Main Content

aryule

Autoregressive all-pole model parameters — Yule-Walker method

Description

example

a = aryule(x,p) returns the normalized autoregressive (AR) parameters corresponding to a model of order p for the input array x.

example

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

Examples

collapse all

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);
noisevar = zeros(1,nrealiz);

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

plot(noisestdz.^2,noisevar,'*')
title('Noise Variance')
xlabel('Input')
ylabel('Estimated')

Repeat the procedure using the function's multichannel syntax.

Y = filter(1,A,noisestdz.*randnoise);

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

hold on
plot(noisestdz.^2,variances,'o')
hold off
legend('Single channel loop','Multichannel','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. Only the first two coefficients lie outside the 95% confidence bounds, indicating that an AR(10) model significantly overestimates the time dependence in the data. See AR Order Selection with Partial Autocorrelation Sequence for more details.

[ar,nvar,rc] = aryule(y,10);

stem(rc)
xlim([0 11])
conf95 = sqrt(2)*erfinv(0.95)/sqrt(1024);
[X,Y] = ndgrid(xlim,conf95*[-1 1]);
hold on
plot(X,Y,'--r')
hold off
title('Reflection Coefficients')

Input Arguments

collapse all

Input array, specified as a vector or matrix.

Example: filter(1,[1 -0.75 0.5],0.2*randn(1024,1)) specifies a second-order autoregressive process.

Data Types: single | double
Complex Number Support: Yes

Model order, specified as a positive integer scalar. p must be less than the number of elements or rows of x.

Data Types: single | double

Output Arguments

collapse all

Normalized autoregressive parameters, returned as a vector or matrix. If x is a matrix, then each row of a corresponds to a column of x. a has p + 1 columns and contains the AR system parameters, A(z), in descending powers of z.

White noise input variance, returned as a scalar or row vector. If x is a matrix, then each element of e corresponds to a column of x.

Reflection coefficients, returned as a column vector or matrix. If x is a matrix, then each column of rc corresponds to a column of x. rc has p rows.

More About

collapse all

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 squared 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

k=0pa(k)y(nk)=x(n).

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.

Algorithms

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

References

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

Introduced before R2006a