# ESTIMATING MA COEFFICIENT FROM ARMA

Started by January 25, 2009
```On 28 Jan, 11:07, "ARMA GUY" <arma_...@yahoo.com> wrote:

>  The only disadvantage i saw in that is that the
> coefficients obtained for both AR and MA are not same with what i used for
> generating the system especially when the variance of additive white
> gaussian noise is very high or the SNR is very low.

Welcome to the world of data analysis. Did you use the correct
orders for the AR and MA steps? What do you think will happen
when you don't know the true orders and parameters before you
compute your estimates? What are the risks involved in getting
those numbers wrong? That is, what consequences will such an
error have for the user of your program? If the answer is
"it doesn't matter to the user" - which might well be the
case - why use the elaborate ARMA model at all? Why not settle
for a simpler, more robust and well-behaved AR model?

Rune
```
```ARMA-guy

Which code do you want? I mentioned several methods. I am not interested
in sending you all of the code that i have for parameteric modeling. From
reading you posts it seems that you need to study this problem further
before blindly using code that is handed to you (It will probably not
help). The built in function to Matlab will be useful for learning. You
should read carefully the last post from Rune about the realities of data
analysis in real aaplications. Also, carefully study the model assumptions
and read through the parameteric model chapters in the texts by both Kay
and Marple. Both books also include code you can use. Then try writing your
own matlab functions so solve for the AR and MA coefficients, etc. Make
sure you understand the direct, or brute force, algorithms and then move on
to the practical, or fast, algorithms (levinson, etc.). For example, can
you explain why your AR and MA coefficients are not the same when when your
SNR is low, etc.? When would they be the same? Read and ponder my
friend...then experiment...then ask for help on specific areas that you do
chapters 5-10 in Modern Spectral Estimation by Steven Kay. It would take me
a very long time to reproduce his work in this discussion group.

Dave

>Dave,
>
>You can send the code to me, I have also implemented something of such
>using MATLAB inbuilt functions (ARYULE, lpc, filter etc). My email is
>arma_ann at yahoo.com. The only disadvantage i saw in that is that the
>coefficients obtained for both AR and MA are not same with what i used
for
>generating the system especially when the variance of additive white
>gaussian noise is very high or the SNR is very low.
>
>Rune,
>
>Thanks for the suggestion on the model order. My own task is just on the
>model coefficients, I am presently evaluating the use of correct model
>order. The next step is to use different model orders for the AR and MA
>part respectively.
>
>
>ARMA GUY.
>>>Hi All,
>>>
>>>I am working on ARMA coefficient determination using a data sequence.
I
>>>have no idea of the underlying system from which the data was
generated
>>.
>>>What i only have is the data and thsi si as shown in the vector form
>>below.
>>>
>>>
>>>If the data is labeled
>>>d = [d(1),d(2),d(3),d(4),d(5),d(6),d(7),d(8),d(9),d(10),d(11),......
>>>d(45)]
>>>
>>>if the ARMA equation is
>>>y(n) = a(1)y(n-1) + a(2)y(n-2) + b(0)w(n) + b(1)w(n-1)
>>>
>>>and the main task is to determine the value of a(1),a(2), b(0) and
>b(1).
>>
>>>
>>>In order to do that, i am trying to model the data as the output of an
>>>ARMA system driving by white noise, w= w(1),w(2),w(3),w(4)...w(n).
>>>
>>>My questions are
>>>
>>>1) Is it logical to assume that w (white noise) above is similar to x
>in
>>>the ARMA equation.
>>>
>>>2) If yes to question (1) above, for example can i assume that using
>the
>>>ARMA equation above, d(5) was generated by
>>>
>>>d(5) = a(1)d(4) + a(2)d(3) + b(0)w(5) + b(1)w(4)
>>>
>>>similarly
>>>
>>>d(25) = a(1)d(24) + a(2)d(23) + b(0)w(25) + b(1)w(24)
>>>
>>>
>>> I have been told not to use any of the least square approach in
>>>determining my ARMA coefficents and not to use any of the sub optimal
>>>technique but to formulate it along that line.
>>>
>>>Thanks for anticipated response.
>>>ARMA-GUY
>>>
>>>
>>>
>>>
>>>
>>
>>ARMA-guy -
>>
>>The w is the driving white noise process. Any system noise, etc.,
should
>>be modeled seperately.
>>
>>To solve for the AR and MA parameters simultaneously (optimal) you need
>to
>>solve nonlinear equations usng an iterative algorithm on the estimated
>>autocorrelation sequence. No convergence is garanteed (see text by Kay
>or
>>the Matlab system identification toolbox by Ljung). Or you can use a
>high
>>order AR approximation that involves linear equations (see text by
>>Marple).
