![93. Statistics in the 21st Century Adrian E. Raftery, Martin A. Tanner, and Martin T. Wells (2001)
94. Accelerated Life Models: Modeling and Statistical Analysis
Vilijandas Bagdonavicius and Mikhail Nikulin (2001)
95. Subset Selection in Regression, Second Edition Alan Miller (2002)
96. Topics in Modelling of Clustered Data Marc Aerts, Helena Geys, Geert Molenberghs, and Louise M. Ryan (2002)
97. Components of Variance D.R. Cox and P.J. Solomon (2002)
98. Design and Analysis of Cross-Over Trials, 2nd Edition Byron Jones and Michael G. Kenward (2003)
99. Extreme Values in Finance, Telecommunications, and the Environment
Bärbel Finkenstädt and Holger Rootzén (2003)
100. Statistical Inference and Simulation for Spatial Point Processes
Jesper Møller and Rasmus Plenge Waagepetersen (2004)
101. Hierarchical Modeling and Analysis for Spatial Data
Sudipto Banerjee, Bradley P. Carlin, and Alan E. Gelfand (2004)
102. Diagnostic Checks in Time Series Wai Keung Li (2004)
103. Stereology for Statisticians Adrian Baddeley and Eva B. Vedel Jensen (2004)
104. Gaussian Markov Random Fields: Theory and Applications +Ý
DYDUG5XHDQG/HRQKDUG+HOG(2005)
105. Measurement Error in Nonlinear Models: A Modern Perspective, Second Edition
Raymond J. Carroll, David Ruppert, Leonard A. Stefanski, and Ciprian M. Crainiceanu (2006)
106. *HQHUDOL]HG/LQHDU0RGHOVZLWK5DQGRP(IIHFWV8QLÀHG$QDOVLVYLD+OLNHOLKRRG
Youngjo Lee, John A. Nelder, and Yudi Pawitan (2006)
107. Statistical Methods for Spatio-Temporal Systems
Bärbel Finkenstädt, Leonhard Held, and Valerie Isham (2007)
108. Nonlinear Time Series: Semiparametric and Nonparametric Methods Jiti Gao (2007)
109. Missing Data in Longitudinal Studies: Strategies for Bayesian Modeling and Sensitivity Analysis
Michael J. Daniels and Joseph W. Hogan (2008)
110. Hidden Markov Models for Time Series: An Introduction Using R
Walter Zucchini and Iain L. MacDonald (2009)
111. ROC Curves for Continuous Data Wojtek J. Krzanowski and David J. Hand (2009)
112. Antedependence Models for Longitudinal Data Dale L. Zimmerman and Vicente A. Núñez-Antón (2009)
113. Mixed Effects Models for Complex Data Lang Wu (2010)
114. Intoduction to Time Series Modeling Genshiro Kitagawa (2010)
115. Expansions and Asymptotics for Statistics Christopher G. Small (2010)
116. Statistical Inference: An Integrated Bayesian/Likelihood Approach Murray Aitkin (2010)
117. Circular and Linear Regression: Fitting Circles and Lines by Least Squares Nikolai Chernov (2010)
118. Simultaneous Inference in Regression Wei Liu (2010)
119. Robust Nonparametric Statistical Methods, Second Edition
Thomas P. Hettmansperger and Joseph W. McKean (2011)
120. Statistical Inference: The Minimum Distance Approach
Ayanendranath Basu, Hiroyuki Shioya, and Chanseok Park (2011)
121. Smoothing Splines: Methods and Applications Yuedong Wang (2011)
122. Extreme Value Methods with Applications to Finance Serguei Y. Novak (2012)
123. Dynamic Prediction in Clinical Survival Analysis Hans C. van Houwelingen and Hein Putter (2012)
124. Statistical Methods for Stochastic Differential Equations
Mathieu Kessler, Alexander Lindner, and Michael Sørensen (2012)
125. Maximum Likelihood Estimation for Sample Surveys
R. L. Chambers, D. G. Steel, Suojin Wang, and A. H. Welsh (2012)
126. Mean Field Simulation for Monte Carlo Integration Pierre Del Moral (2013)
127. Analysis of Variance for Functional Data Jin-Ting Zhang (2013)](https://image.slidesharecdn.com/2171729-250520160946-fea5cbc5/85/Analysis-Of-Variance-For-Functional-Data-Jinting-Zhang-9-320.jpg)
![MOTIVATING FUNCTIONAL DATA 13
University, France). The aim of the experiment was to analyze how muscle
copes with an external perturbation.
The experiment recruited seven young male volunteers. They wore a
spring-loaded orthosis of adjustable stiffness under four experimental con-
ditions: a control condition (without orthosis), an orthosis condition (with
the orthosis only), and two spring conditions (spring 1, spring 2) in which
stepping-in-place was perturbed by fitting a spring-loaded orthosis onto the
right knee joint. All the seven subjects tried all the four conditions ten times
for twenty seconds each while only the central ten seconds were used in the
study in order to avoid possible perturbations in the initial and final parts of
the experiment. The resultant moment of force at the knee is derived by means
of body segment kinematics recorded with a sampling frequency of 200 Hz.
For each stepping-in-place replication, the resultant moment was computed at
256 time points equally spaced and scaled so that a time interval corresponds
to an individual gait cycle. A typical observation is a curve over time t ∈ [0, 1].
Therefore, the orthosis data set consists of
7 (Subjects) × 4 (Treatments) × 10 (Replications) = 280 (Curves).
Figure 1.12 presents the 280 raw orthosis curves with each panel displaying
ten of them.
The orthosis data have been previously studied by a number of authors,
including Abramovich et al. (2004), Abramovich and Angelini (2006), Anto-
niadis and Sapatinas (2007), and Cuesta-Albertos and Febrero-Bande (2010),
among others. In this book, this data set will be used to motivate and illustrate
a heteroscedastic two-way ANOVA model for functional data in Chapter 9.
It will also be used to motivate and illustrate multi-sample equal-covariance
function testing problems in Chapter 10. The orthosis data were kindly pro-
vided by Dr. Brani Vidakovic by email communication.
1.2.8 Ergonomics Data
To study the motion of drivers of automobiles, the researchers at the Center
for Ergonomics at the University of Michigan collected data on the motion
of an automobile driver to twenty locations within a test car. Among other
measures, the researchers measured three times the angle formed at the right
elbow between the upper and lower arms of the driver. The data recorded for
each motion were observed on an equally spaced grid of points over a period of
time, which were rescaled to [0, 1] for convenience, but the number of such time
points varies from observation to observation. See Faraway (1997) and Shen
and Faraway (2004) for detailed descriptions of the data. Figure 1.13 displays
the right elbow angle curves of the ergonomics data downloaded from the
website of the second author of Shen and Faraway (2004). These right elbow
angle curves have been smoothed so that no further smoothing is needed but
some interpolation or smoothing may be needed if one wants to evaluate these
curves at a desired level of resolution.](https://image.slidesharecdn.com/2171729-250520160946-fea5cbc5/85/Analysis-Of-Variance-For-Functional-Data-Jinting-Zhang-42-320.jpg)
![14 INTRODUCTION
0 0.5 1
−50
0
50
100
0 0.5 1
−50
0
50
100
0 0.5 1
−50
0
50
100
0 0.5 1
−50
0
50
100
0 0.5 1
−50
0
50
100
0 0.5 1
−50
0
50
100
0 0.5 1
−50
0
50
100
0 0.5 1
−50
0
50
100
0 0.5 1
−50
0
50
100
0 0.5 1
−50
0
50
100
0 0.5 1
−50
0
50
100
0 0.5 1
−50
0
50
100
0 0.5 1
−50
0
50
100
0 0.5 1
−50
0
50
100
0 0.5 1
−50
0
50
100
0 0.5 1
−50
0
50
100
0 0.5 1
−50
0
50
100
0 0.5 1
−50
0
50
100
0 0.5 1
−50
0
50
100
0 0.5 1
−50
0
50
100
0 0.5 1
−50
0
50
100
0 0.5 1
−50
0
50
100
0 0.5 1
−50
0
50
100
0 0.5 1
−50
0
50
100
0 0.5 1
−50
0
50
100
0 0.5 1
−50
0
50
100
0 0.5 1
−50
0
50
100
0 0.5 1
−50
0
50
100
Figure 1.12 The orthosis data. The row panels are associated with the seven subjects
and the column panels are associated with the four treatment conditions. There are
ten orthosis curves displayed in each panel. An orthosis curve has 256 measurements
of the resultant moment of force at the knee, recorded over a period of time (ten
seconds), scaled to [0, 1].](https://image.slidesharecdn.com/2171729-250520160946-fea5cbc5/85/Analysis-Of-Variance-For-Functional-Data-Jinting-Zhang-43-320.jpg)
![MOTIVATING FUNCTIONAL DATA 15
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
110
120
130
140
150
160
Time
Angle
Figure 1.13 The ergonomics data collected on the motion of an automobile driver
to twenty locations within a test car, measured three times. The measurements of a
curve are the angles formed at the right elbow between the upper and lower arms of
the driver. The measurements for each motion were recorded on an equally spaced
grid of points over a period of time, rescaled to [0, 1] for convenience.
There are totally sixty right elbow angle curves. They can be classified
into a number of groups according to “location” and “area,” where “location”
stores the locations of the targets, taking twenty values while “area” stores the
areas of the targets, taking four values. The categorical variables “location”
and “area” are highly correlated as they are different only in the ways for
grouping the locations of the targets.
One can also find a functional regression model to predict the right elbow
angle curve y(t), t ∈ [0, 1] using the coordinates (a, b, c) of a target, where a
represents the coordinate in the left-to-right direction, b represents the coordi-
nate in the close-to-far direction, and c represents the coordinate in the down-
to-up direction. Shen and Faraway (2004) found that the following quadratic
model is adequate to fit the data:
yi(t) = β0(t) + aiβ1(t) + biβ2(t) + ciβ3(t) + a2
i β4(t)
+b2
i β5(t) + c2
i β6(t) + aibiβ7(t) + aiciβ8(t)
+biciβ9(t) + vi(t), i = 1, · · · , 60,
(1.1)
where yi(t) and vi(t) denote the ith response and location-effect curves over
time, respectively, (ai, bi, ci) denotes the coordinates of the target associated
with the ith angle curve, and βr(t), r = 0, 1, · · · , 9 are unknown coefficient
functions.
Some questions then arise naturally. Is each of the coefficient functions](https://image.slidesharecdn.com/2171729-250520160946-fea5cbc5/85/Analysis-Of-Variance-For-Functional-Data-Jinting-Zhang-44-320.jpg)
![20 NONPARAMETRIC SMOOTHERS FOR A SINGLE CURVE
equally spaced in an interval of interest, or be regarded as a random sample
from a continuous design density, namely, π(t). For simplicity, let us denote
the interval of interest or the support of π(t) as T , which can be a finite
interval [a, b] or the whole real line (−∞, ∞). The responses yi, i = 1, 2, · · · , n
are often observed with measurement errors. For the observed data (2.1), a
standard nonparametric regression model is usually written as
yi = f(ti) + i, i = 1, · · · , n, (2.2)
where f(t) models the underlying regression function and i, i = 1, 2, · · · , n
denote the measurement errors that cannot be explained by the regression
function f(t). Mathematically, f(t) is the conditional expectation of yi, given
ti = t. That is,
f(t) = E(yi|ti = t), i = 1, 2, · · · , n.
For an individual function in a functional data set, the regression function
f(t) is the associated underlying individual function.
There are a number of existing smoothing techniques that can be used
to smooth the regression function f(t) in (2.2). Different smoothing tech-
niques have different strengths in one aspect or another. For example, smooth-
ing splines may be good for handling sparse data, while local polynomial
smoothers may be computationally advantageous for handling dense designs.
In this chapter, as mentioned previously, we review four well-known smooth-
ing techniques, including local polynomial kernel smoothing (Wand and Jones
1995, Fan and Gijbels 1996), regression splines (Eubank 1999), smoothing
splines (Wahba 1990, Green and Silverman 1994), and P-splines (Ruppert,
Wand and Carroll 2003), respectively, in Sections 2.2, 2.3, 2.4, and 2.5. We
conclude this chapter with some bibliographical notes in Section 2.6.
2.2 Local Polynomial Kernel Smoothing
2.2.1 Construction of an LPK Smoother
The main idea of local polynomial kernel (LPK) smoothing is to locally ap-
proximate the underlying function f(t) in (2.2) by a polynomial of some de-
gree. Its foundation is Taylor expansion, which states that any smooth function
can be locally approximated by a polynomial of some degree.
Specifically, let t0 be an arbitrary fixed time point where the function f(t)
in (2.2) will be estimated. Assume f(t) has a (p + 1)st continuous deriva-
tive for some integer p ≥ 0 at t0. By Taylor expansion, f(t) can be locally
approximated by a polynomial of degree p. That is,
f(t) ≈ f(t0)+(t−t0)f(1)
(t0)/1!+(t−t0)2
f(2)
(t0)/2!+· · ·+(t−t0)p
f(p)
(t0)/p!,
in a neighborhood of t0 that allows the above expansion where f(r)
(t0) denotes
the rth derivative of f(t) at t0.](https://image.slidesharecdn.com/2171729-250520160946-fea5cbc5/85/Analysis-Of-Variance-For-Functional-Data-Jinting-Zhang-49-320.jpg)
![LOCAL POLYNOMIAL KERNEL SMOOTHING 21
Set βr = f(r)
(t0)/r!, r = 0, · · · , p. Let β̂r, r = 0, 1, 2, · · · , p be the minimiz-
ers of the following weighted least squares (WLS) criterion:
n
i=1
{yi − [β0 + (ti − t0)β1 + · · · + (ti − t0)p
βp]}
2
Kh(ti − t0), (2.3)
where Kh(·) = K(·/h)/h, obtained by rescaling a kernel function K(·) with a
constant h 0, called the bandwidth or smoothing parameter. The bandwidth
h is mainly used to specify the size of the local neighborhood, namely,
Ih(t0) = [t0 − h, t0 + h], (2.4)
where the local fit is conducted. The kernel function, K(·), determines how
observations within [t0 − h, t0 + h] contribute to the fit at t0.
Remark 2.1 The kernel function K(·) is usually a probability density func-
tion. For example, the uniform density K(t) = 1/2, t ∈ [−1, 1] and the stan-
dard normal density K(t) = exp(−t2
/2)/
√
2π, t ∈ (−∞, ∞) are two well-
known kernels, namely, the uniform kernel and the Gaussian kernel. Other
useful kernels can be found in Gasser, Müller, and Mammitzsch (1985), Mar-
ron and Nolan (1988), and Zhang and Fan (2000), among others.
Denote the estimate of the rth derivative f(r)
(t0) as ˆ
f
(r)
h (t0). Then
ˆ
f
(r)
h (t0) = r!β̂r, r = 0, 1, · · · , p.
In particular, the resulting pth degree LPK estimator of f(t0) is ˆ
fh(t0) = β̂0.
An explicit expression for ˆ
f
(r)
h (t0) is useful and can be made by matrix
notation. Let
X =
⎡
⎢
⎣
1 (t1 − t0) · · · (t1 − t0)p
.
.
.
.
.
.
...
.
.
.
1 (tn − t0) · · · (tn − t0)p
⎤
⎥
⎦ ,
and
W = diag(Kh(t1 − t0), · · · , Kh(tn − t0)),
be the design matrix and the weight matrix for the LPK fit around t0. Then
the WLS criterion (2.3) can be re-expressed as
(y − Xβ)T
W(y − Xβ), (2.5)
where y = (y1, · · · , yn)T
and β = (β0, β1, · · · , βp)T
. It follows that
ˆ
f
(r)
h (t0) = r!eT
r+1,p+1S−1
n Tny,
where er+1,p+1 denotes a (p+1)-dimensional unit vector whose (r+1)st entry
is 1 and the other entries are 0, and
Sn = XT
WX, Tn = XT
W.](https://image.slidesharecdn.com/2171729-250520160946-fea5cbc5/85/Analysis-Of-Variance-For-Functional-Data-Jinting-Zhang-50-320.jpg)
![22 NONPARAMETRIC SMOOTHERS FOR A SINGLE CURVE
When t0 runs over the whole support T of the design time points, a whole
range estimation of f(r)
(t) is obtained. The derivative estimator ˆ
f
(r)
h (t), t ∈
T is usually called the LPK smoother of the underlying derivative function
f(r)
(t). The derivative smoother ˆ
f
(r)
h (t0) is usually calculated on a grid of t’s
in T .
