2, . . . , sr ] ∈ Rr×n
. In order to monitor the non-Gaussian part of the process,
a new statistic variable was defined (Lee et al. 2004; Ge and Song 2007)
I2
= sT
s (3.1)](https://image.slidesharecdn.com/2115521-250518162830-7756bfa1/85/Multivariate-Statistical-Process-Control-Process-Monitoring-Methods-And-Applications-1st-Edition-Zhiqiang-Ge-33-320.jpg)
![22 3 Non-Gaussian Process Monitoring
Assume a dataset containing n samples ŝi ∈ Rr
, i = 1, 2, . . . , n is given, the
mapping from the original space to the feature space : ŝ → F can be simply
done by a given kernel K(si, sj ) =
(si), (ŝj )
, which computes the inner product
in the feature space. Here, the most popular Gaussian kernel is used. To construct
the minimum volume of the hypersphere, SVDD solves the following optimization
problem
min
R,a,ξ
R2
+ C
n
i=1
ξi
s.t. (ŝi) − a
2
≤ R2
+ ξi, ξi ≥ 0 (3.7)
where a is the center of the hypersphere in the feature space. The variable C gives
the trade-off between the volume of the sphere and the number of errors (number of
target objects rejected). ξi represents the slack variable which allows the probability
that some of the training samples can be wrongly classified. The dual form of the
optimization problem can be obtained as follows
min
αi
n
i=1
αiK(ŝi, ŝj ) −
n
i=1
n
j=1
αiαj K(ŝi, ŝj )
s.t. 0 ≤ αi ≤ C,
n
i=1
αi = 1 (3.8)
where αi is Lagrange multipliers. After the hpyersphere in the feature space has
been constructed, the hypothesis that a new sample ŝnew is normal is accepted if the
distance d[(ŝnew)] = (ŝnew) − a ≤ R, which is given by
d[(ŝnew)] =
K(ŝnew, ŝnew) − 2
n
i=1
αiK(ŝi, ŝnew) +
n
i=1
n
j=1
αiαj K(ŝi, ŝj ) (3.9)
3.3.2 Process Monitoring Based on ICA-FA and SVDD
As demonstrated, conventional statistical methods are under the assumption that pro-
cesses are driven by fewer essential variables. These essential variables are either
non-Gaussian or Gaussian distribution. However, under more general circumstances,
processes are driven by non-Gaussian and Gaussian essential variables simulta-
neously, which formulates the key assumption of the mixed essential component
analysis (ICA-FA) method. Figure 3.4 shows the difference between ICA-FA and
the conventional methods (ICA and FA). The relationship of measurement variables
and mixed essential variables is given as follows (Ge et al. 2009a)
x = x1 + x2 = As + Pt + e (3.10)](https://image.slidesharecdn.com/2115521-250518162830-7756bfa1/85/Multivariate-Statistical-Process-Control-Process-Monitoring-Methods-And-Applications-1st-Edition-Zhiqiang-Ge-41-320.jpg)
![3.3 Process Monitoring Based on ICA-FA and SVDD 23
...
...
s2
s1
t1
t2
sr
e
...
s1
...
e
ICA-FA ICA FA
t1
Process
records
Process
records
Process
records
s2
sr
t2
t1
t1
Fig. 3.4 Description of ICA-FA versus ICA and FA
where x1 and x2 represent non-Gaussian and Gaussian parts of the process, respec-
tively, A is the mixing matrix of non-Gaussian essential variable s, P is the loading
matrix of Gaussian essential variables t and e is the Gaussian un-modelled uncer-
tainty, relating to measurement noises, etc., with zero means and different variances
=diag{λi }1, 2,... , m, which permits measurement variables to have different noise levels.
As a general form of PPCA, FA performs better than PCA and PPCA. Since pro-
cess measurements are always corrupted by noise, the ignorance of such information
generation structure will make PCA method leave some essential information in the
discarded space and trigger monitoring errors. In contrast, the noise structure is con-
sidered in PPCA and FA; thus, the useful information can be correctly extracted and
monitored separately.
To estimate the new parameter set = {A, P, } and extract essential variables
EA = {s, t} from measurement variables x, a two-step estimation and extraction
strategy can be developed. First, non-Gaussian essential variables can be extracted
by the efficient FastICA or PSO-based method (Xie and Wu 2006), thus the mixing
matrix A is obtained. Second, the parameter P and can be estimated by the EM
algorithm, which is described as follows. First, the mean and variance of the Gaussian
part of measurement variables x2 can be calculated as
μx2
= PE[t] + E[e] = 0;
x2
= PE[ttT
]PT
+ E[eeT
] = PPT
+
(3.11)
Hence, the distribution of x2 is p(x2) = N{0, PPT
+}. EM is an iterative algorithm,
it consists of two steps: E-step and M-step.
In E-step, two statistics of the Gaussian essential variable t are estimated
t̂i = E[ti|x2i] = Qx2i
E[titT
i |x2i] = I − QP + Qx2i xT
2i
QT
(3.12)
where Q = PT
(PPT
+ )
−1
.
In M-step, the parameters P and are estimated so that the value of the
likelihood function is maximized, the estimated parameters are calculated as](https://image.slidesharecdn.com/2115521-250518162830-7756bfa1/85/Multivariate-Statistical-Process-Control-Process-Monitoring-Methods-And-Applications-1st-Edition-Zhiqiang-Ge-42-320.jpg)
![24 3 Non-Gaussian Process Monitoring
P =
n
i=1
x2i t̂T
i
n
i=1
E[titT
i |x2i]
−1
=
n
i=1
diag(x2i xT
2i
− Pt̂ixT
2i
)
n
(3.13)
where n is the number of samples, diag(Z) means to make a diagonal matrix using
diagonal elements of matrix Z. Equations (3.12) and (3.13) are calculated iteratively
until both of the parameters (P and ) are converged.
In summary, when the reference dataset X = {xi}i=1, 2,..., n is available, with the
ICA-FA model defined by Eq. (3.10), process information can be separated into three
different parts. The systematic information includes non-Gaussian and Gaussian
parts which are driven by their corresponding essential variables, whereas the noise
information and other unmeasured disturbances are represented by e.
As described in the previous section, the entire information of the process is par-
titioned into three parts: non-Gaussian systematic information, Gaussian systematic
information and noise information. Therefore, three different monitoring statistics
can be developed for fault detection. First, for monitoring non-Gaussian informa-
tion, SVDD-based method is used. In contrast to the confidence limit determination
method KDE, the SVDD reduces the complexity to quadratic programming prob-
lem that produces a unique analytical solution. Given the training dataset S (the
extracted independent components), the hypersphere is constructed. The center a
and the radius R can be determined by (Tax and Duin 2004)
a =
n
i=1
αi(ŝi)
R =
1 − 2
n
i=1
αiK(ŝz, ŝj ) +
n
i=1
n
j=1
αiαj K(ŝi, ŝj)
(3.14)
The non-Gaussian statistic (NGS) can be developed as the distance between the data
sample and the center of the hypersphere, thus
NGSi = d2
[(ŝi)] = (ŝi) − a
2
≤ NGSlim = R2
(3.15)
where ŝ i is the extracted independent component of the new sample, d[(ŝi)] is given
in Eq. (3.9). If NGSi ≤ R2
, the non-Gaussian information is regarded as normal,
else, it will be regarded as an outlier or some fault and disturbance has happened.
Since the distribution of Gaussian essential variables t is assumed to be N{0, I},
the Mahalanobis norm of t is identical to the Euclidian norm. Different from the
projection method, the Gaussian essential variables cannot be calculated directly.
However, it can be substituted by their estimation, which is given in Eq. (3.12).](https://image.slidesharecdn.com/2115521-250518162830-7756bfa1/85/Multivariate-Statistical-Process-Control-Process-Monitoring-Methods-And-Applications-1st-Edition-Zhiqiang-Ge-43-320.jpg)
![3.3 Process Monitoring Based on ICA-FA and SVDD 25
It is known that the squared Mahalanobis norm of t follows χ2
distribution, the
confidence limit of T2
statistic can be determined as follows
T 2
i = t̂i
2
= xT
2i
QT
Qx2i ≤ χ2
α,l (3.16)
where χ2
α,l represent the confidence limit with α confidence and l is the corresponding
free parameter. If the value of T2
statistic goes beyond the confidence limit, the
Gaussian systematic process is regarded out-of-control.
Similarly, the noise part of the process information can also be monitored by the
squared Mahalanobis distance. Hence, the Gaussian information matrix x2 can be
represented as
x2 = Pt + e (3.17)
Notice Eqs. (3.12) and (3.17), the noise information e can be estimated as follows
êi = E[ei|x2i] = (I − PQ)x2i (3.18)
Then, the confidence limit of SPE statistic can be determined by
SPEi = −1/2
ei
2
= xT
2i
(I − PQ)T
−1
(I − PQ)x2i ≤ χ2
α,m (3.19)
where χ2
α,m represent the confidence limit with α confidence and m is the correspond-
ing free parameter. If the value of SPE statistic goes beyond the confidence limit,
the noise part of the process is regarded abnormal. This may be due to the change in
noise level, some unmeasured disturbances and other process structure changes.
Note that these two Gaussian parts of information both follow the χ2
distribution;
they can be monitored together. According to Eq. (3.11), the entire Gaussian statistic
(GS) can be constructed as follows
GSi = (PPT
+ )
−1/2
x2i
2
= xT
2i
(PPT
+ )
−1
x2i ≤ χ2
α,m (3.20)
Therefore, if changes happen in Gaussian part, the entire monitored GS should go
beyond its confidence limit. However, if one wants to isolate the change between the
systematic Gaussian part and the noise part, T2
and SPE statistics should be chosen.
In summary, four statistics (NGS, T2
, SPE and GS) have been developed for
process monitoring. By monitoring NGS and GS, the change between non-Gaussian
and Gaussian information can be isolated. Furthermore, the change of Gaussian
information can be isolated by monitoring T2
and SPE separately. Hence, three types
of changes can be monitored separately by their corresponding statistics. Change of
any type or types of information will be reflected in related monitoring charts.
3.3.3 Case Study
In this section, the performance of the method is tested through the TE process. The
variables selected for monitoring are the same as those used for control structure](https://image.slidesharecdn.com/2115521-250518162830-7756bfa1/85/Multivariate-Statistical-Process-Control-Process-Monitoring-Methods-And-Applications-1st-Edition-Zhiqiang-Ge-44-320.jpg)
![4.2 Fault Reconstruction 31
the corresponding fault direction j
zj = z − j fj (4.1)
where fj is a vector storing the estimated fault magnitude such that zj is the closest
to the normal region. Traditionally, the optimal reconstruction is obtained by min-
imizing ||zj − z∗
j ||. For this method, the reconstruction is realized by moving the
corrupted value (ŝ) along the defined fault direction j into the SVDD feature
space. The reconstruction technique therefore leads to the following problem
fj = arg min
f
||(ŝ∗
) − a|| = arg min
f
||(ŝ − Wj f) − a|| ŝ = Wz ŝ∗
= Wz∗
(4.2)
However, the optimization problem of Eq. (4.2) is problematic, since the formulation
of the mapping function ( · ) is unknown. To overcome this deficiency, the iterative
denoising and reconstruction approach developed for kernel PCA (Mika et al. 1998;
TakahashiandKurita2002)canbeinsertedintoEq. (4.2).WithN nsupportvectors,
obtained by the SVDD method, the length of (ŝ) − a is (Tax and Duin 2004)
||(ŝ) − a|| =
K(ŝ, ŝ) − 2
N
i=1
αiK(svi, ŝ) +
N
i=1
N
j=1
αiαj K(svi, svj ) (4.3)
where ŝvi is the ith support vector obtained by the SVDD method and αi is
the corresponding coefficient. As this work utilizes the Gaussian kernel function
K(ŝi, ŝj ) = exp (−||ŝi − ŝj ||2
/δ), K(ŝ, ŝ) = 1, and N
i=1
N
j=1 αiαj K(ŝvi, ŝvj ) is a
constant. By incorporating Eq. (4.3), the optimization problem in Eq. (4.2) finally
becomes
fj = arg max
f
N
i=1
αiK(svi, s∗
j ) = arg max
f
N
i=1
αiK(svi, ŝ − Wj f) (4.4)
Denoting =
N
i=1
αiK(svi, Wj f) and taking the first partial with respect to f yields
∇f =
∂
∂f
= 0 (4.5)
which for Gaussian kernel functions is equal to
∇f = [Wj ]T
N
i=1
αiK(svi, ŝ − Wj f)(ŝ − Wj f − svi) = 0 (4.6)
which can be simplified to
∇f = [Wj ]T
Wj
N
i=1
αiK(svi, ŝ − Wj f)f +
[Wj ]T
N
i=1
αiK(svi, ŝ − Wj f)(svi − ŝ) = 0. (4.7)](https://image.slidesharecdn.com/2115521-250518162830-7756bfa1/85/Multivariate-Statistical-Process-Control-Process-Monitoring-Methods-And-Applications-1st-Edition-Zhiqiang-Ge-49-320.jpg)
![32 4 Fault Reconstruction and Identification
By rewriting Eq. (4.7), the unknown fault vector f can be estimated as follows
f = [j
T
WT
Wj ]
−1
j
T
WT
N
i=1
αiK(svi, ŝ − Wj f)(ŝ − svi)
N
i=1
αiK(svi, ŝ − Wj f)
(4.8)
Appendix shows that the estimated fault vector fj can be found iteratively
f(t + 1) = [j
T
WT
Wj ]
−1
j
T
WT
N
i=1
αiK(svi, ŝ − Wj f(t))(ŝ − svi)
N
i=1
αiK(svi, ŝ − Wj f(t))
(4.9)
such that
z∗
j = z − j f∗
j (4.10)
4.2.2 Fault Diagnosis
It is important to assess the impact of reconstructing each fault condition {Fj ,
j = 1, 2, . . . , J}. After reconstructing each of the possible fault conditions, the
correct fault direction will allow the reconstruction procedure to shift the recon-
structed sample z∗
j closest to the center of the SVDD sphere in the feature space.
This, in turn, implies that the value of the NSGj reduces significantly and shows
an in-statistical-control situation if Fj is the correct fault condition. In contrast, if
an incorrect direction is applied {Fi, i = 1, 2, . . . , J, i = j}, the univariate NGSi
statistic still violates its control limit. Calculating the non-Gaussian statistic value
NGSj by using Eqs. (4.11) and (4.13)
ŝ∗
j = Wẑ∗
j (4.11)
(ŝ∗
j ) − a =
K(ŝ∗
j , ŝ∗
j ) − 2
N
i=1
αiK(svi, ŝ∗
j ) +
N
i=1
N
j=1
αiαj K(svi, svj )
(4.12)
NGSj = (ŝ∗
j ) − a
2
2
(4.13)
the impact of the reconstruction procedure upon the jth fault direction therefore relies
on ξj
ξj =
NGSj
NGSlim
(4.14)](https://image.slidesharecdn.com/2115521-250518162830-7756bfa1/85/Multivariate-Statistical-Process-Control-Process-Monitoring-Methods-And-Applications-1st-Edition-Zhiqiang-Ge-50-320.jpg)
![4.2 Fault Reconstruction 33
which is referred to here as the fault diagnosis index and NGSlim is the control limit of
the NGS statistic. If the reconstruction procedure has been applied using the correct
fault direction, Eq. (4.14) highlights that a fault diagnosis index produces a value that
is smaller than or equal to 1. This, in turn, implies that the faulty sample has been
shifted along the correct fault direction to fall within the normal region. However, if
an incorrect fault direction is applied this value still exceeds 1.
4.2.3 Simulation Example
The six process variables are simulated as linear combinations of a total of
three source variables. In order to account for measurement uncertainty, normally
distributed measurement noise was superimposed onto this matrix vector product
z = As + e. (4.15)
The linear combinations are governed by the mixing matrix A, which was randomly
selected to be
A =
⎡
⎣
0.815 0.906 0.127 0.913 0.632 0.098
0.279 0.547 0.958 0.965 0.158 0.971
0.957 0.485 0.800 0.142 0.422 0.916
⎤
⎦
T
. (4.16)
Thes∈R3
isavectorofsourcesignaleachofwhichfollowsauniformdistributionwith
[0 1] and e are Gaussian distributed i.i.d. sequences of zero mean and variance 0.01,
e ∼ N {0,0.01I}. To test the performance of the fault reconstruction and diagnosis
method, a total of 1,000 samples were simulated. The first 500 samples served as
reference data, whilst the remaining 500 samples were used for model validation
and to inject fault conditions for studying the performance of the fault reconstruction
scheme (test dataset). Selecting the parameters of the Gaussian kernel functions of
the SVDD as C = 0.08 and σ = 6.5 produced the control limit of the NSG statistic
for a significance level of 5 %.
The first fault condition was a bias of magnitude 2 injected to the second sensor
in the test dataset after 250th sample. The conventional ICA-SVDD (Liu et al. 2008)
method could successfully detect this fault. Next, applying the fault identification and
isolation method and ICA contribution plots produced the plots shown in Fig. 4.1.
As discussed in the Sect. 4.2.2, values below or equal to 1 imply this fault condition
is responsible for the detected abnormal event. Values above 1 indicate that this fault
condition cannot be considered as a potential root cause. Given the way in which the
fault was injected the fault identification index for reconstructing along the direction
of variable 2 should be identified as significant, which plot a confirms. However, the
identification results of the contribution plot (ICA) in Fig. 4.1, plots 4.1b and 4.1c,
show that most variables show a significant response to this event for the I2
cont,j and
variables 4 and 5 for SPEcont, j.](https://image.slidesharecdn.com/2115521-250518162830-7756bfa1/85/Multivariate-Statistical-Process-Control-Process-Monitoring-Methods-And-Applications-1st-Edition-Zhiqiang-Ge-51-320.jpg)
![4.2 Fault Reconstruction 35
To analyze this misleading result in more detail, the way in which the variable
contributions are generated need to be reexamined
I2
cont,j ∝ eT
j Q†
B
ˆ
s = eT
j Q†
BBT
Qz = eT
j z. (4.17)
If m* = m and d = m, it follows that = I otherwise reduces to a matrix of rank
min{m*,d}. The variable contributions to the SPE statistic are given by
SPEcont,j =
eT
j [I−ABT
Q]z
2
=
eT
j z
2
. (4.18)
If we now define z by z0 + z where z0 is the measured vector without the impact
of the fault condition and z is a vector that describes the fault impact, Eqs. (4.17)
and (4.18) give rise to
I2
cont,j ∝ eT
j (z0 + z) = I2
cont,j + I2
cont,j
SPE2
cont,j = (eT
j (z0 + z))2
(4.19)
= SPE2
cont,j + SPE0cont,j + 2
SPE0cont,j SPEcont,j .
This separation shows that the fault contributions to the ICA-I2
and -SPE statistics
depend on eT
j z and eT
j z, respectively. Assuming that the ith sensor is faulty,
the contribution of this sensor bias upon the residual variables ej is as follows
I2
cont,j = ωT
j z =
m
i=1
ωijzi SPEcont,j = γ T
j z =
m
i=1
γ ijzi (4.20)
where zi is the magnitude of the sensor bias, ω
ω
ωT
j and γ T
j is the jth row vector of
and whose ith elements are ij and γ ij, respectively. As no assumptions can
generally be imposed on and with respect to its symmetry etc., there is no
guarantee that the residual associated with the faulty sensor is the largest one. The
above equation also suggests the possibility that variable contributions which, by
default, should be insignificant may, in fact, be significant and indicate that these
variables are affected by the fault, which this simulation example confirms. For
d = m = m*, Eqs. (4.19) and (4.20) suggest that I2
cont, j = zj , which produces a
correct variable contribution to the fault condition.
With the application of the reconstruction technique, a different picture emerges.
A total of six fault directions, one for each sensor, were considered. Figure 4.1a
shows the fault reconstruction index for each of the examined fault conditions. As
expected, a bias for the second sensor yielded the smallest value and below 1 and
hence, according to Eq. (4.14) the most significant impact of the reconstruction
approach is for this sensor. In contrast, reconstructing the other fault directions
(sensor faults) had an insignificant impact, leaving the fault reconstruction indices
to exceed 1. It should be noted that plots a–c represent the average values for the
250 faulty samples. For illustrative purposes, Fig. 4.1 also shows the reconstruction
indices and the variable contributions for the first ten samples in plots d–f.](https://image.slidesharecdn.com/2115521-250518162830-7756bfa1/85/Multivariate-Statistical-Process-Control-Process-Monitoring-Methods-And-Applications-1st-Edition-Zhiqiang-Ge-53-320.jpg)
![rare and correspondingly infamous. In any country where there is
abundance of good, free land the phenomenon of interest on money
will disappear, provided labor is free. So it disappeared in the
northern states of this Union in the later part of the 18th century.
These phenomena caused the southerners to adopt slavery though
all their English traditions had declared it immoral for more than
three centuries. The relation of interest to slavery under a condition
of free land is the relation of cause and effect, i.e., the requirement
of interest will produce slavery and the abolition of interest will
abolish slavery.[26] These social phenomena are of importance in our
consideration of the early Christian doctrine of interest. That doctrine
was largely evaded and disobeyed but it still had great effect and
that effect was toward the abolition of slavery. We do not mean that
this economic doctrine alone resulted in the abolition of slavery, or
even that it was a chief cause in the abolition of slavery, it was not
obeyed well enough to be such a chief cause; but so far as it was
obeyed, it tended in that direction.
The net result of all Christian teaching together was to prolong the
existence of the institution of slavery for two centuries, perhaps for
three. The doctrine of the sinfulness of interest however, worked
toward emancipation and forced slavery in its later end to become
almost wholly agricultural, i.e., to yield income as rent. Slaves cannot
be employed in commerce or industry in sufficient numbers to be
profitable where the institution of interest is banned as it was in the
'dark ages.' The Christian concept of interest undermined ancient
civilization by abrogating, slowly but surely, the institution of
property by which such gangs of 'manufacturing slaves' as made the
fortune of Crassus, could alone be made profitable. It is an historical
curiosity that it accomplished this result without any attack on the
institution of slavery itself.
As soon as Christian doctrines became widespread enough to
produce important social results we find Christian slave owners
manumitting their slaves in considerable numbers. It is no
derogation to the influence of the doctrine of human brotherhood or](https://image.slidesharecdn.com/2115521-250518162830-7756bfa1/85/Multivariate-Statistical-Process-Control-Process-Monitoring-Methods-And-Applications-1st-Edition-Zhiqiang-Ge-57-320.jpg)
![to the humanity of the Christian slave owners to mention the fact
that the doctrine of the sinfulness of interest, by tending to make
slavery unprofitable, aided in the process of bringing to light the real
content of the doctrine of human brotherhood, and of making the
humane practice of manumission easier by the removal of certain
economic impediments.
In order to understand properly the working of the prohibition of
interest and its relation to manumission, it is necessary to carry the
analysis one step farther to its ultimate physical basis, which was the
conditioning factor of actual practice and eventually of theory also.
The exhaustion of the soil of western Europe which was the result of
ancient methods of agriculture, together with the rising standard of
living and the competition of other more fertile agricultural regions
like Egypt and North Africa resulted in the substitution of the
latifundi for small landholdings.[27] As the pressure continued the
latifundi in turn became economically unprofitable under forced labor
(slavery) and large tracts of land were abandoned. In order to put
this land under agriculture again the charge upon it had to be
reduced by the substitution of (relatively) free associated labor,
villeinage or serfdom. But this change cut off the economic margin
upon which the structure of ancient civilization was built and is the
ultimate economic reason assignable for the fall of Rome. Of course
the collapse of the empire could, theoretically, have been avoided
had the Romans of the first three centuries A.D. been content to live
the toilsome and frugal life of the Romans of the early republic. But
this was an utter impossibility in practice. This slowly working and
hardly understood decline in the relative and actual ability of ancient
agriculture to sustain the weight imposed upon it, enables us to see
why the sinfulness of interest could be steadily indoctrined even
though steadily evaded, by Christians from the beginning, while
manumission was not taught at all in the beginning and only worked
up to the dignity of a pious action relatively late.[28] It also explains
why manumission of household and personal slaves preceded that of
agricultural slaves. Of course there is nothing peculiarly Christian](https://image.slidesharecdn.com/2115521-250518162830-7756bfa1/85/Multivariate-Statistical-Process-Control-Process-Monitoring-Methods-And-Applications-1st-Edition-Zhiqiang-Ge-58-320.jpg)
![about this later phenomenon and the operation of other causes is
discernable, but it is important for our purpose to observe that
Christian practice, and Christian theory in property matters in the
long run, followed the broad lines of the underlying economic
evolution.[29] The application of this to the origin of Christian
monasticism and to the revival of communistic theories by the later
Church fathers lies at the very outside limit of our study but will be
briefly touched on after we have considered the final overthrow of
the communistic property concept as they appear in the earlier
fathers up to and including Tertullian.
Clement of Alexandria 153-217 A.D. has the distinction of being the
first Christian theological writer who clearly expounds the concept of
private property which has held sway without substantial change in
the Church until the present time. This statement does not apply to
the doctrine of receiving interest on money. In respect to this
doctrine Clement is in perfect accord with all other early Christians
both before and after himself. Indeed he specifically states that the
Mosaic prohibition against taking interest from one's brother extends
in the case of a Christian to all mankind. But in regard to all other
property institutions Clement's attitude is essentially that of any
modern Christian of generous disposition.
