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Time Series I

SyllabusNov 4 Introduction to data mining

Nov 5 Association Rules

Nov 10, 14 Clustering and Data Representation

Nov 17 Exercise session 1 (Homework 1 due)

Nov 19 Classification

Nov 24, 26 Similarity Matching and Model Evaluation

Dec 1 Exercise session 2 (Homework 2 due)

Dec 3 Combining Models

Dec 8, 10 Time Series Analysis

Dec 15 Exercise session 3 (Homework 3 due)

Dec 17 Ranking

Jan 13 Review

Jan 14 EXAM

Feb 23 Re-EXAM

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Why deal with sequential data?• Because all data is sequential

• All data items arrive in the data store in some order

• Examples– transaction data– documents and words

• In some (or many) cases the order does not matter

• In many cases the order is of interest

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Time-series data: example

Financial time series

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Questions

• What is time series?

• How do we compare time series data?

• What is the structure of sequential data?

• Can we represent this structure compactly and accurately?

Time Series• A sequence of observations:

– X = (x1, x2, x3, x4, …, xn)

• Each xi is a real number

– e.g., (2.0, 2.4, 4.8, 5.6, 6.3, 5.6, 4.4, 4.5, 5.8, 7.5)

time axis

valu

e ax

is

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Time Series Databases• A time series is an ordered set of real numbers,

representing the measurements of a real variable at equal time intervals

– Stock prices– Volume of sales over time– Daily temperature readings– ECG data

• A time series database is a large collection of time series

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Time Series Problems

• The Similarity Problem

X = x1, x2, …, xn and Y = y1, y2, …, yn

• Define and compute Sim(X, Y) or Dist(X, Y)– e.g. do stocks X and Y have similar movements?

• Retrieve efficiently similar time series– Indexing for Similarity Queries

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Types of queries

• whole match vs subsequence match

• range query vs nearest neighbor query

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Examples

• Find companies with similar stock prices over a

time interval

• Find products with similar sell cycles

• Cluster users with similar credit card utilization

• Find similar subsequences in DNA sequences

• Find scenes in video streams

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day

$price

1 365

day

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1 365

day

$price

1 365

distance function: by expert

(e.g., Euclidean distance)

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Problems

• Define the similarity (or distance) function• Find an efficient algorithm to retrieve similar

time series from a database– (Faster than sequential scan)

The Similarity function depends on the Application

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Metric Distances

• What properties should a similarity distance have to allow (easy) indexing?

– D(A,B) = D(B,A) Symmetry – D(A,A) = 0 Constancy of Self-Similarity– D(A,B) >= 0 Positivity– D(A,B) D(A,C) + D(B,C) Triangular Inequality

• Some times the distance function that best fits an application is not a metric

• Then indexing becomes interesting and challenging

Euclidean Distance

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• Each time series: a point in the n-dim space

• Euclidean distance– pair-wise point distance

v1

v2

X = x1, x2, …, xn

Y = y1, y2, …, yn

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Euclidean modelQuery Q

n datapoints

S

Q

Euclidean Distance betweentwo time series Q = {q1, q2, …, qn} and X = {x1, x2, …, xn}

Distance

0.98

0.07

0.21

0.43

Rank

4

1

2

3

Database

n datapoints

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• Easy to compute: O(n)• Allows scalable solutions to other problems,

such as– indexing– clustering– etc...

Advantages

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• Query and target lengths should be equal!

• Cannot tolerate noise:– Time shifts– Sequences out of phase– Scaling in the y-axis

Disadvantages

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Limitations of Euclidean Distance

Euclidean DistanceSequences are aligned “one to one”.

“Warped” Time AxisNonlinear alignments are possible.