>>
>>A bootstrap method is the system identification approach; estimate
>>parameters using linear system identification equations, filter to
>produce
>>residuals, use residuals to drive estimated ARMA filter, iterate...This
>is
>>also not garanteed to converge.
>>
>>The sub-optimal approach is to estimate the AR parameters using
>high-order
>>Yule Walker equations, fitting overdetermined least squares equations
by
>>minimizing squared error. Filter the originalsignal with the inverse AR
>>filter to produce MA residuals. Estimate MA parameters of residuals.
>>
>>I hope this helps. I do have Matlab code for this if you are in need.
>Let
>>me know.
>>
>>Dave
>>
>>
>>
>

```
```Dave,

The code on SUB-OPTIMAL APPROACH.

"The sub-optimal approach is to estimate the AR parameters using
high-order
Yule Walker equations, fitting overdetermined least squares equations by
minimizing squared error. Filter the original signal with the inverse AR
filter to produce MA residuals. Estimate MA parameters of residuals ".

Do you mind if i send mine also ?

ARMA GUY

>Dave,
>
>You can send the code to me, I have also implemented something of such
>using MATLAB inbuilt functions (ARYULE, lpc, filter etc). My email is
>arma_ann at yahoo.com. The only disadvantage i saw in that is that the
>coefficients obtained for both AR and MA are not same with what i used
for
>generating the system especially when the variance of additive white
>gaussian noise is very high or the SNR is very low.
>
>Rune,
>
>Thanks for the suggestion on the model order. My own task is just on the
>model coefficients, I am presently evaluating the use of correct model
>order. The next step is to use different model orders for the AR and MA
>part respectively.
>
>
>ARMA GUY.
>>>Hi All,
>>>
>>>I am working on ARMA coefficient determination using a data sequence.
I
>>>have no idea of the underlying system from which the data was
generated
>>.
>>>What i only have is the data and thsi si as shown in the vector form
>>below.
>>>
>>>
>>>If the data is labeled
>>>d = [d(1),d(2),d(3),d(4),d(5),d(6),d(7),d(8),d(9),d(10),d(11),......
>>>d(45)]
>>>
>>>if the ARMA equation is
>>>y(n) = a(1)y(n-1) + a(2)y(n-2) + b(0)w(n) + b(1)w(n-1)
>>>
>>>and the main task is to determine the value of a(1),a(2), b(0) and
>b(1).
>>
>>>
>>>In order to do that, i am trying to model the data as the output of an
>>>ARMA system driving by white noise, w= w(1),w(2),w(3),w(4)...w(n).
>>>
>>>My questions are
>>>
>>>1) Is it logical to assume that w (white noise) above is similar to x
>in
>>>the ARMA equation.
>>>
>>>2) If yes to question (1) above, for example can i assume that using
>the
>>>ARMA equation above, d(5) was generated by
>>>
>>>d(5) = a(1)d(4) + a(2)d(3) + b(0)w(5) + b(1)w(4)
>>>
>>>similarly
>>>
>>>d(25) = a(1)d(24) + a(2)d(23) + b(0)w(25) + b(1)w(24)
>>>
>>>
>>> I have been told not to use any of the least square approach in
>>>determining my ARMA coefficents and not to use any of the sub optimal
>>>technique but to formulate it along that line.
>>>
>>>Thanks for anticipated response.
>>>ARMA-GUY
>>>
>>>
>>>
>>>
>>>
>>
>>ARMA-guy -
>>
>>The w is the driving white noise process. Any system noise, etc.,
should
>>be modeled seperately.
>>
>>To solve for the AR and MA parameters simultaneously (optimal) you need
>to
>>solve nonlinear equations usng an iterative algorithm on the estimated
>>autocorrelation sequence. No convergence is garanteed (see text by Kay
>or
>>the Matlab system identification toolbox by Ljung). Or you can use a
>high
>>order AR approximation that involves linear equations (see text by
>>Marple).
>>
>>A bootstrap method is the system identification approach; estimate
>>parameters using linear system identification equations, filter to
>produce
>>residuals, use residuals to drive estimated ARMA filter, iterate...This
>is
>>also not garanteed to converge.
>>
>>The sub-optimal approach is to estimate the AR parameters using
>high-order
>>Yule Walker equations, fitting overdetermined least squares equations
by
>>minimizing squared error. Filter the originalsignal with the inverse AR
>>filter to produce MA residuals. Estimate MA parameters of residuals.
>>
>>I hope this helps. I do have Matlab code for this if you are in need.
>Let
>>me know.
>>
>>Dave
>>
>>
>>
>
```