In this section, we only focus on the LPK curve smoother
ˆ
fh(t0) = eT
1,p+1S−1
n Tny =
n
i=1
Kn
h (ti − t0)yi, (2.6)
where Kn
h (ti − t0) = eT
1,p+1S−1
n Tnei,n with Kn
(t) known as the empirical
equivalent kernel for the p-order LPK; see Fan and Gijbels (1996). From (2.6),
it is seen that for any t ∈ T ,
ˆ
fh(t) =
n
i=1
Kn
h (ti − t)yi = a(t)T
y, (2.7)
where a(t) = [Kn
h (t1 − t), Kn
h (t2 − t), · · · , Kn
h (tn − t)]T
, which can be obtained
from TT
n S−1
n e1,p+1 after replacing t0 with t. This general formula allows us to
evaluate ˆ
fh(t) at any resolution. Set ŷi = ˆ
fh(ti) as the fitted value of f(ti).
Let ŷh = [ŷ1, · · · , ŷn]T
denote the fitted values at all the design time points.
Then ŷh can be expressed as
ŷh = Ahy, (2.8)
where
Ah = (a(t1), · · · , a(tn))T
(2.9)
is known as the smoother matrix of the LPK smoother.
Remark 2.2 As Ah does not depend on the response vector y, the LPK
smoother ˆ
fh(t) is known as a linear smoother, which is a linear combina-
tion of the responses. The regression spline, smoothing spline, and P-spline
smoothers introduced in the next three sections are also linear smoothers.
2.2.2 Two Special LPK Smoothers
Local constant and linear smoothers are the two simplest and most useful LPK
smoothers. The local constant smoother is known as the Nadaraya-Watson
estimator (Nadaraya 1964, Watson 1964). This smoother results from the
LPK smoother ˆ
fh(t0) (2.6) by simply taking p = 0:
ˆ
fh(t0) =
n
i=1 Kh(ti − t0)yi
n
i=1 Kh(ti − t0)
. (2.10)](https://image.slidesharecdn.com/2171729-250520160946-fea5cbc5/85/Analysis-Of-Variance-For-Functional-Data-Jinting-Zhang-51-320.jpg)
![LOCAL POLYNOMIAL KERNEL SMOOTHING 23
Within a local neighborhood Ih(t0), it fits the data with a constant. That is,
it is the minimizer β̂0 of the following WLS criterion:
n
i=1
(yi − β0)2
Kh(ti − t0).
The Nadaraya-Watson estimator is simple to understand and easy to compute.
Let IA(t) denote the indicator function of some set A.
Remark 2.3 When the kernel function K is the uniform kernel K(t) =
1/2, t ∈ [−1, 1], the Nadaraya-Watson estimator (2.10) is exactly the local
average of yi’s that are within the local neighborhood Ih(t0) (2.4):
ˆ
fh(t0) =
n
i=1 IIh(t0)(ti)yi
n
i=1 IIh(t0)(ti)
=
⎧
⎨
⎩
ti∈Ih(t0)
yi
⎫
⎬
⎭
/mh(t0),
where mh(t0) denotes the number of the observations falling into the local
neighborhood Ih(t0).
Remark 2.4 When t0 is a boundary point of T , fewer design points are within
the neighborhood Ih(t0) so that ˆ
fh(t0) has a slower convergence rate than the
case when t0 is an interior point of T . For a detailed explanation of this
boundary effect, the reader is referred to Müller (1991, 1993), Fan and Gijbels
(1996), Cheng, Fan, and Marron (1997), and Müller and Stadtmuller (1999),
among others.
The local linear smoother (Stone 1984, Fan 1992, 1993) is obtained by
fitting a data set locally with a linear function. Let (β̂0, β̂1) minimize the
following WLS criterion:
n
i=1
[yi − β0 − (ti − t0)β1]2
Kh(ti − t0).
Then the local linear smoother is ˆ
fh(t0) = β̂0. It can be easily obtained from
the LPK smoother ˆ
fh(t0) (2.6) by simply taking p = 1.
Remark 2.5 The local linear smoother is known as a smoother with a free
boundary effect (Cheng, Fan, and Marron 1997). That is, it has the same
convergence rate at any point in T . It also exhibits many good properties that
the other linear smoothers may lack. Good discussions on these properties can
be found in Fan (1992, 1993), Hastie and Loader (1993), and Fan and Gijbels
(1996, Chapter 2), among others.
A local linear smoother can be simply expressed as
ˆ
fh(t0) =
n
i=1 [s2(t0) − s1(t0)(ti − t0)] Kh(ti − t0)yi
s2(t0)s0(t0) − s2
1(t0)
, (2.11)](https://image.slidesharecdn.com/2171729-250520160946-fea5cbc5/85/Analysis-Of-Variance-For-Functional-Data-Jinting-Zhang-52-320.jpg)
![LOCAL POLYNOMIAL KERNEL SMOOTHING 25
−6 −4 −2 0 2 4 6 8 10 12 14
0
0.5
1
1.5
2
2.5
3
Day
Log(Progesterone)
raw data
h=0.32
h=1.3
h=5.22
Figure 2.1 Fitting the first nonconceptive progesterone curve by the LPK method
with p = 2. The circles represent the data. The dot-dashed curve, obtained using
a small bandwidth h = 0.32, nearly interpolates the data while the dashed curve,
obtained using a large bandwidth h = 5.22, does not provide an adequate fit to the
data. The solid curve, obtained using a bandwidth h = 1.3 selected by the GCV rule
(2.12), provides a nice fit to the data.
The solid curve is associated with h = h∗
= 1.3 [log10(1.3) = 0.1139], which is
not too small or too large according to the GCV rule described below and the
solid curve is indeed a nice fit to the data.
Thus, for a given p and a fixed kernel function K(·), a good bandwidth
h should be chosen to trade off between the goodness of fit and the model
complexity of the LPK fit (2.8). Here, better goodness of fit often means
less bias while smaller model complexity usually means less rough or smaller
variance. To better choose the bandwidth h in LPK smoothing, we can use
the following generalized cross-validation (GCV) rule (Wu and Zhang 2006,
Chapter 3):
GCV(h) =
y − ŷh2
(1 − tr(Ah)/n)2
, (2.12)
where the fitted response vector ŷh and the LPK smoother matrix Ah are de-
fined in (2.8) and (2.9), respectively. In the literature, many other bandwidth
selectors are available. Famous bandwidth selectors can be found in Fan and
Gijbels (1992), Herrmann, Gasser, and Kneip (1992), and Ruppert, Sheather,
and Wand (1995), among others.
Remark 2.8 With h increasing, the goodness of fit of the LPK fit measured](https://image.slidesharecdn.com/2171729-250520160946-fea5cbc5/85/Analysis-Of-Variance-For-Functional-Data-Jinting-Zhang-54-320.jpg)
![A WELLINGTON (STREET) MEMORIAL.
General Opinion (Mr. Punch) presents the Medal of the
Highest Order of Histrionic Merit to Henry Irving in
recognition of distinguished service as Corporal Gregory
Brewster in the action of Conan Doyle's Story of
Waterloo.
ON THE NEW STATUE.
[Her Majesty's Government are about to entrust to
one of our first sculptors a great historical statue,
which has too long been wanting to the series of those
who have governed England.—Lord Rosebery at the
Royal Academy Banquet.]
Our Uncrowned King at last to stand
'Midst the legitimate Lord's Anointed?
How will they shrink, that sacred band,](https://image.slidesharecdn.com/2171729-250520160946-fea5cbc5/85/Analysis-Of-Variance-For-Functional-Data-Jinting-Zhang-59-320.jpg)
![And you—you echo the old, old bray
Of Boanerges. A broader greeting
Of brotherhood full, warm hearts, wide eyes
Might lend a meaning to your May Meeting
To gladden the gentle and win the wise.
What's in a Name? A Rossa, c.—Before being ejected from the
House of Commons on Wednesday last, O'Donovan Rossa shouted
out that A stain had been put upon his name. Where is the
ingenious craftsman who did it? He might try his hand next time
at gilding refined gold.
Query.—Can a champagne wine from the vintage of Ay be invariably and
fairly described as Ay 1?
MODES AND METALS.
[Neckties made of aluminium have just been invented in
Germany.—Evening Paper.]
Visited my tailor's puddling works to-day. He has some really neat
new pig-iron fabrics for the season. I am thinking of trying his
Bessemer steel indestructible evening-dress suits.
Really this new plan of mineral clothing comes in very usefully when
one is attacked by roughs on a dark night. Floored an assailant most
satisfactorily with a touch of my lead handkerchief.
The only objection I can find to my aluminium summer suiting is its
tendency to get red hot if I stand in the sun for five minutes.
I think I can now safely defy my laundress to injure my patent safety
ironclad steel shirts.
I find, however, that there is no need of a laundress at all. When
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Analysis Of Variance For Functional Data Jinting Zhang
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- 5. Despite research interest in functional data analysis in the last three decades, few books are available on the subject. Filling this gap, Analysis of Variance for Functional Data presents up-to-date hypothesis testing methods for functional data analysis. The book covers the reconstruction of functional observations, functional ANOVA, functional linear models with functional responses, ill- conditioned functional linear models, diagnostics of functional observations, heteroscedastic ANOVA for functional data, and testing equality of covariance functions. Although the methodologies presented are designed for curve data, they can be extended to surface data. Useful for statistical researchers and practitioners analyzing functional data, this self-contained book gives both a theoretical and applied treatment of functional data analysis supported by easy-to-use MATLAB® code. The author provides a number of simple methods for functional hypothesis testing. He discusses pointwise, L2 -norm- based, F-type, and bootstrap tests. Assuming only basic knowledge of statistics, calculus, and matrix algebra, the book explains the key ideas at a relatively low technical level using real data examples. Each chapter also includes bibliographical notes and exercises. Real functional data sets from the text and MATLAB codes for analyzing the data examples are available for download from the author’s website. K12912 Analysis of Variance for Functional Data Analysis of Variance for Functional Data Jin-Ting Zhang Zhang Monographs on Statistics and Applied Probability 127 127 Statistics K12912_Cover.indd 1 5/9/13 1:45 PM
- 7. MONOGRAPHS ON STATISTICS AND APPLIED PROBABILITY General Editors F. Bunea, V. Isham, N. Keiding, T. Louis, R. L. Smith, and H. Tong 1. Stochastic Population Models in Ecology and Epidemiology M.S. Barlett (1960) 2. Queues D.R. Cox and W.L. Smith (1961) 3. Monte Carlo Methods J.M. Hammersley and D.C. Handscomb (1964) 4. The Statistical Analysis of Series of Events D.R. Cox and P.A.W. Lewis (1966) 5. Population Genetics W.J. Ewens (1969) 6. Probability, Statistics and Time M.S. Barlett (1975) 7. Statistical Inference S.D. Silvey (1975) 8. The Analysis of Contingency Tables B.S. Everitt (1977) 9. Multivariate Analysis in Behavioural Research A.E. Maxwell (1977) 10. Stochastic Abundance Models S. Engen (1978) 11. Some Basic Theory for Statistical Inference E.J.G. Pitman (1979) 12. Point Processes D.R. Cox and V. Isham (1980) 13. ,GHQWLÀFDWLRQRI2XWOLHUVD.M. Hawkins (1980) 14. Optimal Design S.D. Silvey (1980) 15. Finite Mixture Distributions B.S. Everitt and D.J. Hand (1981) 16. ODVVLÀFDWLRQA.D. Gordon (1981) 17. Distribution-Free Statistical Methods, 2nd edition J.S. Maritz (1995) 18. 5HVLGXDOVDQG,QÁXHQFHLQ5HJUHVVLRQR.D. Cook and S. Weisberg (1982) 19. Applications of Queueing Theory, 2nd edition G.F. Newell (1982) 20. Risk Theory, 3rd edition R.E. Beard, T. Pentikäinen and E. Pesonen (1984) 21. Analysis of Survival Data D.R. Cox and D. Oakes (1984) 22. An Introduction to Latent Variable Models B.S. Everitt (1984) 23. Bandit Problems D.A. Berry and B. Fristedt (1985) 24. Stochastic Modelling and Control M.H.A. Davis and R. Vinter (1985) 25. The Statistical Analysis of Composition Data J. Aitchison (1986) 26. Density Estimation for Statistics and Data Analysis B.W. Silverman (1986) 27. Regression Analysis with Applications G.B. Wetherill (1986) 28. Sequential Methods in Statistics, 3rd edition G.B. Wetherill and K.D. Glazebrook (1986) 29. Tensor Methods in Statistics P. McCullagh (1987) 30. Transformation and Weighting in Regression R.J. Carroll and D. Ruppert (1988) 31. Asymptotic Techniques for Use in Statistics O.E. Bandorff-Nielsen and D.R. Cox (1989) 32. Analysis of Binary Data, 2nd edition D.R. Cox and E.J. Snell (1989) 33. Analysis of Infectious Disease Data N.G. Becker (1989) 34. Design and Analysis of Cross-Over Trials B. Jones and M.G. Kenward (1989) 35. Empirical Bayes Methods, 2nd edition J.S. Maritz and T. Lwin (1989) 36. Symmetric Multivariate and Related Distributions K.T. Fang, S. Kotz and K.W. Ng (1990) 37. Generalized Linear Models, 2nd edition P. McCullagh and J.A. Nelder (1989) 38. Cyclic and Computer Generated Designs, 2nd edition J.A. John and E.R. Williams (1995) 39. Analog Estimation Methods in Econometrics C.F. Manski (1988) 40. Subset Selection in Regression A.J. Miller (1990) 41. Analysis of Repeated Measures M.J. Crowder and D.J. Hand (1990) 42. Statistical Reasoning with Imprecise Probabilities P. Walley (1991) 43. Generalized Additive Models T.J. Hastie and R.J. Tibshirani (1990) 44. Inspection Errors for Attributes in Quality Control N.L. Johnson, S. Kotz and X. Wu (1991) 45. The Analysis of Contingency Tables, 2nd edition B.S. Everitt (1992)
- 8. 46. The Analysis of Quantal Response Data B.J.T. Morgan (1992) 47. Longitudinal Data with Serial Correlation—A State-Space Approach R.H. Jones (1993) 48. Differential Geometry and Statistics M.K. Murray and J.W. Rice (1993) 49. Markov Models and Optimization M.H.A. Davis (1993) 50. Networks and Chaos—Statistical and Probabilistic Aspects O.E. Barndorff-Nielsen, J.L. Jensen and W.S. Kendall (1993) 51. Number-Theoretic Methods in Statistics K.-T. Fang and Y. Wang (1994) 52. Inference and Asymptotics O.E. Barndorff-Nielsen and D.R. Cox (1994) 53. Practical Risk Theory for Actuaries C.D. Daykin, T. Pentikäinen and M. Pesonen (1994) 54. Biplots J.C. Gower and D.J. Hand (1996) 55. Predictive Inference—An Introduction S. Geisser (1993) 56. Model-Free Curve Estimation M.E. Tarter and M.D. Lock (1993) 57. An Introduction to the Bootstrap B. Efron and R.J. Tibshirani (1993) 58. Nonparametric Regression and Generalized Linear Models P.J. Green and B.W. Silverman (1994) 59. Multidimensional Scaling T.F. Cox and M.A.A. Cox (1994) 60. Kernel Smoothing M.P. Wand and M.C. Jones (1995) 61. Statistics for Long Memory Processes J. Beran (1995) 62. Nonlinear Models for Repeated Measurement Data M. Davidian and D.M. Giltinan (1995) 63. Measurement Error in Nonlinear Models R.J. Carroll, D. Rupert and L.A. Stefanski (1995) 64. Analyzing and Modeling Rank Data J.J. Marden (1995) 65. Time Series Models—In Econometrics, Finance and Other Fields D.R. Cox, D.V. Hinkley and O.E. Barndorff-Nielsen (1996) 66. Local Polynomial Modeling and its Applications J. Fan and I. Gijbels (1996) 67. Multivariate Dependencies—Models, Analysis and Interpretation D.R. Cox and N. Wermuth (1996) 68. Statistical Inference—Based on the Likelihood A. Azzalini (1996) 69. Bayes and Empirical Bayes Methods for Data Analysis B.P. Carlin and T.A Louis (1996) 70. Hidden Markov and Other Models for Discrete-Valued Time Series I.L. MacDonald and W. Zucchini (1997) 71. Statistical Evidence—A Likelihood Paradigm R. Royall (1997) 72. Analysis of Incomplete Multivariate Data J.L. Schafer (1997) 73. Multivariate Models and Dependence Concepts H. Joe (1997) 74. Theory of Sample Surveys M.E. Thompson (1997) 75. Retrial Queues G. Falin and J.G.C. Templeton (1997) 76. Theory of Dispersion Models B. Jørgensen (1997) 77. Mixed Poisson Processes J. Grandell (1997) 78. Variance Components Estimation—Mixed Models, Methodologies and Applications P.S.R.S. Rao (1997) 79. Bayesian Methods for Finite Population Sampling G. Meeden and M. Ghosh (1997) 80. Stochastic Geometry—Likelihood and computation O.E. Barndorff-Nielsen, W.S. Kendall and M.N.M. van Lieshout (1998) 81. Computer-Assisted Analysis of Mixtures and Applications—Meta-Analysis, Disease Mapping and Others D. Böhning (1999) 82. ODVVLÀFDWLRQQGHGLWLRQA.D. Gordon (1999) 83. Semimartingales and their Statistical Inference B.L.S. Prakasa Rao (1999) 84. Statistical Aspects of BSE and vCJD—Models for Epidemics C.A. Donnelly and N.M. Ferguson (1999) 85. Set-Indexed Martingales G. Ivanoff and E. Merzbach (2000) 86. The Theory of the Design of Experiments D.R. Cox and N. Reid (2000) 87. Complex Stochastic Systems O.E. Barndorff-Nielsen, D.R. Cox and C. Klüppelberg (2001) 88. Multidimensional Scaling, 2nd edition T.F. Cox and M.A.A. Cox (2001) 89. Algebraic Statistics—Computational Commutative Algebra in Statistics G. Pistone, E. Riccomagno and H.P. Wynn (2001) 90. Analysis of Time Series Structure—SSA and Related Techniques N. Golyandina, V. Nekrutkin and A.A. Zhigljavsky (2001) 91. Subjective Probability Models for Lifetimes Fabio Spizzichino (2001) 92. Empirical Likelihood Art B. Owen (2001)
