Deviation-based Dynamic Time Warping for Clustering Human Sleep

Chiying Wang

, Sergio A. Alvarez

, Carolina Ruiz

and Majaz Moonis

Department of Computer Science, Worcester Polytechnic Institute, Worcester, MA 01609 U.S.A.

Department of Computer Science, Boston College, Chestnut Hill, MA 02467 U.S.A.

Department of Neurology, U. of Massachusetts Medical School, Worcester, MA 01655 U.S.A.

Keywords:

Dynamic Time Warping, Deviation, Human Sleep, Clustering.

Abstract:

In this paper, we propose two versions of a modiﬁed dynamic time warping approach for comparing discrete

time series. This approach is motivated by the observation that the distribution of dynamic time warping paths

between pairs of human sleep time series is concentrated around the path of constant slope. Both versions use

a penalty term for the deviation between the warping path and the path of constant slope for a given pair of

time series. In the ﬁrst version, global weighted dynamic time warping, the penalty term is added as a post-

processing step after a standard dynamic time warping computation, yielding a modiﬁed similarity metric that

can be used for time series clustering. The second version, stepwise deviation-based dynamic time warping,

incorporates the penalty term into the dynamic programming optimization itself, yielding modiﬁed optimal

warping paths, together with a similarity metric. Clustering experiments over synthetic data, as well as over

human sleep data, show that the proposed methods yield signiﬁcantly improved accuracy and generative log

likelihood as compared with standard dynamic time warping.

1 INTRODUCTION

Human sleep patterns are closely associated with

overall health and quality of life, making the scientiﬁc

study of sleep an important pursuit. Sleep stage tran-

sitions (Kishi et al., 2008) and bout durations (Chu-

Shore et al., 2010) are essential indicators in charac-

terizing the structure of sleep. Typical patterns of hu-

man sleep have been found (Bianchi et al., 2010), yet

sleep microstructure varies across individuals, being

affected by age, circadian rhythms (Dijk and Lockley,

2002), and other factors.

A substantial challenge in modeling the dynam-

ics of sleep is the scarcity of key dynamical events

such as stage transitions within sleep sequences. This

scarcity yields small samples over which dynami-

cal models are to be trained, leading to high uncer-

tainty in parameter estimates. An approach known

as dynamical modeling-clustering (CDMC) was pro-

posed (Alvarez and Ruiz, 2013) to address this chal-

lenge. CDMC reduces model variance through se-

lective aggregation of instances during a clustering

phase, so that models are learned over collections of

dynamically similar instances rather than individual

instances. The technique of initialization using clus-

tering by Dynamic Time Warping (DTW) similarity

(Oates et al., 2001) yields good convergence proper-

ties for CDMC (Wang et al., 2014).

Despite promising results, standard DTW as a

similarity measure for unsupervised clustering of time

series suffers from certain problems. One of these

is the over-warping problem shown in Fig. 1. Over-

warping refers to unnatural alignment of dissimilar

segments in two time series. In Fig. 1, a subse-

quence of length over 300 in one patient is matched

by dynamic time warping to a subsequence of length

less than 10 in another. The result is unacceptable,

yet the standard dynamic time warping distance be-

tween the two segments is zero. Explicitly penalized

DTW has been developed to address over-warping.

For example, (Clifford et al., 2009) proposes variable

penalty DTW, which reduces nondiagonal moves dur-

ing alignment. However, this approach is heavily de-

pendent on a user-deﬁned penalty function and thus

difﬁcult to apply in practice.

An additional concern with DTW is time com-

plexity. Variants of DTW have been proposed that

focus on improving efﬁciency by globally constrain-

ing the warping path to a predeﬁned geometric re-

gion such as the Sakoe-Chiba band (Sakoe and Chiba,

1978) or the Itakura parallelogram (Itakura, 1975).

However, the use of global constraints alone can lead

Wang, C., Alvarez, S., Ruiz, C. and Moonis, M.

Deviation-based Dynamic Time Warping for Clustering Human Sleep.

