Multi‐Scale Detection of Variance Changes in Renewal Processes in the Presence of Rate Change Points

Abstract

Non-stationarity of the rate or variance of events is a well-known problem in the description and analysis of time series of events, such as neuronal spike trains. A multiple filter test (MFT) for homogeneity of the rate has been proposed earlier that detects change points on multiple time scales simultaneously. It is based on a filtered derivative approach, and the rejection threshold derives from a Gaussian limit process L which is independent of the point process parameters.Here we extend the MFT to the analysis of variance homogeneity of life times. When the rate is constant, the MFT extends directly to the null hypothesis of constant variance and also to higher order moments under the assumption of homogeneity of lower order moments. In the presence of rate change points, we propose an adaptation of the test statistic. In order to ensure that rate changes are not falsely interpreted as variance changes, we propose to estimate the rate change points first, irrespective of potential variance changes, and incorporate these estimates in the statistical test for variance homogeneity. The resulting limit process shows slight deviations from L that depend on unknown process parameters. We use simulations to show that these deviations tend to be small for a wide range of parameter values, suggesting that the use of L does not considerably change the properties of the statistical test. Finally, the procedure is applied to empirical spike trains obtained from dopaminergic neurons in anaesthetized mice, indicating various profiles of rate and variance change points.

Bivariate change point detection: Joint detection of changes in expectation and variance

“…One way to tackle false ‐inference is to incorporate information about : if the empirical variances are centered correctly then ‐estimation will not systematically react falsely to changes in . In the context of stochastic point processes this was studied in Albert et al (2017). A second way of treating this problem is presented in this article: the joint observation of both processes.…”

Bivariate change point detection: joint detection of changes in expectation and variance

“…One way to tackle false variance inference is to incorporate information about the changes in the expectation: if one correctly centers within the empirical variances then variance change detection will not systematically react falsely on changes in expectation. In the context of stochastic point processes this was analyzed in Albert et al (2017). A second way of treating this problem is presented in this paper: the joint observation of both processes.…”