wmtsa (version 1.1-1)

wavMRDSum: Partial summation of a multiresolution decomposition

Description

Forms a multiresolution decomposition (MRD) by taking a specified discrete wavelet transform of the input series and subsequently inverting each level of the transform back to the "time" domain. The resulting components of the MRD form an octave-band decomposition of the original series, and can be summed together to reconstruct the original series. Summing only a subset of these components can be viewed as a denoising operation if the "noisy" components are excluded from the summation.

Usage

wavMRDSum(x, wavelet="s8",
    levels=1, xform="modwt", reflect=TRUE,
    keep.smooth=TRUE, keep.details=TRUE)

Arguments

x

a vector containing a uniformly-sampled real-valued time series.

keep.details

a logical value. If TRUE, the details corresponding to the specified levels are included in the partial summation over the MRD components. The user also has the choice to exclude the smooth in the summation via the keep.smooth op

keep.smooth

a logical value. If TRUE, the smooth at the last decomposition level is added to the partial summation over specified details. The smooth typically contains low-frequency trends present in a series, so removing the smooth (keep.smooth=F

levels

an integer vector of integers denoting the MRD detail(s) to sum over in forming a denoised approximation to the orginal series (the summation is performed across scale and nto across time). All values must be positive integers, and cannot exceed flo

reflect

a logical value. If TRUE, the last $L_J = (2^{\mbox{n.level}} - 1)(L - 1) + 1$ coefficients of the series are reflected (reversed and appended to the end of the series) in order to attenuate the adverse effect of circular filter operations on

wavelet

a character string denoting the filter type. See wavDaubechies for details. Default: "s8".

xform

a character string denoting the wavelet transform type. Choices are "dwt" and "modwt" for the discrete wavelet transform (DWT) and maximal overlap DWT (MODWT), respectively. The DWT is a decimated transform where (at each level)

Value

• a vector containing the denoised series.

concept

noise reductionwavelet

Details

Performs a level $J$ decimated or undecimated discrete wavelet transform on the input series and inverts the transform at each level separately to produce details $D_1,\ldots,D_J$ and smooth $S_J$. The decomposition is additive such that the original series $X$ may be reconstructed ala $X = S_J + \sum_{j=1}^J D_j$. As the effective wavelet filters at level $j$ are nominally associated with approximate band pass filters, the details $D_j$ correspond approximately to normalized frequencies on the interval $[1/2^{j+1}, 1/2^j]$, while the content of the smooth $S_J$ corresponds approximately to normalized frequencies $[0, 1/2^{J+1}]$. The collection of details and smooth form a multiresolution decomposition (MRD).

With the intent of removing unwanted noise events, a summation over a subset of MRD components may be calculated yielding a smooth approximation to the original series. For example, summing all MRD components beyond $D_1$ is tantamount to a low-pass filtering of the original series (whether or not this is a relevant and sufficient noise removal technique is left to the discretion of the practitioner). This function allows the user to specify the decomposition levels they wish to sum over in order to form a multiresolution approximation. The inclusion of the last level's smooth in the summation is controlled by the optional keep.smooth argument.

The user may also select either a decimated wavelet transform (DWT) or an undecimated wavelet transform (MODWT). However, we recommend that the user stick with the MODWT for the following reasons: [object Object],[object Object],[object Object],[object Object]

References

D. B. Percival and A. T. Walden, Wavelet Methods for Time Series Analysis, Cambridge University Press, 2000.

T.W. Randolph and Y. Yasui, Multiscale Processing of Mass Spectrometry Data, Biometrics, 62:589--97, 2006.

See Also

wavDaubechies, wavDWT, wavMODWT, wavMRD.

Examples

## create a MODWT-based MRD of the sunspots series 
## and sum over decomposition levels 4 to 6 
x <- as.vector(sunspots)
z1 <- wavMRDSum(x, levels=4:6)
stackPlot(x=as.vector(time(sunspots)),
    y=list(sunspots=x, "D4+D5+D6"=z1),
    ylim=range(x,z1))