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Fast Subsequence Matching in Time-Series Databases. C. Faloustos, M. Ranganathan, Y. Manolopoulos

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Fast Subsequence Matching in Time-Series Databases. C. Faloustos, M. Ranganathan, Y. Manolopoulos Presented by George Liu / Luis L. Perez. Time series?. Definition Applications Financial markets Weather forecasting Healthcare. What kind of problem are we trying to solve?.

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slide1
Fast Subsequence Matching in Time-Series Databases.

C. Faloustos, M. Ranganathan,

Y. Manolopoulos

Presented by

George Liu / Luis L. Perez

time series
Time series?
  • Definition
  • Applications
    • Financial markets
    • Weather forecasting
    • Healthcare
what kind of problem are we trying to solve
What kind of problem are we trying to solve?
  • Whole sequence matching
    • Given a database S with n sequences, all of them equally long, and a query sequence Q of the same length.
    • Find all sequences in S that match with Q.
  • Subsequence matching
    • Given a database S with n sequences, with potentially different lengths, and a query sequence Q.
    • Find all sequences in S that contain Q.
useful notation
Useful notation
  • Given a sequence S
    • Len(S) denotes the length of the sequence
    • S[i] denotes the ith element
    • S[i:j] denotes the subsequence between S[i] and S[j]
  • Given two sequences, S and Q
    • D(S,Q) denotes the distance between S and Q.
      • Euclidean
  • Distance bound: e
    • Max. distance for two sequences to be considered “equal”
na ve approaches
Naïve approaches
  • Sequential scanning
    • Clearly unfeasible
  • R-tree
    • Might work, but dimensionality is extremely high (proportional to sequence length)‏
    • Poor performance
  • What can we do to improve performance?
dimensionality reduction
Dimensionality reduction
  • Redundant data, lots of patterns
  • Feature extraction
  • Data transformation
    • Cosine
    • Wavelet
    • Fourier <-- we\'ll focus on this.
discrete fourier transformation
Discrete Fourier Transformation
  • Map a sequence x in time-domain to a sequence X in frequency-domain
  • Reversible!
  • Fast and easy-to-implement algorithms
  • Energy preservation property
    • Key concept in dimensionality reduction.
    • Just keep the first 2 or 3 coefficients.
parseval s theorem
Parseval\'s theorem
  • Let S and Q be the original sequences.
  • S\' and Q\' after applying DFT. D(S,Q) = D(S\',Q\')
  • Why is this important?
  • Distance underestimation, remember the bound e.
    • D(S,Q) < e ---> D(S\', Q\') < e
    • We will get no false dismissals.
subsequence matching
Subsequence Matching
  • The problem:
    • You are given a collection of N sequences of real numbers. (S1, S2, .., Sn). Potentially different length.
    • User specifies query subsequence of length Q and the tolerance e, the max. acceptable dis-similarity.
    • You want all to return all the sequences along with the correct offsets k that matches the query and acceptable e.
  • Solutions:
    • many!
possible solutions
Possible Solutions
  • 1) Brute Force method - Sequential scan every possible subsequence of the data sequences for a match.
  • 2) I-Naive - Transform all subsequences to points in feature space and store those points into an R-tree.
  • 3) ST-Index - Transform all subsequences to points in feature space. Store MBRs of sub-trails into an R*-tree.
  • Note: I-Naive and ST-Index are similar in the initial steps.
possible solutions i naive
Possible Solutions I-naive
  • *Assume that the min. query length is w. w changes according to the application. (ie, stock markets have a larger w that are interested in weekly/monthly patterns)‏
  • Procedure:
    • 1) Use the "sliding window" to find every subsequence in a sequence.
    • 2) DFT those subsequences of size w to a point in featured space.
    • 3) A trail is produced of Len(S)-w+1 points.
possible solutions i naive1
Possible Solutions I-naive
  • Procedure cont:
    • 4) Store all the points of the trails in feature space in a spatial access method. (R*-tree)‏
    • 5) When presented with a query of length w and tolerance e, extract the features of the query and perform the spatial access range query with radius e.
    • 6) Discard false alarms by retrieving all those subsequences and calculating their actual distance from the query.
  • Note: Very, very slow approach. Worst that Sequential Scan. You have a large R*-tree (tall and slow).
possible solutions st index
Possible Solutions ST-Index
  • *Assume that the min. query length is w. w changes according to the application. (ie, stock markets have a larger w that are interested in weekly/monthly patterns)‏
  • Procedure:
    • 1) Use the "sliding window" to find every subsequence in a sequence.
    • 2) DFT those subsequences of size w to a point in featured space.
    • 3) A trail is produced of Len(S)-w+1 points.
possible solutions st index1
Possible Solutions ST-Index
  • Procedure cont.
    • 4) Divide the trail of points in feature space into sub-trails. (algorithm mentioned later)‏
    • 5) Represent each of them in a MBR.
    • 6) Store the MBR into a spatial access method. (ie. R*-Tree)‏
insertions
Insertions
  • Problem: How do we divide these trails into sub-trails?
    • Two heuristics:
      • 1) Every sub-trail has a predetermined, fixed number. (I-fixed)‏
      • 2) Every sub-trail has a predetermined, fixed length. (I-fixed)‏
  • Solution: Use an "adaptive heuristic." (I-adaptive)‏
i adaptive algorithm
I-adaptive Algorithm
  • - Based on the idea of the marginal cost of a point in terms of disk accesses.

Marginal cost (mc) = Disk Accesses of a given MBR / k points in a given MBR

  • Algorithm

Assign the first point of the trail in a sub-trail.

FOR each successive point

IF it increase the marginal cost of the current sub-trail

THEN start another sub-trail

ELSE include it in the current sub-trail

searching
Searching
  • Consider the sub-trail length w and distance bound e.
  • Let Q be the query sequence
  • If Len(Q) = w, it\'s all good.
    • Algorithm Search_Short:
      • Use DFT to map Q to a point q in feature space. Make it a sphere with radius e.
      • Retrieve all the sub-trails whose MBRs intersect the query region using our index.
      • Throw away false alarms.
searching1
Searching
  • Now, what if Len(Q) > w?
  • Requires more analysis, but basically we have that Len(Q) = p*w
  • So we can split Q in several subsequences of length p.
  • What about the radius? r = e/sqrt(p)‏
searching2
Searching
  • So we have...
    • Algorithm Search_Long:
      • Break sequence Q in p sub-queries with radius e/sqrt(p)‏
      • Retrieve from the index all the sub-trails whose MBRs insersect at least one of the other sub-query regions.
      • Examine the sub-sequences, discard false alarms.
experimental results1
Experimental results
  • Stock price database with ~300,000 points
  • 1 number = 4 bytes
  • DFT keeping first 3 coefficients (actually 6)
  • w = 512 bytes
  • R*-tree
experimental results2
Experimental results
  • Space
    • Naïve methods: 24mb
    • This method: 5kb
  • Time - “short” queries (Len(Q) = w)‏
    • 3 to 100 times better response times
  • Time - “long” queries (Len(Q) > w)‏
    • 10 to 100 times better response times
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