# Point-wise error estimate in polynomial regression

In our application we wish to estimate the actual path of objects. We have a set of samples of object locations $(t_i, x_i, y_i, P_i)$ where $t_i$ is the sample time, $(x_i, y_i)$ is the 2D location, and $P_i$ is the error covariance matrix.

We estimate the path using polynomial regression as a function of $t$. That is, we have two polynomials $x(t) = a_x + b_x t + c_x t^2 + d_x t^3$ and $y(t) = a_y + b_y t + c_y t^2 + d_y t^3$. We find $\{a, b, c, d\}$ using a log-likelihood estimator. That is, we minimize: $$\Sigma (x(t_i) - x_i, y(t_i) - y_i) P_i^{-1} (x(t_i) - x_i, y(t_i) - y_i)^T$$

I would like to be able to estimate the approximation error for each $t$. After finding our point $(x(t), y(t))$ using the above polynomials, we also want the error covariance matrix $P(t)$. Does anybody know of a method of doing so?

Note that when $P_i$ are diagonal matrices we get the simple linear regression that has known coefficient error estimates that I can use. Does anybody know how can this be done for the non-diagonal case?

Alex.

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I'm puzzled by the $P_i$ notation. A different covariance matrix for each value of the index $i$? That makes it look as if you mean $P_i$ is a scalar for each $i$, maybe the variance of the $i$th error. That would make the sum make sense, that you're trying to minimize. And is there some difference between what you mean by $x_i$ and what you mean by $x(t_i)$? Or do you mean $P_i$ is a $2\times 2$ matrix of covariances: the variance of $x_i$, the variance of $y_i$, and the covariance between them? Is $P_i$ fully known, or is it to be estimated based on the data? –  Michael Hardy Aug 8 '11 at 18:16
@Michael: $x(t_i)$ is the value of the to-be-fitted function $x(t)$ at time $t_i$, whereas $x_i$ is the actually measured value. My understanding of $P_i$ is what you wrote at the end: a known $2\times2$ matrix of covariances between $x_i$ and $y_i$. –  joriki Aug 8 '11 at 18:42
@Alex: I may be missing something, but it seems to me that the coefficient covariance matrix you want is given in this Wikipedia section? –  joriki Aug 8 '11 at 18:50
$P_i$ is indeed a known error covariance matrix for measurement $i$. @joriki, I already read the mentioned wikipedia section. It seems this section is for the "weighed linear least squares". I don't see how I can represent my case with a diagonal weights matrix. –  Alex Aug 8 '11 at 19:13
In addition, as I mentioned in my original post, when $P_i$ are diagonal matrices it reduces to the simple weighed linear least squares - the one mentioned in the wikipedia page. –  Alex Aug 8 '11 at 19:15