Penalized wavelets: Embedding wavelets into semiparametric regression

We introduce the concept of penalized wave lets to facilitate seamless embedding of wave lets into semi parametric regression models. In particular, we show that penalized wavelets are analogous to penalized splines; the latter being the established approach to function estimation in semiparametric regression. They differ only in the type of penalization that is appropriate. This fact is not borne out by the existing wavelet literature, where the regression modelling and fitting issues are over shadowed by computational issues such as efficiency gains afforded by the Discrete Wavelet Transform and partially obscured by a tendency to work in the wavelet coefficient space. With penalized wavelet structure in place, we then show that fitting and inference can be achieved via the same general approaches used for penalized splines: penalized least squares, maximum likelihood and best prediction within a frequentist mixed model framework, and Markov chain Monte Carlo and mean field variational Bayes within a Bayesian framework. Penalized wavelets are also shown have a close relationship with wide data (p >> n) regression and benefit from on going research on that topic.