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PREWHITEN procedure

Filters a time series before spectral analysis (A.W.A. Murray).

Option

`PHI` = scalar Specifies the value of the parameter used in filtering; default 0.99

Parameters

`SERIES` = variates Input series Output series

Description

`PREWHITEN` provides filtering of time series data prior to spectral analysis. Parameters `SERIES` and `FILTERED` specify the input and output series, respectively. The filtered series y is given by

yt = xt – φ × xt-1

where x is the input series. (Thus φ = 1 would give first differencing.) The value of φ is specified by the `PHI` option; the default value of φ=0.99 is often suitable. Alternatively, an empirical approach is to use the value

φ = (1 – 1/L)

where L is the lag at which inspection suggests that the autocorrelation in the series becomes negligible.

To “recolour” the spectrum of the series after estimation, you can multiply by

1/((1 + φ2) – (2×φ×cos(2π×f)))

where f is the frequency at which the spectrum is estimated.

Option: `PHI`.

Parameters: `SERIES`, `FILTERED`.

Method

The procedure uses the `TFILTER` directive with two TSMs defined as follows:

`TSM filter; ORDER=!(1,0,0); PARAM=!(1,0,0,PHI)`

`TSM arima; ORDER=!(0,1,0); PARAM=!(1,0,0)`

`TFILTER SERIES; NEWSERIES=FILTERED; FILTER=filter; ARIMA=arima`

The procedure is based on ideas from Granville Tunnicliffe Wilson, University of Lancaster.

Action with `RESTRICT`

The behaviour is as for the `TFILTER` directive.

Directive: `FOURIER`.

Procedures: `DFOURIER`, `MCROSSPECTRUM`, `PERIODTEST`, `REPPERIODOGRAM`, `SMOOTHSPECTRUM`.

Commands for: Time series.

Example

```CAPTION 'PREWHITEN example',\
'Data are from the Genstat 5 Release 1 Reference Manual page 544.';\
STYLE=meta,plain
VARIATE [NVALUE=26] Auto