DOI: 10.5220/0005729200880095

In Proceedings of the 9th International Joint Conference on Biomedical Engineering Systems and Technologies (BIOSTEC 2016) - Volume 4: BIOSIGNALS, pages 88-95

ISBN: 978-989-758-170-0

0 100 200 300 400 500 600 700 800

Time

Sleep Stage

0 100 200 300 400 500 600 700 800

Time

Sleep Stage

Patient 56

Patient 235

Over−Warping

(a) Over-Warping Occurrence in Time Series.

Patient 235

Patient 56

100 200 300 400 500 600 700 800

100

200

300

400

500

600

700

0.2

0.4

0.6

0.8

Over−Warping

Diagonal Line

(b) Over-Warping Path in Dynamic Time Warping.

Figure 1: Over-warping of sleep stage sequences using dynamic time warping. (Left) Use of standard dynamic time warping

inappropriately matches dissimilar segments (a long one versus a short one) in two sequences. (Right) The same over-

warping problem described in terms of warping search area. The area circled by the dashed line indicates a large deviation of

the standard warping path from the diagonal path of constant slope. The bar graph on the right indicates local cost measure

and the background in the right ﬁgure shows the local cost marix of two discrete time series (patient 56 and patient 235) in

dynamic time warping computation (see section 2.1).

0 200 400 600 800 1000

200

400

600

800

1000

20 Patients

Figure 2: Optimal time warping paths (dashed lines) be-

tween 20 pairs of human sleep recordings. DTW optimal

paths are usually close to the diagonal line from bottom-left

to top-right in the warping space. The varying boundary of

the distribution suggests the desirability of adaptively iden-

tifying warping areas locally instead of using a predeﬁned

global constraint.

to the over-warping problem described above. Some

researchers (Ratanamahatana and Keogh, 2004) have

argued that the effect of warping band width on the

quality of the results is greatly domain dependent and

that a narrow band might be valuable. The distribu-

tion of optimal warping paths between pairs of sleep

time series in Fig. 2 (from the present authors’ own

work) likewise suggests that the use of local search

constraints would be desirable.

Main Contributions of the Present Paper

1. We propose two novel DTW variants, global

weighted dynamic time warping (gwDTW) and

stepwise deviated dynamic time warping (sd-

DTW), that penalize deviations of the warping

path from the path of constant slope. This over-

comes the over-warping issue in Fig. 1, while re-

taining the efﬁciency advantages of approaches

based on global constraints such as the Sakoe-

Chiba band (Sakoe and Chiba, 1978) and Itakura

parallelogram (Itakura, 1975), and without rely-

ing on domain dependent speciﬁcs as in variable

penalty DTW (Clifford et al., 2009).

2. We apply the proposed modiﬁed DTW for clus-

tering initialization within the combined dynami-

cal modeling-clustering (CDMC) framework (Al-

varez and Ruiz, 2013) over human sleep time se-

ries, and show that this approach better captures

the dynamics of human sleep.

Organization of the Paper. Section 2 reviews stan-

dard DTW, and describes the proposed deviation-

based dynamic time warping approach and its appli-

cation to time-series clustering. Section 3 presents

experimental results and analysis on time series clus-

tering using deviation-based dynamic time warping.

Section 4 describes conclusions and future work.

Deviation-based Dynamic Time Warping for Clustering Human Sleep

2 METHODS

We review standard dynamic time warping in sec-

tion 2.1, as that technique will serve as a baseline.

The proposed deviation-based dynamic time warping

approach is described in section 2.2.

2.1 Dynamic Time Warping (DTW)

Dynamic time warping (DTW) is a classical dynamic

programming algorithm for measuring the similarity

of two time series (e.g., (M

uller, 2007)). It performs

an optimal alignment between two time series by non-

linearly warping their time dimensions. DTW has

been applied to speech recognition (e.g., (Sakoe and

Chiba, 1978) and (Itakura, 1975)), time series clas-

siﬁcation (Jeong et al., 2011), and unsupervised time

series clustering (Oates et al., 2001).

The following are the essentials of standard DTW,

as described in (M

uller, 2007).

We consider two time sequences

X = (x

,... ,x

) of length N ∈ N and

Y = (y

,. .. ,y

) of length M ∈ N, with indi-

vidual values x

in some feature space F .

