Time‐series clustering via quasi U‐statistics

Abstract

The problem of time-series discrimination and classification is discussed. We propose a novel clustering algorithm based on a class of quasi U-statistics and subgroup decomposition tests. The decomposition may be applied to any concave time-series distance. The resulting test statistics are proven to be asymptotically normal for either i.i.d. or non-identically distributed groups of time-series under mild conditions. We illustrate its empirical performance on a simulation study and a real data analysis. The simulation setup includes stationary vs. stationary and stationary vs. non-stationary cases. The performance of the proposed method is favourably compared with some of the most common clustering measures available.

Clustering and classification problems in genetics through U-statistics

“…Furthermore, we note that, for larger group sizes, the test achieves adequate power, and we thus recommend its use for homogeneity testing with around 20 samples or more. For smaller group sizes, the overall type I error of the uncorrected multiple U test approach of Valk & Pinheiro (2012) is not largely affected by multiple testing, and should be preferred due to its larger power.…”

“…Pinheiro et al (2009) show that B n is in the class of degenerate U-statistics (called quasi U-statistics) where the asymptotic distribution is normal with convergence rates L and/or n, even if the assumption of stochastic independence between samples does not hold. Adapting the results in Pinheiro et al (2009) to the context of time series, Valk & Pinheiro (2012) develop methods for classification and clustering analysis for stationary time series.…”

“…The procedure for assessing group homogeneity proposed by Valk & Pinheiro (2012) involves applying the U test for all possible group configurations. For large group sizes, when applying this strategy, we must take into account multiple testing issues.…”

“…We are interested in whether a new sample X * would be classified in group G 1 or G 2 . Valk & Pinheiro (2012) suggest a comparative approach based on statistics B 1 and B 2 , where B 1 is the statistics B n of (2.7) when the new sample is classified in group G 1 , and B 2 is defined likewise. Note that if X * is not well classified in G 2 , we might expect the statistic B 2 to be smaller than B n computed without including the new sample, since this increases the distances within group G 2 .…”

“…We consider the homogeneity test which uses the clustering algorithm given in Appendix A.1 to find the configuration with maximum normalized U test statistic and then correct for multiple testing through the max test. We compare these results with the approach of Valk & Pinheiro (2012) of multiple U tests, and with this same approach corrected for multiple testing through the Bonferroni correction. Table 1 presents the size of the homogeneity test, measured as the fraction of simulations under the null hypothesis for which H 0 was rejected, considering the theoretical α = 0.05.…”