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 D-statistic denotes the maximum
deviation of the dj from a hypothetical linear cumulative energy trend. This
D-statistic is then compared to a table of critical D-statistics that defines
the distribution of D for various sample sizes. Comparing the D-statistic 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.
REQUIRED ARGUMENTS:
x
An object of class dwt with convolution style filtering, a corresponding
wavebound object, or a numeric vector. In the latter case, DWT parameters can be
passed to specify the type of wavelet to use and the number of decomposition
levels to perform.
OPTIONAL ARGUMENTS:
wavelet
A character string denoting the filter type. Valid only for input not of class
dwt or wavebound. Default: ``s8".
n.levels
The number of decomposition levels. Valid only for input not of class dwt or
wavebound. Default: the maximum decomposition level that contains at least one
interior wavelet coefficient.
significance
A numeric vector of real values in 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: c(0.1, 0.05, 0.01).
lookup
A logical flag for accessing precalculated critical D-statistics. The critical
D-statistics are calculated for a variety of sample sizes and significances. If
lookup is TRUE, this table is accessed. The table is stored as the matrix object
D.table.critical and is loaded with S+Wavelets. Missing table values are
calculated using the input arguments: n.realization, n.repetition and tolerance.
Default: TRUE.
n.realization
An integer specifying the number of realizations to generate in a Monte Carlo
simulation for calculating the D-statistic(s). This parameter is used either
when lookup is FALSE, or when lookup is TRUE and the table is missing values
corresponding to the specified significances. Default: 10000.
n.repetition
an integer specifying the number of Monte Carlo simulations to perform. This
parameter is coordinated with the n.realization parameter. Default: 3.
tolerance
A numeric real scalar that specifies the amplitude threshold to use in
estimating critical D-statistic(s) via the Inclan-Tiao 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: 1e-6.
VALUE:
result
An object of class WaveletHomogeneity.
DETAILS:
An Inclan-Tiao approximation of critical D-statistics is used for sample sizes N
>= 128 while a Monte Carlo technique is used for N < 128. For the Monte Carlo
technique, the D-statistic for a Gaussian white noise sequence of length N is
calculated. This process is repeated n.realization times, forming a distribution
of the D-statistic. The critical values corresponding to the significances are
calculated a total of n.repetition times, and averaged to form an approximation
to the D-statistic(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.
REFERENCES:
(1) D. B. Percival and A. T. Walden, ``Wavelet Methods for Time Series Analysis'',
Cambridge University Press, 2000.
SEE ALSO:
,
,
,
.
EXAMPLES:
## perform a homogeneity of variance test for a DWT
## decomposition of a long memory process realization
homogeneity <- wavVarianceHomogeneity( fdp045 )