Extending the Scope of Wavelet Regression by Coefficient-dependent Thresholding

Various aspects of the wavelet approach to nonparametric regression are considered, with the overall aim of extending the scope of wavelet techniques, to irregularly-spaced data, to regularly-spaced data sets of arbitrary size, to heteroscedastic and correlated data, and to data that contain outliers.

The core of the methodology is an algorithm for finding all the variances and within-level covariances in the wavelet table of a sequence with given covariance structure. If the original covariance matrix is band limited, then the algorithm is linear in the length of the sequence.

The variance-calculation algorithm allows data on any set of independent variable values to be treated, by first interpolating to a fine regular grid of suitable length, and then constructing a wavelet expansion of the gridded data. Various thresholding methods are discussed and investigated. Exact risk formulae for the mean square error of the methodology for given design are derived. Good performance is obtained by noise-proportional thresholding, with thresholds somewhat smaller than the classical universal threshold.

Outliers in the data can be removed or downweighted, and aspects of such robust techniques are developed and demonstrated in an example. Another natural application is to correlated data, where the covariance of the wavelet coefficients is not due to an initial grid transform but is an intrinsic feature. The use of the method in these circumstances is demonstrated by an application to data synthesized in the study of ion channel gating. The basic approach of the paper has many other potential applications, and some of these are discussed briefly.