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 )