Autoregressive all-pole model parameters — Yule-Walker method


a = aryule(x,p)
[a,e] = aryule(x,p)
[a,e,rc] = aryule(x,p)


a = aryule(x,p) returns the normalized autoregressive (AR) parameters corresponding to a model of order p for the input array, x. If x is a vector, then the output array, a, is a row vector. If x is a matrix, then the parameters along the nth row of a model the nth column of x. a has p + 1 columns. p must be less than the number of elements (or rows) of x.

[a,e] = aryule(x,p) returns the estimated variance, e, of the white noise input.

[a,e,rc] = aryule(x,p) returns the reflection coefficients in rc.


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Use a vector of polynomial coefficients to generate an AR(4) process by filtering 1024 samples of white noise. Reset the random number generator for reproducible results. Use the Yule-Walker method to estimate the coefficients.

rng default

A = [1 -2.7607 3.8106 -2.6535 0.9238];

y = filter(1,A,0.2*randn(1024,1));

arcoeffs = aryule(y,4)
arcoeffs = 1×5

    1.0000   -2.7262    3.7296   -2.5753    0.8927

Generate 50 realizations of the process, changing each time the variance of the input noise. Compare the Yule-Walker-estimated variances to the actual values.

nrealiz = 50;

noisestdz = rand(1,nrealiz)+0.5;

randnoise = randn(1024,nrealiz);

for k = 1:nrealiz
    y = filter(1,A,noisestdz(k) * randnoise(:,k));
    [arcoeffs,noisevar(k)] = aryule(y,4);

title('Noise Variance')

Repeat the procedure using aryule's multichannel syntax.

realiz = bsxfun(@times,noisestdz,randnoise);

Y = filter(1,A,realiz);

[coeffs,variances] = aryule(Y,4);

hold on

q = legend('Single channel loop','Multichannel');
q.Location = 'best';

Use a vector of polynomial coefficients to generate an AR(2) process by filtering 1024 samples of white noise. Reset the random number generator for reproducible results.

rng default

y = filter(1,[1 -0.75 0.5],0.2*randn(1024,1));

Use the Yule-Walker method to fit an AR(10) model to the process. Output and plot the reflection coefficients.

[ar_coeffs,NoiseVariance,reflect_coeffs] = aryule(y,10);

axis([-0.05 10.5 -1 1])
title('Reflection Coefficients by Lag')

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 recursion on the biased estimate of the sample autocorrelation sequence to compute the parameters.


[1] Hayes, Monson H. Statistical Digital Signal Processing and Modeling. New York: John Wiley & Sons, 1996.

Introduced before R2006a