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 crossvalidation 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 ComplexValued 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 nonstationary), 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 HaarFisz transform invented by PF, resulted in many interesting developments (by many others but) including:
HaarFisz for Poisson regression (with PF) and for Binomial sequence proportion estimation (with Matt Nunes)
Datadriven HaarFisz (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 Onecoefficientatatime 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 `nondecimated' version (with MK)
Spectral estimation for time series with missing observations (with MK and MN)
Network time series. (with MK and MN)
