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Pattern Mining in large time series databases

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A literature review on time series representations, distance measures and indexing techniques used for pattern mining in very large time series data.

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M.Tech Seminar Presentation 2014-15 Presented by Jitesh Khandelwal 10211015 Guided by Dr. Durga Toshniwal IIT Roorkee

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Finance e.g. Stock prices Medical e.g. Electro Cardio Grams Marketing e.g. Forecasting product/brand demands Operations e.g. Monitoring control infrastructure at LHC Social Networks e.g. Like count on a profile picture based on gender Almost everything is a time series!

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Value Prediction Pattern Identification 2.1, 9.3, 4.5, 3, 6.7, 4.0, 18.8, 9.2, 5.8, ? 2.1, 9.3, 4.5, 3, 6.7, 4.0, 18.8, 9.2, 5.8 Based on mathematical models Based on human perception What’s next?

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Classification Anomaly DetectionMotif discovery Clustering [16] [16] [16] Source: www.google.com

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Raw time series data Similarity model selection Dimensionality reduction Index construction Mathematical formulation of human perception of similarity High dimensionality makes distance calculation slow Enables efficient querying of big and fast incoming time series data 1 32

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Raw time series data Similarity model selection Dimensionality reduction Index construction 1 32 Symbolic Representation Text Processing Algorithms Double is 4 byte, Char is 1 byte. Hence, lower memory footprint.

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Lp Norms DTW distance Longest common subsequence Landmark Similarity ℒ 𝑝 𝑥 − 𝑦 = 𝑖=1 𝐿 𝑥𝑖 − 𝑦𝑖 𝑝 1 𝑝 ℒ1 ℒ2 - Manhattan distance - Euclidean distance Invariant to amplitude scaling when used with z-score normalization. Source: www.google.com

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Lp Norms DTW distance Longest common subsequence Landmark Similarity 𝐷 𝑖, 𝑗 = 𝑥𝑖 − 𝑦𝑗 + 𝑚𝑖𝑛 𝐷 𝑖 − 1, 𝑗 , 𝐷 𝑖 − 1, 𝑗 − 1 , 𝐷 𝑖, 𝑗 − 1 DTW is Dynamic Time Warping. Allows comparison of variable length time series. Computationally Expensive. Can be optimized using warping window techniques and early abandoning using lower bounds. Source: www.google.com

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Lp Norms DTW distance Longest common subsequence Landmark Similarity Applicable only to symbolic representations of time series. Non-metric because it does not satisfy triangle inequality. 𝑆𝑖𝑚 𝑥, 𝑦 = 𝐿𝐶𝑆 𝑥, 𝑦 A distance measure D is a metric if it satisfies the following properties: 1. Symmetry: D(X, Y) = D(Y, X) 2. Triangle Inequality: D(X, Y) + D(Y, Z) <= D(X, Z) Threshold parameter, matching criteria for 2 points from x and y. Warping threshold, constraint on matching of points along the time axis.

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Works the same ways as human remember patterns. Definition of landmarks vary with application domains. E.g. local minima, local maxima, inflection point etc. Uses MDPP (Minimum Distance/Percentage Principle) technique to eliminate noisy landmarks. 𝑥𝑖+1 − 𝑥𝑖 < 𝐷 𝑦𝑖+1 − 𝑦𝑖 𝑦𝑖 − 𝑦𝑖+1 2 < 𝑃 𝑀𝐷𝑃 𝐷, 𝑃 removes landmarks at and if𝑥𝑖 𝑥𝑖+1 Lp Norms DTW distance Longest common subsequence Landmark Similarity

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𝐷𝑟𝑒𝑑𝑢𝑐𝑒𝑑 𝑠𝑝𝑎𝑐𝑒 𝐴, 𝐵 ≤ 𝐷𝑡𝑟𝑢𝑒 𝐴, 𝐵 False Alarms False Dismissals Objects that appear close in index space are actually distant. Objects appear distant in index space but are actually closer. Removed in post-processing step. Unacceptable.

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DFT DWT PAA eAPCA APCA Discrete Fourier Transform 𝑋 𝑓 = 1 𝑛 𝑡=0 𝑛−1 𝑥𝑡 𝑒 −𝑗2𝜋𝑡𝑓 𝑛 1. Choose coefficients corresponding to a few low values of frequencies. 2. Choose coefficients corresponding to frequencies with higher values of coefficients. Based on Parseval’s Relation, Euclidean distance is preserved in the Frequency domain.