- 9. 93. Statistics in the 21st Century Adrian E. Raftery, Martin A. Tanner, and Martin T. Wells (2001) 94. Accelerated Life Models: Modeling and Statistical Analysis Vilijandas Bagdonavicius and Mikhail Nikulin (2001) 95. Subset Selection in Regression, Second Edition Alan Miller (2002) 96. Topics in Modelling of Clustered Data Marc Aerts, Helena Geys, Geert Molenberghs, and Louise M. Ryan (2002) 97. Components of Variance D.R. Cox and P.J. Solomon (2002) 98. Design and Analysis of Cross-Over Trials, 2nd Edition Byron Jones and Michael G. Kenward (2003) 99. Extreme Values in Finance, Telecommunications, and the Environment Bärbel Finkenstädt and Holger Rootzén (2003) 100. Statistical Inference and Simulation for Spatial Point Processes Jesper Møller and Rasmus Plenge Waagepetersen (2004) 101. Hierarchical Modeling and Analysis for Spatial Data Sudipto Banerjee, Bradley P. Carlin, and Alan E. Gelfand (2004) 102. Diagnostic Checks in Time Series Wai Keung Li (2004) 103. Stereology for Statisticians Adrian Baddeley and Eva B. Vedel Jensen (2004) 104. Gaussian Markov Random Fields: Theory and Applications +Ý DYDUG5XHDQG/HRQKDUG+HOG(2005) 105. Measurement Error in Nonlinear Models: A Modern Perspective, Second Edition Raymond J. Carroll, David Ruppert, Leonard A. Stefanski, and Ciprian M. Crainiceanu (2006) 106. *HQHUDOL]HG/LQHDU0RGHOVZLWK5DQGRP(IIHFWV8QLÀHG$QDOVLVYLD+OLNHOLKRRG Youngjo Lee, John A. Nelder, and Yudi Pawitan (2006) 107. Statistical Methods for Spatio-Temporal Systems Bärbel Finkenstädt, Leonhard Held, and Valerie Isham (2007) 108. Nonlinear Time Series: Semiparametric and Nonparametric Methods Jiti Gao (2007) 109. Missing Data in Longitudinal Studies: Strategies for Bayesian Modeling and Sensitivity Analysis Michael J. Daniels and Joseph W. Hogan (2008) 110. Hidden Markov Models for Time Series: An Introduction Using R Walter Zucchini and Iain L. MacDonald (2009) 111. ROC Curves for Continuous Data Wojtek J. Krzanowski and David J. Hand (2009) 112. Antedependence Models for Longitudinal Data Dale L. Zimmerman and Vicente A. Núñez-Antón (2009) 113. Mixed Effects Models for Complex Data Lang Wu (2010) 114. Intoduction to Time Series Modeling Genshiro Kitagawa (2010) 115. Expansions and Asymptotics for Statistics Christopher G. Small (2010) 116. Statistical Inference: An Integrated Bayesian/Likelihood Approach Murray Aitkin (2010) 117. Circular and Linear Regression: Fitting Circles and Lines by Least Squares Nikolai Chernov (2010) 118. Simultaneous Inference in Regression Wei Liu (2010) 119. Robust Nonparametric Statistical Methods, Second Edition Thomas P. Hettmansperger and Joseph W. McKean (2011) 120. Statistical Inference: The Minimum Distance Approach Ayanendranath Basu, Hiroyuki Shioya, and Chanseok Park (2011) 121. Smoothing Splines: Methods and Applications Yuedong Wang (2011) 122. Extreme Value Methods with Applications to Finance Serguei Y. Novak (2012) 123. Dynamic Prediction in Clinical Survival Analysis Hans C. van Houwelingen and Hein Putter (2012) 124. Statistical Methods for Stochastic Differential Equations Mathieu Kessler, Alexander Lindner, and Michael Sørensen (2012) 125. Maximum Likelihood Estimation for Sample Surveys R. L. Chambers, D. G. Steel, Suojin Wang, and A. H. Welsh (2012) 126. Mean Field Simulation for Monte Carlo Integration Pierre Del Moral (2013) 127. Analysis of Variance for Functional Data Jin-Ting Zhang (2013)
- 10. Monographs on Statistics and Applied Probability 127 Analysis of Variance for Functional Data Jin-Ting Zhang
- 11. MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MAT- LAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software. CRC Press Taylor Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2014 by Taylor Francis Group, LLC CRC Press is an imprint of Taylor Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20130516 International Standard Book Number-13: 978-1-4398-6274-2 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information stor- age or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copy- right.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that pro- vides licenses and registration for a variety of users. For organizations that have been granted a pho- tocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com
- 12. To my Parents and Teachers To Tian-Hui, Tian-Yu, and Yan
- 13. This page intentionally left blank This page intentionally left blank
- 14. Contents List of Figures xv List of Tables xix Preface xxi 1 Introduction 1 1.1 Functional Data 1 1.2 Motivating Functional Data 1 1.2.1 Progesterone Data 2 1.2.2 Berkeley Growth Curve Data 3 1.2.3 Nitrogen Oxide Emission Level Data 6 1.2.4 Canadian Temperature Data 6 1.2.5 Audible Noise Data 9 1.2.6 Left-Cingulum Data 11 1.2.7 Orthosis Data 11 1.2.8 Ergonomics Data 13 1.3 Why Is Functional Data Analysis Needed? 16 1.4 Overview of the Book 17 1.5 Implementation of Methodologies 17 1.6 Options for Reading This Book 18 1.7 Bibliographical Notes 18 2 Nonparametric Smoothers for a Single Curve 19 2.1 Introduction 19 2.2 Local Polynomial Kernel Smoothing 20 2.2.1 Construction of an LPK Smoother 20 2.2.2 Two Special LPK Smoothers 22 2.2.3 Selecting a Good Bandwidth 24 2.2.4 Robust LPK Smoothing 26 2.3 Regression Splines 28 2.3.1 Truncated Power Basis 29 2.3.2 Regression Spline Smoother 30 2.3.3 Knot Locating and Knot Number Selection 30 2.3.4 Robust Regression Splines 34 2.4 Smoothing Splines 35 ix
- 15. x CONTENTS 2.4.1 Smoothing Spline Smoothers 35 2.4.2 Cubic Smoothing Splines 36 2.4.3 Smoothing Parameter Selection 37 2.5 P-Splines 40 2.5.1 P-Spline Smoothers 40 2.5.2 Smoothing Parameter Selection 41 2.6 Concluding Remarks and Bibliographical Notes 44 3 Reconstruction of Functional Data 47 3.1 Introduction 47 3.2 Reconstruction Methods 49 3.2.1 Individual Function Estimators 49 3.2.2 Smoothing Parameter Selection 50 3.2.3 LPK Reconstruction 50 3.2.4 Regression Spline Reconstruction 54 3.2.5 Smoothing Spline Reconstruction 57 3.2.6 P-Spline Reconstruction 59 3.3 Accuracy of LPK Reconstructions 61 3.3.1 Mean and Covariance Function Estimation 63 3.3.2 Noise Variance Function Estimation 65 3.3.3 Effect of LPK Smoothing 66 3.3.4 A Simulation Study 66 3.4 Accuracy of LPK Reconstruction in FLMs 68 3.4.1 Coefficient Function Estimation 69 3.4.2 Significance Tests of Covariate Effects 70 3.4.3 A Real Data Example 72 3.5 Technical Proofs 74 3.6 Concluding Remarks and Bibliographical Notes 80 3.7 Exercises 81 4 Stochastic Processes 83 4.1 Introduction 83 4.2 Stochastic Processes 83 4.2.1 Gaussian Processes 85 4.2.2 Wishart Processes 86 4.2.3 Linear Forms of Stochastic Processes 88 4.2.4 Quadratic Forms of Stochastic Processes 88 4.2.5 Central Limit Theorems for Stochastic Processes 91 4.3 χ2 -Type Mixtures 92 4.3.1 Cumulants 93 4.3.2 Distribution Approximation 94 4.4 F-Type Mixtures 100 4.4.1 Distribution Approximation 101 4.5 One-Sample Problem for Functional Data 107 4.5.1 Pointwise Tests 109
- 16. CONTENTS xi 4.5.2 L2 -Norm-Based Test 111 4.5.3 F-Type Test 114 4.5.4 Bootstrap Test 115 4.5.5 Numerical Implementation 116 4.5.6 Effect of Resolution Number 119 4.6 Technical Proofs 119 4.7 Concluding Remarks and Bibliographical Notes 126 4.8 Exercises 127 5 ANOVA for Functional Data 129 5.1 Introduction 129 5.2 Two-Sample Problem 129 5.2.1 Pivotal Test Function 133 5.2.2 Methods for Two-Sample Problems 134 5.3 One-Way ANOVA 142 5.3.1 Estimation of Group Mean and Covariance Functions 145 5.3.2 Main-Effect Test 148 5.3.3 Tests of Linear Hypotheses 160 5.4 Two-Way ANOVA 164 5.4.1 Estimation of Cell Mean and Covariance Functions 166 5.4.2 Main and Interaction Effect Functions 169 5.4.3 Tests of Linear Hypotheses 171 5.4.4 Balanced Two-Way ANOVA with Interaction 178 5.4.5 Balanced Two-Way ANOVA without Interaction 184 5.5 Technical Proofs 190 5.6 Concluding Remarks and Bibliographical Notes 195 5.7 Exercises 196 6 Linear Models with Functional Responses 197 6.1 Introduction 197 6.2 Linear Models with Time-Independent Covariates 197 6.2.1 Coefficient Function Estimation 200 6.2.2 Properties of the Estimators 202 6.2.3 Multiple Correlation Coefficient 204 6.2.4 Comparing Two Nested FLMs 205 6.2.5 Significance of All the Non-Intercept Coefficient Func- tions 211 6.2.6 Significance of a Single Coefficient Function 212 6.2.7 Tests of Linear Hypotheses 214 6.2.8 Variable Selection 218 6.3 Linear Models with Time-Dependent Covariates 221 6.3.1 Estimation of the Coefficient Functions 221 6.3.2 Compare Two Nested FLMs 222 6.3.3 Tests of Linear Hypotheses 228 6.4 Technical Proofs 232
- 17. xii CONTENTS 6.5 Concluding Remarks and Bibliographical Notes 236 6.6 Exercises 238 7 Ill-Conditioned Functional Linear Models 241 7.1 Introduction 241 7.2 Generalized Inverse Method 245 7.2.1 Estimability of Regression Coefficient Functions 245 7.2.2 Methods for Finding Estimable Linear Functions 247 7.2.3 Estimation of Estimable Linear Functions 252 7.2.4 Tests of Testable Linear Hypotheses 253 7.3 Reparameterization Method 259 7.3.1 The Methodology 259 7.3.2 Determining the Reparameterization Matrices 259 7.3.3 Invariance of the Reparameterization 260 7.3.4 Tests of Testable Linear Hypotheses 261 7.4 Side-Condition Method 261 7.4.1 The Methodology 261 7.4.2 Methods for Specifying the Side-Conditions 262 7.4.3 Invariance of the Side-Condition Method 263 7.4.4 Tests of Testable Linear Hypotheses 264 7.5 Technical Proofs 266 7.6 Concluding Remarks and Bibliographical Notes 271 7.7 Exercises 271 8 Diagnostics of Functional Observations 273 8.1 Introduction 273 8.2 Residual Functions 276 8.2.1 Raw Residual Functions 276 8.2.2 Standardized Residual Functions 277 8.2.3 Jackknife Residual Functions 277 8.3 Functional Outlier Detection 279 8.3.1 Standardized Residual-Based Method 279 8.3.2 Jackknife Residual-Based Method 283 8.3.3 Functional Depth-Based Method 285 8.4 Influential Case Detection 291 8.5 Robust Estimation of Coefficient Functions 292 8.6 Outlier Detection for a Sample of Functions 293 8.6.1 Residual Functions 293 8.6.2 Functional Outlier Detection 294 8.7 Technical Proofs 298 8.8 Concluding Remarks and Bibliographical Notes 298 8.9 Exercises 299
- 18. CONTENTS xiii 9 Heteroscedastic ANOVA for Functional Data 307 9.1 Introduction 307 9.2 Two-Sample Behrens-Fisher Problems 308 9.2.1 Estimation of Mean and Covariance Functions 310 9.2.2 Testing Methods 312 9.3 Heteroscedastic One-Way ANOVA 318 9.3.1 Estimation of Group Mean and Covariance Functions 320 9.3.2 Heteroscedastic Main-Effect Test 322 9.3.3 Tests of Linear Hypotheses under Heteroscedasticity 329 9.4 Heteroscedastic Two-Way ANOVA 330 9.4.1 Estimation of Cell Mean and Covariance Functions 335 9.4.2 Tests of Linear Hypotheses under Heteroscedasticity 337 9.5 Technical Proofs 344 9.6 Concluding Remarks and Bibliographical Notes 348 9.7 Exercises 348 10 Test of Equality of Covariance Functions 351 10.1 Introduction 351 10.2 Two-Sample Case 351 10.2.1 Pivotal Test Function 352 10.2.2 Testing Methods 354 10.3 Multi-Sample Case 356 10.3.1 Estimation of Group Mean and Covariance Functions 358 10.3.2 Testing Methods 360 10.4 Technical Proofs 364 10.5 Concluding Remarks and Bibliographical Notes 366 10.6 Exercises 366 Bibliography 369 Index 381
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- 20. List of Figures 1.1 A simulated functional data set 2 1.2 The raw progesterone data 4 1.3 Four individual nonconceptive progesterone curves 4 1.4 The Berkeley growth curve data 5 1.5 The NOx emission level data 7 1.6 The Canadian temperature data 8 1.7 Four individual temperature curves 8 1.8 The audible noise data 10 1.9 Four individual curves of the audible noise data 10 1.10 The left-cingulum data 12 1.11 Two left-cingulum curves 12 1.12 The orthosis data 14 1.13 The ergonomics data 15 2.1 Fitting the first nonconceptive progesterone curve by the LPK method 25 2.2 Plot of GCV score against bandwidth h (in log10-scale) for fitting the first nonconceptive progesterone curve 26 2.3 Fitting the twentieth nonconceptive progesterone curve by the robust LPK method. 27 2.4 Fitting the fifth nonconceptive progesterone curve by the regression spline method 32 2.5 Plot of GCV score (in log10-scale) against number of knots K for fitting the fifth nonconceptive progesterone curve by the regression spline method 32 2.6 Fitting the fifth Canadian temperature curve by the regression spline method 33 2.7 Plot of GCV score (in log10-scale) against number of knots K for fitting the fifth Canadian temperature curve by the regression spline method 33 2.8 Fitting the twentieth nonconceptive progesterone curve by the robust regression spline method 34 2.9 Fitting the fifth audible noise curve by the cubic smoothing spline method 38 xv
- 21. xvi LIST OF FIGURES 2.10 Plot of GCV score against smoothing parameter λ (in log10- scale) for fitting the fifth audible noise curve by the cubic smoothing spline method 38 2.11 Fitting the fifth Berkeley growth curve by the cubic smoothing spline method 39 2.12 Plot of GCV score against smoothing parameter λ (in log10- scale) for fitting the fifth Berkeley growth curve by the cubic smoothing spline method 39 2.13 Fitting the fifth left-cingulum curve by the P-spline method 42 2.14 Plot of GCV score against smoothing parameter λ (in log10- scale) for fitting the fifth left-cingulum curve by the P-spline method 42 2.15 Fitting the fifth ergonomics curve by the P-spline method 43 2.16 Plot of GCV score against smoothing parameter λ (in log10- scale) for fitting the fifth ergonomics curve by the P-spline method 43 3.1 Plot of GCV score against bandwidth h (in log10-scale) for reconstructing the progesterone curves by the LPK reconstruc- tion method 52 3.2 Reconstructed progesterone curves 52 3.3 Four reconstructed nonconceptive progesterone curves 53 3.4 Four robustly reconstructed nonconceptive progesterone curves 54 3.5 Plot of GCV score against number of knots K for reconstructing the Canadian temperature data by the regression spline method 55 3.6 Reconstructed Canadian temperature curves by the regression spline method 56 3.7 Two reconstructed Canadian temperature curves 57 3.8 Two reconstructed left-cingulum curves obtained by the cubic smoothing spline method 59 3.9 Four reconstructed audible noise curves obtained by the P-spline reconstruction method 61 3.10 Simulation results about the LPK reconstruction for functional data 68 3.11 Estimated nonconceptive and conceptive mean functions with 95% standard deviation bands 73 3.12 Null distribution approximations of Tn 73 4.1 Histograms of two χ2 -type mixtures 95 4.2 Estimated pdfs of the χ2 -type mixtures by the normal approx- imation method 96 4.3 Estimated pdfs of the χ2 -type mixtures by the two-cumulant matched χ2 -approximation method 97 4.4 Estimated pdfs of the χ2 -type mixtures by the three-cumulant matched χ2 -approximation method 98
- 22. LIST OF FIGURES xvii 4.5 Estimated pdfs of Ta and Tb by the four methods 99 4.6 Histograms of two F-type mixtures 102 4.7 Estimated pdfs of the F-type mixtures by the normal approxi- mation method 103 4.8 Estimated pdfs of the F-type mixtures by the two-cumulant matched χ2 -approximation method 104 4.9 Estimated pdfs of the F-type mixtures by the three-cumulant matched χ2 -approximation method 105 4.10 Estimated pdfs of Fa and Fb by the four methods 106 4.11 The conceptive progesterone data example 108 4.12 Pointwise t-test, z-test, and bootstrap test for the conceptive progesterone data 111 4.13 Plots of P-value of the F-type test against resolution number 119 5.1 Reconstructed individual curves of the progesterone data 130 5.2 Sample mean functions of the progesterone data 131 5.3 Sample covariance functions of the progesterone data 132 5.4 Pointwise t-, z- and bootstrap tests for the two-sample problem for the progesterone data 137 5.5 Reconstructed Canadian temperature curves obtained by the local linear kernel smoother 143 5.6 Pointwise tests for the main-effect testing problem with the Canadian temperature data 150 5.7 Sample group mean functions of the Canadian temperature data 152 5.8 Sample cell mean functions of the left-cingulum data 167 5.9 Pooled sample covariance function of the left-cingulum data 167 5.10 Two-way ANOVA for the left-cingulum data by the pointwise F-test 174 6.1 Six selected right elbow angle curves from the ergonomics data 198 6.2 The smoothed right elbow angle curves of the ergonomics data 199 6.3 Four fitted coefficient functions of the ergonomics data with 95% pointwise confidence bands 201 6.4 The estimated covariance function of the ergonomics data 202 6.5 Plots of R2 n(t) and R2 n,adj(t) of the quadratic FLM for the ergonomics data 205 6.6 Four fitted coefficient functions of the quadratic FLM for the ergonomics data with 95% pointwise confidence bands 214 7.1 Estimated main-effect contrast functions of the seven factors of the audible noise data 256 8.1 Reconstructed right elbow angle curves with an unusual curve highlighted 274
- 23. xviii LIST OF FIGURES 8.2 First four fitted coefficient functions with their 95% confidence bands 274 8.3 Outlier detection for the ergonomics data using the standard- ized residual-based method 282 8.4 Reconstructed right elbow angle curves of the ergonomics data with three suspected functional outliers highlighted 283 8.5 Outlier detection for the ergonomics data using three functional depth measures 289 8.6 Outlier detection for the ergonomics data using three functional outlyingness measures 290 8.7 Influential case detection for the ergonomics data using the generalized Cook’s distance 292 8.8 Outlier detection for the NOx emission level curves for seventy- six working days using the standardized residual-based method 297 8.9 NOx emission level curves for seventy-six working days with three detected functional outliers highlighted 297 8.10 Outlier detection for the ergonomics data using the jackknife residual-based method 299 8.11 Outlier detection for the NOx emission level data for seventy- six working days using the jackknife residual-based method 300 8.12 Outlier detection for the NOx emission level curves for thirty- nine non-working days using the standardized residual-based method 301 8.13 NOx emission level curves for thirty-nine non-working days with three detected functional outliers highlighted 302 8.14 Outlier detection for the NOx emission level data for thirty- nine non-working days using the ED-, functional KDE-, and projection-based functional depth measures 303 8.15 Outlier detection for the NOx emission level data using the ED-, functional KDE-, and projection-based functional outlyingness measures 304 9.1 NOx emission level curves for seventy-three working days and thirty-six non-working days with outlying curves removed 308 9.2 Sample covariance functions of NOx emission level curves for working and non-working days with outlying curves removed 309 9.3 Sample covariance functions of the Canadian temperature data 319 9.4 Sample cell covariance functions of the orthosis data 331 9.5 Sample cell covariance functions of the orthosis data for Subject 5 at the four treatment conditions 332
