MAXIMUM CORRENTROPY BASED DICTIONARY LEARNING FOR PHYSICAL ACTIVITY RECOGNITION USING WEARABLE SENSORS

Maximum Correntropy Based Dictionary
Learning Framework for Physical Activity
Recognition Using Wearable Sensors
Sherin M. Mathews1(B)
, Chandra Kambhamettu2
, and Kenneth E. Barner1
1
Department of Electrical and Computer Engineering,
University of Delaware, Newark, DE 19716, USA
{sherinm,barner}@udel.edu
2
Department of Computer and Information Science,
University of Delaware, Newark, DE 19716, USA
chandrak@udel.edu
Abstract. Due to its symbolic role in ubiquitous health monitoring,
physical activity recognition with wearable body sensors has been in the
limelight in both research and industrial communities. Physical activity
recognition is difficult due to the inherent complexity involved with dif-
ferent walking styles and human body movements. Thus we present a
correntropy induced dictionary pair learning framework to achieve this
recognition. Our algorithm for this framework jointly learns a synthesis
dictionary and an analysis dictionary in order to simultaneously perform
signal representation and classification once the time-domain features
have been extracted. In particular, the dictionary pair learning algo-
rithm is developed based on the maximum correntropy criterion, which
is much more insensitive to outliers. In order to develop a more tractable
and practical approach, we employ a combination of alternating direction
method of multipliers and an iteratively reweighted method to approxi-
mately minimize the objective function. We validate the effectiveness of
our proposed model by employing it on an activity recognition problem
and an intensity estimation problem, both of which include a large num-
ber of physical activities from the recently released PAMAP2 dataset.
Experimental results indicate that classifiers built using this correntropy
induced dictionary learning based framework achieve high accuracy by
using simple features, and that this approach gives results competitive
with classical systems built upon features with prior knowledge.
1 Introduction and Related Work
Recognition of basic physical activities (such as walk, run, cycle) and basic pos-
tures (lie, sit, stand) is well researched [1–4], and good recognition performance
can be achieved with a single 3D-accelerometer and simple classifiers. Recent
works have focused on estimating the intensity of an activity (i.e., light, mod-
erate or vigorous) (e.g., in [5,6]) by means of metabolic equivalent (MET), a
parameter that refers to the energy expenditure of a physical activity.
c Springer International Publishing AG 2016
G. Bebis et al. (Eds.): ISVC 2016, Part II, LNCS 10073, pp. 123–132, 2016.
DOI: 10.1007/978-3-319-50832-0 13

124 S.M. Mathews et al.
Over the years, many studies have analysed determining which features in
this activity data are the most informative, and how these data can be most
effectively employed to classify the activities [3,4,7,8]. Other research has inves-
tigated which computational model is the most appropriate to represent human
activity data [9–11]. Despite such research efforts, the scalability of handling
large intra-class variations and the robustness of many existing human-activity
recognition techniques to the model parameters remains limited.
To make physical activity monitoring feasible in everyday life scenarios, an
activity-recognition framework must be robust, i.e., it must handle a wide range
of everyday, household, or sport activities and must manage a variety of potential
users. Two challenges facing these frameworks are dataset size and simultane-
ously occurring background activities. First, using smaller activity datasets con-
sisting of merely a few basic recorded activities limits the scope of the framework
since these methods would only apply to specific scenarios. Although current
research has focused on increasing the number of activities that are recognized,
each increase in the number of activities causes the classification performance to
fall off. Secondly, recording and using only a small set of a few activities in basic
activity recognition, without having simultaneous background activities, limits
the applicability of the developed algorithms. Real-time scenarios might include
such activity switching, thus requiring testing on a wider range of activities than
were used for training. Thus, successful activity-recognition requires frameworks
that are sufficiently robust for classification even with limited training data and
also insensitive to outliers thereby reducing misclassifications.
