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1 High Performance Algorithms for Multiple Streaming Time Series Xiaojian Zhao Advisor: Dennis Shasha Department of Computer Science Courant Institute of Mathematical Sciences New York University Jan. 10 2006

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2 Roadmap: Motivation Incremental Uncooperative Time Series Correlation Incremental Matching Pursuit (MP) (optional) Future Work and Conclusion

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3 Motivation (1) Financial time series streams are watched closely by millions of traders. Which pairs of stocks were correlated with a value of over 0.9 for the last three hours? Report this information every half hour(Incremental pairwise correlation) “Which pairs of stocks were correlated with a value of over 0.9 for the last three hours? Report this information every half hour” (Incremental pairwise correlation) How to form a portfolio consisting of a small set of stocks which replicates the market? Update it every hour” (Incremental matching pursuit) “How to form a portfolio consisting of a small set of stocks which replicates the market? Update it every hour” (Incremental matching pursuit)

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4 Motivation (2) As processors speed up, algorithmic efficiency no longer matters … one might think. True if problem sizes stay same. But they don’t. As processors speed up, sensors improve Satellites spewing out more data a day Magnetic resonance imagers give higher resolution images, etc.

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5 High performance incremental algorithms Incremental Uncooperative Time Series Correlation Monitor and report the correlation information among all time series incrementally (e.g. every half hour) Improve the efficiency from quadratic to super-linear Incremental Matching Pursuit (MP) Monitor and report the approximation vectors of matching pursuit incrementally (e.g. every hour) Improve the efficiency significantly

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6 Incremental Uncooperative Time Series Correlation

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7 Problem statement: Detect and report the correlation incrementally and rapidly Extend the algorithm into a general engine Apply it in practical application domains

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8 Online detection of high correlation Correlated!

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9 Pearson correlation and Euclidean distance Normalized Euclidean distance Pearson correlation Normalization dist 2 =2(1- correlation) From now on, we will not differentiate between correlation and Euclidean distance

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10 Naïve approach: pairwise correlation Given a group of time series, compute the pairwise correlation Time O(WN 2 ), where N : number of streams W: window size (e.g. in the past one hour) Let’s see high performance algorithms!

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11 Technical review Framework: GEMINI Tools: Data Reduction Techniques Deterministic Orthogonal vs. Randomized Fourier Transform, Wavelet Transform, and Random Projection Target: Various Data Cooperative vs. Uncooperative

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12 GEMINI Framework* * Faloutsos, C., Ranganathan, M. & Manolopoulos, Y. (1994). Fast subsequence matching in time-series databases,. SIGMOD, 1994 Data reduction, e.g. DFT, DWT, SVD

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13 GEMINI: an example Objective: find the nearest neighborhood (L 2 -norm) of each time series. Compute the Fourier Transform over each of them, e.g. X and Y; yield two coefficient vectors X f and Y f X f =(a 1, a 2, …a k ) and Y f =(b 1, b 2, …b k ) Original distance vs. coefficient distance (Parseval's Theorem) Because, for some data types, energy concentrates on first a few frequency components, coefficient distance can work as a very good filter and at the same time guarantee no false negatives They may be stored in a tree or grid structure

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14 DFT on random walk

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15 Review: DFT/DWT vs. Random Projection Fourier Transform, Wavelet Transform and SVD A set of orthogonal base (deterministic) Based on Parseval's Theorem Random Projection A set of random base (non-deterministic) Based on Johnson-Lindenstrauss (JL) Lemma Orthogonal BaseRandom Base

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16 Review : Random Projection: Intuition You are walking in a sparse forest and you are lost. You have an outdated cell phone without a GPS (w/o latitude&altitude). You want to know if you are close to your friend. You identify yourself at 100 meters from Bestbuy and 200 meters from a silver building etc. If your friend is at similar distances from several of these landmarks, you might be close to one another. Random projections are analogous to these distances to landmarks.

