Compute Mean Square Error (MSE) for a Problem

[kasraa]

Hi all,

I want to compute mean square error (MSE) for a problem but I'm not sure if I'm doing it right.

Suppose that I want to estimate a variable (e.g. the position of an object) like x. The estimation process depends on the realizations of some specific random variables (i.e. Gaussian noises). In order to get accurate results, I know that I have to perform the estimation process N times with different seeds (i.e. different realizations of noises), right?

Lets show the output (the estimated position) of the i'th trial with x_i.

So I have x_1,...,x_N. Assume that we have access to the true value of x which is showed by x_(true).

How should I compute the MSE?

(1) $$\frac{1}{N} \sum_{i=1}^{N} (x_{true} - x_i)^2$$

or

(2) $$(x_{true} - (\frac{1}{N} \sum_{i=1}^{N} x_i))^2$$

Many thanks.

 

[mathman]

Equation (1) is the correct formula. One problem with equation (2), which indicates how wrong it can be, is that you could have a sample set with an average equal to the true mean, but with wildly fluctuating terms, leading to zero as your variance estimate.

 

[kasraa]

Thanks for your help.