Nonlinear denoising via wavelet shrinkage
Description
Performs a decimated or undecimated discrete wavelet transform on the input series and "shrinks" (decreases the amplitude towards zero) the wavelet coefficients based on a calculated noise threshold and specified shrinkage function. The resulting shrunken set of wavelet transform coefficients is inverted in a synthesis operation, resulting in a denoised version of the original series.
Usage
1 2 3 4 5 
Arguments
x 
a vector containing a uniformlysampled realvalued time series. 
n.level 
the number of decomposition levels, limited to

noise.variance 
a numeric scalar representing (an estimate of) the additive Gaussian white noise variance. If unknown, setting this value to 0.0 (or less) will prompt the function to automatically estimate the noise variance based on the median absolute deviation (MAD) of the scale one wavelet coefficients. Default: 1. 
reflect 
a logical value. If 
shrink.fun 
a character string denoting the shrinkage function.
Choices are 
thresh.fun 
a character string denoting the threshold function to use in calculating the waveshrink thresholds.
Note: if 
thresh.scale 
a positive valued numeric scalar which is used to amplify or attenuate the threshold values at each decomposition level. The use of this argument signifies a departure from a model driven estimate of the thresholds and can be used to tweak the levels to obtain a smoother or rougher result. Default: 1. 
threshold 
explicit setting of the wavelet shrinkage thresholds,
one for each level of the decomposition. If a single threshold is given, it is
replicated appropriately and (if the chosen transform is additionally a MODWT then) these thresholds
are normalized by dividing the threshold at level j by 2\eqn{\mbox{\textasciicircum}}{^}((j1)/2).
If the number of thresholds is equal to the number of decomposition levels, the thresholds
are unaltered prior to use.
Default: 
wavelet 
a character string denoting the filter type.
See 
xform 
a character string denoting the wavelet transform type.
Choices are 
Details
Assume that an appropriate model for our time series is X=D + e where D represents an unknown deterministic signal of interest and e is some undesired stochastic noise that is independent and identically distributed and has a process mean of zero. Waveshrink seeks to eliminate the noise component e of X in hopes of obtaining (a close approximation to) D. The basic algorithm works as follows:
 1
Calculate the DWT of X.
 2
Shrink (reduce towards zero) the wavelet coefficients based on a selected thresholding scheme.
 3
Invert the DWT.
This function support different shrinkage methods and threshold estimation schemes. Let W represent an arbitrary DWT coefficient and W' the correpsonding thresholded coefficient using a threshold of delta. The supported shrinkage methods are
 hard thresholding
W'=0 if W <= delta; W otherwise.
 soft thresholding
W'=sign{W}*f(W  delta) where f(x)=x if x >= 0; 0 otherwise and sign(x)=+1 if x > 0; 0 if x=0; 1 if x < 0.
 mid thresholding
W'=sign{W}*g(W  delta) where g(W)=2*f(Wdelta) if W < 2*delta; W otherwise.
Hard thresholding reduces to zero all coefficients that do not exceed the threshold. Soft thresholding pushes toward zero any coefficient whose magnitude exceeds the threshold, and zeros the coefficient otherwise. Mid thresholding represents a compromise between hard and soft thresholding such that coefficients whose magnitude exceeds twice the threshold are not adjusted, those between the threshold and twice the trhreshold are shrunk, and those below the threshodl are zeroed.
The threshold is selected based on a model of the noise. The supported techniques for estimating the noise threshold are
 universal
delta=sqrt( 2*var{noise}*log(N) ) where is the number of samples in the time series. As the noise variance is typically unknown, it is estimated based on the median absolute deviation of the absolute value of the scale one wavelet coefficients (and scaled by dividing the result by 0.6745 so that if the series were Gaussian white noise, the correct variance would be returned). The universal threshold is defined so that if the original time series was solely comprised of Gaussian noise, then all the wavelet coefficients would be (correctly) set to zero using a hard thresholding scheme. Inasmuch, the universal threshold results in highly smoothed output.
 minimax
These thresholds are used with soft and hard thresholding, and are precomputed based on a minimization of a theoretical upperbound on the asymptotic risk. The minimax thresholds are always smaller than the universal threshold for a given sample size, thus resulting in relatively less smoothing.
 adaptive
These are scaleadaptive thresholds, based on the minimization of Stein's Unbiased Risk Estimator for each level of the DWT. This method is only available with soft shrinkage. As a caveat, this threshold can produce poor results if the data is too sparse (see the references for details).
Finally, the user has the choice of using either a decimated (standard) form of the discrete wavelet transform (DWT) or an undecimated version of the DWT (known as the Maximal Overlap DWT (MODWT)). Unlike the DWT, the MODWT is a (circular) shiftinvariant transform so that a circular shift in the original time series produces an equivalent shift of the MODWT coefficients. In addition, the MODWT can be interpreted as a cyclespun version of the DWT, which is achieved by averaging over all nonredundant DWTs of shifted versions of the original series. The z is a smoother version of the DWT at the cost of an increase in computational complexity (for an Npoint series, the DWT requires O(N) multiplications while the MODWT requires O(N log2(N)) multiplications.
Value
vector containing the denoised series.
References
Donoho, D. and Johnstone, I. Ideal Spatial Adaptation by Wavelet Shrinkage. Technical report, Department of Statistics, Stanford University, 1992.
Donoho, D. and Johnstone, I. Adapting to Unknown Smoothness via Wavelet Shrinkage. Technical report, Department of Statistics, Stanford University, 1992.
D. B. Percival and A. T. Walden, Wavelet Methods for Time Series Analysis, Cambridge University Press, 2000.
See Also
wavDaubechies
, wavDWT
, wavMODWT
.
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24  ## MODWT waveshrinking using various thresh.scale
## values on sunspots series
x < as.vector(sunspots)
tt < as.numeric(time(sunspots))
thresh < seq(0.5,2,length=4)
ws < lapply(thresh, function(k,x)
wavShrink(x, wavelet="s8",
shrink.fun="hard", thresh.fun="universal",
thresh.scale=k, xform="modwt"), x=x)
stackPlot(x=tt, y=data.frame(x, ws),
ylab=c("sunspots",thresh),
xlab="Time")
## DWT waveshrinking using various threshold
## functions
threshfuns < c("universal", "minimax", "adaptive")
ws < lapply(threshfuns, function(k,x)
wavShrink(x, wavelet="s8",
thresh.fun=k, xform="dwt"), x=x)
stackPlot(x=tt, y=data.frame(x, ws),
ylab=c("original", threshfuns),
xlab="Normalized Time")
