# spectrum0

##### 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
spec The 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.

##### See Also

*Documentation reproduced from package coda, version 0.17-1, License: GPL (>= 2)*