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 |
---|---|

`FILTERED` = variates |
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

*y _{t}* =

*x*– φ ×

_{t}*x*

_{t-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 + φ^{2}) – (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.

### See also

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 READ Auto 1.00000000 0.75529144 0.51519220 0.20418942 0.09391165 -0.00993248 -0.03836911 -0.05863828 -0.01256090 0.01050724 -0.03169708 -0.11680566 -0.19236910 -0.21314260 -0.21102816 -0.13031514 -0.05516818 0.05541264 0.09524165 0.11746112 0.08982970 0.08212313 0.04734724 -0.00584709 -0.06916435 -0.10306836 : PREWHITEN [PHI=0.98] SERIES=Auto; FILTERED=wAuto PRINT Auto,wAuto