AR model

SELECTION OF AR MODEL ORDER

Presented by:

Naveen KumarM.E. ECERoll No. : 112610

IntroductionIn 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

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.

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.

AR Model Equation

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.

Need for selection of model order

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.

Need…

Two of the better known criteria for selection the model

order have been proposed by Akaike –(1969,1974.)

1)Known as Finite Prediction Error (FPE) criterion.

= estimated variance of the linear prediction error.

N = number of samples.

p = is the order of model.

AR Model Order Selection

2) 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.

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.

3) An alternative information criterion, proposed by

Rissanen (1983),is based on selecting the order that

minimizes the description length :-

4) 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)

ApplicationsTexture modelling of visual content.Speech processing.Models for future sample predictions

DrawbackAR models linearly relate the signal samples which is

not valid for many real-life applications, where there

may be many non-linearity.

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.

Conclusion

ReferencesProakis 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/