TOWARDS MORE ACCURATE CLUSTERING METHOD BY USING DYNAMIC TIME WARPING

International Journal of Data Mining & Knowledge Management Process (IJDKP) Vol.3, No.2, March 2013

TOWARDS MORE ACCURATE CLUSTERING METHOD
BY USING DYNAMIC TIME WARPING

Khadoudja Ghanem

MISC Laboratory, Constantine 2 University , Nouvelle ville, Constantine, Algeria ;
gkhadoudja@yahoo.fr

ABSTRACT
An intrinsic problem of classifiers based on machine learning (ML) methods is that their learning time
grows as the size and complexity of the training dataset increases. For this reason, it is important to have
efficient computational methods and algorithms that can be applied on large datasets, such that it is still
possible to complete the machine learning tasks in reasonable time. In this context, we present in this paper
a more accurate simple process to speed up ML methods. An unsupervised clustering algorithm is
combined with Expectation, Maximization (EM) algorithm to develop an efficient Hidden Markov Model
(HMM) training. The idea of the proposed process consists of two steps. In the first step, training instances
with similar inputs are clustered and a weight factor which represents the frequency of these instances is
assigned to each representative cluster. Dynamic Time Warping technique is used as a dissimilarity
function to cluster similar examples. In the second step, all formulas in the classical HMM training
algorithm (EM) associated with the number of training instances are modified to include the weight factor
in appropriate terms. This process significantly accelerates HMM training while maintaining the same
initial, transition and emission probabilities matrixes as those obtained with the classical HMM training
algorithm. Accordingly, the classification accuracy is preserved. Depending on the size of the training set,
speedups of up to 2200 times is possible when the size is about 100.000 instances. The proposed approach
is not limited to training HMMs, but it can be employed for a large variety of MLs methods.

KEYWORDS
Dynamic Time Warping, clustering, Hidden Markov Models.

1. INTRODUCTION
The main problem of classifiers based on machine learning (ML) methods is that their learning
time grows as the size and complexity of the training dataset increases because training with only
a few data will lead to an unreliable performance. Indeed, due to advances in technology, the size
and dimensionality of data sets used in machine learning tasks have grown very large and
continue to grow by the day. For this reason, it is important to have efficient computational
methods and algorithms that can be applied on very large data sets, such that it is still possible to
complete the machine learning tasks in reasonable time.

The speeding up of the estimation process of the built model parameters is a common problem of
ML methods, although it raises several challenges. Many methods have been employed to
overcome this problem. These methods can be classified into two categories: the first category
consists of methods which compress data, either the number of instances [2, 3, 4, 5], or the set of
features (attributes) which characterize instances [16,20,21]. The second category consists of
methods which reduce running time in the execution and compilation level [16, 22, 23].
DOI : 10.5121/ijdkp.2013.3207 107


This paper is an extended version of our previous work described in [1]. In that paper a simple
process to speed up ML methods is presented. An unsupervised clustering algorithm is combined
with Expectation, Maximization (EM) algorithm to develop an efficient Hidden Markov Model
(HMM) training. The idea of the proposed process consists of two steps. The first one involves a
preprocessing step to reduce the number of training instances with any information loss. In this
step, training instances with similar inputs are clustered and a weight factor which represents the
frequency of these instances is assigned to each representative cluster (index). The Euclidean
distance is used as a metric to compare two observation sequences. Increasing evidence with the
use of this metric is its poor accuracy for classification and clustering of time dependent
sequences. The Euclidean distance metric is widely known to be very sensitive to distortion in time axis
[6][7]. In spite of that it is used in many fields because of its ease of implementation and its time
and space efficiency.

In this paper, we introduce Dynamic Time Warping (DTW) as a dissimilarity function. The
problem of distortion in the time axis can be addressed by DTW. This method allows non-linear
alignments between two sequences with different length to accommodate those that are similar,
but locally out of phase, as shown in Figure 1.