A local cost measure is a function

c : F × F → R

≥0

(1)

The value of c (x

) is small if x

are close to each

other, and otherwise not. For discrete time series, one

can use a cost matrix to deﬁne the values c(x, y) for all

pairs of values x,y; the simplest possibility is to use

the identity matrix, that is, to let c(x,y) = 0 if x = y,

otherwise c(x,y) = 1.

A warping path between X and Y is a sequence

p = (p

, p

,. .. , p

) (2)

where p

= (n

) ∈ [1 : N] × [1 : M] for l ∈ [1 : L]

(the diagonal line from bottom-left to top-right in

Fig. 1(b)), and max{N, M} ≤ L ≤ N + M. It must sat-

isfy the following three conditions:

• Boundary condition: p

= (1,1) and p

= (N,M)

are the start and end points respectively.

• Monotonicity condition: horizontal and vertical

components increase monotonically: n

≤ n

≤

.. .n

and m

≤ m

≤ ...m

• Step size condition: for each l < L, the difference

l+1

− p

is one of (1,0), (0,1),(1,1).

The total cost of a warping path p between X and

Y is

(X,Y ) =

∑

l=1

c(x

) (3)

Time series 1

ψ(Xn

l-1

,Ym

l-1

)

ψ(Xn

l-1

,Ym

)

ψ(Xn

,Ym

l-1

)

ψ(Xn

,Ym

)

Time series 2

Figure 3: Deviation (e.g., gray shaded area) of warping

path (squares with directional arrows) from path of constant

slope (solid red line). Given two warping paths (1 and 2),

the path with smaller deviation (the green one) is better.

An optimal warping path p

∗

is one having min-

imum total cost Φ

∗

(X,Y ) among all warping paths

from p

to p

. Φ

∗

(X,Y ) is referred to as the DTW

distance between sequences X and Y .

2.2 Proposed Approach

2.2.1 Deviation Measure

We address the standard DTW concerns of over-

warping and time complexity described in the In-

troduction, by penalizing nondiagonal moves in the

search for an optimal warping path. This is done by

using the measure of deviation discussed below.

Deviation refers to the area ∆

(shaded areas in

Fig. 3) bounded by the warping path p between two

time series and the diagonal path of constant slope.

Deviation is computed by the procedure described

in Algorithm 1. The following are the main steps:

• Initialize the deviation ∆

to be 0. (step 1)

• Calculate the slope k and the intercept b of the

diagonal path deﬁned as y = k ∗ x + b. (step 2-3)

• Repeat until the end point p

is reached. (step 5)

- Add absolute vertical distance between current

point p

and diagonal to the deviation ∆

, if p

and p

i−1

differ vertically. (step 6-7)

- update the counter, i. (step 8)

• Return the deviation ∆

(step 9)

BIOSIGNALS 2016 - 9th International Conference on Bio-inspired Systems and Signal Processing

Algorithm 1: Deviation Calculation.

Input: A warping path p = {p

, p

,. . . , p

}, each

= (n

); L: total number of points in path p.

Output: The deviation ∆

of p from the diagonal

line through p

and p

in warping space.

ComputeDeviation(p)

1. ∆

= 0

2. k = (M − 1)/(N − 1)

3. b = M − k ∗ N

4. i = 2

5. While (p

6= p

)

6. if (m

6= m

i−1

)

7. ∆

= ∆

+ |m

− (k · n

+ b)|

8. i = i + 1

9. return ∆

Time Series 1

Time Series 2

5 10 15 20 25 30

DTW

sdDTW

Diagonal

Figure 4: Optimal warping paths for standard DTW and

stepwise deviation-based DTW (sdDTW). Standard DTW

allows large deviations in searching for a path of minimum

total cost. sdDTW aligns time series closer to the diagonal

line. Background shading indicates local cost in sdDTW,

which increases with distance to the diagonal.

