Equivalent degrees of freedom (EDOF) estimates for a chisquared 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 chisquare 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 chisquare 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 nonnegative 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,...,Mj1]s(j,tau)^2.
 EDOF 2
Large sample approximation where the SDF is known a priori.
n2 = 2 * (sum[k=1,...,floor((Mj1)/2)] Cj(f(k)))^2 / sum[k=1,...,floor((Mj1)/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 bandpass approximation for the SDF.
n3 = max(Mj/2, 1)
.
See references for more details.
Usage
1 2 
Arguments
x 
an object of class 
levels 
a vector containing the decomposition levels. Default: when

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.freq1)/P]
where P=2*(n.freq1)/sampling.interval. Only used when the input SDF is not 
sampling.interval 
sampling interval of the time series. Default: 
sdf 
a spectral density function of the process corresponding to the input time series.
This input must be a function whose first argument is 
sdfargs 
a list of arguments passed directly to the SDF function ala

wavelet 
a character string denoting the filter type.
See 
Value
a list containing the EDOF estimates for modes 1, 2 and 3 as well as the blockdependent 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
, mutilsSDF
.
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25  ## 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))

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