Documentation

of the application "eventsystem1"

 

If you use any of the applications in an academic or other type of work, please refer to the relating papers of the section References!

 

Overview

 

It generates an event system randomly based on the pattern of Example 5.2 and 5.3 in Mádi-Nagy (2009) and writes the probabilities of the intersections of the events up to the given order m into the file “prob.txt”. The prob.txt file is an appropriate input of application “bonferroni1”.

 

Manual

 

1.      Download the binary file

eventsystem1.exe (Windows)

eventsystem1 (Linux)

 (Alternatively you can download and compile the C++ source file eventsystem1.cpp)

 

2.      Run the binary file. You will be asked about the number of events (n), the probability (density) of the 1 entry in the generator matrix R (see Technical details), the probability of the union of the events and the maximum order of the intersections (m). 

 

3.      Then the program writes to the file “prob.txt” the probabilities of the intersections of the events of the generated event system up to the order m.

 

The probabilities in “prob.txt” are listed in individual lines, in the following (subscript) order. E.g., in case of n=4, m=3:

P(A1)

P(A2)

P(A3)

P(A4)

P(A1∩A2)

P(A1∩A3)

P(A1∩A4)

P(A2∩A3)

P(A2∩A4)

P(A3∩A4)

P(A1∩A2∩A3)

P(A1∩A2∩A4)

P(A1∩A3∩A4)

P(A2∩A3∩A4)

 

 

Technical details

 

The program uses the generation framework of Example 5.2 and 5.3 of the paper Mádi-Nagy (2009) for the case of m=mj.

 

In case of n events 2n outcomes will be considered, let us denote them by x0,x1,…,x2n. They have the probabilities P(x0),P(x1),…,P(x2n), respectively.  The outcomes are partitions of the sample space. The events of the system are defined by the matrix  R=(rij), where rij =1, if Aj occurs in outcome xi, otherwise  rij =0. Let  x0 be the outcome that all the n events do not occur. Then the probability of the union of the events will be 1-P(x0).

 

In the application P(x0):=1-(probability of the union) and the other probabilities are set as P(x1)=…=P(x2n). The the probabilities of the intersections can be calculated easily.

 

 

References

 

Mádi-Nagy, G.  (2009). On Multivariate Discrete Moment Problems: Generalization of the Bivariate Min Algorithm for Higher Dimensions. SIAM Journal on Optimization 19(4) 1781-1806.

click here to see the paper (.pdf)