Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?


share|cite|improve this question
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.