Mining Time Series Data Carlo ZanioloUCLA CS Dept

A Tutorial on Indexing and Mining Time Series Data ICDM '01

The 2001 IEEE International Conference on Data Mining

Dr Eamonn KeoghComputer Science & Engineering Department

University of California - RiversideRiverside,CA 92521eamonn@cs.ucr.edu

With Slides from:

• Introduction, Motivation• Similarity Measures

• Properties of distance measures• Preprocessing the data• Time-warped measures

• Indexing Time Series• Dimensionality reduction

• Discrete Fourier Transform• Discrete Wavelet Transform• Singular Value Decomposition• Piecewise Linear Approximation• Symbolic Approximation • Piecewise Aggregate Approximation • Adaptive Piecewise Constant Approximation

• Summary, Conclusions

Outline

What are Time Series?

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.. .. 24.6250 24.6750 24.6750 24.6250 24.6250 24.6250 24.6750 24.7500

A time series is a collection of observations made sequentially in time.

Note that virtually all similarity measurements, indexing and dimensionality reduction techniques discussed in this tutorial can be used with other data types.

Time Series are Ubiquitous! I

People measure things...• The presidents approval rating.• Their blood pressure.• The annual rainfall in Riverside.• The value of their Yahoo stock.• The number of web hits per second.

… and things change over time.

Thus time series occur in virtually every medical, scientific and businesses domain.

Time Series are Ubiquitous! II

A random sample of 4,000 graphics from 15 of the world’s newspapers published from 1974 to 1989 found that more than 75% of all graphics were time series (Tufte, 1983).

Defining the similarity between two time series is at the heart of most time series data mining applications/tasks

Thus time series similarity will be the primary focus of this tutorial.

10s = 0.5

c = 0.3

Database C

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Query Q(template)

Time Series Similarity

Classification

Clustering

Rule Discovery

Query by Content

Why is Working With Time Series so Difficult? Part I

1 Hour of EKG data: 1 Gigabyte.

Typical Weblog: 5 Gigabytes per week.

Space Shuttle Database: 158 Gigabytes and growing.

Macho Database: 2 Terabytes, updated with 3 gigabytes per day.

Answer: How do we work with very large databases?

Since most of the data lives on disk (or tape), we need a representation of the data we can efficiently manipulate.

Why is Working With Time Series so Difficult? Part II

The definition of similarity depends on the user, the domain and the task at hand. We need to be able to handle this subjectivity.

Answer: We are dealing with subjective notions of similarity.

Why is working with time series so difficult? Part III

Answer: Miscellaneous data handling problems.

• Differing data formats.• Differing sampling rates.• Noise, missing values, etc.

Similarity Matching Problem: Flavors 1

Database C

Query Q(template)

Given a Query Q, a reference database C and a distance measure, find the

Ci that best matches Q.

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C6 is the best match.

1: Whole Matching

Similarity matching problem: flavor 2

Database C

Query Q(template)

Given a Query Q, a reference database C and a distance measure, find the location that best matches Q.

2: Subsequence Matching

The best matching subsection.

Note that we can always convert subsequence matching to whole matching by sliding a window across the long sequence, and copying the window contents.

After all that background we might have forgotten what we are doing and why we care!

So here is a simple motivator and review..

You go to the doctor because of chest pains. Your ECG looks strange…

You doctor wants to search a database to find similar ECGS, in the hope that they will offer clues about your condition...

Two questions:• How do we define similar?• How do we search quickly?

Similarity is always subjective.(i.e. it depends on the application)

All models are wrong, but some are useful…

This slide was taken from: A practical Time-Series Tutorial with MATLAB—presented at ECLM

PAKDD 2005, by Michalis Vlachos.

Distance functions

Metric distancesi.e., those that satisfy the Triangle Inequality: d(x,z) ≤ d(x,y) + d(y,z)

E.g., Correlation, Euclidean Distance

• Assume: d(Q,bestMatch) = 35, d(Q,B) =150, d(A,B)=20: then• d(Q,A) ≥ d(Q,B) – d(B,A) thus

d(Q,A) ≥ 150 – 20 = 130We do not need to get A from disk

Non-Metric DistancesExamples:

• Time Warping

• LCSS: longest common sub-sequence

Preprocessing the data before distance calculations

If we naively try to measure the distance between two “raw” time series, we may get very unintuitive results.