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17 inner product random vector sketches time series Random Projection Sketch: A vector of output returned by random projection

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18 Review: Sketch Guarantees * Johnson-Lindenstrauss ( JL) Lemma: For any and any integer n, let k be a positive integer such that Then for any set V of n points in, there is a map such that for all Further this map can be found in randomized polynomial time W.B.Johnson and J.Lindenstrauss. “Extensions of Lipshitz mapping into hilbert space”. Contemp. Math.,26:189-206,1984

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19 Empirical study : sketch approximation Time series length=256 and sketch size=30

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20 Empirical study : sketch approximation

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21 Empirical study: sketch distance/real distance Sketch=30 Sketch=80 Sketch=1000

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22 Data classification Cooperative Time series exhibiting a fundamental degree of regularity, allowing them to be represented by the first few coefficients in the spectral space with little loss of information Example: Stock Price (random walk) Tools: Fourier Transform, Wavelet Transform, SVD Uncooperative Time series whose energy is not concentrated in only a few frequency components, e.g. Example: Stock Return (= ) Tool: Random Projection

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23 DFT on random walk and white noise Cooperative Uncooperative

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24 Approximation Power: SVD Distance vs. Sketch Distance Note: SVD is superior to DFT and DWT in approximation power. But all of them are all bad for uncooperative data. Here sketch size = 32 and SVD coefficient number =30

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25 Our new algorithm * The big picture of the system Structured random vector (New) Compute sketch by structured convolution (New) Optimize in the parameter space (New) Empirical study Richard Cole, Dennis Shasha and Xiaojian Zhao. “Fast Window Correlations Over Uncooperative Time Series”. SIGKDD 2005

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26 Big Picture Random Projection time series 1 time series 2 time series 3 … time series n … sketch 1 sketch 2 … sketch n … Grid structure Correlated pairs Data ReductionFiltering

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27 Our objective reminded Monitor and report the correlation periodically e.g. “every half hour” We chose Random Projection as a means to reduce the data dimension The time series needs to be looked at in a time window. This time window should slide forward as time goes on.

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28 Definitions: Sliding window and Basic window Time axis … Stock 1 Stock 2 Stock 3 Stock n Sliding window (sw) Sliding window size=8 Basic window size=2 Basic window (bw) Tim e poin t Example Example: Every half hour (bw) report the correlation of the last three hours (sw)

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29 Random vector and naïve random projection Choose randomly sw random numbers to form a random vector R=(r 1, r 2, r 3, r 4, r 5, r 6, r 7, r 8, r 9, r 10, r 11, r 12 ) Inner product starts from each data point X sk1 =(x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12 )*R X sk2 =(x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12 x 13 )*R X sk3 =(x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12 x 13 x 14 )*R …… We improve it in two ways Partition a random vector of length sw into several basic windows Use convolution instead of inner product

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30 How to construct a random vector Construct a random vector of 1/-1 of length sw. Suppose sliding window size=12, and basic window size=4 The random vector within a basic window is A control vector A final complete random vector for a sliding window may look like: (1 1 -1 1; -1 -1 1 -1; 1 1 -1 1) Here R bw =(1 1 -1 1) b=(1 -1 1) R bw -R bw R bw

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31 Naive algorithm and hope for improvement There is redundancy in the second dot product given the first one. We will eliminate the repeated computation to save time dot product r=( 1 1 -1 1 ; -1 -1 1 -1; 1 1 -1 1 ) x=(x1 x2 x3 x4; x5 x6 x7 x8; x9 x10 x11 x12) xsk=r*x= x1+x2-x3+x4-x5-x6+x7-x8+x9+x10-x11+x12 With new data point arrival, this operation will be done again r= ( 1 1 -1 1 ; -1 -1 1 -1; 1 1 -1 1 ) x’=(x5 x6 x7 x8 ; x9 x10 x11 x12; x13 x14 x15 x16) xsk=r*x’= x5+x6-x7+x8-x9-x10+x11+x12+x13+x14+x15- x16 *

