Exact mean computation in dynamic time war** spaces

Abstract

Averaging time series under dynamic time war** is an important tool for improving nearest-neighbor classifiers and formulating centroid-based clustering. The most promising approach poses time series averaging as the problem of minimizing a Fréchet function. Minimizing the Fréchet function is NP-hard and so far solved by several heuristics and inexact strategies. Our contributions are as follows: we first discuss some inaccuracies in the literature on exact mean computation in dynamic time war** spaces. Then we propose an exponential-time dynamic program for computing a global minimum of the Fréchet function. The proposed algorithm is useful for benchmarking and evaluating known heuristics. In addition, we present an exact polynomial-time algorithm for the special case of binary time series. Based on the proposed exponential-time dynamic program, we empirically study properties like uniqueness and length of a mean, which are of interest for devising better heuristics. Experimental evaluations indicate substantial deficits of state-of-the-art heuristics in terms of their output quality.

Appendices

A Performance profiles

To compare the performance of the mean algorithms, we used a slight variation of the performance profiles proposed by Dolan and Moré (2002). A performance profile is a cumulative distribution function for a performance metric. Here, the chosen performance metric is the error percentage from the exact solution.

To define a performance profile, we assume that ${\mathbb {A}}$ is a set of mean algorithms and ${\mathbb {S}}$ is a set of samples each of which consists of k time series. For each sample ${\mathcal {X}} \in {\mathbb {X}}$ and each mean algorithm $A \in {\mathbb {A}}$, we define $E_{A,{\mathcal {S}}}$ as the error percentage obtained by applying algorithm A on sample ${\mathcal {S}}$. The performance profile of algorithm A over all samples ${\mathcal {S}} \in {\mathbb {S}}$ is the empirical cumulative distribution function defined by

$$\begin{aligned} P_A(\tau ) = \frac{1}{\mathop {\left|{\mathbb {S}} \right|}} \mathop {\left|\mathop {\left\{ {\mathcal {S}} \in {\mathbb {S}}{:}\, E_{A,{\mathcal {S}}} \le \tau \right\} } \right|} \end{aligned}$$

for all $\tau \ge 0$. Thus, $P_A(\tau )$ is the estimated probability that the error percentage of algorithm A is at most $\tau$. The value $P_A(0)$ is the estimated probability that algorithm A finds an exact solution.

B Detailed results

See Tables 5, 6 and 7.

Table 5 Average error percentage of the five heuristics on samples ${\mathcal {S}}_{\text {ucr}}$ of size $k = 2$ grouped by UCR data set

Table 6 Average error percentage of the five heuristics on ${\mathcal {S}}_{\text {rw}}$-samples of size $k = 2$ grouped by length n of random walks

Table 7 Average error percentage of the five heuristics on ${\mathcal {S}}_{\text {rw}}^k$-samples of varying sample size k