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High Performance Correlation Techniques For Time Series

High Performance Correlation Techniques For Time Series. Xiaojian Zhao Department of Computer Science Courant Institute of Mathematical Sciences New York university 25 Oct. 2004. Roadmap. Section 1: Introduction Motivation Problem Statement Section 2 : Background GEMINI framework

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High Performance Correlation Techniques For Time Series

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  1. High Performance Correlation Techniques For Time Series Xiaojian Zhao Department of Computer Science Courant Institute of Mathematical Sciences New York university 25 Oct. 2004

  2. Roadmap Section 1: Introduction • Motivation • Problem Statement Section 2 : Background • GEMINI framework • Random Projection • Grid Structure • Some Definitions • Naive method and Yunyue’s Approach Section 3 : Sketch based StatStream • Efficient Sketch Computation • Sketch technique as a filter • Parameter selection • Grid structure • System Integration Section 4 : Empirical Study Section 5 : Future Work Section 6 : Conclusion

  3. Section 1: Introduction

  4. Motivation • Stock prices streams • The New York Stock Exchange (NYSE) • 50,000 securities (streams); 100,000 ticks (trade and quote) • Pairs Trading, a.k.a. Correlation Trading • Query:“which pairs of stocks were correlated with a value of over 0.9 for the last three hours?” XYZ and ABC have been correlated with a correlation of 0.95 for the last three hours. Now XYZ and ABC become less correlated as XYZ goes up and ABC goes down. They should converge back later. I will sell XYZ and buy ABC …

  5. Correlated! Correlated! Online Detection of High Correlation

  6. Why speed is important • 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 a terabyte a day, magnetic resonance imagers give higher resolution images, etc.

  7. Problem Statement • Detect and report the correlation rapidly and accurately • Expand the algorithm into a general engine • Apply them in many practical application domains

  8. Big Picture time series 1 time series 2 time series 3 … time series n … sketch 1 sketch 2 … sketch n … Correlatedpairs Random Projection Grid structure

  9. Section 2: Background

  10. GEMINI framework* DFT, DWT, etc * Faloutsos, C., Ranganathan, M. & Manolopoulos, Y. (1994). Fast subsequence matching in time-series databases. In proceedings of the ACM SIGMOD Int'l Conference on Management of Data. Minneapolis, MN, May 25-27. pp 419-429.

  11. Goals of GEMINI framework • High performance Operations on synopses will save time such as distance computation • Guarantee no false negative Feature Space shrinks the original distances in the raw data space .

  12. Random Projection: Intuition • You are walking in a sparse forest and you are lost. • You have an outdated cell phone without a GPS. • You want to know if you are close to your friend. • You identify yourself at 100 meters from the pointy rock and 200 meters from the giant oak etc. • If your friend is at similar distances from several of these landmarks, you might be close to one another. • The sketches are the set of distances to landmarks.

  13. How to make Random Projection* • Sketch pool: A list of random vectors drawn from stable distribution (like the landmarks) • Project the time series into the space spanned by these random vectors • The Euclidean distance (correlation) between time series is approximated by the distance between their sketches with a probabilistic guarantee. • W.B.Johnson and J.Lindenstrauss. “Extensions of Lipshitz mapping into hilbert space”. Contemp. Math.,26:189-206,1984

  14. Random Projection X’ relative distances X’ current position Rocks, buildings… inner product Y’ relative distances Y’ current position random vector sketches raw time series

  15. Sketch Guarantees • Note: Sketches do not provide approximations of individual time series window but help make comparisons. Johnson-Lindenstrauss 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

  16. Sketches : Random Projection Why we use sketches or random projections? To reduce the dimensionality! For example: The original time series x is of the length 256, we may represent it with a sketch vector of length 30. First step to removing “the curse of dimensionality”

  17. Achliptas’s lemma • Dimitris Achliptas proved that Let P be an arbitrary set of n points in , represented as an matrix A. Given , let For integer , let R be a random matrix with R(i;j)= , where { } are independent random variables from either one of the following two probability distributions shown in next slide: *Idea from Dimitris Achlioptas, “Database-friendly Random Projections”, Proceedings of the twentieth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems

  18. Achliptas’s lemma or Let Let map the row of A to the row of E. With a probability at least , for all *Idea from Dimitris Achlioptas, “Database-friendly Random Projections”, Proceedings of the twentieth ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems

  19. Definition: Sketch Distance Note: DFT, DWT distance are analogous. For those measures, the difference between the original vectors is approximated by the difference between the first Fourier/Wavelet coefficients of those vectors.

