# Why would prefiltering measurement data affect the least squares estimate?

In estimating parameters in a discrete time model I've often seen the use of filters applied to the input data, before its applied to least squares processing. I've been told that the filters are useful if one wishes to remove bias or high frequency noise that's not considered a part of the model.

But I'm currently working with a problem where I'm using the Moore Penrose pseudoinverse to calculate the least squares. I'm filtering the input data with a lowpass filter to remove high frequency noise well beyond the model dynamics, but the choice of the filter pole seems to have a significant affect on the outcome of the estimate. And in any case the residues are near zero (they don't seem to be better or worse according to pole selection).

How should one chose to filter input data, and should it affect the estimate?

• do you mean LMS adaptive filtering? what parameters are you trying to estimate in a least-squares manner? – robert bristow-johnson Nov 5 '15 at 1:15
• @robertbristow-johnson No - least squares. I have a linear parametric model $z = \phi \theta^T$ and I'm estimating the parameter vector $\theta$ using pseudoinverse. – docscience Nov 5 '15 at 1:39
• $z$ is the output and $\phi$ is the input and you're measuring both and trying to determine the values of the elements of the $\theta$ matrix? – robert bristow-johnson Nov 5 '15 at 1:42
• @robertbristow-johnson yes - basically that's it. So if I filter $z$ and $phi$ with the same filter - but such that it does not step on the dynamics of the model, then I should always get the same $\theta$ regardless of the filter poles - right? – docscience Nov 5 '15 at 1:51
• it seems to me that it should be $$z = \theta^T \phi$$. i would prefer "$x$" for input and "$y$" for output and "$h$" for what goes in-between. can you form the question more rigorously? at least for the sake of those of us that haven't done System Identification since grad school? – robert bristow-johnson Nov 5 '15 at 1:56