Testing for the absence of correlation between two spatial or temporal sequences

The purpose of this paper is to elucidate the problem of testing for the absence of correlation between the trajectories of two stochastic processes. It is assumed that the process is homogeneous on a pre-specified partition of the index set. The hypothesis testing methodology developed in this article consists in estimating codispersion coefficients on each subset of the partition, and in testing for the simultaneous nullity of the coefficients. To this aim, the Mahalanobis distance between the observed and theoretical codispersion vectors is used to define a test statistic, which converges to a chi-square distribution under the null hypothesis. Three examples in the context of signal processing and spatial models are discussed to point out the advantages and limitations of our proposal. Simulation studies are carried out to explore both the distribution of the test statistic under the null hypothesis and its power function. The method introduced in this paper has potential applications in time series where it is of interest to measure the comovement of two temporal sequences. The proposed test is illustrated with a real data set. Two signals are compared in terms of comovement to validate two confocal sensors in the context of biotechnology. The analysis carried out using this technique is more appropriate than previous validation tests where the mean values were compared via t test and Wilcoxon signed rank test ignoring the correlation within and across the series.