X

Y

Figure 1, a warping between two temporal signals (Matching by stretching & scaling along time
axis).
.
The Euclidean distance between two sequences can be seen as a special case of DTW, it is only
defined in the special case where the two sequences have the same length. It is a way to solve a
vast range of time dependant sequence problems, and it is widely used in various disciplines:- In
bioinformatics, Aach and Church successfully applied DTW to cluster RNA expression data [8].
In chemical engineering, it has been used for the synchronization and monitoring of batch
processes [9]. DTW has been effectively used to align biometric data, such as gait [10], signatures
[11], fingerprints [12], and ECGs [13]. Rath and Manmatha have successfully applied DTW to
the problem of indexing repositories of handwritten historical documents [14] and is also used in
indexing video motion streams [15]. In robotics, Schmill et al. demonstrate a technique that
utilizes DTW to cluster robots sensory outputs [17]. And in music, Zhu and Shasha have
exploited DTW to query music databases with snippets of hummed phrases [18,19].

The superiority of DTW over Euclidean distance for these tasks has been demonstrated by many
authors [8, 6, 13]

However, the greater accuracy of DTW comes at a cost. Depending on the length of the
sequences, DTW is typically hundreds or thousands of times slower than Euclidean distance. But,

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extensive research has been performed on how to accelerate DTW computations, in particular for
one-dimensional (or low-dimensional) real-valued sequences, often referred to as time series.
The main contributions of this paper are summarized as follows:

- We investigate the DTW technique in clustering time dependant sequences. Compared to
the previous work [1], DTW technique provide better accuracy in formed clusters than
Euclidian metric;
- We evaluate DTW performance in term of speed. Compared to Euclidian metric, DTW is
many of times slower than Euclidean distance. In spite of that the global time taken by
the clustering process with DTW technique is less many of times than the time taken
without clustering;
- We also evaluate the generalization ability of DTW in clustering cross other databases.

The remainder of this paper is structured as follows. The two next sections briefly review the
HMM on which we apply our proposed technique and the Dynamic time warping technique.
Section 4 describes the proposed technique for reduced time learning algorithm. Obtained results
are discussed in section 4. Section 5 discusses the possibility of an extended usability of the
proposed technique on other database. Finally, section 6 concludes the paper.

2. HIDDEN MARKOV MODELS[24]
A Hidden Markov Model is a model for sequential data. It is a doubly embedded stochastic
process with an underlying stochastic process that is not observable (hidden) , but can only be
observed through another set of stochastic process that produce the sequence of observations. The
performance of a generative model like the HMM depends heavily on the availability of an
adequate amount of representative training data to estimate its parameters, and in some cases its
topology.

2.1 Elements of an HMM:
An HMM is characterized by the following:

1- N : the number of hidden states in the model. For many practical applications there is often
some physical significance attached to the states or to the sets of states of the model. We denote
the individual state as S= {S1, S2…Sn} and the state at time t as qt.

2- M : the number of distinct observation symbols. The observation symbols correspond to the
physical output of the system being modeled. We denote the individual symbols as V={V1,
V2…VM}

3- The state transition probability distribution A={aij}where: aij=P[qt+1=Sj|qt=Si] 1<=i, j<=N.

4- The observation symbol probability distribution in state j, B={bj(k)}, where:
Bj(k)=P[Vk at t|qt=Sj] 1<=j<=N; 1<=k<=M.

5- The initial state distribution π ={ πi} where π =P[q1=Si] 1<=i<=N.
Given appropriate values of N, M, A, B and π the HMM can be used as generator to give an
observation sequence : O= O1O2….OT.
The compact notation: λ =(A,B,π) is used to indicate the complete parameter set of the model.

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2.2 The Three Basic Problems for HMMs:
Given the form of HMM there are three basic problems of interest that must be solved for the
model to be useful in real world applications. These problems are the following:

1- Given the observation sequence O=O1O2…OT, and a model λ =(A,B,π) how do we
efficiently compute P(O/λ) , the probability of the observation sequence given the model?
2- Given the observation sequence O=O1O2…OT, and a model λ =(A,B,π) how do we
choose a corresponding state sequence Q=q1q2…qT which is optimal in some meaningful
sense .
3- How do we adjust the model parameters λ=(A,B,π) to maximize P(O/λ)?

