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

USAGE:

wavEDOF(x, wavelet = "d6", levels = 3:5 )

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.

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

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

See references for details.

REQUIRED ARGUMENTS:

x
An object of class WaveletTransform or a vector containing a uniformly-sampled real-valued time series.

OPTIONAL ARGUMENTS:

wavelet
A character string denoting the filter type. See wavDaubechies for details. Only used if input x is a time series. Default: ``s8".
levels
A vector containing the decomposition levels. Default: when x is of class WaveletTransform then levels = 1:x.nlevel, otherwise levels = 1:J where J is the maximum wavelet transform level in which there exists at least one interior wavelet coefficient.
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, send in an empty vector for this argument and the EDOF mode 2 and corresponding confidence intervals will not be calculated. Default: empty vector.

VALUE:

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

REFERENCES:

(1) D. B. Percival and A. T. Walden, ``Wavelet Methods for Time Series Analysis'', Cambridge University Press, 2000.

SEE ALSO:

.

EXAMPLES:

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