But that’s just least squares…

August 9th, 2006 ~ Posted in: Mathematics

My latest task has been to look at a formula Bernard derived which measures the error in using a given FIR filter to predict the values of a certain type of random field. In the code, we’ve been using least squares prediction– just calculating the relevant quantities for a given type of tissue, and solving the normal equations to get the filter taps– but this was without justification. Or at least Papadakis seems to think it is (at least, that is the only reason I see for asking me to look at the formula). I think it is perfectly justified: after all, the first step in getting the normal equations is to write an error expression, and then find the coefficients which minimize them; this assumes only that the field is wide sense stationary (which we do).

To set the stage, let c \in C^\infty(\Pi^d) be the power spectral density (fourier transform of the autocorrelation sequence) of the random field, and let p = \sum_{k=1}^\Lambda p_k \cos(2 \pi k \cdot \xi) be a trigonometric polynomial of ‘length’ \Lambda with all real coefficients p_k. Since this trig poly is really just the fourier transform of the filter tap sequence, we are assuming implicitly that our filter is symmetric (\frac{p_{-k}}{2} = \frac{p_k}{2}) and real. The error expression he derived is

\displaystyle Q(p) = \int_{\Pi^d} |1 - p(\xi)|^2 c(\xi)\, d\xi.

But, if you consider Q as a function of the filter taps and minimize it accordingly, you get \Lambda equations

\displaystyle \int_{\Pi^d} \cos(2\pi n \cdot \xi) c(\xi) \, d\xi = \sum_{k=1}^\Lambda p_k \int_{\Pi^d} \cos(2\pi\xi \cdot k) \cos(2 \pi \xi \cdot n) c(\xi)\, d\xi,

which look suspiciously like Parseval’s equality applied to the usual normal equations. They aren’t quite the normal equations, (there’s a \sin(2\pi \xi \cdot k) \sin(2\pi \xi \cdot l) term mysteriously missing from under the right hand integral), but they are pretty close.

Mathematics ~

This entry was posted on Wednesday, August 9th, 2006 at 10:32 am and is filed under Mathematics. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

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