spectrum0: Estimate spectral density at zero

In coda: Output Analysis and Diagnostics for MCMC

Estimate spectral density at zero

Description

The spectral density at frequency zero is estimated by fitting a glm to the low-frequency end of the periodogram. spectrum0(x)/length(x) estimates the variance of mean(x).

Usage

spectrum0(x, max.freq = 0.5, order = 1, max.length = 200) 

Arguments

x	A time series.
max.freq	The glm is fitted on the frequency range (0, max.freq]
order	Order of the polynomial to fit to the periodogram.
max.length	The data x is aggregated if necessary by taking batch means so that the length of the series is less than max.length. If this is set to NULL no aggregation occurs.

Details

The raw periodogram is calculated for the series x and a generalized linear model with family Gamma and log link is fitted to the periodogram.

The linear predictor is a polynomial in terms of the frequency. The degree of the polynomial is determined by the parameter order.

Value

A list with the following values

spec	The predicted value of the spectral density at frequency zero.

Theory

Heidelberger and Welch (1991) observed that the usual non-parametric estimator of the spectral density, obtained by smoothing the periodogram, is not appropriate for frequency zero. They proposed an alternative parametric method which consisted of fitting a linear model to the log periodogram of the batched time series. Some technical problems with model fitting in their original proposal can be overcome by using a generalized linear model.

Batching of the data, originally proposed in order to save space, has the side effect of flattening the spectral density and making a polynomial fit more reasonable. Fitting a polynomial of degree zero is equivalent to using the ‘batched means’ method.

Note

The definition of the spectral density used here differs from that used by spec.pgram. We consider the frequency range to be between 0 and 0.5, not between 0 and frequency(x)/2.

The model fitting may fail on chains with very high autocorrelation.

References

Heidelberger, P and Welch, P.D. A spectral method for confidence interval generation and run length control in simulations. Communications of the ACM, Vol 24, pp233-245, 1981.

See Also

spectrum, spectrum0.ar, glm.

coda documentation built on May 29, 2024, 11:23 a.m.