- 24. List of Tables 3.1 Test results for comparing the two group mean functions of the progesterone data 74 4.1 Testing (4.39) for the conceptive progesterone data by the L2 -norm-based test 113 4.2 Testing (4.39) for the conceptive progesterone data by the F-type test 115 4.3 Testing (4.39) for the conceptive progesterone data by the L2 -norm-based and F-type bootstrap tests 116 5.1 Traces of the sample covariance functions γ̂1(s, t) and γ̂2(s, t) and their cross-square functions γ̂⊗2 1 (s, t) and γ̂⊗2 2 (s, t) over various periods 132 5.2 Traces of the pooled sample covariance function γ̂(s, t) and its cross-square function γ̂⊗2 (s, t) over various periods 132 5.3 The L2 -norm-based test for the two-sample problem for the progesterone data 140 5.4 The F-type test for the two-sample problem for the proges- terone data 141 5.5 The bootstrap tests for the two-sample problem for the progesterone data 142 5.6 Traces of the pooled sample covariance functions γ̂(s, t) and its cross-square function γ̂⊗2 (s, t) of the Canadian temperature data over various seasons 153 5.7 The L2 -norm-based test for the one-way ANOVA problem for the Canadian temperature data 154 5.8 The F-type test for the one-way ANOVA problem for the Canadian temperature data 155 5.9 The L2 -norm-based and F-type bootstrap tests for the one-way ANOVA problem with the Canadian temperature data 159 5.10 Two-way ANOVA for the left-cingulum data by the L2 -norm- based test 176 5.11 Two-way ANOVA for the left-cingulum data by the F-type test 177 xix
- 25. xx LIST OF TABLES 6.1 Individual coefficient tests of the quadratic FLM for the ergonomics data 213 6.2 Functional variable selection for the ergonomics data by the forward selection method 220 6.3 Functional variable selection for the ergonomics data by the backward elimination method 220 7.1 L2 -norm-based test for the audible noise data 258 9.1 Various quantities of the Canadian temperature data over various seasons 320 9.2 Heteroscedastic one-way ANOVA of the Canadian temperature data by the L2 -norm-based test 325 9.3 Heteroscedastic one-way ANOVA of the Canadian temperature data by the F-type test 328 9.4 Heteroscedastic two-way ANOVA for the orthosis data by the L2 -norm-based test 340 9.5 Heteroscedastic two-way ANOVA of the orthosis data by the F-type test 343 10.1 Tests of the equality of the covariance functions for the Canadian temperature data 363
- 26. Preface Functional data analysis has been a popular statistical research topic for the past three decades. Functional data are often obtained by observing a num- ber of subjects over time, space, or other continua densely. They are fre- quently collected from various research areas, including audiology, biology, children’s growth studies, ergonomics, environmentology, meteorology, and women’s health studies among others in the form of curves, surfaces, or other complex objects. Statistical inference for functional data generally refers to estimation and hypothesis testing about functional data. There are at least two monographs available in the literature that are devoted to estimation and classification of functional data. This book mainly focuses on hypothe- sis testing problems about functional data, with topics including reconstruc- tion of functional observations, functional ANOVA, functional linear models with functional responses, ill-conditioned functional linear models, diagnostics of functional observations, heteroscedastic ANOVA for functional data, and testing equality of covariance functions, among others. Although the method- ologies proposed and studied in this book are designed for curve data only, it is straightforward to extend them to the analysis of surface data. The main purpose of this book is to provide the reader with a number of simple methodologies for functional hypothesis testing. Pointwise, L2 -norm based, F-type, and bootstrap tests are discussed. The key ideas of these methodologies are stated at a relatively low technical level. The book is self- contained and assumes only a basic knowledge of statistics, calculus, and ma- trix algebra. Real data examples from the aforementioned research areas are provided to motivate and illustrate the methodologies. Some bibliographical notes and exercises are provided at the end of each chapter. A supporting web- site is provided where most of the real data sets analyzed in this book may be downloaded. Some related MATLABR codes for analyzing these real data examples are also available and will be updated whenever necessary. Statisti- cal researchers or practitioners analyzing functional data may find this book useful. Chapter 1 provides a brief overview of the book, and in particular, it presents real functional data examples from various research areas that have motivated various models and methodologies for statistical inferences about functional data. Chapters 2 and 3 review four popular nonparametric smooth- ing techniques for reconstructing or interpolating a single curve or a group of curves in a functional data set. Chapters 4–8 present the core contents of this xxi
- 27. xxii PREFACE book. Chapters 9 and 10 are devoted to heteroscedastic ANOVA and tests of equality of covariance functions. Most of the contents of this book should be comprehensible to readers with some basic statistical training. Advanced mathematics and technical skills are not necessary for understanding the key ideas of the methodologies and for applying them to real data analysis. The materials in Chapters 1–8 may be used in a lower- or medium-level graduate course in statistics or biostatistics. Chapters 9 and 10 may be used in a higher-level graduate course or as reference materials for those who intend to do research in this field. I have tried my best to acknowledge the work of the many investigators who have contributed to the development of the models and methodologies for hypothesis testing in the context of analysis of variance for functional data. However, it is beyond the scope of this book to prepare an exhaustive review of the vast literature in this active research field, and I regret any oversight or omissions of particular authors or publications. I am grateful to Rob Calver, Rachel Holt, Jennifer Ahringer, Karen Simon, and Mimi Williams at Chapman and Hall who have made great efforts in co- ordinating the editing, review, and finally the publishing of this book. I would like to thank my colleagues, collaborators, and friends, Mingyen Cheng, Jeng- min Chiou, Jianqing Fan, James S. Marron, Heng Peng, Naisyin Wang, and Chongqi Zhang, for their help and encouragement. Special thanks go to Pro- fessor Wenyang Zhang, Department of Mathematics, the University of York, United Kingdom and Professor Jeff Goldsmith, Department of Biostatistics, Columbia University, New York and an anonymous reviewer who went through the first draft of the book and made some insightful comments and suggestions which helped improve the book substantially. Thanks also go to some of my PhD students, including Xuehua Liang, Xuefeng Liu, and Shengning Xiao, for their reading some chapters of the book. I thank my family who provided strong support and encouragement during the writing process of this book. The book was partially supported by the National University of Singapore Academic Research grants R-155-000-108-112 and R-155-000-128-112. Most of the chapters were written when I visited the Department of Operational Research and Financial Engineering, Princeton University. I thank Professor Jianqing Fan for his hospitality and partial financial support. Some chapters of the book were written when I visited Professor Jengmin Chiou at the Institute of Statistical Science, Academia Sinica, Taipei, Taiwan; and Professor Ming- Yen Cheng at the Department of Mathematics, National Taiwan University, Taipei, Taiwan. Jin-Ting Zhang Department of Statistics Applied Probability National University of Singapore Singapore
- 28. PREFACE xxiii MATLABR is a registered trademark of The MathWorks, Inc. For product information, please contact: The MathWorks, Inc. 3 Apple Hill Drive Natick, MA 01760-2098 USA Tel: 508 647 7000 Fax: 508-647-7001 E-mail: info @ mathworks.com Web: www.mathworks.com
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- 30. Chapter 1 Introduction This chapter aims to give an introduction to the book. We first introduce the concept of functional data in Section 1.1. We then present several motivat- ing real functional data sets in Section 1.2. The difference between classical multivariate data analysis and functional data analysis is briefly discussed in Section 1.3. Section 1.4 gives an overview of the book. The implementation of the methodologies, the options for reading this book, and some bibliographical notes are given in Sections 1.5, 1.6, and 1.7, respectively. 1.1 Functional Data Functional data are a natural generalization of multivariate data from finite dimensional to infinite dimensional. To get some feeling about functional data, Figure 1.1(a) displays a functional data set, consisting of twenty curves, and Figure 1.1(b) displays one of the curves. These curves were simulated without adding measurement errors and they look smooth and continuous. In practice, functional data are obtained by observing a number of subjects over time, space, or other continua. The resulting functional data can be curves, surfaces, or other complex objects; see next section for several real curve data sets and see Zhang (1999) for some surface data sets. One may note that real functional data can be accurately observed so that the measurement errors can be ignored, or can be observed subject to substantial measurement errors. For example, measurements of the heights of children over a wide range of ages in the Berkeley growth data presented in Figure 1.4 in Section 1.2.2 have an error level so small as to be ignorable. However, the progesterone data presented in Figures 1.2 and 1.3 in Section 1.2.1 and the daily records of the Canadian temperature data at a weather station presented in Figures 1.6 and 1.7 in Section 1.2.4 are very noisy. 1.2 Motivating Functional Data Functional data arise naturally in many research areas. Classical examples are provided by growth curves in longitudinal studies. Other examples can be found in Rice and Silverman (1991), Kneip (1994), Faraway (1997), Zhang (1999), Shen and Faraway (2004), Ramsay and Silverman (2002, 2005), Fer- 1
- 31. 2 INTRODUCTION 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −15 −10 −5 0 5 10 (a) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −20 −15 −10 −5 0 5 10 (b) Figure 1.1 A simulated functional data set with (a) twenty simulated curves and (b) one of the simulated curves. raty and Vieu (2006), and references therein. In this section, we present a number of curve data examples. For surface data examples, see Zhang (1999). These curve data sets are from various research areas, including audiology, biology, children’s growth studies, ergonomics, environmentology, meteorology, and women’s health studies, among others. They will be used throughout this book for motivating and illustrating the functional hypoth- esis testing techniques described in this book. Although these functional data examples are from only a few research areas, the proposed method- ologies in this book are also applicable to functional data collected from other scientific fields. All these curve data sets and the computer codes for the corresponding analysis in this book are accessible at the website: http://www.stat.nus.edu.sg/∼zhangjt/books/Chapman/FANOVA.htm. 1.2.1 Progesterone Data The progesterone data were collected in a study of early pregnancy loss con- ducted by the Institute for Toxicology and Environmental Health at the Re- productive Epidemiology Section of the California Department of Health Ser- vices, Berkeley. Each observation shows the levels of urinary metabolite pro- gesterone over the course of the women’s menstrual cycles (in days). The ob- servations came from patients with healthy reproductive function enrolled in an artificial insemination clinic where insemination attempts were well-timed for each menstrual cycle. The data had been aligned by the day of ovulation (Day 0), determined by serum luteinizing hormone, and truncated at each end
- 32. MOTIVATING FUNCTIONAL DATA 3 to present curves of equal length. Measurements were recorded once per day per cycle from eight days before the day of ovulation and until fifteen days after the ovulation. A woman may have one or several cycles. The length of the observation period was twenty-four days. Some measurements from some subjects are missing due to various reasons. For more details about the pro- gesterone data, see Munro et al. (1991), Yen and Jaffe (1991), Brumback and Rice (1998), and Fan and Zhang (2000), among others. The data set consisted of two groups: nonconceptive progesterone curves (sixty-nine menstrual cycles) and conceptive progesterone curves (twenty-two menstrual cycles), as displayed in Figures 1.2 (a) and (b), respectively. The nonconceptive and conceptive progesterone curves came from those women without or with viable zygotes formed. From Figures 1.2 (a) and (b), it is seen that the raw progesterone data are rather noisy, showing that some non- parametric smoothing techniques may be needed to remove some amount of measurement errors and to reconstruct the progesterone curves. Figure 1.3 displays four individual nonconceptive progesterone curves. It is seen that some outlying measurements appear in the progesterone curves 10 and 20. To reduce the effects of these outliers, some robust smoothing techniques may be desirable. To this end, four major nonparametric smoothing techniques for a single curve and for all the curves in a functional data set will be briefly reviewed in Chapters 2 and 3, respectively. From Figure 1.2 (a), it is also seen that the nonconceptive progesterone curves are quite flat before the ovulation day. A question arises naturally. Is the mean nonconceptive curve a constant before the ovulation day? This is a one-sample problem for functional data. It will be handled in Chapter 4. The statement “the mean conceptive curve is a constant before the ovulation day” can be tested similarly. Other questions also arise. For example, is there a conception effect? That is, is there a significant difference between the nonconceptive and conceptive mean progesterone curves? Over what period? These are two-sample problems for functional data and will be handled in Chapter 5. The progesterone data have been used for illustrations of nonparametric regression methods by several authors. Brumback and Rice (1998) used them to illustrate a smoothing spline modeling technique for estimating both mean and individual functions; Fan and Zhang (2000) used them to illustrate their two-step method for estimating the underlying mean function for functional data; and Wu and Zhang (2002) used them to illustrate a local polynomial mixed-effects modeling approach. 1.2.2 Berkeley Growth Curve Data The Berkeley growth curve data were collected in the Berkeley growth study (Tuddenham and Snyder 1954). The heights of thirty-nine boys and fifty-four girls were recorded at thirty-one not equally spaced ages from Year 1 to Year 18. The growth curves of girls and boys are displayed in Figures 1.4 (a) and
- 33. 4 INTRODUCTION −6 −4 −2 0 2 4 6 8 10 12 14 −4 −3 −2 −1 0 1 2 3 (a) Day Log(Progesterone) −6 −4 −2 0 2 4 6 8 10 12 14 −4 −2 0 2 4 (b) Day Log(Progesterone) Figure 1.2 The raw progesterone data with (a) nonconceptive progesterone curves and (b) conceptive progesterone curves. The progesterone curves are the levels of urinary metabolite progesterone over the course of the women’s menstrual cycles (in days). They are rather noisy so that some smoothing may be needed. −10 −5 0 5 10 15 −1 0 1 2 3 4 Noncon. Prog. Curve 1 −10 −5 0 5 10 15 −1 −0.5 0 0.5 1 1.5 2 2.5 Noncon. Prog. Curve 5 −10 −5 0 5 10 15 −2 −1 0 1 2 3 Noncon. Prog. Curve 10 −10 −5 0 5 10 15 −4 −3 −2 −1 0 1 Noncon. Prog. Curve 20 Day Log(Progesterone) Figure 1.3 Four individual nonconceptive progesterone curves. Outlying measure- ments appear in the progesterone curves 10 and 20.
