(Inverse) Discrete Wavelet Packet Transforms

All possible filtering combinations (low- and high-pass) are performed to decompose a vector or time series. The resulting coefficients are associated with a binary tree structure corresponding to a partitioning of the frequency axis.

dwpt(x, wf="la8", n.levels=4, boundary="periodic")
idwpt(y, y.basis)
modwpt(x, wf = "la8", n.levels = 4, boundary = "periodic")

Arguments

x

a vector or time series containing the data be to decomposed. This must be a dyadic length vector (power of 2).
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 \(\log(\mbox{length}(x),2)\).
boundary

Character string specifying the boundary condition. If boundary=="periodic" the default, 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.
y

Object of S3 class dwpt.
y.basis

Vector of character strings that describe leaves on the DWPT basis tree.

Value

Basically, a list with the following components

w?.?

Wavelet coefficient vectors. The first index is associated with the scale of the decomposition while the second is associated with the frequency partition within that level.

wavelet

Name of the wavelet filter used.

boundary

How the boundaries were handled.

Details

The code implements the one-dimensional DWPT using the pyramid algorithm (Mallat, 1989).

References

Mallat, S. G. (1989) A theory for multiresolution signal decomposition: the wavelet representation, IEEE Transactions on Pattern Analysis and Machine Intelligence, 11, No. 7, 674-693.

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

Wickerhauser, M. V. (1994) Adapted Wavelet Analysis from Theory to Software, A K Peters.

See also

dwt, modwpt, wave.filter.

Author

B. Whitcher

Examples

data(mexm)
J <- 4
mexm.mra <- mra(log(mexm), "mb8", J, "modwt", "reflection")
mexm.nomean <- ts(
  apply(matrix(unlist(mexm.mra), ncol=J+1, byrow=FALSE)[,-(J+1)], 1, sum), 
  start=1957, freq=12)
mexm.dwpt <- dwpt(mexm.nomean[-c(1:4)], "mb8", 7, "reflection")