In all that Clement has to say about property, and the 'bulk' of his
'property passages' is as great as that of all previous Christian
writers together, he speaks like a man on the defensive. Indeed
there has come down to us no other Christian writing earlier than his
time which presents his view, with the dubious exception of some
passages in Hermas. The fact seems to be that while Clement is
undoubtedly presenting an apologetic for the existing practice in the
Church of his day, that practice was felt to be more or less open to
attack in the light of certain scripture passages. Communism as an
existential reality was gone by the time of Clement—whatever may
have been the extent—probably a limited one—to which it had
existed in the earlier ages. But while communism as a fact was
dead, communism as an idea or ideal of Christian economy was not](https://image.slidesharecdn.com/2115521-250518162830-7756bfa1/85/Multivariate-Statistical-Process-Control-Process-Monitoring-Methods-And-Applications-1st-Edition-Zhiqiang-Ge-59-320.jpg)
![dead. Indeed Clement's views about the morality of wealth were so
different from those of previous writers that a great modern
economist[30] in treating of this subject ventures the opinion, though
doubtfully, that the reason why Clement, alone among the great
early theologians, was never canonized by the Church was that he
ran counter to popular belief on this subject. This opinion is probably
erroneous. Clement's theological opinions have a semi-Gnostic tinge
quite sufficient to explain the absence of his name from the calendar
of saints.
Clement justifies the institution of private property. He justifies, on
the highest ethical and philosophical principles, the possession by
Christians of even the most enormous wealth. His apologetic is not
an original one. He borrows it bodily from Plato. Indeed he quotes
Plato verbatim, invocation to Pan and the other heathen gods
included.[31] The originality lies in applying this Platonic doctrine to
the exposition of Christian scripture. Clement's method is strictly that
of Biblical exegesis. In the well known sermon or essay on: Who is
the Rich Man that shall be saved he takes up practically all of the
scriptural passages which seem opposed to the institutions of private
property and explains them in so modern a spirit that the whole
sermon might be delivered today in any ordinary Church and would
be readily accepted as sound and reliable doctrine. His thesis is that
wealth or poverty are matters in themselves indifferent. That riches
are not to be bodily gotten rid of, but are to be wisely conserved and
treated as a stewardship intrusted to the owner by God. That charity
to the poor should be in proportion to one's wealth and that a right
use of wealth will secure salvation to the upright Christian even
though he possesses great riches all his life and leaves them to his
heirs. The wealth that is dangerous to the soul is not physical
possessions, but spiritual qualities of greed and avarice.
His views can be best expressed by himself. We give two
characteristic passages from the sermon above referred to.[32] Rich
men that shall with difficulty enter into the kingdom, is to be
apprehended in a scholarly way, not awkwardly, or rustically, or](https://image.slidesharecdn.com/2115521-250518162830-7756bfa1/85/Multivariate-Statistical-Process-Control-Process-Monitoring-Methods-And-Applications-1st-Edition-Zhiqiang-Ge-60-320.jpg)
![carnally. For if the expression is used thus, salvation does not
depend upon external things, whether they be many or few, small or
great, or illustrious or obscure or esteemed or disesteemed; but on
the virtue of the soul, on faith and hope and love and brotherliness,
and knowledge, and meekness and humility and truth the reward of
which is salvation. Sell thy possessions. What is this? He does not,
as some off hand conceive, bid him throw away the substance he
possesses and abandon his property; but he bids him banish from
his soul his notions about wealth, his excitement and morbid feeling
about it, the anxieties, which are the thorns of existence which
choke the seed of life. And what peculiar thing is it that the new
creature, the Son of God intimates and teaches? It is not the
outward act which others have done, but something else indicated
by it, greater, more godlike, more perfect, the stripping off of the
passions from the soul itself and from the disposition, and the
cutting up by the roots and casting out of what is alien to the mind.
One, after ridding himself of the burden of wealth, may none the
less have still the lust and desire for money innate and living; and
may have abandoned the use of it, but being at once destitute of
and desiring what he spent may doubly grieve both on account of
the absence of attendance and the presence of regret.[33]
We have now come to the beginning of what is in many respects the
most interesting period in the history of property concepts. It is a
period in which everything is upside down and wrong end to. In that
strange age we find a famous archbishop, one of the world's noblest
orators, a man of the most spotless integrity and the most saintly
life, publicly preaching in the foremost pulpit of Christendom
doctrines of property, the implications of which, the most hardened
criminal would scarcely venture to breathe to a gang of thieves.[34]
We find the most learned scholar of the century, in the weightiest
expositions of Christian Scripture, penning the most powerful
apologetic of anarchy that is to be found in the literature of the
world.[35] We find one of the greatest of the popes, a man whose
genius as a statesman will go down to the latest ages of history,](https://image.slidesharecdn.com/2115521-250518162830-7756bfa1/85/Multivariate-Statistical-Process-Control-Process-Monitoring-Methods-And-Applications-1st-Edition-Zhiqiang-Ge-61-320.jpg)
![setting forth in a manual for the instruction of Christian bishops,
property concepts more radical than those of the fiercest Jacobins in
the bloodiest period of the Terror.[36]
Stranger still, these incredible performances are the strongest proofs
of the wisdom and piety of the men responsible for them. These
men are today honored as the saviors of civilized religion and their
images in bronze and marble and painted glass adorn the proudest
temples of the most conservative denominations of Christians. The
strange history of these famous men: Athanasius, the two Gregories,
Basil and Chrysostom in the East; Augustine, Ambrose, Jerome and
Gregory in the West, lies outside the limits of our study. But the
explanation of their desperate and uncompromising communism can
be given in a word. It was the communism of crisis: the communism
of shipwrecked sailors forced to trust their lives to a frail lifeboat
with an insufficient supply of provisions. These great Christian
scholars, enriched by all the accumulated culture of their civilization,
saw that culture falling into ruin all around them; they felt the
foundations of that civilization trembling beneath their feet. To vary
the figure, they beheld the rising tide of ignorance and barbarism
rapidly engulfing the world and with desperate haste they set to
work rebuilding and strengthening the ark of the Church that in it,
religion, and so much of civilization as possible, might be saved till
the flood subsided. Their task, perhaps the most important and most
urgent, that men have ever had to perform, was of such a nature
that they cared not what they wrecked in order to accomplish it.
They ripped up the floor of the bridal chamber for timber and took
the doors of the bank-safe for iron.
These rhetorical figures are violent; but they are less violent than
the reality they are intended to express. Monasticism was the last
desperate hope of civilized Christianity and these men knew it. To
establish monasticism they degraded the sanctity of marriage and
denounced the sacredness of property. They conferred the most
sacred honors upon the lowliest drudgery;[37] they turned princes
into plowmen and nobles into breakers of the soil. Some historians,](https://image.slidesharecdn.com/2115521-250518162830-7756bfa1/85/Multivariate-Statistical-Process-Control-Process-Monitoring-Methods-And-Applications-1st-Edition-Zhiqiang-Ge-62-320.jpg)
![judging them by the different standards of a later age, have
pronounced them fanatics led astray by vulgar superstition. But
judged by the needs of their own age, judged by the inestimable
services rendered to the world by the monastic system they
instituted, they are entitled to a place far up in the list of the wisest
and the ablest of the human kind.
Sketchy and imperfect as the above study necessarily is, it
nevertheless gives the primary facts which are essential to an
understanding of the important part played by property concepts
and property institutions in the transformation of early Christianity
from a predominantly eschatological to a practically socialized
movement.
We have seen,[38] that the earliest generations of Christians took
over from contemporary Judaism a strongly Chiliastic eschatology.
The logical consequence of such an eschatology is an indifference to,
or undervaluation of, the existing social arrangements including the
property concepts and institutions. One form easily taken by this
indifference and undervaluation is that of practical communism. We
accordingly find in the Acts and in such early writings as the Didache
and the Epistle of Barnabas a distinctly communistic theory and the
traces of more or less effort to put this theory into some degree of
practical effect. Chiliasm and communism in these writers go
together naturally.
Pari passu with this logical, communistic Chiliasm we can trace the
development of an illogical, individualistic Chiliasm in St. Paul,
Clement of Rome and Hermas. It is already manifest even at this
early stage, that the weight of influence and power of control in the
Christian societies is on the side of the individualists. This is due to
two causes. In the first place the communists among the Christians
worked under a great handicap. The underlying economic
institutions of society can indeed be changed. But they can be
changed—on any considerable scale—only very slowly and by
enormous effort. At any attempt to change them a thousand](https://image.slidesharecdn.com/2115521-250518162830-7756bfa1/85/Multivariate-Statistical-Process-Control-Process-Monitoring-Methods-And-Applications-1st-Edition-Zhiqiang-Ge-63-320.jpg)
![interested and determined antagonists at once arise. It is not too
much to say that had all Christians insisted upon communism as an
essential element of the Christian faith and practice, Christianity in
the Roman world could never have developed into anything more
than an unimportant sect. The very fact that Christianity spread as
rapidly as it did in the first century of its existence is proof that the
communists in the Church made very little headway. It was hard
enough to combat pagan religion and philosophy. Had the property
institutions been attacked also, the primary religious objects would
have been lost sight of in the conflict.
In the second place the more practical minded Christian leaders
would be antagonistic to a doctrine and practice which alienated
many persons who might otherwise be won to the Church, and
practically minded persons outside the Church regarded the
individualists with more favor and were more easily influenced by
them to become Christians themselves. The early importance
attained by the Church of Rome is to be largely ascribed to the
predominance in its councils of such practical persons.[39]
Communism had no hold there at all and Chiliasm was never allowed
to interfere with the practical workings of society.
By the time of Justin the three concepts; Chiliasm, Communism, and
Individualism had arrived at a modus vivendi. According to this
arrangement Chiliasm and Communism held sway as theories while
individualism ruled in the world of fact. This agreement proved very
satisfactory and for more than half a century was the the accepted
thing. It is seen in full force in Tertullian.
There is a general tendency, due to the natural effects of use and
disuse, for theories which do not correspond to realities to become
discredited, even as theories. Conversely realities which at first lack
theoretical justification tend to accumulate such justification with the
lapse of time. It is therefore not surprising to find by the beginning
of the Third Century, a movement to discard theoretical Chiliasm and
communism and to validate by theoretical apologetic the actually](https://image.slidesharecdn.com/2115521-250518162830-7756bfa1/85/Multivariate-Statistical-Process-Control-Process-Monitoring-Methods-And-Applications-1st-Edition-Zhiqiang-Ge-64-320.jpg)
![The Christian slave did not rob his master. These facts gave
Christianity an enormous leverage in its efforts to force its way into
social recognition. It went far toward securing a favorable disposition
toward the new religion on the part of the influential, wealthy, and
conservative elements in the population.
Into the general economic changes which began to operate toward
the end of our period it is not our purpose to enter, but it is worth
notice that the efforts made by the Church to save itself in the
general ruin which overtook the ancient world, chiefly the institution
of monasticism, were such as to secure more firmly than ever the
hold of the Church upon society. The Church rapidly became an
economic factor of the first importance. The only secure basis of
lasting social influence is economic. Christianity by teaching the
virtues of honesty, frugality, simplicity, and charity laid the
foundations of her subsequent triumph, and when she had great
societies of men and women working hard and living plainly and
adding all their accumulations to institutions belonging to the Church
and directly under the supervision and control of the ecclesiastical
authority, the Church paved the way for her subsequent domination
of the civil government. Monastic communism, being economically
superior to Chiliastic Communism, inevitably superseded it.
FOOTNOTES:
[1] Cf. Plato, Laws, V, 742. Aristotle, Politics, 1:X, XI. Cicero, De
Officus, II, XXV. Seneca, De Beneficus, VII, X.
[2] Acts IV.
[3] I. Cor. vii 30.
[4] Rom. xiii 3.
[5] Jas. Chap. V.
[6] Chaps. 21-22.
[7] Chap. xxxviii.](https://image.slidesharecdn.com/2115521-250518162830-7756bfa1/85/Multivariate-Statistical-Process-Control-Process-Monitoring-Methods-And-Applications-1st-Edition-Zhiqiang-Ge-67-320.jpg)
![[8] Did. IV. 8.
[9] Barn. XIV. 16.
[10] Schaff, Vol. 1.
[11] Past. V. vi. 6.
[12] Past. S. IX. XXX. 5.
[13] Past III. 2.
[14] Apol. I. IV.
[15] Apol. I. xiv.
[16] De Mort. Per. XIV.
[17] Apol. XXXIX.
[18] Eus., E. H., V. 18.
[19] De Lapsis, VI.
[20] See Pronouncement of the Sacred Penitentiary, 11 Feb.,
1832.
[21] Sir James Macintosh.
[22] Civil Disabilities of the Jews.
[23] Lourie, Monuments of the Early Church, Chap. II.
[24] Lourie, ibid.
[25] Cf. Hypolytus.
[26] A. Loria. Cf. Economic Basis of Society. (Int.)
[27] Cf. A. Loria, Economic Foundations of Society. (Int.)
[28] Circa 200(?).
[29] Cf. K. Marx, Das Kapital, Vol. 1.
[30] F. Nitti in Catholic Socialism.
[31] Phaedus, The Laws, in Strom. II, 6.
[32] Chap. XIV.
[33] Chap. XXXI.](https://image.slidesharecdn.com/2115521-250518162830-7756bfa1/85/Multivariate-Statistical-Process-Control-Process-Monitoring-Methods-And-Applications-1st-Edition-Zhiqiang-Ge-68-320.jpg)
![[34] Chrysostom, Sermons Rich Man and Lazarus, etc.
[35] Jerome, Commentaries.
[36] Gregory, Pastoralis Cura.
[37] Laborare est orare.
[38] Chap. I.
[39] E.g., Clement and Hermas.](https://image.slidesharecdn.com/2115521-250518162830-7756bfa1/85/Multivariate-Statistical-Process-Control-Process-Monitoring-Methods-And-Applications-1st-Edition-Zhiqiang-Ge-69-320.jpg)
![accommodation has been accomplished in less than three
generations. It is the purpose of this chapter to trace, so far as the
surviving source material permits, the steps of this accommodation
in the case of early Christianity.
For some time before Christ the Jewish people had been restless.
Their desires and aspirations for national and religious greatness had
been repressed and inhibited. The unrest thus generated took
various forms; patriotic uprisings, religious revivals, etc. Christ was
at first considered merely as another Theudas or Judas of Galilee or
John the Baptist. In the pagan world the pax Romana produced a
somewhat similar restlessness. Travel increased; wandering, much of
it aimless, characterized whole classes of people;[1] there was a
marked increase in crime, vice, insanity, and suicide which alarmed
all the moralists. This condition of affairs was eminently suitable for
the first beginnings of a crowd movement; indeed no great crowd
movement can begin except under such circumstances. The
wanderings of St. Paul and the other Christians apostles—called
missionary journeys—were really only particular cases of a general
condition. The same organic demand for new stimulation, the same
sense of shattered religious and philosophic ideals prevailed in the
pagan as in the Jewish world. It would be hard to find a greater
contrast of character than Christ and Lucian. Yet the fiery
earnestness with which Christ denounces contemporary Jewish
religiosity and the cool cynicism with which Lucian mocks at the
pagan piety of the same age have a like cause. Economic pressure
on the lower strata of society contributed to the unrest. The slave,
the small shopkeeper, and the free artisan had a hard time of it in
the Roman world. Economically oppressed classes are material ready
to the hand of the agitator, religious or other. In the crowd
movements recorded in the Acts we can trace the first beginnings of
the Christian populace.[2] In Iconium a great multitude both of
Jews and of Greeks believed but the Jews that were disobedient
stirred up the souls of the Gentiles and made them evil affected
against the brethren. But the multitude of the city was divided and](https://image.slidesharecdn.com/2115521-250518162830-7756bfa1/85/Multivariate-Statistical-Process-Control-Process-Monitoring-Methods-And-Applications-1st-Edition-Zhiqiang-Ge-72-320.jpg)
![part held with the Jews and part with the apostles. At Lytra there
was a typical case of mob action where the apostles were first
worshipped and then stoned. In the cases of the mobs at Philippi
and Ephesus we see the economic motive, the threatened loss of
livelihood, entering along with anger at an attack on the received
religion. In the case of the Jerusalem and Athenian crowds we see
acceptance, or at least acquiescence, on the part of the crowd up to
the point where Christianity breaks with their tradition. In general
we see anger on the part of the crowds only after agitation
deliberately stirred up by interested parties; priests, sorcerers,
craftsmen or the like. Generally speaking the antipathy is no part of
the crowd psychology, and on occasion the crowd may be on the
side of the missionaries of the new religion. In general also the
Christians were not sufficiently numerous to make a counter crowd
demonstration of their own.
In Pliny's letter to Trojan, although it is a generation later than the
Acts and refers to a region where Christianity had been preached for
a considerable period of time, we find a marked instability in the
attitude of the public: Many of every age, every rank and even of
both sexes are brought into danger and will be in the future. The
contagion of that superstition has penetrated not only the cities but
also the villages and country places and yet it seems possible to stop
it and set it right. At any rate it is certain enough that the temples
deserted until quite recently begin to be frequented, that the
ceremonies of religion, long disused, are restored and that fodder for
the victims comes to market, whereas buyers for it were until now
very few. From this it may easily be supposed that a multitude of
men can be reclaimed if there be a place of repentence.[3]
There seems no reasonable ground for doubting that Pliny's
judgment was correct. While the blood of the martyrs is doubtless
the seed of the church, a continuous, general, and relentless
persecution can extirpate a religion in a given nation; as the history
of the Inquisition abundantly proves. Still more easily can
propaganda for the older religion win back its former adherents of](https://image.slidesharecdn.com/2115521-250518162830-7756bfa1/85/Multivariate-Statistical-Process-Control-Process-Monitoring-Methods-And-Applications-1st-Edition-Zhiqiang-Ge-73-320.jpg)
![the first and second generations. It is not, in general, till a
generation has grown up entirely inside a new religion that such a
religion is well established. The generation which at maturity makes
the rupture with the older faith can be brought back to it by less
expenditure of energy than was expended by them in breaking away
in the first place. The success of the Jesuits e.g., is quite inexplicable
on any other hypothesis. The generation who are children at the
time their parents make the break with the old religion are
notoriously undependable in the religious matters. It was in all
probability these people that Pliny had to deal with. It is at least
permissable to hazard the guess that the Laodiceans who aroused
the wrath of the author of the Revelation were of this generation. It
is certain that many of the 'Lapsi' who caused so much trouble to
Christian apologists and church councils belonged in this
chronological class.
In Justin Martyr we have a hint of a further development in the
crowd attitude toward the Christians. Justin says: When you (Jews)
knew that He had risen from the dead and ascended to heaven as
the prophets foretold He would, you not only did not repent of the
wickedness you had committed, but at that time you selected and
sent out from Jerusalem chosen men through all the land to tell that
the godless heresy of the Christians had sprung up and to publish
those things which all they, who knew us not, speak against us. So
that you are the cause not only of your own unrighteousness but
that of all other men.[4]
Irrespective of the exact historical accuracy of this statement, it is
indicative of the process, technically known as 'circular interaction,'
which is so essential a step in the development of popular opinion
and the building up of crowd sentiment. Before any group of people
can become either popular or unpopular there must be a focusing
and fixation of public attention upon them. Even in the new
Testament we find the Jews sending emissaries from city to city to
call attention to the Christian propaganda. Prejudice against the
Christians was thus aroused in persons who had never either seen or](https://image.slidesharecdn.com/2115521-250518162830-7756bfa1/85/Multivariate-Statistical-Process-Control-Process-Monitoring-Methods-And-Applications-1st-Edition-Zhiqiang-Ge-74-320.jpg)
![heard them. The basis of 'circular interaction' is unconscious or
subconscious emotional reaction. A's frown brings a frown to the
face of B. B's frown in turn intensifies A's. This simple process is the
source of all expressions of crowd emotion. By multiplication of
numbers and increase in the stimuli employed it is capable of
provoking a vicious circle of feeling which eventually causes
individuals in a crowd to do things and feel things which no
individual in the crowd would do or feel when outside the circle. It is
to the credit or discredit of the Jews that they first set this 'vicious
circle' in operation against the Christians. Of course the same
psychological principle operated to produce zeal and enthusiasm and
contempt of pain and death in the Christian 'crowd'. By this process
of 'circular interaction' the name, 'Christian,' had already in the time
of Justin become a mob shibboleth. It seems to have operated
precisely as the shibboleth 'traitor' operates on a patriotic crowd in
war time, or 'scab' on a labor group. It became a shibboleth of
exactly opposite significance in the Christian 'crowd'. The way was
thus prepared for the next step in the process of developing the
ultimate crisis. This step—the disparate 'universe of discourse'—is
exhibited in process of formation in the account of the martyrdom of
Polycarp. The account, as we have it, undoubtedly contains later
additions, but these additions even of miraculous elements, do not
necessarily invalidate those portions of the story with which we are
alone concerned. The martyrologist certainly had no intention of
writing his story for the purpose of illustrating the principles of group
psychology and the undesigned and incidental statements of crowd
reactions are precisely the ones of value for our purpose. A few brief
excerpts are sufficient to illustrate the stage reached in the growth
of the disparate 'universe of discourse.' The whole multitude,
marvelling at the nobility of mind displayed by the devout and godly
race of Christians cried out: Away with the Atheists: let Polycarp be
sought out.[5] He went eagerly forward with all haste and was
conducted to the Stadium where the tumult was so great that there
was no possibility of being heard.[6]](https://image.slidesharecdn.com/2115521-250518162830-7756bfa1/85/Multivariate-Statistical-Process-Control-Process-Monitoring-Methods-And-Applications-1st-Edition-Zhiqiang-Ge-75-320.jpg)
![Polycarp has confessed that he is Christian. This proclamation
having been made by the herald, the whole multitude both of the
heathen and Jews who dwelt in Smyrna cried out with uncontrollable
fury and in a loud voice: This is the teacher of Asia, the father of
the Christians and the overthrower of our gods, he who has been
teaching many not to sacrifice or to worship the gods. Speaking
thus they cried out and besought Phillip, the Asiarch, to let loose a
lion upon Polycarp. But Philip answered that it was not lawful for him
to do so seeing the shows of beasts were already finished. Then it
seemed good to them to cry out with one voice that Polycarp should
be burned alive.[7]
This then was carried into effect with greater speed than it was
spoken, the multitude immediately gathering together wood and
fagots out of the shops and baths, the Jews especially, according to
custom eagerly assisting them in it.[8]
We afterwards took up his bones, as being more precious than the
most exquisite jewels and more purified than gold and deposited
them in a fitting place, whither, being gathered together as
opportunity is allowed us, with joy and rejoicing the Lord shall grant
us to celebrate the anniversary of his martyrdom both in memory of
those who have already finished their course and for the exercising
and preparation of those yet to walk in their steps.[9]
In the disparate universe of discourse in its complete form common
shibboleths produce entirely different mental reactions—usually
antagonistic ones. There is also complete accord as to the
shibboleths. The cry here is at one time against the Atheists, then
against the Christians. But the Christians could and did deny the
charge of Atheism. They were as antagonistic to Atheism as the
Pagans. An incomplete development of crowd feeling is evident on
the part of the pagans. The Jews are still the inciters and leading
spirits of the mob. The very statement that the Jews acted
'according to custom' shows that mobbing Christians was still looked
upon as a peculiarly Jewish trait. It was not yet entirely spontaneous](https://image.slidesharecdn.com/2115521-250518162830-7756bfa1/85/Multivariate-Statistical-Process-Control-Process-Monitoring-Methods-And-Applications-1st-Edition-Zhiqiang-Ge-76-320.jpg)
![on the part of the pagan public. Most noticeable of all is the
indifference of the mob toward the Christians' adoration of relics of
the martyrs. No effort was made to prevent the Christians from
obtaining the bones of Polycarp. Either the cult of relics was not
known to the pagans and Jews—though it seems to be firmly
established among the Christians—or else, the effect of the cult in
perpetuating Christianity had not yet had time to make itself
manifest to the pagan public—or to the Jewish. In any case we have
here the plain evidence of the imperfectly developed condition of the
crowd mind, owing perhaps to a too short tradition.
Our next evidence is the martyrdoms of Lyons and Vienne preserved
in a letter quoted by Eusebius. They (the Christians) endured nobly
the injuries inflicted upon them by the populace, clamor and blows
and draggings and robberies and stonings and imprisonments and all
things which an infuriated mob delight in inflicting on enemies and
adversaries.[10]
When these accusations were reported all the people raged like wild
beasts against us, so that even if any had before been moderate on
account of friendship, they were now exceedingly furious and
gnashed their teeth against us.
When he (Bishop Pothinus) was brought to the tribunal
accompanied by a multitude who shouted against him in every
manner as if he were Christ himself, he bore noble witness. Then he
was dragged away harshly and received blows of every kind. Those
men near him struck him with their hands and feet, regardless of his
age, and those at a distance hurled at him whatever they could
seize, all of them thinking that they would be guilty of great
wickedness and impiety if any possible abuse were omitted. For thus
they thought to avenge their own deities.[11]
But not even thus was their madness and cruelty toward the saints
satisfied. Wild and barbarous tribes were not easily appeased and
their violence found another peculiar opportunity in the dead bodies.