2.2.2 Deviation-based Dynamic Time Warping

Global Weighted Dynamic Time Warping: The global

weighted dynamic time warping (gwDTW ) distance

between sequences X and Y is deﬁned as

gwDTW (X,Y ) = λ

· Φ

∗

(X,Y )

+(1 − λ

) ·

∆

∗

(X,Y )

(4)

where Φ

∗

(X,Y ) is standard DTW distance, and the

deviation ∆

∗

(X,Y )

(Algorithm 1) is added as a post-

processing penalty to standard DTW. p

∗

(X,Y ) is the

optimal path between X,Y in standard DTW. λ

con-

trols the balance between standard DTW cost and the

deviation ∆

∗

(X,Y )

. λ

ranges from 0 to 1. The square

root of the deviation ∆

∗

(X,Y )

is used because it scales

linearly with Euclidean distance.

Stepwise Deviation-based Dynamic Time Warping:

The stepwise deviation-based dynamic time warping

(sdDTW ) distance between sequences X and Y is de-

ﬁned as

sdDTW (X,Y ) = Ψ

∗

(X,Y )

(5)

where Ψ

∗

(X,Y ) is the minimum total cost of a warp-

ing path p

∗

in Eq. 3 obtained by replacing the local

cost measure in Eq. 1 by the modiﬁed measure ϕ(x, y)

in Eq. 6. Thus, sdDTW optimal warping paths are

minimizers of a different cost measure than standard

DTW paths.

ϕ(x,y) = λ

· c(x,y) + (1 − λ

) ·

∆(x,y) (6)

where c(x,y) denotes the original local cost measure

in equation 1. ∆(x,y) is the deviation of a position

(x,y) relative to the diagonal path of constant slope

for sequences X and Y . See section 2.2.1. λ

is a

parameter that determines the relative weights of the

standard local cost measure and the deviation.

We use dynamic programming as in standard

DTW to compute the sdDTW optimal warping path,

based on the modiﬁed accumulated cost matrix

D(n,m) = Ψ(X(1 : n),Y (1 : m)) (7)

where X(1 : n) = (x

,. .. ,x

) and Y (1 : m) =

,. .. ,y

). n ∈ [1 : N] and m ∈ [1 : M]. That is

X(1 : n) and Y (1 : m) are subsequences of X and Y .

The procedure is as follows:

• Initially,

D(n,1) =

∑

k=1

ϕ(x

) and

D(1,m) =

∑

k=1

ϕ(x

• Iteratively, take the minimum accumulated cost

from three immediately adjacent directions:

D(n − 1, m) + ϕ(x

D(n,m − 1) + ϕ(x

D(n − 1, m − 1) + ϕ(x

• Until the ﬁnal position (N,M) is reached.

D(N,M) is the optimal dynamic time warping dis-

tance with respect to stepwise deviation-based dy-

namic time warping.

As illustrated in Fig. 4, optimal warping paths for

the sdDTW distance metric exhibit less over-warping

than the corresponding standard DTW paths.

2.2.3 Deviation-based Dynamic Time Warping

Clustering

Deviation-based dynamic time warping clustering

(dDTWC) performs unsupervised agglomerative hier-

archical clustering of time series using the deviation-

based DTW approaches in section 2.2.2 to calculate

Deviation-based Dynamic Time Warping for Clustering Human Sleep

distances. The proposed approach is described in

pseudocode in Algorithm 2. The main steps are:

• Initially each time series instance X is in its own

cluster (steps 1-2).

• Repeat until only k clusters remain (steps 3-6):

- Merge the closest clusters, C and C

; the dis-

tance between two instances (X and Y ) is de-

ﬁned by gwDTW or sdDTW in section 2.2.2;

the distance between two clusters is the average

distance between pairs of instances.

• Return clustering of dataset in k clusters (step 7).

3 EXPERIMENTAL EVALUATION

Deviation-based DTW clustering as described in sec-

tion 2.2.3 was compared with clustering using the

standard DTW distance metric. For all dynamic time

warping computations, the local cost measure in Eq. 1

was deﬁned as c(x,y) = 1 if the elements x and y are

different, otherwise c(x, y) = 0. The weight values

and λ

in Eqs. 4 and 5, respectively, were de-

termined empirically in order to maximize mean ac-

curacy over a sample of labeled synthetic data gener-

ated as in section 3.2 (but separate from the synthetic

data sample used for performance evaluation in sec-

tion 3.2.3): λ

was set to 0.67. λ

was set to 0.83.

