A review on distance based time series classification

Semi-Metrification of the Dynamic Time Warping Distance

“…Equation (1) shows the first property of a prime factorization and equation (2) shows that pk is a factor of x. From the induction hypothesis follows that…”

“…( 1) and the transitivity of expansions follows that µ r 2 and µ s 2 are warping identical for all r, s ∈ N. Equation (2) yields δ(µ 1 2 , y) ≤ • • • ≤ δ(µ r 2 , y) for all time series y ∈ T and all r ∈ N. Hence, with increasing number r of replications, the Voronoi cell of site µ r 2 decreases, whereas the Voronoi cell of site µ 1 increases. (1) , x (2) . The means µ and µ are expansions of µ * .…”

“…Consequently, dtw-spaces are mathematically less structured than metric spaces. To overcome the inherent structural limitations of the dtw-distance, metric distances and positive semi-definite alignment kernels have been proposed [1,8,19]. The resulting (dis)similarities, however, differ from the original dtw-distance and therefore do not contribute to a better understanding of dtw-spaces.…”