[Paper Review] Bayesian Change Point Detection for Functional Data

Xiuqi Li, Subhashis Ghosal|arXiv (Cornell University)|Aug 3, 2018

Metabolomics and Mass Spectrometry Studies12 references3 citations

TL;DR

This paper proposes a Bayesian change point detection method for functional data using discrete wavelet transforms (DWT) to extract features, modeling each feature's change points independently with wavelet-specific priors. The method identifies change points by minimizing the ratio of between-group to within-group similarity in a similarity matrix, successfully detecting climate data change points at 1914, 1839, and 1969 with high accuracy under varying noise levels.

ABSTRACT

We propose a Bayesian method to detect change points for functional data. We extract the features of a sequence of functional data by the discrete wavelet transform (DWT), and treat each sequence of feature independently. We believe there is potentially a change in each feature at possibly different time points. The functional data evolves through such changes throughout the sequences of observations. The change point for this sequence of functional data is the cumulative effect of changes in all features. We assign the features with priors which incorporate the characteristic of the wavelet coefficients. Then we compute the posterior distribution of change point for each sequence of feature, and define a matrix where each entry is a measure of similarity between two functional data in this sequence. We compute the ratio of the mean similarity between groups and within groups for all possible partitions, and the change point is where the ratio reaches the minimum. We demonstrate this method using a dataset on climate change.

Motivation & Objective

• To develop a Bayesian framework for detecting change points in functional data, particularly when changes occur gradually across multiple features.
• To address the challenge of detecting structural changes in functional data where the overall change is a cumulative effect of shifts in individual features.
• To incorporate wavelet coefficient characteristics into priors to improve sensitivity and specificity in change point detection.
• To enable recursive detection of multiple change points by partitioning data after each detection, allowing hierarchical segmentation of functional sequences.
• To validate the method on real-world climate data, demonstrating robustness under increasing noise levels and identifying significant climatic transition periods.

Proposed method

• Apply the discrete wavelet transform (DWT) to functional data to extract wavelet coefficients, treating each coefficient sequence independently as a univariate time series.
• Assign wavelet-specific priors to the wavelet coefficients to reflect their known sparsity and localization properties, enhancing detection sensitivity.
• Compute the posterior distribution of change points for each feature sequence using Bayesian inference, incorporating uncertainty in the estimates.
• Define a similarity matrix between functional observations based on the wavelet coefficient sequences, measuring pairwise similarity across the data.
• For each potential change point, compute the ratio of mean similarity between groups (before and after the point) to within-group similarity; the change point is selected where this ratio is minimized.
• Apply the method recursively to subgroups formed by detected change points, enabling detection of multiple change points until a stopping criterion (e.g., minimal similarity difference) is met.

Experimental results

Research questions

• RQ1 Can a Bayesian approach effectively detect change points in functional data by modeling feature-level changes through wavelet transforms?
• RQ2 How does the method perform in detecting change points when noise levels vary, particularly in real-world climate data?
• RQ3 To what extent does the cumulative effect of changes across multiple wavelet features lead to a more accurate overall change point detection than single-feature methods?
• RQ4 Can the method recursively detect multiple change points in a hierarchical manner, and how does it perform on long sequences with multiple structural shifts?
• RQ5 How does the use of wavelet-specific priors improve the detection accuracy compared to non-informative or standard priors?

Key findings

• The method successfully detected the true change points at 1914, 1839, and 1969 in the Berkeley Earth climate data, with the first detection corresponding to a major shift in global temperature patterns.
• Under low noise (variance 0.01), the method detected all three true change points exactly, matching the ground truth in simulation studies.
• With moderate noise (variance 0.1), the method detected most change points correctly, including 26, 49, and 68–75, while the E-Divisive method showed some deviation.
• Even under high noise (variance 1), the method detected multiple change points (e.g., 5, 24, 40, 46, 94, 99), indicating robustness to noise, though with increased false positives.
• The hierarchical application of the method identified 15 subgroups in the climate data, revealing a complex, multi-level structure of climatic shifts over time.
• The similarity ratio criterion effectively identified change points, with the minimum ratio corresponding to the most significant structural break in the data sequence.