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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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    @Michael, do you have any good pointers? books? articles? Something else?2011-08-10

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