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DFT DWT PAA eAPCA APCA Discrete Fourier Transform [6]

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DFT DWT PAA eAPCA APCA Discrete Fourier Transform [6]

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DFT DWT PAA eAPCA APCA Discrete Wavelet Transform 𝜓𝑗,𝑘 = 2 𝑗 2 𝜓 2𝑗 𝑡 − 𝑘 Used with Haar wavelets as the basis function. Applicable only for time series with lengths which are a power of 2. Lower bound is tighter than DFT.

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DFT DWT PAA eAPCA APCA Discrete Wavelet Transform Using Haar Wavelet as the basis function. [6]

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DFT DWT PAA eAPCA APCA Piecewise Aggregate Approximation 𝑥𝑖 = 𝑁 𝑛 𝑗= 𝑛 𝑁 𝑖−1 +1 𝑛 𝑁 𝑖 𝑥𝑗 Reconstruction quality and estimated distance in index space is same as DWT with the Haar Wavelet. With no restriction on length. [6]

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DFT DWT PAA eAPCA APCA Piecewise Aggregate Approximation 𝐷 𝑋, 𝑌 = 𝑛 𝑁 𝑖=1 𝑁 𝑥𝑖 − 𝑦𝑖 2 A lower bound on the Euclidean distance in the PAA space. N = actual number of points n = number of PAA segments

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DFT DWT PAA eAPCA APCA Adaptive Piecewise Constant Approximation Data adaptive. Shorter segments for areas of high activity. An extension of PAA. 𝑋 = < 𝑥1, 𝑟1 >, < 𝑥2, 𝑟2 > ⋯ < 𝑥 𝑛, 𝑟 𝑛 > 𝑥𝑖 = 𝑚𝑒𝑎𝑛(𝑥 𝑟𝑖−1+1 … 𝑥 𝑟𝑖 ) [7]

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DFT DWT PAA eAPCA APCA Adaptive Piecewise Constant Approximation 𝐷 𝐶, 𝑄 = 𝑖=1 𝑀 𝑐𝑟𝑖 − 𝑐𝑟𝑖−1 𝑞𝑥𝑖 − 𝑐𝑥𝑖 2 M = number of APCA segments A lower bound on the Euclidean distance in the APCA space.

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DFT DWT PAA eAPCA APCA Extended APCA 𝑆 = 𝜇1, 𝜎1, 𝑟1 , … , 𝜇 𝑚, 𝜎 𝑚, 𝑟 𝑚 Also stores variance along with mean for the segments. As a result, it gives both a lower and upper bound on the Euclidean distance. Formulas are very ugly!

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SAX iSAX SFA Based on PAA. Symbolic Aggregate Approximation Static alphabet size. “Desirable to have a discretization technique that produce symbols with equal probability.” Can leverage run length encoding compression. Breakpoints [9]

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SAX iSAX SFA Symbolic Aggregate Approximation Supports a lower bound distance measure to Euclidean distance. 𝐷𝑆𝐴𝑋 ≤ 𝐷 𝑃𝐴𝐴 ≤ 𝐷𝑡𝑟𝑢𝑒 Can be calculated in a streaming fashion. [9]

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SAX iSAX SFA Indexable SAX a, b, c, d (SAX) 00, 01, 10, 11 (iSAX) 0 00, 01, 10, 11 1 00, 01, 10, 11 1 00, 01, 0 10, 11 1 00, 01, 1 10, 11 Fixed number of segments. Dynamic alphabet size. iSAX Notation: iSAX(T, segment count, alphabet size) e.g. iSAX(T, 4, 8)

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SAX iSAX SFA Indexable SAX Comparison of iSAX words with different alphabet size. iSAX(A, 4, 8) = { 110, 110, 011, 000 } iSAX(B, 4, 2) = { 0 , 0 , 1 , 1 } Replace 0 with either of { 0 00, 0 01, 0 10, 0 11 } whichever is closest to 110. Similarly for all segments. { 011, 011, 100, 100} != iSAX(B, 4, 8) Just a lower bound estimate. We cannot undo lossy compression.

