# Defining a function as a sum of several Gaussians

I have a function that is a summation of several Gaussians. Working with a 1D Gaussian here, there are 3 variables for each Gaussian: A, mx, and sigma:

$A \exp \left ( - \frac{\left ( x - mx \right )^{2}}{2 \times sigma^{2}} \right )$

A*Exp[-((x - mx)^2/(2 sigma^2))]


The number of Gaussians in the final function will be vary each time the function is called, so my question is: What is the best way to define a function in Mathematica that can handle this variation, rather than hard-coding each Gaussian?

I was thinking along the lines of providing a list of {A,mx,sigma} to the function, so that if I want one Gaussian, I provide:

f[{{A,mx,sigma}}]


And if I want two Gaussians, I provide

f[{{A,mx,sigma},{A2,mx2,sigma2}}]


which would give:

A*Exp[-((x-mx)^2/(2sigma^2))] + A2*Exp[-((x-mx2)^2/(2sigma2^2))]


and so on.

But I'm not at all sure how to design the function f[] to do this efficiently (for example, can it be done without a For[] loop? Can it be compiled in future if necessary?).

Any help much appreciated - I did several searches on here and couldn't find anything, but I realise that might be because I'm not sure how to define my problem succinctly, so apologies if it has been asked before and I've missed it.

• It is not clear to me that this question is statistically valid. You have a random variable $X_i \sim N(\mu_i, \sigma_i^2)$ with pdf $f_i(x_i)$. Why are you adding the pdfs $f_i(x_i)$? Are you aware that your density has to integrate to 1 in order to be well-defined? Are you intending to find the pdf of $X_1 + X_2 + ...$? Or something else? It is usually better to first express your problem as a math stats problem, before diving into the mma code. Commented Jun 23, 2014 at 15:44
• It's for some Gaussian peak-fitting. Commented Jun 23, 2014 at 15:46
• Or rather, some peak-fitting that happen to look a lot like Gaussians. Commented Jun 23, 2014 at 15:47
• For fitting Gaussians, you might be interested in the answers here. Commented Jun 23, 2014 at 16:18
• Yep, thanks Kenny - I'd seen that before (and in fact looking again it does actually include an answer to this question...oops!) and am using some of the ideas from it. Commented Jun 23, 2014 at 21:45

Since all of the component functions are Listable

f[{a_, m_, s_}, x_] := Total[a*Exp[-(x - m)^2/(2*s^2)]]

n = 5;

amp = Array[a, n];

mean = Array[m, n];

sigma = Array[s, n];

f[{amp, mean, sigma}, x]


a[1]/E^((x - m[1])^2/ (2*s[1]^2)) + a[2]/E^((x - m[2])^2/ (2*s[2]^2)) + a[3]/E^((x - m[3])^2/ (2*s[3]^2)) + a[4]/E^((x - m[4])^2/ (2*s[4]^2)) + a[5]/E^((x - m[5])^2/ (2*s[5]^2))

f[data_] := Total[#1*Exp[-((x - #2)^2/(2 #3^2))] & @@@ data];
f[{{A, mx, sigma}}]
f[{{A, mx, sigma}, {A2, mx2, sigma2}}]


Use Function, Apply and Total.

The solution by Apple was very helpful. However, I found it did not work for me (I had a similar situation). No function of x is created in that example. I found the following would work:

f[data_] := Total[#1*Exp[-((x - #2)^2/(2 #3^2))] & @@@ data];
(* a specific example might be *)
g[x_] := Evaluate[f[{{x, 1.0, 0.5, 0.5}, {x, 2.0, 0.7, 1.5}}]]


g now acts as a function of x, but of course the parameters needed to be specified before numerical values are obtained. This is done above.