Q

Q

C

C

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Dynamic Time Warping [Berndt, Clifford, 1994]

• DTW allows sequences to be stretched along the time axis– Insert ‘stutters’ into a sequence– THEN compute the (Euclidean) distance

‘stutters’original

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Computation

)1,1(

),1(

)1,(

min),(

),(),(

jif

jif

jif

qpjif

MNfQPD

ji

dtw

q-stutterno stutter

p-stutter

• DTW is computed by dynamic programming• Given two sequences

– P = {p1, p2, …, pi}

– Q = {q1, q2, …, qj}

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DTW: Dynamic time warping (1/2)

• Each cell c = (i, j) is a pair of indices

whose corresponding values will be

computed, (xi–yj)2, and included in

the sum for the distance.

• Euclidean path:

– i = j always.

– Ignores off-diagonal cells.

X

Y

xi

yj

(x2–y2)2 + (x1–y1)2 (x1–y1)2

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(i, j)

DTW: Dynamic time warping (2/2)

• DTW allows any path.• Examine all paths:

• Standard dynamic programming to fill in the table.

• The top-right cell contains final result.

(i, j)(i-1, j)

(i-1, j-1) (i, j-1)

shrink x / stretch y

stretch x / shrink y

X

Y

a

b

• Warping path W: – set of grid cells in the time warping matrix

• DTW finds the optimum warping path W:– the path with the smallest matching score

Optimum warping path W(the best alignment) Properties of a DTW legal path

I. Boundary conditions

W1=(1,1) and WK=(n,m)

II. ContinuityGiven Wk = (a, b), then Wk-1 = (c, d), where a-c ≤ 1, b-d

≤ 1III. Monotonicity

Given Wk = (a, b), then Wk-1 = (c, d), where a-c ≥ 0, b-d ≥ 0

Properties of DTW

X

Y

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Properties of DTW

I. Boundary conditions

W1=(1,1) and WK=(n,m)

II. ContinuityGiven Wk = (a, b), then Wk-1 = (c, d), where a-c ≤ 1, b-d

≤ 1III. Monotonicity

Given Wk = (a, b), then Wk-1 = (c, d), where a-c ≥ 0, b-d ≥ 0 24

C. S. Myers and L. R. Rabiner. A comparative study of several dynamic time-warping algorithms for connected word recognition. The Bell System Technical Journal, 60(7):1389-1409, Sept. 1981.

Sakoe, H. and Chiba, S., Dynamic programming algorithm optimization for spoken word recognition, IEEE Transactions on Acoustics, Speech and Signal Processing, 26(1) pp. 43– 49, 1978, ISSN: 0096-3518

• Query and target lengths may not be of equal length

• Can tolerate noise:– time shifts– sequences out of phase– scaling in the y-axis

Advantages

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• Computational complexity: O(nm)

• May not be able to handle some types of noise...

• It is not metric (triangle inequality does not hold)

Disadvantages

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Sakoe-Chiba Band Itakura Parallelogram

r =

Global Constraints Slightly speed up the calculations and prevent pathological warpings A global constraint limits the indices of the warping path

wk = (i, j)k such that j-r i j+r Where r is a term defining allowed range of warping for a given point in a

sequence

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Complexity of DTW

• Basic implementation = O(n2) where n is the length of the sequences– will have to solve the problem for each (i, j) pair

• If warping window is specified, then O(nr)– only solve for the (i, j) pairs where | i – j | <= r

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Longest Common Subsequence Measures

(Allowing for Gaps in Sequences)

Gap skipped

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Longest Common Subsequence (LCSS)

ignore majority of noise

match

match

Advantages of LCSS:

A. Outlying values not matched

B. Distance/Similarity distorted less

Disadvantages of DTW:

A. All points are matched

B. Outliers can distort distance

C. One-to-many mapping

LCSS is more resilient to noise than DTW.

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Longest Common SubsequenceSimilar dynamic programming solution as DTW, but now we measure similarity not distance.

