wmtsa (version 1.1-1)

wavEDOF: Equivalent degrees of freedom (EDOF) estimates for a chi-squared distribution

Description

Let X be a collection of M uncorrelated zero mean Gaussian random variables (RVs). The sum of the squares of the RVs in X will obey a scaled chi-square distribution with M degrees of freedom (DOF). If, however, the original Gaussian RVs are (partially) correlated, we can approximate the distribution of the sum of the squares of (correlated Gaussian) RVs using a scaled chi-square distribution with the DOF adjusted for the correlation in the RVs. These adjusted DOF estimates are known as the equivalent degrees of freedom (EDOF). In the context of unbiased wavelet variance analysis, the EDOF can be used to estimate confidence intervals that are guaranteed to have non-negative bounds.

This program calculates three estimates of the EDOF for each level of a discrete wavelet transform. The three modes are described as follows for the MODWT of an an input sequence ${X_t}_{t=0}^{N-1}$:

[object Object],[object Object],[object Object]

See references for more details.

Usage

wavEDOF(x, wavelet="s8", levels=NULL, sdf=NULL, sdfargs=NULL,
    sampling.interval=1, n.fft=1024)

Arguments

x

an object of class wavTransform or a vector containing a uniformly-sampled real-valued time series.

levels

a vector containing the decomposition levels. Default: when x is of class wavTransform then levels is set to 1:n.level, otherwise levels is set to 1:J, where J is t

n.fft

a positive integer (greater than one) defining the number of frequencies to use in evaluating the SDF for EDOF 2 calculations. The frequencies are uniformly distributed over the interval [0, Nyquist] ala f=[0, 1/P , 2/P, 3/P, ..., (n.freq-1)/P] where P=2*

sampling.interval

sampling interval of the time series. Default: 1.

sdf

a spectral density function of the process corresponding to the input time series. This input must be a function whose first argument is f (representing frequency). At a minimum, the SDF must be defined over frequencies [0, Nyquist] where

sdfargs

a list of arguments passed directly to the SDF function ala do.call. Default: NULL (no additional arguments).

wavelet

a character string denoting the filter type. See wavDaubechies for details. Only used if input x is a time series. Default: "s8".

Value

• a list containing the EDOF estimates for modes 1, 2 and 3 as well as the block-dependent unbiased wavelet variance estimates.

concept

waveletwavelet varianceequivalent degrees of freedom

References

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

See Also

wavVar, wavVarConfidence, mutilsSDF.

Examples

## initialize variables 
n.level <- 9
wavelet <- "d6"
N       <- 4096
phi     <- 0.9

## define input SDF 
S <- function(f, phi)  1/(1 + phi^2 - 2*phi*cos(2*pi*f))
sdfarg <- list(phi=phi)

## create series and MODWT 
set.seed(100)
x <- rnorm(N)
W <- wavMODWT(x, wavelet=wavelet, n.level=n.level)

## calculate EDOF using the wavTransform object 
z1 <- wavEDOF(W, sdf=S, sdfarg=sdfarg)
print(z1)

## calculate EDOF using original time series 
z2 <- wavEDOF(x, wavelet=wavelet, levels=seq(n.level), sdf=S, sdfarg=sdfarg)
print(z2)

## compare the two approaches 
print(all.equal(z1,z2))