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AR model

This document discusses methods for selecting the order of an autoregressive (AR) model. It explains that AR models depend only on previous outputs and have poles but no zeros. Several criteria for selecting the optimal AR model order are presented, including the Akaike Information Criterion (AIC) and Finite Prediction Error (FPE) criterion. Higher order models fit the data better but can introduce spurious peaks, so the goal is to minimize criteria like AIC or FPE to find the best balance. The document concludes that while these criteria provide guidance, the optimal order depends on the specific data, and inconsistencies can exist between the different methods.

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SELECTION OF AR MODEL ORDER Presented by: Naveen Kumar M.E. ECE Roll No. : 112610
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
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.
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.
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.
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.
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.
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.
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.
1) An alternative information criterion, proposed by Rissanen (1983),is based on selecting the order that minimizes the description length :-
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)
Applications  Texture modelling of visual content.  Speech processing.  Models for future sample predictions
Drawback  AR models linearly relate the signal samples which is not valid for many real-life applications, where there may be many non- linearity.
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.
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/
AR model