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Interpretation of Second Moment in MC simulations

Mar 29, 2013 at 7:14 PM
Hi,

I was just playing around with the RadianceOfXAndYAndZAndThetaAndPhiDetectorInput and I (only now) noticed that the results of the MC simulations are exported as mean and second moments (and a derived standard deviation). Assuming that these are the means over all runs, how should I interpret the standard deviations?

Am I correct that the variance should approach the mean as we are dealing with a Poisson distribution? Or does that not hold because of the weights associated with the photons along the trajectories?

Thanks,

Martijn
Mar 29, 2013 at 8:33 PM
Hi Martijn,
The calculated mean is determined by mean = E[xi] = (1/N) sum_{i=1}^{N} xi(beta_i) where E is expectation, xi is the random variable or estimator, beta_i is the ith random walk, and N is the number of photons launched in the simulation. The N-sample variance is var[xi] = (1/N) {E[xi^2]-(E[xi])^2} where E[xi^2] is the second moment estimate. The standard deviation sigma = sqrt(var[xi]). This sigma provides the 1-sigma confidence interval about the mean assumed in a normal or Gaussian distribution. So the variance goes to zero as O(1/N) and error as O(1/sqrt(N)).
I hope this helps.
Carole
Apr 2, 2013 at 3:57 AM
Hi Carole,

Thanks for the explanation. I still do not completely get why the variance should go to zero, given the inherent variability between random walks, but hopefully it becomes clear when I played with it a bit more.

Thanks!

Martijn
Apr 2, 2013 at 4:10 AM
The way I think about it, qualitatively, is that the variance and the standard deviation are quantities to understand the probability of the next MEAN value (expectation value of solution, averaging ALL photons' trajectories) to be similar to the one calculated as part of this solution. So, more photons (or a better algorithm) allows you to reduce that uncertainty, down to zero at the limit.


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Apr 3, 2013 at 6:03 PM
Ah yes, I think what went wrong in my head is that I didn't realize that the result of the MC simulation is normalized. Then the figures Carole gave (O(1/N) for the variance and O(1/sqrt(N) for the SD) make perfect sense.

Thanks!

Martijn