The wavelet scaling approach to forecasting: Verification on a large set of Noisy data

• DOI: http://doi.org/10.1002/for.2634
• Author: Joanna Bruzda
• Published date: 01 April 2020
• Date: 01 April 2020

RESEARCH ARTICLE

The wavelet scaling approach to forecasting: Verification on

a large set of Noisy data

Joanna Bruzda

Faculty of Economic Sciences and

Management, Department of Logistics,

Nicolaus Copernicus University, Gagarina

11, 87‐100 Toruń, Poland

Correspondence

Joanna Bruzda, Nicolaus Copernicus

University, Faculty of Economic Sciences

and Management, Department of

Logistics, Gagarina 11, 87‐100. Toruń,

Poland.

Email: joanna.bruzda@uni.torun.pl

Funding information

National Science Center, Grant/Award

Number: 2017/27/B/HS4/01025

Abstract

In the paper, we undertake a detailed empirical verification of wavelet scaling

as a forecasting method through its application to a large set of noisy data. The

method consists of two steps. In the first, the data are smoothed with the help

of wavelet estimators of stochastic signals based on the idea of scaling, and, in

the second, an AR(I)MA model is built on the estimated signal. This procedure

is compared with some alternative approaches encompassing exponential

smoothing, moving average, AR(I)MA and regularized AR models. Special

attention is given to the ways of treating boundary regions in the wavelet signal

estimation and to the use of biased, weakly biased and unbiased estimators of

the wavelet variance. According to a collection of popular forecast accuracy

measures, when applied to noisy time series with a high level of noise, wavelet

scaling is able to outperform the other forecasting procedures, although this

conclusion applies mainly to longer time series and not uniformly across all

the examined accuracy measures.

1|INTRODUCTION

Wavelet filters are popular mathematical tools which are

used in different areas of statistics and are known for

their good localization, approximation and decorrelation

properties. Wavelet scaling, which is also called wavelet

smoothing, has been suggested in Percival and Walden

(2000) as an approach to estimate stochastic signals and

was further analyzed in Bruzda (2015) in the context of

its use with the nondecimated discrete wavelet transform.

Wavelet scaling is a denoising procedure consisting in

reshaping the spectrum of the analyzed stochastic

process under the assumption that its decimated wavelet

coefficients at each dyadic scale constitute white noise

processes or, alternatively, that its nondecimated wavelet

coefficients at each scale can be approximately treated as

bandpass white noises. As discussed in Bruzda (2015),

such an approach can be of use in business cycle studies,

where it can serve to smooth growth cycles or business

cycle indicators based on survey data, or in solving the

so‐called errors‐in‐variables problem. It was also shown

in Bruzda (2015) that the signal estimation based on the

nondecimated variant of wavelet scaling outperforms its

decimated counterpart in terms of the Mean Squared

Error (MSE) and that, in smaller samples of length 100,

wavelet scaling often leads to better signal estimates than

the parametric maximum likelihood estimation.

For the reasons mentioned above, it is tempting to

determine if wavelet scaling can be of interest in forecast-

ing, where –combined with some parametric models –it

can constitute an alternative to popular smoothing

methods, which dominate in the practice of forecasting

noisy microlevel economic data. Such an attempt has

been undertaken in Bruzda (2014) on the basis of a

small‐scale empirical study encompassing 16 time series

and one accuracy measure. This initial examination has

shown rather modest forecasting gains from wavelet scal-

ing, although simulation experiments pointed out that

applying wavelet estimators prior to building AR(I)MA

models may lead to even ca. 8% reductions in the forecast

Received: 31 October 2018 Revised: 8 August 2019 Accepted: 28 October 2019

DOI: 10.1002/for.2634

Journal of Forecasting. 2020;39:353–367. © 2019 John Wiley & Sons, Ltd.wileyonlinelibrary.com/journal/for 353