Spatial and Temporal Databases

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Spatial and Temporal Databases
Efficiently Time Series Matching by Wavelets
(ICDE 98) Kin-pong Chan and Ada Wai-chee Fu
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Table of Contents

• Introduction
• Related Works
• The Proposed Approach
• Overall Strategy
• Performance Evaluation
• Conclusion

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Introduction

• Time-series a sequence of real numbers, each
  number representing a value at a time point
  (financial data, scientific observation data, )
• Time-series databases supporting fast retrieval
  of data and similarity query are desired

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Introduction (cont)

• Similarity Search
• Finds data sequences that differ only slightly
  from
• the given query sequence
• Example) One may want to find all companies whose
  stock price
• fluctuations behave similarly with IBM during a
  year.
• Similarity matching process
• Given
• compute

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Introduction (cont.)

• Indexing
• Dimensionality reduction
• Transformation is applied to reduce dimension
• Completeness
• Nature of data
• Effectiveness of power concentration of a
  particular
• transformation depends on the nature of the time
  series

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Related Works

• Discrete Fourier Transform (Agrawal et al)
• Parsevals theorem
• F-index may raise false alarm, but guarantee no
  false dismissal
• Disadvantage misses the important feature of
  time localization

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Related Works (cont.)

• Singular Value Decomposition decompose a matrix
  X of size NM into
• Restriction
• X is not updated
• X can be updated daily or monthly. In that case,
  SVD has to be recomputed the whole matrix again
  to update

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The proposed Approach Similarity Model

• Define new similarity model used in sequence
  matching

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Proposed Approach Haar Wavelet

• Haar wavelet
• Allows a good approximation with a subset of
  coefficients
• Fast to compute and requires little storage
• It preserves Euclidean distance

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Proposed Approach Haar Wavelet (cont)

• Example of Wavelet Computation

Assume Original time sequence is f(x) (9 7 3 5)
4 (9 7 3 5)
2 (8 4) (1 1)
1 (6) (2)
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Proposed Approach Haar Wavelet (cont)

• Instead of storing 6,2,1 and -1, assume we store
  first two coefficient, 6 and 2
• Reconstruction Process

Resolution Average Coefficients
4 (8 8 4 4)
2 (8 4)
(0 0)
1 (6) (2)
Original (9 7 3 5), Reconstructed (8 8 4 4)
We can reduce dimension of the data
with sacrificing the accuracy
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Proposed Approach DFT versus Haar (cont)

• Motivation of replacing DFT with DWT
• Pruning power less false alarm appear in DWT
  than DFT
• Complexity consideration
• Complexity of Haar is O(n) while O(nlogn) for
  Fast Fourier Transform
• Note DWT does not require massive index
  reorganization in case of update, which is a
  major drawback of SVD

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Proposed ApproachGuarantee of no False Dismissal

• No qualified time sequence will be rejected, thus
  no false dismissal
• They show that this property holds for the Haar
  wavelet
• where

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The Overall Strategy

• Pre-processing
• Similarity Model Selection
• User can select Euclidean distance or v-shift
  similarity
• Haar wavelet transform is applied to time-series
• Index Construction
• Index structure such as R-tree is built using
  first few coefficients
• Range Query
• Nearest Neighbor Query

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Experimental Results
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Experimental Results (cont.)

• Scalability Test

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Conclusion

• Efficient time series matching through dimension
  reduction by Haar wavelet transform
• Outperforms DFT in terms of pruning power,
  scalability and complexity