wmtsa (version 1.1-1)

wavMODWT: The maximal overlap discrete wavelet transform (MODWT)

Description

Let $j, t$ be the decomposition level, and time index, respectively, and $s_{0,t}=X_{t=0}^{N-1}$ where $X_t$ is a real-valued uniformly-sampled time series. The $j^{th}$ level MODWT wavelet coefficients $\tilde{d}_{j,t}$ and scaling coefficients $\tilde{s}_{j,t}$ are defined as $\tilde{d}_{j,t} \equiv \sum_{l=0}^{L-1}\widetilde{h}_l \tilde{s}_{j-1, t - 2^{j-1}\;l \mbox{ mod }N,}$ and $\tilde{s}_{j,t} \equiv \sum_{l=0}^{L-1}\widetilde{g}_l \tilde{s}_{j-1, t - 2^{j-1}\;l \mbox{ mod }N}$ The variable $L$ is the length of both the scaling filter $(g)$ and wavelet filter $(h)$. The $\tilde{d}_{j,t}$ and $\tilde{s}_{j,t}$ are the wavelet and scaling coefficients, respectively, at decomposition level $j$ and time index $t$. The MODWT is a collection of all wavelet coefficients and the scaling coefficients at the last level: $\mathbf{\tilde{d}_1,\tilde{d}_2},\ldots,\mathbf{\tilde{d}_J,\tilde{s}_J}$ where $\mathbf{\tilde{d}_j}$ and $\mathbf{\tilde{s}_j}$ denote a collection of wavelet and scaling coefficients, respectively, at level $j$.

Usage

wavMODWT(x, wavelet="s8", n.levels=ilogb(length(x), base=2),
    position=list(from=1,by=1,units=character()), units=character(),
    title.data=character(), documentation=character(), keep.series=FALSE)

Arguments

x

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

documentation

a character string used to describe the input data. Default: character().

keep.series

a logical value. If TRUE, the original series is preserved in the output object. Default: FALSE.

n.levels

the number of decomposition levels. Default: as.integer(floor(logb(length(x),base=2))).

position

a list containing the arguments from, by and to which describe the position(s) of the input data. All position arguments need not be specified as missing members will be filled in by their default values

title.data

a character string representing the name of the input data. Default: character().

units

a string denoting the units of the time series. Default: character() (no units).

wavelet

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

Value

• an object of class wavTransform.

concept

wavelettransforms, maximal overlap discrete wavelet

Details

The MODWT is a non-decimated form of the discrete wavelet transform (DWT) having many advantages over the DWT including the ability to handle arbitrary length sequences and shift invariance (while the wavDWT function can handle arbitrary length sequences, it does so by means of an ad hoc storage sytem for odd length scaling coefficient crystals. The MODWT needs no such scheme and is more robust in this respect). The cost of the MODWT is in its redundancy. For an $N$ point input sequence, there are $N$ wavelet coefficients per scale. However, the number of multiplication operations is $O(N \log_2(N))$ which is the same as the fast Fourier transform, and is acceptably fast for most situations.

References

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

See Also

reconstruct, wavDaubechies, wavDWT, wavMODWPT, wavDictionary, wavIndex, wavTitle, wavBoundary.

Examples

## calculate the MODWT of linear chirp 
linchirp <- make.signal("linchirp", n=1024)
result   <- wavMODWT(linchirp, wavelet="s8", n.levels=5, keep.series=TRUE)

## plot the transform shifted for approximate zero 
## phase alignment 
plot(wavShift(result))

## plot summary 
eda.plot(result)

## summarize the transform 
summary(result)