# Detecting periodicities with Gaussian processes

Nicolas Durrande, Ecole des Mines, St-Etienne
James Hensman, University of Lancaster
Magnus Rattray, University of Manchester
Neil D. Lawrence, University of Sheffield

PeerJ Computer Science 4, pp 0-0

#### 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 Matérn 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.

  @Article{durrande-periodicities16, title = {Detecting periodicities with {G}aussian processes}, journal = {PeerJ Computer Science}, author = {Nicolas Durrande and James Hensman and Magnus Rattray and Neil D. Lawrence}, pages = {0}, year = {2016}, volume = {4}, month = {00}, edit = {https://github.com/lawrennd//publications/edit/gh-pages/_posts/2016-04-13-durrande-periodicities16.md}, url = {http://inverseprobability.com/publications/durrande-periodicities16.html}, 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 Matérn 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.}, key = {Durrande-periodicities16}, doi = {10.7717/peerj-cs.50}, linkpdf = {https://peerj.com/articles/cs-50.pdf}, OPTgroup = {} }
 %T Detecting periodicities with Gaussian processes %A Nicolas Durrande and James Hensman and Magnus Rattray and Neil D. Lawrence %B %C PeerJ Computer Science %D %F durrande-periodicities16 %J PeerJ Computer Science %P 0--0 %R 10.7717/peerj-cs.50 %U http://inverseprobability.com/publications/durrande-periodicities16.html %V 4 %X 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 Matérn 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. 
 TY - CPAPER TI - Detecting periodicities with Gaussian processes AU - Nicolas Durrande AU - James Hensman AU - Magnus Rattray AU - Neil D. Lawrence PY - 2016/04/13 DA - 2016/04/13 ID - durrande-periodicities16 SP - 0 EP - 0 DO - 10.7717/peerj-cs.50 L1 - https://peerj.com/articles/cs-50.pdf UR - http://inverseprobability.com/publications/durrande-periodicities16.html AB - 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 Matérn 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. ER - 
 Durrande, N., Hensman, J., Rattray, M. & Lawrence, N.D.. (2016). Detecting periodicities with Gaussian processes. PeerJ Computer Science 4:0-0