Dimension Reduction in Dissimilarity Spaces for Time Series Classification

Abstract

Time series classification in the dissimilarity space combines the advantages of elastic dissimilarity functions such as the dynamic time warping distance and the rich mathematical structure of Euclidean spaces. We applied dimension reduction using PCA followed by support vector learning on dissimilarity representations to 42 UCR datasets. The results suggest that time series classification in dissimilarity space has potential to complement the state-of-the-art, because the SVM classifiers perform better on the 42 datasets with higher confidence than the nearest-neighbor classifier based on the dynamic time warping distance.

A Performance Profiles

Performance profiles have been introduced by Dolan to compare the efficiency of algorithms [7]. Here, we use performance profiles to compare differences in the classification accuracy of a collection of classifiers on a set of classification problems. The comparison is summarized by one curve per classifier, which is easier to read than a table of classification accuracies.

To define a performance profile, we assume that $\mathbb {C}$ is a set of classifiers to be compared and $\mathbb {P}$ is the set of all classification problems. For each classification problem $p \in \mathbb {P}$ and each classifier $c \in \mathbb {C}$, we define

$$ \rho _{c,p} = \text {accuracy of classifier } c \in \mathbb {C} \text { on problem } p \in \mathbb {P} $$

as the performance of classifier c for problem p. In performance profiles, we do not consider the absolute performance of a classifier in terms of its classification accuracy, but its relative performance with respect to the best performing classifier. The classifier with the best performance on problem p has classification accuracy

$$ \rho _p^* = \max \mathop {\left\{ \rho _{\kappa , p} \,:\, \kappa \in \mathbb {C} \right\} }. $$

Then the relative performance of classifier c on problem p is given by

$$ r_{c, p} = 1-\frac{\rho _{c, p}}{\rho _p^*}. $$

The relative performance $r_{c, p}$ takes values from the interval [0, 1]. The better the performance of a classifier for a given problem, the lower is its relative performance. Thus, the lower the relative performance, the better the classifier. Moreover, from

$$ r_{c, p} \cdot \rho _p^* = \mathop {\left( 1-\frac{\rho _{c, p}}{\rho _p^*} \right) }\rho _p^* = \rho _p^* -\rho _{c, p} $$

follows that $r_{c,p}$ is the factor by which the classification accuracy $\rho _{c,p}$ deviates from the best classification accuracy $\rho _p^*$.

Finally, the performance profile of classifier $c \in \mathbb {C}$ over all problems $p \in \mathbb {P}$ is an empirical cumulative distribution function

$$ P_c(\tau ) = \frac{1}{\mathop {\left|\mathbb {P} \right|}} \mathop {\left|\mathop {\left\{ p \in \mathbb {P} \,:\, r_{c, p} \le \tau \right\} } \right|}. $$

It is sufficient to keep three three facts in mind to interpret performance profiles:

• 1. The value $P_c(0)$ is the fraction of problems on which classifier c is best.

• 2. $P_c(\tau )$ is the fraction of problems on which the performance of classifier c deviates at most by factor $\tau$ from the best performance.

• 3. $\tau _{\max }$ with $P_c(\tau _{\max }) = 1$ is the maximum factor by which classifier c deviates from the best performance.