# spectrum0

0th

Percentile

##### Estimate spectral density at zero

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).

Keywords
ts
##### 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
• specThe predicted value of the spectral density at frequency zero.

##### 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.

##### 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.

##### 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.

spectrum, spectrum0.ar, glm`.