A wavelet-based approach for imputation in nonstationary multivariate time series

Matthew A Nunes,Idris A Eckley,Timothy Park,Rebecca E Wilson

doi:10.1007/s11222-021-09998-2

Matthew A Nunes, Idris A Eckley + Show 2 more

Open Access

https://doi.org/10.1007/s11222-021-09998-2

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Abstract

Many multivariate time series observed in practice are second order nonstationary, i.e. their covariance properties vary over time. In addition, missing observations in such data are encountered in many applications of interest, due to recording failures or sensor dropout, hindering successful analysis. This article introduces a novel method for data imputation in multivariate nonstationary time series, based on the so-called locally stationary wavelet modelling paradigm. Our methodology is shown to perform well across a range of simulation scenarios, with a variety of missingness structures, as well as being competitive in the stationary time series setting. We also demonstrate our technique on data arising in a health monitoring application.

Highlights

Time series data arise in a variety of different areas including finance (Taylor 2007), biology (Bar-Joseph et al 2003) and energy (Alvarez et al 2011; Doucoure et al 2016)
The local wavelet spectral (LWS) matrix and the local auto- and cross-covariance structure are important quantities within the imputation method we propose in Sect. 3 as they are used within the prediction step to estimate missing values
In order to assess the performance of the imputation methods, we consider a modified version of the root-mean-square error (RMSE) and mean absolute error (MAE)

Summary

Introduction

Time series data arise in a variety of different areas including finance (Taylor 2007), biology (Bar-Joseph et al 2003) and energy (Alvarez et al 2011; Doucoure et al 2016). Regardless of the type of missingness present, further analysis of the time series such as autocovariance or spectral estimation can be difficult without first replacing the missing data with appropriate estimates. The section is organised as follows; we first review existing methods for modelling locally stationary time series using the LSW framework both in a univariate and multivariate context in Sect. The locally stationary wavelet (LSW) framework introduced by Nason et al (2000) provides a flexible model for nonstationary time series that captures the changing second-order structure of such series.

Methods

Results

Conclusion