ParX

Statistical Models

ParX is often used for estimating the parameters of statistical distributions of events, or to test for normality. Here the cumulative normal distribution is calculated as a Chebyshev approximation (to within an error margin of 1e-7): 


                                        
statistical-erf.parx

                                    
                                        
Alternatively, we can use the build-in error-function:

                                    
                                        
R = p - (1 + erf(x.s))/2

                                    
                                        

                                    
                                        
This model checks the “normality” of a histogram of events.
                                        

                                    
                                        

                                        
statistical-normhist.parx

                                    
                                        
As an example, we present the histogram data collected from a radiation detector.
                                            The energy from the particle is converted to a voltage at the output of the detector electronics.
                                            The output voltage is between 50 and 250 (mV),
                                            and collected with a bucket-size w of 4 (mV).
                                            The total number of events Q is close to 500k.
                                            The spread parameter σ that is extracted is a measure of
                                            the noise in the detection system,
                                            in relation to the gain μ.
                                        

                                    
                                        
Histogram Data

                                    
                                        

                                            When the model does not describe a monotonous function,
                                            the initial values for the parameters must
                                            be chosen with considerable care.
                                            The existence of multiple solutions is almost guaranteed.
                                            The optimization must be started within de “collection region” of the intended solution.