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) for t=0,...,N1 where X(t) is a realvalued uniformlysampled time series. The jth level MODWT wavelet coefficients d(j,t) and scaling coefficients s(j,t) are defined as d(j,t)=sum(h(l) s(j1, t  2^(j1) l mod N)) and s(j,t)=sum(g(l) s(j1, t  2^(j1) l mod N)) The variable L is the length of both the scaling filter (g) and wavelet filter (h). The d(j,t) and 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: d(1),d(2),...,d(J),s(J) where d(j) and s(j) denote a collection of wavelet and scaling coefficients, respectively, at level j.
Usage
1 2 3 
Arguments
x 
a vector containing a uniformlysampled realvalued time series. 
documentation 
a character string used to describe the input

keep.series 
a logical value. If 
n.levels 
the number of decomposition levels. Default: 
position 
a 
title.data 
a character string representing the name of the input

units 
a string denoting the units of the time series. Default: 
wavelet 
a character string denoting the filter type.
See 
Details
The MODWT is a nondecimated 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 log2(N)) which is the same as
the fast Fourier transform, and is acceptably fast for most situations.
Value
an object of class wavTransform
.
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
1 2 3 4 5 6 7 8 9 10 11 12 13  ## 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)
