## 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.