spectrum0: 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

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.

Value

A list with the following values
specThe 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.

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.

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.