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SAX iSAX SFA Symbolic Fourier Approximation Uses MCB (multiple coefficient binning) discretization. Based on DFT. SAX - assumes a common distribution for all the coefficients of the reduced representation MCB – histograms are built for all the coefficients and then equi-width binning is used. Tighter lower bound than iSAX [12]

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SAX iSAX SFA Symbolic Fourier Approximation Every SFA symbol has some global information since it is based on DFT. Cannot be calculated in a streaming fashion. Unlike iSAX, Fixed alphabet size. Dynamic segment count. Quality of representation improves with segment count.

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Pruning Power Tightness of Lower bound number of data points examined to answer the query total number of data points in the database Intuitively captures the measure of the quality of representation. Free from implementation bias. On a random walk dataset with query lengths 256 to 1024 and dimension of representation 16 to 64 [7] Lower is better.

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Pruning Power Tightness of Lower bound lower bounding distance true distance Tightness of lower bounds for various time series representations on the Koski ECG dataset [10] Higher is better.

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R/R* Trees SFA Trie DS Tree ADS index iSAX Tree R trees are multi-dimensional index structures. Encloses close objects in a MBR (Minimum Bounding Rectangle). Individual objects are at the leaves and intermediate nodes are MBRs enclosing other MBRs or the objects. Used for indexing time series after dimensionality reduction using DFT, PAA, APCA and other numeric representations. Unlike R trees, In R* trees, there is no overlap between the different MBRs due to which it also works for range queries rather than only point queries.

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R/R* Trees SFA Trie DS Tree ADS index iSAX Tree Based on the dynamic alphabet size of iSAX representation. Given the segment count, say d. The root node has 2^d children. A Leaf node, when overflows, is converted to an intermediate node. A segment is selected and its cardinality is increased to produce 2 child leaf nodes that contain the iSAX representations of the time series. iSAX 2.0 is an improvement over iSAX where the segment on which split occurs is selected based on the distribution of time series so that the splitting is balanced.

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R/R* Trees SFA Trie DS Tree ADS index iSAX Tree Based on the SFA representation. Time series with common SFA prefix lie in common sub-tree. SFA is computed for more number of Fourier coefficients. But not all are used in the index. Hence, small index size. Example: SFA( T1 ) = abaacde | SFA( T2 ) = abbadef SFA( T1 ) = abaacde | SFA( T2 ) = abbadef | SFA( T3 ) = abaagef

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R/R* Trees SFA Trie DS Tree ADS index iSAX Tree Based on the Extended APCA reduction method. Intermediate nodes store 𝜇𝑖 𝑚𝑖𝑛 , 𝜇𝑖 𝑚𝑎𝑥 , 𝜎𝑖 𝑚𝑖𝑛 , 𝜎𝑖 𝑚𝑎𝑥 for all segments i = 1 to m. It also stores the splitting strategy chose during splitting. Dynamic Segmentation Tree

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R/R* Trees SFA Trie DS Tree ADS index iSAX Tree Dynamic Segmentation Tree Splitting strategies are of 2 types: Horizontal and Vertical. Horizontal: using mean and variance. Vertical: using segment splitting. Splitting strategy is chosen based on the value of a Quality Measure. The one with maximum value is selected. 𝑄 = 𝑖=1 𝑚 𝑟𝑖 − 𝑟𝑖−1 𝜇𝑖 𝑚𝑎𝑥 − 𝜇𝑖 𝑚𝑖𝑛 2 + 𝜎𝑖 𝑚𝑎𝑥2 𝑆𝑝𝑙𝑖𝑡𝑡𝑖𝑛𝑔 𝐵𝑒𝑛𝑒𝑓𝑖𝑡 = 𝑄 𝑝𝑎𝑟𝑒𝑛𝑡 − 𝑄𝑙 + 𝑄 𝑟 2

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R/R* Trees SFA Trie DS Tree ADS index iSAX Tree Dynamic Segmentation Tree Apart from similarity search as supported by other indices, it also allows distance histogram computation for a given query. For e.g. Given a query Q, a list L = [ ([10, 20], 10), ([15, 30], 15), ([40, 50], 2) ] means that there are 3 leaf nodes: N1, N2 and N3. N1 includes 10 time series, and their distance from Q is between [10, 20]. Similarly for N2 and N3. This is due to the lower and upper bounds provided by eAPCA.

