Hvolume

Calculates the H-volume (generalised volume) of an n-box in a space with some measure H. Usefull in probability theory, the H-volume computes the probability measure within some n-box from a multivartiate cdf H.

An n-box J is a hyper-rectangle made from the carteesian product of n intervals:

The two dimensional calculation is:

The general formula is:

where the sum is taken over every vertex of J.

Code

Vol = Hvolume(measure, J1, J2, .. , Jn)

measure is an anonymous function of dimension n and J1 to Jn are n intervals defining the n-box.

>> C1 = @(x) unifcdf(x,0,1);
>> H1 = Hvolume(C1,[0,0.5])

H1 =

    0.5000

>> Rho2 =[1, 0; 0, 1];
>> C2 = @(x) copulacdf('Gaussian', [x(1), x(2)],Rho2);
>> H2 = Hvolume(C2,[0,0.5],[0,0.5])

H2 =

    0.2500

>> Rho3 =eye(3);
>> C3 = @(x) copulacdf('Gaussian', [x(1),x(2),x(3)],Rho3);
>> H3 = Hvolume(C3,[0,0.5], [0,0.5], [0,0.5])

H3 =

    0.1250

>> Rho4 =eye(4);
>> C4 = @(x) copulacdf('Gaussian', [x(1),x(2),x(3),x(4)],Rho4);
>> H4 = Hvolume(C4,[0,0.5], [0,0.5], [0,0.5], [0,0.5])

H4 =

    0.0625

References

Schweizer, B. and Sklar, A., 2011. Probabilistic metric spaces. Courier Corporation.
Ferson, S., R. Nelsen, J. Hajagos, D. Berleant, J. Zhang, W.T. Tucker, L. Ginzburg and W.L. Oberkampf. 2004. Dependence in Probabilistic Modeling, Dempster-Shafer Theory, and Probability Bounds Analysis. Sandia National Laboratories, SAND2004-3072, Albuquerque, NM

AnderGray/Hvolume-Matlab

Hvolume

Code

References