For they cast to the dogs those who had died of suffocation in the](https://image.slidesharecdn.com/2115521-250518162830-7756bfa1/85/Multivariate-Statistical-Process-Control-Process-Monitoring-Methods-And-Applications-1st-Edition-Zhiqiang-Ge-77-320.jpg)
![prison and they exposed the remains left by the wild beasts and by
fire mangled and charred. And some gnashed their teeth against
them, but others mocked at them. The bodies of the martyrs having
thus in every manner been exposed for six days were afterwards
burned and reduced to ashes and swept into the Rhone so that no
trace of them might appear on the earth. And this they did as if able
to conquer God and prevent their new birth; 'that', as they said,
'they may have no hope of a resurrection through trust in which they
bring to us this foreign and new religion.' [12]
We have in this account a marked advance, as regards the
development of the mob mind, over what is found in the martyrdom
of Polycarp. Many of the 'crowd' phenomena are indeed the same
but the differences are even more striking than the similarities. We
find in Lyons no body of Jews or other especially interested persons
leading the mob on by manifestations of peculiar zeal and
forwardness. When the accounts are compared in their entirety it
becomes at once manifest that there is a consistency of attitude, a
whole heartedness in the actions of the Lyons mob that is lacking in
the case of the Syrmnaens. There is a degree of familiarity with
Christian doctrine—especially the doctrine of the resurrection—which
denotes a much more thorough permeation of the public mind by
Christianity. There may be no difference in the hatred of the two
mobs for the new faith, but it had more content in the mind of the
Gallic crowd. The degree of thought and pains taken by the Lyonese
persecutors—the guards placed to prevent the Christians from
stealing the relics of the martyrs, the elaborate efforts to nullify the
possibility of a resurrection—the very extent and thoroughness and
duration of the persecution are different from anything to be found
in the other martyrdom.
The difficulty to be explained—if it is a difficulty—from the point of
view of crowd psychology is that there is difference of only eleven
years—taking the ordinary chronology—between the two
persecutions. It is true that the Lyons persecution is the later, but
the difference in the mob behavior is such as might well demand the](https://image.slidesharecdn.com/2115521-250518162830-7756bfa1/85/Multivariate-Statistical-Process-Control-Process-Monitoring-Methods-And-Applications-1st-Edition-Zhiqiang-Ge-78-320.jpg)
![lapse of a generation had the phenomena been exhibited by the
public of the same city. There must unquestionably have been a
great difference in the demotic composition of the populations of
Lyons and Smyrna; the reference to barbarians in Lyons shows as
much, but the behavior of mobs as controlled by the time needed for
the focusing and fixation of attention and the development of a
disparate universe of discourse is very little effected by difference of
demotic composition. It has indeed been suggested by one critic,[13]
that the persecution at Lyons belongs in the reign of Septimus
Severus instead of that of Marcus Aurelius. This would explain away
the difficulty, but there seems no necessary reason for adopting this
opinion. It would rather appear that there existed peculiar conditions
in Lyons and vicinity which account for the fact that the persecution,
so far as we know, was confined to that locality and also for the fact
that the mob mind was in a maturer state of antagonism to
Christianity. Just what these peculiar conditions were, it is impossible
to say with entire certainty. However there is at least a very
suggestive hint in a paragraph by the greatest modern authority on
Roman Gaul[14] contained in his well known volume on Ancient
France.[15] The paragraph is also worth quoting as giving a valuable
insight into the psychology of the peoples of the ancient Roman
World. The Roman Empire was in no wise maintained by force but
by the religious admiration it inspired. It would be without a parallel
in the history of the world that a form of government held in popular
detestation should have lasted for five centuries. It would be
inexplicable that the thirty legions of the Empire should have
constrained a hundred million men to obedience. The reason of their
obedience was that the Emperor, who personified the greatness of
Rome was worshipped like a divinity by unanimous consent. There
were altars in honor of the Emperor in the smallest townships of his
realm. From one end of the Empire to the other a new religion was
seen to arise in those days which had for its divinities the Emperors
themselves. Some years before the Christian era the whole of Gaul,
represented by sixty cities, built in common a temple near the city of
Lyons in honor of Augustus. Its priests, elected by the united Gallic](https://image.slidesharecdn.com/2115521-250518162830-7756bfa1/85/Multivariate-Statistical-Process-Control-Process-Monitoring-Methods-And-Applications-1st-Edition-Zhiqiang-Ge-79-320.jpg)
![cities, were the principal personages in their country. It is impossible
to attribute all this to fear and servility. Whole nations are not servile
and especially for three centuries. It was not the courtiers who
worshipped the prince, it was Rome, and it was not Rome merely
but it was Gaul, it was Spain. It was Greece and Asia.
While no dogmatic assertion is justified, it does not, perhaps, exceed
the limits of reasonable inference to suppose that the existence of
this noted center of Emperor worship in the immediate
neighborhood of Lyons may account, in part at least, for the especial
hatred of the populace of that city for persons who refused to
sacrifice to the Emperor and also for the maturity of their feeling
against the Christians, who were as far as we are aware, probably
the only persons who refused thus to sacrifice. This stray bit of
evidence is admittedly not conclusive. It is offered merely for what it
may be worth. There is evidence that by the middle of the second
Century popular opinion was sufficiently inflamed against the
Christians to render the administration of justice precarious because
of mob violence. Edicts of Hadrian and Antonius Pious specifically
declared that the clamor of the multitude should not be received as
legal evidence to convict or to punish them, as such tumultuous
accusations were repugnant both to the firmness and the equity of
the law.[16]
This attitude seems to have persisted with relatively little change for
about a century. During this period the official 'persecutions' were
neither numerous nor severe. From the very few scattered and
incidental references which have alone survived regarding the mob
feeling of the time, we can assert no more than that it was an
exasperated one, likely to break out upon provocation but under
ordinary circumstances more or less in abeyance. On the whole it
was undoubtedly more violent at the end of the period than at the
beginning.
Fortunately from the middle of the third Century onwards we have a
fairly continuous history of a single 'public' (Alexandria) which is](https://image.slidesharecdn.com/2115521-250518162830-7756bfa1/85/Multivariate-Statistical-Process-Control-Process-Monitoring-Methods-And-Applications-1st-Edition-Zhiqiang-Ge-80-320.jpg)
![upper story.[17] And there was no street, nor public read, nor lane
open to us night or day but always and everywhere all them cried
out that if anyone would not repeat their impious words, he should
be immediately dragged away and burned. And matters continued
thus for a considerable time. But a sedition and civil war came upon
the wretched people and turned their cruelty toward us against one
another. So we breathed for a while as they ceased from their rage
against us.[18]
The mob broke loose against the Christians again the following year,
but there is no object in cataloguing the grewsome exhibitions of
crowd brutality. It is evident that what we have in this account is no
exhibition of political oppression by a tyrannical government, but a
genuine outbreak of group animosity which had been long
incubating in the popular mind. All the phenomena which are
characteristic of fully matured public feeling are found complete;
circular interaction, shibboleths, sect isolation devices and the rest.
When public feeling has developed to such a degree of intensity as
this, the accumulated sentiment and social unrest must of necessity
discharge themselves in some form of direct group action. This
direct action however may take the from either of physical violence
or, under certain conditions, of some sort of mystical experience;
conversion, dancing, rolling on the ground, etc. In exceptional cases
the two forms are combined. An illustration of this latter
phenomenon is given by Bishop Dionysius in this same letter; In
Cephus, a large assembly gathered with us and God opened for us a
door for the word. At first we were persecuted and stoned but
afterward not a few of the heathen forsook their idols and turned to
God.[19] It is necessary to mention perhaps the largest, and
certainly the most dignified and respectable crowd that is to be met
with in connection with this persecution—that of Carthage on the
occasion of the martyrdom of Bishop Cyprian. We find here neither
rage on one side nor unseemly exaltation on the other. Pagans and
Christians alike behaved with decent seriousness at the death of that
famous man who was equally respected by all classes of the](https://image.slidesharecdn.com/2115521-250518162830-7756bfa1/85/Multivariate-Statistical-Process-Control-Process-Monitoring-Methods-And-Applications-1st-Edition-Zhiqiang-Ge-82-320.jpg)
![period that mob had to be most carefully considered by the
government in other than religious matters. But as a religious power
it did not exist. Had the persecution of Diocletian happened a
generation earlier it could have counted on a very considerable
degree of popular support, had it happened a generation later it
would have caused a revolt that could only have been put down by a
large army. Happening at the precise time it did, it provoked no
popular reaction at all.
This strange apathy is not peculiar to Alexandria. Practically without
exception the authentic acts of the martyrs of this persecution are
court records taken down by the official stenographers in the
ordinary course of the day's work. They are dry, mechanical, and
repetitious to a degree. They exhibit, in general, harrassed and
exasperated judges driven to the infliction of extreme penalties in
the face of a cold and skeptical public. One imperial decree ordered
that all men, women, and children, even infants at the breast,
should sacrifice and offer oblations, that guards should be placed in
the markets and at the baths in order to enforce sacrifices there.
The popular reaction in Caesarea is thus recorded: The heathen
blamed the severity and exceeding absurdity of what was done for
these things appeared to them extreme and burdensome.[20] He
(the Judge) ordered the dead to be exposed in the open air as food
for wild beasts; and beasts and birds of prey scattered the human
limbs here and there, so that nothing appeared more horrible even
to those who formerly hated us, though they bewailed not so much
the calamity of those against whom these things were done as the
outrage against themselves and the common nature of man.[21]
The one thing to be said of this type of mob mind is manifestly that
it is transitional. The pendulum has swung through exactly half its
arc and for the brief instant presents the fallacious appearance of
quiescence. How transitory this quiet was on the part of the
Alexandrian mob is evidenced by the history of Athanasius. That
great statesman conciliated and consolidated public opinion in Egypt.
Backed by this opinion he practically cancelled the power of the civil](https://image.slidesharecdn.com/2115521-250518162830-7756bfa1/85/Multivariate-Statistical-Process-Control-Process-Monitoring-Methods-And-Applications-1st-Edition-Zhiqiang-Ge-84-320.jpg)
![authorities of the country and negotiated as an equal with the
emperors. For the first time in more than three centuries the will of
the common people again became a power able to limit the military
despotism which dominated the civilized world.
The re-birth of popular government in the Fourth century through
the agency of Christian mobs is the most important preliminary step
in the growth of the political power of the Catholic Church. A study
of the mobs of Alexandria, Rome, Constantinople and other great
cities shows beyond question that the political power of the Church
had its origin in no alliance with imperial authority, but was
independent of and generally antagonistic to that authority. The
history of these Christian mobs lies outside the limits of our study
but it is worth while in the case of the Alexandrian populace to give
two or three brief extracts illustrating the final steps of the process
which changed a fanatically pagan mob into an equally fanatical
Christian one. What we have to consider is only the last stage of an
evolution already more than half complete at the time of the Nicene
Council. Under extreme provocation and certain of imperial
complacency at their excesses, the pagan mob during the reign of
Julian indulged in one last outburst against the exceedingly
unpopular George of Cappadocia who had been forcibly intruded into
the seat of Athanasius. To quote the Historian Socrates: The
Christians on discovering these abominations went forth eagerly to
expose them to the view and execration of all and therefore carried
the skulls throughout the city in a kind of triumphal procession for
the inspection of the people. When the pagans of Alexandria beheld
this, unable to bear the insulting character of the act, they became
so exasperated that they assailed the Christians with whatever
weapons chanced to come to hand, in their fury destroying numbers
of them in a variety of ways and, as it generally happens in such a
case, neither friends or relations were spared but friends, brothers,
parents, and children imbued their hands in each others blood. The
pagans having dragged George out of the church, fastened him to a
camel and when they had torn him to pieces they burned him
together with the camel.[22] In this account we see the last expiring](https://image.slidesharecdn.com/2115521-250518162830-7756bfa1/85/Multivariate-Statistical-Process-Control-Process-Monitoring-Methods-And-Applications-1st-Edition-Zhiqiang-Ge-85-320.jpg)
![efforts of the pagan mob movement. Any mob movement collapses
rapidly when it turns in upon itself, and the evil results of its violence
react immediately upon the members of the mob. By this time it is
evident that the number of Christians in Alexandria was so large that
any public persecution of them brought serious and unendurable
consequences upon the populace generally. Then the movement
ended.
But in the two centuries or more that the pagan movement lasted, a
contrary Christian mob movement had been developing along the
same general lines as the other. This movement, being later in its
inception, came to a head correspondingly later and reached its
crisis under the patriarch Cyril. Its violence was first directed against
the Jews whom the Christians appear to have hated even more than
they hated the pagans. The Jews were the weaker and less
numerous faction opposed to the Christians and as the Pagans seem
to have liked them too little to support them against the Christians, it
is not surprising that the Christian mob, which had pretty well
reduced the political authorities to impotence, should vent its rage
against the Jews and their synagogues. Cyril accompanied by an
immense crowd of people, going to their synagogues, took them
away from them and drove the Jews out of the city, permitting the
multitude to plunder their goods. Thus the Jews who had inhabited
the city from the time of Alexander were expelled from it.[23]
Sometime after the expulsion of the Jews, the Christian mob, now
directing its spite against the rapidly disappearing paganism,
perpetrated perhaps the most atrocious crime that stains the history
of Alexandria—the murder of Hypatia. This beautiful, learned, and
virtuous woman, 'the fairest flower of paganism' is one of the very
few members of her sex who has attained high eminence in the
realm philosophical speculation. She enjoyed the deserved esteem of
all the intellectual leaders of her age—Christian as well as pagan—
and to the latest ages her name will be mentioned with respect by
all those speculative thinkers whose respect can confer honor.
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- 5. Advances in Industrial Control For further volumes: http//:www.springer.com/series/1412
- 6. Zhiqiang Ge · Zhihuan Song Multivariate Statistical Process Control Process Monitoring Methods and Applications 2123
- 7. Zhiqiang Ge Zhihuan Song Department of Control Department of Control Science and Engineering Science and Engineering Institute of Industrial Process Control Institute of Industrial Process Control Zhejiang University Zhejiang University Hangzhou, Zhejiang, Hangzhou, Zhejiang, People’s Republic of China People’s Republic of China ISSN 1430-9491 ISSN 2193-1577 (Electronic) ISBN 978-1-4471-4512-7 ISBN 978-1-4471-4513-4 (eBook) DOI 10.1007/978-1-4471-4513-4 Springer London Dordrecht Heidelberg New York Library of Congress Control Number: 2012947424 © Springer-Verlag London 2013 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored, or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright LicensingAgency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered names, trademarks, etc., in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
- 8. Series Editors’ Foreword The series Advances in Industrial Control aims to report and encourage technol- ogy transfer in control engineering. The rapid development of control technology has an impact on all areas of the control discipline. New theory, new controllers, actuators, sensors, new industrial processes, computer methods, new applications, new philosophies. . . , new challenges. Much of this development work resides in industrial reports, feasibility study papers, and the reports of advanced collabora- tive projects. The series offers an opportunity for researchers to present an extended exposition of such new work in all aspects of industrial control for wider and rapid dissemination. Statistical process control (SPC) has now evolved into a group of statistical tech- niques for monitoring process performance and product quality.An important feature of these methods is that they are used to monitor performance and are not a control method per se since they contain no automatic feedback mechanism that defines a control action to be taken once a fault condition has been detected. The classical statistical process control tools are the Shewhart control charts that monitor operational process means and use the assumption of the Gaussian dis- tribution to design upper and lower control limits. The interval of process output variation between these limits defines the normal operating region for the process. If the process output strays outside these limits, this is taken as an indication that the process is operating abnormally and that a process fault or disturbance has occurred. Diagnosis and interpretation of what is causing the process upset is a much more complicated issue and tools like the “cause and effect” chart or a root-cause analysis were developed for this aspect of the performance monitoring problem. However, the complexity of large-scale industrial operations and the ease with which supervisory control and data acquisition systems and distributed computer control systems can accumulate large quantities of online process data provided an impetus to develop new statistical concepts and tools in the field of SPC. One influential monograph that created interest in extending the techniques of SPC to exploit the information in these industrial datasets was the Advances in Industrial Control series monograph Data-Driven Techniques for Fault Detection and Diagno- sis in Chemical Processes by E. L. Russell, L. H. Chiang, and R. D. Braatz (ISBN 978-1-85233-258-7, 2000). v
- 9. vi Series Editors’ Foreword In the field of feedback control systems, a similar data-driven approach emerged in the late 1980s, but for controller performance assessment. TheAdvances in Industrial Control series published two much-cited monographs on these new approaches, namely, Performance Assessment of Control Loops by B. Huang and S. L. Shah (ISBN 978-1-85233-639-4, 1999) and Process Control Performance Assessment by A. W. Ordys, D. Uduehi, M. A. Johnson (Eds.) (ISBN 978-1-84628-623-0, 2007). Recent developments in SPC have focussed on procedures that use principal com- ponent analysis, partial least squares, independent component analysis, and factor analysis, and these techniques began entering the published literature from the 1990s onward. Now, researchers Zhiqiang Ge and Zhihuan Song have brought together the advanced SPC methods they have developed in this Advances in Industrial Control monograph. The volume clearly demonstrates how the methods of SPC have evolved to explore different features of industrial data for performance monitoring. Firstly, the multivariate nature of processes has been absorbed into the field to yield multivariate statistical process control methods. In moving away from the classical assumption of wholly Gaussian variations in the data, Drs. Ge and Song show how to analyse non-Gaussian process situations (Chaps. 3 and 4). The focus of the monograph then moves on to nonlinear process monitoring (Chaps. 5 and 6). To extend the techniques introduced to time-varying processes, the authors introduce adaptive monitoring for nonlinear and multimode processes (Chap. 7). Multimode processes are those that evolve through a range of operating conditions. Monitoring process performance as the process moves through different operating conditions requires methods that min- imize the numbers of false alarms and yet are still able to detect faults. The authors devote two chapters to these problems (Chaps. 8 and 9). In Chap. 10, they consider fault detection in vibration signals from an experimental gearbox rig. The penulti- mate chapter of the monograph is on probabilistic process monitoring (Chap. 11). Plant-wide fault detection and monitoring for large-scale industrial processes have been of considerable interest to the process control community in recent years, and this monograph closes with a chapter that reports the authors’ research into using multiblock methods for performance monitoring in such applications. Throughout the monograph, the authors demonstrate their procedures using some academic examples, along with the quite frequent use of the Tennessee Eastman benchmark chemical process network. Experimental gearbox data is used in the vibration signal analysis and another chapter uses data from a polypropylene production process. Consequently, the reader will find illustrative demonstration examples on a range of industrial processes to study. The monograph should be of interest to engineering and academic readers from the process control and chemical engineering community. Control engineers, industrial statisticians, and graduates from the control community will also find the monograph a valuable contribution to the multivariate statistical process control literature. Industrial Control Centre Michael J. Grimble Glasgow Michael A. Johnson Scotland, UK 2012
- 10. Preface Process safety and product quality are two important issues for modern industrial processes. As one of the key technologies in the process system engineering and control area, process monitoring methods can be used to improve product quality and enhance process safety. If a process fault can be anticipated at an early stage and corrected in time, product loss can be greatly reduced. Timely identification of faults can also be used to initiate the removal of out of spec products thereby preserving high standards of product quality. On the other hand, the decisions and expert advice obtained from process monitoring procedures can also be used for process improvement. In general, process monitoring methods can be divided into three categories: model-based methods, knowledge-based methods, and data-based methods. Model- based methods can be traced to 1970s, at which time they were mainly used in aerospace, engine and power systems. Since model-based methods are based on exact process models, they tend to give more accurate monitoring decisions than the other two method categories, and this is the main advantage of the technique. However, due to the complexity of modern industrial processes, it is very costly to obtain an accurate process model, and in some situations it is not actually possible to develop a process model, per se. In contrast, knowledge-based methods are often more satisfactory because they are based on the available real-time knowledge of the process behavior and the experience of expert plant operators. Consequently, the monitoring results provided by these methods tend to be more intuitive. However, the creation of the process knowledge base is always a time-consuming and difficult operation requiring the long-term accumulation of expert knowledge and experiences. Although there are limitations to the model-based and knowledge-based methods, they are still popular in particular areas, especially those in which process models can be easily obtained or the process knowledge readily accumulated. Compared to the model-based and knowledge-based methods, data-driven process monitoring methods have no restrictions on the process model and the associ- ated knowledge and consequently have become more and more popular in recent vii
- 11. viii Preface years. The application areas for the data-driven methods include the important com- plex industrial processes of the chemical, petrochemical, and biological process industries. The widespread use of distributed control systems and SCADA technology in the process industries means that large amounts of real-time process data are read- ily available and this has greatly accelerated the development of data-based process monitoring methods. In addition, the progress made in developing data-mining pro- cedures also provides new technologies for use in process monitoring systems. The data-driven method of Multivariate Statistical Process Control (MSPC) has been the subject of considerable interest from both industry and the academic community as an important tool in the process monitoring area. Three books using Multivari- ate Statistical Process Control for process monitoring have been published in 1999 (Wang), 2000 (Russell, et al.) and 2001 (Chiang, et al.). Whilst Wang introduced some data-mining technology into MSPC for process monitoring and control, Russell et al. (2000) are mainly concerned with the traditional MSPC methods for process monitoring. Later, Chiang et al. (2001) extended data-based process monitoring to incorporate model-based and knowledge-based methods. It should be noted that the traditional MSPC method is limited to Gaussian, linear, stationary, and single-mode processes. During the last decade, the MSPC-based process monitoring approach has been intensively researched, and many papers have been published. However, to our best knowledge, there is no book on the MSPC-based process monitoring topic that has been published since 2001. The purpose of this book is to report recent developments of the MSPC method for process monitoring purpose. The book first reviews the recent research works on MSPC and then the shortcomings of the existing approaches are exposed and demonstrated. This provides the motivation for our research and applications work. In our opinion, this book can be assimilated by advanced undergraduates and grad- uate (Ph.D.) students, as well as industrial and process engineering researchers and practitioners. However, the knowledge of basic multivariate statistical analysis is required, and some familiarity with pattern recognition and machine learning would be helpful.
