spectrum0 {coda} | R Documentation |

## 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 = NULL)

### 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`

.

[Package

*coda* version 0.8-3

Index]