Modelling Time-Varying First and Second-Order Structure of Time Series via Wavelets and Differencing

Euan T. McGonigle; Rebecca Killick; Matthew A> Nunes

doi:10.1214/22-EJS2044

Modelling Time-Varying First and Second-Order Structure of Time Series via Wavelets and Differencing

Euan T. McGonigle, Rebecca Killick, Matthew A> Nunes

Research output: Contribution to journal › Article › peer-review

2 Citations (SciVal)

Abstract

Most time series observed in practice exhibit time-varying trend
(first-order) and autocovariance (second-order) behaviour. Differencing is
a commonly-used technique to remove the trend in such series, in order
to estimate the time-varying second-order structure (of the differenced se-
ries). However, often we require inference on the second-order behaviour
of the original series, for example, when performing trend estimation. In
this article, we propose a method, using differencing, to jointly estimate
the time-varying trend and second-order structure of a nonstationary time
series, within the locally stationary wavelet modelling framework. We de-
velop a wavelet-based estimator of the second-order structure of the original
time series based on the differenced estimate, and show how this can be
incorporated into the estimation of the trend of the time series. We perform
a simulation study to investigate the performance of the methodology, and
demonstrate the utility of the method by analysing data examples from
environmental and biomedical science.

Original language	English
Pages (from-to)	4398-4448
Number of pages	51
Journal	Electronic Journal of Statistics
Volume	16
Issue number	2
Early online date	22 Aug 2022
DOIs	https://doi.org/10.1214/22-EJS2044
Publication status	Published - 31 Dec 2022

Bibliographical note

Funding Information:
arXiv: 2108.07550 ∗E.T. McGonigle gratefully acknowledges financial support from EPSRC and Numerical Algorithms Group Ltd. via The Smith Institute i-CASE award No. EP/R511997/1. R. Killick gratefully acknowledges funding from EP/R01860X/1.

Keywords

Differencing
locally stationary time series
trend estimation
wavelet spectrum
wavelet thresholding

ASJC Scopus subject areas

Statistics and Probability
Statistics, Probability and Uncertainty

Access to Document

10.1214/22-EJS2044Licence: CC BY

Cite this

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title = "Modelling Time-Varying First and Second-Order Structure of Time Series via Wavelets and Differencing",

abstract = "Most time series observed in practice exhibit time-varying trend(first-order) and autocovariance (second-order) behaviour. Differencing isa commonly-used technique to remove the trend in such series, in orderto estimate the time-varying second-order structure (of the differenced se-ries). However, often we require inference on the second-order behaviourof the original series, for example, when performing trend estimation. Inthis article, we propose a method, using differencing, to jointly estimatethe time-varying trend and second-order structure of a nonstationary timeseries, within the locally stationary wavelet modelling framework. We de-velop a wavelet-based estimator of the second-order structure of the originaltime series based on the differenced estimate, and show how this can beincorporated into the estimation of the trend of the time series. We performa simulation study to investigate the performance of the methodology, anddemonstrate the utility of the method by analysing data examples fromenvironmental and biomedical science.",

keywords = "Differencing, locally stationary time series, trend estimation, wavelet spectrum, wavelet thresholding",

author = "McGonigle, {Euan T.} and Rebecca Killick and Nunes, {Matthew A>}",

note = "Funding Information: arXiv: 2108.07550 ∗E.T. McGonigle gratefully acknowledges financial support from EPSRC and Numerical Algorithms Group Ltd. via The Smith Institute i-CASE award No. EP/R511997/1. R. Killick gratefully acknowledges funding from EP/R01860X/1. ",

year = "2022",

month = dec,

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doi = "10.1214/22-EJS2044",

language = "English",

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AU - Nunes, Matthew A>

N1 - Funding Information: arXiv: 2108.07550 ∗E.T. McGonigle gratefully acknowledges financial support from EPSRC and Numerical Algorithms Group Ltd. via The Smith Institute i-CASE award No. EP/R511997/1. R. Killick gratefully acknowledges funding from EP/R01860X/1.

PY - 2022/12/31

Y1 - 2022/12/31

N2 - Most time series observed in practice exhibit time-varying trend(first-order) and autocovariance (second-order) behaviour. Differencing isa commonly-used technique to remove the trend in such series, in orderto estimate the time-varying second-order structure (of the differenced se-ries). However, often we require inference on the second-order behaviourof the original series, for example, when performing trend estimation. Inthis article, we propose a method, using differencing, to jointly estimatethe time-varying trend and second-order structure of a nonstationary timeseries, within the locally stationary wavelet modelling framework. We de-velop a wavelet-based estimator of the second-order structure of the originaltime series based on the differenced estimate, and show how this can beincorporated into the estimation of the trend of the time series. We performa simulation study to investigate the performance of the methodology, anddemonstrate the utility of the method by analysing data examples fromenvironmental and biomedical science.

AB - Most time series observed in practice exhibit time-varying trend(first-order) and autocovariance (second-order) behaviour. Differencing isa commonly-used technique to remove the trend in such series, in orderto estimate the time-varying second-order structure (of the differenced se-ries). However, often we require inference on the second-order behaviourof the original series, for example, when performing trend estimation. Inthis article, we propose a method, using differencing, to jointly estimatethe time-varying trend and second-order structure of a nonstationary timeseries, within the locally stationary wavelet modelling framework. We de-velop a wavelet-based estimator of the second-order structure of the originaltime series based on the differenced estimate, and show how this can beincorporated into the estimation of the trend of the time series. We performa simulation study to investigate the performance of the methodology, anddemonstrate the utility of the method by analysing data examples fromenvironmental and biomedical science.

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KW - wavelet thresholding

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Modelling Time-Varying First and Second-Order Structure of Time Series via Wavelets and Differencing

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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Multiscale Machine Learning in Resource-constrained Environments

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Modelling Time-Varying First and Second-Order Structure of Time Series via Wavelets and Differencing

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Embargoed Document

Fingerprint

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Multiscale Machine Learning in Resource-constrained Environments

Cite this