# Statistical significance test in polygaussian fitting, using Levenberg-Marquardt

I have a set of dihedral angle values that I have fitted using a polygaussian function via the Levenberg-Marquardt algorithm http://en.wikipedia.org/wiki/Levenberg-Marquardt. Specifically, the function $$f(x) = G_1(x) + G_2(x) + ... + G_n(x)$$ where $$Gi(x) = \frac{\omega_i}{\sqrt(2\pi)\sigma_i}e^{-\frac{(x - \mu_i)^2}{2\sigma_i²}}$$

was defined. The values of $\mu_i,\sigma_i,\omega_i$ are declared as the respective mean, standard deviation and weight constants for that particular subgaussian. It is also true that $\sum^n_{i=1}{\omega_i} = 1$

Given that the original values were a dihedral angles vector, I plotted a histogram from them, using a particular bin size. Then, I extracted the $(x, y)$ pair values from the final bars. $x$ are the x-axis values in the middle of the bin, and $y$ are the frequencies for that bin, within the range $(x - binsize/2,x + binsize/2)$.

I am trying to figure out a way to assess the statistical significance of the fitting. I know that from this analysis a goodness of fit value (Q) can be obtained. However, I am aware that the values that I plugged in where modified to create the $(x, y)$ pairs, and there are no estimated errors for them.

I can currently measure the fitting quality via the area difference of the final function and the area from an averaged shifted histogram of the same dataset (http://sfb649.wiwi.hu-berlin.de/fedc_homepage/xplore/ebooks/html/spm/spmhtmlnode11.html). A low area difference value shows that the fittings are good, but not that they are statistically significant.

I would also like to ask if there exist a Normality Test available for multiple Gaussians (http://en.wikipedia.org/wiki/Normality_test)

Thanks for any advice, Ignacio

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