All experiments were performed in MATLAB



(The

MathWorks, 2015).

Two sets of experiments were carried out, corre-

sponding to synthetic data and human sleep data, re-

spectively. Details speciﬁc to each of these are de-

scribed in sections 3.2.2 and 3.3.2 below.

3.1 Statistical Signiﬁcance

Pairwise comparisons of median classiﬁcation accu-

racy values (see section 3.2.2) of gwDTW and sd-

DTW clustering against the accuracy of standard

DTW clustering (for synthetic data) and of negative

log likelihood values of gwDTW and sdDTW cluster-

ing against that of standard DTW clustering (for hu-

man sleep data) were carried out by a non-parametric

two-sided Wilcoxon rank sum test, since a Lilliefors

normality test rejected normality at the p < 0.05 sig-

niﬁcance level in each case. A Bonferroni correction

was performed jointly on the accuracy and log likeli-

hood Wilcoxon p-values to ensure a familywise error

rate less than 0.05.

3.2 Synthetic Markov Mixture Data

3.2.1 Dataset Generation

A synthetic dataset of discrete sequences was gener-

ated as in (Alvarez and Ruiz, 2013), from two distinct

Markov models, each with two states. The two mod-

els differ in their transition probability matrices. Self-

transition probabilities of 0.6 in one model and 0.8 in

the other were selected. The probabilities of transi-

tioning between states were 0.4 and 0.2, respectively.

One of the two models is selected randomly and used

to generate a sequence of the desired length, L. This

process is iterated until a predetermined number of

sequences, N, is obtained. The present paper uses the

values N = 100 and L = 300 in all trials.

3.2.2 Experimental Procedure

Clustering Classiﬁcation Accuracy. Supervised

classiﬁcation via clustering was performed with the

generating model label as the classiﬁcation target.

Each cluster was associated with the class c that oc-

curs most frequently among its members. The evalu-

ation metric was classiﬁcation accuracy, equal to the

fraction of labeled instances (X , c) that are assigned

to a cluster in which c is the majority class. Statistical

hypothesis testing was performed using a Wilcoxon

rank sum test to compare median accuracies as de-

scribed in section 3.1. Experimental procedure was

as in the following pseudocode:

Experimental Procedure, Synthetic Data Classiﬁ-

cation:

begin

for i := 1 to Tr ialNum

SD = generateSyntheticDataset(N, L);

Accuracy(1, i) = evaluateByDTW(SD);

Accuracy(2, i) = evaluateBygwDTW(λ

, SD);

Accuracy(3, i) = evaluateBysdDTW(λ

, SD);

end

Perform Wilcoxon rank sum test over Accuracy.

end

Note:

• generateSyntheticDataset followed the descrip-

tion in section 3.2.

• evaluateByDTW refers to the clustering proce-

dure in section 2.2.3 and clustering evaluation by

classiﬁcation accuracy (see above); likewise for

evaluateBygwDTW and evaluateBysdDTW.

• The total number of sequences, N, was set to 100.

• The length of a sequence, L, was set to be 300.

• The number of trials, TrialNum, was set to 100.

BIOSIGNALS 2016 - 9th International Conference on Bio-inspired Systems and Signal Processing

Algorithm 2: Deviation-Based DTW Clustering (dDTWC).

Input: An unlabeled time series dataset D = {X | X is a time series}; a positive integer, k, the desired number

of clusters; a predeﬁned local cost measure c : F × F → R

≥0

where F is the feature space in which the time

series in D take their values. dDTW denotes the total cost measure associated with c, which is deﬁned by Eq. 4

in the case of gwDTW and by Eq. 5 in the case of sdDTW.