In this work, we propose to develop a dictionary pair learning framework
based on the maximum correntropy criterion [12] to solve the wearable sensor
based classification problem. Correntropy has demonstrated to obtain robust
inferences in information theory learning (ITL) [13] and effectively handle
non-Gaussian noise and large outliers [12]. Inspired by dictionary learning exper-
iments that achieved highly successful recognition rates using a few representa-
tive samples on high dimensional data, we propose a unified dictionary pair
learning-based framework based on maximum correntropy for human physical
activity monitoring and recognition. To optimize the non-convex correntropy
objective function, a new alternate minimization algorithm incorporated with
an iteratively reweighted method is developed to facilitate convergence. To the
best of our knowledge, dictionary learning frameworks, and specifically corren-
tropy based dictionary pair learning frameworks, have not to date been used in
wearable sensor-based applications. Consequently, our novel dictionary learning-
based framework algorithm will engender future research on this method’s poten-
tial applicability for accurate sensor-based data classifications and for other
physiological-signal classifications.
2 Proposed Methodology
The proposed framework consists of two stages: data processing and recognition.
The data processing stage consists of preprocessing, segmentation and feature
extraction. During the preprocessing step, raw sensory data is synchronized,

Maximum Correntropy Based Dictionary Learning Framework 125
timestamped, and labeled; 3D-acceleration and heart rate data are collected.
During the segmentation step, this data is segmented with a sliding window,
using a window size of 512 samples. For the feature extraction step, signal fea-
tures extracted from the segmented 3D-acceleration data are computed for each
axis separately and for the 3 axes together. Including the Heart Rate(HR) moni-
tor with the commonly used inertial sensors proved especially useful for physical
activity intensity estimation. Hence, mean and feature gradient features are cal-
culated on both the raw and normalized heart rate signals from the HR data.
In addition, derived features were extracted as they better distinguish activities
involving both upper and lower body movements, thereby improving recognition
of physical activities. Overall, a total of 137 basic features from each data win-
dow of 512 samples are extracted: 133 features from Inertial Measurement Unit
(IMU) acceleration data and 4 features from HR data.
2.1 Correntropy Based Dictionary Learning Criterion
Maximum Correntropy Criterion (MCC)
Recognition against outliers and noise is critically challenging, mainly due to
the unpredictable nature of the errors (bias) caused by noise and outliers. The
concept of correntropy was proposed in ITL [13] to process non-Gaussian noise.
Correntropy is directly relevant to Renyis quadratic entropy [12] wherin the
Parzen windowing method is employed to estimate the data distribution. Maxi-
mization of correntropy criterion cost function (MCC) is defined by maximizing
V (X, Y ) =
1
N
N
i=1
kσ(xi − yi) =
1
N
N
i=1
kσ(ei) (1)
where X = [x1, x2, ..xN ] is the desired signal, Y = [y1, y2, ..yN ] is the system
output, E = [e1, e2, ..eN ] is the error signal, each of them being N-dimensional
vectors, where N is the training data size and kσ(x) is the Gaussian kernel with
bandwidth σ given by:
kσ(x) =
1
√
2πσ
exp−x2
/2σ2
(2)
In a general framework of M-estimation, MCC can be defined as
ρ(e) = (1 − exp−e2
/2σ2
)/
√
2πσ (3)
MCC cost function has proved to satisfy the properties of non-negativity, trans-
lation invariant, triangle inequality and symmetry, thus is a well-defined metric.
Adoption of MCC to train adaptive systems actually makes the output signal
close to the desired signal. By analyzing the contour maps [12], it can be inferred
that when the error vector is close to zero, it acts like l2 distance; when the error
gets larger, it is equivalent to l1 distance; and for cases when the error is large
the cost metric levels off and is very insensitive to the large-value of error vector,
thereby intuitively explaining the robustness of MCC.