  20. Empirical Study : Sketch Approximation

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

  22. Grid Structure

  23. Correlation and Distance • There is relationship between Euclidean distance and Pearson correlation • Normalization • dist2=2(1- correlation)

  24. How to compute the correlation efficiently? Goal: To find the most highly correlated stream pairs over sliding windows • Naive method • Statstream method • Our method

  25. Naïve Approach • Space and time cost • Space O(N) and time O(N2sw) • N : number of streams • sw : size of sliding window. • Let’s see Statstream approach

  26. Definitions: Sliding window and Basic window Time point Basic window Stock 1 Stock 2 Stock 3 …… Stock n Sliding window size=8 Basic window size=2 Sliding window Time axis

  27. StatStream Idea • Use Discrete Fourier Transform(DFT) to approximate correlation as in the GEMINI approach discussed earlier. • Every two minutes (“basic window size”), update the DFT for each time series over the last hour (“sliding window size”) • Use a grid structure to filter out unlikely pairs

  28. Basic window digests: sum DFT coefs Basic window digests: sum DFT coefs Time point Basic window StatStream: Stream synoptic data structure Sliding window

  29. Section 3: Sketch based StatStream

  30. Problem not yet solved • DFT approximates the price-like data type very well. Gives a poor approximation for returns(today’s price – yesterday’s price)/yesterday’s price. • Return is more like white noise which contains all frequency components. • DFT uses the first n (e.g. 10) coefficients in approximating data, which is insufficient in the case of white noise.

  31. Big Picture Revisited time series 1 time series 2 time series 3 … time series n … sketch 1 sketch 2 … sketch n … Correlatedpairs Random Projection Grid structure Random Projection: inner product between Data Vector and random vector

  32. How to compute the sketch efficiently We will not compute the inner product at each data point because the computation is expensive. A new strategy, in joint work with Richard Cole, is used to compute the sketch. Here the random variable will be drawn from:

  33. How to construct the random vector: Given time series , compute its sketch for a window of size sw=12. Partition to smaller basic windows of size bw = 4. The random vector within a basic window is R and a control vector b is used to determine which basic window will be multiplied with –1 or 1 (Why? Wait…) A final complete random vector may look like: Here bw=(1 1 -1 1) b=(1 -1 1) (1 1 -1 1; -1 -1 1 -1; 1 1 -1 1)

  34. Naive algorithm and hope for improvement 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) • There is redundancy in the second dot product given the first one. • We will eliminate the repeated computation to save time dot product xsk=r*x= x1+x2-x3+x4-x5-x6+x7-x8+x9+x10-x11+x12 With new data point arrival, such operations 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

  35. conv1:(1 1 -1 1 0 0 0 0) (x1,x2,x3,x4) conv2:(1 1 -1 1 0 0 0 0) (x5,x6,x7,x8) conv3:(1 1 -1 1 0 0 0 0) (x9,x10,x11,x12) Our algorithm(Pointwise version) Convolve with corresponding after padding with |bw| zeros. x4 x4+x3 Animation shows convolution in action: -x4+x3+x2 1 1 -1 1 0 0 0 0 x4-x3+x2+x1 x1 x2 x3 x4 x1 x2 x3 x4 x1 x2 x3 x4 x1 x2 x3 x4 x1 x2 x3 x4 x1 x2 x3 x4 x1 x2 x3 x4 x3-x2+x1 x2-x1 x1

  36. Our algorithm: example First Convolution Second Convolution Third Convolution 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 x4 x4+x3 x2+x3-x4 x1+x2-x3+x4 x1-x2+x3 x2-x1 x1 + +

  37. Our algorithm: example 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 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)+(x9+x10-x11+x12) b=( 1 -1 1) Second sliding window (Sk1 Sk5 Sk9)*(b1 b2 b3) * is inner product

  38. Our algorithm • The projection of a sliding window is decomposed into operations over basic windows • Each basic window is convolved with each random vector only once • We may provide the sketches incrementally starting from each data point. • There is no redundancy.

  39. x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 1 1 –1 1 1 1 –1 1 1 1 –1 1 x5+x6-x7+x8 x9+x10-x11+x12 x1+x2-x3+x4 Jump by a basic window (basic window version) • Or if time series are highly correlated between two consecutive data points, we may compute the sketch every other basic window. • That is, we update the sketch for each time series only when data of a complete basic window arrive.

  40. Online Version • We take the basic window version for instance • Review: To have the same baseline we normalize the time series within each siding window. • Challenge: The normalization of the time series change over each basic window

  41. Online Version • Its incremental computation nature results in a update of the average and variance whenever a new basic window enters • Do we have to compute the normalization and thus the sketch whenever a new basic window enters? • Of course not. Otherwise our algorithm will degrade into the trivial computation

  42. Sum of the whole sliding window Sum of the square of each data in a sliding window Sum of the whole basic window Sum of the square of each data in a basic window Dot Product of random vector with each basic window Online Version • Then how? After mathematical manipulation, we claim that we only need store and maintain the following quantities

  43. Performance comparison • Naïve algorithm For each datum and random vector 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 basic window and random vector (1) O(|sw|/|bw|) integer additions (2) O(|bw|) floating point operations

  44. Sketch distance filter quality • We may use the sketch distance to filter the unlikely data pairs • How accurate is it? • How is it compared to DFT and DWT distance in terms of the approximation ability?

  45. Empirical Study: Sketch sketch compared to DFT and DWT distance • Data length=256 • DFT: the first 14 DFT coefficients are used in the distance computation, • DWT: db2 wavelet is used with coefficient size=16 • Sketch: the random vector number is 64

  46. Empirical Comparison: DFT, DWT and Sketch

  47. Empirical Comparison : DFT, DWT and Sketch

  48. Use the sketch distance as a filter • We may compute the sketch distance: • c could be 1.2 or larger to reduce the number of false negatives. • Finally any possible data point will be double checked with the raw data.

  49. Use the sketch distance as a filter • But we will 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

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