I have an ARMA process with known parameters. Are there general expressions for variance and possibly autocorrelation of the output for such processes? I can find the above for an order-(1,1) process in for example Shanmugan & Breipohl's "Random Signals" (Wiley, 1988), but is it possible for higher orders? If it helps, I would be happy to confine the processess in question to order (p,p). Thomas Arildsen -- All email to sender address is lost. My real adress is at es dot aau dot dk for user tha.

# ARMA model statistical properties

Started by ●October 2, 2007

Reply by ●October 2, 20072007-10-02

On Oct 3, 1:02 am, Thomas Arildsen <comp-dsp.es- aau...@spamgourmet.com> wrote:> I have an ARMA process with known parameters. Are there general > expressions for variance and possibly autocorrelation of the output for > such processes? > I can find the above for an order-(1,1) process in for example Shanmugan > & Breipohl's "Random Signals" (Wiley, 1988), but is it possible for > higher orders? If it helps, I would be happy to confine the processess > in question to order (p,p). > > Thomas Arildsen > -- > All email to sender address is lost. > My real adress is at es dot aau dot dk for user tha.There is a set of tables in a book by Karl Astrom - the old 1970s book. Hardy

Reply by ●October 3, 20072007-10-03

HardySpicer skrev:> On Oct 3, 1:02 am, Thomas Arildsen <comp-dsp.es- > aau...@spamgourmet.com> wrote: >> I have an ARMA process with known parameters. Are there general >> expressions for variance and possibly autocorrelation of the output for >> such processes? >> I can find the above for an order-(1,1) process in for example Shanmugan >> & Breipohl's "Random Signals" (Wiley, 1988), but is it possible for >> higher orders? If it helps, I would be happy to confine the processess >> in question to order (p,p). >> >> Thomas Arildsen > > There is a set of tables in a book by Karl Astrom - the old 1970s > book. > > Hardy >Would that be "Introduction to Stochastic Control Theory" from 1970? What is meant by 'a set of tables'? Are they tables of calculated variances and/or correlation values for specific ARMA configurations or expressions for calculating these? I am looking for the latter. Thomas Arildsen -- All email to sender address is lost. My real adress is at es dot aau dot dk for user tha.

Reply by ●October 3, 20072007-10-03

On Oct 3, 8:43 pm, Thomas Arildsen <comp-dsp.es- aau...@spamgourmet.com> wrote:> HardySpicer skrev: > > > > > On Oct 3, 1:02 am, Thomas Arildsen <comp-dsp.es- > > aau...@spamgourmet.com> wrote: > >> I have an ARMA process with known parameters. Are there general > >> expressions for variance and possibly autocorrelation of the output for > >> such processes? > >> I can find the above for an order-(1,1) process in for example Shanmugan > >> & Breipohl's "Random Signals" (Wiley, 1988), but is it possible for > >> higher orders? If it helps, I would be happy to confine the processess > >> in question to order (p,p). > > >> Thomas Arildsen > > > There is a set of tables in a book by Karl Astrom - the old 1970s > > book. > > > Hardy > > Would that be "Introduction to Stochastic Control Theory" from 1970? > What is meant by 'a set of tables'? Are they tables of calculated > variances and/or correlation values for specific ARMA configurations or > expressions for calculating these? I am looking for the latter. > > Thomas Arildsen > > -- > All email to sender address is lost. > My real adress is at es dot aau dot dk for user tha.Yes! It's formula based on the ARMA coefficients for variance orders and yes it's variances. Hardy

Reply by ●October 3, 20072007-10-03

Thomas Arildsen wrote:> I have an ARMA process with known parameters. Are there general > expressions for variance and possibly autocorrelation of the output for > such processes?Yes. If x[n] is a WSS stochastic process, and h[n] the impulse response of your (stable) ARMA filter, and y[n] the output of the ARMA filter, ie. y[n] = x[n] * h[n], then the autocorrelation r_y[n] is r_y[n] = r_x[n] * r_h[n], where r_h[n] = h[n] * h[-n] is the autocorrelation of the impulse response. The mean m_y of the process y[n] is the mean m_x of x[n] multiplied by the DC response of the ARMA filter, m_y = m_x sum_k h[k]. The variance is sigma_y^2 = r_y[0]. Regards, Andor

Reply by ●October 3, 20072007-10-03

HardySpicer skrev:> On Oct 3, 8:43 pm, Thomas Arildsen <comp-dsp.es- > aau...@spamgourmet.com> wrote:>>> Would that be "Introduction to Stochastic Control Theory" from 1970? >> What is meant by 'a set of tables'? Are they tables of calculated >> variances and/or correlation values for specific ARMA configurations or >> expressions for calculating these? I am looking for the latter. >> >> Thomas Arildsen > > Yes! It's formula based on the ARMA coefficients for variance orders > and yes it's variances. > > Hardy >Great, thank's. It seems to have been reprinted on Dover in 2006 and is available pretty cheap on Amazon. I'll give it a try. Thomas Arildsen -- All email to sender address is lost. My real adress is at es dot aau dot dk for user tha.

Reply by ●October 3, 20072007-10-03

Andor skrev:> Thomas Arildsen wrote: >> I have an ARMA process with known parameters. Are there general >> expressions for variance and possibly autocorrelation of the output for >> such processes? > > Yes. If x[n] is a WSS stochastic process, and h[n] the impulse > response of your (stable) ARMA filter, and y[n] the output of the ARMA > filter, ie. > > y[n] = x[n] * h[n], > > then the autocorrelation r_y[n] is > > r_y[n] = r_x[n] * r_h[n], > > where r_h[n] = h[n] * h[-n] is the autocorrelation of the impulse > response. The mean m_y of the process y[n] is the mean m_x of x[n] > multiplied by the DC response of the ARMA filter, > > m_y = m_x sum_k h[k]. > > The variance is sigma_y^2 = r_y[0]. > > Regards, > Andor >I have the PSD expression for the filter; I think I will try to inverse transform from that to the autocorrelation in stead. Thomas Arildsen -- All email to sender address is lost. My real adress is at es dot aau dot dk for user tha.

Reply by ●October 3, 20072007-10-03

Thomas Arildsen wrote:> I have an ARMA process with known parameters. Are there general > expressions for variance and possibly autocorrelation of the output for > such processes? > I can find the above for an order-(1,1) process in for example Shanmugan > & Breipohl's "Random Signals" (Wiley, 1988), but is it possible for > higher orders? If it helps, I would be happy to confine the processess > in question to order (p,p).Would http://www.polyx.com/demo_covf.htm help? Jerry -- Engineering is the art of making what you want from things you can get. ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

Reply by ●October 4, 20072007-10-04

Jerry Avins skrev:> Thomas Arildsen wrote: >> I have an ARMA process with known parameters. Are there general >> expressions for variance and possibly autocorrelation of the output >> for such processes? >> I can find the above for an order-(1,1) process in for example >> Shanmugan & Breipohl's "Random Signals" (Wiley, 1988), but is it >> possible for higher orders? If it helps, I would be happy to confine >> the processess in question to order (p,p). > > Would http://www.polyx.com/demo_covf.htm help? >Thanks, I think there might be some ideas to help me solve the problem as well. I actually already have one method, but I would like to check it against some other known method. Thomas Arildsen -- All email to sender address is lost. My real adress is at es dot aau dot dk for user tha.