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R/R* Trees SFA Trie DS Tree ADS index iSAX Tree Adaptive data series index Based in iSAX representation. Delays the construction of leaf nodes to query time. Also, leaf nodes contain only the iSAX representations and the actual data series remain in the disk. Even during splits, only the iSAX representations are shuffled. Trade off: Small leaf size require splits that costs disk IO time, whereas Big leaf size leads to increased query time for linear scan. So, ADS+ uses adaptive leaf size. A bigger build time leaf size and a much smaller query time leaf size.

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[1] Agrawal, R., Faloutsos, C., & Swami, A. (1993). “Efficient similarity search in sequence databases”. Proceedings of the 4th Conference on Foundations of Data Organization and Algorithms. [2] Antonin Guttman, (1984). “R-trees: a dynamic index structure for spatial searching”. Proceedings of the 1984 ACM SIGMOD international conference on Management of data. [3] Yi, B.K., Faloutsos, C. (2000) “Fast Time Sequence Indexing for Arbitrary Lp-Norms”. Proceedings of the 26th International Conference on Very Large Data Bases. [4] Keogh, E. (2002) “Exact Indexing of Dynamic Time Warping”. Proceedings of the 28th international conference on Very Large Data Bases. [5] Perng, C., Wang H., Zhang S. R., Parker, D.S. (2000). “Landmarks: A New Model for Similarity-Based Pattern Querying in Time Series Databases”. Proceedings of the 16th International Conference on Data Engineering [6] Keogh, E., Chakrabarti, K., Pazzani, M. & Mehrotra, S. (2000). “Dimensionality Reduction for Fast Similarity Search in Large Time Series Databases”. Published in Journal Knowledge and Information Systems.

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[7] Keogh, E., Chakrabarti, K., Mehrotra, S., Pazzani, M. (2002) “Locally Adaptive Dimensionality Reduction for Indexing Large Time Series Databases”. Published in Journal ACM Transactions on Database Systems. [8] Wang, Y., Wang, P., Pei, J., Wang, W., Huang, S. (2013) “A Data-adaptive and Dynamic Segmentation Index for Whole Matching on Time Series”. Proceedings of the VLDB Endowment [9] Lin, J., Keogh, E., Lonardi, S., Chiu, B. (2003). “A symbolic representation of time series, with implications for streaming algorithms”. Proceedings of the 8th ACM SIGMOD workshop on Research issues in data mining and knowledge discovery. [10] Shieh, J., Keogh, E., (2008) “iSAX: Indexing and Mining Terabyte Sized Time Series”. Proceedings of the 14th ACM SIGKDD international conference on Knowledge discovery and data mining. [11] Camerra, A., Palpanas, T., Shieh, J., Keogh, E. (2010) “iSAX 2.0: Indexing and Mining One Billion Time Series”. Proceedings of the IEEE International Conference on Data Mining.

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[12] Schäfer, P., Högqvist, M. (2012) “SFA: A Symbolic Fourier Approximation and Index for Similarity Search in High Dimensional Datasets”. Proceedings of the 15th International Conference on Extending Database Technology. [13] Beckmann, N., Kriegel, H., Schneider, R., Seeger, B. (1990) “The R*-tree: an efficient and robust access method for points and rectangles”, Proceedings of the ACM SIGMOD international conference on Management of data. [14] Zoumpatianos, K., Idreos, S., Palpanas, T. (2014) “Indexing for Interactive Exploration of Big Data Series”. Proceedings of the ACM SIGMOD International Conference on Data Management. [15] Wu, Y.L., Agrawal, D., Abbadi, A.E., (2000) “A comparison of DFT and DWT based Similarity Search in Time-Series Databases”. Proceedings of the ninth international conference on Information and knowledge management. [16] Esling, P., Agon, C. (2012) “Time-Series Data Mining”. Published in Journal ACM Computing Surveys (CSUR).