- 12. Acknowledgement The material presented in this book has been the outcome of several years of research efforts by the authors. Many chapters of the book have gone through several rounds of revisions by discussions with a number of colleagues, collaborators, graduate and Ph.D. students. We would like specifically to thank our colleagues and collaborators, Prof.Youxian Sun for his great supports of the work; Prof. Lei Xie, Prof. Uwe Kruger, Prof. Furong Gao, Prof. Haiqing Wang, Prof. Chunjie Yang, Prof. Jun Liang, Prof. Shuqing Wang, and Prof. Tao Chen who have inspired many discussions in this or related topics over the past years; former and current graduate and Ph. D. students, Dr. Muguang Zhang, Dr. Yingjian Ye, Dr. Kun Chen, Ms. Ruowei Fu, Mr. Zhibo Zhu, Mr. Hong Wang, Mr. Qiaojun Wen, Ms. Aimin Miao, Mr. Le Zhou, and Mr. Hongwei Zhang for their participations in the discussions; and other supporting staffs of the Department of Control Science and Engineering at Zhejiang University. The financial supports from the National Natural Science Foundation of China (NSFC) (60774067, 60974056, 61004134), and the National Project 973 (2012CB720500) are gratefully acknowledged. Last but not least, we would also like to acknowledge Mr. Oliver Jackson (Springer) and other supporting staffs for their editorial comments and detailed examinations of the book. ix
- 13. Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 An Overview of This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Main Features of This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Organization of This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 An Overview of Conventional MSPC Methods . . . . . . . . . . . . . . . . . . . . . 5 2.1 Principal Component Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Partial Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Factor Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 Independent Component Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.5 Kernal Principal Component Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3 Non-Gaussian Process Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Two-step ICA-PCA Information Extraction Strategy for Process Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2.1 Process Monitoring Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2.2 TE Benchmark Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.3 Process Monitoring Based on ICA-FA and SVDD . . . . . . . . . . . . . . . 21 3.3.1 Support Vector Data Description . . . . . . . . . . . . . . . . . . . . . . . . 21 3.3.2 Process Monitoring Based on ICA-FA and SVDD . . . . . . . . . 22 3.3.3 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4 Fault Reconstruction and Identification. . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.2 Fault Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.2.1 Non-Gaussisan Fault Reconstruction Based on SVDD . . . . . 30 4.2.2 Fault Diagnosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.2.3 Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 xi
- 14. xii Contents 4.3 Fault Identification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.3.1 PCA-Based Similarity Factor . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.3.2 Similarity Factors Based on ICA, FA, and Noise Variance . . 38 4.3.3 Implementation Procedures of Fault Identification Based on Similarity Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.3.4 Case Study of TE Benchmark Process . . . . . . . . . . . . . . . . . . . 41 4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.5 Appendix: Iterative Reconstruction Procedure in the Feature Space 43 5 Nonlinear Process Monitoring: Part 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.2 Incorporation of Statistical Local Approach and KPCA . . . . . . . . . . . 46 5.2.1 Statistical Local Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5.2.2 Introducing Statistical Local Approach into KPCA Monitoring Framework . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.3 Nonlinear Process Monitoring Based on Improved KPCA Method 49 5.3.1 Process Monitoring Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.3.2 Outline and Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.3.2.1 Normal Operating Condition (NOC) Model Development . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.3.2.2 Online Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 5.4 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 5.4.1 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 5.4.2 TE Benchmark Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.6 Appendix: Proof of Theorem 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 6 Nonlinear Process Monitoring: Part 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 6.2 Linear Subspace Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 6.3 Nonlinear Process Monitoring Based on Linear Subspace Method 65 6.3.1 Process Monitoring Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 65 6.3.2 Fault Diagnosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 6.4 Method Implementation and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . 69 6.5 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 6.5.1 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 6.5.2 TE Benchmark Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 6.7 Appendix: Algorithm Complexity Analyses of PCA, KPCA, and BSPCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 7 Time-Varying Process Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 7.2 Local Modeling Strategy Based on JITL and LSSVR . . . . . . . . . . . . . 83 7.2.1 LSSVR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 7.2.2 Local Modeling Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
- 15. Contents xiii 7.3 Real-Time Problem Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 7.4 Time-Varying Process Monitoring Based on Local LSSVR Model and ICA-PCA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 7.5 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 7.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 8 Multimode Process Monitoring: Part 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 8.2 Mode Clustering and Feature Extraction. . . . . . . . . . . . . . . . . . . . . . . . 96 8.2.1 Fuzzy C-Mean Method for Operation Mode Clustering . . . . . 97 8.2.2 Two-Step ICA-PCA Feature Extraction for Each Operation Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 8.3 Bayesian-Based Method for Process Monitoring and Fault Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 8.3.1 Multimode Process Monitoring Through the Bayesian Viewpoint. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 8.3.2 Fault Identification and New Operation Mode Determination 101 8.3.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 8.4 Illustrations and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 8.4.1 A Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 8.4.2 TE Benchmark Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 8.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 9 Multimode Process Monitoring: Part 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 9.2 Mode Partition and Linear Subspace Construction . . . . . . . . . . . . . . . 114 9.2.1 Data Partition Through Sample Direction . . . . . . . . . . . . . . . . 114 9.2.2 Two-StepVariable Selection for Linear Subspace Construction 115 9.3 Two-Dimensional Bayesian Inference for Process Monitoring . . . . . 117 9.3.1 Method Description and Implementation . . . . . . . . . . . . . . . . . 117 9.3.2 Algorithm Complexity Analysis . . . . . . . . . . . . . . . . . . . . . . . . 121 9.3.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 9.4 Case Study of TE Benchmark Process . . . . . . . . . . . . . . . . . . . . . . . . . 122 9.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 10 Dynamic Process Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 10.2 Two Dynamical Process Monitoring Methods . . . . . . . . . . . . . . . . . . . 132 10.2.1 First Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 10.2.2 Second Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 10.3 Illustration of the Gearbox System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 10.3.1 Description of the Gearbox System . . . . . . . . . . . . . . . . . . . . . 133 10.3.2 Analysis of Reference Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 10.3.3 Generation of Fault Condition . . . . . . . . . . . . . . . . . . . . . . . . . . 136
- 16. xiv Contents 10.4 Overview of Existing Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 10.4.1 Independent Component Analysis. . . . . . . . . . . . . . . . . . . . . . . 137 10.4.2 Subspace Model Identification . . . . . . . . . . . . . . . . . . . . . . . . . 138 10.5 Dynamic Monitoring Schemes for Non-Gaussian Vibration Systems 138 10.5.1 DICA-SVDD Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 10.5.2 SMI Technique Using the Statistical LA (SMILA) . . . . . . . . . 140 10.6 Application Study in a Gearbox System . . . . . . . . . . . . . . . . . . . . . . . . 141 10.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 11 Probabilistic Process Monitoring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 11.2 PPCA-Based Method for Process Monitoring . . . . . . . . . . . . . . . . . . . 148 11.3 Bayesian Regularization of PPCA Monitoring Method. . . . . . . . . . . . 150 11.4 Multimode Process Monitoring Based on MBPCA . . . . . . . . . . . . . . . 152 11.4.1 Mixture Bayesian Regularization of PPCA . . . . . . . . . . . . . . . 152 11.4.2 Multimode Process Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . 154 11.5 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 11.5.1 A Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 11.5.2 Polypropylene Production Process Application Study . . . . . . 160 11.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 11.7 Appendix: Derivation of the EM Algorithm for the Mixture Bayesian PCA Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 12 Plant-Wide Process Monitoring: Multiblock Method . . . . . . . . . . . . . . . 169 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 12.2 Multiblock ICA-PCA (MBICA-PCA). . . . . . . . . . . . . . . . . . . . . . . . . . 170 12.3 Process Monitoring Based on Two-Level MBICA-PCA . . . . . . . . . . . 171 12.3.1 Fault Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 12.3.2 Fault Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 12.4 Case Study of TE Benchmark Process . . . . . . . . . . . . . . . . . . . . . . . . . 175 12.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
- 17. Notation A mixture matrix of the ICA model ai coefficient a center of the hypersphere in the SVDD model B othogonal matrix of the ICA model BICT2 (x) Bayeaisn inference monitoring statistic based on the T2 BICSPE(x) Bayeaisn inference monitoring statistic based on the SPE comb{·} combination operator CFM critical fault magnitude C tuning parameter in the SVDD model c sample mean in the feature space cα normal deviate corresponding to the upper 1−α percentile cos θ cosine value of the main angle d(•) distance calculator diag(·) make a diagonal matrix E residual matrix E{·} expectation FDIs T 2 (x) fault detection index in each linear subspace under T2 statistic FDIs SPE(x) fault detection index in each linear subspace under SPE statistic FDI fault detection index FDP fault detection performance F feature space fj fault magnitude GS Gaussian statistic h dimension of feature space H bandwidth matrix H0 hypothesis test H1 hypothesis test Ind(i) selected variable index for subspace i I identity matrix I2 monitoring statistic of the independent component part I2 glob global monitoring statistic of the independent component part I2 Contblock,b block contribution of the I2 statistic xv
- 18. xvi Notation I2 Contglob,i global contribution of the I2 statistic ICS independent component subspace JITL just-in-time-learning JP joint probability k number of principal components in the PCA model K kernel matrix KDE kernel density estimation L(P,β) likelihood function of PPCA lopt optimal number of principal factors mean(•) mean value of the elements m variable number N(•,•) normal distribution NFDIT 2 (x) nonlinear fault detection index under T2 statistic NFDISPE(x) nonlinear fault detection index under SPE statistic NFDI nonlinear fault detection index n sample number nc current sample number nkpca number of kernel principal components NGS non-Gaussian statistic O(·) Computational complexity calculator P loading matrix of the PCA model PCS principal component subspace PT 2 (N) prior probability of the process being normal PT 2 (F) prior probability of the process being abnormal PT 2 (F|x) fault probability based on the T2 statistic PSPE(F|x) fault probability based on the SPE statistic PT 2 (x|N) condition probability under normal situation based on the T2 statistic PT 2 (x|F) condition probability under abnormal situation based on the T2 statistic PSPE(x|N) condition probability under a situation based on the SPE statistic PSPE(x|F) condition probability under abnormal situation based on the SPE statistic PI2 (f |xfault) posterior probability of the faulty sample belongs to a specific faulty mode f under I2 statistic PT 2 (f |xfault) posterior probability of the faulty sample belongs to a specific faulty mode f under T2 statistic PSPE(f |xfault) posterior probability of the faulty sample belongs to a specific faulty mode f under SPE statistic R radius of the hypersphere in the SVDD model RBC reconstruction-based contribution RS residual subspace r number of independent components in the ICA model S independent component matrix of the ICA model Smix mixed independent component matrix of the multiblock model
- 19. Notation xvii Sglobal global independent component matrix of the multiblock model s independent component vector SPE monitoring statistic of the residual space SPEglob global monitoring statistic of the residual space SPEContblock,b block contribution of the SPE statistic SPEContglob,i global contribution of the SPE statistic SPEα control limit of the SPE monitoring statistic SPCA similarity factor of the principal component space SICA similarity factor of the independent component space SFA similarity factor of the FA SN similarity factor of the noisy part Savg weighted average of similarity factors SICA mixed similarity factor SPC_dist distance similarity factor of the principal component space SIC_dist distance similarity factor of the independent component space SubCIPCS(i, j) contribution of variable i in subspace j constructed in PCS SubCIRS contribution of variable i in subspace constructed in RS subspace{·} linear subspace TDB two-dimensional Bayesian T score matrix of the PCA model Tmix mixed score matrix of the multiblock model Tglob global score matrix of the multiblock model Tnl nonlinear latent variable matrix TPCA time complexity of PCA TKPCA time complexity of KPCA TSPCA time complexity of subspace PCA TTDB time complexity of two-dimensional Bayesian method TOPCA online time complexity of PCA TOKPCA online time complexity of KPCA TOSPCA online time complexity of subspace PCA TOTDB online time complexity of two-dimensional Bayesian method T r(·) trace value calculator t score vector T2 monitoring statistic of the schematic part T 2 glob Global monitoring statistic of the schematic part T 2 Contblock,b block contribution of the T2 statistic T 2 Contglob,i global contribution of the T2 statistic VI(b) independent component number extracted in the bth sub-block VP(b) principal component number extracted in the bth sub-block v load vector of KPCA W weight matrix of the ICA model Wc ij weighted index of each variable in corresponding linear sub- space w size of moving window
- 20. xviii Notation x(k) sample vector at time point k Zp past arrangement of Hankel matrices Zf future arrangement of Hankel matrices α hyperparameter vector in the mixture Bayesian regularization PPCA method α significance level β reverse of the noise variance αi Lagrange multipliers mapping function (·) nonlinear relationship among the process variables primary residual n primary residual of noisy part s primary residual of systematic part σmax maximum singular value λ eigenvalue parameter set j fault direction opt c optimal parameter of each BPCA model θ process parameter monitoring function n monitoring function for noisy part s monitoring function for systematic part covariance value ρ linear correlation between process variables ζ improved residual ζn improved residual of noisy part ζs improved residual of systematic part ξi slack variable in the SVDD model ξj fault identification index ηICS identification index in IC subspace ηPCS identification index in PC subspace ηRS identification index in Residual subspace μ mean value χ2 a Chi distribution
- 21. Chapter 1 Introduction 1.1 An Overview of This Book With the wide use of the distributed control systems in modern industrial processes, a large amount of data has been recorded and collected. How to efficiently use these datasets for process modelling, monitoring and control is of particular interest, as the traditional first-principle model-based method is difficult to use in modern complex processes, which is mainly due to the high human and resource costs or special environments. Different from the first-principle model-based method, the data-based method rarely needs any prior knowledge of the process. By extracting the useful information from the recorded process data, data-based models are also able to model the relationship between different process variables. Particularly, for process monitoring purpose, the multivariable statistical process control (MSPC)- based method has received much attention since the 1990s. The main idea of the MSPC-based monitoring approach is to extract the useful data information from the original dataset, and construct some statistics for monitoring. Most MSPC-based methods can successfully handle the high-dimensional and correlated variables in the process because they are able to reduce the dimension of the process variables and decompose the correlations between them. Therefore, MSPC has become very popular in industrial processes, especially when used for process monitoring. So far, the most widely used MSPC method for process monitoring may be the principal component analysis (PCA) and the partial least squares (PLS) methods. By extracting the principal components from the process data, and then constructing T2 and Squared Prediction Error (SPE) statistics for process monitoring, PCA and PLS can both handle the high-dimensional and correlated process variables, and provide detailed monitoring results for each of the data sample in the process. Along past several decades, different improvements and modifications have been made to the traditional PCA and PLS methods. Additionally, some new techniques have also been introduced into the process monitoring area, such as probabilistic PCA, factor analysis, independent component analysis (ICA), kernel PCA, support vector data description (SVDD), etc. most of which were originally proposed in other areas. On the basis of these newly developed and introduced methods, the monitoring per- formance has been improved for processes under specific conditions. For example, Z. Ge, Z. Song, Multivariate Statistical Process Control, Advances in Industrial Control, 1 DOI 10.1007/978-1-4471-4513-4_1, © Springer-Verlag London 2013
- 22. 2 1 Introduction when the process data are not strictly Gaussian distribution, the traditional PCA method is not sufficient to provide a good monitoring performance. With the intro- duction of the ICA algorithm, the non-Gaussian data information can be efficiently extracted, on the basis of which new monitoring statistics have been constructed for process monitoring. Nonlinearity is a common data behaviour in many industrial processes; although the conventional MSPC methods fail to describe nonlinear rela- tionship among process variables, some new nonlinear modelling approaches have recently been developed for process monitoring purposes, such as neural network- based MSPC, kernel PCA, linear subspace method, etc. Another common behaviour of modern industrial processes is, their operation condition may be time-varying or they may have multiple operation modes. In these cases, the conventional MSPC methods may not provide satisfactory results. Fortunately, a lot of improvements and new approaches have been developed for monitoring those processes which are time-varying or have multiple operation modes. Some representatives of these meth- ods are adaptive PCA, recursive PLS, moving-window PCA, multimodel method, local model approach, Bayesian inference method, etc. While most of the exist- ing MSPC methods focused on static industrial processes, only a few works have explored the topic of dynamic process monitoring, especially for dynamic non- Gaussian processes. In order to handle the noisy data information in the process, several probabilistic modelling methods have been proposed, e.g. probabilistic PCA and factor analysis. As each process variable is inherently a random variable, the probabilistic model may be more appropriate to describe the relationships among them. Actually, the monitoring performance of most processes has been improved with the introduction of the probabilistic models. Recently, research attention has also focused on plant-wide process monitoring, which is also known as large-scale process monitoring problem in related references. For those processes, the tradi- tional MSPC methods have been extended to the multiblock counterparts, such as multiblock PCA and multiblock PLS. In addition, several new approaches have also been developed on the basis of these multiblock monitoring methods. The aim of this book is to provide a recent overview of the MSPC method and it introduces some new techniques for the process monitoring purpose. Specif- ically, this book gives an overview of recently developed methods in different aspects, namely non-Gaussian process monitoring, nonlinear process monitoring, time-varying process monitoring, multimode process monitoring, dynamic process monitoring, probabilistic process monitoring, and plant-wide process monitoring. However, due to the limited space, only some methods have been selected for detailed description in this book. 1.2 Main Features of This Book The key features of this book are given as follows: 1. According to the complex distribution of given process data, a two-step ICA-PCA based information extraction and process monitoring strategy is introduced. This
- 23. 1.3 Organization of This Book 3 method was subsequently improved by using SVDD and factor analysis. For fault diagnosis, the SVDD reconstruction-based non-Gaussian fault diagnosis method is introduced, which can be considered as a complement of reconstruction-based methods for fault diagnosis in the non-Gaussian case. Besides, a similarity-based method is introduced for fault identification. 2. For nonlinear process monitoring, the statistical local approach (LA) is introduced to the traditional kernel PCA modelling structure. This effectively eliminates the restriction of the process data to the Gaussian distribution. Due to the offline modelling and online implementation difficulties of existing methods, a new viewpoint for nonlinear process monitoring, which is based on linear subspace integration and Bayesian inference, is illustrated. Compared to existing nonlinear methods, the new method can both improve the monitoring performance and reduce the algorithm complexity. 3. In order to improve the monitoring performance for time-varying and multimode processes, three new methods are given in detail. First, a local least squares support vector regression-based method is introduced to improve the deficiency of the traditional recursive method for time-varying processes. This greatly en- hances the real-time performance for monitoring purpose. Second, a Bayesian inference method is introduced for multimode process monitoring. Third, a two- dimensional Bayesian-based method, which greatly alleviates the lean of the monitoring method to process knowledge and experiences, is also introduced for monitoring nonlinear multimode processes. 4. Very few works have been reported on process monitoring for non-Gaussian dynamic processes. A new monitoring method is introduced in this book for these special processes, which is based on subspace model identification and LA. In contrast to other methods, the new method is more efficient in monitoring non-Gaussian dynamic processes. 5. While the probabilistic PCA method has recently been introduced for process monitoring, it has an inherent limitation that it cannot determine the effective dimensionality of latent variables. A Bayesian treatment of the PCA method is developed and introduced. For multimode process monitoring, this Bayesian regularization method is extended to its mixture form. 6. Based on the traditional multiblock method that has been proposed for plant-wide process monitoring, a two-level MultiBlock ICA-PCA method is introduced. Through this method, the process monitoring task can be reduced and the inter- pretation of the process can be made more efficiently. Detailed descriptions of fault detection, fault reconstruction and fault diagnosis tasks are provided. 1.3 Organization of This Book The chapter organization of this book is detailed as follows. Chapter 1 gives an overview of this book, and illustrates the main features of this book.
- 24. 4 1 Introduction Chapter 2 introduces some conventionally used MSPC methods that have been widely used and developed in the past years. Chapter 3 gives introductions of several non-Gaussian process monitoring meth- ods, including the two-step ICA-PCA strategy, ICA-factor analysis strategy, and the SVDD algorithm. Chapter 4 introduces the SVDD-based non-Gaussian fault reconstruction and diagnosis algorithm, and the similarity-based fault identification method. Chapters 5 and 6 introduce the nonlinear process monitoring topic through two different viewpoints. Chapter 7 gives a detailed description of the local model-based approach for time- varying process monitoring. Chapters 8 and 9 are two parts which are dedicated on the topic of multimode process monitoring. Chapter 10 focuses on dynamic process monitoring, which specifically introduces two methods for monitoring non-Gaussian dynamic processes. Chapter 11 introduces a Bayesian regularization of the probabilistic PCA method and its mixture form, on the basis of which a probabilistic multimode process monitoring approach is also demonstrated. Chapter 12 gives an introduction of a two-level multiblock method for plant-wide process monitoring. Some of the materials presented in this book have been published in academic journals by the authors, and are included after necessary modification and updates to ensure accuracy and coherence.
- 25. Chapter 2 An Overview of Conventional MSPC Methods 2.1 Principal Component Analysis Consider an n × m data matrix X, for m process variables with n measurements of each variable. Here, we assume that the data for each variable have been scaled to zero mean and unit variance. The covariance matrix of X is defined as = XT · X n − 1 (2.1) Let λi(i = 1, 2, . . . , m) be the eigenvalues of the matrix of which are arranged in descending order to determine the principal components (PCs), and with the principal component loadings pi to be their corresponding eigenvectors. Then, the first k PCs are selected to build the PCA model, so the data matrix X can be expanded using principal component loadings pi, score vectors ti, and a residual matrix E (Chiang et al. 2001; Qin 2003) X = k i=1 tipT i + E (2.2) For a new observation xnew, the prediction of the PCA model is given by x̂new = tnewPT = xnewPPT (2.3) where tnew, is the score vector. The resulting residual is defined as enew = xnew − x̂new (2.4) For monitoring purpose, two statistical variables T2 and SPE can be calculated based on tnew and enew, respectively, T 2 new = tnew tT new (2.5) SPEnew = eneweT new (2.6) Z. Ge, Z. Song, Multivariate Statistical Process Control, Advances in Industrial Control, 5 DOI 10.1007/978-1-4471-4513-4_2, © Springer-Verlag London 2013
- 26. 6 2 An Overview of Conventional MSPC Methods where is the eigenvalue matrix of the PCA model. The confidence limit of the T2 and SPE monitoring statistics are determined as follows (Chiang et al. 2001) T 2 ≤ T 2 lim = k(n − 1) n − k Fk,(n−k),α (2.7) SPE ≤ SPElim = gχ2 h,α g = v/(2m) (2.8) h = 2m2 /v where k is the number of PCs, α is significance level, cα is the normal deviate corresponding to the upper 1 − α percentile, m and v are the mean and variance values of SPE of the training dataset. When the statistic values of the new data sample exceed their corresponding control limits, T 2 new T 2 lim or SPEnew SPElim, a fault is considered to be detected, and further actions should be taken to get the process back in its good condition. 2.2 Partial Least Squares The principle of the PLS method is similar to that of PCA, except that the PLS method incorporates the information of the quality variables in the process. Given a pair of process and quality dataset {X, Y}, PLS intends to decompose X and Y into a combination of scores matrix T, loading matrices P and Q, and weight matrix W. The relationship between X and Y can be described by the following equations X = TT P + E (2.9) Y = TQT + F (2.10) The regression matrix of the PLS model between the process and quality variables can be determined as follows R = W(PT W) −1 QT (2.11) Given a new process data sample, (xnew, ynew), the principal components are calculated in the first step, which is given as tnew = xnewW(PT W) −1 (2.12) In the next step, the T2 and SPE monitoring statistics can be constructed for pro- cess monitoring. Different from the PCA approach, a total of four monitoring statistics can be used for processs monitoring in the PLS model, two of which correspond to the data information of the quality variables. Detailed description of the PLS model and its monitoring strategy can be found in Chiang et al. (2001).
- 27. 2.3 Factor Analysis 7 2.3 Factor Analysis Similar to the probabilistic PCA method, factor analysis concentrates on the latent variables t whose distribution are Gaussian, and the original measurement variables x are treated as linear combination of t plus small additive white noise e. The aim of FA is to find the most probable parameter set = {P, } in the model structure, which is described as follows (Bishop 2006) x = Pt + e (2.13) where P = [p1, p2, . . . , pl] ∈ Rm×l is the loading matrix, just as that in PCA. The variances matrix of measurement noise e is represented by = diag{λi}1,2,...,m, in which different noise levels of measurement variables are assumed. If all the λ are assumed to be of the same value, then FA is equivalent to PPCA. If all λi, i = 1, 2, . . . , m are assumed to be zero, then FA becomes PCA. Therefore, FA is the general description of Gaussian latent model structure. PPCA and PCA are just two special cases of FA. In the FA model, the latent variable t is assumed to be Gaussian distribution with zero mean and unity variance, which is t ∈ N(0, I), where I is the unity matrix with appropriate dimension. Precisely, the distribution of the latent variable t is p(t) = (2π)−k/2 exp − 1 2 tT t (2.14) Then, the conditioned probability of the measured process variable x is given as p(x|t) = (2π)−m/2 |e|−1/2 exp − 1 2 (x − Pt)T −1 e (x − Pt) (2.15) Based on Eqs. (2.14) and (2.15), the probability of x can be calculated as p(x) = p(x|t)p(t)dt = (2π)−m/2 |C|−1/2 exp − 1 2 xT C−1 x (2.16) where C = e +PPT is the variance matrix of the process data, thus the distribution of x can be represented as x ∈ N(0, e + PPT ), |C| is to calculate the discriminant value for the C. The parameter set = {P, e} can be estimated by the expectation and maxi- mization (EM) algorithm, which is an iterative likelihood maximization algorithm. EM is a widely used algorithm for parameter estimation due to its simplicity and ef- ficiency. Besides, it can also handle incomplete datasets, which is also very attractive to the application in process industry. The EM algorithm can be partitioned into two steps: the E-step and the M-step. The maximum likelihood function of the process data can be given as L = n i=1 ln{p(xi, ti)} = n i=1 ln{p(ti|xi)p(xi)} (2.17)
- 28. 8 2 An Overview of Conventional MSPC Methods where the posterior distribution of each latent variable can be calculated as p(ti|xi) = (2π)−k/2 |M|−1/2 exp − 1 2 (ti − Qxi)T M−1 (ti − Qxi) (2.18) where Q = PT (PPT + e) −1 , M−1 = I − QP. In the E-step of the EM algorithm, the expectation value of the likelihood function L is calculated as follows, which is based on the first and second statistics of the latent variable E(L) = − n i=1 1 2 xT i −1 e xi − xT i −1 e PE(ti|xi) + 1 2 tr[P−1 e PE(titT i |xi)] − (n/2) ln e + cont (2.19) where E(·) and tr(·) are expectation and trace calculators, cont represents a con- stant value, and the first and second statistics of the latent variables are given as follows t̂i = E{ti|xi} = Qxi (2.20) E{titT i |xi} = I − QP + QxixT i QT (2.21) In the M-step of the EM algorithm, by maximizing the expectation value of the likelihood function, taking the partial derivatives of L with respect to the parameter set = {P, e}, and setting them to zero, the optimal values of the two parameters can be determined, which are given as P = n i=1 xit̂T i n i=1 E{titT i |xi} −1 (2.22) e = n i=1 diag xixT i − Pt̂ixT i n (2.23) By calculating the E-step and M-step iteratively, the final optimal parameters for the FA model can be obtained. For process monitoring, the T2 and SPE monitoring statistics based on the FA model can be constructed as follows T 2 i = t̂i 2 = xT i QT Qxi ≤ χ2 α,k (2.24) êi = E{ei|xi} = (I − PQ)xi (2.25) SPEi = −1/2 e ei 2 = xT i (I − PQ)T −1 e (I − PQ)xi ≤ χ2 α,m (2.26)
- 29. 2.4 Independent Component Analysis 9 2.4 Independent Component Analysis Independent component analysis (ICA) is a statistical and computational technique for revealing hidden factors that underlie sets of random variables, measurements, or signals. ICA was originally proposed to solve the blind source separation prob- lem. To introduce the ICA algorithm, it is assumed that m measured variables, x(k) = [x1(k), x2(k), . . . , xm(k)] at sample k can be expressed as linear combina- tions of r( ≤ m)unknown independent components [s1, s2, . . . , sr ]T , the relationship between them is given by Hyvarinen and Oja (2000) XT = AS + E (2.27) where n is the number of measurements, X ∈ Rn×m is the data matrix, A = [a1, a2, . . . , ar ] ∈ Rm×r is the mixing matrix, S = [s1, s2, . . . , sr ] ∈ Rr×n is the independent component matrix, and E ∈ Rl×n is the residual matrix. The basic problem of ICA is to estimate the original component S and the mixing matrix A with X, therefore, the objective of ICA is to calculate a separating matrix W so that the components of the reconstructed data matrix Ŝ become as independent of each other as possible, given as Ŝ = WXT (2.28) Before applying the ICA algorithm, the data matrix X should be whitened, in order to eliminate the cross-correlations among random variables. One popular method for whitening is to use the eigenvalue decomposition, considering x(k) with its covariance Rx = E{x(k)x(k)T }, the eigenvalue decomposition of Rx is given by Rx = U UT (2.29) The whitening transformation is expressed as z(k) = Qx(k) (2.30) where Q = −1/2 UT . One can easily verify that Rz = E{z(k)z(k)T } is the identity matrix under this transformation. After the whitening transformation, we have z(k) = Qx(k) = QAs(k) = Bs(k) (2.31) where B is an orthogonal matrix, giving the verification as follows E{z(k)z(k)T } = BE{s(k)s(k)T }BT = BBT = I (2.32) Therefore, we have reduced the problem of finding an arbitrary full-rank matrix A to the simple problem of finding an orthogonal matrix B. Then, we can estimate as follows ŝ(k) = BT z(k) = BT Qx(k) (2.33)
- 30. 10 2 An Overview of Conventional MSPC Methods From Eqs. (2.28) and (2.33), we can get the relation between W and B W = BT Q (2.34) There are two classic measures of non-Gaussianity: kurtosis and negentropy (Hy- varinen and Oja 2000). Kurtosis is the fourth-order cumulant of a random variable, unfortunately, it is sensitive to outliers. On the other hand, negentropy is based on the information-theoretic quantity of differential entropy. When we have obtained the approximate forms for negentropy, Hyvarinen (1997) introduced a very simple and efficient fixed-point algorithm for ICA. This algorithm calculates the independent components one by one through iterative steps. After all vectors bi (i = 1, . . . , m) have been calculated and put together to form the orthogonal mixing matrix B, then we can obtain ŝ(k) and demixing matrix W form Eqs. (2.33) and (2.34), respectively. Dimension reduction of ICA is based on the idea that these measured variables are the mixture of some independent component variables. The performance and interpretation of ICA monitoring depend on the correct choice of the ordering and dimension of the ICA model. Because negentropy is used for measurement of non- Gaussianity, we can select suitable number of ICs by checking their negentropy values. If the negentropy value of current IC is zero or approximately zero, it indicates that the non-Gaussian information of the process has already been extracted, and the rest of the process is Gaussian, which can be analyzed by conventional MSPC methods such as PCA. Based on the identified ICA model, two monitoring statistics can be constructed for monitoring, which are defined as I2 and SPE, given as (Lee et al. 2004; Ge and Song 2007) I2 = sT s (2.35) SPE = eT e (2.36) where s and e are independent components and residuals, respectively. 2.5 Kernal Principal Component Analysis As a simple linear transformation technique, PCA compresses high-dimensional data into low dimension with minimum loss of data information. When the algo- rithm is carried out in the feature space, KPCA is obtained. This algorithm was originally proposed by Schölkopf et al. (1998). Suppose the original training dataset is x1, x2, . . . , xn ∈ Rm , where n is the sample number and m is the number of process variables. The feature space is constructed by using a nonlinear mapping: Rm (·) − − → Fh , where ( · ) is a nonlinear mapping function and h is the dimension in feature space, which is assumed to be a large value. Similar to PCA, the covariance matrix in the feature space F can be calculated as F = 1 n n i=1 [(xi) − c][(xi) − c]T (2.37)
- 31. 2.6 Conclusion 11 where c is the sample mean in the feature space. Conveniently, denote ¯ (xi) = (xi) − c as the centered feature space sample. The kernel principal component can be obtained by the eigenvalue problem below λv = F v = 1 n n i=1 [ ¯ (xi)T v] ¯ (xi) (2.38) v = n i=1 αi ¯ (xi) (2.39) where λ and v denote eigenvalue and eigenvector of the covariance matrix F , respectively. The αi is the coefficient which indicated that the kernel principal com- ponent is spanned by feature space training samples. Multiplying ¯ (xj ) with both sides of Eq. (2.38), the following equation is obtained λ[ ¯ (xj )v]= ¯ (xj )Fv (2.40) The problem can be transformed to Eq. (2.41) by introducing a kernel matrix K̄ij = ¯ (xi) · ¯ T (xj ), where the kernel matrix is centered, α should be scaled as α2 = 1/nλ to ensure the normality of v (Choi et al. 2005) λα = 1 n ¯ Kα (2.41) The score vector of qth observation is calculated as tq,k = [vk · ¯ (xq)] = n i=1 αk i [ ¯ (xq) ¯ T (xi)] = n i=1 αk i K̄qi (2.42) Then, T2 and SPE statistics can be developed for process monitoring. 2.6 Conclusion In this chapter, several typical multivariate statistical process control methods have been introduced, including PCA, PLS, ICA, FA, and KPCA. Based on these basic MSPC methods, more specific and complex process monioring methods can be further developed, some of which will be introduced in the following chapters.