Problem three is the crucial one for most applications of the HMMs, since it allows us to
optimally adapt model parameters to observed training data to create best models for real
phenomena. The iterative procedure like Baum-Welch method or equivalently the EM
(Expectation, maximization) [25] method is used to solve this problem. One associated problem
to this solution is the time taken by the training process. In this paper, we aim to reduce training
time by reducing the original training set via an unsupervised clustering. Any information loss
during clustering is performed, we introduce a weight factor that is assigned to each training
instance which represent each cluster, and modify all expectation formulas to include the weight
factor in appropriate terms. The weight factor is the “frequency” of observing an instance with
similar behavior (characteristics) in the training set. Consequently the obtained model parameters:
A, B, π are equal to those obtained with the classical HMM training algorithm but in less time.

3. DYNAMIC TIME WARPING (DTW)
DTW algorithms have been proposed around 1970 in the context of speech recognition, when it
was mainly applied to isolated word recognition [25, 29].
DTW is a well-known technique to find an optimal alignment between two given (time-
dependent) sequences:
X :=(x1, x2, . . . , xN) of length N ∈ N and;
Y := (y1, y2, . . . , yM) of length M ∈ N
which may vary in time or speed. These sequences may be discrete signals (time-series) or, more
generally, feature sequences sampled at equidistant points in time.
To align these two sequences using DTW, one needs a local cost measure, sometimes also
referred to as local distance measure : C(x,y).
Typically, C(x, y) is small (low cost) if x and y are similar to each other, and otherwise C(x, y) is
large (high cost). Then, the cost matrix C(N,M) ∈ RN×M defined by C(n,m) := C(xn, ym) is
constructed. To find the best match between the two sequences, we can find a path through the
obtained matrix. A warping path, p, is a contiguous set of matrix elements that characterizes a
mapping between X and Y by assigning the element xnl of X to the element yml of Y .
The total cost Cp(X, Y ) of a warping path p between X and Y with respect to the local cost
measure c is defined as:

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Furthermore, an optimal warping path between X and Y is a warping path p* having minimal
total cost among all possible warping paths. The DTW distance DTW(X, Y ) between X and Y is
then defined as the total cost of p*:

4. TRAINING SET CLUSTERING USING DTW
In many real world classification problems there is significant data redundancy in training data,
consequently, the time and memory complexity grows with sequence length, the number of
training sequences, and with the number of HMM states N. For such problems one might be able
to reduce the learning time by preprocessing the data.

Training HMM means optimizing the model parameters (A,B,π) to maximize the probability of
the observation sequence P(O|λ). In order to obtain a good estimation of HMM; a big amount of
computation and a large database is required.

HMM profiles are usually trained with a set of sequences that are known to belong to a single
category. Consequently this set of sequences is often redundant in the sense that most of these
sequences have the same values taken by the different features describing this category.

Many real world applications are sensitive to these phenomena as instance, in medicine, and for
many patients reached with the same disease can have the same symptoms associated with this
disease. In gender recognition or skin recognition, many subjects can have same characteristics
(color, texture…). In facial expression recognition, many subjects display the same description of
any facial expression (e.g Joy : the distance between mouth corners increases, the distance
between lips relaxes or increases and the distance between lids decreases or relaxes) [26].

In image analysis, in many times, the extracted wavelets or FFT coefficients from some image
parts are the same when the images are belonging to the same category, etc…

Our aim is to reduce the size of the training set, so, the time of the training process without
reducing generalization (classification) accuracy. In this context, our approach involves a
preprocessing step to reduce the number of training samples, so we do not require that all data to
be resident in main memory (minimize storage and time training requirements). To this end, we
apply an unsupervised clustering on the original training data set to extract representative clusters.
To preserve training data from any information loss, we associate a weight factor to each cluster.
This weight factor is the frequency of observing instances with similar behavior (characteristics).
This approach is suitable for many real world applications where there is significant data
redundancy. After that, we modify all expectation formulas computed with the classical HMM
learning method (EM) to include the weight factor in appropriate terms.