Can also be expressed as distance

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Similarity Retrieval

• Range Query– Find all time series X where

• Nearest Neighbor query– Find all the k most similar time series to Q

• A method to answer the above queries: – Linear scan

• A better approach – GEMINI [next time]

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Lower Bounding – NN search

Intuition Try to use a cheap lower bounding calculation as often as possible Do the expensive, full calculations when absolutely necessary

We can speed up similarity search by using a lower bounding function D: distance measure

LB: lower bounding function s.t.: LB(Q, X) ≤ D(Q, X)

Set best = ∞ For each Xi:

if LB(Xi, Q) < bestif D(Xi, Q) < best best = D(Xi, Q)

1-NN Search Using LB

We assume a database of time series: DB = {X1, X2, …, XN}

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Lower Bounding – NN search

Intuition Try to use a cheap lower bounding calculation as often as possible Do the expensive, full calculations when absolutely necessary

We can speed up similarity search by using a lower bounding function D: distance measure

LB: lower bounding function s.t.: LB(Q, X) ≤ D(Q, X)

Range Query Using LB

For each Xi:if LB(Xi, Q) ≤ ε

if D(Xi, Q) < ε report Xi

We assume a database of time series: DB = {X1, X2, …, XN}

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Problems• How to define Lower bounds for different distance

measures?

• How to extract the features? How to define the

feature space?– Fourier transform

– Wavelets transform

– Averages of segments (Histograms or APCA)

– Chebyshev polynomials

– .... your favorite curve approximation...

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Some Lower Bounds on DTW

A

B

C

D

Each sequence is represented by 4 features: <First, Last, Min, Max>

LB_Kim = maximum squared difference of the corresponding features

LB_Kim

max(Q)

min(Q)

LB_Yi

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LB_Keogh [Keogh 2004]

L

U

Q

U

LQ

C

Q

C

Q

Sakoe-Chiba Band

Itakura Parallelogram

Ui = max(qi-r : qi+r)Li = min(qi-r : qi+r)

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CU

LQ

CU

LQ

C

Q

C

Q

Sakoe-Chiba Band

Itakura Parallelogram

n

iiiii

iiii

otherwise

LcifLc

UcifUc

CQKeoghLB1

2

2

0

)(

)(

),(_

LB_Keogh

LB_Keogh

),(),(_ CQDTWCQKeoghLB

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LB_KeoghSakoe-Chiba

LB_KeoghItakura

LB_Yi

LB_Kim

…proportional to the length of gray lines used in the illustrations

Tightness of LB

nceDistaWarpTimeDynamicTruenceDistaWarpTimeDynamicofEstimateBoundLowerT

0 T 1The larger the

better

Lower Bounding

distanceQ

we want to find the 1-NN to our query data series, Q

Lower Bounding

distanceQ true S1

we compute the distance to the first data series in our dataset, D(S1,Q)

this becomes the best so far (BSF)

Lower Bounding

distanceQ true S1

BSF

LB S2

we compute the distance LB(S2,Q) and it is greater than the BSF

we can safely prune it, since D(S2,Q) LB(S2,Q)

Lower Bounding

distanceQ true S1

BSF

LB S2

we compute the distance LB(S3,Q) and it is smaller than the BSFwe have to compute D(S3,Q) LB(S3,Q), since it may still be smaller

than BSF

LB S3

Lower Bounding

distanceQ true S1

BSF

LB S2

it turns out that D(S3,Q) BSF, so we can safely prune S3

true S3

Lower Bounding

distanceQ true S1

BSF

LB S2true S3

Lower Bounding

distanceQ true S1

BSF

LB S2true S3

we compute the distance LB(S4,Q) and it is smaller than the BSFwe have to compute D(S4,Q) LB(S4,Q), since it may still be smaller

than BSF

LB S4

Lower Bounding

distanceQ true S1

BSF

LB S2true S3true S4

it turns out that D(S4,Q) BSF, so S4 becomes the new BSF

Lower Bounding

distanceQ true S1

S1 cannot be the 1-NN, because S4 is closer to Q

LB S2true S3true S4

BSF

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How about subsequence matching?

