Autoregressive processes : learning with an interactive tool





Bits of a non-interactive study



Creating an interactive tool


screen capture from the Interactive AR2 demo



My feeling about interactive plotting with Matplotlib


Author: pierre

assistant professor in Electrical Engineering & Control

3 thoughts on “Autoregressive processes : learning with an interactive tool”

  1. Your study is very interesting, and intuitive~

    Nowadays I'm also working with AR models,  actually trying to find the coefficients' behavior with respect to the signal's frequency content. Then I found out, the roots of  AR characteristic polynomial, are the the main components for the autocorrelation function of this AR process. And the frequency spectral density function is then the fourier transform of our autocorrelation function. So it seemed to me that the coefficients do have some influence on the frequency domain~

    As for the capture of oscillations, the real root of AR characteristic polynomial contribute to the autocorrelation function that has a exponential decaying shape (linear signal), and the complex root gave a exponentially damped oscillation. So I guess, while the order of AR model is 2 or above, there is the possibility that the characteristic polynomial will have a complex root, and therefore has the ability to describe a wave shaped signal, or say, a signal which has some more frequency content.

    But then, the behavior of 2nd order AR is hard enough to interpret，and I'm working with 4,5 or 8th order AR... kind of frustrated to find some one to one relation for AR coefficients and frequency components.


    1. The AR coefficients have of course an influence on the spectral density : there are directly linked. I just found a quick reference on which gives the formula for the spectrum of an ARMA.

      About the characteristic polynomial, you can indeed see on the plot that the two roots are complex conjugate when the behavior is oscillatory. That's why an order of at least two is required to get oscillations.

      The physical interpretation of coefficients for a discrete time model is alway difficult, but much better with continuous time models. Also, for higher order models, it is always possible to factorize the polynomials as a product of first or second order terms.

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