- 32. Chapter 3 Non-Gaussian Process Monitoring 3.1 Introduction Many successful applications have shown the practicability of multivariable statisti- cal process control (MSPC). However, the achievable performance of the MSPC method is limited due to the assumption that monitored variables are Gaussian distribution. In fact, most of the process variables do not strictly form Gaussian distribution. In contrast, it is well known that many of the variables monitored in process systems are not independent; chemical processes are usually driven by fewer essential variables which may not be measured and the measured process variables may be the combinations of these independent latent variables. Independent com- ponent analysis (ICA) is an emerging technique for finding several independent variables as linear combinations of measured variables. What distinguished ICA from principal component analysis (PCA) is that it looks for components that are bothstatisticallyindependentandnon-Gaussian. PCAcanonlyimposeindependence up to second-order statistics information, and the direction vectors are constrained to be orthogonal; whereas, ICA has no orthogonal constraint and also involves higher order statistics. ICA is able to extract the non-Gaussian information while PCA is looking for the Gaussian one. In other words, ICA may reveal more meaningful information in the non-Gaussian data than PCA. A number of applications of ICA have been reported in speech processing, biomedical signal processing, machine vibration analysis, nuclear magnetic resonance spectroscopy, infrared optical source separation, radio communications, etc. (Hyvarinen and Oja 2000). Recently, ICA has been introduced for the process monitoring purpose (Li and Wang 2002; Kano et al. 2003). Kano et al. (2004a) developed a unified framework for MSPC, which combined PCA-based SPC and ICA-based SPC. The proposed combined MSPC could monitor both of the Gaussian and non-Gaussian information of the process, good performance was shown in a multivariate system and a continuous stirred-tank reactor (CSTR) process. Several applications of the ICA method for process mon- itoring have also been reported (Lee et al. 2004a, b; Lee et al. 2006a; Zhang and Qin 2007; Ge and Song 2007, 2008; Zhao et al. 2008a; Hsu et al. 2010; Zhang and Zhang 2010; Odiowe and Cao 2010; Stefatos and Ben Hamza 2010). Z. Ge, Z. Song, Multivariate Statistical Process Control, Advances in Industrial Control, 13 DOI 10.1007/978-1-4471-4513-4_3, © Springer-Verlag London 2013
- 33. 14 3 Non-Gaussian Process Monitoring It is also important to note that all the reported ICA-based monitoring methods are based on the assumption that the essential variables which drive the process are non-Gaussian and discarded Gaussian parts are utilized to set up a single squared prediction error (SPE) monitoring chart. Actually, complex multivariate processes are always driven by non-Gaussian and Gaussian essential variables simultaneously. Separating the Gaussian information from the un-modelled Gaussian uncertainty will facilitate the diagnosis of faults which occur in different sources. Therefore, a two-step ICA-PCA information extraction strategy has been proposed and used for process monitoring. Due to complicated confidence limit determination of the traditional ICA-based monitoring method, which is based on the kernel density estimation (KDE), support vector data description (SVDD) has been introduced to improve the determination of confidence limits for the non-Gaussian components (Tax and Duin 1999, 2004). Compared to KDE (Lee et al. 2004), the SVDD-based method is more computation- ally efficient, which relies on a quadratic programming cost function. To monitor discarded Gaussian parts of the process information, PCA might be the straight- forward but not the optimal approach in which the projected directions have the maximum variations irrespective of the underlying information generating structure. More precisely, the process recordings always consist of inevitable measurement noises, while PCA is not capable of characterizing the different influences from common process Gaussian impelling factors and random variations due to measure- ment equipments. A probabilistic generative model (probabilistic PCA, PPCA) was employed to address this issue via the expectation and maximization (EM) algorithm (Kim and Lee 2003). However, PPCA assumes the same noise level of all measure- ment variables, which cannot be easily satisfied in practice. To this end, a general extension of PPCA model, factor analysis (FA) can model different noise levels of measurement variables. To determine the number of essential variables, lots of use- ful approaches have been reported, including Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), SCREE test, PRESS cross validation test (Chiang et al. 2001) and more recent fault detection criteria (Valle et al. 1999; Wang et al. 2002, 2004). 3.2 Two-step ICA-PCA Information Extraction Strategy for Process Monitoring 3.2.1 Process Monitoring Method Giving the data matrix X, assume r independent components (ICs) are extracted, S = [s1, s2, . . . , sr ] ∈ Rr×n . In order to monitor the non-Gaussian part of the process, a new statistic variable was defined (Lee et al. 2004; Ge and Song 2007) I2 = sT s (3.1)
- 34. 3.2 Two-step ICA-PCA Information Extraction Strategy for Process Monitoring 15 After the non-Gaussian information has been extracted, the residual matrix E is obtained. Then PCA is used to analyse it, expanding E as follows E = k i=1 tipT i + F (3.2) where F is the residual resulting from the PCA model. Here, we define the limits of T2 and SPE statistics to monitor the remaining Gaussian part of the process T 2 = k i=1 t2 i λi ≤ k(n − 1) n − k Fk,(n−k),α, (3.3) SPE = ffT = e(I − PPT )eT ≤ SPElim (3.4) In the PCA monitoring method, the confidence limits are based on a specified dis- tribution, based upon the assumption that the latent variables follow a Gaussian distribution. However, in ICA monitoring, the independent component do not con- form to a specific distribution, hence, the confidence limit of the I2 statistic cannot be determined directly from a particular approximate distribution. An alternative ap- proach to define the nominal operating region of the I2 statistic is to use KDE (Chen et al. 2000, 2004). Here we only need a univariate kernel estimator, which is defined by ˆ f (I2 , H) = 1 n n i=1 K H−1/2 (I2 − I2 i ) (3.5) where H is the bandwidth matrix and K is a kernel function, which satisfies the following condition K(I2 ) ≥ 0, Rp K(I2 )dI2 = 1 (3.6) There are a number of kernel functions, among which the Gaussian kernel function is the most commonly used. Many methods have been proposed for the estimation of the window width or the smoothing parameter, which is of crucial importance in density estimation. One efficient method is called mean integrated squared error (MISE) (Hyvarinen and Oja 2000). There are three choices for H 1. For a process with p variables, a full symmetrical bandwidth matrix H = ⎡ ⎢ ⎢ ⎢ ⎣ h2 11 h2 12 · · · h2 1p h2 21 h2 22 · · · h2 2p . . . . . . . . . h2 p1 h2 p2 · · · h2 pp ⎤ ⎥ ⎥ ⎥ ⎦
- 35. 16 3 Non-Gaussian Process Monitoring which is a positive-definite matrix with p(p + 1)/2 parameters, in which h2 ik = h2 ki; 2. A diagonal matrix with only p parameters, H = diag(h2 1, h2 2, . . . , h2 p); 3. A diagonal matrix with one parameter, H = h2 I, where I is an identity matrix. Approach one is the most accurate but is unrealistic in terms of computational load and time.Approach three is the simplest, and the method used in the case of univariate data, and can be adopted with slight modification. However, it can cause problems in some situations due to the loss of accuracy by forcing the bandwidths in all dimensions to be the same. This can introduce significant error in the density function shape. Hence, approach two appears to be the appropriate choice, as a compromise, but the computational load is still unacceptable if the dimensionality of the problem is high and the size of the sample is large. The implementation of the monitoring method, as mentioned in the previous section, consists of two procedures: off-line modelling and on-line monitoring. In the off-line modelling procedure, an ICA-PCA monitoring model is developed in the normal operating condition. Then, the fault detection is executed by using this monitoring model in the on-line monitoring procedure. The detailed algorithm flow of the monitoring procedure is summarized as follows (1) Off-line modelling procedure Step 1 Acquire an operating dataset X during normal process; Step 2 Normalize and whiten the data matrix X; Step 3 Carry out the ICA algorithm to obtain the matrix W, B and Ŝ, so that Ŝ has great non-Gaussianity. We can also obtain A = (QT Q) −1 QT B and W = BT Q; Step 4 For each sample, calculate its I2 value, I2 (k) = ŝ(k)T ŝ(k), (k = 1, 2, . . . , n), using KDE to decide its 99 or 95 % confidence limit I2 lim; Step 5 Carry out the PCA algorithm to obtain the score matrix T and the load matrix P, decide the confidence limits T 2 lim and SPElim for T2 and SPE. (2) On-line monitoring procedure Step 1 For a new sample data xnew, using the same scaling method used in the modelling steps; Step 2 Calculate the ICs of the new sample data, xnew, ŝnew = Wxnew; Step 3 Calculate the I2 statistic value for this new sample data, I2 new = Ŝ T newŜnew; Step 4 The remaining part is given by enew = xnew − Aŝnew, calculate the score vector, tnew = enewP, and also obtain the residual, fnew = enew − ênew; Step 5 Calculate T 2 new and SPEnew using Eqs. (3.3) and (3.4); Step 6 If the statistics reject their corresponding limits, some kind of fault is detected, otherwise, go to step 1 and continue monitoring. The on-line process fault detection scheme of the ICA-PCA based method is shown in Fig. 3.1.
- 36. 3.2 Two-step ICA-PCA Information Extraction Strategy for Process Monitoring 17 Data scaling and whitening Monitoring three statistics: I2 -T2 and Fault identification, isolation and repair New sample data from process Any of the statistical value exceeds confidence limit? YES Extract ICs and calculate the I2 statistic value the T2 and statistic values for the current sample data Calculate the remain matrix E for PCA analysis ICA-PCA model no Calculate SPE SPE Fig. 3.1 Process monitoring strategy for the ICA-PCA method 3.2.2 TE Benchmark Case Study As a benchmark simulation, the Tennessee Eastman (TE) process has been widely used to test the performance of various monitoring approaches (Downs and Vogel 1993; Chiang et al. 2001; Raich and Cinar 1995; Singhai and Seborg 2006). This process consists of five major unit operations: a reactor, a condenser, a compressor, a separator and a stripper. The control structure is shown schematically in Fig. 3.2 which is the second structure listed in Lyman and Georgakist (1995). The TE process has 41 measured variables (22 continuous process measurements and 19 composition measurements) and 12 manipulated variables, a set of 21 programmed faults are introduced to the process. The details on the process description are well explained in a book of Chiang et al. (2001). In this process, the variables which are selected for monitoring are listed in Table 3.1. There are 33 variables in the table, 22 continuous process measurements and 11 manipulated variables. The agitation speed is not included because it is not manipulated. Besides, we also exclude 19 composition measurements as they are difficult to measure on-line in real processes. The simulation data which we have collected were separated into two parts: the training datasets and the testing datasets;
- 37. 18 3 Non-Gaussian Process Monitoring XG XH E FI D A FI Condense CWS CWR TI Purge FI Stin FI Cond Product FI C FI LI TI FI Compressor S E P A R A T O R PI S T R I P P E R FC XC FC XC LC XC LC FC CWS CWR TI Reactor PI XE XA XD FC FC FC TC TC FC XC XC FI XC LC FI TI LI TC XE FC TI PI PC FI LI Fig. 3.2 Control system of the Tennessee Eastman process Table 3.1 Monitoring variables in the TE process No. Measured variables No. Measured variables No. Manipulated variables 1 A feed 12 Product separator level 23 D feed flow valve 2 D feed 13 Product separator pressure 24 E feed flow valve 3 E feed 14 Product separator underflow 25 A feed flow valve 4 Total feed 15 Stripper level 26 Total feed flow valve 5 Recycle flow 16 Stripper pressure 27 Compressor recycle valve 6 Reactor feed rate 17 Stripper underflow 28 Purge valve 7 Reactor pressure 18 Stripper temperature 29 Separator pot liquid flow valve 8 Reactor level 19 Stripper steam flow 30 Stripper liquid product flow valve 9 Reactor temperature 20 Compressor work 31 Stripper steam valve 10 Purge rate 21 Reactor cooling water outlet temperature 32 Reactor cooling water flow 11 Product separator temperature 22 Separator cooling water outlet temperature 33 Condenser cooling water flow they consisted of 960 observations for each mode (1 normal and 21 fault), respec- tively, and their sampling interval was 3 min. All faults in the testing datasets were introduced in the process at sample 160, which are tabulated in Table 3.2. Faults 1–7 are step changes of process variables; faults 8–12 are random changes of variables; fault 13 is slow shift of Reaction kinetics; faults 14, 15 and 21 are related to valve sticking and faults 16–20 are types of unknown faults. Among these faults, some faults are easy to detect as they greatly affect the process and change the relations
- 38. 3.2 Two-step ICA-PCA Information Extraction Strategy for Process Monitoring 19 Table 3.2 Process disturbances Fault number Process variable Type 1 A/C feed ratio, B composition constant (stream 4) Step 2 B composition, A/C ratio constant (stream 4) Step 3 D feed temperature (stream 2) Step 4 Reactor cooling water inlet temperature step 5 Condenser cooling water inlet temperature Step 6 A feed loss (stream 1) Step 7 C header pressure loss-reduced availability (stream 4) Step 8 A, B, C feed composition (stream 4) Random variation 9 D feed temperature (stream 2) Random variation 10 C feed temperature (stream 4) Random variation 11 Reactor cooling water inlet temperature Random variation 12 Condenser cooling water inlet temperature Random variation 13 Reaction kinetics Slow drift 14 Reactor cooling water valve Sticking 15 Condenser cooling water valve Sticking 16 Unknown Unknown 17 Unknown Unknown 18 Unknown Unknown 19 Unknown Unknown 20 Unknown Unknown 21 Valve position constant (stream 4) Constant position between process variables. However, there are also faults that are difficult to detect (faults 3, 9 and 15) because they are very small and have little influence to the process. Sofar, 22datasetshavebeengenerated, correspondingtothe22differentoperation modes (1 normal and 21 fault) in the TE process. Before the application of PCA, ICA and ICA-PCA, all the datasets were auto-scaled. A total of 9 ICs and 15 principal components (PCs) were selected for ICA and PCA by cross-validation, respectively. In the study of the ICA-PCA method, we select the same number of ICs (9) and PCs (15) to compare the monitoring performance with ICA and PCA. The 99 % confidence limits of all the statistic variables were determined by KDE. For each statistic, the detection rates for all the 21 fault modes were calculated and tabulated in Table 3.3. The minimum missing detection rate achieved for each mode is marked with a bold number except the modes of faults 3 and 9; as these faults are quite small and have almost no effect on the overall process so they were excluded from our research. As shown in Table 3.3, ICA outperforms PCA for most fault modes, and particularly the missing detection rates of ICA for the modes of faults 5, 10 and 16 are much lower than that of PCA, which indicates that ICA can detect small events that are difficult to detect by PCA. However, the missing detection rates of ICA-PCA for most modes are even lower than that of ICA, as ICA-PCA not only monitors the non-Gaussian information of the process by ICA, but also the Gaussian information is monitored by PCA. As a result of the simulation, the best monitoring performance is found in the case of ICA-PCA. In most modes, the missing detection rate of ICA-PCA is the lowest. The monitoring results of fault 5 are shown in Fig. 3.3. The condenser cooling water inlet temperature is step changed in the mode of fault 5. In that mode, the flow
- 39. 20 3 Non-Gaussian Process Monitoring Table 3.3 Monitoring results of the TE process Fault modes PCA PCA ICA ICA ICA-PCA ICA-PCA ICA-PCA T2 SPE I2 SPE I2 T2 SPE 1 0.007 0.004 0.006 0.002 0.005 0.003 0 2 0.022 0.015 0.020 0.010 0.018 0.015 0.008 3 1.000 0.998 0.995 0.990 0.996 0.993 0.985 4 0.975 0.045 0.035 0 0.032 0.020 0 5 0.750 0.745 0 0 0 0 0 6 0.010 0 0 0 0 0 0 7 0.090 0 0.065 0 0.050 0.035 0 8 0.035 0.025 0.030 0.021 0.028 0.022 0.011 9 1.000 0.997 0.995 0.996 0.994 0.993 0.991 10 0.710 0.651 0.215 0.200 0.212 0.156 0.162 11 0.811 0.367 0.756 0.182 0.712 0.526 0.182 12 0.031 0.027 0.025 0 0.024 0.005 0 13 0.065 0.050 0.025 0 0.016 0.011 0 14 0.162 0 0.045 0 0.042 0.013 0 15 0.999 0.985 0.426 0.328 0.412 0.268 0.196 16 0.812 0.755 0.213 0.125 0.185 0.146 0.125 17 0.241 0.112 0.211 0.196 0.206 0.107 0.195 18 0.112 0.100 0.102 0.085 0.100 0.085 0.098 19 0.998 0.762 0.351 0.322 0.346 0.238 0.208 20 0.712 0.524 0.426 0.358 0.412 0.247 0.306 21 0.725 0.621 0.635 0.528 0.622 0.549 0.526 rate of the outlet stream from the condenser to the separator also increases, which causes an increase in temperature in the separator, and thus affects the separator cooling water outlet temperature. As we have established a control structure for this process, the control loops will act to compensate for the change and the temperature in the separator will return to its set-point. It takes about 10 h to reach the steady state again.As shown in Fig. 3.3a, PCA has detected the fault at sample 160 approximately. However, the fault cannot be detected after approximately sample 350 as most of the variables returned to their set-points. In this case, if a process operator judges the status of the process based on PCA, they would probably conclude that a fault entered the process and then corrected itself in about 10 h. However, as a matter of fact, the condenser cooling water inlet temperature is still high than normal condition after sample 350, and the condenser cooling water flow rate is continuously manipulated while most variables return to their set-points. This indicates that a fault still remains in the process. As PCA is based on only second-order statistics information, the effect of the condenser cooling water flow may be neglected compared to the effect of other variables, thus, the problem occurs. Alternatively, as shown in Fig. 3.3b and 3.3c, all the ICA and ICA-PCA statistic variables stayed above their confidence limit, which indicate that a fault remains in the process. What distinguished ICA from PCA is that ICA involves higher order statistics information, which is non- Gaussian and, thus, can correctly reflect the effect of condenser cooling water flow rate. Such a persistent fault detection statistic will continue to inform the operator that a process abnormality remains in the process; although, all the process variables will appear to have returned to their normal values through control loops. In the
- 40. 3.3 Process Monitoring Based on ICA-FA and SVDD 21 0 100 200 300 400 500 600 700 800 900 1000 0 50 100 150 200 T 2 0 100 200 300 400 500 600 700 800 900 1000 0 20 40 60 sample number SPE 0 100 200 300 400 500 600 700 800 900 1000 0 500 1000 1500 I 2 0 100 200 300 400 500 600 700 800 900 1000 0 500 1000 1500 2000 2500 sample number SPE a b c 0 100 200 300 400 500 600 700 800 900 1000 0 5000 10000 I 2 0 100 200 300 400 500 600 700 800 900 1000 0 1000 2000 3000 T 2 0 100 200 300 400 500 600 700 800 900 1000 0 500 1000 1500 sample number SPE Fig. 3.3 Monitoring results for mode of fault 5 of TE process: a result based on PCA, b result based on ICA, c result based on ICA-PCA method of ICA-PCA, as ICA has extracted essential components that underlie the process,the following procedure of PCA can reflect the effect of condenser cooling water flow rate. So, ICA and ICA-PCA may provide the process operator with more reliable information. Additionally, comparing the result of ICA-PCA with that of ICA, we can conclude that the effect of the fault is magnified more in the former statistic variables. 3.3 Process Monitoring Based on ICA-FA and SVDD 3.3.1 Support Vector Data Description The main idea of SVDD is to map the input vectors to a feature space and to find hypersphere with the minimum volume which separates the transferred data from the rest of the feature space. Applications have shown a high generalization performance of SVDD if large reference dataset with few abnormal samples is available (Tax and Duin 1999).
