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I have some data which I want to analyze by fitting a function to it. To do that, I have two functions, one being a gaussian, and one the sum of two gaussians. To test the goodness of these fits, I test the with scipy's ks-2samp test. The result of both tests are that the KS-statistic is $0.15$, and the P-value is $0.476635$.

figure

There is clearly visible that the fit with two gaussians is better (as it should be), but this doesn't reflect in the KS-test.

I am sure I dont output the same value twice, as the included code outputs the following: (hist_cm is the cumulative list of the histogram points, plotted in the upper frames)

KStest= stats.ks_2samp(hist_cm,hist_fit_cm)
KStest_1gauss = stats.ks_2samp(hist_cm,hist_fit_1gauss_cm)
print KStest
print KStest_1gauss
#output
#(0.15000000000000002, 0.47663525071642981)
#(0.15000000000000002, 0.47663525071642981)

I know the tested list are not the same, as you can clearly see they are not the same in the lower frames.

So i've got two question: Why is the P-value and KS-statistic the same? (this might be a programming question). And how to interpret these values?

edit: I calculate radial velocities from a model of N-bodies, and should be normally distributed. In the figure I showed I've got 1043 entries, roughly between $-300$ and $300$. I then make a (normalized) histogram of these values, with a bin-width of 10. To this histogram I make my two fits (and eventually plot them, but that would be too much code)

bins = np.arange(-300,301,10)
hist, bin_edges = np.histogram(v[i], normed=True, bins=bins)
hist_cm=np.cumsum(hist)
bin_centres = (bin_edges[:-1] + bin_edges[1:])/2
coeff, pcov2 = leastsq(residuals, x0=(0.01,0.,60.,0.01,150.,40.) ,args=(bin_centres, hist))
coeff, pcov2 = leastsq(residuals, x0=(0.01,0.,60.,0.01,150.,40.) ,args=(bin_centres, hist))
hist_fit = gauss2(bin_centres, *coeff)
hist_fit_cm=np.cumsum(hist_fit)
coeff_1gaus, pcov2_1gauss = leastsq(residualsOneGauss, x0=(0.01,0,100), args=(bin_centres, hist))
hist_fit_1gauss = gauss(bin_centres, *coeff_1gaus)
hist_fit_1gauss_cm = np.cumsum(hist_fit_1gauss)

#and the used functions:
def gauss(x, A, mu, sigma):
  return A*np.exp(-(x-mu)**2/(2.*sigma**2))

def gauss2(x,A, mu, sigma, A2, mu2, sigma2):
  if A2<0:
    return 1000
  return A*np.exp(-(x-mu)**2/(2.*sigma**2))+ A2*np.exp(-(x-mu2)**2/(2.*sigma2**2))

def residuals(p, x,y):
  integral = quad( gauss2, -500, 500, args= (p[0],p[1],p[2],p[3],p[4],p[5]))[0]
  penalization = abs(1-integral)*1000
  return y - gauss2(x, p[0],p[1],p[2],p[3],p[4],p[5] ) - penalization

def residualsOneGauss(p,x,y):
  integral = quad( gauss, -500, 500, args= (p[0],p[1],p[2]))[0]
  penalization = abs(1-integral)*1000
  return y - gauss(x, p[0],p[1],p[2]) - penalization
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  • $\begingroup$ Are your distributions fixed, or do you estimate their parameters from the sample data? In the latter case, there shouldn't be a difference at all, since the sum of two normally distributed random variables is again normally distributed. Also, why are you using the two-sample KS test? That's meant to test whether two populations have the same distribution (independent from what the specific distribution is), not to check whether one polulations has a specific distributions. For the latter case, the one-sample KS test is appropriate, but only if the distribution to compare to is fixed. $\endgroup$ – fgp Apr 9 '14 at 11:38
  • $\begingroup$ I estimate the variables (for the three different gaussians) using scipy.leastsq with a penalty on the integrand. I dont get why I get the same results with both functions. What is another way to test my fit with the provided data? $\endgroup$ – Mathias711 Apr 9 '14 at 11:43
  • $\begingroup$ That doesn't make sense. If you estimate the parameters, i.e. determine the gaussian's mean and variance, from the sample data, than what's the difference between your three different gaussians? $\endgroup$ – fgp Apr 9 '14 at 11:46
  • $\begingroup$ Because I have two functions: func1 is the sum of two, uncorrelated (the parameters arent, the function of course is), gaussians. And func2, that is just ` gaussian. In my picture, in the lower left frame, you see in green and red the two indepent gaussians from func1, and in teal the sum of these two, which is func1. I hope this made it clearer $\endgroup$ – Mathias711 Apr 9 '14 at 11:50
  • $\begingroup$ I've said it, and say it again: The sum of two independent gaussian random variables is again a gaussian random variable! If $X_1,X_2$ have means $\mu_1,\mu_2$ and variables $\sigma_1^2,\sigma_2^2$, then $X_1 + X_2$ has mean $\mu_1 + \mu_2$ and variance $\sigma_1^2 + \sigma_2^2$. The fact that you're see a difference at all (even if not in the KS test statistic) is just an artifact of the way you estimate the parameters, I think. $\endgroup$ – fgp Apr 9 '14 at 11:53

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