2. Introduction
In the model-based approach, the spectrum estimation
procedure consists of two steps.
(i) We estimate the parameters{ak}and{bk} of the model.
(ii) From these estimates, we compute the power spectrum
estimate.
There are three types of models :-
AR Model
MA Model
ARMA Model
3. What is AR Model?
A model which depends only on the previous outputs of the
system is called an autoregressive model (AR).
Note that:-
AR model is based on frequency-domain analysis.
AR model has only poles while the MA model has only zeros.
4. AR Model Equation
The AR-model of a random process in discrete time is defined
by the following expression:
where a1,a2…..,ap coefficients of the recursive filter;
p is the order of the model;
Є(t) are output uncorrelated errors or simply White noise.
5. Need for selection of model order
An order selection criterion is used to determine the
appropriate order for the AR model.
The model parameters are found by solving a set of linear
equation obtained by minimizing the mean squared error.
The characteristic of this error is that it decreases as the order
of the AR model is increased.
6. Need…
One of the most important consideration is the choice of the
number of terms in the AR model, this is known as its order p.
If a model with too low an order, We obtain a highly smoothed
spectrum.
If a model with too high an order, There is risk of introducing
spurious low-level peaks in the spectrum.
7. AR Model Order Selection
Two of the better known criteria for selection the model order
have been proposed by Akaike –(1969,1974.)
2) Known as Finite Prediction Error (FPE) criterion.
= estimated variance of the linear prediction error.
N = number of samples.
p = is the order of model.
8. 1) The second criterion proposed by Akaike (1974),called the
Akaike Information Criterion (AIC)
decreases & therefore also decreases as the order of
the AR model is increased.
increases with increases in p.
9. Difference between FPE & AIC
(i) FPE (p)
Is recommended for longer data records.
It never exceeds model order selected by AIC
(ii) AIC (p)
Is recommended for short data records.
10. 1) An alternative information criterion, proposed by Rissanen
(1983),is based on selecting the order that minimizes the
description length :-
11. 1) A fourth criterion has been proposed by Parzen(1974).
This is called the Criterion Autoregressive Transfer (CAT)
function & defined as
The order p is selected to minimize CAT(p)
13. Drawback
AR models linearly relate the signal samples which is not valid
for many real-life applications, where there may be many non-
linearity.
14. Conclusion
The experimental results, just indicate that the model-order
selection criteria do not yields definitive results.
The FPE(p) criterion tends to underestimate the model order.
The AIC criterion is statistically inconsistent as N→∞.
The MDL information criterion is statistically consistent.
15. References
Proakis John G. , “ Digital Signal Processing “ 4rd edition
Comparison of Criteria for Estimating the Order of
Autoregressive Process: www.eurojournals.com/ejsr.htm
http://www.hindawi.com/journals/asp/2009/475147/