- 41. 22 3 Non-Gaussian Process Monitoring Assume a dataset containing n samples ŝi ∈ Rr , i = 1, 2, . . . , n is given, the mapping from the original space to the feature space : ŝ → F can be simply done by a given kernel K(si, sj ) = (si), (ŝj ) , which computes the inner product in the feature space. Here, the most popular Gaussian kernel is used. To construct the minimum volume of the hypersphere, SVDD solves the following optimization problem min R,a,ξ R2 + C n i=1 ξi s.t. (ŝi) − a 2 ≤ R2 + ξi, ξi ≥ 0 (3.7) where a is the center of the hypersphere in the feature space. The variable C gives the trade-off between the volume of the sphere and the number of errors (number of target objects rejected). ξi represents the slack variable which allows the probability that some of the training samples can be wrongly classified. The dual form of the optimization problem can be obtained as follows min αi n i=1 αiK(ŝi, ŝj ) − n i=1 n j=1 αiαj K(ŝi, ŝj ) s.t. 0 ≤ αi ≤ C, n i=1 αi = 1 (3.8) where αi is Lagrange multipliers. After the hpyersphere in the feature space has been constructed, the hypothesis that a new sample ŝnew is normal is accepted if the distance d[(ŝnew)] = (ŝnew) − a ≤ R, which is given by d[(ŝnew)] = K(ŝnew, ŝnew) − 2 n i=1 αiK(ŝi, ŝnew) + n i=1 n j=1 αiαj K(ŝi, ŝj ) (3.9) 3.3.2 Process Monitoring Based on ICA-FA and SVDD As demonstrated, conventional statistical methods are under the assumption that pro- cesses are driven by fewer essential variables. These essential variables are either non-Gaussian or Gaussian distribution. However, under more general circumstances, processes are driven by non-Gaussian and Gaussian essential variables simulta- neously, which formulates the key assumption of the mixed essential component analysis (ICA-FA) method. Figure 3.4 shows the difference between ICA-FA and the conventional methods (ICA and FA). The relationship of measurement variables and mixed essential variables is given as follows (Ge et al. 2009a) x = x1 + x2 = As + Pt + e (3.10)
- 42. 3.3 Process Monitoring Based on ICA-FA and SVDD 23 ... ... s2 s1 t1 t2 sr e ... s1 ... e ICA-FA ICA FA t1 Process records Process records Process records s2 sr t2 t1 t1 Fig. 3.4 Description of ICA-FA versus ICA and FA where x1 and x2 represent non-Gaussian and Gaussian parts of the process, respec- tively, A is the mixing matrix of non-Gaussian essential variable s, P is the loading matrix of Gaussian essential variables t and e is the Gaussian un-modelled uncer- tainty, relating to measurement noises, etc., with zero means and different variances =diag{λi }1, 2,... , m, which permits measurement variables to have different noise levels. As a general form of PPCA, FA performs better than PCA and PPCA. Since pro- cess measurements are always corrupted by noise, the ignorance of such information generation structure will make PCA method leave some essential information in the discarded space and trigger monitoring errors. In contrast, the noise structure is con- sidered in PPCA and FA; thus, the useful information can be correctly extracted and monitored separately. To estimate the new parameter set = {A, P, } and extract essential variables EA = {s, t} from measurement variables x, a two-step estimation and extraction strategy can be developed. First, non-Gaussian essential variables can be extracted by the efficient FastICA or PSO-based method (Xie and Wu 2006), thus the mixing matrix A is obtained. Second, the parameter P and can be estimated by the EM algorithm, which is described as follows. First, the mean and variance of the Gaussian part of measurement variables x2 can be calculated as μx2 = PE[t] + E[e] = 0; x2 = PE[ttT ]PT + E[eeT ] = PPT + (3.11) Hence, the distribution of x2 is p(x2) = N{0, PPT +}. EM is an iterative algorithm, it consists of two steps: E-step and M-step. In E-step, two statistics of the Gaussian essential variable t are estimated t̂i = E[ti|x2i] = Qx2i E[titT i |x2i] = I − QP + Qx2i xT 2i QT (3.12) where Q = PT (PPT + ) −1 . In M-step, the parameters P and are estimated so that the value of the likelihood function is maximized, the estimated parameters are calculated as
- 43. 24 3 Non-Gaussian Process Monitoring P = n i=1 x2i t̂T i n i=1 E[titT i |x2i] −1 = n i=1 diag(x2i xT 2i − Pt̂ixT 2i ) n (3.13) where n is the number of samples, diag(Z) means to make a diagonal matrix using diagonal elements of matrix Z. Equations (3.12) and (3.13) are calculated iteratively until both of the parameters (P and ) are converged. In summary, when the reference dataset X = {xi}i=1, 2,..., n is available, with the ICA-FA model defined by Eq. (3.10), process information can be separated into three different parts. The systematic information includes non-Gaussian and Gaussian parts which are driven by their corresponding essential variables, whereas the noise information and other unmeasured disturbances are represented by e. As described in the previous section, the entire information of the process is par- titioned into three parts: non-Gaussian systematic information, Gaussian systematic information and noise information. Therefore, three different monitoring statistics can be developed for fault detection. First, for monitoring non-Gaussian informa- tion, SVDD-based method is used. In contrast to the confidence limit determination method KDE, the SVDD reduces the complexity to quadratic programming prob- lem that produces a unique analytical solution. Given the training dataset S (the extracted independent components), the hypersphere is constructed. The center a and the radius R can be determined by (Tax and Duin 2004) a = n i=1 αi(ŝi) R = 1 − 2 n i=1 αiK(ŝz, ŝj ) + n i=1 n j=1 αiαj K(ŝi, ŝj) (3.14) The non-Gaussian statistic (NGS) can be developed as the distance between the data sample and the center of the hypersphere, thus NGSi = d2 [(ŝi)] = (ŝi) − a 2 ≤ NGSlim = R2 (3.15) where ŝ i is the extracted independent component of the new sample, d[(ŝi)] is given in Eq. (3.9). If NGSi ≤ R2 , the non-Gaussian information is regarded as normal, else, it will be regarded as an outlier or some fault and disturbance has happened. Since the distribution of Gaussian essential variables t is assumed to be N{0, I}, the Mahalanobis norm of t is identical to the Euclidian norm. Different from the projection method, the Gaussian essential variables cannot be calculated directly. However, it can be substituted by their estimation, which is given in Eq. (3.12).
- 44. 3.3 Process Monitoring Based on ICA-FA and SVDD 25 It is known that the squared Mahalanobis norm of t follows χ2 distribution, the confidence limit of T2 statistic can be determined as follows T 2 i = t̂i 2 = xT 2i QT Qx2i ≤ χ2 α,l (3.16) where χ2 α,l represent the confidence limit with α confidence and l is the corresponding free parameter. If the value of T2 statistic goes beyond the confidence limit, the Gaussian systematic process is regarded out-of-control. Similarly, the noise part of the process information can also be monitored by the squared Mahalanobis distance. Hence, the Gaussian information matrix x2 can be represented as x2 = Pt + e (3.17) Notice Eqs. (3.12) and (3.17), the noise information e can be estimated as follows êi = E[ei|x2i] = (I − PQ)x2i (3.18) Then, the confidence limit of SPE statistic can be determined by SPEi = −1/2 ei 2 = xT 2i (I − PQ)T −1 (I − PQ)x2i ≤ χ2 α,m (3.19) where χ2 α,m represent the confidence limit with α confidence and m is the correspond- ing free parameter. If the value of SPE statistic goes beyond the confidence limit, the noise part of the process is regarded abnormal. This may be due to the change in noise level, some unmeasured disturbances and other process structure changes. Note that these two Gaussian parts of information both follow the χ2 distribution; they can be monitored together. According to Eq. (3.11), the entire Gaussian statistic (GS) can be constructed as follows GSi = (PPT + ) −1/2 x2i 2 = xT 2i (PPT + ) −1 x2i ≤ χ2 α,m (3.20) Therefore, if changes happen in Gaussian part, the entire monitored GS should go beyond its confidence limit. However, if one wants to isolate the change between the systematic Gaussian part and the noise part, T2 and SPE statistics should be chosen. In summary, four statistics (NGS, T2 , SPE and GS) have been developed for process monitoring. By monitoring NGS and GS, the change between non-Gaussian and Gaussian information can be isolated. Furthermore, the change of Gaussian information can be isolated by monitoring T2 and SPE separately. Hence, three types of changes can be monitored separately by their corresponding statistics. Change of any type or types of information will be reflected in related monitoring charts. 3.3.3 Case Study In this section, the performance of the method is tested through the TE process. The variables selected for monitoring are the same as those used for control structure
- 45. 26 3 Non-Gaussian Process Monitoring Fig. 3.5 Scatter plot of TE normal operation condition -2 0 2 4 -3 -2 -1 0 1 2 3 s1 s 2 development in Lyman and Georgakist (1995). Simulation data which are acquired in the normal process consist of 960 samples, and their sampling interval was 3 min. In order to evaluate fault detection and identification performance of the method, simulation data under 21 fault processes are also obtained. During the simulation, each fault is introduced in the 161st sample. For monitoring of the TE process, the number of ICs and principal factor (PFs) should be chosen appropriately. In this study, 3 ICs and 6 PFs are selected. Then the ICA-FA model can be developed. The kernel parameter for the Gaussian function is chosen as σ = 2.0, the variable C is selected as C = 0.0625. For comparison, the ICA, PCA and ICA-PCA algorithms are also carried out. The number of ICs for ICA is selected as 3, whereas the PCs number for PCA is chosen as 6. All confidence limits are set as 99 %. The scatter plot of the first independent component versus the second independent component and two confidence limits of NGS and I2 is given in Fig. 3.5. Under normal operation, both of the confidence limits indicate that 99 % samples are inside their corresponding regions. However, as discussed previously, the confidence limit of NGS is tighter than that of I2 , which is shown correctly in the figure. Therefore, when some fault happens, NGS is more sensitive to detect this fault. As samples lie in the gap between the NGS confidence limit and the I2 confidence limit, only NGS can detect them. The I2 statistic regards it as a normal operation because the confidence limit has not been rejected. For each statistic, the monitoring results (type II error) are tabulated in Table 3.4. The minimum value achieved for each mode is marked with a bold number. As shown in the table, ICA-FA outperforms ICA, PCA and ICA-PCA for most of the fault modes. Particularly, the missing detection rates of ICA-FA for faults 10, 15, 16 and 19–21 are much lower than those of the other three methods. As indicated in the 2–4 columns of the table, best monitoring performance of some faults are revealed by the NGS, some are gained by T2 , whereas others are gained by the SPE statistic, which shows the fault isolation ability of the ICA-FA method.
- 46. 3.4 Conclusions 27 Table 3.4 Monitoring results of TE process (type II error) Mode ICA-FA ICA PCA ICA-PCA NGS T2 SPE I2 SPE T2 SPE I2 T2 SPE 0 0.990 0.994 0.994 0.988 0.991 0.986 0.988 0.988 0.991 0.989 1 0.015 0.003 0.005 0.003 0.005 0.008 0.005 0.003 0.011 0.003 2 0.018 0.018 0.021 0.023 0.018 0.034 0.015 0.023 0.018 0.016 3 0.911 0.993 0.995 0.974 0.983 0.984 0.974 0.974 0.990 0.979 4 0.963 0.986 0.994 0.990 0.989 0.990 0.981 0.990 0.993 0.984 5 0.764 0.778 0.855 0.790 0.773 0.773 0.814 0.790 0.785 0.868 6 0.008 0.000 0.003 0.014 0.000 0.001 0.000 0.014 0.007 0.000 7 0.596 0.664 0.811 0.641 0.670 0.618 0.735 0.641 0.681 0.751 8 0.050 0.029 0.118 0.089 0.031 0.059 0.039 0.089 0.064 0.069 9 0.933 0.985 0.995 0.981 0.976 0.980 0.974 0.981 0.988 0.975 10 0.385 0.336 0.271 0.579 0.596 0.685 0.728 0.579 0.841 0.519 11 0.886 0.789 0.556 0.961 0.605 0.859 0.579 0.961 0.815 0.538 12 0.124 0.009 0.155 0.081 0.024 0.029 0.088 0.081 0.059 0.100 13 0.060 0.074 0.083 0.068 0.074 0.064 0.056 0.068 0.098 0.079 14 0.061 0.001 0.000 0.100 0.001 0.058 0.000 0.100 0.001 0.000 15 0.900 0.978 0.993 0.976 0.975 0.968 0.975 0.976 0.984 0.980 16 0.664 0.853 0.841 0.748 0.886 0.788 0.850 0.748 0.913 0.929 17 0.261 0.103 0.039 0.243 0.059 0.184 0.046 0.243 0.179 0.041 18 0.091 0.101 0.106 0.105 0.104 0.108 0.101 0.105 0.109 0.098 19 0.869 0.981 0.809 0.971 0.814 0.905 0.851 0.971 0.829 0.866 20 0.449 0.490 0.454 0.594 0.594 0.780 0.684 0.594 0.818 0.524 21 0.374 0.806 0.755 0.434 0.853 0.689 0.554 0.434 0.921 0.874 3.4 Conclusions In this chapter, different non-Gaussian process monitoring methods have been re- viewed, especially the ICA-based methods. For processes which are simultaneously driven by non-Gaussian and Gaussian components, a two-step ICA-PCA informa- tion extraction strategy has been demonstrated, based on which a corresponding monitoring scheme has been developed. Compared to the traditional ICA and PCA monitoring methods, the monitoring performance has been improved by the ICA-PCA based approach. Furthermore, the SVDD method has been introduced for modelling the distribution of the independent components, based on which a non-Gaussian monitoring statistic has been developed. Compared to the conven- tional I2 monitoring statistic, the monitoring performance has been improved by this non-Gaussian monitoring statistic. Besides, for probabilistic monitoring of the Gaussian information of the process, the FA method has been incorporated. Two additional Gaussian monitoring statistics have been constructed for process moni- toring. Through the simulation study of the TE benchmark process, the efficiency of both ICA-PCA and ICA-FA with SVDD methods have been demonstrated.
- 47. Chapter 4 Fault Reconstruction and Identification 4.1 Introduction Fault detection and diagnosis are of ever growing importance for guaranteeing a safe and efficient operation of complex industrial processes. Over the past few decades, statistical-based techniques have been intensely researched, as they directly address thechallengesoflargerecordedvariablesetsresultingfromtheincreasingcomplexity of distributed control systems. Notable algorithmic developments have been firmly made based on principal component analysis (PCA) and partial least squares (PLS; Qin 2003; Venkatasubramanian et al. 2003). Whilst statistically based fault detection schemes are well-established, several fault diagnosis methods have also been proposed, such as contribution plot and the backward elimination sensor identification (BESI) algorithm (Westerhuis et al. 2000; Storketal. 1997). However, thesemethodsaredifficulttoimplementinpracticegiven that the contribution method may produce misleading results (Lieftucht et al. 2006) and the BESI algorithm requires a new PCA model to be calculated for each step (Stork et al. 1997). For enhanced fault isolation, Gertler et al. (1999) introduced a structure residual-based approach, which was later improved upon by Qin and Li (2001). Dunia and Qin (1998a, b) proposed a uniform geometric method for both unidimensional and multidimensional fault identification and reconstruction. This projection-based variable reconstruction work relied on the conventional SPE statis- tic. More recent work by Wang et al. (2002) discussed the detectability of faults for the PCA method using the T2 statistic and Yue and Qin (2001) suggested the use of a combined index for reconstruction-based fault identification and isolation method. Other extensions of these methods include Lieftucht et al. (2006) and Li and Rong (2006). However, these methods rely on PCA, which is applicable under the assumption that the recorded process variables follow a multivariate Gaussian distribution. For monitoring processes that exhibit non-Gaussian behavior, recent work on in- dependent component analysis (ICA) has shown its potential (Lee et al. 2004a, b; Ge and Song 2007; Liu et al. 2008). Although ICA has been successfully employed Z. Ge, Z. Song, Multivariate Statistical Process Control, Advances in Industrial Control, 29 DOI 10.1007/978-1-4471-4513-4_4, © Springer-Verlag London 2013
- 48. 30 4 Fault Reconstruction and Identification to extract non-Gaussian source signal that can be used to construct univariate moni- toring statistics for fault detection, the issue of fault identification and isolation has only been sporadically touched upon. A notable exception is discussed in Lee et al. (2004a), where contribution plots for fault identification were developed for ICA models. However, as mentioned above, contribution plot may produce misleading and erroneous fault diagnosis results. This chapter introduces a technique that relies on PCA-based fault reconstruction to address the important issue of fault diagnosis for multivariate non-Gaussian pro- cesses (Ge and Song 2009b). This method was developed for a description of the independent components in the feature space determined by the support vector data description (SVDD). In a similar fashion to PCA-based reconstruction, this scheme relies on predefined fault directions, where the effect of a fault upon these direction is evaluated using a developed fault diagnosis index. For fault identification, Zhang et al. (1996) proposed the characteristic direction method to identify faults, which took the first loading vector as the characteristic direction. Although the first principal component extracted most of the fault infor- mation, lot of other information has been omitted. Krzanowski (1979) developed a method for measuring the similarity of two datasets using a PCA similarity factor SPCA. Johannesmeyer et al. (2002) used that PCA similarity factor to identify the similarity between different operation modes. Within that method, all the informa- tion that the PCA model had covered was used, so the power of fault identification has been improved. However, these similarity factors are all based on the assump- tion that the data formed Gaussian distribution. For non-Gaussian process data, an ICA similarity factor has been defined (Ge and Song 2007). Based on the ICA-FA method in Chap. 3 a further similarity factor for the noise subspace of the process data has also been defined. These similarity indices are used to classify various different process faults. 4.2 Fault Reconstruction 4.2.1 Non-Gaussisan Fault Reconstruction Based on SVDD For the introduction of the fault reconstruction method, we assume that the fault subspace is known a priori. This is not a restriction of generality, as the application of a singular value decomposition can estimate this subspace (Qin 2003). Assuming there are a total of J possible fault conditions, denoted here by Fj , j = 1, 2, . . . , J , each fault condition is described by an individual fault subspace. These subspaces are of dimension dim (j ) ≥ 1, j = 1, 2, . . . , J , which implies that unidirec- tional, dim(j ) = 1, and multidimensional fault conditions dim(j ) 1 can be reconstructed. After defining the fault subspace, z∗ can be reconstructed from the corrupted data vector z using the ICA model and the SVDD method. Suppose a fault Fj has occurred, a reconstructed zj represents an adjustment of the corrupted value z moving along
- 49. 4.2 Fault Reconstruction 31 the corresponding fault direction j zj = z − j fj (4.1) where fj is a vector storing the estimated fault magnitude such that zj is the closest to the normal region. Traditionally, the optimal reconstruction is obtained by min- imizing ||zj − z∗ j ||. For this method, the reconstruction is realized by moving the corrupted value (ŝ) along the defined fault direction j into the SVDD feature space. The reconstruction technique therefore leads to the following problem fj = arg min f ||(ŝ∗ ) − a|| = arg min f ||(ŝ − Wj f) − a|| ŝ = Wz ŝ∗ = Wz∗ (4.2) However, the optimization problem of Eq. (4.2) is problematic, since the formulation of the mapping function ( · ) is unknown. To overcome this deficiency, the iterative denoising and reconstruction approach developed for kernel PCA (Mika et al. 1998; TakahashiandKurita2002)canbeinsertedintoEq. (4.2).WithN nsupportvectors, obtained by the SVDD method, the length of (ŝ) − a is (Tax and Duin 2004) ||(ŝ) − a|| = K(ŝ, ŝ) − 2 N i=1 αiK(svi, ŝ) + N i=1 N j=1 αiαj K(svi, svj ) (4.3) where ŝvi is the ith support vector obtained by the SVDD method and αi is the corresponding coefficient. As this work utilizes the Gaussian kernel function K(ŝi, ŝj ) = exp (−||ŝi − ŝj ||2 /δ), K(ŝ, ŝ) = 1, and N i=1 N j=1 αiαj K(ŝvi, ŝvj ) is a constant. By incorporating Eq. (4.3), the optimization problem in Eq. (4.2) finally becomes fj = arg max f N i=1 αiK(svi, s∗ j ) = arg max f N i=1 αiK(svi, ŝ − Wj f) (4.4) Denoting = N i=1 αiK(svi, Wj f) and taking the first partial with respect to f yields ∇f = ∂ ∂f = 0 (4.5) which for Gaussian kernel functions is equal to ∇f = [Wj ]T N i=1 αiK(svi, ŝ − Wj f)(ŝ − Wj f − svi) = 0 (4.6) which can be simplified to ∇f = [Wj ]T Wj N i=1 αiK(svi, ŝ − Wj f)f + [Wj ]T N i=1 αiK(svi, ŝ − Wj f)(svi − ŝ) = 0. (4.7)
- 50. 32 4 Fault Reconstruction and Identification By rewriting Eq. (4.7), the unknown fault vector f can be estimated as follows f = [j T WT Wj ] −1 j T WT N i=1 αiK(svi, ŝ − Wj f)(ŝ − svi) N i=1 αiK(svi, ŝ − Wj f) (4.8) Appendix shows that the estimated fault vector fj can be found iteratively f(t + 1) = [j T WT Wj ] −1 j T WT N i=1 αiK(svi, ŝ − Wj f(t))(ŝ − svi) N i=1 αiK(svi, ŝ − Wj f(t)) (4.9) such that z∗ j = z − j f∗ j (4.10) 4.2.2 Fault Diagnosis It is important to assess the impact of reconstructing each fault condition {Fj , j = 1, 2, . . . , J}. After reconstructing each of the possible fault conditions, the correct fault direction will allow the reconstruction procedure to shift the recon- structed sample z∗ j closest to the center of the SVDD sphere in the feature space. This, in turn, implies that the value of the NSGj reduces significantly and shows an in-statistical-control situation if Fj is the correct fault condition. In contrast, if an incorrect direction is applied {Fi, i = 1, 2, . . . , J, i = j}, the univariate NGSi statistic still violates its control limit. Calculating the non-Gaussian statistic value NGSj by using Eqs. (4.11) and (4.13) ŝ∗ j = Wẑ∗ j (4.11) (ŝ∗ j ) − a = K(ŝ∗ j , ŝ∗ j ) − 2 N i=1 αiK(svi, ŝ∗ j ) + N i=1 N j=1 αiαj K(svi, svj ) (4.12) NGSj = (ŝ∗ j ) − a 2 2 (4.13) the impact of the reconstruction procedure upon the jth fault direction therefore relies on ξj ξj = NGSj NGSlim (4.14)
- 51. 4.2 Fault Reconstruction 33 which is referred to here as the fault diagnosis index and NGSlim is the control limit of the NGS statistic. If the reconstruction procedure has been applied using the correct fault direction, Eq. (4.14) highlights that a fault diagnosis index produces a value that is smaller than or equal to 1. This, in turn, implies that the faulty sample has been shifted along the correct fault direction to fall within the normal region. However, if an incorrect fault direction is applied this value still exceeds 1. 4.2.3 Simulation Example The six process variables are simulated as linear combinations of a total of three source variables. In order to account for measurement uncertainty, normally distributed measurement noise was superimposed onto this matrix vector product z = As + e. (4.15) The linear combinations are governed by the mixing matrix A, which was randomly selected to be A = ⎡ ⎣ 0.815 0.906 0.127 0.913 0.632 0.098 0.279 0.547 0.958 0.965 0.158 0.971 0.957 0.485 0.800 0.142 0.422 0.916 ⎤ ⎦ T . (4.16) Thes∈R3 isavectorofsourcesignaleachofwhichfollowsauniformdistributionwith [0 1] and e are Gaussian distributed i.i.d. sequences of zero mean and variance 0.01, e ∼ N {0,0.01I}. To test the performance of the fault reconstruction and diagnosis method, a total of 1,000 samples were simulated. The first 500 samples served as reference data, whilst the remaining 500 samples were used for model validation and to inject fault conditions for studying the performance of the fault reconstruction scheme (test dataset). Selecting the parameters of the Gaussian kernel functions of the SVDD as C = 0.08 and σ = 6.5 produced the control limit of the NSG statistic for a significance level of 5 %. The first fault condition was a bias of magnitude 2 injected to the second sensor in the test dataset after 250th sample. The conventional ICA-SVDD (Liu et al. 2008) method could successfully detect this fault. Next, applying the fault identification and isolation method and ICA contribution plots produced the plots shown in Fig. 4.1. As discussed in the Sect. 4.2.2, values below or equal to 1 imply this fault condition is responsible for the detected abnormal event. Values above 1 indicate that this fault condition cannot be considered as a potential root cause. Given the way in which the fault was injected the fault identification index for reconstructing along the direction of variable 2 should be identified as significant, which plot a confirms. However, the identification results of the contribution plot (ICA) in Fig. 4.1, plots 4.1b and 4.1c, show that most variables show a significant response to this event for the I2 cont,j and variables 4 and 5 for SPEcont, j.
- 52. 34 4 Fault Reconstruction and Identification 1 2 3 4 5 6 0 0.5 1 1.5 Assumed Fault Condition 1 2 3 4 5 6 0 5 10 15 20 25 Variables I 2 cont a b c d e f 1 2 3 4 5 6 0 5 10 15 20 25 Variables SPE cont 1 2 3 4 5 6 251 253 255 257 259 0 0.5 1 1.5 Assumed Fault Condition S a m p l e N u m b e r 1 2 3 4 5 6 251 253 255 257 259 0 10 20 30 Variables Sam ple Num ber I 2 cont 1 2 3 4 5 6 251 253 255 257 259 0 10 20 30 Variables Sample Number SPE cont Fig. 4.1 Average results for 250 samples describing bias in sensor 2. a SVDD reconstruction. b ICA contribution plot. c SPE contribution plot and individual plots for first ten samples. d SVDD reconstruction. e ICA contribution plot. f SPE contribution plot
- 53. 4.2 Fault Reconstruction 35 To analyze this misleading result in more detail, the way in which the variable contributions are generated need to be reexamined I2 cont,j ∝ eT j Q† B ˆ s = eT j Q† BBT Qz = eT j z. (4.17) If m* = m and d = m, it follows that = I otherwise reduces to a matrix of rank min{m*,d}. The variable contributions to the SPE statistic are given by SPEcont,j = eT j [I−ABT Q]z 2 = eT j z 2 . (4.18) If we now define z by z0 + z where z0 is the measured vector without the impact of the fault condition and z is a vector that describes the fault impact, Eqs. (4.17) and (4.18) give rise to I2 cont,j ∝ eT j (z0 + z) = I2 cont,j + I2 cont,j SPE2 cont,j = (eT j (z0 + z))2 (4.19) = SPE2 cont,j + SPE0cont,j + 2 SPE0cont,j SPEcont,j . This separation shows that the fault contributions to the ICA-I2 and -SPE statistics depend on eT j z and eT j z, respectively. Assuming that the ith sensor is faulty, the contribution of this sensor bias upon the residual variables ej is as follows I2 cont,j = ωT j z = m i=1 ωijzi SPEcont,j = γ T j z = m i=1 γ ijzi (4.20) where zi is the magnitude of the sensor bias, ω ω ωT j and γ T j is the jth row vector of and whose ith elements are ij and γ ij, respectively. As no assumptions can generally be imposed on and with respect to its symmetry etc., there is no guarantee that the residual associated with the faulty sensor is the largest one. The above equation also suggests the possibility that variable contributions which, by default, should be insignificant may, in fact, be significant and indicate that these variables are affected by the fault, which this simulation example confirms. For d = m = m*, Eqs. (4.19) and (4.20) suggest that I2 cont, j = zj , which produces a correct variable contribution to the fault condition. With the application of the reconstruction technique, a different picture emerges. A total of six fault directions, one for each sensor, were considered. Figure 4.1a shows the fault reconstruction index for each of the examined fault conditions. As expected, a bias for the second sensor yielded the smallest value and below 1 and hence, according to Eq. (4.14) the most significant impact of the reconstruction approach is for this sensor. In contrast, reconstructing the other fault directions (sensor faults) had an insignificant impact, leaving the fault reconstruction indices to exceed 1. It should be noted that plots a–c represent the average values for the 250 faulty samples. For illustrative purposes, Fig. 4.1 also shows the reconstruction indices and the variable contributions for the first ten samples in plots d–f.