This is because Euclidean distance is very sensitive to some distortions in the data. For most problems these distortions are not meaningful, and thus we can and should remove them.

In the next four slides, we discuss the 4 most common forms of distortion, and how to remove them.

• Offset Translation• Amplitude Scaling• Linear Trend• Noise

Transformation I: Offset Translation

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Q = Q - mean(Q)

C = C - mean(C)

D(Q,C)

D(Q,C)

Operations Performed:

Transformation II: Amplitude Scaling

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Q = (Q - mean(Q)) / std(Q)

C = (C - mean(C)) / std(C)D(Q,C)

Operations Performed:

Transformation III: Linear Trend

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Operation performed:

1. offset translation 2. amplitude scaling

2. offset translation 3. amplitude scaling

Operation Performed

1. Removed linear trend Removing linear trend :

Fit the best fitting straight line to the time series, then subtract that line from the time series.

Transformation IIII: Noise

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Q = smooth(Q)

C = smooth(C)D(Q,C)

The intuition behind removing noise is this.

Average each datapoints value with its neighbors.

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A Quick Experiment to Demonstrate the

Utility of Preprocessing the Data

Clustered using Euclidean distance

on the raw data

Clustered using Euclidean distance on the raw data, after removing noise, linear trend, offset translation and amplitude

scaling.

Summary of PreprocessingThe “raw” time series may have distortions which we should remove before clustering, classification etc.

Of course, sometimes the distortions are the most interesting thing about the data, the above is only a general rule.

We should keep in mind these problems as we consider the high level representations of time series which we will encounter later (Fourier transforms, Wavelets etc). Since these representations often allow us to handle distortions in elegant ways.

Fixed Time AxisSequences are aligned “one to one”.

“Warped” Time AxisNonlinear alignments are possible.

Dynamic Time Warping

Note: We will first see the utility of DTW, then see how it is calculated.

Utility of Dynamic Time Warping: Example II, Data Mining

Power-Demand Time Series.Each sequence corresponds to a week’s demand for power in a Dutch research facility in 1997 [van Selow 1999].

Monday

Tuesday

Wednesday

Thursday

Friday

Saturday

Sunday

Wednesday was a national holiday

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Hierarchical clustering with Euclidean Distance.

<Group Average Linkage>

The two 5-day weeks are correctly grouped.

Note however, that the three 4-day weeks are not clustered together.

Also, the two 3-day weeks are also not

clustered together.

Hierarchical clustering with Dynamic Time Warping.

<Group Average Linkage>

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The two 5-day weeks are correctly grouped.

The three 4-day weeks are clustered together.

The two 3-day weeks are also clustered

together.

Dynamic Time-Warping

• (how does it work?)The intuition is that we copy an element multiple times so as to achieve a better matching

Euclidean distance: d = 1T1 = [1, 1, 2, 2] | | | |T2 = [1, 2, 2, 2]

Warping distance: d = 0T1 = [1, 1, 2, 2] | | |T2 = [1, 2, 2, 2]

Q

C

n11

p

w1

wk

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Computing the Dynamic Time Warp Distance I

|n||p|

Note that the input sequences can be of

different lengths Q

C

Computing the Dynamic Time Warp Distance II

|n||p|

KwCQDTWK

k k1min),(

(i,j) = d(qi,cj) + min{ (i-1,j-1) , (i-1,j ) , (i,j-1) }

Q

C

Every possible mapping from Q to C can be represented as a warping path in the search matrix.

We simply want to find the cheapest one…

Although there are exponentially many such paths, we can find one in only quadratic time using dynamic programming.

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Time taken to create hierarchical clustering of power-demand time series.

• Time to create dendrogram using Euclidean Distance 1.2 seconds

• Time to create dendrogram using Dynamic Time Warping 3.40 hours

How to speed it up.

• Approach 1: Complexity is O(n2). We can reduce it to O(δn) simply by restricting the warping path.

• Approach 2: Approximate the time series with some compressed or downsampled representation, and do DTW on the new representation.

Complexity of Time Warping

Fast Approximations to Dynamic Time Warp Distance II

.. strong visual evidence to suggests it works well.

Good experimental evidence the utility of the approach on clustering, classification and query by content problems also has been demonstrated.

1.3 sec

22.7 sec

Weighted Distance Measures I

Weighting features is a well known technique in the machine learning community to improve classification

and the quality of clustering.

