Scalable multiple changepoint detection for functional data sequences

Abstract

We propose the multiple changepoint isolation (MCI) method for detecting multiple changes in the mean and covariance of a functional process. We first introduce a pair of projections to represent the variability "between" and "within" the functional observations. We then present an augmented fused lasso procedure to split the projections into multiple regions robustly. These regions act to isolate each changepoint away from the others so that the powerful univariate CUSUM statistic can be applied region-wise to identify the changepoints. Simulations show that our method accurately detects the number and locations of changepoints under many different scenarios. These include light and heavy tailed data, data with symmetric and skewed distributions, sparsely and densely sampled changepoints, and mean and covariance changes. We show that our method outperforms a recent multiple functional changepoint detector and several univariate changepoint detectors applied to our proposed projections. We also show that MCI is more robust than existing approaches and scales linearly with sample size. Finally, we demonstrate our method on a large time series of water vapor mixing ratio profiles from atmospheric emitted radiance interferometer measurements.

Elastic functional changepoint detection of climate impacts from localized sources

“…For future work, we look to extend the method to the multiple change‐point problem. This extension will not be trivial as scalability is a problem as shown in Harris et al (2022). Multiple changepoints can be detected in $$$ {S}_{n,k} $$$ if using a multiple peak detection scheme, but determining the $$$ p $$$‐value is not straightforward, and the test statistics will have to be defined carefully.…”

“…where Z l,i ind ∼  (0, 𝜎 2 l ), with (𝜎 1 , 𝜎 2 , 𝜎 3 ) = (0.14, 0.10, 0.08). Finally, each function was warped with an independent, random warping function with variance 0.2, generated similarly to Section 5.1 of Harris et al (2021). We consider results over 100 simulations for each value of n and each signal strength Δ ∈ {0, 0.04, 0.08, 0.12, 0.16}.…”

Break point detection for functional covariance

“…Aue et al (2020) dealt with analyzing structural break of spectrum and trace of covariance operator. Harris et al (2022) proposed a scalable multiple change point detection procedure which also handles changes in variance. In contrast to these methods, we study the structural break for the complete covariance structure.…”

Functional forecasting of dissolved oxygen in high‐frequency vertical lake profiles

“…FDA is an established framework for modeling environmental data. For example, Fortuna et al (2020) use basis function expansions to estimate ecological diversity profiles; Qu et al (2021) apply functional multivariate analysis of variance to temperature curves and bivariate wind speed data; and Harris et al (2022) detect change points in water vapor profiles using functional principal component analysis (FPCA). The particular family of FDA that we use in this work forecasts realizations from a functional time series (FTS) by estimating the functional principal components (FPCs) and corresponding FPC scores from the data.…”