- 34. MOTIVATING FUNCTIONAL DATA 5 2 4 6 8 10 12 14 16 18 60 80 100 120 140 160 180 200 (a) Age Height 2 4 6 8 10 12 14 16 18 60 80 100 120 140 160 180 200 (b) Age Height Figure 1.4 The Berkeley growth curve data with (a) growth curves for fifty-four girls and (b) growth curves for thirty-nine boys. The growth curves are the heights of the children recorded at thirty-one not equally spaced ages from Year 1 to Year 18. The measurement errors of the heights of the children are rather small and may be ignored for many purposes. (b), respectively. It is seen that these growth curves are rather smooth, indi- cating that the measurement errors of the heights are rather small and may be ignored for many purposes. However, to evaluate the individual growth curves at a desired resolution, these growth curves may be interpolated or smoothed using some nonparametric smoothing techniques reviewed in Chapter 2. It is of interest to check if gender difference has some impact on the growth process of a child. That is, one may want to test if there is any significant difference between the mean growth curves of girls and boys. This is a two- sample problem for functional data. It can be handled using the methods developed in Chapter 5. Another question also arises naturally. Is there any significant difference between the covariance functions of girls and boys? This problem will be handled in an exercise problem in Chapter 10. Then comes a related question. How to test if there is any significant difference between the mean growth curves of girls and boys when there is no knowledge about if the covariance functions of girls and boys are equal. This latter problem may be referred to as a two-sample Behrens-Fisher problem that can be handled using the methodologies developed in Chapter 9. The Berkeley growth curve data have been used by several authors. Ram- say and Li (1998) and Ramsay and Silverman (2005) used it to illustrate their curve registration techniques. Chiou and Li (2007) used it to illustrate their k-centers functional clustering approach. Zhang, Liang, and Xiao (2010)
- 35. 6 INTRODUCTION used it to motivate and illustrate their testing approaches for a two-sample Behrens-Fisher problem for functional data. 1.2.3 Nitrogen Oxide Emission Level Data Nitrogen oxides (NOx) are known to be among the most important pollutants, precursors of ozone formation, and contributors to global warming (Febrero, Galeano, and Gonzalez-Manteiga 2008). NOx is primarily caused by combus- tion processes in sources that burn fuels such as motor vehicles, electric utili- ties, and industries among others. Figures 1.5 (a) and (b) show NOx emission levels for seventy-six working days and thirty-nine non-working days, respec- tively, measured by an environmental control station close to an industrial area in Poblenou, Barcelona, Spain. The control station measured NOx emis- sion levels in μg/m3 every hour per day from February 23 to June 26 in 2005. The hourly measurements in a day (twenty-four hours) form a natural NOx emission level curve of the day. It is seen that within a day, the NOx levels increase in morning, attain their extreme values around 8 a.m., then decrease until 2 p.m. and increase again in evening. The influence of traffic on the NOx emission levels is not ignorable as the control station is located at the city cen- ter. It is not difficult to notice that the NOx emission levels of working days are generally higher than those of non-working days. This is why these NOx emission level curves were divided into two groups as pointed out by Febrero, Galeano, and Gonzalez-Manteiga (2008). In both the upper and lower panels, we highlighted one NOx emission level curve, respectively, to emphasize that they have very large NOx emission levels compared with the remaining curves in each group. It is important to identify these and other curves whose NOx emission levels are significantly large and try to figure out the sources that produced these abnormally large NOx emissions. This task will be handled in Chapter 8. The NOx emission level data were kindly provided by the authors of Febrero, Galeano, and Gonzalez-Manteiga (2008) who used these data to il- lustrate their functional outlier detection methods. 1.2.4 Canadian Temperature Data The Canadian temperature data (Canadian Climate Program 1982) were downloaded from “ftp://ego.psych.mcgill.ca/pub/ramsay/FDAfuns/Matlab/” at the book website of Ramsay and Silverman (2002, 2005). The data are the daily temperature records of thirty-five Canadian weather stations over a year (365 days), among which fifteen are in Eastern, another fifteen in Western, and the remaining five in Northern Canada. Panels (a), (b), and (c) of Figure 1.6 present the raw Canadian temperature curves for these thirty-five weather stations. From these figures, we see that at least at the middle of the year, the tem- peratures at the Eastern weather stations are comparable with those at the
- 36. MOTIVATING FUNCTIONAL DATA 7 2 4 6 8 10 12 14 16 18 20 22 24 0 100 200 300 400 Hours NOx level (a) 2 4 6 8 10 12 14 16 18 20 22 24 0 100 200 300 400 Hours NOx level (b) 18/03/05 Other days 30/04/05 Other days Figure 1.5 The NOx emission level data for (a) seventy-six working days and (b) thirty-nine non-working days, measured by an environmental control station close to an industrial area in Poblenou, Barcelona, Spain. Each of the NOx emission level curve consists of the hourly measurements in a day (24 hours). In each panel, an unusual NOx emission level curve is highlighted. Western weather stations, but they are generally higher than those temper- atures at the Northern weather stations. This observation seems reasonable as the Eastern and Western weather stations are located at about the same latitudes while the Northern weather stations are located at higher latitudes. Statistically, we can ask the following question. Is there a location effect among the mean temperature curves of the Eastern, Western, and Northern weather stations? This is a three-sample or one-way ANOVA problem for functional data. This problem will be treated in Chapters 5 and 9, respectively, when the three samples of the temperature curves are assumed to have the same and different covariance functions. The figures in Figure 1.6 may indicate that the covariance functions of the temperature curves of the Eastern, Western, and Northern weather stations are different. In fact, we can see that at the beginning or at the end of the year, the temperatures at the Eastern weather stations are less variable than those at the Western weather stations, but they are generally more variable than those temperatures at the Northern weather stations. To statistically verify if the covariance functions of the temperature curves of the Eastern, Western, and Northern weather stations are the same, we will develop some tests about the equality of the covariance functions in Chapter 10. Four individual temperature curves are presented in Figure 1.7. It is seen that these temperature curves are quite similar in shape and the temperature
- 37. 8 INTRODUCTION 0 50 100 150 200 250 300 350 −20 0 20 (a) 0 50 100 150 200 250 300 350 −20 0 20 (b) 0 50 100 150 200 250 300 350 −20 0 20 (c) Day Temperature Figure 1.6 The Canadian temperature data for (a) fifteen Eastern weather stations, (b) fifteen Western weather stations, and (c) five Northern weather stations. It is seen that at least at the middle of the year, the temperatures recorded at the Eastern weather stations are comparable with those recorded at the Western weather sta- tions, but they are generally higher than those temperatures recorded at the Northern weather stations. 0 100 200 300 −30 −20 −10 0 10 20 Weather station 1 0 100 200 300 −30 −20 −10 0 10 20 Weather station 5 0 100 200 300 −30 −20 −10 0 10 20 Weather station 7 0 100 200 300 −30 −20 −10 0 10 20 Weather station 13 Day Temperature Figure 1.7 Four individual Canadian temperature curves. They are similar in shape. It seems that the temperature records have large measurement errors.
- 38. MOTIVATING FUNCTIONAL DATA 9 records are with large measurement errors. To remove some of these measure- ment errors, some smoothing techniques may be needed. As mentioned earlier, we will introduce four major smoothing techniques for a single curve and for all the curves in a functional data set in Chapters 2 and 3, respectively. The Canadian temperature data have been analyzed in a number of papers and books in the literature, including Ramsay and Silverman (2002, 2005), Zhang and Chen (2007), and Zhang and Liang (2013) among others. 1.2.5 Audible Noise Data The audible noise data were collected in a study to reduce audible noise levels of alternators, as reported in Nair et al. (2002). When an alternator rotates, it generates some amount of audible noise. With advances in technology, the engine noise can be reduced greatly so that the alternator noise levels be- come more remarkable, leading to an increasing quality concern. A robust design study conducted by an engineering team aiming to investigate the ef- fects of seven process assembly factors and their potential for reducing noise levels. The seven factors under consideration are “Through Bolt Torque,” “Ro- tor Balance,” “Stator Varnish,” “Air Gap Variation,” “Stator Orientation,” “Housing Stator Slip Fit,” and “Shaft Radial Alignment.” For simplicity, these seven factors are represented by A, B, C, D, E, F, and G respectively. In these seven factors, D is the noise factor while the others are control factors. Each factor has only two levels: low and high, denoted as “−1” and “+1,” respec- tively. The study adopted a 27−2 IV design, supplemented by four additional replica- tions at the high levels of all factors, resulting in an experiment with thirty-six runs. Nair et al. (2002) pointed out that the fractional factorial design was a combined array, allowing estimation of the main effects of the noise and con- trol factors and all two-factor control-by-noise interactions by assuming that all three factors and higher-order interactions are negligible. For each run, au- dible noise levels were measured over a range of rotating speeds. The audible sound was recorded by the microphones located at several positions near the alternator. The response was a transformed pressure measurement, known as sound pressure level. For each response curve, forty-three measurements of sound pressure levels (in decibels) were recorded with rotating speeds ranging from 1, 000 to 2, 500 revolutions per minute. Figure 1.8 displays the thirty-six response curves of the audible noise data. It appears that the sound pressure levels were accurately measured with the measurement errors ignorable as shown by the four individual curves of the audible noise data presented in Figure 1.9. Nevertheless, Figure 1.8 indicates that these response curves are rather noisy although the measurement errors are rather small. It is of interest to test if the main effects and the control-by-noise inter- action effects of the seven factors on the response curves are significant. This problem will be handled in Chapter 7.
- 39. 10 INTRODUCTION 1000 1500 2000 2500 40 45 50 55 60 65 70 75 80 85 Speed Sound pressure level Figure 1.8 The audible noise data with thirty-six sound pressure level curves. Each curve has forty-three measurements of sound pressure levels (in decibels), recorded with rotating speeds ranging from 1, 000 to 2, 500 revolutions per minute. These sound pressure level curves are rather noisy although the measurement errors of each curve are rather small. 1000 1500 2000 2500 45 50 55 60 65 70 75 80 Run 1 1000 1500 2000 2500 50 55 60 65 70 75 80 Run 5 1000 1500 2000 2500 45 50 55 60 65 70 Run 10 1000 1500 2000 2500 50 55 60 65 70 75 80 Run 30 Speed Sound pressure level Figure 1.9 Four individual curves of the audible noise data. It appears that the sound pressure levels were accurately measured with the measurement errors ignorable.
