Vladimir Kurbalija, Milo

1
Analysis of Constrained Time-Series Similarity
Measures

• Vladimir Kurbalija, Miloš Radovanovic, Zoltan
  Geler, Mirjana Ivanovic
• Department of Mathematics and Informatics
• Faculty of Science
• University of Novi Sad
• Serbia

2
Agenda

• Introduction
• Related Work
• Experimental Evaluation
• Computational Times
• The Change of 1NN Graph
• Conclusions and Future Work

3
Time Series

• Time-series (TS) consists of sequence of values
  or events obtained over repeated measurements of
  time
• Time-series analysis (TSA) comprises methods that
  attempt to understand such time series
• To understand the underlying context of the data
  points, or to make forecasts

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Applications and Task Types

• Applications
• stock market analysis,
• economic and sales forecasting,
• observation of natural phenomena,
• scientific and engineering experiments,
• medical treatments etc.
• Task Types
• indexing,
• classification,
• clustering,
• prediction,
• segmentation,
• anomaly detection, etc.

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Important Concepts

• Pre-processing transformation,
• Time-series representation
• Similarity/distance measure

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Pre-processing Transformation

• Raw time series usually contain some
  distortions
• The presence of distortions can seriously
  deteriorate the indexing problem
• Some of the most common pre-processing tasks are
  
• offset translation,
• amplitude scaling,
• removing linear trend,
• removing noise etc.

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Time-series Representation

• Time series are generally high-dimensional data
• Many techniques have been proposed
• Discrete Fourier Transformation (DFT)
• Singular Value Decomposition (SVD)
• Discrete Wavelet Transf. (DWT)
• Piecewise Aggregate Approximation (PAA)
• Adaptive Piecewise Constant Approx. (APCA)
• Symbolic Aggregate approX. (SAX)
• Indexable Piecewise Linear Approx. (IPLA)
• Spline Representation
• etc.

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Similarity/distance Measure

• Similarity-based retrieval is used in all a fore
  mentioned task types
• The distance between time series needs to be
  carefully defined in order to reflect the
  underlying (dis)similarity (based on shapes and
  patterns).
• There is a number of distance measures
• Lp distance (Lp) - Eucledian Distance (for p2)
• Dynamic Time Warping (DTW)
• distance based on Longest Common Subsequence
  (LCS)
• Edit Distance with Real Penalty (ERP)
• Edit Distance on Real sequence (EDR)
• Sequence Weighted Alignment model (Swale) 31,
  etc.

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SimilarityMeasures

• Many of these similarity measures are based on
  dynamic programming (DTW, LCS, ERP, EDR...)
• The computational complexity of dynamic
  programming algorithms is quadratic
• The usage of global constraints such as the
  Sakoe-Chiba band and the Itakura parallelogram
  can significantly speed up the calculation of
  similarities
• The usage of global constraints can improve the
  accuracy of classification

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Our Research

• Dynamic Time Warping (DTW) and Longest Common
  Subsequence measure (LCS)
• the speed-up gained from these constraints
• the change of the 1-nearest neighbor graph with
  respect to the change of the constraint size
• FAP (Framework for Analysis and Prediction)
  http//perun.pmf.uns.ac.rs/fap/
• UCR Time Series Repository http//www.cs.ucr.edu/
  eamonn/time_series_data/

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Agenda

• Introduction
• Related Work
• Experimental Evaluation
• Computational Times
• The Change of 1NN Graph
• Conclusions and Future Work

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Euclidean Metric

• Most intuitive metric for time series, and as a
  consequence very commonly used
• Very fast computation complexity is linear
• Very brittle and sensitive to small translations
  across the time axis

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Dynamic Time Warping (DTW)

• Generalization of Euclidian measure
• Allows elastic shifting of the time axis where in
  some points time warps
• Computes the distance by finding an optimal path
  in the matrix of distances of two time series

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Longest Common Subsequence (LCS)