The proposed approach reduces significantly the run-time of HMM training and the storage
requirement, while providing accurate discovery and discrimination of different category
behaviors when applying the unsupervised clustering algorithm.

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4.1 Clustering Process
Given an original training set, we apply an unsupervised clustering on all training instances that
belong to a particular class. That is, unsupervised clustering is performed independently on each
class. There are numerous clustering techniques including the K-means, fuzzy C-means,
hierarchical clustering, and self-organizing maps. The common problem with most of these
methods is the choice of the number of clusters or the number of neighbors. In our approach we
propose a simpler clustering algorithm where, it is important to note that the number of clusters is
not known a priori.

Clustering algorithm:
Begin
N: is the number of categories;
ni: is the number of instances (sequence observation) associated to category i; i=1..N
Ck(i): cluster number k of category i; 1<=k<=ni; j=sequence length
&
Ck (i,j+1)= frequency of sequence observation
(instance with same features values)
For all categories(N)
Put the first instance from the training set of the category i in table C of clusters;
Initialize the frequency field by 1;
For all instances belonging to each category(ni)
Compare the new instance(inst_nouv) with all instances (inst_in) in the cluster table;
If DTW(Inst_nouv,inst_in)=0
Update frequency field(+1);
Else
Add a new cluster with the new instance;
Initialize the frequency field with 1;
End
End
End
End.

In our previous work [1], the Euclidian distance was used to compare two sequences. In this
paper we investigate the DTW technique to compare sequences. Indeed, with Euclidean distance,
the two compared sequences must have the same length, only corresponding Time Points are
compared, but: In many real world applications, time dependant sequences are out-of-
synchronization and produced sequences are not always in the same length. As instance in our
experiments when studying facial expressions, one can display an expression (Joy) slowly and
another one displays the same expression (Joy) rapidly. Hence, the first recorded video produces
more frames than the second one, so the two sequences have not the same length, also, facial
features deformations in the first video (which are the same in the two videos) are not
synchronized with those of the second one. In spite of that the two sequences can easily be
aligned then clustered in the same cluster because both represent the same expression (Joy).
In another hand, generally, when modelling a situation with HMM, produced sequences count
multiple redundant consecutive emitted observations, because of the structure of HMM which can
stay in the same state or transit to another one and emit a set of observations with given
probabilities.

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Example:

Suppose we have two HMM with Two states. The possible observations which can be emitted by
each state are: O={1,2,3,4,5,6,7,8}. Some possible produced sequences from the two HMMs can
be: {1234567, 1222234, 1123334, 1222344...} as we can see, the first sequence represents one
occurrence of each observation; the next ones represent multiple occurrences of different
observations.

When applying Euclidean metric to cluster these sequences, we get four clusters, but when using
DTW technique we get only two clusters, the distance between the three last sequences is equal to
“0”. Consequently, the first sequence can be assigned to the first HMM and the three other
sequences can be assigned to the second HMM. The maximum likelihood confirms the decision.
Thus, DTW technique is most accurate than the Euclidean distance since it produces meaningful
clusters.

Once clustering is completed, we use each instance in the cluster table as a new training instance.
Its weight factor can be found in the associated frequency field.

In many studies, clusters with small weight are considered as noisy instances so they are
removed. It is not the case in all studies, in our experiments, each cluster describe a specific
behavior associated to the considered class (e.g Anger or disgust are two universal facial
expressions which can be displayed in different ways, each cluster of each class (facial
expression) can be a specific description of each facial expression so it is not a noisy instance), in
this case only a human expert can remove these clusters to improve classification accuracy.

4.2 The Modified EM Algorithm for HMM Training
Generally, the training of Hidden Markov Models is done by the EM Algorithm to create the best
fitting HMM from a given set of sequences F.