Intuition: For some queries different parts of the sequence are more important.

Note: In this example we are using a piecewise linear approximation of the data. We will learn more about this representation later.

Relevance Feedback for Time SeriesThe original query

The weigh vector. Initially, all weighs are the same.

One by one the 5 best matching sequences will appear, and the user will rank them from between very bad (-3) to very good (+3)

The initial query is executed, and the five best matches are shown (in the dendrogram)

Based on the user feedback, both the shape and the weigh vector of the query are changed.

The new query can be executed.The hope is that the query shape and weights will converge to the optimal query.

Two paper consider relevance feedback for time series.

L Wu, C Faloutsos, K Sycara, T. Payne: FALCON: Feedback Adaptive Loop for Content-Based Retrieval. VLDB 2000: 297-306

Motivating Example Revisited...

You go to the doctor because of chest pains. Your ECG looks strange…

You doctor wants to search a database to find similar ECGS, in the hope that they will offer clues about your condition...

Two questions:• How do we define similar?• How do we search quickly?

Indexing Time Series

• We have seen techniques for assessing the similarity of two time series.

• However we have not addressed the problem of finding the best match to a query in a large database...

• We need someway to index the data...• A topics extensively discussed in

topical literature that we will not discuss here for lack of time

Database C

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Query QFind shapes like this

In this DB

Compression – Dimensionality Reduction

• Project all sequences into a new space, and search this space instead.

Keogh, C

hakrabarti, Pazzani

& M

ehrotra KA

IS 2000

Yi &

Faloutsos VLD

B 2000

Keogh, C

hakrabarti, Pazzani

& M

ehrotra SIGM

OD

2001

Agraw

al, Faloutsos, &.

Swam

i. FOD

O 1993

Faloutsos, Ranganathan, &

Manolopoulos. SIG

MO

D

1994

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Chan &

Fu. ICD

E 1999

Korn, Jagadish &

Faloutsos. SIGM

OD

1997

Morinaka,

Yoshikawa, A

magasa, &

Uem

ura, PAK

DD

2001

DFT DWT SVD APCA PAA PLA

Compressed Representations … Compressed Representations

Fourier Transforms and Discrete Fourier Transforms: Represent functions as sums of a series of sine&cosine terms of increasing frequency.

Sampling the function at regular intervals

FFTs

Jean Fourier

1768-1830

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Discrete Fourier Transform I

Excellent free Fourier Primer

Hagit Shatkay, The Fourier Transform - a Primer'', Technical Report CS-95-37, Department of Computer Science, Brown University, 1995.

http://www.ncbi.nlm.nih.gov/CBBresearch/Postdocs/Shatkay/

Basic Idea: Represent the time series as a linear combination of sines and cosines, but keep only the first n/2 coefficients.

Why n/2 coefficients? Because each sine wave requires 2 numbers, for the phase (w) and amplitude or (A,B).

n

kkkkk twBtwAtC

1

))2sin()2cos(()(

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C

An Example of a Dimensionality Reduction

Technique 0.4995 0.5264 0.5523 0.5761 0.5973 0.6153 0.6301 0.6420 0.6515 0.6596 0.6672 0.6751 0.6843 0.6954 0.7086 0.7240 0.7412 0.7595 0.7780 0.7956 0.8115 0.8247 0.8345 0.8407 0.8431 0.8423 0.8387 …

RawData

The graphic shows a time series with 128 points.

The raw data used to produce the graphic is also reproduced as a column of numbers (just the first 30 or so points are shown).

n = 128

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C

. . . . . . . . . . . . . .

Dimensionality Reduction (cont.)

1.5698 1.0485 0.7160 0.8406 0.3709 0.4670 0.2667 0.1928 0.1635 0.1602 0.0992 0.1282 0.1438 0.1416 0.1400 0.1412 0.1530 0.0795 0.1013 0.1150 0.1801 0.1082 0.0812 0.0347 0.0052 0.0017 0.0002 ...

FourierCoefficients

0.4995 0.5264 0.5523 0.5761 0.5973 0.6153 0.6301 0.6420 0.6515 0.6596 0.6672 0.6751 0.6843 0.6954 0.7086 0.7240 0.7412 0.7595 0.7780 0.7956 0.8115 0.8247 0.8345 0.8407 0.8431 0.8423 0.8387 …

RawData

We can decompose the data into pure sine waves using the Discrete Fourier Transform (just the first few sine waves are shown).