• Different methodology
• Similarity between two time series is expressed
  as a length of the longest common subsequence of
  both time series

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Global Constraints

• DTW and LCS are based on dynamic programming
  the algorithms search for the optimal path in the
  search matrix
• Global constraints narrow the search path in the
  matrix which results in a significant decrease in
  the number of performed calculations

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Agenda

• Introduction
• Related Work
• Experimental Evaluation
• Computational Times
• The Change of 1NN Graph
• Conclusions and Future Work

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Quality of Similarity Measures

• Quality of similarity measures is usually
  evaluated indirectly
• By assessment of different classifier accuracy
• Simple 1-nearest classifier (1NN) gives among the
  best results for time-series data
• The accuracy of 1NN directly reflects the quality
  of a similarity measure
• We report the calculation times for unconstrained
  and constrained DTW and LCS
• We focus on the 1NN graph and its change with
  regard to the change of constraints

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Experimental Evaluation

• The unconstrained measure and a measure with the
  following constraints 75, 50, 25, 20, 15,
  10, 5, 1 and 0 of the size of the time series
• Smaller constraints have more interesting
  behavior
• Set of experiments was conducted on 38 datasets
  from UCR Time Series Repository
• The length of time series varies from 24 to 1882
  depending of the data set
• The number of time series per data set varies
  from 60 to 9236.

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Computational Times

• The efficiency of calculating the distance matrix
• The distance matrix for one data set is the
  matrix where element (i,j) contains the distance
  between i-th and j-th time series
• The calculation of the distance matrix is a
  time-consuming operation
• All experiments are performed on AMD Phenom II X4
  945 with 3GB RAM

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Computational Times - DTW
21
Computational Times - LCS
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Computational Times

• Introduction of global constraints in both
  measures significantly speeds up the process of
  distance matrix computation
• Direct consequence of a faster similarity measure
• It is known for DTW that smaller values of
  constraints can give more accurate classification
• The average constraint size, which gives the best
  accuracy, for all datasets is 4 of the
  time-series length
• LCS measure is still not well investigated

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The Change of 1NN Graph

• The nearest neighbor graph is a directed graph
  where each time series is connected with its
  nearest neighbor
• graph for unconstrained measures (DTW and LCS)
  and for measures with the following constraints
  75, 50, 25, 20, 15, 10, 5, 1 and 0 of
  the length of time series
• The change of nearest neighbor graphs is tracked
  as the percentage of time series (nodes in the
  graph) that changed their nearest neighbor
  compared to the nearest neighbor in the
  unconstrained measure

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The Change of 1NN Graph - DTW
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The Change of 1NN Graph - LCS
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The Change of 1NN Graph

• Both measures behave in a similar manner when the
  constraint is narrowed
• 1NN graph remains the same until the size of the
  constraint is narrowed to approximately 20, and
  after that the graph starts to change
  significantly
• All datasets (for both measures) reach high
  percentages of difference (over 50) for small
  constraint sizes (5-10)
• Constrained measures represent qualitatively
  different measures than the unconstrained ones

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Agenda

• Introduction
• Related Work
• Experimental Evaluation
• Computational Times
• The Change of 1NN Graph
• Conclusions and Future Work

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Conclusions

• We examined the influence of global constraints
  on two most representative elastic measures for
  time series DTW and LCS
• Through an extensive set of experiments we showed
  that the usage of global constraints can
  significantly reduce the computation time
• We demonstrated that the constrained measures are
  qualitatively different than their unconstrained
  counterparts
• For DTW it is known that the constrained measures
  are more accurate, while for LCS this issue is
  still open.

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Future Work

• To investigate the accuracy of the constrained
  LCS measure for different values of constraints
• To explore the influence of global constraints on
  the computation time and 1NN graphs of other
  elastic measures like ERP, EDR, Swale, etc.
• The constrained variants of these elastic
  measures should also be tested with respect to
  classification accuracy

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Thank you for your attention

• FAP site
• http//perun.pmf.uns.ac.rs/fap/