Testing nonparametrically for dependence between nonstationary time series with very few replicates

Abstract

AbstractMany processes of scientific interest are nonstationary, meaning that they experience systematic changes over time. These processes pose a myriad of challenges to data analysis. One such challenge is the problem of testing for statistical dependence between two nonstationary time series. Existing tests mostly require strong modeling assumptions and/or are largely heuristic. If multiple independent and statistically identical replicates are available, a trial-swapping permutation test can be used. That is, within-replicate correlations (between time series ofXandYfrom the same replicate) can be compared to between-replicate correlations (betweenXfrom one replicate andYfrom another). Although this method is simple and largely assumption-free, it is severely limited by the number of replicates. In particular, the lowest attainablep-value is 1/n! wherenis the number of replicates. We describe a modified permutation test that partially alleviates this issue. Our test reports a lowerp-value of 1/nnwhen there is particularly strong evidence of dependence, and otherwise defaults to a regular trial-swapping permutation test. We use this method to confirm the observation that groups of zebrafish swim faster when they are aligned, using an existing dataset with only 3 biological replicates.