The Fourier Coefficients are reproduced as a column of numbers (just the first 30 or so coefficients are shown).

Note that at this stage we have not done dimensionality reduction, we have merely changed the representation...

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C

An Example of a Dimensionality Reduction

Technique III 1.5698 1.0485 0.7160 0.8406 0.3709 0.4670 0.2667 0.1928

TruncatedFourier

Coefficients

C’

We have

discarded

of the data.16

15

1.5698 1.0485 0.7160 0.8406 0.3709 0.4670 0.2667 0.1928 0.1635 0.1602 0.0992 0.1282 0.1438 0.1416 0.1400 0.1412 0.1530 0.0795 0.1013 0.1150 0.1801 0.1082 0.0812 0.0347 0.0052 0.0017 0.0002 ...

FourierCoefficients

0.4995 0.5264 0.5523 0.5761 0.5973 0.6153 0.6301 0.6420 0.6515 0.6596 0.6672 0.6751 0.6843 0.6954 0.7086 0.7240 0.7412 0.7595 0.7780 0.7956 0.8115 0.8247 0.8345 0.8407 0.8431 0.8423 0.8387 …

RawData

… however, note that the first few sine waves tend to be the largest (equivalently, the magnitude of the Fourier coefficients tend to decrease as you move down the column).

We can therefore truncate most of the small coefficients with little effect.

n = 128N = 8Cratio = 1/16

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An Example of a Dimensionality Reduction

Technique IIII

SortedTruncated

FourierCoefficients

C’

1.5698 1.0485 0.7160 0.8406 0.3709 0.1670 0.4667 0.1928 0.1635 0.1302 0.0992 0.1282 0.2438 0.2316 0.1400 0.1412 0.1530 0.0795 0.1013 0.1150 0.1801 0.1082 0.0812 0.0347 0.0052 0.0017 0.0002 ...

FourierCoefficients

0.4995 0.5264 0.5523 0.5761 0.5973 0.6153 0.6301 0.6420 0.6515 0.6596 0.6672 0.6751 0.6843 0.6954 0.7086 0.7240 0.7412 0.7595 0.7780 0.7956 0.8115 0.8247 0.8345 0.8407 0.8431 0.8423 0.8387 …

RawData

1.5698 1.0485 0.7160 0.8406 0.2667 0.1928 0.1438 0.1416

Instead of taking the first few coefficients, we could take the best coefficients

This can help greatly in terms of approximation quality, but makes indexing hard (impossible?).

Note this applies also to Wavelets

Keogh, C

hakrabarti, Pazzani

& M

ehrotra KA

IS 2000

Yi &

Faloutsos VLD

B 2000

Keogh, C

hakrabarti, Pazzani

& M

ehrotra SIGM

OD

2001

Agraw

al, Faloutsos, &.

Swam

i. FOD

O 1993

Faloutsos, Ranganathan, &

Manolopoulos. SIG

MO

D

1994

0 20 40 60 80 100 120 0 20 40 60 80 100 120 0 20 40 60 80 100 120 0 20 40 60 80 100 1200 20 40 60 80 100 120 0 20 40 60 80 100 120

Chan &

Fu. ICD

E 1999

Korn, Jagadish &

Faloutsos. SIGM

OD

1997

Morinaka,

Yoshikawa, A

magasa, &

Uem

ura, PAK

DD

2001

DFT DWT SVD APCA PAA PLA

Compressed Representations … Compressed Representations

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DWT

Discrete Wavelet Transform I

Alfred Haar

1885-1933

Excellent free Wavelets Primer

Stollnitz, E., DeRose, T., & Salesin, D. (1995). Wavelets for computer graphics A primer: IEEE Computer Graphics and Applications.

Basic Idea: Represent the time series as a linear combination of Wavelet basis functions, but keep only the first N coefficients.

Although there are many different types of wavelets, researchers in time series mining/indexing generally use Haar wavelets.

Haar wavelets seem to be as powerful as the other wavelets for most problems and are very easy to code.

