Multiscale (Wavelet) Methods


The main theory of wavelets was developed during the 1980s, although special cases had been around for a long time (for example, the Haar system). Orthonormal wavelet bases are perhaps the most familiar, acting as a basis for many function spaces (e.g. the spaced of square integrable functions). Wavelet representations of functions possess several interesting properties. Wavelet coefficients of those representations encapsulate highly local information about the function at a given scale and location. Some wavelets have the `vanishing moments' property which means that coefficients of low-order polynomial behaviour of functions are exactly zero. These properties enable wavelet systems to sparsely represent a wide range of functions which gives such systems particular advantages when carrying out statistical estimation or for compression (such as for fingerprints or images).


wavethresh: an R package containing implementations of many wavelet based ideas.

Contains one-, two- and three-dimensional discrete wavelet transforms, one- and two-dimensional stationary wavelet transforms, wavelet packet transforms, tensor product wavelet transform, wavelet shrinkage/thresholding methods, local wavelet spectral analysis and LSW processes.

Work in wavelets

I've been fortunate to work in this fascinating area for many years. Also, by mathematical standards wavelets and multiscale is a relatively young field and there is still much to discover! Most of my early work was in collaboration with Bernard Silverman mainly in understanding wavelets and using them for regression problems, such as:

The Discrete Wavelet Transform in S (with BWS): which gave birth to wavethresh

Ways to use cross-validation for wavelet regression

Wavelets for survival function estimation (with Anestis Antoniadis and Gerard Gregoire)

The Stationary Wavelet Transform (with BWS)

Posterior Probability (Confidence) Intervals for Wavelet Thresolding (with Stuart Barber and BWS)

Wavelet Regression using Complex-Valued Wavelets (with Stuart Barber) and for Image Denoising (with Norbert Remenyi, Orietta Nicholis and Brani Vidakovic)


A few years after "getting into" wavelets I became (and still am) interested in the potential for using wavelets in time series analysis such as in

Locally Stationary Wavelet Process: a new wavelet time series model for nonstationary processes (with Rainer von Sachs and Gerald Kroisandt)

Wavelet packet transfer function modelling of nonstationary time series (with Theofanis Sapatinas)

Estimating Spectra (stationary and non-stationary), with Piotr Fryzlewicz and RvS

Costationarity of Locally Stationary Time Series (with Alessandro Cardinali)

Locally Stationary Wavelet Fields (with Idris Eckley and Rob Treloar)

Tests for Nonstationarity and for White Noise (with Delyan Savchev)


I have been fortunate to have been able to work with a number of gifted colleagues (as the authorship lists on the papers attest to). One particular exciting idea, the Haar-Fisz transform invented by PF, resulted in many interesting developments (by many others but) including:

Haar-Fisz for Poisson regression (with PF) and for Binomial sequence proportion estimation (with Matt Nunes)

Data-driven Haar-Fisz (with PF and Veronique Delouille)

Spectral Estimation (with PF)

Multiscale variance stabilization via Maximum Likelihood.


More recently, I have been interested in looking at how to extend multiscale methods to data which are not regularly spaced or subject to missing observations. A key tool here is the lifting transform introduced by Wim Sweldens.

Maarten Jansen, BWS and myself created a variant of the lifting scheme, useful for many problems in statistics called Lifting One-coefficient-at-a-time or LOCAAT.

We've used this in the following situations:

Adaptive lifting (changing the wavelet as you go!) with Marina Knight and MN)

Developing a `non-decimated' version (with MK)

Spectral estimation for time series with missing observations (with MK and MN)

Network time series. (with MK and MN)


The book "Wavelet Methods in Statistics with R" was written for upper level undergraduates, postgraduates and researchers who might be interewmiswRsted in learning about and using wavelets to solve real problems. The book meshes closely with the wavethresh software (above) to enable people to DO things, rather than just read about it!


Your car can identify you!

A WIRED report "A Car's Computer Can `Fingerprint' You in Minutes Based on How You Drive" shows that information that your car is already This car can't id you!recording can be used to accurately identify the driver. The researchers behind this study used MATLAB toolboxes and the stationary wavelet transform to denoise the signal stream as part of their procedure.

© Guy Nason 2016