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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?

- 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

- 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”

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

- Redundant data, lots of patterns
- Feature extraction
- Data transformation
- Cosine
- Wavelet
- Fourier <-- we\'ll focus on this.

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

- 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

- 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

- 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

- *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-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

- *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-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

- 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

- - 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

- 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.

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)

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 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 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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