Output: A partition of D into k clusters

dDTWC(D, k, d)

1. for each i, let C

= a cluster that contains only the i-th time series in D

2. s = the number of time series in D (initial number of clusters )

3. while s > k

4. (i

∗

, j

∗

) = arg min

i, j∈{1,...,s}

¯c(C

) (where ¯c is mean cost for instance pairs in the two clusters)

= arg min

i, j∈{1,...,s}



∑

X∈C

,Y ∈C

dDTW(X,Y,c)

|·|C



5. Merge C

∗

and C

∗

to reduce the number of clusters to s − 1

6. s = s − 1

7. return {C

,. .. ,C

}

DTW gwDTW sdDTW

0.5

0.6

0.7

0.8

0.9

Figure 5: Clustering accuracies using standard DTW,

gwDTW, and sdDTW as similarity measures over hidden

Markov mixture data. Non-overlapping notches indicate

signiﬁcant difference in medians (p < 0.05). gwDTW and

sdDTW are signiﬁcantly more accurate than standard DTW.

3.2.3 Synthetic Data Results

This section evaluates performance of clustering

over synthetic data using globally weighted DTW

(gwDTW) (Eq. 4) or stepwise deviation-based DTW

(sdDTW) as the similarity measure, as compared with

standard DTW similarity.

Clustering accuracies over the synthetic dataset

appear in Fig. 5. Both gwDTW and sdDTW perform

signiﬁcantly better than standard DTW, proving the

beneﬁt of incorporating deviation into the DTW com-

putation for clustering of synthetic time series data.

Median accuracies appear in Table 1.

Table 1: Median accuracies of clusterings based on

DTW, gwDTW, and sdDTW. Asterisks denote Bonferroni-

corrected statistical signiﬁcance of differences with stan-

dard DTW in Wilcoxon rank sum test (p < 0.05).

DTW gwDTW sdDTW

0.65 0.92

∗

0.91

∗

3.3 Human Sleep Data

3.3.1 Datasets

A collection of 244 fully anonymized human

polysomnographic recordings was extracted from

polysomnographic overnight sleep studies performed

in the Sleep Clinic at Day Kimball Hospital in Put-

nam, Connecticut, USA. Each polysomnographic

recording is split into 30-second epochs. Lab tech-

nicians staged each 30-second epoch into one of the

sleep stages Wake, stage 1, stage 2, stage 3, and REM

(Rapid Eye Movement). Three versions of the human

sleep dataset are considered, depending on whether

these stage labels are grouped in some way:

• (W5) uses the ﬁve standard stage labels Wake, 1,

2, 3, REM.

• (WNR) uses the three stage labels Wake, NREM

(stages 1, 2, and 3), REM.

• (WDL) uses the three stage labels Wake, Deep

(stage 3), Light (stages 1,2,REM).

3.3.2 Experimental Procedure

Unsupervised clustering was performed over the

human sleep datasets described in section 3.3.1.

Deviation-based Dynamic Time Warping for Clustering Human Sleep

The collective dynamic modeling clustering algo-

rithm (Alvarez and Ruiz, 2013) was used for cluster-

ing, with two-state hidden semi-Markov chain models

as the dynamical models. Initial cluster labels were

computed by deviation-based DTW clustering as de-

scribed in Algorithm 2, with either gwDTW (Eq. 4)

or sdDTW (Eq. 5) as the distance metric. Clustering

driven by the standard DTW distance metric was used

as a basis for comparison.

Generative negative log likelihood was used to

measure the quality of model ﬁt for unsupervised

clustering. Given a hidden semi-Markov model,

M, built over a group of sequences such as hu-

man sleep sequences, the generative negative log-

likelihood −log(P(s|M)) of a sequence, s, is a mea-

sure of the probability that the sequence, s, would

be produced by the model, M. Lower negative log-

likelihood values (higher generative probabilities) im-

ply a better model ﬁt. The goal of clustering was

to minimize the generative negative log likelihood.

Comparison of median negative log likelihoods for

different models was measured by a Wilcoxon rank

sum test as described in section 3.1.