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32 conv1:(1 1 -1 1 0 0 0) (x1,x2,x3,x4) conv2:(1 1 -1 1 0 0 0) (x5,x6,x7,x8) conv3:(1 1 -1 1 0 0 0) (x9,x10,x11,x12) Our algorithm ● All the operations are over the basic window; ● Pad with |bw-1| zeros, then convolve with X bw Animation shows convolution in action: 1 1 -1 1 0 0 0 x1 x2 x3 x4 x4 x4+x3 -x4+x3+x2 x4-x3+x2+x1 x3-x2+x1 x2-x1 x1 x1 x2 x3 x4

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33 Our algorithm: example + First ConvolutionSecond ConvolutionThird Convolution x4 x4+x3 x2+x3-x4 x1+x2-x3+x4 x1-x2+x3 x2-x1 x1 x8 x8+x7 x6+x7-x8 x5+x6-x7+x8 x5-x6+x7 x6-x5 x5 x12 x12-x11 x10+x11-x12 x9+x10-x11+x12 x9-x10+x11 x10-x9 x9 + xsk1= (x1+x2-x3+x4)-(x5+x6-x7+x8)+(x9+x10-x11+x12) xsk2=(x2+x3-x4+x5)-(x6+x7-x8+x9)+(x10+x11-x12+x13)

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34 Our algorithm: example (Sk1 Sk5 Sk9)*(b1 b2 b3) * is inner product sk2=(x2+x3-x4) + (x5) sk6=(x6+x7-x8) + (x9) sk10=(x10+x11-x12) + (x13) Then sum up and we have xsk2=(x2+x3-x4+x5)-(x6+x7-x8+x9)+(x10+x11-x12+x13) b=( 1 -1 1) sk1=(x1+x2-x3+x4) sk5=(x5+x6-x7+x8) sk9=(x9+x10-x11+x12) xsk1= (x1+x2-x3+x4)-(x5+x6-x7+x8)+(x9+x10-x11+x12) b= ( 1 -1 1) First sliding window Second sliding window

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35 Basic window version Or if time series are highly correlated between two consecutive data points, we may compute the sketch every basic window. That is, we update the sketch for each time series only when data of a complete basic window arrive. No convolution, only inner product. 1 1 –1 1 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 1 1 –1 1 x1+x2-x3+x4 x5+x6-x7+x8x9+x10-x11+x12

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36 Overview of our new algorithm The projection of a sliding window is decomposed into operations over basic windows Each basic window is convolved/inner product with each random vector only once We may provide the sketches starting from each data point or starts from the beginning of each basic window. There is no redundancy.

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37 Performance comparison Naïve algorithm For each datum and random vector (1) O(|sw|) integer additions Pointwise version Asymptotically for each datum and random vector (1) O(|sw|/|bw|) integer additions (2) O(log |bw|) floating point operations (use FFT in computing convolutions) Basic window version Asymptotically for each datum and random vector O(|sw|/|bw| 2 ) integer additions

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38 Big picture revisited Random Projection time series 1 time series 2 time series 3 … time series n … sketch 1 sketch 2 … sketch n … Grid structure Correlated pairs Filtering So far we reduce the data dimension efficiently. Next, how can it be used as a filter?

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39 How to use the sketch distance as a filter Naive method: compute the sketch distance: Being close by sketch distance are likely to be close by original distance (JL Lemma) Finally any close data pair will be double checked with the original data.

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40 Use the sketch distance as a filter But we do not use it, why? Expensive. Since we still have to do the pairwise comparison between each pair of stocks which is, k is the size of the sketches, e.g. typically 30, 40, etc Let’s see our new strategy

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41 Our method: sketch unit distance Given sketches: If f distance chunks have we may say where: f: 30%, 40%, 50%, 60% … c: 0.8, 0.9, 1.1… We have

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42 Further: sketch groups We may compute the sketch group: For example If f sketch groups have we may say Remind us of a grid structure

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43 Grid structure To avoid checking all pairs, we can use a grid structure and look in the neighborhood, this will return a super set of highly correlated pairs. The data labeled as “potential” will be double checked using the raw data vectors.

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44 Optimization in parameter space We will choose the best one to be applied to the practical data. But how? --- an engineering problem Combinatorial Design (CD) Bootstrapping How to choose the parameters g, c, f, N? N: total number of the sketches g: group size c: the factor of distance f: the fraction of groups which are necessary to claim that two time series are close enough Now, Let’s put all together.

