wavMRDSum: Partial summation of a multiresolution decomposition
In wconstan/wmtsa: Wavelet Methods for Time Series Analysis

Description Usage Arguments Details Value References See Also Examples

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.

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

`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` option, but one of `keep.details` and `keep.smooth` must be `TRUE`. Default: `TRUE`.
`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=FALSE`) will result in removing the trend in such cases. The user also has the choice to exclude the details in the summation via the `keep.details` option, but one of `keep.details` and `keep.smooth` must be `TRUE`. Default: `TRUE`.
`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 `floor(logb(length(x),2))` if `reflect=FALSE` and, if `reflect=TRUE`, cannot exceed `floor(logb((length(x)-1)/(L-1) + 1, b=2))` where L is the length of the wavelet filter. Use the `keep.smooth` option to also include the last level's smooth in the summation. Default: `1`.
`reflect`	a logical value. If `TRUE`, the last Lj = (2^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 transform coefficients for series whose endpoint levels are (highly) mismatched. The variable Lj represents the effective filter length at decomposition level `n.level`, where L is the length of the wavelet (or scaling) filter. A similar operation is performed at the beginning of the series. After synthesis and (partial) summation of the resulting details and smooth, the middle N points of the result are returned, where N is the length of the original time series. Default: `TRUE`.
`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) the number of transform coefficients is halved. Given N is the length of the original time series, the total number of DWT transform coefficients is N. The MODWT is a non-decimated transform where the number of coefficients at each level is N and the total number of transform coefficients is N*`n.level`. Unlike the DWT, the MODWT is shift-invariant and is seen as a weighted average of all possible non-redundant shifts of the DWT. See the references for details. Default: `"modwt"`.

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 D1,...,DJ and smooth SJ. The decomposition is additive such that the original series X may be reconstructed ala X=D1 + D2 + ... DJ + SJ. As the effective wavelet filters at level j are nominally associated with approximate band pass filters, the details Dj correspond approximately to normalized frequencies on the interval [1/2^(j+1), 1/2^j], while the content of the smooth SJ 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 D1 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:

Translation invariance: Unlike the DWT, the MODWT is translation invariant, meaning that a (circular) shift of the input series will result in a corresponding (circular) shift of the transform coefficients.
Smoothness: The MODWT coefficients are a result of cycle-spinning, where averages are taken over all unique DWTs resulting from various circular shifts of the original series. The resulting MODWT MRD is relatively more smooth than the corresponding DWT MRD.
Zero phase aligment: Unlike the DWT MRD, the MODWT MRD produces components that are associated with exactly zero phase filter operations such that events (such as peaks) in the details and smooth line up exactly with those of the original series.
Computational speed: The DWT is faster than the MODWT, but the MODWT is still quite fast, requiring multiplication and summation operations on the same order as the popular Fast Fourier Transform.

a vector containing the denoised series.

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.

wavDaubechies, wavDWT, wavMODWT, wavMRD.

## 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)
ifultools::stackPlot(x=as.vector(time(sunspots)),
    y=list(sunspots=x, "D4+D5+D6"=z1),
    ylim=range(x,z1))