The EM algorithm for HMMs
- Initialization : set initial parameters λ0 to some value;
- For t = 1..num iterations
o Use the forward_backward algorithm to compute all expected counts :
Number of transitions from state Si to Sj using the precedent set of parameters λt-1
Number of emissions of symbol Vk from state Sj using the precedent set of
parameters λt-1
Update the parameters based on the expected counts using formulas (2) and (3).
End.

From the initial HMM M and the initial parameter set λ we can generate a new optimized
parameter set λ’. Using the concept of counting event occurrences, the following formulas are
given for re_estimation of the new HMM parameters:
- πi‘= The expected frequency in state Si at time t=1 (1) ;
- aij‘= the expected number of state Si to state Sj transitions/ the expected number of transitions
from state Si (2);
- bj(k)’= The expected number of times the observation Vk is generated from state Sj (3);

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These computations are performed for all training instances, in our case, these computations are
performed for the new set of training instances, it consists on the clustered training set instances
so they are iterated for m=1..ki (ki is the number of clusters of class i), multiplying each formula
by Ck(i,j+1) (number of similar instances);

As a result the transition probability from state Si to state Sj and the emission probability of the
observation Vk from state Sj increase with the number of instances (redundant and non
redundant). More than that, time running in the computation of all these expectations and for all
iterations is significantly reduced.

5. EXPERIMENTAL RESULTS
The experiments were conducted on both (i) extracted information from (MMI database[27])
videos and (ii) synthetically generated instances. Our aim is to analyze the capability of the
proposed method in reducing training set instances without information loss and reducing training
time. Our code source is written in Matlab, and, our experiments are done on: Intel Dual-Core
1.47GHz machine with 2G memory.

5.1. MMI database
The MMI database is a resource for Action unit (AU) and basic emotion recognition from face
video. It consists of V parts; each part is recorded in different conditions. The second set of data
recorded was posed displays of the six basic facial expressions: Joy, Sadness, Anger, Disgust,
Fear and Surprise. In total, 238 videos of 28 subjects were recorded. All expressions were
recorded twice. In our experiments [28], we aim to discover novel clusters with different behavior
relative to each class (generate for each facial expression a set of clusters which allow describing
all possible behaviors associated with each studied class). To this end, facial characteristic points
were detected on the first frame of each video and tracked, then, characteristic distances were
computed leading to several time series. Obtained time series were analyzed to produce a smaller
set of time series which represent occurrence order of facial features deformations of each subject
with each facial expression (This is what we call data preparation).

5.2. Synthetic Time Series
To generate synthetic data, the Markov Model with its transition and emission probabilities
matrixes constructed with the MMI database was used. Accordingly, for a specific alphabet size
M=10 observations, several training sequences were produced.

With both data, we evaluated our experiments on an HMM with three states(3), sequence length
equal to five (5) from where we have extracted the most interesting subsequences which length is
three (3), the number of possible observations is ten (10) and the number of iterations is fifty (50).
The estimated time training of the EM and EM based cluster training algorithms are presented on
table 1 according to each experiment:

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Table 1. Time training of EM and EM based cluster training algorithms.

Number of Number Time of clustering /s Time of EM Time of DIFF in
sequence of training/s EM number
observations clusters Euclidean DTW technique based of times
distance cluster
training/s
100 10 0 0.0780 0.5304 0.0936 5.6
1.000 19 0.0312 1.0764 5.234 0.162 32.3
10.000 24 0.2652 7.9717 44.3979 0.1945 228.26
100.000 25 4.6020 135.4245=2mn25s 497.4092=8mn29s 0.2106 2361.87

Results on this table are computed as the mean of ten (10) executions of the two algorithms. In
term of processing speed, the EM training takes 0.5304s to learn 100 training sequences. With the
EM based cluster training, it takes 0.0936s, it means that it is 5.6 times faster than EM training.
When the number of training sequences is equal to 1.000, the EM based cluster training is 32.3
times faster than the EM training. When the number of training sequences is equal to 10.000, the
EM based cluster training is 228.26 times faster than the EM training. Finally, when the number
of training sequences is equal to 100.000, the EM based cluster training is 2361.87 times faster
than the EM training.

