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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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)‏
• - 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.
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