# multivariate cepstrum

Started by July 13, 2004
```Hi

I know what the cepstrum of a univariate ARMA model is. Does anyone know how
to extend the idea to multivariate ARMA processes? Recomendations for a good
textbook?

MODEL:

x(t) = SUM( k=1..P, a(k)x(t-k) ) + SUM( k=1..Q, b(k)w(t-k) )

where w(t) ~ N(0,S) are normally distributed multivariate random variables,
and x(t) is a z by 1 vector.

DEFINITION FOR UNIVARIATE CASE, z=1:

Simply put, it's the fourier transform of the power spectrum.

A(z) = "Z"-transf. of a(k) = SUM( k=1..P, a(k)z^(-k) );
B(z) = "Z"-transf. of b(k) = SUM( k=1..Q, b(k)z^(-k) );

H(z) = system function = B(z) / A(z)

log H(z) = SUM( k=0..infinity, h(k)z^(-k) )

and h(k) are the cepstrum coefficients!

USES OF CEPSTRA

You can define a "cepstral distance", eg in natural language processing.
It's a measure of disparity
between ARMA models (in this case).

ANY help at all would be grately appreciated. Thanks very much -- Franco.

(also posted to: sci.stat.math, sci.stat.consult, comp.dsp,
comp.ai.nat-lang)

```
```"Francis Woolfe" <franco.woolfe@ntlworld.com> wrote in message
news:UyXIc.146\$3E6.69@newsfe2-win.ntli.net...
> Hi
>
> I know what the cepstrum of a univariate ARMA model is. Does anyone know
how
> to extend the idea to multivariate ARMA processes? Recomendations for a
good
> textbook?
>
> MODEL:
>
> x(t) = SUM( k=1..P, a(k)x(t-k) ) + SUM( k=1..Q, b(k)w(t-k) )
>
> where w(t) ~ N(0,S) are normally distributed multivariate random
variables,
> and x(t) is a z by 1 vector.
>
> DEFINITION FOR UNIVARIATE CASE, z=1:
>
> Simply put, it's the fourier transform of the power spectrum.
>
> A(z) = "Z"-transf. of a(k) = SUM( k=1..P, a(k)z^(-k) );
> B(z) = "Z"-transf. of b(k) = SUM( k=1..Q, b(k)z^(-k) );
>
> H(z) = system function = B(z) / A(z)
>
> log H(z) = SUM( k=0..infinity, h(k)z^(-k) )
>
> and h(k) are the cepstrum coefficients!
>
> USES OF CEPSTRA
>
> You can define a "cepstral distance", eg in natural language processing.
> It's a measure of disparity
> between ARMA models (in this case).
>
>
> ANY help at all would be grately appreciated. Thanks very much -- Franco.
>
> (also posted to: sci.stat.math, sci.stat.consult, comp.dsp,
> comp.ai.nat-lang)
>
>
They are going to be Cepstrum matrices - not coefficients with your ARMA
model.

Tom

```
```"Francis Woolfe" <franco.woolfe@ntlworld.com> wrote in message news:<UyXIc.146\$3E6.69@newsfe2-win.ntli.net>...
> Hi
>
> I know what the cepstrum of a univariate ARMA model is. Does anyone know how
> to extend the idea to multivariate ARMA processes? Recomendations for a good
> textbook?

...

> (also posted to: sci.stat.math, sci.stat.consult, comp.dsp,
> comp.ai.nat-lang)

I have seen the 2D cepstrum mentioned in the context of image
processing. You might want to at sci.image.processing as well.

Rune
```