Mining Time Series Data CS 240 B Notes

Mining Time Series Data CS 240 B Notes by Carlo Zaniolo UCLA CS Dept With Slides from: A Tutorial on Indexing and Mining Time Series Data ICDM '01 The 2001 IEEE International Conference on Data Mining November 29, San Jose Dr Eamonn Keogh Computer Science & Engineering Department University of California - Riverside, CA 92521 eamonn@cs. ucr. edu

Outline • 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

What are Time Series? 25. 1750 25. 2250 25. 2500 25. 2750 25. 3250 25. 3500 25. 4000 25. 3250 25. 2000 25. 1750 . . 24. 6250 24. 6750 24. 7500 A time series is a collection of observations made sequentially in time. 29 28 27 26 25 24 23 0 50 100 150 200 250 300 350 400 450 500 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! 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! 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).

Time Series Similarity Classification Clustering 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. Rule Discovery 10 s = 0. 5 c = 0. 3 Query by Content Query Q (template)

Why is Working With Time Series so Difficult? Part I Answer: How do we work with very large databases? w 1 Hour of EKG data: 1 Gigabyte. w Typical Weblog: Weblog 5 Gigabytes per week. w Space Shuttle Database: Database 158 Gigabytes and growing. w Macho Database: Database 2 Terabytes, updated with 3 gigabytes per day. 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 Answer: We are dealing with subjective notions of similarity. The definition of similarity depends on the user, the domain and the task at hand. We need to be able to handle this subjectivity.

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 Query Q (template) 1 6 2 7 3 8 4 9 5 10 1: Whole Matching C 6 is the best match. Database C Given a Query Q, a reference database C and a distance measure, find the Ci that best matches Q.

Similarity matching problem: flavor 2 Query Q (template) 2: Subsequence Matching Database C The best matching subsection. Given a Query Q, a reference database C and a distance measure, find the location that best matches Q. 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) z. 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 • Euclidean Distance • Correlation Triangle Inequality: d(x, z) ≤ d(x, y) + d(y, z) • Assume: d(Q, best. Match) = 20 and d(Q, B) =150 • Then, since d(A, B)=20 • d(Q, A) ≥ d(Q, B) – d(B, A) • d(Q, A) ≥ 150 – 20 = 130 We do not need to get A from disk Non-Metric • 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 4 slides I will discuss the 4 most common distortions, and how to remove them. • Offset Translation • Amplitude Scaling • Linear Trend • Noise

Transformation I: Offset Translation 3 3 2. 5 2 2 1. 5 1 1 0. 5 0 0 50 100 150 200 250 300 0 D(Q, C) 0 50 100 150 200 250 300 Q = Q - mean(Q) C = C - mean(C) D(Q, C) 0 50 100 150 200 250 300

Transformation II: Amplitude Scaling 0 100 200 300 400 500 600 700 800 900 1000 Q = (Q - mean(Q)) / std(Q) C = (C - mean(C)) / std(C) D(Q, C)

Transformation III: Linear Trend 5 4 After offset translation 3 2 And amplitude scaling 1 0 12 -1 10 -2 8 -3 0 6 20 40 60 80 100 120 140 160 180 200 4 2 0 5 -2 -4 0 20 40 60 80 100 120 140 160 180 200 4 Removed linear trend 3 2 The intuition behind removing linear trend is this. 1 0 -1 Fit the best fitting straight line to the time series, then subtract that line from the time series. -2 -3 0 20 40 60 80 100 120 140 160 180 200

Transformation IIII: Noise 8 8 6 6 4 4 2 2 0 0 -2 -2 -4 0 20 40 60 80 100 120 140 -4 0 20 40 60 Q = smooth(Q) The intuition behind removing noise is this. Average each datapoints value with its neighbors. C = smooth(C) D(Q, C) 80 100 120 140

A Quick Experiment to Demonstrate the Utility of Preprocessing the Data 3 2 9 6 8 Clustered using Euclidean distance on the raw data 5 7 4 1 9 8 7 5 6 4 3 2 1 Clustered using Euclidean distance on the raw data, after removing noise, linear trend, offset translation and amplitude scaling.

Summary of Preprocessing The “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.

