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Estimate autoregressive (AR) all-pole model using Yule-Walker method


ar_coeffs = aryule(data,order)
[ar_coeffs,NoiseVariance] = aryule(data,order)
[ar_coeffs,NoiseVariance,reflect_coeffs] = aryule(data,order)


ar_coeffs = aryule(data,order) returns the AR coefficients for the input data and model order. The elements of ar_coeffs are normalized by ar_coeffs(1). order is a positive integer that cannot exceed the length of the input data.

[ar_coeffs,NoiseVariance] = aryule(data,order) returns the estimated variance NoiseVariance of the white noise input.

[ar_coeffs,NoiseVariance,reflect_coeffs] = aryule(data,order) returns the reflection coefficients reflect_coeffs.


Create an AR(4) process and estimate the coefficients:

A=[1 -2.7607 3.8106 -2.6535 0.9238];
% AR(4) coefficients
%filter a white noise input to create AR(4) process
%compare the results in ar_coeffs to the vector A.

Estimate model order using decay of reflection coefficients:

rng default;
y=filter(1,[1 -0.75 0.5],0.2*randn(1024,1));
%create AR(2) process
% Fit AR(10) model
stem(reflect_coeffs); axis([-0.05 10.5 -1 1]);
title('Reflection Coefficients by Lag'); xlabel('Lag');
ylabel('Reflection Coefficent');

The reflection coefficients decay to zero after lag 2, which indicates that an AR(10) model significantly overestimates the time dependence in the data.

More About

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AR(p) Model

In an AR model of order p, the current output is a linear combination of the past p outputs plus a white noise input. The weights on the p past outputs minimize the mean-square prediction error of the autoregression. If y[n] is the current value of the output and x[n] is a zero-mean white noise input, the AR(p) model is:

Reflection Coefficients

The reflection coefficients are the partial autocorrelation coefficients scaled by (–1). The reflection coefficients indicate the time dependence between y[n] and y[n-k] after subtracting the prediction based on the intervening k-1 time steps.


aryule uses the Levinson-Durbin recursions on the biased estimate of the sample autocorrelation sequence to compute the coefficients.


Monson,H. Statistical Digital Signal Processing and Modeling, John Wiley & Sons, 1996.

See Also

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