Fast Subsequence Matching in Time-Series Databases

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Fast Subsequence Matching in Time-Series Databases

• Christos Faloutsos
• M. Ranganathan
• Yannis Manolopoulos
• Department of Computer Science and ISR
• University of Maryland at College Park

Presented by Rui Li
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Abstract

• Goal To find an efficient indexing method to
  locate time series in a database
• Main Idea
• Map each time series into a small set of
  multidimensional rectangles in feature space
• Rectangles can be readily indexed using
  traditional spatial access methods, e.g., R-tree

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Introduction

• Hot Problem Searching similar patterns in
  time-series databases
• Applications
• financial, marketing and production time series,
  e.g. stock prices
• scientific databases, e.g. weather, geological,
  environmental data

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Introduction (cont.)

• Similarity Queries
• Whole Matching
• Subsequence Matching
• partial matching
• report time series along with offset

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Introduction (cont.)

• Whole Matching (Previous Work)
• Use a distance-preserving transform (e.g., DFT)
  to extract f features from time series (e.g., the
  first f DFT coefficients), and then map them into
  points in the f-dimensional feature space
• Spatial access method (e.g., R-trees) can be
  used to search for approximate queries

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Introduction (cont.)

• Subsequence Matching (Goal)
• Map time series into rectangles in feature space
• Spatial access methods as the eventual indexing
  mechanism

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Background

• To guarantee no false dismissals for range
  queries, the feature extraction function F()
  should satisfy the following formula
• Parseval Theorem
• The DFT preserves the Euclidean distance between
  two time series

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Proposed Method

• Mapping each time series to a trail in feature
  space
• Use a sliding window of size w and place it at
  every possible offset
• For each such placement of the window, extract
  the features of the subsequence inside the window
• A time series of length L is mapped to a trail in
  feature space, consisting of
• L-w1 points one point for each offset

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• Example1

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• Example2
• (a) a sample stock-price time series
• (b) its trail in the feature space of the 0-th
  and 1-st DFT coefficients
• (c) its trail of the 1-st and 2-nd DFT
  coefficients

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Proposed Method (cont.)

• Indexing the trails
• Simply storing the individual points of the trail
  in an R-tree is inefficient
• Exploit the fact that successive points of the
  trail tend to be similar, i.e., the contents of
  the sliding window in nearby offsets tend to be
  similar
• Divide the trail into sub-trails and represent
  each of them with its minimum bounding
  (hyper)-rectangle (MBR)
• Store only a few MBRs

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Proposed Method (cont.)

• Indexing the trails (cont.)
• Can guarantee no false dismissals when a query
  arrives, all the MBRs that intersect the query
  region are retrieved, i.e., all the qualifying
  sub-trails are retrieved, plus some false alarms

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• Return to example1

e
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Proposed Method (cont.)

• Indexing the trails (cont.)
• Map a time series into a set of rectangles in
  feature space
• Each MBR corresponds to a sub-trail

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Proposed Method (cont.)

• For each MBR we have to store
• , which are the offsets of the first
  and last such positionings
• A unique identifier for each time series
• The extent of the MBR in each dimension, i.e.,
• Store the MBRs in an R-tree
• Recursively group the MBRs into parent MBRs,
  grandparent MBRs, etc.

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• Example1 (cont.)
• assuming a fan-out of 4

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Proposed Method (cont.)

• The structure of a leaf node and a non-leaf node

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Proposed Method (cont.)

• Two questions
• Insertions when a new time series is inserted,
  what is a good way to divide its trail into
  sub-trails
• Queries how to handle queries, especially the
  ones that are longer than the sliding window

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Proposed Method (cont.)

• Insertion Dividing trails into sub-trails
• Goal Optimal division so that the number of disk
  accesses is minimized

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• Example3fixed heuristic adaptive
  heuristic

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Proposed Method (cont.)

• Insertion (cont.)
• Group trail-points into sub-trails by means of an
  adaptive heuristic
• Based on a greedy algorithm, using a cost
  function to estimate the number of disk accesses
  for each of the options

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Proposed Method (cont.)

• Insertion (cont.)
• The cost functionwhere is the
  sides of the n-dimensional MBR of a node in an
  R-tree
• The marginal cost of each point
  where k is the number of points in this MBR

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Proposed Method (cont.)

• Insertion (cont.)
• AlgorithmAssign the first point of the trail to
  a sub-trail (would be a predefined small MBR)FOR
  each successive point IF it increases the
  marginal cost of the current sub-trail THEN
  start a new sub-trail ELSE include it into the
  current sub-trail

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Proposed Method (cont.)

• Insertion (cont.)
• The algorithm may not work well under certain
  circumstances
• The algorithms goal is to minimize the size of
  each MBR, why dont we use clustering techniques!

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Proposed Method (cont.)

• Searching Queries longer than w
• If Len(Q)w, the searching algorithm goes like
• Map Q to a point q in the feature space the
  query corresponds to a sphere with center q and
  radius e
• Retrieve the sub-trails whose MBRs intersect the
  query region
• Examine the corresponding time series, and
  discard the false alarms

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Proposed Method (cont.)

• Searching (cont.)
• If Len(Q)gtw, consider the following Lemma
• Consider two sequences Q and S of the same length
  Len(Q)Len(S)pw
• Consider their p disjoint subsequences
• andwhere
• If Q AND S agree within tolerance e, then at
  least one of the pairs of corresponding
  subsequence agree within tolerance

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Proposed Method (cont.)

• Searching (cont.)
• If Len(Q)gtw, the searching algorithm goes like
• The query time series Q is broken into p
  sub-queries which correspond to p spheres in the
  feature space with radius
• Retrieve the sub-trails whose MBRs intersect at
  least one of the sub-query regions
• Examine the corresponding subsequences of the
  time series, and discard the false alarms

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Experiments

• Experiments are ran on a stock prices database of
  329,000 points
• Only the first 3 frequencies of the DFT are used
  thus the feature space has 6 dimensions (real and
  imaginary parts of each retained DFT coefficient)
• Sliding window size w512

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Experiments (cont.)

• Query time series were generated by taking random
  offsets into the time series and obtaining
  subsequences of length Len(Q) from those offsets

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Experiments (cont.)

• For groups of experiments were carried out
• Comparison of the proposed method against the
  method that has sub-trails with only one point
  each
• Experiments to compare the response time
• Experiments with queries longer than w
• Experiments with larger databases

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Related Works (citations)

• Continuous queries over data streams
• Similarity indexing with M-tree/SS-tree, etc.
• Efficient time series matching by wavelets
• Fast similarity search in the presence of noise,
  scaling, and translation in time-series databases

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Thank you!