- 40. MOTIVATING FUNCTIONAL DATA 11 The audible noise data have been analyzed by Nair et al. (2002) who computed the pointwise estimators of the main effects and control-by-noise interaction-effects, and constructed the pointwise confident intervals with standard errors estimated by the four additional replications at the high lev- els of all factors. They were also analyzed by Shen and Xu (2007) using their diagnostic methodologies. 1.2.6 Left-Cingulum Data The cingulum is a collection of white-matter fibers projecting from the cingu- late gyrus to the entorhinal cortex in the brain, allowing for communication between components of the limbic system. It is a prominent white-matter fiber track in the brain that is involved in emotion, attention, and memory, among many other functions. To study if the Radial Diffusibility (RD) in the left- cingulumn is affected by age and family of a child, the left-cingulum data were collected for thirty-nine children from 9 to 19 years old over arc length from −60 to 60. The response variable is “RD” while the covariates include “GHR” and “AGE” where GHR stands for Genetic High Risk and AGE is the age of a child in the study. The GHR variable is a categorical variable, taking two values. When GHR = 1, it means that the child is from a family with at least 1 direct relative with schizophrenia disease and when GHR = 0, the child is from a normal family. The AGE variable is a continuous variable. Figure 1.10 displays the thirty-nine left-cingulum curves over arc length. For each curve, there are 119 measurements. It is seen that the left-cingulum curves are very noisy and an outlying left-cingulum curve is also seen. Figure 1.11 displays two individual left-cingulum curves, showing that the left-cingulum curve was measured with very small measurement errors. The main variations of the left-cingulum curves come from the between-curve variations. In Chapter 5, the left-cingulum data will be used to motivate and illus- trate unbalanced two-way ANOVA models for functional data by transform- ing the AGE variable into a categorical variable with two levels: children with AGE 15 and children with AGE ≥ 15. The left-cingulum data were kindly provided by Dr. Hongbin Gu, De- partment of Psychiatry, School of Medicine, University of North Carolina at Chapel Hill during email discussions. 1.2.7 Orthosis Data An orthotic is an orthopedic device applied externally to limb or body to pro- vide support, stability, prevention of deformity from getting worse, or replace- ment of lost function. Depending on the diagnosis and physical needs of the individual, a large variety of orthosis is available. According to Abramovich, Antoniadis, Sapatinas, and Vidakovic (2004), the orthosis data were acquired and computed in an experiment by Dr. Amarantini David and Dr. Martin Luc (Laboratoire Sport et Performance Motrice, EA 597, UFRAPS, Grenoble
- 41. 12 INTRODUCTION −60 −40 −20 0 20 40 60 2 4 6 8 10 12 14 Arc Length Radial Diffusibility Figure 1.10 The left-cingulumn data for thirty-nine children from 9 to 19 years old. Each curve has 119 measurements of the Radial Diffusibility (RD) in the left- cingulum of a child, recorded over arc length from −60 to 60. −60 −40 −20 0 20 40 60 4.5 5 5.5 6 6.5 7 7.5 8 Curve 1 −60 −40 −20 0 20 40 60 4 6 8 10 12 14 Curve 10 Arc Length Radial Diffusibility Figure 1.11 Two left-cingulum curves. It appears that the measurements of the left- cingulum curve were measured with very small measurement errors.
- 42. MOTIVATING FUNCTIONAL DATA 13 University, France). The aim of the experiment was to analyze how muscle copes with an external perturbation. The experiment recruited seven young male volunteers. They wore a spring-loaded orthosis of adjustable stiffness under four experimental con- ditions: a control condition (without orthosis), an orthosis condition (with the orthosis only), and two spring conditions (spring 1, spring 2) in which stepping-in-place was perturbed by fitting a spring-loaded orthosis onto the right knee joint. All the seven subjects tried all the four conditions ten times for twenty seconds each while only the central ten seconds were used in the study in order to avoid possible perturbations in the initial and final parts of the experiment. The resultant moment of force at the knee is derived by means of body segment kinematics recorded with a sampling frequency of 200 Hz. For each stepping-in-place replication, the resultant moment was computed at 256 time points equally spaced and scaled so that a time interval corresponds to an individual gait cycle. A typical observation is a curve over time t ∈ [0, 1]. Therefore, the orthosis data set consists of 7 (Subjects) × 4 (Treatments) × 10 (Replications) = 280 (Curves). Figure 1.12 presents the 280 raw orthosis curves with each panel displaying ten of them. The orthosis data have been previously studied by a number of authors, including Abramovich et al. (2004), Abramovich and Angelini (2006), Anto- niadis and Sapatinas (2007), and Cuesta-Albertos and Febrero-Bande (2010), among others. In this book, this data set will be used to motivate and illustrate a heteroscedastic two-way ANOVA model for functional data in Chapter 9. It will also be used to motivate and illustrate multi-sample equal-covariance function testing problems in Chapter 10. The orthosis data were kindly pro- vided by Dr. Brani Vidakovic by email communication. 1.2.8 Ergonomics Data To study the motion of drivers of automobiles, the researchers at the Center for Ergonomics at the University of Michigan collected data on the motion of an automobile driver to twenty locations within a test car. Among other measures, the researchers measured three times the angle formed at the right elbow between the upper and lower arms of the driver. The data recorded for each motion were observed on an equally spaced grid of points over a period of time, which were rescaled to [0, 1] for convenience, but the number of such time points varies from observation to observation. See Faraway (1997) and Shen and Faraway (2004) for detailed descriptions of the data. Figure 1.13 displays the right elbow angle curves of the ergonomics data downloaded from the website of the second author of Shen and Faraway (2004). These right elbow angle curves have been smoothed so that no further smoothing is needed but some interpolation or smoothing may be needed if one wants to evaluate these curves at a desired level of resolution.
- 43. 14 INTRODUCTION 0 0.5 1 −50 0 50 100 0 0.5 1 −50 0 50 100 0 0.5 1 −50 0 50 100 0 0.5 1 −50 0 50 100 0 0.5 1 −50 0 50 100 0 0.5 1 −50 0 50 100 0 0.5 1 −50 0 50 100 0 0.5 1 −50 0 50 100 0 0.5 1 −50 0 50 100 0 0.5 1 −50 0 50 100 0 0.5 1 −50 0 50 100 0 0.5 1 −50 0 50 100 0 0.5 1 −50 0 50 100 0 0.5 1 −50 0 50 100 0 0.5 1 −50 0 50 100 0 0.5 1 −50 0 50 100 0 0.5 1 −50 0 50 100 0 0.5 1 −50 0 50 100 0 0.5 1 −50 0 50 100 0 0.5 1 −50 0 50 100 0 0.5 1 −50 0 50 100 0 0.5 1 −50 0 50 100 0 0.5 1 −50 0 50 100 0 0.5 1 −50 0 50 100 0 0.5 1 −50 0 50 100 0 0.5 1 −50 0 50 100 0 0.5 1 −50 0 50 100 0 0.5 1 −50 0 50 100 Figure 1.12 The orthosis data. The row panels are associated with the seven subjects and the column panels are associated with the four treatment conditions. There are ten orthosis curves displayed in each panel. An orthosis curve has 256 measurements of the resultant moment of force at the knee, recorded over a period of time (ten seconds), scaled to [0, 1].
- 44. MOTIVATING FUNCTIONAL DATA 15 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 110 120 130 140 150 160 Time Angle Figure 1.13 The ergonomics data collected on the motion of an automobile driver to twenty locations within a test car, measured three times. The measurements of a curve are the angles formed at the right elbow between the upper and lower arms of the driver. The measurements for each motion were recorded on an equally spaced grid of points over a period of time, rescaled to [0, 1] for convenience. There are totally sixty right elbow angle curves. They can be classified into a number of groups according to “location” and “area,” where “location” stores the locations of the targets, taking twenty values while “area” stores the areas of the targets, taking four values. The categorical variables “location” and “area” are highly correlated as they are different only in the ways for grouping the locations of the targets. One can also find a functional regression model to predict the right elbow angle curve y(t), t ∈ [0, 1] using the coordinates (a, b, c) of a target, where a represents the coordinate in the left-to-right direction, b represents the coordi- nate in the close-to-far direction, and c represents the coordinate in the down- to-up direction. Shen and Faraway (2004) found that the following quadratic model is adequate to fit the data: yi(t) = β0(t) + aiβ1(t) + biβ2(t) + ciβ3(t) + a2 i β4(t) +b2 i β5(t) + c2 i β6(t) + aibiβ7(t) + aiciβ8(t) +biciβ9(t) + vi(t), i = 1, · · · , 60, (1.1) where yi(t) and vi(t) denote the ith response and location-effect curves over time, respectively, (ai, bi, ci) denotes the coordinates of the target associated with the ith angle curve, and βr(t), r = 0, 1, · · · , 9 are unknown coefficient functions. Some questions then arise naturally. Is each of the coefficient functions
- 45. 16 INTRODUCTION significant? Can the quadratic model (1.1) be further reduced? These two questions and other similar questions can be answered easily after we study Chapter 6 where this ergonomics data set will be used to motivate and illus- trate linear models for functional data with functional responses. 1.3 Why Is Functional Data Analysis Needed? In many situations curve or surface data can be treated as multivariate data so that classical multivariate data analysis (MDA) tools (for example, Anderson 2003) can be applied directly. This treatment, however, ignores the fact that the underlying object of the measurements of a subject is a curve or a surface or any continuum as indicated by the real functional data sets presented in Section 1.2. In addition, direct MDA treatment may encounter difficulties in many other situations, including • The sampling time points of the observed functional data are not the same across various subjects. For example, the nonconceptive and con- ceptive progesterone curves have different numbers of measurements for different curves. • The sampling time points are not equally spaced. For example, the sampling time points of the Berkeley growth curve data are not equally spaced. • The number of sampling time points is larger than the number of subjects in a sample of functional data. For example, the number of sampling time points for a Canadian temperature curve is 365 while the total number of the Canadian temperature curves is only fifteen. In the first situation, direct MDA treatment may not be possible or rea- sonable; in the second situation, MDA inferences may be applied directly to the data but wether the observed data really represent the underlying curves or surfaces may be questionable; and in the third situation, standard MDA treatment fails as the associated sample covariance matrix is degenerated so that most of the inference tools in MDA, for example, the Hotelling T2 -test or the Lawley-Hotelling trace test (Anderson 2003), will not be well-defined. A remedy is to reduce the dimension of the functional data using some dimen- sion reduction techniques. These dimension reduction techniques often work well. However, in many situations, dimension reduction techniques may also fail to reduce the dimension of the data sufficiently without loss of too much information. Therefore, in these situations, functional data analysis (FDA) is more nat- ural. In fact, Ramsay and Silverman (2002, 2005) provide many nice FDA tools to solve the aforementioned problems. In Chapters 2 and 3 of this book, we also provide some tools to overcome difficulties encountered in the first two situations while other chapters of the book provide methodologies to overcome difficulties encountered in the third situation.
- 46. OVERVIEW OF THE BOOK 17 1.4 Overview of the Book In this book, we aim to conduct a thorough survey on the topics of hypothesis testing in the context of analysis of variance for functional data and give a systematic treatment of the methodologies. For this purpose, the remaining chapters are arranged as follows. With science and technology development, functional data can be observed densely. However, they are still discrete observations. Fortunately, continuous versions of functional data can be reconstructed from discrete functional data by some smoothing techniques. For this purpose, we review four major non- parametric smoothing techniques for a single curve in Chapter 2 and present the methods for reconstructing a whole functional data set in Chapter 3. In classical MDA, inferences are often conducted based on normal distri- bution and Mahalanobis distance. In FDA, the Mahalanobis distance often cannot be well-defined as the sample covariance matrices of discretized func- tional data are often degenerated. Instead we have to use the L2 -norm distance (and its modifications). In fact, the Gaussian process and the L2 -norm dis- tance in FDA play the roles of normal distribution and Mahalanobis distance in classical MDA. Thus, we shall discuss the properties of the L2 -norm of a Gaussian process in Chapter 4. We also discuss the properties of Wishart pro- cesses (a natural extension of Wishart matrices), chi-squared mixtures, and F-type mixtures there. These properties are very important for this book and will be used in successive chapters. ANOVA models for functional data are handled in Chapter 5, including two-sample problems, one-way ANOVA, and two-way ANOVA for functional data. Chapters 6 and 7 study functional linear models with functional re- sponses when the design matrices are full rank and ill-conditioned, respec- tively. Chapter 8 studies how to detect unusual functions. In all these chap- ters, different groups are assumed to have the same covariance function. When this assumption is not satisfied, we deal with the heteroscedastic ANOVA for functional data in Chapter 9. The last chapter (Chapter 10) is devoted to test- ing the equality of the covariance functions. Examples are provided in each chapter. Technical proofs and exercises are given in almost every chapter. 1.5 Implementation of Methodologies Most methodologies introduced in this book can be implemented using existing software such as S-PLUS, SAS, and MATLABR , among oth- ers. We shall publish our MATLAB codes for most of the methodologies proposed in this book and the data analysis examples on our website: http://www.stat.nus.edu.sg/∼zhangjt/books/Chapman/FANOVA.htm. We will keep updating the codes when necessary. We shall also make the data sets used in this book available through our website. The reader may try to apply the methodologies to the related data sets. We thank all the people who kindly provided us with these functional data sets.
- 47. 18 INTRODUCTION 1.6 Options for Reading This Book Readers who are particularly interested in one or two of nonparametric smoothing techniques for functional data analysis may read Chapters 2 and 3. For a lower-level graduate course, Chapters 4–8 are recommended. If students already have some background in nonparametric smoothing techniques, Chap- ters 2 and 3 may be briefly reviewed or even skipped. Chapters 9 and 10 may be included in a higher-level graduate course or can be used as individual research materials for those who want to do research in a related field. 1.7 Bibliographical Notes There are two major monographs about FDA methodologies currently avail- able. One is by Ramsay and Silverman (2005). This book mainly focuses on description statistics in FDA, with topics such as curve registration, princi- pal components analysis, canonical correlation analysis, pointwise t-test and F-test, fitting functional linear models, among others. Another book is by Ferraty and Vieu (2006). This book mainly focuses on nonparametric kernel estimation, functional prediction, and classification of functional observations. There are another three books, accompanying the above two books, by Ram- say and Silverman (2002), Ramsay et al. (2009), and Ferraty and Romain (2011), respectively. In recent years, there has been a prosperous period in the development of functional hypothesis testing procedures. In terms of test statistics con- structed, these testing procedures include the L2 -norm-based tests (Faraway 1997, Zhang and Chen 2007, Zhang, Peng, and Zhang 2010, Zhang, Liang, and Xiao 2010, etc.), F-type tests (Shen and Faraway 2004, Zhang 2011a, Zhang and Liang 2013, etc.), basis-based tests (Zhang 1999), thresholding tests (Fan and Lin 1998, Yang and Nie 2008), bootstrap tests (Faraway 1997, Cuevas, Febrero, and Fraiman 2004, Zhang, Peng, and Zhang 2010, Zhang and Sun 2010, etc.), penalized tests (Mas 2007), among others. In terms of the regression models considered, these testing procedures study the mod- els including one-sample problems (Mas 2007), two-sample problems (Zhang, Peng, and Zhang 2010), one-way ANOVA (Cuevas, Febrero, and Fraiman 2004), functional linear models with functional responses (Faraway 1997, Shen and Faraway 2004, Zhang and Chen 2007, Zhang 2011a), two-sample Behrens- Fisher problems for functional data (Zhang, Peng, and Zhang 2010, Zhang, Liang, and Xiao 2010), k-sample Behrens-Fisher problems (Cuevas, Febrero, and Fraiman 2004), analysis of surface data (Zhang 1999), among others. However, all these important results and recent developments in this area are spread out in various statistical journals. This book aims to conduct a sys- tematic survey of these results, to develop a common framework for the newly developed models and methods, and to provide a comprehensive guideline for applied statisticians to use the methods. However, the book is not intended to and is actually impossible to cover all the statistical inference methodologies available in the literature.
