# Estimating point coordinates given its coordinates and the distance (norm)

What would be the MLE (Maximum Likelihood Estimator) of the given data:

Assuming the i-th point ${p}_{i} = ({x}_{i}, {y}_{i})$.
We have the following measurements of this specific point - $(x, y, r = \sqrt{{x}_{i}^{2} + {y}_{i}^{2}})$.
Namely, we have a measurement of the x, y coordinate and the norm of the point with the following distributions:

$$x \sim N(0, {\sigma}_{x}), y \sim N(0, {\sigma}_{y}), r \sim N(0, {\sigma}_{r})$$

Given this measurement vector, what would be the MLE of ${p}_{i}$ coordinates (${x}_{i}, {y}_{i})$?

I know the brute force to calculate it, is there any efficient method for this specific formulation?

What about given a set of points and their corresponding measurements?
Assuming the model of the set of points is linear (polynomial of degree N, etc...).

• You're absolutely right. I didn't mention the quantity to estimate (The Parameter Vector) which is the point coordinate (${x}_{i}, {y}_{i})$. – Royi Nov 4 '13 at 5:29
• Sorry, I thought $(x_i,y_i)$ was data. If that's parameters, what's the data? I think perhaps I am confused by the notation. – Glen_b -Reinstate Monica Nov 4 '13 at 11:38