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Pattern Mining in large time series databases

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Pattern Mining in large time series databases

  • 1. M.Tech Seminar Presentation 2014-15 Presented by Jitesh Khandelwal 10211015 Guided by Dr. Durga Toshniwal IIT Roorkee
  • 2. Finance e.g. Stock prices Medical e.g. Electro Cardio Grams Marketing e.g. Forecasting product/brand demands Operations e.g. Monitoring control infrastructure at LHC Social Networks e.g. Like count on a profile picture based on gender Almost everything is a time series!
  • 3. Value Prediction Pattern Identification 2.1, 9.3, 4.5, 3, 6.7, 4.0, 18.8, 9.2, 5.8, ? 2.1, 9.3, 4.5, 3, 6.7, 4.0, 18.8, 9.2, 5.8 Based on mathematical models Based on human perception What’s next?
  • 5. Raw time series data Similarity model selection Dimensionality reduction Index construction Mathematical formulation of human perception of similarity High dimensionality makes distance calculation slow Enables efficient querying of big and fast incoming time series data 1 32
  • 6. Raw time series data Similarity model selection Dimensionality reduction Index construction 1 32 Symbolic Representation Text Processing Algorithms Double is 4 byte, Char is 1 byte. Hence, lower memory footprint.
  • 7. Lp Norms DTW distance Longest common subsequence Landmark Similarity ℒ 𝑝 𝑥 − 𝑦 = 𝑖=1 𝐿 𝑥𝑖 − 𝑦𝑖 𝑝 1 𝑝 ℒ1 ℒ2 - Manhattan distance - Euclidean distance Invariant to amplitude scaling when used with z-score normalization. Source: www.google.com
  • 8. Lp Norms DTW distance Longest common subsequence Landmark Similarity 𝐷 𝑖, 𝑗 = 𝑥𝑖 − 𝑦𝑗 + 𝑚𝑖𝑛 𝐷 𝑖 − 1, 𝑗 , 𝐷 𝑖 − 1, 𝑗 − 1 , 𝐷 𝑖, 𝑗 − 1 DTW is Dynamic Time Warping. Allows comparison of variable length time series. Computationally Expensive. Can be optimized using warping window techniques and early abandoning using lower bounds. Source: www.google.com
  • 9. Lp Norms DTW distance Longest common subsequence Landmark Similarity Applicable only to symbolic representations of time series. Non-metric because it does not satisfy triangle inequality. 𝑆𝑖𝑚 𝑥, 𝑦 = 𝐿𝐶𝑆 𝑥, 𝑦 A distance measure D is a metric if it satisfies the following properties: 1. Symmetry: D(X, Y) = D(Y, X) 2. Triangle Inequality: D(X, Y) + D(Y, Z) <= D(X, Z) Threshold parameter, matching criteria for 2 points from x and y. Warping threshold, constraint on matching of points along the time axis.
  • 10. Works the same ways as human remember patterns. Definition of landmarks vary with application domains. E.g. local minima, local maxima, inflection point etc. Uses MDPP (Minimum Distance/Percentage Principle) technique to eliminate noisy landmarks. 𝑥𝑖+1 − 𝑥𝑖 < 𝐷 𝑦𝑖+1 − 𝑦𝑖 𝑦𝑖 − 𝑦𝑖+1 2 < 𝑃 𝑀𝐷𝑃 𝐷, 𝑃 removes landmarks at and if𝑥𝑖 𝑥𝑖+1 Lp Norms DTW distance Longest common subsequence Landmark Similarity
  • 11. 𝐷𝑟𝑒𝑑𝑢𝑐𝑒𝑑 𝑠𝑝𝑎𝑐𝑒 𝐴, 𝐵 ≤ 𝐷𝑡𝑟𝑢𝑒 𝐴, 𝐵 False Alarms False Dismissals Objects that appear close in index space are actually distant. Objects appear distant in index space but are actually closer. Removed in post-processing step. Unacceptable.
  • 12. DFT DWT PAA eAPCA APCA Discrete Fourier Transform 𝑋 𝑓 = 1 𝑛 𝑡=0 𝑛−1 𝑥𝑡 𝑒 −𝑗2𝜋𝑡𝑓 𝑛 1. Choose coefficients corresponding to a few low values of frequencies. 2. Choose coefficients corresponding to frequencies with higher values of coefficients. Based on Parseval’s Relation, Euclidean distance is preserved in the Frequency domain.
