Homogeneity test for discrete wavelet transform crystals
Description
Tests for homogeneity of variance for each scale of a discrete wavelet transform (DWT) decomposition. Based on the assumption that the DWT decorrelates colored noise processes, the interior wavelet coefficients in a given scale (dj) can be regarded as a zero mean Gaussian white noise process. For a homogeneous distribution of dj, there is an expected linear increase in the cumulative energy as a function of time. The so called Dstatistic denotes the maximum deviation of the dj from a hypothetical linear cumulative energy trend. This Dstatistic is then compared to a table of critical Dstatistics that defines the distribution of D for various sample sizes. Comparing the Dstatistic of dj to the corresponding critical values provides a means of quantitatively rejecting or accepting the linear cumulative energy hypothesis. This function performs this test for an ensemble of distribution probabilities.
Usage
1 2 3  wavVarTest(x, wavelet="s8", n.levels=NULL,
significance=c(0.1,0.05,0.01), lookup=TRUE, n.realization=10000,
n.repetition=3, tolerance=1e6)

Arguments
x 
an object of class 
lookup 
a logical flag for accessing precalculated critical Dstatistics. The
critical Dstatistics are calculated for a variety of sample sizes
and significances. If 
n.levels 
the number of decomposition levels. Valid only for input not of class

n.realization 
an integer specifying the number of realizations to generate in a
Monte Carlo simulation for calculating the Dstatistic(s). This
parameter is used either when 
n.repetition 
an integer specifying the number of Monte Carlo simulations to
perform. This parameter is coordinated with the 
significance 
a numeric vector of real values on the interval (0,1).
Qualitatively the significance is the fraction of times that the
linear cumulative energy hypothesis is incorrectly rejected. It is
equal to the difference of the distribution probability (p) and unity. Default: 
tolerance 
a numeric real scalar that specifies the amplitude threshold to use in
estimating critical Dstatistic(s) via the InclanTiao approximation.
Setting this parameter to a higher value
results in a lesser number of summation terms at the expense of obtaining
a less accurate approximation. Default: 
wavelet 
a character string denoting the filter type. Valid only for input not of class

Details
An InclanTiao approximation of critical Dstatistics is used for sample
sizes N >= 128 while a
Monte Carlo technique is used for
N < 128.
For the Monte Carlo technique, the Dstatistic for a
Gaussian white noise sequence of length N is calculated. This
process is repeated n.realization
times,
forming a distribution of the Dstatistic.
The critical values corresponding to the significances
are calculated a total of n.repetition
times, and averaged to form
an approximation to the Dstatistic(s).
Because the Monte Carlo study can be both computationally and memory
intensive, it is highly recommended that lookup be set to
TRUE
, its default value.
Value
an object of class wavVarTest
.
References
D. B. Percival and A. T. Walden, Wavelet Methods for Time Series Analysis, Cambridge University Press, 2000.
See Also
wavVar
, wavDWT
, D.table
.
Examples
1 2 3 4  ## perform a homogeneity of variance test for a
## DWT decomposition of a long memory process
## realization
homogeneity < wavVarTest(fdp045)
