Naveen (AR Model)

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SELECTION OFAR MODEL ORDER

Presented by:

Naveen KumarM.E. ECERoll No. : 112610


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Introductiony In the model-based approach, the spectrum estimationprocedure 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.

y There are three types of models :-

y AR Model

y MA Model

y ARMA Model


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What is AR Model?y A model which depends only on the previous outputs of the

system is called an autoregressive model (AR).

y Note that:-

AR model is based on frequency-domain analysis.

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


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The AR-model of a random process in discrete time is defined by

the following expression:

y where a1,a2..,ap coefficients of the recursive filter;

y p is the order of the model;

y (t) are output uncorrelated errors or simply White noise.

AR Model Equation


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y An order selection criterion is used to determine the appropriate

order for the AR model.

y The model parameters are found by solving a set of linear

equation obtained by minimizing the mean squared error.

y The characteristic of this error is that it decreases as the order of

the AR model is increased.

Need for selection of model order


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y 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.

y If a model with too low an order, We obtain a highly smoothed

spectrum.

y If a model with too high an order, There is risk of introducing

spurious low-level peaks in the spectrum.

Need


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y 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


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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.


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Difference between FPE & AIC(i) FPE (p)

y Is recommended for longer data records.

y It never exceeds model order selected by AIC

(ii)AIC (p)

y Is recommended for short data records.


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3) An alternative information criterion, proposed by Rissanen

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

description length :-


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

y This is called the Criterion Autoregressive Transfer (CAT)

function & defined as

y The order p is selected to minimize CAT(p)


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Applicationsy Texture modelling of visual content.

y Speech processing.

y Models for future sample predictions


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Drawbacky AR models linearly relate the signal samples which is not valid

for many real-life applications, where there may be many non-

linearity.


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y The experimental results, just indicate that the model-order

selection criteria do not yields definitive results.

y The FPE(p) criterion tends to underestimate the model order.

y The AIC criterion is statistically inconsistent as N.

y The MDL information criterion is statistically consistent.

Conclusion


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Referencesy Proakis John G. , Digital Signal Processing 4rd edition

y Comparison of Criteria for Estimating theOrder of

Autoregressive Process: www.eurojournals.com/ejsr.htm

y http://www.hindawi.com/journals/asp/2009/475147/


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Naveen (AR Model)