 |
|
|
|
|
 | We attempt to make a composite noise model. One component will be normal gaussian, the other will attempt to fit the tail. A Weibull distribution follows the functional form 1 - exp(-m xg). The parameters for m and g can be found by taking log(log(1 - H)), where H is the cumulative histogram, and fitting a line against log x in the (tail) data. The resulting fit looks like:
|
|
|
|
|
|
| The gaussian can be better fit by finding the straight line section at the 50% point in its distribution, the slope of which identifies sigma. Here is a plot showing the cumulative histogram data (dots), the fitted normal distribution (N, solid), and the Weibull model fit (W, dashed). So far, these are independent models for different sections of the observed distribution.
|
|
|
|
|
|
| It is not clear how to best combine these separate models. It does not appear that both noise mechanisms are active everywhere (the Weibull model especially has strong numbers near zero, but we are interested in its behavior far away from there). Without deeper physical guidance, a sigmoid weighting that blends the two functions was used. The crossover occurs where the Weibull and normal distributions each account for half of the tail residual. The dashed line in the figure is the blending weight for W, the solid line is the net distribution obtained.
|
|
|
|
|
|
| Here is a log view of the residual 1-H, and the fitted function.
|
|
|
|
|
|
| Finally, here is a review of the original histogram itself, along with the fitted components, normal gaussian (solid) and Weibull (dashed).
|
|
|
|
|
 |
|
