AR MODEL ORDER

PRESENTED BY:- Sarbjeet Singh

NITTTR- Chandigarh

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 ModelMA ModelARMA Model

Of these three models the AR model is by far the most widely used.

Reasons are twofold:-(i) The AR model is suitable for representing

spectra with narrow peaks.(ii) The AR model result in very simple linear

equations

What is AR ModelA 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 and should be windowed. (We use the hamming.)

AR model has only poles while the MA model has only zeros.

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.

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.

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.

Much work has been done by various researchers on this problem and many experimental results have been given:-

Two of the better known criteria for selection the model order have been proposed by Akaike –(1969,1974.)Known as (FPE) criterion.

= estimated variance of the linear prediction error.

N = number of samples. p = is the order of model.

The second criterion proposed by Akaike (1974),called the AIC,

AIC(p)=

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.

(ii) AIC (p) Is recommended for short data records.

.

An alternative information criterion, proposed by Rissanen (1983),is based on selecting the order that minimizes the description length :-

A fourth criterion has been proposed by Parzen(1974).

This is called the CAT function & defined as

The order P is selected to minimize CAT(p)

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.

Other experimental results indicate that:-

For small data lengths, the order of the AR model should be selected in the range N/3 to N/2 for good results.

The computational complexity of sequential estimation method is generally proportional to p, the order of the AR process

ReferencesProakis John G. , “ Digital Signal Processing “

3rd edition