WaveThresh Help

ipndacw

Compute inner product matrix of discrete non-decimated autocorrelation wavelets.

DESCRIPTION

This function computes the inner product matrix of discrete non-decimated autocorrelation wavelets.

USAGE

ipndacw(J, filter.number = 10, family = "DaubLeAsymm", tol = 1e-100, verbose
         = F)

REQUIRED ARGUMENTS

J
Dimension of inner product matrix required. This number should be a negative integer.

OPTIONAL ARGUMENTS

filter.number
The index of the wavelet used to compute the inner product matrix.
family
The family of wavelet used to compute the inner product matrix.
tol
In the brute force computation for Daubechies compactly supported wavelets many inner product computations are performed. This tolerance discounts any results which are smaller than tol which effectively defines how long the inner product/autocorrelation products are.
verbose
If T then informative messages are printed. Some of these can be quite fun as the function tells you whether precomputed matrices are being used, how much computation needs to be done and so forth.

VALUE

A matrix of order (-J)x(-J) containing the inner product matrix of the discrete non-decimated autocorrelation matrices.

SIDE EFFECTS

None

DETAILS

This function computes the inner product matrix of the discrete non-decimated autocorrelation wavelets. This matrix is used to correct the wavelet periodogram as a step to turning it into a evolutionary wavelet spectral estimate. The matrix returned by ipndacw is the one called A in the paper by Nason, von Sachs and Kroisandt.

For the Haar wavelet the matrix is computed by using the analytical formulae in the paper by Nason, von Sachs and Kroisandt and is hence very fast and efficient and can be used for large values of -J.

For other Daubechies compactly supported wavelets the matrix is computed directly by autocorrelating discrete non-decimated wavelets at different scales and then forming the inner products of these. A function that computes the autocorrelation wavelets themselves is PsiJ. This brute force computation is slow and memory inefficient hence ipndacw contains a mechanism that stores any inner product matrix that it creates according to a naming scheme defined by the convention defined in rmname. The stored matrices are assigned to the current data directory. These stored matrices can be used in future computations by the following automatic procedure:

  1. The whole list of functions (returned by the S function search()) is searched for any previously computed inner product matrices. This search is performed by the rmget function.
  2. If a matrix of higher order is discovered then the appropriate top-left submatrix is returned, otherwise...
  3. If the right order of matrix is found it is returned, otherwise ...
  4. If a matrix of smaller order is found it is used as the top-left submatrix of the answer. The remaining elements to the right of and below the submatrix are computed and then the whole matrix is returned, otherwise...
  5. If none are found then the whole matrix is computed in C and returned.
In this way a particular matrix for a given wavelet need only be computed once.

RELEASE

Version 3.9 Copyright Guy Nason 1998

REFERENCES

Nason, G.P., von Sachs, R. and Kroisandt, G. (1998). Wavelet processes and adaptive estimation of the evolutionary wavelet spectrum. Technical Report, Department of Mathematics University of Bristol/ Fachbereich Mathematik, Kaiserslautern.

SEE ALSO

ewspec, PsiJ, rmname, rmget, filter.select.

EXAMPLES

#
# Let us create the 4x4 inner product matrix for the Haar wavelet.
# We'll turn on the jolly verbose messages as well. 
#
> ipndacw(-4, filter.number=1, family="DaubExPhase", verbose=T)
Computing ipndacw
Calling haarmat
Took  0.0699999  seconds
       -1     -2     -3     -4 
-1 1.5000 0.7500 0.3750 0.1875
-2 0.7500 1.7500 1.1250 0.5625
-3 0.3750 1.1250 2.8750 2.0625
-4 0.1875 0.5625 2.0625 5.4375
#
# If we do this again it will use the precomputed version
#
>  ipndacw(-4, filter.number=1, family="DaubExPhase", verbose=T)
Computing ipndacw
Returning precomputed version: using  4 
Took  0.08  seconds
       -1     -2     -3     -4 
-1 1.5000 0.7500 0.3750 0.1875
-2 0.7500 1.7500 1.1250 0.5625
-3 0.3750 1.1250 2.8750 2.0625
-4 0.1875 0.5625 2.0625 5.4375
#
# Let's use a smoother wavelet from the least-asymmetric family
# and generate the 6x6 version.
#
>  ipndacw(-6, filter.number=10, family="DaubLeAsymm", verbose=T)
Computing ipndacw
Took  0.95  seconds
             -1           -2           -3           -4           -5 
-1 1.839101e+00 3.215934e-01 4.058155e-04 8.460063e-06 4.522125e-08
-2 3.215934e-01 3.035353e+00 6.425188e-01 7.947454e-04 1.683209e-05
-3 4.058155e-04 6.425188e-01 6.070419e+00 1.285038e+00 1.589486e-03
-4 8.460063e-06 7.947454e-04 1.285038e+00 1.214084e+01 2.570075e+00
-5 4.522125e-08 1.683209e-05 1.589486e-03 2.570075e+00 2.428168e+01
-6 5.161675e-10 8.941666e-08 3.366416e-05 3.178972e-03 5.140150e+00
             -6 
-1 5.161675e-10
-2 8.941666e-08
-3 3.366416e-05
-4 3.178972e-03
-5 5.140150e+00
-6 4.856335e+01