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:

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

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

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

 

[CellCoree]

Hello,

Thank you for sharing your dataset and question with me. I understand the importance of accurately fitting a probability density function to a dataset. In this case, it seems that your dataset can be best modeled as a sum of N normal distributions.

To accurately determine the values of \mu_1, \sigma_1, \mu_2, \sigma_2, and so on, I would recommend using a curve fitting method such as the least squares method or maximum likelihood estimation. These methods are commonly used in statistics and can provide robust estimates for the parameters of a probability density function.

It is important to note that this is not the same as convolution of normal distributions. Convolution is a mathematical operation that combines two probability density functions, while curve fitting aims to find the best-fitting function for a given dataset.

I hope this helps and good luck with your analysis.

Sincerely,