The D-statistic denotes the maximum deviation of sequence from a hypothetical
linear cumulative energy trend. The critical D-statistics define the
distribution of D for a zero mean Gaussian white noise process. Comparing the
sequence D-statistic to the corresponding critical values provides a means of
quantitatively rejecting or accepting the linear cumulative energy hypothesis.
The table is generated for an ensemble of distribution probabilities and sample
sizes.
OPTIONAL ARGUMENTS:
n.sample
A vector of integers denoting the sample sizes for which critical D-statistics
are created. Default: c(127,130).
significance
A numeric vector of real values in the interval (0,1). 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 (recommended), this table is accessed. The table is stored as the
matrix object D.table.critical on the S+Wavelets 2.0 workspace. 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 coordinates 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
A matrix containing the critical D-statistics corresponding to the supplied
sample sizes and significances.
DETAILS:
A precalculated critical D-statistics object (D.table.critical) exists on the
S+Wavelets 2.0 workspace and was built for a variety of sample sizes and
significances using 3 repetitions and 10000 realizations/repetition. This
D.table function should be used in cases where specific D-statistics are missing
from D.table.critical. Note: the results of the D.table value should not be
returned to a variable named D.table.critical as it will override the
precalculated table that exists on the S+Wavelets 2.0 workspace. An Inclan-Tiao
approximation of critical D-statistics is used for sample sizes n.sample >= 128
while a Monte Carlo technique is used for n.sample < 128. For the Monte Carlo
technique, the D-statistic for a Gaussian white noise sequence of length
n.sample 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).
REFERENCES:
(1) D. B. Percival and A. T. Walden, ``Wavelet Methods for Time Series Analysis'',
Cambridge University Press, 2000. Chapter 9.