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Dynamic Time Warping (DTW) J.-S Roger Jang ( 張智星 ) MIR LabMIR Lab, CSIE Dept National Taiwan University

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Dynamic Time Warping Goal To align two sequences under certain constraints, such that the distance between these two sequences is as small as possible. Method Dynamic programming

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Distance between Same-length Sequences Distance between Alignment

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Distance between Different-length Sequences

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Alignment Constraints: Type 1 Temporal constraints Other alignment constraints One-to-one mapping No consecutive skip-over x1x1 x2x2 x3x3 y1y1 x4x4 x5x5 y2y2 y3y3 y4y4 y5y5 y6y6 y7y7 y8y8

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Alignment Constraints: Type 2 Temporal constraints Other alignment constraints 1-to-1, 1-to-many, or many-to-1 mapping No skip-over x1x1 x2x2 x3x3 y1y1 x4x4 x5x5 y2y2 y3y3 y4y4 y5y5 y6y6 y7y7 y8y8

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Type-1 DTW: Table Fillup i j x(i-1) y(j) x, y: input vector/matrix Local paths: degrees DTW formulation: y(j-1) x(i)

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Type-2 DTW: Table Fillup x, y: input vector/matrix Local paths: degrees DTW formulation: i j x(i-1) y(j) y(j-1) x(i)

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Local Path Constraints zType 1: y local paths zType 2: y local paths

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-10- Path Penalty for Type-1 DTW zAlignment path of type-1 DTW y45-degree paths are likely to be avoided since we are minimizing the total distance. ySo we can add penalty for paths deviated from 45-degree.

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-11- Path Penalty for Type-2 DTW zAlignment path of type-1 DTW y45-degree paths are likely to be taken since we are minimizing the total distance. ySo we can add penalty for paths of 45-degree.

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-12- Other Minutes about DTW zTypical applications ySpeech recognition: MFCC matrices as inputs (where x(i) is the MFCC vector of frame i) yQuery by singing/humming: Pitch vectors as inputs (where x(i) is the pitch value of frame i) zAbundant variants for various applications yRecurrent formulas yLocal path constraints

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-13- Applications zApplications of DTW yDTW for speech recognitionDTW for speech recognition yDTW for query by singing/hummingDTW for query by singing/humming

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