(Inverse) Maximal Overlap Discrete Wavelet Transform

This function performs a level \(J\) decomposition of the input vector using the non-decimated discrete wavelet transform. The inverse transform performs the reconstruction of a vector or time series from its maximal overlap discrete wavelet transform.

modwt(x, wf = "la8", n.levels = 4, boundary = "periodic")
imodwt(y)

Arguments

x	a vector or time series containing the data be to decomposed. There is no restriction on its length.
y	Object of class "modwt".
wf	Name of the wavelet filter to use in the decomposition. By default this is set to "la8", the Daubechies orthonormal compactly supported wavelet of length \(L=8\) (Daubechies, 1992), least asymmetric family.
n.levels	Specifies the depth of the decomposition. This must be a number less than or equal to \(\log_2(\mbox{length}(x))\).
boundary	Character string specifying the boundary condition. If boundary=="periodic" the defaulTRUE, then the vector you decompose is assumed to be periodic on its defined interval, if boundary=="reflection", the vector beyond its boundaries is assumed to be a symmetric reflection of itself.

Value

Object of class "modwt", basically, a list with the following components

d?

Wavelet coefficient vectors.

s?

Scaling coefficient vector.

wavelet

Name of the wavelet filter used.

boundary

How the boundaries were handled.

Details

The code implements the one-dimensional non-decimated DWT using the pyramid algorithm. The actual transform is performed in C using pseudocode from Percival and Walden (2001). That means convolutions, not inner products, are used to apply the wavelet filters.

The MODWT goes by several names in the statistical and engineering literature, such as, the ``stationary DWT'', ``translation-invariant DWT'', and ``time-invariant DWT''.

The inverse MODWT implements the one-dimensional inverse transform using the pyramid algorithm (Mallat, 1989).

References

Gencay, R., F. Selcuk and B. Whitcher (2001) An Introduction to Wavelets and Other Filtering Methods in Finance and Economics, Academic Press.

Percival, D. B. and P. Guttorp (1994) Long-memory processes, the Allan variance and wavelets, In Wavelets and Geophysics, pages 325-344, Academic Press.

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

See also

dwt, idwt, mra.

Author

B. Whitcher

Examples

## Figure 4.23 in Gencay, Selcuk and Whitcher (2001)
data(ibm)     
ibm.returns <- diff(log(ibm))
# Haar
ibmr.haar <- modwt(ibm.returns, "haar")
names(ibmr.haar) <- c("w1", "w2", "w3", "w4", "v4")
# LA(8)
ibmr.la8 <- modwt(ibm.returns, "la8")
names(ibmr.la8) <- c("w1", "w2", "w3", "w4", "v4")
# shift the MODWT vectors
ibmr.la8 <- phase.shift(ibmr.la8, "la8")
## plot partial MODWT for IBM data
par(mfcol=c(6,1), pty="m", mar=c(5-2,4,4-2,2))
plot.ts(ibm.returns, axes=FALSE, ylab="", main="(a)")
for(i in 1:5)
  plot.ts(ibmr.haar[[i]], axes=FALSE, ylab=names(ibmr.haar)[i])
axis(side=1, at=seq(0,368,by=23), 
  labels=c(0,"",46,"",92,"",138,"",184,"",230,"",276,"",322,"",368))

par(mfcol=c(6,1), pty="m", mar=c(5-2,4,4-2,2))
plot.ts(ibm.returns, axes=FALSE, ylab="", main="(b)")
for(i in 1:5)
  plot.ts(ibmr.la8[[i]], axes=FALSE, ylab=names(ibmr.la8)[i])
axis(side=1, at=seq(0,368,by=23), 
  labels=c(0,"",46,"",92,"",138,"",184,"",230,"",276,"",322,"",368))