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Fast Subsequence Matching in TimeSeries Databases

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Fast Subsequence Matching in Time-Series Databases. Author: Christos Faloutsos etc. ... False alarm: non-qualified sequence not discarded, Not so bad. Discrete ... – PowerPoint PPT presentation

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Title: Fast Subsequence Matching in TimeSeries Databases


1
Fast Subsequence Matching in Time-Series Databases
  • Author Christos Faloutsos etc.
  • Speaker Weijun He

2
What is the problem?
  • What is Time Series 1-dimensional data
  • e.g. Daily stock market price,
  • Daily temperature, etc
  • Our goal
  • Design fast searching methods that will locate
    subsequence that match a query subsequence,
    exactly or approximately

3
Motivation/Application
  • Financial, marketing, production
  • Typical query
  • find companies whose stock prices move
    similarly
  • Scientific databases
  • Typical query
  • find past days in which solar magnetic wind
    showed similar patterns as todays

4
Some notational conventions
  • If S and Q are two sequences, then
  • Len(S) length of S
  • Sij subsequence including i and j
  • Si i-th entry of S
  • D(S,Q) distance of two equal length
  • sequence S and Q

5
Queries
  • Two categories for queries
  • Whole Mathing len(data) len(query)
  • Subsequence Matching
  • len(data) gt len(query)
  • Remark
  • The distance function D(S,Q) is defined, e.g. D()
    can be the Euclidean distance
  • Matching means D(S,Q) lt ?, i.e., approximately

6
Whole Matching
  • Any distance-preserving transform(e.g., Discrete
    Fourier Transform(DFT),extract f features from
    sequences(e.g., the first f DFT coefficients)
    f-dimensional feature space
  • Any spatial access method(e.g., R-tree) can be
    used for range/approximate queries

7
Mathematical Background
  • Lemma 1
  • To guarantee no false dismissals for range
    queries, the feature extraction function F()
    should satisfy the following formula
  • Dfeature(F(O1),F(O2))ltDobject(O1,O2)
  • False dismissal discard the qualified sequence,
    BAD
  • False alarm non-qualified sequence not
    discarded, Not so bad

8
Discrete Fourier Transform
  • Theorem(Parseval)
  • ?i0,..,n-1Xi2 ?f0,..,n-1Xf2 (distance
    preserving)
  • DFT is a linear transform, so it can be proved
    that
  • DFT satisfy Lemma 1.
  • We Keep the first few(2-3) coefficients as
    features
  • Properties 1. Only false alarm, no false
    dismissal
  • 2. Practically, false alarms
    are few

9
From Whole to Subsequence matching
  • Question
  • How to generalize the method to approximate
    match queries for subsequences of arbitrary
    length?

10
Subsequence MatchingCriterion
  • Some criterion
  • Fast sequential scanning and distance
    calculation at each and every possible offset is
    too slow for large databases
  • Correct No false dismissals, but false
    alarms are acceptable
  • Small space overhead
  • Dynamic
  • Varying length for data and query sequences

11
Proposed Method
  • Using Sliding window of w, minimum query length.
  • A data sequence of length Len(S) is mapped to
    a trail in feature space, consisting of
    len(S)-w1 points.
  • Sub-Trail-index

12
I-naïve method
  • The straightforward way is
  • keep track of the individual points of each
    trail, storing them in spatial access method
  • Disadvantage
  • Inefficient since almost every point in a data
    sequence will correspond to a point in the
    f-dimensional feature space.

13
I-naïve method Contd.
  • How to improve
  • Observation the content of the sliding window in
    nearby offset will be similar.
  • Solution Divide the trail into sub-trails and
    represent each of them with its Minimum Bounding
    Rectangle (MBR), thus we only need to store a few
    MBRs, no false dismissals are guaranteed.

14
Illustration
15
MBR Property
  • Each MBR corresponds to a whole sub-trail, i.e.,
    points in feature space that correspond to
    successive positions of the sliding window.
  • Each leaf-MBR has tstart, tend which are the
    offsets of the first and last such positions,
    also has a unique identifier for the data
    sequence (sequence_id)
  • The extent of the MBR in each dimension is
    denoted as
  • (F1low,F1high, F2low,F2high,……)
  • MBR are stored in R tree.

16
Figure2 Structure of a leaf node and a non-leaf
node index node layout for the last two levels
17
ST-index
  • There are two questions for ST-index
  • Insertion (Dynamic requirement)
  • when new data sequence is inserted, what is a
    good way to divide its trail into sub-trail?
  • Queries longer than w
  • how to handle queries, especially the ones
    longer than w.

18
ST-index Insertion
19
Illustration
20
I-adaptive heuristic
  • Cost function
  • DA(L)?(Li0.5) where L(L1,L2,..Ln), 1ltiltn.
  • Marginal cost of a point
  • Consider a sub-trail of K points with a MBR of
    sizes L1,…Ln, each point in this sub-trail has
  • mcDA(L) /k

21
I-adaptive heuristic algorithm
  • / Algorithm Divide-to-Subtrails /
  • Assign the first point of the trail in a
    (trivial) 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

22
Searching-Queries longer than W
  • Two methods
  • PrefixSearch
  • select the prefix of Q of length w, match the
    prefix within tolerance e
  • MultiPiece Search
  • Suppose the query sequence has length pw,
  • Break Q into p sub-queries which correspond to p
    sphere in feature space with raius e/sqrt(p)
  • Use ST-index to retrieve the sub-trails whose
    MBRs intersect at least one of the sub-query
    region.

23
Prefix vs. MultiPiece search
  • Volume required in feature space(K is a
    constant)
  • Prefixsearch K ef
  • Multipiece Kp(e/sqrt(p))f
  • Multipiece is likely to produce fewer false
    alarms

24
Conclusions
  • The main contribution is I-adaptive method
  • achieves orders of magnitude savings over the
    sequential scanning.
  • Small space overhead
  • It is dynamic
  • No false dismissal
  • Future work
  • Extend this method for 2-dimensional gray scale
    images, and in general for n-dimensional
    vector-fields(e.g. 3-d MRI brain scans)

25
The End
  • Thank you for your attention!
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