wmtsa (version 2.0-3)

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 \(\{X_t\}_{t=0}^{N-1}\):

EDOF 1

Large sample approximation that requires an SDF estimation via wavelet coefficients. $$\hat\eta_1 ={ M_j (\hat{s}_{j,0})^2 \over \hat{A}_j},$$ where \(\hat{s}_{j,\tau}\) is the autocovariance sequence defined by $$ \hat{s}_{j,\tau} \equiv{1 \over M_j} \sum_{t=0}^{M_j - 1} \tilde{d}_{j,t}^{(int)} \tilde{d}_{j,t + |\tau|}^{(int)} \;\; 0\le|\tau|\le M_j-1, $$ and \(\tilde{d}_{j,t}^{(int)}\) are the \(M_j$ $j^{th}\) level interior MODWT wavelet coefficients and \(\hat{A}_j\) is defined as $$ \hat{A}_j \equiv{ ( \hat{s}_{j,0} ) \over 2 } + \sum_{\tau=1}^{M_j-1} ( \hat{s}_{j,\tau} )^2.$$

EDOF 2

Large sample approximation where the SDF is known a priori. $$\hat\eta_2 ={ 2{\biggl( \sum_{k=1}^{\lfloor (M_j - 1)/ 2 \rfloor} C_j(f_k) \biggr)}^2 \over \sum_{k=1}^{\lfloor (M_j - 1) / 2 \rfloor} C_j^2(f_k)},$$ where \(f_k \equiv k /M_j\) and \(C_j \equiv \widetilde{\mathcal{H}}_j^{(D)}(f) S_X(f)\) 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. $$\hat\eta_3=\mbox{max}\{M_j/2^j, 1\}$$.

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 the maximum wavelet transform level in which there exists at least one interior wavelet coefficient.

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*(n.freq-1)/sampling.interval. Only used when the input SDF is not NULL. Default: 1024.

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 Nyquist=1/(2*sampling.interval). Additional arguments that are needed to calculate the SDF should be passed via the sdfargs parameter. 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. See the mutilsSDF function for more details. 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, mutilsSDF.

Examples

# NOT RUN {
## 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))
# }