Correntropy Based Dictionary Pair Learning Framework
The dictionary pair learning classification algorithm initially jointly learns a
synthesis dictionary and an analysis dictionary to achieve the goal of signal
representation and discrimination. To define discriminative dictionary learn-
ing, we denote a set of p-dimensional training samples from K classes by
X = [X1, ........, Xk, ...., XK], where Xk ∈ Rp×n
is the training sample set of
class k, and n is the number of samples of each class. Discriminative dictionary
learning (DL) methods aim to learn an effective data representation model from
X for classification tasks by exploiting the class label information of training
data, and can be formulated under the following framework:
min
D,A
X − DA 2
F + λ||A||p + Ψ(D, A, Y) (4)
Here λ 0 is a scalar constant; Y represents the class label matrix of samples
in X; D is the synthesis dictionary to be learned; and A is the coding coefficient
matrix of X over D. In the training model (Eq. 4), the data fidelity term X −
DA 2
F ensures the representation ability of D; ||A||p is the lp-norm regularizer
on A; and Ψ(D, A, Y) stands for some discrimination promotion function that
ensures the discrimination power of D and A [14].
If when using an analysis dictionary denoted by P ∈ RmK×p
, the code A
can be analytically obtained as A = PX, then the representation of X becomes
efficient. Based on this idea, the DPL model learns an analysis dictionary P
together with the synthesis dictionary D, leading to the following DPL model
[14]:
P∗
, D∗
= arg min
P,D
X − DPX 2
F + Ψ(D, P, X, Y) (5)
Here Ψ (D, P, X, Y) is the discrimination function; and D and P form a
dictionary pair where the analysis dictionary P is used to analytically code X,
and the synthesis dictionary D is used to reconstruct X. The learned structured
synthesis dictionary D = [D1, ..Dk, ., DK] and the structured analysis dictionary
P = [P1, ....., Pk......PK] form a sub-dictionary pair corresponding to class k.
Thus using the structured analysis dictionary P, we want the sub-dictionary Pk
to project the samples from class i,i = k to a nearly null space thereby making
the coefficient matrix PX nearly block diagonal. Using variable matrix A to relax
the non-convex problem, we readily have the following DPL model:
P∗
, A∗
, D∗
= arg min
P,A,D
K
k=1
Xk−DkAk
2
F +τ PkXk−Ak
2
F +λ||Pk, Xi||2
F (6)
Here Xk denotes the complementary data matrix of Xk in the whole training set
X, λ > 0 is a scalar constant, and di denotes the ith atom of synthesis dictionary
D. Incorporating correntropy based criteria (Eq. 2), Eq. 6 can be rewritten as:
P∗
, A∗
, D∗
=arg min
P,A,D
K
k=1
1+τ−λ exp Xk−DkAk
2
F /σ2
−τ exp PkXk−Ak
2
F /σ2
−λ exp||Pk,Xi||2
F /σ2
(7)

In order to solve this equation, we employ alternating direction method of
multipliers with iteratively reweighted method to facilitate convergence. A gen-
eral maximization problem solved by an iteratively reweighted method can be
described as follows: Consider a general equivalent maximization problem
max f(x) +
i
hi(gi(x)) (8)
where f(x) and gi(x) are arbitrary functions, x denotes an arbitrary constant
and hi(x) is an arbitrary convex function in domain of gi(x). The details to solve
general maximization problem (Eq. 8) using iteratively reweighted optimization
is described in Algorithm 1 [15], where hi(gi(x)) denotes any supergradient of
the concave function hi at point gi(x).
Problem Transformation: hi(gi(x)) → Tr(DT
i gi(x))
Algorithm 1. Optimization algorithm for a general maximization problem
0: Initialize Di = I
1: Update x by optimal solution to the problem
max f(x) + i Tr(DT
i gi(x))
2: Calculate Di = hi(gi(x)) for each i
3: Iteratively perform 1–2 until convergence
Now consider maximum correntropy criterion problem:
min f(x) +
i
−exp−l2
i (x)/2σ2
(9)
where f(x) and li(x) are arbitrary functions, x denotes an arbitrary constant.