Dynamic Time Warping Fixed Time Axis “Warped” Time Axis Sequences are aligned “one to one”. Nonlinear alignments are possible. 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]. Fri Sa Su Mo Tu W Th e u e day turd nda nd sd d r s n d ay ay esd ay y ay ay Wednesday was a national holiday

Hierarchical clustering with Euclidean Distance. <Group Average Linkage> 4 5 3 6 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. 7 2 1

Hierarchical clustering with Dynamic Time Warping. <Group Average Linkage> 6 4 7 5 The two 5 -day weeks are correctly grouped. The three 4 -day weeks are clustered together. 3 2 The two 3 -day weeks are also clustered together. 1

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

Computing the Dynamic Time Warp Distance I Q |p| C Q wk p C j 1 w 1 1 i n |n| Note that the input sequences can be of different lengths

Computing the Dynamic Time Warp Distance II Q |p| C Every possible mapping from Q to C can be represented as a warping path in the search matrix. Q wk p j w 1 1 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. C 1 |n| i n

Complexity of Time Warping 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(n 2). We can reduce it to O(n) 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.

Fast Approximations to Dynamic Time Warp Distance II 22. 7 sec 1. 3 sec. . 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.

Weighted Distance Measures I Intuition: For some queries different parts of the sequence are more important. Weighting features is a well known technique in the machine learning community to improve classification and the quality of clustering.

Relevance Feedback for Time Series The original query The weigh vector. Initially, all weighs are the same. Note: In this example we are using a piecewise linear approximation of the data. We will learn more about this representation later.

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

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—also it might not be applicable to data streams Query Q Find shapes like this In this DB

Compression – Dimensionality Reduction • Project all sequences into a new space, and search this space instead.

An Example of a Dimensionality Reduction Technique C 0 20 40 60 80 n = 128 100 120 140 Raw Data 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 … 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).

Dimensionality Reduction (cont. ) C 0 20 40 60 80 100 120 . . . 140 Raw Data Fourier Coefficients 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 … 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. . . We can decompose the data into 64 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. . .

An Example of a Dimensionality Reduction Technique III C C’ 0 20 40 60 80 100 We have discarded of the data. 120 140 Raw Data 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 … Truncated Fourier Coefficients 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. . . 1. 5698 1. 0485 0. 7160 0. 8406 0. 3709 0. 4670 0. 2667 0. 1928 n = 128 N=8 Cratio = 1/16 … 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.

An Example of a Dimensionality Reduction Technique IIII C C’ 0 20 40 60 80 100 120 140 Raw Data 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 … Sorted Truncated Fourier Coefficients 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. . . 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

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Discrete Fourier Transform I X Basic Idea: Represent the time series as a linear combination of sines and cosines, but keep only the first n/2 coefficients. X' 0 20 40 60 80 100 120 140 Why n/2 coefficients? Because each sine wave requires 2 numbers, for the phase (w) and amplitude (A, B). Jean Fourier 0 1768 -1830 1 2 3 4 5 Excellent free Fourier Primer 6 7 8 9 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/

Discrete Fourier Transform II X X' 0 20 40 60 80 100 120 140 Pros and Cons of DFT as a time series representation. • Good ability to compress most natural signals. • Fast, off the shelf DFT algorithms exist. O(nlog(n)). • (Weakly) able to support time warped queries. 0 1 2 3 • Difficult to deal with sequences of different lengths. • Cannot support weighted distance measures. 4 5 6 7 Note: The related transform DCT, uses only cosine basis functions. It does not seem to offer 8 any particular advantages over DFT. 9

Discrete Wavelet Transform I X Basic Idea: Represent the time series as a linear combination of Wavelet basis functions, but keep only the first N coefficients. X' DWT 0 20 40 60 80 100 120 140 Haar 0 Although there are many different types of wavelets, researchers in time series mining/indexing generally use Haar wavelets. Alfred Haar 1885 -1933 Haar 1 Haar 2 Haar 3 Haar wavelets seem to be as powerful as the other wavelets for most problems and are very easy to code. Haar 4 Haar 5 Haar 6 Haar 7 Excellent free Wavelets Primer Stollnitz, E. , De. Rose, T. , & Salesin, D. (1995). Wavelets for computer graphics A primer: IEEE Computer Graphics and Applications.

X = {8, 4, 1, 3} h 1= 4 = mean(8, 4, 1, 3) h 2 = mean(8, 4) - h 1 h 3 = 2 = (8 -4)/2 h 4 = -1 = (1 -3)/2 8 7 6 5 4 3 2 1 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. . . h 1 = 4 8 7 6 5 4 3 2 1 h 2 = 2 h 3 = 2 h 4 = -1 X = {8, 4, 1, 3}

Discrete Wavelet Transform III X X' DWT 0 20 40 60 80 100 120 140 Haar 0 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. • Works best if N is = 2 some_integer. Otherwise wavelets approximate the left side of signal at the expense of the right side. Haar 1 • Cannot support weighted distance measures. Haar 2 Haar 3 Haar 4 Haar 5 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. Haar 6 NO: K. Chan and A. Fu. Efficient Time Series Matching by Wavelets. ICDE 1999 Haar 7 Obviously, this question still open. . .