• DTW is defined for full-sequence matching:– All points of the query sequence are matched to all points of

the target sequence

• Subsequence matching:– The query is matched to a part (subsequence) of the target

sequence

Query sequence Data stream

X: long sequence

Q: short sequence

What subsequence of X is the best match for Q?

Subsequence Matching

X: long sequence

Q: short sequence

What subsequence of X is the best match for Q …such that the match ends at position j?

position j

J-Position Subsequence Match

X: long sequence

Q: short sequence

X: long sequence

Q: short sequence

position j

J-Position Subsequence Match

X: long sequence

Q: short sequence

Naïve Solution: DTWExamine all possible subsequences

X: long sequence

Q: short sequence

position j

J-Position Subsequence Match

Naïve Solution: DTWExamine all possible subsequences

X: long sequence

Q: short sequence

X: long sequence

Q: short sequence

Naïve Solution: DTWExamine all possible subsequences

X: long sequence

Q: short sequence

position j

J-Position Subsequence Match

Naïve Solution: DTWExamine all possible subsequences

X: long sequence

Q: short sequence

X: long sequence

Q: short sequence

Naïve Solution: DTWExamine all possible subsequences

X: long sequence

Q: short sequence

position j

J-Position Subsequence Match

Too costly!

Naïve Solution: DTWExamine all possible subsequences

X: long sequence

Q: short sequence

X: long sequence

Q: short sequence

Naïve Solution: DTWExamine all possible subsequences

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• Compute the time warping matrices starting from every database frame– Need O(n) matrices, O(nm) time per frame

Q

Xxtstartxtend

x1

Why not ‘naive’?

Capture the optimal subsequence starting

from t = tstart

n

m

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Key Idea• Star-padding

– Use only a single matrix (the naïve solution uses n matrices)

– Prefix Q with ‘*’, that always gives zero distance

– Instead of Q=(q1 , q2 , …, qm), compute distances with Q’

– O(m) time and space (the naïve requires O(nm))

(*)

),,,,('

0

210

q

qqqqQ m

SPRING: dynamic programming

Initialization Insert a “dummy” state ‘*’ at the beginning of the query ‘*’ matches every value in X with score 0

database sequence X

quer

y Q

* 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Computation Perform dynamic programming computation in a similar

manner as standard DTW

database sequence X

* 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

quer

y Q

SPRING: dynamic programming

(i, j)(i, j)(i-1, j)

(i-1, j-1) (i, j-1)

Q[1:i] is

matched

with X[s,j]

database sequence X

* 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

quer

y Q

i

js

For each (i, j): compute the j-position subsequence match of the first i

items of Q to X[s:j]

SPRING: dynamic programming

For each (i, j): compute the j-position subsequence match of the first i

items of Q to X[s:j] Top row: j-position subsequence match of Q for all j’s Final answer: best among j-position matches

Look at answers stored at the top row of the table

database sequence X

* 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

quer

y Q

SPRING: dynamic programming

database sequence X

* 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Subsequence vs. full matchingqu

ery

Q

Q

p1 pi pN

q1

qj

qM

Assume that the database is one very long sequence Concatenate all sequences into one sequence

O (|Q| * |X|) But can be computed faster by looking at only two

adjacent columns

Computational complexity

database sequence X

* 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

quer

y Q

STWM (Subsequence Time Warping Matrix)

• Problem of the star-padding: we lose the information about

the starting frame of the match

• After the scan, “which is the optimal subsequence?”

• Elements of STWM

– Distance value of each subsequence

– Starting position !!

• Combination of star-padding and STWM

– Efficiently identify the optimal subsequence in a stream

fashion

Up next…

• Time series summarizations

– Discrete Fourier Transform (DFT)

– Discrete Wavelet Transform (DWT)

– Piecewise Aggregate Approximation (PAA)

– Symbolic ApproXimation (SAX)

• Time series classification

– Lazy learners

– Shapelets