Comparing with equation (8), in hi(gi(x)), let hi(z) = exp−z/2σ2
(z > 0)
and gi(x) = l2
i (x)(z > 0), then hi(gi(x)) = exp−l2
i (x)/2σ2
where hi(z) =
1 − exp−z/2σ2
(z > 0) is concave function. Using iteratively reweighted method
(Algorithm 1)[15], the problem transformation and steps to solve the maximum
correntropy criterion problem can be described as:
Problem Transformation: 1−exp−l2
i (x)/2σ2
→ dil2
i (x)(di = 1
2σ2 exp−l2
i (xt)/2σ2
)
Algorithm 2. Optimization algorithm for maximum correntropy criterion
0: Initialize di = 1
1: Update x by optimal solution to the problem
min f(x) + i dil2
i (x)
2: Calculate di = 1
2σ2 exp−l2
i (xt)/2σ2
for each i
3: Iteratively perform 1–2 until convergence

The original objective function (Eq. 7) can now be easily solved by a com-
bination of ADMM and an iteratively re-weighted algorithm. The alternating
direction method of multipliers (ADMM) allows to solve convex optimization
problems by fixing some variables and solving for the other variable, thereby
breaking the problem into smaller pieces making each of them easier to handle.
The minimization can be alternated between the following two steps:
1: Update A
A∗
= arg min
A
K
k=1
−λ exp Xk−DkAk
2
F /σ2
−τ exp PkXk−Ak
2
F /σ2
(10)
Using the problem transformation defined in Algorithm 2, we have the closed-
form solution:
A∗
= (d1DT
k Dk + d2τI)−1
(d1DT
k Xk + τd2PkXk) (11)
where d1 =
1
2σ2
exp− Xk−DkAk
2
F /2σ2
and d2 =
1
2σ2
exp− PkXk−Ak
2
F /2σ2
(12)
2: For updating P
P∗
= arg min
P
K
k=1
−τ exp PkXk−Ak
2
F /σ2
−λ exp||Pk,Xi||2
F /σ2
(13)
The closed-form solutions of P can be obtained as:
P∗
= d2τAkXT
k (d2τXkXT
k + d3λ ¯Xk
¯XT
k + Y I)−1
(14)
where
d2 =
1
2σ2
exp− PkXk−Ak
2
F /2σ2
and d3 =
1
2σ2
exp− Pk
¯Xk
2
F /2σ2
(15)
Iterate between the steps until it converges.
In the testing phase, the analysis sub-dictionary Pk is trained to produce
small coefficients for samples from classes other than k, and thus can only gen-
erate significant coding coefficients for samples from class k. Meanwhile, the
synthesis sub-dictionary Dk is trained to reconstruct the samples of class k from
their projective coefficients PkXk, i.e. residual will be small. Conversely, since
PkXi will be small and Dk is not trained to reconstruct Xi, the residual ai will
be much larger. Thus, if the query sample y is from class k, its projective coding
vector Pk will more likely be large, while its projective coding vectors Pi will be
small. Therefore, class-specific reconstruction residual are used to identify the
class label of testing sample.
3 Experimental Results and Discussion
For our evaluation experiments, we used the recently released database PAMAP2
Physical Activity Monitoring Data Set available at the UCI machine learn-
ing repository [16]. Briefly, this database not only incorporates basic physical

activities and postures, but also includes a wide range of everyday, household,
and fitness activities. The dataset captures 18 physical activities performed by
9 subjects wearing 3 IMUs (Inertial measurement unit) and a HR (heart rate)
monitor. Each subject followed a predefined data collection protocol of 12 activ-
ities (lie, sit, stand, walk, run, cycle, Nordic walk, iron, vacuum, jump rope,
ascend and descend stairs), and optionally performed 6 other activities (watch
TV, work at a computer, drive a car, fold laundry, clean house, play soccer),
added to enrich the range.