Experimental Procedure, Sleep Data Clustering:

begin

D1, D2, D3 = W5, WNR, WDL datasets (3.3.1)

m1, m2, m3 = DTW, gwDTW, sdDTW

for j = 1 to 3

for k = 1 to 3

(M, nlogll(Dj, mk, s1...s244)) = CDMC(Dj, mk)

end

Perform pairwise Wilcoxon rank sum tests on

nlogll(Dj, m1...m3, s1...s244)

end

Notes:

• The W5, WNR, WDL datasets are as in sec-

tion 3.3.1.

• DTW refers to clustering using standard DTW as

the similarity metric; gwDTW and sdDTW refer

to the deviation-based clustering techniques de-

scribed in section 2.2.3.

• CDMC(D j,mk) refers to CDMC clustering (Al-

varez and Ruiz, 2013) with semi-Markov cluster

models, using the given method, mk, for cluster-

ing initialization, and is assumed to return a set of

dynamical models together with negative genera-

tive log likelihoods nlogll(Dj,mk,sl) for all input

sequences, sl, l = 1,···244.

Figure 6: Visualization of clusters over human sleep dataset

using gwDTW as similarity measure. Coordinates are

Weibull shape and scale parameters for Wake stage. Red

circles and blue triangles denote gwDTW clusters; back-

ground colors represent DTW clusters.

3.3.3 Human Sleep Data Results

Fig. 6 shows the two CDMC clusters (circles and tri-

angles) with coordinates equal to the Weibull scale

and shape parameters for the wake stage in the WNR

dataset. gwDTW clusters better capture the bound-

ary between the natural Weibull dynamical clusters in

the human sleep dataset, as compared with standard

DTW clusters.

Model ﬁt was signiﬁcantly better for both global

weighted DTW (gwDTW) clustering and stepwise

deviation-based DTW (sdDTW) clustering as com-

pared with standard DTW-driven clustering, as shown

in Table 2. This shows that deviation-based DTW is

superior to standard DTW as a similarity metric for

initialization of CDMC clustering over human sleep

data, as well as for standalone clustering over syn-

thetic data as shown in section 3.2.3.

Table 2: Median negative log likelihoods of gwDTW, sd-

DTW, and standard DTW clusterings over WNR, WDL, and

W5 human sleep datasets in section 3.3.1. Asterisks indi-

cate Bonferroni-corrected signiﬁcance of differences with

standard DTW in Wilcoxon rank sum test (p < 0.05).

DTW gwDTW sdDTW

WNR 150.7 148.2

∗

147.9

∗

WDL 159.4 157.9

∗

158.1

∗

W5 194.1 192.3

∗

191.9

∗

4 CONCLUSIONS

This paper proposes two versions of a modiﬁed dy-

namic time warping (DTW) approach for comparing

BIOSIGNALS 2016 - 9th International Conference on Bio-inspired Systems and Signal Processing

discrete time series such as human sleep sequences:

global weighted dynamic time warping (gwDTW)

and stepwise deviation-based dynamic time warping

(sdDTW). Both versions penalize deviations from the

path of constant slope in the warping space, yielding

the efﬁciency advantages of DTW approaches based

on global constraints such as the Itakura parallelo-

gram or the Sakoe-Chiba band, while better account-

ing for local deviations. gwDTW adds a deviation-

based term to the standard DTW distance metric.

sDTW adds a deviation term into the local cost func-

tion that drives the DTW dynamic programming op-

timization itself, yielding an improved warping path

together with a similarity metric. Both gwDTW and

sdDTW lead to signiﬁcantly better clustering results

than DTW in a classiﬁcation task over labeled syn-

thetic semi-Markov data, as well as in unsupervised

clustering of human sleep data. The authors learned

of an interesting “salient feature” approach to con-

strained DTW (Candan et al., 2012) after completing

the work reported in the present paper. The salient

feature approach extracts features of the input se-

quences that are then used to deﬁne locally adaptive

constraints on the warping path. In future work, it

would be desirable to pursue a performance compar-

ison of the salient feature approach of (Candan et al.,

2012) with that of the present paper.

ACKNOWLEDGEMENTS

The authors thank the anonymous referees for com-

ments that helped improve the legibility of the paper,

and for making us aware of (Candan et al., 2012).

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Deviation-based Dynamic Time Warping for Clustering Human Sleep