arcov

Autoregressive all-pole model parameters — covariance method

Syntax

a = arcov(x,p)

[a,e] = arcov(x,p)

Description

a = arcov(x,p) returns the normalized autoregressive (AR) parameters corresponding to a model of order p for the input array x, where x is assumed to be the output of an AR system driven by white noise. This method minimizes the forward prediction error in the least-squares sense.

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

Examples

Parameter Estimation Using the Covariance Method

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 covariance 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 = arcov(y,4)

arcoeffs = 1×5

    1.0000   -2.7746    3.8419   -2.6857    0.9367

Generate 50 realizations of the process, changing each time the variance of the input noise. Compare the covariance-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)] = arcov(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] = arcov(Y,4);

hold on
plot(noisestdz.^2,variances,'o')
hold off
legend('Single channel loop','Multichannel','Location',"best")

Input Arguments

x — Input array

vector | matrix

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

p — Model order

positive integer scalar

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

a — Normalized autoregressive parameters

row vector | matrix

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.

e — White noise input variance

scalar | row vector

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.

More About

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.

Let y(n) be a wide-sense stationary random process obtained by filtering white noise of variance e with the system function A(z). If P_y(e^jω) is the power spectral density of y(n), then

$$P_y(e^{j\omega })=\frac{e}{{\left|A(e^{j\omega })\right|}^2}=\frac{e}{{\left|1+{\displaystyle \sum _{k=1}^{p}a}(k)e^{-j\omega k}\right|}^2}.$$

Because the covariance method characterizes the input data using an all-pole model, the correct choice of the model order, p, is important.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Version History

Introduced before R2006a