To evaluate the proposed framework, we employed activity recognition and
intensity estimation classification problems defined on the PAMAP2 dataset. In
the PAMAP2 dataset, the activity classification task is extended to 15 activi-
ties including 3 additional activities from the optional activity list (fold laundry,
clean house, play soccer). The complete activity recognition task consisting of
these 15 different activity classes is referred to as the PAMAP2 Activity Recog-
nition (PAMAP2-AR) task. The intensity-estimation classification task aims
to distinguish activities of light, moderate, and vigorous effort, and is referred
to as the PAMAP2 Intensity Estimation (PAMAP2 - IE) task. Differentiating
these levels of effort is based on the MET of the various physical activities, as
provided by [17]. Therefore, intensity classes are defined as activities of light
effort (<3.0 METs) (lie, sit, stand, drive a car, iron, fold laundry, clean house,
watch TV, work at a computer), moderate effort (3.0–6.0 METs) (walk, cycle,
descend stairs, vacuum and Nordic walk), or vigorous effort (>6.0 METs) (run,
ascend stairs, jump rope, play soccer). Thus two classification tasks were defined:
(1) activity recognition and (2) intensity estimation. Using these two defined clas-
sification tasks, we evaluated different boosting methods and compared them to
our proposed correntropy based dictionary pair learning-based approach. For
the evaluation procedure, we randomly selected 75% of the data for training and
25% for testing. The final result is the averaged value over all 10 runs.
Results on Intensity Estimation Task. In [18], base-level classifiers were
used for activity monitoring classification tasks. Within various classification
approaches, the C4.5 decision tree algorithm was tested on the PAMAP2 dataset
and demonstrated an accuracy of 70.07%. Also, the results obtained in [18] indi-
cated that further improvement in classification accuracy is possible since even
the best result using Adaboost classifier was only 73.93% accuracy. The overall
confusion matrix using the proposed framework for the three intensity estima-
tion tasks (i.e., light (Class 1), moderate (Class 2) and vigorous (Class 3) tasks)
is given in Table 1. An independent performance assessment of the proposed
framework results in an accuracy of 87.6% on the AR-IE task, demonstrating
that our framework outperforms the C4.5 decision tree and AdaBoost classifiers
(Table 2).
Results on Activity Recognition Task. To study our framework’s classifica-
tion performance on the PAMAP2−AR task, we determined its performance on
15 classes based activity-recognition tasks on the PAMAP2 dataset. Table 3 enu-
merates the overall confusion matrix for these tasks. We find that our framework
outperforms the C4.5 decision tree with an accuracy of 74.12% versus 71.59%,

Table 1. Overall confusion matrix using dictionary pair learning framework on
PAMAP2-IE dataset. The table shows how different annotated activities are classi-
fied into different classes.
Annotated activity Class 1 Class 2 Class 3
Predicted Class 1 12893 422 9
Table 2. Comparison of proposed approach on PAMAP2-IE dataset to state-of-the-art
methods in terms of accuracy (calculated in %)
Methodology C4.5 [18] Adaboost [18] Proposed approach
Accuracy 70.07% 73.93% 87.59%
and it gives competitive results when compared to an AdaBoost classifier on the
PAMAP2 AR task (Table 4). In addition to competitive accuracy, our framework
provides the additional advantages of lower training and testing times.
Further examination of our results indicate that averaged over 10 test runs,
the confusion matrix of the best-performing classifier on the PAMAP2-AR task
yields an overall accuracy of 74.12%, showing that some activities are recog-
nized with high accuracy, such as lying, walking, or even distinguishing between
ascending and descending. Overall, misclassifications, where activities belong-
ing to one class are mistakenly classified as belonging to its neighboring classes,
are lower. One example of overlapping activity characteristics is the over 5%
confusion between nordic walk (class 7) and cycle (class 6) and ascend (class 9)
Table 3. Confusion matrix using proposed framework on PAMAP2 -AR dataset.
