[Paper Review] Gaussian process models for periodicity detection

Nicolas Durrande, James Hensman+2 more|arXiv (Cornell University)|Mar 28, 2013

Gaussian Processes and Bayesian Inference26 references8 citations

TL;DR

This paper proposes a Gaussian process framework for detecting periodic components in noisy, sparse time series by decomposing the covariance kernel into periodic and aperiodic sub-kernels using reproducing kernel Hilbert space (RKHS) theory. The method leverages Matérn kernels to derive a periodicity ratio that quantifies the proportion of signal variance explained by periodicity, validated on circadian gene expression data with improved detection of periodic genes compared to standard methods.

ABSTRACT

We consider the problem of detecting and quantifying the periodic component of a function given noise-corrupted observations of a limited number of input/output tuples. Our approach is based on Gaussian process regression which provides a flexible non-parametric framework for modelling periodic data. We introduce a novel decomposition of the covariance function as the sum of periodic and aperiodic kernels. This decomposition allows for the creation of sub-models which capture the periodic nature of the signal and its complement. To quantify the periodicity of the signal, we derive a periodicity ratio which reflects the uncertainty in the fitted sub-models. Although the method can be applied to many kernels, we give a special emphasis to the Mat\'ern family, from the expression of the reproducing kernel Hilbert space inner product to the implementation of the associated periodic kernels in a Gaussian process toolkit. The proposed method is illustrated by considering the detection of periodically expressed genes in the arabidopsis genome.

Motivation & Objective

• To develop a principled method for detecting periodic components in functions from sparse, noisy observations.
• To address the limitation of traditional harmonic analysis by using RKHS inner products instead of L2 inner products for periodic decomposition.
• To quantify periodicity with uncertainty-aware metrics, enabling robust detection in noisy biological data.
• To extend Gaussian process models with periodic and aperiodic sub-models for improved signal decomposition and prediction.
• To apply the method to detect periodically expressed genes in circadian clock studies, improving on existing detection methods.

Proposed method

• The method decomposes a Gaussian process covariance function k into periodic kp and aperiodic ka components using RKHS theory.
• It derives the RKHS inner product for Matérn kernels, enabling orthogonal projection of signals onto periodic subspaces.
• The periodic kernel kp is constructed using the spectral representation of Matérn kernels, ensuring smoothness and periodicity.
• A periodicity ratio is defined as the ratio of the variance explained by the periodic sub-model to the total variance, incorporating uncertainty via GP marginal likelihood.
• The approach uses a probabilistic framework to estimate the periodic component and its uncertainty, avoiding overfitting.
• The implementation is integrated into the GPy GP toolbox, enabling scalable inference on large datasets.

Experimental results

Research questions

• RQ1 Can a Gaussian process model be decomposed into periodic and aperiodic sub-models using RKHS theory to isolate periodic components?
• RQ2 How can the periodicity of a signal be quantified with uncertainty, rather than just detecting a single dominant frequency?
• RQ3 What is the impact of using RKHS inner products instead of L2 inner products for periodic signal extraction?
• RQ4 How does the proposed method compare to traditional harmonic analysis and folding methods in detecting periodicity in noisy, sparse data?
• RQ5 Can the method detect biologically relevant periodically expressed genes in real-world genomics data?

Key findings

• The proposed method successfully detects periodic components in time series with high accuracy, even under noisy and sparse observation conditions.
• The periodicity ratio provides a statistically sound, uncertainty-aware measure of periodicity, outperforming traditional methods in signal decomposition.
• The method identified four new genes with strong periodic expression patterns, consistent with circadian clock regulation.
• The decomposition into periodic and aperiodic sub-models improves predictive performance on the Mauna Loa CO2 dataset, demonstrating robustness.
• The derived RKHS inner products for Matérn kernels enable precise and efficient computation of periodic components in the GP framework.
• The implementation in GPy allows for scalable, reproducible analysis of large-scale biological time series.