Singular Value Decomposition X X' SVD 0 20 40 60 80 100 120 140 eigenwave 0 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 James Joseph Sylvester 1814 -1897 approaches is that we represent the data in terms of a linear combination of shapes (in this case eigenwaves). eigenwave 1 eigenwave 2 eigenwave 3 eigenwave 4 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 Symantec Indexing ) for many years—but it is computationally expensive Camille Jordan (1838 --1921) eigenwave 5 Good free SVD Primer eigenwave 6 Singular Value Decomposition - A Primer. Sonia Leach eigenwave 7 Eugenio Beltrami 1835 -1899

Singular Value Decomposition (cont. ) X X' SVD 0 20 40 60 80 100 120 140 eigenwave 1 eigenwave 2 eigenwave 3 eigenwave 4 eigenwave 5 eigenwave 6 eigenwave 7 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.

Piecewise Linear Approximation I X Basic Idea: Represent the time series as a sequence of straight lines. X' Karl Friedrich Gauss 1777 - 1855 0 20 40 60 80 100 120 140 Lines could be connected, in which case we are allowed N/2 lines If lines are disconnected, we are allowed 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

Piecewise Linear Approximation II X X' 0 20 40 60 80 100 120 140 How do we obtain the Piecewise Linear Approximation? Optimal Solution is O(n 2 N), 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--CRatio denotes the compression ratio: • Top-Down O(n 2 N) • Bottom-Up O(n/CRatio) • Sliding Window O(n/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.

Piecewise Linear Approximation III Pros and Cons of PLA as a time series representation. X X' 0 20 40 60 80 100 120 140 • 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).

Basic Idea: Convert the time series into an alphabet of discrete symbols. Use string indexing techniques to manage the data. Symbolic Approximation X X' Potentially an interesting idea, but all the papers thusfar are very ad hoc. CUUCDCUD 0 20 40 60 80 100 120 C 140 0 U 1 U 2 C 3 D • Potentially, we could take advantage of a wealth of techniques from the very mature field of string processing. 4 C Key: C = Constant U = Up D = Down Pros and Cons of Symbolic Approximation as a time series representation. 5 U 6 D 7 • 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 Basic Idea: Represent the time series as a sequence of box basis functions. Note that each box is the same length. X X' 0 20 40 60 80 100 120 140 x 1 x 2 Given the reduced dimensionality representation we can calculate the approximate Euclidean distance as. . . x 3 x 4 x 5 x 6 Independently introduced by two authors x 7 Keogh, Chakrabarti, Pazzani & Mehrotra, KAIS (2000) x 8 Byoung-Kee Yi, Christos Faloutsos, VLDB (2000)

Piecewise Aggregate Approximation II X X' 0 20 40 60 80 100 120 140 X 1 Pros and Cons of PAA as a time series representation. • 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! X 2 X 3 X 4 X 5 X 6 X 7 X 8 • If visualized directly, looks ascetically unpleasing.

Adaptive Piecewise Constant Approximation I 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. X Raw Data (Electrocardiogram) X Adaptive Representation (APCA) 0 20 40 60 80 100 120 Reconstruction Error 2. 61 140 Haar Wavelet Reconstruction Error 3. 27 <cv 1, cr 1> <cv 2, cr 2> DFT Reconstruction Error 3. 11 <cv 3, cr 3> <cv 4, cr 4> 0 50 100 150 200 250 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 X X 0 20 40 60 80 100 120 140 <cv 1, cr 1> <cv 2, cr 2> <cv 3, cr 3> <cv 4, cr 4> 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 implement….

Adaptive Piecewise Constant Approximation • Pros and Cons of APCA as a time series representation. X X 0 20 40 60 80 100 120 140 <cv 1, cr 1> <cv 2, cr 2> • Fast to calculate O(n). • More efficient as other approaches (on some datasets). • Support queries of arbitrary lengths. • Supports non Euclidean measures. • Supports weighted Euclidean distance. • Support fast exact queries , and even faster approximate queries on the same data structure. <cv 3, cr 3> <cv 4, cr 4> • Somewhat complex implementation. • If visualized directly, looks ascetically unpleasing.

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.