  • 15. DFT DWT PAA eAPCA APCA Discrete Wavelet Transform 𝜓𝑗,𝑘 = 2 𝑗 2 𝜓 2𝑗 𝑡 − 𝑘 Used with Haar wavelets as the basis function. Applicable only for time series with lengths which are a power of 2. Lower bound is tighter than DFT.
  • 16. DFT DWT PAA eAPCA APCA Discrete Wavelet Transform Using Haar Wavelet as the basis function. [6]
  • 17. DFT DWT PAA eAPCA APCA Piecewise Aggregate Approximation 𝑥𝑖 = 𝑁 𝑛 𝑗= 𝑛 𝑁 𝑖−1 +1 𝑛 𝑁 𝑖 𝑥𝑗 Reconstruction quality and estimated distance in index space is same as DWT with the Haar Wavelet. With no restriction on length. [6]
  • 18. DFT DWT PAA eAPCA APCA Piecewise Aggregate Approximation 𝐷 𝑋, 𝑌 = 𝑛 𝑁 𝑖=1 𝑁 𝑥𝑖 − 𝑦𝑖 2 A lower bound on the Euclidean distance in the PAA space. N = actual number of points n = number of PAA segments
  • 19. DFT DWT PAA eAPCA APCA Adaptive Piecewise Constant Approximation Data adaptive. Shorter segments for areas of high activity. An extension of PAA. 𝑋 = < 𝑥1, 𝑟1 >, < 𝑥2, 𝑟2 > ⋯ < 𝑥 𝑛, 𝑟 𝑛 > 𝑥𝑖 = 𝑚𝑒𝑎𝑛(𝑥 𝑟𝑖−1+1 … 𝑥 𝑟𝑖 ) [7]
  • 20. DFT DWT PAA eAPCA APCA Adaptive Piecewise Constant Approximation 𝐷 𝐶, 𝑄 = 𝑖=1 𝑀 𝑐𝑟𝑖 − 𝑐𝑟𝑖−1 𝑞𝑥𝑖 − 𝑐𝑥𝑖 2 M = number of APCA segments A lower bound on the Euclidean distance in the APCA space.
  • 21. DFT DWT PAA eAPCA APCA Extended APCA 𝑆 = 𝜇1, 𝜎1, 𝑟1 , … , 𝜇 𝑚, 𝜎 𝑚, 𝑟 𝑚 Also stores variance along with mean for the segments. As a result, it gives both a lower and upper bound on the Euclidean distance. Formulas are very ugly!
  • 22. SAX iSAX SFA Based on PAA. Symbolic Aggregate Approximation Static alphabet size. “Desirable to have a discretization technique that produce symbols with equal probability.” Can leverage run length encoding compression. Breakpoints [9]
  • 23. SAX iSAX SFA Symbolic Aggregate Approximation Supports a lower bound distance measure to Euclidean distance. 𝐷𝑆𝐴𝑋 ≤ 𝐷 𝑃𝐴𝐴 ≤ 𝐷𝑡𝑟𝑢𝑒 Can be calculated in a streaming fashion. [9]
  • 24. SAX iSAX SFA Indexable SAX a, b, c, d (SAX) 00, 01, 10, 11 (iSAX) 0 00, 01, 10, 11 1 00, 01, 10, 11 1 00, 01, 0 10, 11 1 00, 01, 1 10, 11 Fixed number of segments. Dynamic alphabet size. iSAX Notation: iSAX(T, segment count, alphabet size) e.g. iSAX(T, 4, 8)
  • 25. SAX iSAX SFA Indexable SAX Comparison of iSAX words with different alphabet size. iSAX(A, 4, 8) = { 110, 110, 011, 000 } iSAX(B, 4, 2) = { 0 , 0 , 1 , 1 } Replace 0 with either of { 0 00, 0 01, 0 10, 0 11 } whichever is closest to 110. Similarly for all segments. { 011, 011, 100, 100} != iSAX(B, 4, 8) Just a lower bound estimate. We cannot undo lossy compression.
  • 26. SAX iSAX SFA Symbolic Fourier Approximation Uses MCB (multiple coefficient binning) discretization. Based on DFT. SAX - assumes a common distribution for all the coefficients of the reduced representation MCB – histograms are built for all the coefficients and then equi-width binning is used. Tighter lower bound than iSAX [12]
  • 27. SAX iSAX SFA Symbolic Fourier Approximation Every SFA symbol has some global information since it is based on DFT. Cannot be calculated in a streaming fashion. Unlike iSAX, Fixed alphabet size. Dynamic segment count. Quality of representation improves with segment count.
  • 28. Pruning Power Tightness of Lower bound number of data points examined to answer the query total number of data points in the database Intuitively captures the measure of the quality of representation. Free from implementation bias. On a random walk dataset with query lengths 256 to 1024 and dimension of representation 16 to 64 [7] Lower is better.
