wavEDOF {wmtsa}R Documentation

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):

EDOF 1
Large sample approximation that requires an SDF estimation via wavelet coefficients.

n1 = s(j,0)^2 / Aj,

where s(j,tau) is the autocovariance sequence defined by

s(j,tau) = (1 / Mj) * sum[t=0,..., Mj - 1]{d(j,t)}

and d(j,t) are the Mj jth level interior MODWT wavelet coefficients and Aj is defined as

Aj = s(j,0)/2 + sum[tau=1,...,Mj-1]s(j,tau)^2.

EDOF 2
Large sample approximation where the SDF is known a priori.

n2 = 2 * (sum[k=1,...,floor((Mj-1)/2)] Cj(f(k)))^2 / sum[k=1,...,floor((Mj-1)/2)] (Cj(f(k)))^2,

where f(k)=k/Mj and Cj = Hj(f) Sx(f) is the product of Daubechies wavelet filter squared gain function and the spectral density function of X(t).

EDOF 3
Large sample approximation using a band-pass approximation for the SDF.

n3 = max(Mj/2, 1)

.

See references for more details.

Usage

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

Arguments

x an object of class WaveletTransform or a vector containing a uniformly-sampled real-valued time series.
levels a vector containing the decomposition levels. Default: when x is of class WaveletTransform then levels is set to 1:n.level, otherwise levels is set to 1:J, where J is the maximum wavelet transform level in which there exists at least one interior wavelet coefficient.
sampling.interval sampling interval of the time series. Default: 1.
sdf. a vector containing a discretized approximation of the process spectral density function (SDF). The coefficients of this argument should correspond exactly with the normalized Fourier frequencies f=(0, 1/P , 2/P, 3/P, ..., (M-1)/P), where P=2*(M-1) and M is the number of points in the SDF vector. For example, if the sdf vector contains five elements, the corresponding frequencies will be f=(0, 1/8, 1/4, 3/8, 1/2). This argument is used only for calculating mode 2 EDOF. If the EDOF mode 2 estimates are not desired, specify this this argument as NULL and the EDOF mode 2 and corresponding confidence intervals will not be calculated. Default: NULL.
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.

References

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

See Also

wavVar, wavVarConfidence.

Examples

## calculate the EDOF estimates for the ocean 
## series 
wavEDOF(ocean)

[Package wmtsa version 1.0-2 Index]