- 54. 36 4 Fault Reconstruction and Identification Next, we consider a multidimensional fault case. A multiple sensor fault is gen- erated by injecting sensor bias of magnitude 2 to sensors 2 and 5 after the 250th sample of the test dataset. Along with the six fault conditions describing a bias for each sensor, there are now seven conditions and consequently seven different fault directions where the direction for the multidimensional sensor bias is 7 = 0 1 0 0 0 0 0 0 0 0 1 0 !T (4.21) Figure 4.2 summarizes the results of analyzing the 250 faulty samples. Plot a in this figure confirms that this event was correctly diagnosed as fault condition no. 7. In contrast, none of the remaining fault conditions produced a significant effect upon the corresponding fault identification index and hence produced values that exceeded 1. As before, plots b and c show that contribution charts for the ICA model, which again, produce a misleading picture. This can be explained through the use of Eqs. (4.17) to (4.20). Plots d–f show the reconstruction indices and the variable contribution charts for the first ten samples, respectively. 4.3 Fault Identification 4.3.1 PCA-Based Similarity Factor Krzanowski (1979) developed a method for measuring the similarity of two datasets using a PCA similarity factor, SPCA. Let us consider two datasets, D1 and D2 that have thesamevariablesbutnotnecessarilythesamenumberofmeasurements. Supposewe have selected k principal components for the two PCA models. The PCA similarity factor is a measure of the similarity between datasets D1 and D2 and to be exact, it is between the two reduced subspaces. Denoted L and M, the two subspaces, the PCA similarity factor can be calculated from the angles between principal components SPCA = 1 k k i=1 k j=1 cos2 Θij (4.22) where Θij is the angle between the ith principal component of dataset D1 and the jth principal component of dataset D2 which is defined as cos Θij = LT i Mj ||Li ||2||Mj ||2 , here ||Li||2=||Mj ||2 = 1 and 0 ≤ Θ ≤ π/2 SPCA can also be expressed by subspaces L and M as SPCA = trace(LT MMT L) k (4.23) If the value of SPCA exceeds a specified cutoff, the two datasets D1 and D2 are con- sidered to be similar. By introducing different weight for each principal component, this similarity factor has been modified by Singhai and Seborg (2002). Besides, Zhao et al. (2004) also modified the similarity factor by introducing the principal angle, in order to mine the most similar information between two datasets.
- 55. Another Random Document on Scribd Without Any Related Topics
- 56. convenience and economy with which it serves these ends. If any other property institution can, in a given situation, serve a given end more easily and more cheaply than the institution of interest, then, in that situation, the institution of interest—other things being equal —is immoral and should be abolished. If, in the given situation, no other property institution can serve the given end more easily and more cheaply than the institution of interest, then that institution is moral and should be retained. That is, from the modern sociological point of view, the institution of interest is inconceivable except as a means to some end outside itself. As a means it is to be judged in a purely objective and pragmatic manner by the ordinary standards of cost price, economic, social, and other. The method of the ancients is entirely otherwise. Assuming still the correctness of the modern viewpoint, which viewpoint be it said is not unassailable and indeed is assailed by divers radicals, socialists and others, but for the most part persons lacking in pecuniary reputability; the mistake then, that the Early Church fathers make is that of taking the means for an end. They have many arguments against interest but all these arguments can be criticised for this one error. The fathers elevate interest to the dignity of an end in itself. Interest, qua interest, is condemned. It is taking advantage of a brother's necessity. It is grinding the face of the poor. It is producing pride, luxury, and vice. As soon as moral value is attached to anything, it of course, is viewed as an end in itself. If it be true that interest is an end in itself, then the fiercest diatribes of the fathers are none too severe. Assuming their premises, their conclusions follow inevitably. The modern man—he is not unknown—who talks about the sacred rights of private property is guilty of the same error as the ancient Christians, the error of mistaking means for ends. The early Christians could not see that the property institution of interest is neither good nor bad except as it is good or bad for something. The something determines the judgment. As a matter of historical fact the condemnation of interest developed in certain early stages of human civilization and at those stages interest was socially detrimental. At those stages, however, it was exceedingly
- 57. rare and correspondingly infamous. In any country where there is abundance of good, free land the phenomenon of interest on money will disappear, provided labor is free. So it disappeared in the northern states of this Union in the later part of the 18th century. These phenomena caused the southerners to adopt slavery though all their English traditions had declared it immoral for more than three centuries. The relation of interest to slavery under a condition of free land is the relation of cause and effect, i.e., the requirement of interest will produce slavery and the abolition of interest will abolish slavery.[26] These social phenomena are of importance in our consideration of the early Christian doctrine of interest. That doctrine was largely evaded and disobeyed but it still had great effect and that effect was toward the abolition of slavery. We do not mean that this economic doctrine alone resulted in the abolition of slavery, or even that it was a chief cause in the abolition of slavery, it was not obeyed well enough to be such a chief cause; but so far as it was obeyed, it tended in that direction. The net result of all Christian teaching together was to prolong the existence of the institution of slavery for two centuries, perhaps for three. The doctrine of the sinfulness of interest however, worked toward emancipation and forced slavery in its later end to become almost wholly agricultural, i.e., to yield income as rent. Slaves cannot be employed in commerce or industry in sufficient numbers to be profitable where the institution of interest is banned as it was in the 'dark ages.' The Christian concept of interest undermined ancient civilization by abrogating, slowly but surely, the institution of property by which such gangs of 'manufacturing slaves' as made the fortune of Crassus, could alone be made profitable. It is an historical curiosity that it accomplished this result without any attack on the institution of slavery itself. As soon as Christian doctrines became widespread enough to produce important social results we find Christian slave owners manumitting their slaves in considerable numbers. It is no derogation to the influence of the doctrine of human brotherhood or
- 58. to the humanity of the Christian slave owners to mention the fact that the doctrine of the sinfulness of interest, by tending to make slavery unprofitable, aided in the process of bringing to light the real content of the doctrine of human brotherhood, and of making the humane practice of manumission easier by the removal of certain economic impediments. In order to understand properly the working of the prohibition of interest and its relation to manumission, it is necessary to carry the analysis one step farther to its ultimate physical basis, which was the conditioning factor of actual practice and eventually of theory also. The exhaustion of the soil of western Europe which was the result of ancient methods of agriculture, together with the rising standard of living and the competition of other more fertile agricultural regions like Egypt and North Africa resulted in the substitution of the latifundi for small landholdings.[27] As the pressure continued the latifundi in turn became economically unprofitable under forced labor (slavery) and large tracts of land were abandoned. In order to put this land under agriculture again the charge upon it had to be reduced by the substitution of (relatively) free associated labor, villeinage or serfdom. But this change cut off the economic margin upon which the structure of ancient civilization was built and is the ultimate economic reason assignable for the fall of Rome. Of course the collapse of the empire could, theoretically, have been avoided had the Romans of the first three centuries A.D. been content to live the toilsome and frugal life of the Romans of the early republic. But this was an utter impossibility in practice. This slowly working and hardly understood decline in the relative and actual ability of ancient agriculture to sustain the weight imposed upon it, enables us to see why the sinfulness of interest could be steadily indoctrined even though steadily evaded, by Christians from the beginning, while manumission was not taught at all in the beginning and only worked up to the dignity of a pious action relatively late.[28] It also explains why manumission of household and personal slaves preceded that of agricultural slaves. Of course there is nothing peculiarly Christian
- 59. about this later phenomenon and the operation of other causes is discernable, but it is important for our purpose to observe that Christian practice, and Christian theory in property matters in the long run, followed the broad lines of the underlying economic evolution.[29] The application of this to the origin of Christian monasticism and to the revival of communistic theories by the later Church fathers lies at the very outside limit of our study but will be briefly touched on after we have considered the final overthrow of the communistic property concept as they appear in the earlier fathers up to and including Tertullian. Clement of Alexandria 153-217 A.D. has the distinction of being the first Christian theological writer who clearly expounds the concept of private property which has held sway without substantial change in the Church until the present time. This statement does not apply to the doctrine of receiving interest on money. In respect to this doctrine Clement is in perfect accord with all other early Christians both before and after himself. Indeed he specifically states that the Mosaic prohibition against taking interest from one's brother extends in the case of a Christian to all mankind. But in regard to all other property institutions Clement's attitude is essentially that of any modern Christian of generous disposition. In all that Clement has to say about property, and the 'bulk' of his 'property passages' is as great as that of all previous Christian writers together, he speaks like a man on the defensive. Indeed there has come down to us no other Christian writing earlier than his time which presents his view, with the dubious exception of some passages in Hermas. The fact seems to be that while Clement is undoubtedly presenting an apologetic for the existing practice in the Church of his day, that practice was felt to be more or less open to attack in the light of certain scripture passages. Communism as an existential reality was gone by the time of Clement—whatever may have been the extent—probably a limited one—to which it had existed in the earlier ages. But while communism as a fact was dead, communism as an idea or ideal of Christian economy was not
- 60. dead. Indeed Clement's views about the morality of wealth were so different from those of previous writers that a great modern economist[30] in treating of this subject ventures the opinion, though doubtfully, that the reason why Clement, alone among the great early theologians, was never canonized by the Church was that he ran counter to popular belief on this subject. This opinion is probably erroneous. Clement's theological opinions have a semi-Gnostic tinge quite sufficient to explain the absence of his name from the calendar of saints. Clement justifies the institution of private property. He justifies, on the highest ethical and philosophical principles, the possession by Christians of even the most enormous wealth. His apologetic is not an original one. He borrows it bodily from Plato. Indeed he quotes Plato verbatim, invocation to Pan and the other heathen gods included.[31] The originality lies in applying this Platonic doctrine to the exposition of Christian scripture. Clement's method is strictly that of Biblical exegesis. In the well known sermon or essay on: Who is the Rich Man that shall be saved he takes up practically all of the scriptural passages which seem opposed to the institutions of private property and explains them in so modern a spirit that the whole sermon might be delivered today in any ordinary Church and would be readily accepted as sound and reliable doctrine. His thesis is that wealth or poverty are matters in themselves indifferent. That riches are not to be bodily gotten rid of, but are to be wisely conserved and treated as a stewardship intrusted to the owner by God. That charity to the poor should be in proportion to one's wealth and that a right use of wealth will secure salvation to the upright Christian even though he possesses great riches all his life and leaves them to his heirs. The wealth that is dangerous to the soul is not physical possessions, but spiritual qualities of greed and avarice. His views can be best expressed by himself. We give two characteristic passages from the sermon above referred to.[32] Rich men that shall with difficulty enter into the kingdom, is to be apprehended in a scholarly way, not awkwardly, or rustically, or
- 61. carnally. For if the expression is used thus, salvation does not depend upon external things, whether they be many or few, small or great, or illustrious or obscure or esteemed or disesteemed; but on the virtue of the soul, on faith and hope and love and brotherliness, and knowledge, and meekness and humility and truth the reward of which is salvation. Sell thy possessions. What is this? He does not, as some off hand conceive, bid him throw away the substance he possesses and abandon his property; but he bids him banish from his soul his notions about wealth, his excitement and morbid feeling about it, the anxieties, which are the thorns of existence which choke the seed of life. And what peculiar thing is it that the new creature, the Son of God intimates and teaches? It is not the outward act which others have done, but something else indicated by it, greater, more godlike, more perfect, the stripping off of the passions from the soul itself and from the disposition, and the cutting up by the roots and casting out of what is alien to the mind. One, after ridding himself of the burden of wealth, may none the less have still the lust and desire for money innate and living; and may have abandoned the use of it, but being at once destitute of and desiring what he spent may doubly grieve both on account of the absence of attendance and the presence of regret.[33] We have now come to the beginning of what is in many respects the most interesting period in the history of property concepts. It is a period in which everything is upside down and wrong end to. In that strange age we find a famous archbishop, one of the world's noblest orators, a man of the most spotless integrity and the most saintly life, publicly preaching in the foremost pulpit of Christendom doctrines of property, the implications of which, the most hardened criminal would scarcely venture to breathe to a gang of thieves.[34] We find the most learned scholar of the century, in the weightiest expositions of Christian Scripture, penning the most powerful apologetic of anarchy that is to be found in the literature of the world.[35] We find one of the greatest of the popes, a man whose genius as a statesman will go down to the latest ages of history,
- 62. setting forth in a manual for the instruction of Christian bishops, property concepts more radical than those of the fiercest Jacobins in the bloodiest period of the Terror.[36] Stranger still, these incredible performances are the strongest proofs of the wisdom and piety of the men responsible for them. These men are today honored as the saviors of civilized religion and their images in bronze and marble and painted glass adorn the proudest temples of the most conservative denominations of Christians. The strange history of these famous men: Athanasius, the two Gregories, Basil and Chrysostom in the East; Augustine, Ambrose, Jerome and Gregory in the West, lies outside the limits of our study. But the explanation of their desperate and uncompromising communism can be given in a word. It was the communism of crisis: the communism of shipwrecked sailors forced to trust their lives to a frail lifeboat with an insufficient supply of provisions. These great Christian scholars, enriched by all the accumulated culture of their civilization, saw that culture falling into ruin all around them; they felt the foundations of that civilization trembling beneath their feet. To vary the figure, they beheld the rising tide of ignorance and barbarism rapidly engulfing the world and with desperate haste they set to work rebuilding and strengthening the ark of the Church that in it, religion, and so much of civilization as possible, might be saved till the flood subsided. Their task, perhaps the most important and most urgent, that men have ever had to perform, was of such a nature that they cared not what they wrecked in order to accomplish it. They ripped up the floor of the bridal chamber for timber and took the doors of the bank-safe for iron. These rhetorical figures are violent; but they are less violent than the reality they are intended to express. Monasticism was the last desperate hope of civilized Christianity and these men knew it. To establish monasticism they degraded the sanctity of marriage and denounced the sacredness of property. They conferred the most sacred honors upon the lowliest drudgery;[37] they turned princes into plowmen and nobles into breakers of the soil. Some historians,
- 63. judging them by the different standards of a later age, have pronounced them fanatics led astray by vulgar superstition. But judged by the needs of their own age, judged by the inestimable services rendered to the world by the monastic system they instituted, they are entitled to a place far up in the list of the wisest and the ablest of the human kind. Sketchy and imperfect as the above study necessarily is, it nevertheless gives the primary facts which are essential to an understanding of the important part played by property concepts and property institutions in the transformation of early Christianity from a predominantly eschatological to a practically socialized movement. We have seen,[38] that the earliest generations of Christians took over from contemporary Judaism a strongly Chiliastic eschatology. The logical consequence of such an eschatology is an indifference to, or undervaluation of, the existing social arrangements including the property concepts and institutions. One form easily taken by this indifference and undervaluation is that of practical communism. We accordingly find in the Acts and in such early writings as the Didache and the Epistle of Barnabas a distinctly communistic theory and the traces of more or less effort to put this theory into some degree of practical effect. Chiliasm and communism in these writers go together naturally. Pari passu with this logical, communistic Chiliasm we can trace the development of an illogical, individualistic Chiliasm in St. Paul, Clement of Rome and Hermas. It is already manifest even at this early stage, that the weight of influence and power of control in the Christian societies is on the side of the individualists. This is due to two causes. In the first place the communists among the Christians worked under a great handicap. The underlying economic institutions of society can indeed be changed. But they can be changed—on any considerable scale—only very slowly and by enormous effort. At any attempt to change them a thousand
- 64. interested and determined antagonists at once arise. It is not too much to say that had all Christians insisted upon communism as an essential element of the Christian faith and practice, Christianity in the Roman world could never have developed into anything more than an unimportant sect. The very fact that Christianity spread as rapidly as it did in the first century of its existence is proof that the communists in the Church made very little headway. It was hard enough to combat pagan religion and philosophy. Had the property institutions been attacked also, the primary religious objects would have been lost sight of in the conflict. In the second place the more practical minded Christian leaders would be antagonistic to a doctrine and practice which alienated many persons who might otherwise be won to the Church, and practically minded persons outside the Church regarded the individualists with more favor and were more easily influenced by them to become Christians themselves. The early importance attained by the Church of Rome is to be largely ascribed to the predominance in its councils of such practical persons.[39] Communism had no hold there at all and Chiliasm was never allowed to interfere with the practical workings of society. By the time of Justin the three concepts; Chiliasm, Communism, and Individualism had arrived at a modus vivendi. According to this arrangement Chiliasm and Communism held sway as theories while individualism ruled in the world of fact. This agreement proved very satisfactory and for more than half a century was the the accepted thing. It is seen in full force in Tertullian. There is a general tendency, due to the natural effects of use and disuse, for theories which do not correspond to realities to become discredited, even as theories. Conversely realities which at first lack theoretical justification tend to accumulate such justification with the lapse of time. It is therefore not surprising to find by the beginning of the Third Century, a movement to discard theoretical Chiliasm and communism and to validate by theoretical apologetic the actually
- 65. existing individualism. These two processes in the nature of the case are closely connected with one another and it is not by mere chance that they find a common exponent in Clement of Alexandria. That famous opponent of Chiliasm is equally well known as the justifier of an extreme individualism. He greatly facilitated the spread of Christian theology by liberating it from the burden of an eschatological theory increasingly hard to reconcile with reality and also by bringing the economic teachings of Christianity into conformity with current practice. As noted above, there was one economic doctrine which neither he nor any other early Christian teacher ever attempted to reconcile with the facts, and it is undoubtedly true that the doctrine of the sinfulness of interest was alike detrimental to the spread of Christianity and to the general well being of society as it then existed. The reasons why this particular reality i.e., interest on money, was so slow in receiving its theoretical justification are numerous. The only ones that need concern us here are that the opposition to be overcome in this case was much more formidable than in the cases of Chiliasm and communism and the fact that this inconsistency on the part of the Christians did not in reality offer any very serious obstacle to the growth of the Church. Communism had no great body of Biblical authority at its back. There are indeed some texts in its favor but there are plenty of an opposite nature. The doctrine had no great popular prejudice in its favor. In addition it was insuperably difficult of realization in fact. It was otherwise with interest. The theoretical prejudice against interest was almost as great among the Jews and Pagans as among the Christians themselves. The Scriptures were unequivocal in their denunciation of it. Furthermore the correlative institutions of rent and profit offered so many opportunities to disguise the fact of interest that it was exceedingly easy to retain the theoretical opposition without ceasing the actual practice. Although Clement's condemnation of interest was probably merely an inherited prejudice it is by no means impossible that he considered that an attempt to justify it would endanger his defense of the more fundamental institution of private property. At any rate his course can be defended as a practical one under the circumstances. Whatever may
- 66. be said of its consistency, the Christian custom of condemning the theory and winking at the practice of interest worked well. The inconsistency which seems so glaring to us, was probably very largely unperceived by the ancient pagans—they had exactly the same inconsistency themselves. In regard to Chiliasm and property, practically the same attitude prevailed. It worked indeed even more easily. In the West there seems to have been a considerable Chiliastic tradition. So long as this tradition did not result in any practices which interfered with the actual progress of the Church, the Fathers were content to let it alone. It did not, till at least the Third Century, hinder the acceptance of Christian doctrine by the pagans and may even have aided the process among some of the lower classes. Its long survival can be taken as sure proof that it did not effect either the development of the hierarchy or the institution of property. As regards property of man in man, the superior power of the Christian religion to keep slaves in subjection accounts in no small measure for its relatively rapid rise to power in the ancient world. The pagan religion was inferior in usefulness to the Christian religion because it could not keep the slave contented with his position. The next world in the pagan theology was only a worse copy of this world. Christianity, in glaring contrast to paganism, proclaimed that the despised and afflicted were to sit on golden thrones in the next life. The more they were exploited in this life, the brighter their crown in the next one. The pagan slave was dangerous. The whole pre-Christian literature of Classical antiquity shows the ever present fear of a servile outbreak. There were good grounds for that fear. Outbreaks were frequent and of a most ferocious character. On more than one occasion they threatened the very existence of the ancient civilization. Christianity was able to make the slave contented to be a slave. It was economically an enormous advance over paganism. A master whose slaves were Christians was not afraid of being murdered by them. Not only was the master's life secure, his property was secure also. The pagan slaves were notorious thieves.
- 67. The Christian slave did not rob his master. These facts gave Christianity an enormous leverage in its efforts to force its way into social recognition. It went far toward securing a favorable disposition toward the new religion on the part of the influential, wealthy, and conservative elements in the population. Into the general economic changes which began to operate toward the end of our period it is not our purpose to enter, but it is worth notice that the efforts made by the Church to save itself in the general ruin which overtook the ancient world, chiefly the institution of monasticism, were such as to secure more firmly than ever the hold of the Church upon society. The Church rapidly became an economic factor of the first importance. The only secure basis of lasting social influence is economic. Christianity by teaching the virtues of honesty, frugality, simplicity, and charity laid the foundations of her subsequent triumph, and when she had great societies of men and women working hard and living plainly and adding all their accumulations to institutions belonging to the Church and directly under the supervision and control of the ecclesiastical authority, the Church paved the way for her subsequent domination of the civil government. Monastic communism, being economically superior to Chiliastic Communism, inevitably superseded it. FOOTNOTES: [1] Cf. Plato, Laws, V, 742. Aristotle, Politics, 1:X, XI. Cicero, De Officus, II, XXV. Seneca, De Beneficus, VII, X. [2] Acts IV. [3] I. Cor. vii 30. [4] Rom. xiii 3. [5] Jas. Chap. V. [6] Chaps. 21-22. [7] Chap. xxxviii.
- 68. [8] Did. IV. 8. [9] Barn. XIV. 16. [10] Schaff, Vol. 1. [11] Past. V. vi. 6. [12] Past. S. IX. XXX. 5. [13] Past III. 2. [14] Apol. I. IV. [15] Apol. I. xiv. [16] De Mort. Per. XIV. [17] Apol. XXXIX. [18] Eus., E. H., V. 18. [19] De Lapsis, VI. [20] See Pronouncement of the Sacred Penitentiary, 11 Feb., 1832. [21] Sir James Macintosh. [22] Civil Disabilities of the Jews. [23] Lourie, Monuments of the Early Church, Chap. II. [24] Lourie, ibid. [25] Cf. Hypolytus. [26] A. Loria. Cf. Economic Basis of Society. (Int.) [27] Cf. A. Loria, Economic Foundations of Society. (Int.) [28] Circa 200(?). [29] Cf. K. Marx, Das Kapital, Vol. 1. [30] F. Nitti in Catholic Socialism. [31] Phaedus, The Laws, in Strom. II, 6. [32] Chap. XIV. [33] Chap. XXXI.
- 69. [34] Chrysostom, Sermons Rich Man and Lazarus, etc. [35] Jerome, Commentaries. [36] Gregory, Pastoralis Cura. [37] Laborare est orare. [38] Chap. I. [39] E.g., Clement and Hermas.