  • 29. Pruning Power Tightness of Lower bound lower bounding distance true distance Tightness of lower bounds for various time series representations on the Koski ECG dataset [10] Higher is better.
  • 30. R/R* Trees SFA Trie DS Tree ADS index iSAX Tree R trees are multi-dimensional index structures. Encloses close objects in a MBR (Minimum Bounding Rectangle). Individual objects are at the leaves and intermediate nodes are MBRs enclosing other MBRs or the objects. Used for indexing time series after dimensionality reduction using DFT, PAA, APCA and other numeric representations. Unlike R trees, In R* trees, there is no overlap between the different MBRs due to which it also works for range queries rather than only point queries.
  • 31. R/R* Trees SFA Trie DS Tree ADS index iSAX Tree Based on the dynamic alphabet size of iSAX representation. Given the segment count, say d. The root node has 2^d children. A Leaf node, when overflows, is converted to an intermediate node. A segment is selected and its cardinality is increased to produce 2 child leaf nodes that contain the iSAX representations of the time series. iSAX 2.0 is an improvement over iSAX where the segment on which split occurs is selected based on the distribution of time series so that the splitting is balanced.
  • 32. R/R* Trees SFA Trie DS Tree ADS index iSAX Tree Based on the SFA representation. Time series with common SFA prefix lie in common sub-tree. SFA is computed for more number of Fourier coefficients. But not all are used in the index. Hence, small index size. Example: SFA( T1 ) = abaacde | SFA( T2 ) = abbadef SFA( T1 ) = abaacde | SFA( T2 ) = abbadef | SFA( T3 ) = abaagef
  • 33. R/R* Trees SFA Trie DS Tree ADS index iSAX Tree Based on the Extended APCA reduction method. Intermediate nodes store 𝜇𝑖 𝑚𝑖𝑛 , 𝜇𝑖 𝑚𝑎𝑥 , 𝜎𝑖 𝑚𝑖𝑛 , 𝜎𝑖 𝑚𝑎𝑥 for all segments i = 1 to m. It also stores the splitting strategy chose during splitting. Dynamic Segmentation Tree
  • 34. R/R* Trees SFA Trie DS Tree ADS index iSAX Tree Dynamic Segmentation Tree Splitting strategies are of 2 types: Horizontal and Vertical. Horizontal: using mean and variance. Vertical: using segment splitting. Splitting strategy is chosen based on the value of a Quality Measure. The one with maximum value is selected. 𝑄 = 𝑖=1 𝑚 𝑟𝑖 − 𝑟𝑖−1 𝜇𝑖 𝑚𝑎𝑥 − 𝜇𝑖 𝑚𝑖𝑛 2 + 𝜎𝑖 𝑚𝑎𝑥2 𝑆𝑝𝑙𝑖𝑡𝑡𝑖𝑛𝑔 𝐵𝑒𝑛𝑒𝑓𝑖𝑡 = 𝑄 𝑝𝑎𝑟𝑒𝑛𝑡 − 𝑄𝑙 + 𝑄 𝑟 2
  • 35. R/R* Trees SFA Trie DS Tree ADS index iSAX Tree Dynamic Segmentation Tree Apart from similarity search as supported by other indices, it also allows distance histogram computation for a given query. For e.g. Given a query Q, a list L = [ ([10, 20], 10), ([15, 30], 15), ([40, 50], 2) ] means that there are 3 leaf nodes: N1, N2 and N3. N1 includes 10 time series, and their distance from Q is between [10, 20]. Similarly for N2 and N3. This is due to the lower and upper bounds provided by eAPCA.
  • 36. R/R* Trees SFA Trie DS Tree ADS index iSAX Tree Adaptive data series index Based in iSAX representation. Delays the construction of leaf nodes to query time. Also, leaf nodes contain only the iSAX representations and the actual data series remain in the disk. Even during splits, only the iSAX representations are shuffled. Trade off: Small leaf size require splits that costs disk IO time, whereas Big leaf size leads to increased query time for linear scan. So, ADS+ uses adaptive leaf size. A bigger build time leaf size and a much smaller query time leaf size.
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