WaveThresh Help

Use `threshold()' on a `wd' object

DESCRIPTION

Applies hard or soft thresholding to wavelet decomposition object "wd.object".

USAGE

threshold.wd(wd, levels = 3:(wd$nlevels - 1), type = "hard", policy = "sure", by.level = F, value = 0, dev = var, boundary = F, verbose = F, return.threshold = F, force.sure = F, cvtol = 0.01)

REQUIRED ARGUMENTS

wd: Object of class "wd", maybe from a wavelet decomposition using the wd function.

OPTIONAL ARGUMENTS

levels: vector that determines which levels are thresholded in the decomposition
type: Determines the type of thresholding, this can be "hard" or "soft"
policy: which threshold to use, this can be "sure", "cv", "universal", "manual", "mannum" or "probability",
by.level: If F then a global threshold is applied to all the levels specified by "levels", otherwise a threshold is computed and applied separately to each level.
value: This is the user-supplied: threshold for the manual policy; number of ordered coefficients for the mannum policy; and the user-supplied quantile level for the probability policy.
dev: deviance measure. The default is to use the var() function, but you can insert your own measure of deviance.
boundary: If this is true then the boundary correction values are included for thresholding, if it is false then they are not.
verbose: If true then threshold prints informative messages at you.
return.threshold: If this is true then the actual threshold is returned, otherwise the thresholded object is returned.
force.sure: If true then the SURE procedure is applied to all levels. If false then if there is not enough signal in a level universal thresholding is sometimes used for that level.
cvtol: The termination tolerance for the cross-validation optimisation. This is the relative tolerance for convergence of the threshold value.

VALUE

An object of class "wd" that has been thresholded.

SIDE EFFECTS

None

DETAILS

Thresholding modifies the coefficients within a wd wavelet decomposition object. The modification can be performed either with a "hard" or "soft" thresholding selected by the "type" argument.

Hard thresholding simply compares a coefficient with a threshold. If it is larger in absolute magnitude it is left alone, if it is smaller it is set to zero. The "soft threshold" formula is

soft(w) = sgn(w)*max(|w| - t, 0)

where "w" is the wavelet coefficient, "t" is the threshold and "sgn" is the sign of w. Soft thresholding causes w to be replaced by soft(w).

There are many ways that the threshold can be computed, we term this the "policy" and these are described next.

Universal

A universal policy computes a threshold based on Donoho and Johnstone's "universal thresholds". The threshold is sqrt(2*log(n))*noise, where noise is computed as sqrt(dev(w)), i.e. some measure of the variability of the coefficients, and n is the number of data points (or number of wavelet coefficients). By default "dev" is "var", so the noise level is estimated using the sample standard deviation. You can use any other such estimate by writing your own function and supplying it as the "dev" argument. For example you could create the function "myvar" by

myvar <- function(d) mad(d)^2

(Actually this is implemented in WaveThresh, and called madmad). This computes the square of the mean absolute deviation of the data. It is squared because "dev" should be a measure on the variance scale, not the standard deviation (you know what I mean). If you make the "by.levels" argument T, then a separate threshold is computed for each level in the "levels" vector. This means that the variance is estimated separately for each level.

Sure

The "sure" policy is described by Donoho and Johnstone (1993).

Cross-validation

Choosing the threshold using by cross-validation is described in Nason (1996). Discussion of both these methods here is beyond the scope of this help.

Manual

The "manual" policy is simple. You supply a threshold value ("value") and hard or soft thresholding is performed using that value. The "value" argument is a vector. If it is of length 1 then it is replicated to be the same length as the "levels" vector, otherwise it is repeated as many times as necessary to be the length vector's length. In this way, different thresholds can be given for different levels. Note that the "by.level" argument has no effect with this policy.

Mannum

The "mannum" policy is also very simple. You decide how many coefficients you want to keep (e.g. say you want to keep the smallest coefficients [smallest in absolute magnitude]) and supply that number to the value argument.

Probability

The "probability" policy works as follows. All coefficients that are smaller than the "value"th quantile of the coefficients are set to zero. If "by.level" is false, then the quantile is computed for all coefficients in the levels specified by the "levels" vector; if "by.level" is true, then each level's quantile is estimated separately. The probability policy is pretty stupid - do not use it.

This function is a method for the generic function threshold for class "wd". It can be invoked by calling threshold for an object of the appropriate class, or directly by calling threshold.wd regardless of the class of the object.

RELEASE

REFERENCES

Donoho, D.L. and Johnstone, I. M. (1992) Ideal Spatial Adaptation via Wavelet Shrinkage. Stanford University, Department of Statistics Technical Report, Number 400.

Donoho, D.L. and Johnstone, I. M. (1993) Adapting to Unknown smoothness via Wavelet Shrinkage. Stanford University, Department of Statistics Technical Report, Number 425.

Nason, G.P. (1996)Wavelet shrinkage using cross-validation. Journal of the Royal Statistical Society, Series B, 58, 463--479

BUGS

There should be an optimal policy as well, although universal is meant to come close.

EXAMPLES

See example for threshold.

Wavelets Home Page

G.P.Nason@bristol.ac.uk