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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/
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