Annotated Class Class 1 Class 2 Class 3 Class 4 Class 5 Class 6 Class 7 Class 8 Class 9 Class 10 Class 11 Class 12 Class 13 Class 14 Class 15
Predicted Class 1 1660 2 2 19 14 0 0 0 1 0 0 0 28 2 0
Predicted Class 2 0 1387 34 126 24 4 0 5 0 2 0 0 68 4 1
Predicted Class 3 0 23 1364 144 102 1 2 1 6 0 0 0 57 1 2
Predicted Class 4 0 10 55 1779 90 0 0 0 0 0 0 0 236 19 0
Predicted Class 5 0 0 13 194 1086 5 0 6 0 2 0 0 243 0 8
Predicted Class 6 0 0 0 0 2 474 13 210 69 2 0 0 9 0 0
Predicted Class 7 0 0 0 0 0 159 295 56 81 1 0 1 38 0 0
Predicted Class 8 0 0 0 0 0 73 16 1824 275 2 0 0 3 0 0
Predicted Class 9 0 0 0 0 0 5 6 98 1600 0 0 0 0 0 0
Predicted Class 10 0 0 0 3 10 8 9 49 90 1271 0 0 27 0 5
Predicted Class 11 0 0 0 0 199 0 0 0 6 0 624 0 0 0 6
Predicted Class 12 0 0 0 0 25 0 49 0 38 0 0 253 3 0 0
Predicted Class 13 0 9 23 320 478 10 6 1 8 11 0 0 860 16 6
Predicted Class 14 0 4 16 439 209 0 0 0 0 0 0 0 145 88 0
Predicted Class 15 0 0 0 0 15 50 1 22 52 2 8 3 55 0 187
Table 4. Comparison of proposed approach on PAMAP2-AR dataset to state-of-the-
art methods in terms of accuracy (calculated in %)
Methodology C4.5 [18] Adaboost [18] Proposed approach
Accuracy 71.59% 71.78% 74.12%

that is function of the positioning of the sensors; thus an IMU on the thigh would
help reliably differentiate these postures.
Another example comes from the playing soccer activity, because playing
soccer (class 14) is a composite activity. Thus it becomes problematic to distin-
guish running with a ball from just running (class 4). Arguably, however, the
main reason for these misclassifications is the diversity inherent in the subject’s
performance of physical activities. Therefore, further increasing the accuracy of
physical activity recognition will require the introduction and investigation of
personalized approaches.
In [18], the evaluation technique was leave-one-activity-out (LOAO) where
an activity monitoring system is used on a previously unknown activity. Our
framework takes in data randomly. Thus evaluation is based on a completely
unknown activity from an unknown user, while training is performed using a dif-
ferent activity with a different user in our random 75–25% validation approach.
These types of subject-independent and activity-independent validation tech-
niques are preferred for physical activity monitoring since they provide results
with more practical meaning. Using our framework, we can not only achieve
good classifier performance but also eliminate the need of pre-training a partic-
ular activity for classification. Thus, our proposed method makes it possible to
design a robust physical activity monitoring system that has the desired gener-
alization characteristics.
4 Conclusion
In this paper, we presented an effective dictionary pair learning-based frame-
work based on the maximum correntropy criterion to evaluate the robustness of
activity recognition and intensity estimation of aerobic activities using data from
wearable sensors. The proposed objective function is robust to outliers and can
be efficiently optimized by combination of iteratively reweighted technique and
alternating direction method of multipliers. Experimental results indicate that
classifiers built in this framework not only provide competitive performance but
also demonstrate subject-independent activity classification using accelerome-
ters. Both of these are important considerations for developing systems that
must be robust, scalable and must perform well in real world settings. Based
on our promising results, we will further extend this work by incorporating tree
structure and smooth constraint to the classification framework.
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