1. Dynamic Time Warping Applications and Derivation
Charles Tappert
Seidenberg School of CSIS, Pace University

2. Dynamic Time Warping (DTW) non-linear/elastic matching, Viterbi algorithm
Many Applications
Speech recognition
Speech sound alignment
Speech sound generation
Online handwriting recognition

3. Dynamic Time Warping (DTW) non-linear/elastic matching, Viterbi algorithm
Derivation of a DTW algorithm variation (speech recognition)
A speech utterance is represented as a time sequence of feature vectors
Example

4. Dynamic Time Warping (DTW) non-linear/elastic matching, Viterbi algorithm
Consider a finite state machine model of a speech utterance prototype where the observable output from transitions between states is an acoustic feature vector which is a probabilistic function of the origin state of each transition
Note: some transitions cause stretching and others cause compression of the sequence of feature vectors produced

5. Dynamic Time Warping (DTW) non-linear/elastic matching, Viterbi algorithm
Background information
Univariate (one-dimensional) normal density function

6. Dynamic Time Warping (DTW) non-linear/elastic matching, Viterbi algorithm
Background information
Multivariate normal density function

7. Dynamic Time Warping (DTW) non-linear/elastic matching, Viterbi algorithm
In traversing each arc in this model a feature vector is produced with assumed underlying normal distribution
where i is the unknown, j the prototype, Vi are feature vectors of the unknown, Mj are mean feature vectors and sigma the covariance matrix of the prototype
This statistical characterization of prototypes would require multiple repetitions of the vocabulary to be recognized

8. Dynamic Time Warping (DTW) non-linear/elastic matching, Viterbi algorithm
To find the optimal overall probability of the model (prototype) generating the candidate, we estimate the maximum value of the cumulative probability over the possible paths through the model
Assuming statistical independence of the feature vectors, the best path to any point (i, j) and probability P(i, j) can be computed, starting with P(0, 0) = Prob(0, 0) and P(i, j) = 0 elsewhere, using the recursion relation

9. Dynamic Time Warping (DTW) non-linear/elastic matching, Viterbi algorithm
Taking the log of terms in previous equation, dropping constant terms, multiplying by -2, and assuming zero covariance terms yields the recursion relation
where D(i, j) is considered a cumulative distance measure

10. Dynamic Time Warping (DTW) non-linear/elastic matching, Viterbi algorithm
Further, assume equal variances and transition probabilities, and include an index k indicating the prototype
d(i, j; k) is the distance between feature vectors i and j
Note: since the log function is a monotonically increasing function of its argument and changing sign converts a maximizing relation into a minimizing one, this distance relation leads to the same decisions as the probability recursion relation, except for the simplifying assumptions

11. Dynamic Time Warping (DTW) non-linear/elastic matching, Viterbi algorithm
This derivation shows the simplifying assumptions made in going from a probabilistic model to a greatly simplified distance model
Currently, most string matching commercial and research systems use probabilistic models, and the Hidden Markov Model (HMM) is probably the dominant one
As computing power has increased over the years, more complex and primarily probabilistic models requiring large training corpuses have been used

12. Dynamic Time Warping (DTW) non-linear/elastic matching, Viterbi algorithm
In his research at IBM’s T.J. Watson Research Center, your instructor worked in both the speech recognition and the pen computing/handwriting recognition groups
founding member of the speech group (once over 50 workers)
spearheaded development of ThinkWrite handwriting recognizer in IBM’s pen-enabled ThinkPad product in the early 1990s
The data in both the speech and online handwriting problems are time sequences
Speech is recorded as a time waveform and usually transformed via frequency analysis into a sequence of spectral time samples
Online handwriting is captured as a time sequence of x-y coordinates describing the trajectory of the handwriting