- 70. CHAPTER III THE EARLY CHURCH AND THE POPULACE The transformation of early Christianity from an eschatological to a socialized movement was the result of the interaction of three social groups—three 'publics'—the Jewish, the Pagan, and the Christian. It was a single movement, working itself out through these three 'crowds'. Christianity, like all other great religions, was in its first beginnings essentially a mob phenomenon—that is to say it was a very slow movement which had a long history back of it. Perhaps no current opinion is more unfounded than the notion that mob movements are sudden and unpredictable. They are almost incredibly slow of development. The range of action found in the mob is more narrowly and rigidly circumscribed than in almost any other social group. A crowd is open to suggestions that are in line with its previous experience, and to no others. The initial success of Christ with the Jewish crowds was only possible because for generations the whole Jewish public had been looking forward to a Messiah and a Messianic kingdom. In so far as Christ appeared to fulfill this preconceived expectation he gained popular support. When he disappointed it, he lost his popularity and his life. The early and enormous success of the apostles on the day of Pentecost and immediately afterwards was due primarily to the fact that the Chiliastic expectation preached to the Jerusalem crowds was very closely in line with their inherited beliefs. As soon as Christianity began to develop doctrines and practices even slightly at variance with those traditional to Judaism it lost the support of the Jewish public. Beginning as a strictly Jewish sect, it alienated practically the whole Jewish race within little more than a generation. This
- 71. alienation was the inevitable effect of an idea of universalism opposed to the hereditary Jewish nationalism. This idea of universalism was not a new thing. It was to be found in the ancient Jewish scriptures. But it had never become popularized. It formed no part of the content of contemporary public opinion among the Jews. Christianity met with success in the great cosmopolitan centers, like Antioch and Alexandria, where universalism was a tradition and had become a part of the crowd sentiment. It succeeded best of all in Rome where universalism reached its highest development. Yet even here a limitation is to be noted. Christianity was universal in its willingness to receive people of all races and nations. It was not universal in its willingness to acknowledge the validity of other religions. This variation from the traditional Greek and Roman universalism had momentous results. It made the propagation of the Christian Gospel much more difficult and involved the church, at least temporarily, in the current syncretism which was a popular movement. So e.g., we find Justin calling Socrates a Christian and asserting that the stories of Noah and Deucalion are merely versions of the same event. The main characteristics of crowd psychology are familiar enough. Crowds do not reason. They accept or reject ideas as a whole. They are governed by phrases, symbols, and shibboleths. They tolerate neither discussion nor contradiction. The suggestions brought to bear on them invade the whole of their understanding and tend to transform themselves into acts. Crowds entertain only violent and extreme sentiments and they unconsciously accord a mysterious power to the formula or leader that for the moment arouses their enthusiasm. Any movement in order to become popular, in order to 'get over' to the general public, has to operate within the limits set by this psychology. The amount of change, adaptation, and development necessary before a movement can fit into these limitations and express itself powerfully within them is so considerable that no historical example can probably be found where the required
- 72. accommodation has been accomplished in less than three generations. It is the purpose of this chapter to trace, so far as the surviving source material permits, the steps of this accommodation in the case of early Christianity. For some time before Christ the Jewish people had been restless. Their desires and aspirations for national and religious greatness had been repressed and inhibited. The unrest thus generated took various forms; patriotic uprisings, religious revivals, etc. Christ was at first considered merely as another Theudas or Judas of Galilee or John the Baptist. In the pagan world the pax Romana produced a somewhat similar restlessness. Travel increased; wandering, much of it aimless, characterized whole classes of people;[1] there was a marked increase in crime, vice, insanity, and suicide which alarmed all the moralists. This condition of affairs was eminently suitable for the first beginnings of a crowd movement; indeed no great crowd movement can begin except under such circumstances. The wanderings of St. Paul and the other Christians apostles—called missionary journeys—were really only particular cases of a general condition. The same organic demand for new stimulation, the same sense of shattered religious and philosophic ideals prevailed in the pagan as in the Jewish world. It would be hard to find a greater contrast of character than Christ and Lucian. Yet the fiery earnestness with which Christ denounces contemporary Jewish religiosity and the cool cynicism with which Lucian mocks at the pagan piety of the same age have a like cause. Economic pressure on the lower strata of society contributed to the unrest. The slave, the small shopkeeper, and the free artisan had a hard time of it in the Roman world. Economically oppressed classes are material ready to the hand of the agitator, religious or other. In the crowd movements recorded in the Acts we can trace the first beginnings of the Christian populace.[2] In Iconium a great multitude both of Jews and of Greeks believed but the Jews that were disobedient stirred up the souls of the Gentiles and made them evil affected against the brethren. But the multitude of the city was divided and
- 73. part held with the Jews and part with the apostles. At Lytra there was a typical case of mob action where the apostles were first worshipped and then stoned. In the cases of the mobs at Philippi and Ephesus we see the economic motive, the threatened loss of livelihood, entering along with anger at an attack on the received religion. In the case of the Jerusalem and Athenian crowds we see acceptance, or at least acquiescence, on the part of the crowd up to the point where Christianity breaks with their tradition. In general we see anger on the part of the crowds only after agitation deliberately stirred up by interested parties; priests, sorcerers, craftsmen or the like. Generally speaking the antipathy is no part of the crowd psychology, and on occasion the crowd may be on the side of the missionaries of the new religion. In general also the Christians were not sufficiently numerous to make a counter crowd demonstration of their own. In Pliny's letter to Trojan, although it is a generation later than the Acts and refers to a region where Christianity had been preached for a considerable period of time, we find a marked instability in the attitude of the public: Many of every age, every rank and even of both sexes are brought into danger and will be in the future. The contagion of that superstition has penetrated not only the cities but also the villages and country places and yet it seems possible to stop it and set it right. At any rate it is certain enough that the temples deserted until quite recently begin to be frequented, that the ceremonies of religion, long disused, are restored and that fodder for the victims comes to market, whereas buyers for it were until now very few. From this it may easily be supposed that a multitude of men can be reclaimed if there be a place of repentence.[3] There seems no reasonable ground for doubting that Pliny's judgment was correct. While the blood of the martyrs is doubtless the seed of the church, a continuous, general, and relentless persecution can extirpate a religion in a given nation; as the history of the Inquisition abundantly proves. Still more easily can propaganda for the older religion win back its former adherents of
- 74. the first and second generations. It is not, in general, till a generation has grown up entirely inside a new religion that such a religion is well established. The generation which at maturity makes the rupture with the older faith can be brought back to it by less expenditure of energy than was expended by them in breaking away in the first place. The success of the Jesuits e.g., is quite inexplicable on any other hypothesis. The generation who are children at the time their parents make the break with the old religion are notoriously undependable in the religious matters. It was in all probability these people that Pliny had to deal with. It is at least permissable to hazard the guess that the Laodiceans who aroused the wrath of the author of the Revelation were of this generation. It is certain that many of the 'Lapsi' who caused so much trouble to Christian apologists and church councils belonged in this chronological class. In Justin Martyr we have a hint of a further development in the crowd attitude toward the Christians. Justin says: When you (Jews) knew that He had risen from the dead and ascended to heaven as the prophets foretold He would, you not only did not repent of the wickedness you had committed, but at that time you selected and sent out from Jerusalem chosen men through all the land to tell that the godless heresy of the Christians had sprung up and to publish those things which all they, who knew us not, speak against us. So that you are the cause not only of your own unrighteousness but that of all other men.[4] Irrespective of the exact historical accuracy of this statement, it is indicative of the process, technically known as 'circular interaction,' which is so essential a step in the development of popular opinion and the building up of crowd sentiment. Before any group of people can become either popular or unpopular there must be a focusing and fixation of public attention upon them. Even in the new Testament we find the Jews sending emissaries from city to city to call attention to the Christian propaganda. Prejudice against the Christians was thus aroused in persons who had never either seen or
- 75. heard them. The basis of 'circular interaction' is unconscious or subconscious emotional reaction. A's frown brings a frown to the face of B. B's frown in turn intensifies A's. This simple process is the source of all expressions of crowd emotion. By multiplication of numbers and increase in the stimuli employed it is capable of provoking a vicious circle of feeling which eventually causes individuals in a crowd to do things and feel things which no individual in the crowd would do or feel when outside the circle. It is to the credit or discredit of the Jews that they first set this 'vicious circle' in operation against the Christians. Of course the same psychological principle operated to produce zeal and enthusiasm and contempt of pain and death in the Christian 'crowd'. By this process of 'circular interaction' the name, 'Christian,' had already in the time of Justin become a mob shibboleth. It seems to have operated precisely as the shibboleth 'traitor' operates on a patriotic crowd in war time, or 'scab' on a labor group. It became a shibboleth of exactly opposite significance in the Christian 'crowd'. The way was thus prepared for the next step in the process of developing the ultimate crisis. This step—the disparate 'universe of discourse'—is exhibited in process of formation in the account of the martyrdom of Polycarp. The account, as we have it, undoubtedly contains later additions, but these additions even of miraculous elements, do not necessarily invalidate those portions of the story with which we are alone concerned. The martyrologist certainly had no intention of writing his story for the purpose of illustrating the principles of group psychology and the undesigned and incidental statements of crowd reactions are precisely the ones of value for our purpose. A few brief excerpts are sufficient to illustrate the stage reached in the growth of the disparate 'universe of discourse.' The whole multitude, marvelling at the nobility of mind displayed by the devout and godly race of Christians cried out: Away with the Atheists: let Polycarp be sought out.[5] He went eagerly forward with all haste and was conducted to the Stadium where the tumult was so great that there was no possibility of being heard.[6]
- 76. Polycarp has confessed that he is Christian. This proclamation having been made by the herald, the whole multitude both of the heathen and Jews who dwelt in Smyrna cried out with uncontrollable fury and in a loud voice: This is the teacher of Asia, the father of the Christians and the overthrower of our gods, he who has been teaching many not to sacrifice or to worship the gods. Speaking thus they cried out and besought Phillip, the Asiarch, to let loose a lion upon Polycarp. But Philip answered that it was not lawful for him to do so seeing the shows of beasts were already finished. Then it seemed good to them to cry out with one voice that Polycarp should be burned alive.[7] This then was carried into effect with greater speed than it was spoken, the multitude immediately gathering together wood and fagots out of the shops and baths, the Jews especially, according to custom eagerly assisting them in it.[8] We afterwards took up his bones, as being more precious than the most exquisite jewels and more purified than gold and deposited them in a fitting place, whither, being gathered together as opportunity is allowed us, with joy and rejoicing the Lord shall grant us to celebrate the anniversary of his martyrdom both in memory of those who have already finished their course and for the exercising and preparation of those yet to walk in their steps.[9] In the disparate universe of discourse in its complete form common shibboleths produce entirely different mental reactions—usually antagonistic ones. There is also complete accord as to the shibboleths. The cry here is at one time against the Atheists, then against the Christians. But the Christians could and did deny the charge of Atheism. They were as antagonistic to Atheism as the Pagans. An incomplete development of crowd feeling is evident on the part of the pagans. The Jews are still the inciters and leading spirits of the mob. The very statement that the Jews acted 'according to custom' shows that mobbing Christians was still looked upon as a peculiarly Jewish trait. It was not yet entirely spontaneous
- 77. on the part of the pagan public. Most noticeable of all is the indifference of the mob toward the Christians' adoration of relics of the martyrs. No effort was made to prevent the Christians from obtaining the bones of Polycarp. Either the cult of relics was not known to the pagans and Jews—though it seems to be firmly established among the Christians—or else, the effect of the cult in perpetuating Christianity had not yet had time to make itself manifest to the pagan public—or to the Jewish. In any case we have here the plain evidence of the imperfectly developed condition of the crowd mind, owing perhaps to a too short tradition. Our next evidence is the martyrdoms of Lyons and Vienne preserved in a letter quoted by Eusebius. They (the Christians) endured nobly the injuries inflicted upon them by the populace, clamor and blows and draggings and robberies and stonings and imprisonments and all things which an infuriated mob delight in inflicting on enemies and adversaries.[10] When these accusations were reported all the people raged like wild beasts against us, so that even if any had before been moderate on account of friendship, they were now exceedingly furious and gnashed their teeth against us. When he (Bishop Pothinus) was brought to the tribunal accompanied by a multitude who shouted against him in every manner as if he were Christ himself, he bore noble witness. Then he was dragged away harshly and received blows of every kind. Those men near him struck him with their hands and feet, regardless of his age, and those at a distance hurled at him whatever they could seize, all of them thinking that they would be guilty of great wickedness and impiety if any possible abuse were omitted. For thus they thought to avenge their own deities.[11] But not even thus was their madness and cruelty toward the saints satisfied. Wild and barbarous tribes were not easily appeased and their violence found another peculiar opportunity in the dead bodies. For they cast to the dogs those who had died of suffocation in the
- 78. prison and they exposed the remains left by the wild beasts and by fire mangled and charred. And some gnashed their teeth against them, but others mocked at them. The bodies of the martyrs having thus in every manner been exposed for six days were afterwards burned and reduced to ashes and swept into the Rhone so that no trace of them might appear on the earth. And this they did as if able to conquer God and prevent their new birth; 'that', as they said, 'they may have no hope of a resurrection through trust in which they bring to us this foreign and new religion.' [12] We have in this account a marked advance, as regards the development of the mob mind, over what is found in the martyrdom of Polycarp. Many of the 'crowd' phenomena are indeed the same but the differences are even more striking than the similarities. We find in Lyons no body of Jews or other especially interested persons leading the mob on by manifestations of peculiar zeal and forwardness. When the accounts are compared in their entirety it becomes at once manifest that there is a consistency of attitude, a whole heartedness in the actions of the Lyons mob that is lacking in the case of the Syrmnaens. There is a degree of familiarity with Christian doctrine—especially the doctrine of the resurrection—which denotes a much more thorough permeation of the public mind by Christianity. There may be no difference in the hatred of the two mobs for the new faith, but it had more content in the mind of the Gallic crowd. The degree of thought and pains taken by the Lyonese persecutors—the guards placed to prevent the Christians from stealing the relics of the martyrs, the elaborate efforts to nullify the possibility of a resurrection—the very extent and thoroughness and duration of the persecution are different from anything to be found in the other martyrdom. The difficulty to be explained—if it is a difficulty—from the point of view of crowd psychology is that there is difference of only eleven years—taking the ordinary chronology—between the two persecutions. It is true that the Lyons persecution is the later, but the difference in the mob behavior is such as might well demand the
- 79. lapse of a generation had the phenomena been exhibited by the public of the same city. There must unquestionably have been a great difference in the demotic composition of the populations of Lyons and Smyrna; the reference to barbarians in Lyons shows as much, but the behavior of mobs as controlled by the time needed for the focusing and fixation of attention and the development of a disparate universe of discourse is very little effected by difference of demotic composition. It has indeed been suggested by one critic,[13] that the persecution at Lyons belongs in the reign of Septimus Severus instead of that of Marcus Aurelius. This would explain away the difficulty, but there seems no necessary reason for adopting this opinion. It would rather appear that there existed peculiar conditions in Lyons and vicinity which account for the fact that the persecution, so far as we know, was confined to that locality and also for the fact that the mob mind was in a maturer state of antagonism to Christianity. Just what these peculiar conditions were, it is impossible to say with entire certainty. However there is at least a very suggestive hint in a paragraph by the greatest modern authority on Roman Gaul[14] contained in his well known volume on Ancient France.[15] The paragraph is also worth quoting as giving a valuable insight into the psychology of the peoples of the ancient Roman World. The Roman Empire was in no wise maintained by force but by the religious admiration it inspired. It would be without a parallel in the history of the world that a form of government held in popular detestation should have lasted for five centuries. It would be inexplicable that the thirty legions of the Empire should have constrained a hundred million men to obedience. The reason of their obedience was that the Emperor, who personified the greatness of Rome was worshipped like a divinity by unanimous consent. There were altars in honor of the Emperor in the smallest townships of his realm. From one end of the Empire to the other a new religion was seen to arise in those days which had for its divinities the Emperors themselves. Some years before the Christian era the whole of Gaul, represented by sixty cities, built in common a temple near the city of Lyons in honor of Augustus. Its priests, elected by the united Gallic
- 80. cities, were the principal personages in their country. It is impossible to attribute all this to fear and servility. Whole nations are not servile and especially for three centuries. It was not the courtiers who worshipped the prince, it was Rome, and it was not Rome merely but it was Gaul, it was Spain. It was Greece and Asia. While no dogmatic assertion is justified, it does not, perhaps, exceed the limits of reasonable inference to suppose that the existence of this noted center of Emperor worship in the immediate neighborhood of Lyons may account, in part at least, for the especial hatred of the populace of that city for persons who refused to sacrifice to the Emperor and also for the maturity of their feeling against the Christians, who were as far as we are aware, probably the only persons who refused thus to sacrifice. This stray bit of evidence is admittedly not conclusive. It is offered merely for what it may be worth. There is evidence that by the middle of the second Century popular opinion was sufficiently inflamed against the Christians to render the administration of justice precarious because of mob violence. Edicts of Hadrian and Antonius Pious specifically declared that the clamor of the multitude should not be received as legal evidence to convict or to punish them, as such tumultuous accusations were repugnant both to the firmness and the equity of the law.[16] This attitude seems to have persisted with relatively little change for about a century. During this period the official 'persecutions' were neither numerous nor severe. From the very few scattered and incidental references which have alone survived regarding the mob feeling of the time, we can assert no more than that it was an exasperated one, likely to break out upon provocation but under ordinary circumstances more or less in abeyance. On the whole it was undoubtedly more violent at the end of the period than at the beginning. Fortunately from the middle of the third Century onwards we have a fairly continuous history of a single 'public' (Alexandria) which is
- 81. lacking before this time. The Alexandrian populace were noted for their tumultuous disposition, but we have no reliable account of their behavior towards the Christians until the time of Severus, 202 A.D. In the account given by Eusebius of the martyrdom of the beautiful virgin, Potamiaena, it is stated that: the people attempted to annoy and insult her with abusive words. As however the intervention of a single officer sufficed to protect her from the people on this occasion, the public sentiment cannot have been inflamed to any alarming extent. If we may trust Palladius, her martyrdom was the result of a plot of a would-be ravisher and in any case it was not the product of any spontaneous popular movement. In the period between 202 A.D. and 249 A.D. a well developed tradition of hatred and violence grew up in the popular mind. We have no record of the steps in the process but the extant accounts of the Decian and Valerian persecutions in Alexandria leave no doubt of the fact. These persecutions can only be called 'legal' by a violent stretch of verbal usage. They were mob lynchings, sometimes sanctioned by the forms of law, but quite as often without even the barest pretense of judicial execution. They were quite as frequent and as savage in the later part of the reign of Philip, as in the time of Decius. They were not called forth by any imperial edict—they preceded the edict by at least a year and were of a character such as no merely governmental, legal process would ever, or could ever, take on. Mobbing Christians had become a form of popular sport, a generally shared sort of public amusement—exciting and not dangerous. The letter of Bishop Dionysius makes this very clear. To quote: The persecution among us did not begin with the royal decree but preceded it an entire year. The prophet and author of evils to this city moved and aroused against us the masses of the heathen rekindling among them the superstition of their country and finding full opportunity for any wickedness. They considered this the only pious service of their demons that they should slay us. Then follows a long list of mob lynchings of which we take a single specimen: They seized Serapion in his own house and tortured him and having broken all his limbs, they threw him headlong from an
- 82. upper story.[17] And there was no street, nor public read, nor lane open to us night or day but always and everywhere all them cried out that if anyone would not repeat their impious words, he should be immediately dragged away and burned. And matters continued thus for a considerable time. But a sedition and civil war came upon the wretched people and turned their cruelty toward us against one another. So we breathed for a while as they ceased from their rage against us.[18] The mob broke loose against the Christians again the following year, but there is no object in cataloguing the grewsome exhibitions of crowd brutality. It is evident that what we have in this account is no exhibition of political oppression by a tyrannical government, but a genuine outbreak of group animosity which had been long incubating in the popular mind. All the phenomena which are characteristic of fully matured public feeling are found complete; circular interaction, shibboleths, sect isolation devices and the rest. When public feeling has developed to such a degree of intensity as this, the accumulated sentiment and social unrest must of necessity discharge themselves in some form of direct group action. This direct action however may take the from either of physical violence or, under certain conditions, of some sort of mystical experience; conversion, dancing, rolling on the ground, etc. In exceptional cases the two forms are combined. An illustration of this latter phenomenon is given by Bishop Dionysius in this same letter; In Cephus, a large assembly gathered with us and God opened for us a door for the word. At first we were persecuted and stoned but afterward not a few of the heathen forsook their idols and turned to God.[19] It is necessary to mention perhaps the largest, and certainly the most dignified and respectable crowd that is to be met with in connection with this persecution—that of Carthage on the occasion of the martyrdom of Bishop Cyprian. We find here neither rage on one side nor unseemly exaltation on the other. Pagans and Christians alike behaved with decent seriousness at the death of that famous man who was equally respected by all classes of the
- 83. population. But martyrs of the social eminence of Cyprian were very rare, and orderly behaviour in such a vast multitude as witnessed his end was still rarer. To return to the populace of Alexandria. The long peace of the Church which intervened between the persecution of Valerian and that of Diocletian witnessed in Alexandria, as elsewhere, a great growth of Christianity in numbers, influence, and wealth. It would perhaps be going beyond the evidence to say that in this interval, the majority of the population of the city were won over to the new faith, but it is certain that the number of Christians became so great as to intimidate the pagan portion of the people. The Alexandrian mob was still very much in evidence but it gradually ceased to harrass the Christians except under the most exceptional circumstances. The dangers of such action became so considerable and the chances of success so problematical that we find a period when a practice of mutual forbearance governed the behavior of the hostile groups. The study of crowd psychology presents no more impressive contrast than that exhibited by the people of Alexandria during the Diocletian persecution compared with their behavior during that of Decius. In the last and greatest of the persecutions, in the most tumultuous city of the empire, the mob took no part. Like the famous image of Brutus, it is more conspicuous by its absence than it would be by its presence. The persecution was a purely governmental measure officially carried out by judges and executioners in accordance with orders. In one obscure and doubtful instance we are told that the bystanders beat certain martyrs when legal permission was given to the people to treat them so. In another case we are told that the cruelty of the punishments filled the spectators with fear. These are the only references to the public that occur in the long and minute account of an eye witness of famous events extending over a considerable number of years. Both before and after this period the mob of the Egyptian metropolis exhibits the utmost extreme of religious fanaticism. During this
- 84. period that mob had to be most carefully considered by the government in other than religious matters. But as a religious power it did not exist. Had the persecution of Diocletian happened a generation earlier it could have counted on a very considerable degree of popular support, had it happened a generation later it would have caused a revolt that could only have been put down by a large army. Happening at the precise time it did, it provoked no popular reaction at all. This strange apathy is not peculiar to Alexandria. Practically without exception the authentic acts of the martyrs of this persecution are court records taken down by the official stenographers in the ordinary course of the day's work. They are dry, mechanical, and repetitious to a degree. They exhibit, in general, harrassed and exasperated judges driven to the infliction of extreme penalties in the face of a cold and skeptical public. One imperial decree ordered that all men, women, and children, even infants at the breast, should sacrifice and offer oblations, that guards should be placed in the markets and at the baths in order to enforce sacrifices there. The popular reaction in Caesarea is thus recorded: The heathen blamed the severity and exceeding absurdity of what was done for these things appeared to them extreme and burdensome.[20] He (the Judge) ordered the dead to be exposed in the open air as food for wild beasts; and beasts and birds of prey scattered the human limbs here and there, so that nothing appeared more horrible even to those who formerly hated us, though they bewailed not so much the calamity of those against whom these things were done as the outrage against themselves and the common nature of man.[21] The one thing to be said of this type of mob mind is manifestly that it is transitional. The pendulum has swung through exactly half its arc and for the brief instant presents the fallacious appearance of quiescence. How transitory this quiet was on the part of the Alexandrian mob is evidenced by the history of Athanasius. That great statesman conciliated and consolidated public opinion in Egypt. Backed by this opinion he practically cancelled the power of the civil
- 85. authorities of the country and negotiated as an equal with the emperors. For the first time in more than three centuries the will of the common people again became a power able to limit the military despotism which dominated the civilized world. The re-birth of popular government in the Fourth century through the agency of Christian mobs is the most important preliminary step in the growth of the political power of the Catholic Church. A study of the mobs of Alexandria, Rome, Constantinople and other great cities shows beyond question that the political power of the Church had its origin in no alliance with imperial authority, but was independent of and generally antagonistic to that authority. The history of these Christian mobs lies outside the limits of our study but it is worth while in the case of the Alexandrian populace to give two or three brief extracts illustrating the final steps of the process which changed a fanatically pagan mob into an equally fanatical Christian one. What we have to consider is only the last stage of an evolution already more than half complete at the time of the Nicene Council. Under extreme provocation and certain of imperial complacency at their excesses, the pagan mob during the reign of Julian indulged in one last outburst against the exceedingly unpopular George of Cappadocia who had been forcibly intruded into the seat of Athanasius. To quote the Historian Socrates: The Christians on discovering these abominations went forth eagerly to expose them to the view and execration of all and therefore carried the skulls throughout the city in a kind of triumphal procession for the inspection of the people. When the pagans of Alexandria beheld this, unable to bear the insulting character of the act, they became so exasperated that they assailed the Christians with whatever weapons chanced to come to hand, in their fury destroying numbers of them in a variety of ways and, as it generally happens in such a case, neither friends or relations were spared but friends, brothers, parents, and children imbued their hands in each others blood. The pagans having dragged George out of the church, fastened him to a camel and when they had torn him to pieces they burned him together with the camel.[22] In this account we see the last expiring
- 86. efforts of the pagan mob movement. Any mob movement collapses rapidly when it turns in upon itself, and the evil results of its violence react immediately upon the members of the mob. By this time it is evident that the number of Christians in Alexandria was so large that any public persecution of them brought serious and unendurable consequences upon the populace generally. Then the movement ended. But in the two centuries or more that the pagan movement lasted, a contrary Christian mob movement had been developing along the same general lines as the other. This movement, being later in its inception, came to a head correspondingly later and reached its crisis under the patriarch Cyril. Its violence was first directed against the Jews whom the Christians appear to have hated even more than they hated the pagans. The Jews were the weaker and less numerous faction opposed to the Christians and as the Pagans seem to have liked them too little to support them against the Christians, it is not surprising that the Christian mob, which had pretty well reduced the political authorities to impotence, should vent its rage against the Jews and their synagogues. Cyril accompanied by an immense crowd of people, going to their synagogues, took them away from them and drove the Jews out of the city, permitting the multitude to plunder their goods. Thus the Jews who had inhabited the city from the time of Alexander were expelled from it.[23] Sometime after the expulsion of the Jews, the Christian mob, now directing its spite against the rapidly disappearing paganism, perpetrated perhaps the most atrocious crime that stains the history of Alexandria—the murder of Hypatia. This beautiful, learned, and virtuous woman, 'the fairest flower of paganism' is one of the very few members of her sex who has attained high eminence in the realm philosophical speculation. She enjoyed the deserved esteem of all the intellectual leaders of her age—Christian as well as pagan— and to the latest ages her name will be mentioned with respect by all those speculative thinkers whose respect can confer honor. Socrates describes her murder as follows: It was calumniously
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