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45 Inner product with random vectors r1,r2,r3,r4,r5,r6 X Y Z

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46 Grid structure

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47 Empirical study: various data sources Cstr: Continuous stirred tank reactor Fortal_ecg: Cutaneous potential recordings of a pregnant woman Steamgen: Model of a steam generator at Abbott Power Plant in Champaign IL Winding: Data from a test setup of an industrial winding process Evaporator: Data from an industrial evaporator Wind: Daily average wind speeds for 1961-1978 at 12 synoptic meteorological stations in the Republic of Ireland Spot_exrates: The spot foreign currency exchange rates EEG: Electroencepholgram

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48 Empirical study: performance comparison Sliding window=3616, basic window=32 and sketch size=60

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49 Section conclusion How to perform data reduction over uncooperative time series efficiently in contrast to well-established methods for cooperative data How to cope with middle-size sketch vectors systematically. Sketch vector partition, grid structure Parameter space optimization by combinatorial design and bootstrapping Many ideas can be extended to other applications

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50 Incremental Matching Pursuit (MP)

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51 Problem Statement: Imagine a scenario where a group of representative stocks will be chosen to form an index e.g. for the Standard and Poor’s (S&P) 500. Target vector: The summation of all the vectors weighted by their capitalization. Candidate pool: All the stock price vectors in the market Objective: Find from candidate pool a small group of vectors representing the target vectors

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52 1.Set i=1; 2.Search the pool V and find the vector v i whose angle with respect to target vector v t is maximal; 3.Compute the residue r = v t -c i v i where c i = ; V A = V A {v i } ; 4.If r < error tolerance, then terminate and return V A ; 5.Else set i = i + 1 and v t = r, go back to 2 ; Vanilla Matching Pursuit (MP) Greedily select a linear combination of vectors from a dictionary to approximate a target vector

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53 v1v1 v2v2 v3v3 vtvt

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54 The incremental setting * Time granularity revisited Recomputing the representative vectors entirely for each sliding window is wasteful since there may be a trend between consecutive sliding windows Basic window=a sequence of unit time points Sliding window=several consecutive basic windows Sliding window “slides” once per basic window Xiaojian Zhao and Xin Zhang and Tyler Neylon and Dennis Shasha. “Incremental Methods for Simple Problems in Time Series: algorithms and experiments”, IDEAS 2005

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55 First idea: reuse vectors The representative vectors may change only slightly in both components and their order True only if basic window is sufficiently small e.g. 2, 3 time points However, any newly introduced representative vector may alter the entire tail of the approximation path The relative importance of the same representative vector may differ a lot from one sliding window to the next

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56 Two insightful observations The representative vectors are likely to remain the same within a few sliding windows, though the order may change The vector of angles keeps quite consistent, i.e. (,,,…). Here is the cosine of angle between the i th residue and the selected vector at that round. An example is (0.9, 0.8, 0.7, 0.7, 0.6, 0.6, 0.6,…..)

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57 Angle space exploration ( ) Whenever a vector is found whose is larger than some threshold, choose that vector. If there is no such vector, the vector with largest is selected as the representative vector at this round.

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58 Second idea: cache good vectors Those representative vectors appearing in the last several sliding windows form a cache C The search for a representative vector starts from C. If not found then go to whole pool V Works well in practice.

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59 Empirical study: time comparison

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60 Empirical study: approximation power comparison

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61 Future Work and Conclusion

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62 Future work: Anomaly Detection Measure the relative distance of each point from its nearest neighbors Our approach may serve as a monitor by reporting those points far from any normal points

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63 Conclusion 1.Motivation 2.Introduce the concept of cooperative vs. uncooperative time series 3.Propose a set of strategies dealing with different data (Random projection, Structured Convolution, Combinatorial Design, Bootstrapping, Grid Structure) 4.Explore various incremental schemes Filter away obvious irrelevancies Reuse previous results. 5.Future Work

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64 Thanks a lot!

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