However, we can observe that time clustering with Euclidean distance is less than time clustering
with DTW technique in all cases. Anyway, we can explore one of the proposed methods
presented in the literature to speed up DTW computation. The complexity of this technique is
O(N.M), it can be reduced to O(N+M) like in [30].

With conducted experiments on these databases, we can observe that the number of constructed
clusters when using Euclidean metric is the same when using DTW technique this can be
explained by the fact that the new training set sequences (representative/index clusters) are really
different (no multiple occurrences of some observations in the same sequence). To evaluate the
ability of DTW in the construction of more accurate fewer clusters, we use another database (see
section 5).

Although clustering based DTW technique consumes more time than clustering based Euclidean
metric, the overall time learning (clustering + training) based clustering with Euclidean/DTW is
less than time learning without clustering.

The experiments show that the difference in time training between the two algorithms increases
dramatically with the number of training sequences in favor of the proposed modified EM
training algorithm (it reduces). These results suggest that the proposed method can be used
especially with very large datasets.

Obtained results show that the proposed process reduces not only time computation requirements,
but, it also reduces significantly storage requirements. Because HMM profiles are usually trained
with a set of sequences that are known to belong to a single category, the clustering process
produce a small number of clusters. These clusters form indexes of constructed clusters and will
represent the new set of training instances.

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Experimental results show that both EM and EM based cluster training algorithms achieve
similar: Initial (π), Transition (A) and Emission (B) probabilities matrixes. Consequently the
classification accuracy is maintained.

6. GENERALIZATION TO OTHER DATABASE
To evaluate the ability of DTW in more accurate clustering, we conduct our experiments on
another database. The new database count 30.000 sequences obtained from MMI facial
expressions database, the length of each sequence is about nine (9) observation instead of three
and many sequences have multiple occurrences of the same observation. The evaluation of the
proposed approach on this database is summarized on table 2:

Table 2. Time clustering, time training of EM and EM based cluster training algorithms with
Euclidean and DTW distances.

Number Number Euclidean distance DTW technique Time
of of real Training
Sequences clusters Number Time Time Number Time Time without
of Clustering Training of Clustering Training clustering
construc based construc based
ted clustering ted clustering
clusters clusters

30.000 70 70 2.26 0.41 25 79.7477 0.18 163.45

As we have explained before, we can note that the number of constructed clusters when using
DTW is less than the one obtained when using Euclidean metric, it match very well the real
number of clusters. This can be explained by the fact that many sequences with Euclidean metric
are considered as different in spite that they are similar but not synchronized. This means that
DTW are more accurate in the field of clustering than Euclidean metric.

Also, we can observe that time training based clustering with DTW is less than the one with
Euclidean metric, this can be explained by the number of constructed clusters which is greater
when using Euclidean metric.

Finally we can say that in general, the proposed idea can be adapted to all ML methods which
involve a number of iterations equal to the number of instances in their processing. The proposed
process reduces the number of iterations and consequently reduces the time training of these ML
methods.

7. CONCLUSION
In this paper, a more accurate simple process to speed up ML methods is studied. An
unsupervised clustering algorithm is combined with EM algorithm to develop an efficient HMM
training to tackle the problem of HMM time training of large datasets. The idea of the proposed
process consists of two steps. In the first step, training instances with similar inputs are clustered
in one cluster and each cluster is weighted with the number of similar instances. DTW technique
is used to accurately cluster training time dependant sequences. In the second step, all formulas
associated with the number of training instances are modified to include the weight factor in
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appropriate terms. Through empirical experiments, we demonstrated that the EM based cluster
training algorithm reduces significantly the HMM time training and storage requirements.
Furthermore, the proposed approach is not limited to HMMs it can be employed for a large
variety of ML methods. In a future work, we plan to explore new techniques to cluster more
accurately time dependant sequences, to reduce more and more time training and improve more
and more classification accuracy and test these methods on different ML methods.

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TOWARDS MORE ACCURATE CLUSTERING METHOD BY USING DYNAMIC TIME WARPING

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