# D.table: Critical D-statistic table generation In wmtsa: Wavelet Methods for Time Series Analysis

## Description

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.

## Usage

 ```1 2 3``` ```D.table(n.sample=c(127, 130), significance=c(0.1, 0.05, 0.01), lookup=TRUE, n.realization=10000, n.repetition=3, tolerance=1e-6) ```

## Arguments

 `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`. 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`. `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)`. `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`.

## Details

A precalculated critical `D-statistics` object (`D.table.critical`) exists on the package 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-statistic`s 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 available in the package.

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)`.

## Value

a matrix containing the critical `D-statistics` corresponding to the supplied sample sizes and significances.

## References

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

`D.table.critical`.
 ```1 2``` ```D.lookup <- D.table(significance=c(10,5,1)/100, n.realization=100, n.sample=125:130, lookup=FALSE) ```