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Mining Time Series Data CS240B Notes by Carlo Zaniolo UCLA CS Dept 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 Riverside,CA 92521 eamonn@cs.ucr.edu With Slides from:

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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

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What are Time Series? 050100150200250300350400450500 23 24 25 26 27 28 29 25.1750 25.2250 25.2500 25.2750 25.3250 25.3500 25.4000 25.3250 25.2250 25.2000 25.1750.. 24.6250 24.6750 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.

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Time Series are Ubiquitous 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.

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Time Series are Ubiquitous 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).

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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. 10 s = 0.5 c = 0.3 Query Q (template) Time Series Similarity Classification Clustering Rule Discovery Query by Content

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Why is Working With Time Series so Difficult? Part I 1 Hour of EKG data: w 1 Hour of EKG data: 1 Gigabyte. Typical Weblog w Typical Weblog: 5 Gigabytes per week. Space Shuttle Database w Space Shuttle Database: 158 Gigabytes and growing. Macho Database w 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.

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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.

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Why is working with time series so difficult? Part III Answer: Miscellaneous data handling problems. Differing data formats. Differing data formats. Differing sampling rates. Differing sampling rates. Noise, missing values, etc. Noise, missing values, etc.

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

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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.

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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?

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

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Distance functions Metric Euclidean Distance Correlation Triangle Inequality: d(x,z) ≤ d(x,y) + d(y,z) Assume: d(Q,bestMatch) = 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

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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

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Transformation I: Offset Translation 050100150200250300 0 0.5 1 1.5 2 2.5 3 050100150200250300 0 0.5 1 1.5 2 2.5 3 050100150200250300 050100150200250300 Q = Q - mean(Q) C = C - mean(C) D(Q,C)

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Transformation II: Amplitude Scaling 01002003004005006007008009001000 01002003004005006007008009001000 Q = (Q - mean(Q)) / std(Q) C = (C - mean(C)) / std(C) D(Q,C)

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Transformation III: Linear Trend 020406080100120140160180200 -4 -2 0 2 4 6 8 10 12 020406080100120140160180200 -3 -2 0 1 2 3 4 5 020406080100120140160180200 -3 -2 0 1 2 3 4 5 After offset translation And amplitude scaling Removed linear trend The intuition behind removing linear trend is this. Fit the best fitting straight line to the time series, then subtract that line from the time series.

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Transformation IIII: Noise 020406080100120140 -4 -2 0 2 4 6 8 020406080100120140 -4 -2 0 2 4 6 8 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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1 2 3 4 6 5 7 8 9 1 4 7 5 8 6 9 2 3 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.

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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.

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Fixed Time Axis Sequences are aligned “one to one”. “ Warped” Time Axis Nonlinear alignments are possible. Dynamic Time Warping Note: We will first see the utility of DTW, then see how it is calculated.

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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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1 2 7 6 3 5 4 Hierarchical clustering with Euclidean Distance. 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.

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Hierarchical clustering with Dynamic Time Warping. 1 2 3 5 7 4 6 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.

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

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

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Computing the Dynamic Time Warp Distance II |n||n| |p||p| 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 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. Complexity of Time Warping

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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

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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.

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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 Series The original query The weigh vector. Initially, all weighs are the same.

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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)

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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

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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?

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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

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

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020406080100120140 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 … Raw Data 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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020406080100120140 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... 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 … Raw Data 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...

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020406080100120140 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 Truncated Fourier Coefficients C’ We have discarded of the data. 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... 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 … Raw Data … 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 = 128 N = 8 C ratio = 1/16

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020406080100120140 C An Example of a Dimensionality Reduction Technique IIII Sorted Truncated Fourier Coefficients 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... 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 … Raw Data 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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Keogh, Chakrabarti, Pazzani & Mehrotra KAIS 2000 Yi & Faloutsos VLDB 2000 Keogh, Chakrabarti, Pazzani & Mehrotra SIGMOD 2001 Agrawal, Faloutsos, &. Swami. FODO 1993 Faloutsos, Ranganathan, & Manolopoulos. SIGMOD 1994 Compressed Representations

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Jean Fourier 1768-1830 020406080100120140 0 1 2 3 X X' 4 5 6 7 8 9 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 (A,B).

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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 exist. O(nlog(n)). (Weakly) able to support time warped queries. Difficult to deal with sequences of different lengths. Cannot support weighted distance measures. 020406080100120140 0 1 2 3 X X' 4 5 6 7 8 9 Note: The related transform DCT, uses only cosine basis functions. It does not seem to offer any particular advantages over DFT.

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020406080100120140 Haar 0 Haar 1 Haar 2 Haar 3 Haar 4 Haar 5 Haar 6 Haar 7 X X' 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.

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X = {8, 4, 1, 3} 1 2 3 4 5 6 7 8 h 1 = 4 = mean(8,4,1,3) h 2 = 2 = mean(8,4) - h 1 h 3 = 2 = (8-4)/2 h 4 = -1 = (1-3)/2 h 1 = 4 1 2 3 4 5 6 7 8 h 2 = 2 h 3 = 2 h 4 = -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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020406080100120140 Haar 0 Haar 1 Haar 2 Haar 3 Haar 4 Haar 5 Haar 6 Haar 7 X X' DWT 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. Works best if N is = 2 some_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 Obviously, this question still open...

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020406080100120140 X X' 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 Symantec Indexing ) for many years—but it is computationally expensive Good free SVD Primer Singular Value Decomposition - A Primer. Sonia Leach

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020406080100120140 X X' 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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020406080100120140 X X' Piecewise Linear Approximation I Basic Idea: Represent the time series as a sequence of straight lines. 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 Karl Friedrich Gauss 1777 - 1855

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020406080100120140 X X' Piecewise Linear Approximation II 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-- C Ratio denotes the compression ratio: Top-Down O(n 2 N) Bottom-Up O(n/ C Ratio ) Sliding Window O(n/ C Ratio ) 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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020406080100120140 X X' 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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020406080100120140 X X' 0 1 2 3 4 5 6 7 C U U C D C U D C U C D C U D U Symbolic Approximation Key: C = Constant U = Up D = 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)

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Piecewise Aggregate Approximation I 020406080100120140 X X' x1x1 x2x2 x3x3 x4x4 x5x5 x6x6 x7x7 x8x8 Given the reduced dimensionality representation we can calculate the approximate Euclidean distance as... Basic Idea: Represent the time series as a sequence of box basis functions. Note that each box is the same length. Independently introduced by two authors Keogh, Chakrabarti, Pazzani & Mehrotra, KAIS (2000) Byoung-Kee Yi, Christos Faloutsos, VLDB (2000)

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Piecewise Aggregate Approximation II 020406080100120140 X X' X1X1 X2X2 X3X3 X4X4 X5X5 X6X6 X7X7 X8X8 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.

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Adaptive Piecewise Constant Approximation I 020406080100120140 X X 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. 50100150200250 0 Raw Data (Electrocardiogram) Adaptive Representation (APCA) Reconstruction Error 2.61 Haar Wavelet Reconstruction Error 3.27 DFT Reconstruction 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.

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Adaptive Piecewise Constant Approximation II 020406080100120140 X X 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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Adaptive Piecewise Constant Approximation 020406080100120140 X X Pros and Cons of APCA as a time series representation.Pros and Cons of APCA as a time series representation. 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. Somewhat complex implementation. If visualized directly, looks ascetically unpleasing.

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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.

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