X = {8, 4, 1, 3}

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h1= 4 = mean(8,4,1,3) h2 = 2 = mean(8,4) - h1 h3 = 2 = (8-4)/2 h4 = -1 = (1-3)/2

h1 = 4

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h2 = 2 h3 = 2 h4 = -1 X = {8, 4, 1, 3}

I have converted a raw time series X = {8, 4, 1, 3}, into the Haar Wavelet representation H = [4, 2 , 2, 1]We can covert the Haar representation back to raw signal with no loss of information...

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Discrete Wavelet Transform II We have only considered one type of

wavelet, there are many others.Are the other wavelets better for indexing?

YES: I. Popivanov, R. Miller. Similarity Search Over Time Series Data Using Wavelets. ICDE 2002.

NO: K. Chan and A. Fu. Efficient Time Series Matching by Wavelets. ICDE 1999

I consider this an open question...

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Discrete Wavelet Transform III

Pros and Cons of Wavelets as a time series representation.

• Good ability to compress stationary signals.• Fast linear time algorithms for DWT exist.• Able to support some interesting non-Euclidean

similarity measures.

• Signals must have a length n = 2some_integer • Works best if N is = 2some_integer. Otherwise wavelets

approximate the left side of signal at the expense of the right side.• Cannot support weighted distance measures.

Open Question: We have only considered one type of wavelet, there are many others. Are the other wavelets better for indexing?

YES: I. Popivanov, R. Miller. Similarity Search Over Time Series Data Using Wavelets. ICDE 2002.

NO: K. Chan and A. Fu. Efficient Time Series Matching by Wavelets. ICDE 1999

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eigenwave 0

eigenwave 1

eigenwave 2

eigenwave 3

eigenwave 4

eigenwave 5

eigenwave 6

eigenwave 7

SVD

Singular Value Decomposition

Eugenio Beltrami

1835-1899

Camille Jordan (1838--1921)

James Joseph Sylvester 1814-1897

Basic Idea: Represent the time series as a linear combination of eigenwaves but keep only the first N coefficients.

SVD is similar to Fourier and Wavelet approaches is that we represent the data in terms of a linear combination of shapes (in this case eigenwaves).

SVD differs in that the eigenwaves are data dependent.

SVD has been successfully used in the text processing community (where it is known as Latent Semantic Indexing ) for many years—but it is computationally expensive

Good free SVD Primer

Singular Value Decomposition - A Primer. Sonia Leach

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eigenwave 0

eigenwave 1

eigenwave 2

eigenwave 3

eigenwave 4

eigenwave 5

eigenwave 6

eigenwave 7

SVD

Singular Value Decomposition (cont.)

How do we create the eigenwaves?

We have previously seen that we can regard time series as points in high dimensional space.

We can rotate the axes such that axis 1 is aligned with the direction of maximum variance, axis 2 is aligned with the direction of maximum variance orthogonal to axis 1 etc.

Since the first few eigenwaves contain most of the variance of the signal, the rest can be truncated with little loss.

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Piecewise Linear Approximation I

Basic Idea: Represent the time series as a sequence of straight lines.

If we have a n N-vector then:

If lines are connected, we can represent N/2 lines

If lines are disconnected, we can represent only N/3 lines

Personal experience on dozens of datasets suggest disconnected is better. Also only disconnected allows a lower bounding Euclidean approximation

Each line segment has • length • left_height (right_height can be inferred by looking at the next segment)

Each line segment has • length • left_height • right_height

Karl Friedrich Gauss

1777 - 1855

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Piecewise Linear Approximation II

How do we obtain the Piecewise Linear Approximation?

Optimal Solution is O(n2N), which is too slow for data mining.

A vast body on work on faster heuristic solutions to the problem can be classified into the following classes:• Top-Down O(n2N)• Bottom-Up O(n(1/CRatio))

• Sliding Window O(n(1/CRatio))• Other (genetic algorithms, randomized algorithms,

Bspline wavelets, MDL etc)

Recent extensive empirical evaluation of all approaches suggest that Bottom-Up is the best

approach overall.

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Piecewise Linear Approximation III

Pros and Cons of PLA as a time series representation.

• Good ability to compress natural signals.• Fast linear time algorithms for PLA exist.• Able to support some interesting non-Euclidean

similarity measures. Including weighted measures, relevance feedback, fuzzy queries…

• Already widely accepted in some communities (ie, biomedical)

• Not (currently) indexable by any data structure (but does allows fast sequential scanning).

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C U U C D C U D

C

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Symbolic Approximation

Key: C = ConstantU = UpD = Down

Basic Idea: Convert the time series into an alphabet of discrete symbols. Use string indexing techniques to manage the data.