- 48. Chapter 2 Nonparametric Smoothers for a Single Curve 2.1 Introduction In the main body of this book, we aim to survey various hypothesis test- ing methodologies for functional data analysis. In the development of these methodologies, we essentially assume that continuous functional data are available or can be evaluated at any desired resolution. In practice, however, the observed functional data are discrete, probably with large measurement errors as indicated by the curve data sets presented in the previous chapter. To overcome this difficulty, in this chapter, we briefly review four well-known nonparametric smoothing techniques for a single curve. These smoothing tech- niques may allow us to achieve the following goals: • To reconstruct the individual functions in a real functional data set so that any reconstructed individual function can be evaluated at any desired resolution. • To remove measurement errors as much as possible so that the vari- ations of the reconstructed functional data mainly come from the between-subject variations. • To improve the power of a proposed hypothesis testing procedure by the removal of measurement errors. In Chapter 3, we focus on how to apply these smoothing techniques to reconstruct the functions in a functional data set so that the reconstructed functional data can be used directly by the methodologies surveyed in this book and study the effects of the substitution of the underlying individual functions with the reconstructed individual functions in a functional data set. In this chapter, we focus on smoothing an individual function in a func- tional data set. For convenience, we generally denote the observed data of an individual function in a functional data set as (ti, yi), i = 1, 2, · · · , n, (2.1) where ti, i = 1, 2, · · · , n denote the design time points, and yi, i = 1, 2, · · · , n are the responses at the design time points. The design time points may be 19
- 49. 20 NONPARAMETRIC SMOOTHERS FOR A SINGLE CURVE equally spaced in an interval of interest, or be regarded as a random sample from a continuous design density, namely, π(t). For simplicity, let us denote the interval of interest or the support of π(t) as T , which can be a finite interval [a, b] or the whole real line (−∞, ∞). The responses yi, i = 1, 2, · · · , n are often observed with measurement errors. For the observed data (2.1), a standard nonparametric regression model is usually written as yi = f(ti) + i, i = 1, · · · , n, (2.2) where f(t) models the underlying regression function and i, i = 1, 2, · · · , n denote the measurement errors that cannot be explained by the regression function f(t). Mathematically, f(t) is the conditional expectation of yi, given ti = t. That is, f(t) = E(yi|ti = t), i = 1, 2, · · · , n. For an individual function in a functional data set, the regression function f(t) is the associated underlying individual function. There are a number of existing smoothing techniques that can be used to smooth the regression function f(t) in (2.2). Different smoothing tech- niques have different strengths in one aspect or another. For example, smooth- ing splines may be good for handling sparse data, while local polynomial smoothers may be computationally advantageous for handling dense designs. In this chapter, as mentioned previously, we review four well-known smooth- ing techniques, including local polynomial kernel smoothing (Wand and Jones 1995, Fan and Gijbels 1996), regression splines (Eubank 1999), smoothing splines (Wahba 1990, Green and Silverman 1994), and P-splines (Ruppert, Wand and Carroll 2003), respectively, in Sections 2.2, 2.3, 2.4, and 2.5. We conclude this chapter with some bibliographical notes in Section 2.6. 2.2 Local Polynomial Kernel Smoothing 2.2.1 Construction of an LPK Smoother The main idea of local polynomial kernel (LPK) smoothing is to locally ap- proximate the underlying function f(t) in (2.2) by a polynomial of some de- gree. Its foundation is Taylor expansion, which states that any smooth function can be locally approximated by a polynomial of some degree. Specifically, let t0 be an arbitrary fixed time point where the function f(t) in (2.2) will be estimated. Assume f(t) has a (p + 1)st continuous deriva- tive for some integer p ≥ 0 at t0. By Taylor expansion, f(t) can be locally approximated by a polynomial of degree p. That is, f(t) ≈ f(t0)+(t−t0)f(1) (t0)/1!+(t−t0)2 f(2) (t0)/2!+· · ·+(t−t0)p f(p) (t0)/p!, in a neighborhood of t0 that allows the above expansion where f(r) (t0) denotes the rth derivative of f(t) at t0.
- 50. LOCAL POLYNOMIAL KERNEL SMOOTHING 21 Set βr = f(r) (t0)/r!, r = 0, · · · , p. Let β̂r, r = 0, 1, 2, · · · , p be the minimiz- ers of the following weighted least squares (WLS) criterion: n i=1 {yi − [β0 + (ti − t0)β1 + · · · + (ti − t0)p βp]} 2 Kh(ti − t0), (2.3) where Kh(·) = K(·/h)/h, obtained by rescaling a kernel function K(·) with a constant h 0, called the bandwidth or smoothing parameter. The bandwidth h is mainly used to specify the size of the local neighborhood, namely, Ih(t0) = [t0 − h, t0 + h], (2.4) where the local fit is conducted. The kernel function, K(·), determines how observations within [t0 − h, t0 + h] contribute to the fit at t0. Remark 2.1 The kernel function K(·) is usually a probability density func- tion. For example, the uniform density K(t) = 1/2, t ∈ [−1, 1] and the stan- dard normal density K(t) = exp(−t2 /2)/ √ 2π, t ∈ (−∞, ∞) are two well- known kernels, namely, the uniform kernel and the Gaussian kernel. Other useful kernels can be found in Gasser, Müller, and Mammitzsch (1985), Mar- ron and Nolan (1988), and Zhang and Fan (2000), among others. Denote the estimate of the rth derivative f(r) (t0) as ˆ f (r) h (t0). Then ˆ f (r) h (t0) = r!β̂r, r = 0, 1, · · · , p. In particular, the resulting pth degree LPK estimator of f(t0) is ˆ fh(t0) = β̂0. An explicit expression for ˆ f (r) h (t0) is useful and can be made by matrix notation. Let X = ⎡ ⎢ ⎣ 1 (t1 − t0) · · · (t1 − t0)p . . . . . . ... . . . 1 (tn − t0) · · · (tn − t0)p ⎤ ⎥ ⎦ , and W = diag(Kh(t1 − t0), · · · , Kh(tn − t0)), be the design matrix and the weight matrix for the LPK fit around t0. Then the WLS criterion (2.3) can be re-expressed as (y − Xβ)T W(y − Xβ), (2.5) where y = (y1, · · · , yn)T and β = (β0, β1, · · · , βp)T . It follows that ˆ f (r) h (t0) = r!eT r+1,p+1S−1 n Tny, where er+1,p+1 denotes a (p+1)-dimensional unit vector whose (r+1)st entry is 1 and the other entries are 0, and Sn = XT WX, Tn = XT W.
- 51. 22 NONPARAMETRIC SMOOTHERS FOR A SINGLE CURVE When t0 runs over the whole support T of the design time points, a whole range estimation of f(r) (t) is obtained. The derivative estimator ˆ f (r) h (t), t ∈ T is usually called the LPK smoother of the underlying derivative function f(r) (t). The derivative smoother ˆ f (r) h (t0) is usually calculated on a grid of t’s in T . In this section, we only focus on the LPK curve smoother ˆ fh(t0) = eT 1,p+1S−1 n Tny = n i=1 Kn h (ti − t0)yi, (2.6) where Kn h (ti − t0) = eT 1,p+1S−1 n Tnei,n with Kn (t) known as the empirical equivalent kernel for the p-order LPK; see Fan and Gijbels (1996). From (2.6), it is seen that for any t ∈ T , ˆ fh(t) = n i=1 Kn h (ti − t)yi = a(t)T y, (2.7) where a(t) = [Kn h (t1 − t), Kn h (t2 − t), · · · , Kn h (tn − t)]T , which can be obtained from TT n S−1 n e1,p+1 after replacing t0 with t. This general formula allows us to evaluate ˆ fh(t) at any resolution. Set ŷi = ˆ fh(ti) as the fitted value of f(ti). Let ŷh = [ŷ1, · · · , ŷn]T denote the fitted values at all the design time points. Then ŷh can be expressed as ŷh = Ahy, (2.8) where Ah = (a(t1), · · · , a(tn))T (2.9) is known as the smoother matrix of the LPK smoother. Remark 2.2 As Ah does not depend on the response vector y, the LPK smoother ˆ fh(t) is known as a linear smoother, which is a linear combina- tion of the responses. The regression spline, smoothing spline, and P-spline smoothers introduced in the next three sections are also linear smoothers. 2.2.2 Two Special LPK Smoothers Local constant and linear smoothers are the two simplest and most useful LPK smoothers. The local constant smoother is known as the Nadaraya-Watson estimator (Nadaraya 1964, Watson 1964). This smoother results from the LPK smoother ˆ fh(t0) (2.6) by simply taking p = 0: ˆ fh(t0) = n i=1 Kh(ti − t0)yi n i=1 Kh(ti − t0) . (2.10)
- 52. LOCAL POLYNOMIAL KERNEL SMOOTHING 23 Within a local neighborhood Ih(t0), it fits the data with a constant. That is, it is the minimizer β̂0 of the following WLS criterion: n i=1 (yi − β0)2 Kh(ti − t0). The Nadaraya-Watson estimator is simple to understand and easy to compute. Let IA(t) denote the indicator function of some set A. Remark 2.3 When the kernel function K is the uniform kernel K(t) = 1/2, t ∈ [−1, 1], the Nadaraya-Watson estimator (2.10) is exactly the local average of yi’s that are within the local neighborhood Ih(t0) (2.4): ˆ fh(t0) = n i=1 IIh(t0)(ti)yi n i=1 IIh(t0)(ti) = ⎧ ⎨ ⎩ ti∈Ih(t0) yi ⎫ ⎬ ⎭ /mh(t0), where mh(t0) denotes the number of the observations falling into the local neighborhood Ih(t0). Remark 2.4 When t0 is a boundary point of T , fewer design points are within the neighborhood Ih(t0) so that ˆ fh(t0) has a slower convergence rate than the case when t0 is an interior point of T . For a detailed explanation of this boundary effect, the reader is referred to Müller (1991, 1993), Fan and Gijbels (1996), Cheng, Fan, and Marron (1997), and Müller and Stadtmuller (1999), among others. The local linear smoother (Stone 1984, Fan 1992, 1993) is obtained by fitting a data set locally with a linear function. Let (β̂0, β̂1) minimize the following WLS criterion: n i=1 [yi − β0 − (ti − t0)β1]2 Kh(ti − t0). Then the local linear smoother is ˆ fh(t0) = β̂0. It can be easily obtained from the LPK smoother ˆ fh(t0) (2.6) by simply taking p = 1. Remark 2.5 The local linear smoother is known as a smoother with a free boundary effect (Cheng, Fan, and Marron 1997). That is, it has the same convergence rate at any point in T . It also exhibits many good properties that the other linear smoothers may lack. Good discussions on these properties can be found in Fan (1992, 1993), Hastie and Loader (1993), and Fan and Gijbels (1996, Chapter 2), among others. A local linear smoother can be simply expressed as ˆ fh(t0) = n i=1 [s2(t0) − s1(t0)(ti − t0)] Kh(ti − t0)yi s2(t0)s0(t0) − s2 1(t0) , (2.11)
- 53. 24 NONPARAMETRIC SMOOTHERS FOR A SINGLE CURVE where sr(t0) = n i=1 Kh(ti − t0)(ti − t0)r , r = 0, 1, 2. Remark 2.6 The choice of the degree of the local polynomial used in LPK smoothing, p, is usually not as important as the choice of the bandwidth, h. A local constant (p = 0) or a local linear (p = 1) smoother is often good enough for most applications if the kernel function K and the bandwidth h are properly determined. Remark 2.7 Fan and Gijbels (1996, Chapter 3) pointed out that for curve estimation (not valid for derivative estimation), an odd p is preferable. This is true as an LPK fit with p = 2q + 1 introduces an extra parameter compared to an LPK fit with p = 2q, but does not increase the variance of the associ- ated LPK estimator. However, the associated bias may be significantly reduced especially in the boundary regions (Fan 1992, 1993; Hastie and Loader 1993; Fan and Gijbels 1996; Cheng, Fan, and Marron 1997). Thus, the local linear smoother is strongly recommended for most non- parametric smoothing problems in practice. For fast computation of the local linear smoother, the reader is referred to Fan and Marron (1994). 2.2.3 Selecting a Good Bandwidth In most applications, the kernel function K(·) can be simply chosen as the uni- form kernel or the Gaussian kernel. It is more crucial to choose the bandwidth h (Fan and Gijbels 1996). Example 2.1 We now employ the LPK smoother (2.6) to smooth the first nonconceptive progesterone curve by the LPK method with p = 2 and K(t) = exp(−t2 /2)/ √ 2π. That is, we employ the local quadratic kernel smoother with the Gaussian kernel. The data are shown as circles in Figure 2.1, which are the same as those circles presented in the left upper panel of Figure 1.3. Three dif- ferent bandwidths h∗ /4, h∗ , 4h∗ are used to show the effect of different choices of bandwidth, where h∗ = 1.3 is the optimal bandwidth selected by the GCV rule defined below in (2.12). It is seen that the three different local quadratic kernel fits are very different from each other, as shown in Figure 2.1. The dot- dashed curve is associated with h = h∗ /4 = 0.32 and it is rather rough and almost interpolates the data. Therefore, when the bandwidth h is too small, the fitting bias is small; but as only a few data points are involved in the local fit, the associated fitting variance is large. The dashed curve is associated with h = 4h∗ = 5.22 and is rather smooth but it does not fit the data adequately. Therefore, when the bandwidth h is too large, the fitting variance is small; but as the local neighborhood is large, the associated fitting bias is also large. A good fit will be obtained with a bandwidth that is not too small or too large.
- 54. LOCAL POLYNOMIAL KERNEL SMOOTHING 25 −6 −4 −2 0 2 4 6 8 10 12 14 0 0.5 1 1.5 2 2.5 3 Day Log(Progesterone) raw data h=0.32 h=1.3 h=5.22 Figure 2.1 Fitting the first nonconceptive progesterone curve by the LPK method with p = 2. The circles represent the data. The dot-dashed curve, obtained using a small bandwidth h = 0.32, nearly interpolates the data while the dashed curve, obtained using a large bandwidth h = 5.22, does not provide an adequate fit to the data. The solid curve, obtained using a bandwidth h = 1.3 selected by the GCV rule (2.12), provides a nice fit to the data. The solid curve is associated with h = h∗ = 1.3 [log10(1.3) = 0.1139], which is not too small or too large according to the GCV rule described below and the solid curve is indeed a nice fit to the data. Thus, for a given p and a fixed kernel function K(·), a good bandwidth h should be chosen to trade off between the goodness of fit and the model complexity of the LPK fit (2.8). Here, better goodness of fit often means less bias while smaller model complexity usually means less rough or smaller variance. To better choose the bandwidth h in LPK smoothing, we can use the following generalized cross-validation (GCV) rule (Wu and Zhang 2006, Chapter 3): GCV(h) = y − ŷh2 (1 − tr(Ah)/n)2 , (2.12) where the fitted response vector ŷh and the LPK smoother matrix Ah are de- fined in (2.8) and (2.9), respectively. In the literature, many other bandwidth selectors are available. Famous bandwidth selectors can be found in Fan and Gijbels (1992), Herrmann, Gasser, and Kneip (1992), and Ruppert, Sheather, and Wand (1995), among others. Remark 2.8 With h increasing, the goodness of fit of the LPK fit measured
- 55. Another Random Document on Scribd Without Any Related Topics
- 56. A LITTLE CHANGE. Hang it all! They have blocked the street and are laying it with asphalte; just in May, as usual. From early morning the quiet of my rooms is disturbed by the noise of the work, when I go out I scramble over heaps of rubbish, past smoking cauldrons of pitch, and when I come home at night my cab drops me nearly a quarter of a mile away. Moreover, one neighbouring house is being painted, and the other is being rebuilt. I fly from falling dust and brickbats, only to run against ladders and paint-pots. It is awful. And now my Aunt Jane is coming up from Bath, and has invited herself to tea at my chambers. Her rheumatism prevents her from walking more than a yard or two, she cannot bear any noise, and the smell of paint makes her ill. She is very rich, and could leave all she has to the poor. Accurately speaking, that class includes me, but in my aunt's opinion it does not. She is very suspicious, and, if I made excuses and invited her to tea anywhere else, she would feel convinced that I was hiding some guilty secret in my dull, quiet, respectable rooms. She is very prim, and the mere suggestion of such a thing would alienate her from me for ever. Why on earth can't she stop in Bath? And I shall have to go with her to May meetings! It is impossible; I must fly. But where? She has a horror and suspicion of all foreign nations, except perhaps the steady, industrious Swiss. Good idea—Switzerland. But what reason can I give for rushing off just now? Someone must send me. I have it. She knows I try to write a little, so I will say my editor requires me to go at once to Geneva to write a series of articles in the Jardin Alpin d'Acclimatation on Alpine botany. Botany, how respectable! Geneva, how sedate! Makes one think at once of Calvin and Geneva bands. These sound rather frivolous, something like German bands, but they are not really so, only, I believe, a sort of clerical cravat. Then I will start off to Paris, the direct way to Geneva. Perhaps I shall never reach Geneva. Paris will do well enough. No streets there taken up in the Spring. No painting on the clean stone houses. No rebuilding on the Boulevards. No aunt of mine anywhere near. I shall escape all my troubles. I shall be able to smoke my cigarette lazily in the pleasant courtyard of the Grand Hôtel, and try to imagine that I see some of the people in Trilby—Little Billee, or Taffy, or the Laird—amongst the animated, cosmopolitan crowd. And the stately giant in the gilt chain will solemnly arrange the newspapers in all languages, and will supply me with note-paper.
