ebayesthresh.wavelet {EbayesThresh}

Empirical Bayes thresholding on the levels of a wavelet transform.

Description

Apply an Empirical Bayes thresholding approach level by level to the detail coefficients in a wavelet transform.

Usage

ebayesthresh.wavelet(xtr, vscale = "independent", smooth.levels = Inf, 
      prior = "laplace", a = 0.5, bayesfac = FALSE, 
      threshrule = "median")

ebayesthresh.wavelet.dwt(x.dwt, vscale = "independent", smooth.levels = Inf, 
      prior = "laplace", a = 0.5, bayesfac = FALSE,
      threshrule = "median")

ebayesthresh.wavelet.wd(x.wd, vscale = "independent", smooth.levels = Inf,
      prior = "laplace", a = 0.5, bayesfac = FALSE,
      threshrule = "median")

ebayesthresh.wavelet.splus(x.dwt, vscale = "independent", smooth.levels = Inf, 
      prior = "laplace", a = 0.5, bayesfac = FALSE,
      threshrule = "median")

Arguments

xtr

The wavelet transform of a vector of data. The transform is obtained using one of the wavelet transform routines in R or in S+WAVELETS. Any choice of wavelet, boundary condition, etc provided by these routines can be used.

x.dwt

Wavelet transform input for ebayesthresh.wavelet.dwt.

x.wd

Wavelet transform input for ebayesthresh.wavelet.wd.

vscale

Controls the scale used at different levels of the transform. If vscale is a scalar quantity, then it will be assumed that the wavelet coefficients at every level have this standard deviation. If vscale = "independent", the standard deviation will be estimated from the highest level of the wavelet transform and will then be used for all levels processed. If vscale="level", then the standard deviation will be estimated separately for each level processed, allowing standard deviation that is level-dependent.

smooth.levels

The number of levels to be processed, if less than the number of levels of detail calculated by the wavelet transform.

prior

Specification of prior to be used for the coefficients at each level, conditional on their mean being nonzero; can be cauchy or laplace.

a

Scale factor if Laplace prior is used. Ignored if Cauchy prior is used. If, on entry, a=NA and prior="laplace", then the scale parameter will also be estimated at each level by marginal maximum likelihood. If a is not specified then the default value 0.5 will be used.

bayesfac

If bayesfac=TRUE, then whenever a threshold is explicitly calculated, the Bayes factor threshold will be used.

threshrule

Specifies the thresholding rule to be applied to the coefficients. Possible values are median (use the posterior median); mean (use the posterior mean); hard (carry out hard thresholding); soft (carry out soft thresholding).

Details

The routine ebayesthresh.wavelet can process a wavelet transform obtained using the routine wd in the WaveThresh R package, the routines dwt or modwt in the waveslim R package, or one of the routines (either dwt or nd.dwt) in S+WAVELETS.

Note that the wavelet transform must be calculated before the routine ebayesthresh.wavelet is called; the choice of wavelet, boundary conditions, decimated vs nondecimated wavelet, and so on, are made when the wavelet transform is calculated.

Apart from some housekeeping to estimate the standard deviation if necessary, and to determine the number of levels to be processed, the main part of the routine is a call, for each level, to the smoothing routine ebayesthresh.

The basic notion of processing each level of detail coefficients is easily transferred to transforms constructed using other wavelet software. Similarly, it is straightforward to modify the routine to give other details of the wavelet transform, if necessary using the option verbose = TRUE in the calls to ebayesthresh.

The main routine ebayesthresh.wavelet calls the relevant one of the routines ebayesthresh.wavelet.wd (for a transform obtained from WaveThresh), ebayesthresh.wavelet.dwt (for transforms obtained from either dwt or modwt in waveslim) or ebayesthresh.wavelet.splus (for transforms obtained from S+WAVELETS.

Value

The wavelet transform (in the same format as that supplied to the routine) of the values of the estimated regression function underlying the original data.

Author(s)

Bernard Silverman

References

Johnstone, I. M. and Silverman, B. W. (2005) Empirical Bayes selection of wavelet thresholds. Annals of Statistics, 33, 1700–1752.

See also the other references given for ebayesthresh and at http://www.bernardsilverman.com.

See Also

ebayesthresh