Potentially an interesting idea, but all the papers thusfar are very ad hoc.

Pros and Cons of Symbolic Approximation as a time series representation.

• Potentially, we could take advantage of a wealth of techniques from the very mature field of string processing.

• There is no known technique to allow the support of Euclidean queries.

• It is not clear how we should discretize the times series (discretize the values, the slope, shapes? How big of an alphabet? etc)

Piecewise Aggregate Approximation I

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Given the reduced dimensionality representation we can calculate the approximate Euclidean distance (a lower bound)

Basic Idea: Represent the time series of n points as a sequence of box basis functions. Each box is the same length w (simple case: assume n multiple of w)

Independently introduced by:Keogh, Chakrabarti, Pazzani & Mehrotra, KAIS (2000) and Byoung-Kee Yi, Christos Faloutsos, VLDB (2000)

is the mean value for points in the ith segment

Piecewise Aggregate Approximation II

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• Extremely fast to calculate• As efficient as other approaches (empirically)• Support queries of arbitrary lengths• Can support any Minkowski metric• Supports non Euclidean measures• Supports weighted Euclidean distance• Simple! Intuitive!

• If visualized directly, looks ascetically unpleasing.

Pros and Cons of PAA as a time series representation.

Adaptive Piecewise Constant

Approximation I

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X

X

<cv1,cr1>

<cv2,cr2>

<cv3,cr3>

<cv4,cr4>

Basic Idea: Generalize PAA to allow the piecewise constant segments to have arbitrary lengths. Note that we now need 2 coefficients to represent each segment, its value and its length.

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Raw Data (Electrocardiogram)

Adaptive Representation (APCA)Reconstruction Error 2.61

Haar Wavelet Reconstruction Error 3.27

DFTReconstruction Error 3.11

The intuition is this, many signals have little detail in some places, and high detail in other places. APCA can adaptively fit itself to the data achieving better approximation.

Adaptive Piecewise Constant

Approximation II

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X

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<cv1,cr1>

<cv2,cr2>

<cv3,cr3>

<cv4,cr4>

The high quality of the APCA had been noted by many researchers. However it was believed that the representation could not be indexed because some coefficients represent values, and some represent lengths.

However an indexing method was discovered! (SIGMOD 2001 best paper award)

Unfortunately, it is non-trivial to understand and implement….

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SAX: Symbol Mapping• Each average value from the PAA vector is replaced by a

symbol from an alphabet• An alphabet size of 5 to 8 is recommended

– a,b,c,d,e– a,b,c,d,e,f– a,b,c,d,e,f,g– a,b,c,d,e,f,g,h

• Given an average value we need a symbol• This is achieved by using the normal distribution from

statistics– Because our input series is normalized we can use normal

distribution as the data model– We divide the area under the normal distribution into ‘a’

equal sized areas where a is the alphabet size– Each such area is bounded by breakpoints

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SAX Computation – in pictures

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C

C

0

-

-

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bbb

a

cc

c

a

baabccbc

This slide taken from Eamonn’s Tutorial on SAX

Conclusion

This is just an introduction, with many unavoidable omissions:• There are dozens of papers that offer new

distance measures. • Hidden Markov models do have a sound basis,

but don’t scale well. • Time series analysis remains a hot area of

research and the most recent papers have not been discussed here.

References

• Eamonn J. Keogh: Indexing and Mining Time Series Data. Encyclopedia of GIS 2008: 493-497.

• Jin Shieh, Eamonn J. Keogh: iSAX: indexing and mining terabyte sized time series. KDD 2008: 623-631.

• Jessica Lin, Michail Vlachos, Eamonn J. Keogh, Dimitrios Gunopulos: Iterative Incremental Clustering of Time Series. EDBT 2004: 106-122

• Chiu, E. Keogh, S. Lonardi, Probabilistic discovery of time series motifs, 9th ACM SIGKDD international conference, 2003, 493-498.

Discrete Fourier Transform II Pros and Cons of DFT as a time series

representation.

• Good ability to compress most natural signals.• Fast, off the shelf DFT algorithms. FFT O(nlog(n)).• (Weakly) able to support time warped queries.

• Difficult to deal with sequences of different lengths.• Cannot support weighted distance measures.

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0

1

2

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X

X'

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9