## Zt

(a) Minimum slope

(b) Maximum slope

Figure 6.22. Sakoe-Chiba slope constraints . Figure 6.23. An example of local continuity constraint expressed in terms of coordinate increments (Sakoe-Chiba local transition constraint with slope intensity of 1) [406, 484, 533],

w(k). This function depends only on the local path and controls the contribution of each local time distortion d[z(fc), j(k)].

Based on the local continuity constraint used, many slope weighting functions are possible. Sakoe and Chiba  have proposed the following four types of slope weighting functions and their effects on       Figure 6.24. Local continuity constraints studied by Myers et al. [406, 533].

DTW performance were extensively studied by Myers et al. :

w(k) = min[i(fc) - i(k - 1 ),j(k) - j(k - 1)] (6.63)

w{k) = max[i(k) - i{k - 1 ),j(k) - j(k - 1)] (6.64)

where it is assumed that ¿(0) = j(0) = 0 for initialization. Figure 6.25 illustrates the effects of weighting functions on Type III local continuity constraints . The numbers refer to particular weighting coefficient (calculated by the relevant formula) associated with each local path. Note that, since the increase in distortion will cause a decrease in the likelihood of proper matching, larger weightings will lead to less preferable paths. For instance, in Figure 6.25(b), Type (b) weighting will promote diagonal moves in the search grid.  (a) w(k)=min(i(k)-i(k-l)j(k)-j(k-l)) (b) w(k)=max(i(k)-i(k-l),j(k)-j(k-l))

(c) w(k)=i(k)-i(k-l) (d) w(k)=i(k)-i(k-l)+j(k)-j(k-l)

Figure 6.25. Sakoe-Chiba slope weightings for Type III local continuity constraint studied by Myers et al. [406, 533].

When these four types of weighting functions are applied to different types of local continuity constraints, sometimes 0 weight can be assigned to certain local paths. When this happens, weightings of inconsistent paths are smoothed by redistributing the existing weight equally on each move. This situation is illustrated using Sakoe and Chiba's Type II local constraint on Figure 6.26. Similar smoothing can be applied to different local continuity constraints given in Figure 6.24 if needed.

The calculation of accumulated distance in Eq. 6.55 requires an overall normalization to provide an average path distortion independent of the lengths of the two patterns being synchronized. The normalization factor N(w) is a function of the slope weighting type chosen such that