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Can anyone explain what are Autoregressive Coefficients? What is their meaning that is.

Consider a method:

public double[] calculateARCoefficients(double[] inputseries, int order)

When this method is called on 256 values, lets say these values represent some kind of signal, I will get an array of n numbers, where n=order. What do these numbers represent?

I mean for example variance tells me how spread out the values of the signal are. The higher variance value is, the more spread out values are. The smaller variance is, the more unified signal is.

Are you able to say, in a similar fashion to above paragraph about variance, what value of AR coefficients says about the signal?

LucasSeveryn
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1 Answers1

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Briefly: A signal is often modeled as a zero mean stationary stochastic process, and its main characteristics of interest are described by its second-order statistics: the autocorrelation function, or its Fourier transform: the spectral density.

Now, it can be shown that we can construct an AR process of sufficiently high order $N$ that has (for $N->\infty$) the same spectrum that our process (and then, according to our assumption that we are interested only in second-order moments, is statistically "the same" as our signal).

An AR process is described by the recursive difference equation $x_n = a_1 x_{n-1} + \cdots + a_N x_{n-M} + e_n$ where $e_n$ is white noise (think of it as a random number generator with zero mean).

To gain some insight, consider the AR(1) process, with, say $a_1= 0.9$, so that $x_n = 0.9 x_{n-1} + e_n$. In this case, the signal value in the "present time" will tend to be quite similar to the previous value, which means that the signal will be highly correlated (i.e., more smooth). By contrast, if $a_1=0.1$, the signal will be much more noisy.

In your example, your method calculateARCoefficients will compute some coefficients so that the corresponding AR signal will have (approximately) the same second order statistics (same spectrum) as your signal.

leonbloy
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