Curve fitting of summed normal distributions

[exmachina]

Hi,

I have a dataset of a random variable whose probability density function can be fitted/modelled as a sum of N probability density functions of normal distributions:

$F_X(x) = p(\mu_1,\sigma_1^2)+p(\mu_2,\sigma_2^2)+\ldots+p({\mu}_x,\sigma_x^2)$

I am interested in a fitting method can robustly determine the values of $\mu_1,\sigma_1,\mu_2,\sigma_2,$ etc

Note this is NOT convolution of normal distributions.

 

[Bill Simpson]

These folks have put a lot of time and thought into your problem

http://www.sigmaplot.com/products/peakfit/peakfit.php

and they have free 30 day trial evaluations.

 

[exmachina]

Interesting, any idea what method they use? Expectation-Maximization?

 

[exmachina]

Edit: I guess in particular, this is the equation I'm trying to maximize, given an input vector:$X = (x_1,x_2,...,x_n)$

Maximize:

$$\begin{equation}<br /> \prod_{j=1}^n\sum_{i=1}^k \frac{p_i}{\sqrt{2\pi} \sigma_i} \exp(-\frac{(x_j-\mu_i)^2}{2\sigma_i^2})<br /> <br /> Edit: I found a nice paper tackling this exact problem using EM.<br /> \end{equation}$$

Subject to $$\sum_{i=1}^{k} p_i = 1$$

When I say maximize, I mean to find the model parameters $$\mu_i, \sigma_i, p_i$$