- 57. I must be careful not to write to my aunt a long description of the Jardin Alpin d'Acclimatation de Geneve on paper stamped Grand Hôtel, Paris. And the attentive Joseph, with those long grey whiskers, sacred to the elderly French waiter and the elderly French lawyer, will exclaim, V'là, M'sieu! in all those varied tones which make the two syllables mean Yessir! Coming, Sir! Here is your coffee, Sir! In a minute, Sir! and so many things besides. And I shall be able to watch, assembled from all parts of the world, some younger and prettier faces than my Aunt Jane's. That settles it. A regretful letter to my aunt. And to-morrow en route! Change of Spelling?—Our dramatic friend known to the public through Mr. Punch as Enry Hauthor Jones appears to have recently altered the spelling of his name. He has left the Jones and the Henry alone, but in the Times of Friday he appears as Henry Arther Jones, U out of it; and what was E doing there? Presentation to the Rev. Guinness Rogers.—Last week this worthy minister was presented by his Congregationalists with an address and a cheque for a thousand guineas, Mr. Gladstone, ex- minister, being among the subscribers. In future the bénéficiaire will be remembered as the Reverend Thousand Guinness Rogers. Music Note (after hearing Mr. J. M. Coward's performance on the Orchestral Harmonium).—It would be high praise to say of any organist that he attacks his instrument in a Cowardly manner. Very Appropriate.—Last Wednesday the Right Hon. A. W. Peel became a Skinner. A COMING CHARGE. (Prematurely Communicated by our Prophetic Reporter.)
- 58. Gentlemen of the Jury, for the last couple of years or so you have no doubt read any number of denunciations of the conduct of the man whose actions you are now about to investigate. You have heard him abused right and left. You have seen pictures of him, in which he has been held up to scorn and public ridicule. You have heard it announced in all quarters that he is a scoundrel and a thief. And as this has been the case, Gentlemen of the Jury, it is my duty to tell you that you must put aside the recollection of these attacks. You must treat the prisoner before you as if he were immaculate. In fact you must lay aside all prejudice, and give the man a fair trial; and, Gentlemen, it is my duty (sanctioned by precedent) to have the pleasure of informing you that I am sure you will! Yes, Gentlemen of the Jury, having regard to all the circumstances of the case, I repeat, I am sure you will! At the National Liberal Club, on Wednesday, Lord Rosebery told the company they were not dancing on a volcano. That may be true, but it is equally true that the Government, in proposing to remit the sixpenny duty on whisky, are riding for a fall in (or, shall we say, a drop of) the crater.
- 59. A WELLINGTON (STREET) MEMORIAL. General Opinion (Mr. Punch) presents the Medal of the Highest Order of Histrionic Merit to Henry Irving in recognition of distinguished service as Corporal Gregory Brewster in the action of Conan Doyle's Story of Waterloo. ON THE NEW STATUE. [Her Majesty's Government are about to entrust to one of our first sculptors a great historical statue, which has too long been wanting to the series of those who have governed England.—Lord Rosebery at the Royal Academy Banquet.] Our Uncrowned King at last to stand 'Midst the legitimate Lord's Anointed? How will they shrink, that sacred band,
- 60. Dismayed, disgusted, disappointed! The parvenu Protector thrust Amidst the true Porphyrogeniti? How will it stir right royal dust! The mutton-eating king's amenity Were hardly proof against this slur. William the thief, Rufus the bully, The traitor John, and James the cur,— Their royal purple how 'twill sully To rub against the brewer's buff! Harry, old Mother Church's glory Meet this Conventicler?—Enough! The Butcher dimmed not England's story But rather brightened her renown. In camp and court it must be said, And if he did not win a crown, At least he never lost his head! Among Mr. Le Gallienne's new poems there is one entitled Tree Worship. It is not dedicated to the lessee of the Haymarket Theatre by an Admirer. A MAY MEETING. They met in a cake-shop hard by the Strand, He in black broadcloth, and she in silk. She had a glass of fizz in her hand, He had a bun and a cup of milk. She had a sunshade of burnished crimson, He had a brolly imperfectly furled, And a pair of pince-nez with tortoiseshell rims on. He looked the Church, and she seemed the World. They sat on each side of a marble table, His legs were curled round the legs of his chair. Around them babbled a miniature Babel; The sunlight gleamed on her coppery hair. She held a crumpled Academy Guide, Scored with crosses in bold blacklead; A pile of leaflets lay at his side,
- 61. And he grasped a Report, which he gravely read. His shaven lip was pendulous, long, Her mouth was a cherry-hued moue mutine, His complacent, uncomely, strong, Hers soft appetence sharpened with spleen. Her eyes scale-glitter, his oyster-dim, His huge mouth hardened, her small lips curled As he gazed at her and she glanced at him; He looked the Church, and she seemed the World. A holy spouter from Exeter Hall! (So she mused as she sipped her wine.) A butterfly in the Belial thrall Of Vanity Fair, all tinkle and shine! So thought he as he crumbled his bun With clumsy fingers in loose black cloth; And the impish spirit of genial fun Hovered about them and mocked them both. Mutual ignorance, mutual scorn, Revealed in glances aflame though fleeting; Such, in the glow of this glad May morn, The inhuman spirit of mortal meeting. The worm must disparage the butterfly, The butterfly must despise the worm; And Scorn, the purblind, will ne'er descry A common bond, or a middle term. Modish folly, factitious Art? True, grave homilist, sadly true! But Boanerges truculent, tart, What of the part that is played by you? You denouncing the Snare of Beauty, She affecting to feel its spell,— Which falls shortest of human duty? Shallow censor, can you quite tell? Meanwhile the lilac is blithely budding, And sweetly breatheth the nutty May, The golden sunshine the earth is flooding,
- 62. And you—you echo the old, old bray Of Boanerges. A broader greeting Of brotherhood full, warm hearts, wide eyes Might lend a meaning to your May Meeting To gladden the gentle and win the wise. What's in a Name? A Rossa, c.—Before being ejected from the House of Commons on Wednesday last, O'Donovan Rossa shouted out that A stain had been put upon his name. Where is the ingenious craftsman who did it? He might try his hand next time at gilding refined gold. Query.—Can a champagne wine from the vintage of Ay be invariably and fairly described as Ay 1? MODES AND METALS. [Neckties made of aluminium have just been invented in Germany.—Evening Paper.] Visited my tailor's puddling works to-day. He has some really neat new pig-iron fabrics for the season. I am thinking of trying his Bessemer steel indestructible evening-dress suits. Really this new plan of mineral clothing comes in very usefully when one is attacked by roughs on a dark night. Floored an assailant most satisfactorily with a touch of my lead handkerchief. The only objection I can find to my aluminium summer suiting is its tendency to get red hot if I stand in the sun for five minutes. I think I can now safely defy my laundress to injure my patent safety ironclad steel shirts. I find, however, that there is no need of a laundress at all. When one's linen is soiled, sand-paper and a mop will clean it in no time.
- 63. My frock-coat has got a nasty kink in it; must send it to be repaired at the smelting furnace. Once Cut don't Come Again!—It was said by The Figaro last week that Japan would demand an extra payment of one hundred millions of taels by China. But surely a hundred million Chinamen would evince a pig-headed obstinacy in parting with, or being parted from, their tails on any consideration. A Lightship Sunk.—Impossible! couldn't have been a lightship, it must have been a very heavy ship. Daughter (enthusiastically). Oh, Mamma! I must Learn Bicycling! So delightful to go at such a pace! Mamma (severely). No thank you, my dear; you are quite 'fast' enough already!
- 64. The Joys of Office. Speaker! Hats off, Strangers! The Cares of Office. Mr. Cawmel-Bannerman crosses the Lobby. ESSENCE OF PARLIAMENT. EXTRACTED FROM THE DIARY OF TOBY, M.P. House of Commons, Monday, May 6.—Welsh Disestablishment Bill on. So is The Man from Shropshire. Stanley Leighton, as George Trevelyan pointed out long ago, is irresistibly like the ruined Chancery Suitor of Bleak House. Always dashing into debate as The Man from Shropshire broke in on the business of the Court of Chancery. Mr. Chairman! he shouts, and waves his arms, as The Man from Shropshire cried aloud, My lord! My lord! and tried to seize the Lord Chancellor by wig or neck. After first ebullition, our Man from Shropshire quietens down. Argues with gravity of tone and manner that seem to imply he has something to say. Turns out he hasn't; but, on the Welsh Disestablishment Bill, that no matter. Curious how this Church Bill brings to the front men who, if heard at all, certainly do not speak in chorus on any other question. After The Man from Shropshire comes Tomlinson, who, early in proceedings, displays irresistible tendency to discuss points of order with Speaker. New Speaker has, however, already got hand in, and, before Tomlinson, who remembers being on his feet addressing Chair, quite knows where he is, he finds himself sitting down again, Cranborne also on warpath, his very hair bristling with indignation at this fresh attack on the Church. Glib Griffith-Boscawen has a field-night; makes
- 65. long speech on moving Instruction standing in his own name. His obvious, unaffected enjoyment of his own oratory should be infectious; but isn't. Colonel Lockwood, that pillar of the Church, was the first called on in Committee to move amendment. Colonel not in his place. Report has it the devout man is in library reading Thomas À Kempis, or Drelincourt on Death. Here is opportunity for Glib-Griffith to make another speech. Dashes in; starting off with promise of good half-hour; desire for Lockwood's appearance irresistible. As Addison says, with hereditary disposition to drop into poetry, and the belief that he is quoting Tennyson, Better fifty words from Lockwood Than a thousand from Boscawen. Scouts sent out in all directions. The Colonel discovered in sort of oratory he has contrived in far recess of library. Brought back to House; found Boscawen bowling along. This is my show, said the Colonel as he passed Boscawen on his way to his seat. More fierceness in his eye than befit the man or the occasion. Boscawen stared over his head, and went on with his speech. Opportunity too precious to be lost. If Lockwood meant to move his amendment he should have been there when called upon. He wasn't: Boscawen found it, so to speak, by roadside. Now it was his; would make the most of it; pegged along whilst the Colonel muttered remarks as he glared upon him. Some who sat by said it was a prayer. Others, catching a word here and there, said it was a quotation from Thomas À Kempis. Whatever it might have been, Colonel seemed much moved. Hardly pacified when, at end of twenty minutes, Glib-Griffith sat down, and Lockwood, finding himself in peculiar position of seconding his own amendment, delivered the speech he had prepared for moving it. Business done.—Got into Committee on Welsh Disestablishment Bill. Tuesday.—Pretty to see Prince Arthur drop down on George Russell just now for speaking disrespectfully of Silomio. That eminent patriot, having in his newly-assumed character of Patron Saint of Japan, cross-examined Edward Grey upon latest Treaty negotiations, accused Asquith of nothing less than stealing a county. Filching was precise word, which has its equivalent in Slang Dictionary in sneaking. Idea of Home Secretary hovering over the Marches in dead of night, and, when he thought no one was looking, picking up Monmouthshire, and putting it in his coat-tail pocket, amused scanty audience. But Silomio really wrath. Always Anti-English this Government, he
- 66. exclaimed, with scornful sweep of red right hand along line of smiling faces on Treasury Bench. A stirring burst of British patriotism, George Russell characterised it. John Bull in excelsis. The more notable since, on reference to official record, he found the Knight from Sheffield was born in the United States, and descended from the Pilgrim Fathers. Which one? inquired voice from back bench, an inquiry very properly disregarded. (A new phrase this, Sark notes, for use by retired tradesmen, setting up to spend rest of useful lives in retirement at Clapham or Camberwell. To trace their family tree back to transplantation at period of Conquest, played out. Instead of Came over with the Conqueror, newer, more picturesque, equally historical to say, Came over with the Pilgrim Fathers.) Prince Arthur not in mood for speculation of this kind. Cut to the heart by remarks he suspected of slighting intent towards his friend and colleague. In Silomio Prince Arthur has long learned to recognise all the graces and all the talents. Apart from personal consideration, he feels how much the Party owe to him for having raised within its ranks the standard of culture and conduct. To have him attacked, even in fun, by an Under Secretary, was more than he could stand. So, in gravest tone, with no flicker of a smile on his expressive countenance, he declared that a more unfortunate speech he had never heard. If the hon. gentleman intends, he added, to take a considerable part in debate, I would earnestly recommend him either to change the character of his humour, or entirely to repress the exhibition. Beautiful! In its way, all things considered, best thing Prince Arthur has done this Session. House grinned; but two big hot tears coursed down cheek of Silomio, making deep furrows in the war paint. That's tit for tat with Georgie Russell, said Herbert Gardner to Solicitor- General, with vague recollection of a historic phrase. Quite perfect, said Lockwood. But what a loss the stage has sustained by Prince Arthur taking to politics? Tried both myself and know something about it. Business done.—An eight hours day with Welsh Disestablishment Bill.
- 67. Piling Peeler upon Rossa! Thursday.—Tanner's curiosity inconveniently uncontrollable. At end of sitting given up to Scotland no one thinking about Commander-in-Chief or Tanner either. Successive divisions had carried sitting far beyond midnight, that blessed hour at which, in ordinary circumstances, debate stands adjourned. Quarter of an hour occupied in dividing on question whether they should divide on amendment. Proposal affirmed; another quarter of an hour spent in fresh division. Nothing possible further to be done, Members streamed forth, scrambling for cabs in Palace Yard. Conybeare in charge of a Bill dealing with false alarms of fire, managed to get it through Committee unopposed. Members little recked how near they were to real alarm of worse than fire. Twenty minutes earlier, when last division taken, over 330 Members filled House. Now the tide ebbed; only the thirty odd Members in their places jealously watching Speaker running through Orders of the Day. Tanner bobbing up and down on bench like parched pea. Heard it somewhere whispered that Duke of Cambridge, worn out with long campaign, about to unhelm, unbuckle his sword, hang up his dinted armour. Tanner feels he can't go to bed leaving
- 68. unsettled the problem of truth or phantasy. Not a moment to be lost. Speaker risen to put question That this House do now adjourn. Then Tanner blurts out the inquiry, Is it true? Order! order! says the Speaker. Well, if they didn't like the question in the form he had first put it, he would try again. I would ask, he said, adopting conditional mood as least likely to hurt anyone's feelings, whether a member of the Royal Family who has really (most desirous of not putting it too strongly, but really you know) been drawing public money too long is going to retire? Order! order! roared the few Members present. I would ask that question, repeated Tanner, still in the conditional mood, but nodding confidentially all round. The Blameless Bartley happily at post of duty. Broke in with protest. Speaker ruled question out of order. But the good Tanner came back like a bad sixpence. Is his Royal Highness going to retire? he insisted, getting redder than ever in the face. Order! order! shouted Members in chorus. Thus encouraged, Tanner sang out the solo again, Is his Royal Highness going to retire? That was his question. The Speaker, distinctly differing, affirmed The question is that the House do now adjourn; which it did straightway, leaving Dr. Tanner to go to a sleepless bed haunted by an unanswered question. What I should like, said Lieut.-General Sir Frederick Wellington Fitz Wygram, who served in the Crimea with H.R.H., has been in command of the Cavalry Brigade at Aldershot, and in other positions come in personal contact with the Commander-in-Chief, What I should like, he repeated reflectively, stroking his chin, would be the opportunity, enjoyed from a safe distance, of hearing the Dook personally reply to Tanner's interrogation. Business done.—Wrangle all night round Scotch Committee. Friday.—Squire sat through dull morning sitting listening with air of pathetic resignation to Members talking round Budget. Quilter led off with prodigiously long paper on the Art of Brewing Beer. Seems they fill up the cup with all kinds of mysterious ingredients. Brookfield, looking round and observing both Joseph and Jesse absent, whispered in ear of sympathetic Chairman that Birmingham has reputation in the Trade of making and drinking beer
- 69. containing minimum of malt, maximum of sugar, and warranted to do the greatest damage to the system. Squire, momentarily waking up from mournful mood, observed that Birmingham is also headquarters of Liberal Unionism. Might be nothing in coincidence, but there it was. Rasch posed as the distressed agriculturist. Jokim tried to walk on both sides of road at same time, and Government got majority of 24. Business done.—Budget Resolutions agreed to. TO YVETTE GUILBERT AT THE EMPIRE. Yvette! your praise resounds on every hand. And those laugh loudest who least understand. Transcriber's Note Page 229: 'visistor' corrected to 'visitor'. (Knock.) Ah, here comes my visitor. (Enter stranger.) The illustration for 'The Old Crusaders' originally covered 2 pages, pp. 234 and 235 (centrefold/centerfold), with a blank page on either side.
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