How can I compute the volume under a three class ROC surface

Hi, I am working on a three class diagnosis problem where I need to compute the volume under a three class receiver operator characteristic (ROC) surface. An example of such a surface is provided with the link below.

https://docs.google.com/file/d/0B7YLR6pCjnzjdzlJZVFlQmpvbE0/edit?usp=sharing

I am having problem with the way of computing the volume under this non-uniform complex surface. As a resource, I have a set of triplets (x,y,z points), but these are not organized as meshgrid.

If anyone can provide me a solution of this problem,this would be really helpful for me. Thanks in advance.

Answer 1

[Roger Stafford]

Based on what I see in your link, your ROC volume can be subdividing into individual five-faced polyhedrons each of whose base is a triangle in the x-y plane, whose three sides are trapezoids orthogonal to this plane and whose top is another (slanted) triangle. Your volume is the sum of the volumes of these separate polyhedrons.

Call the cartesian coordinates of the vertices of a top triangle (x1,y1,z1), (x2,y2,z2), and (x3,y3,z3) where the ordering from points 1 to 2 to 3 is counterclockwise with respect to the x-y plane. Then the volume of the polyhedron underneath this top triangle is:

 V = 1/6*det([x1,x2,x3;y1,y2,y3;1,1,1])*(z1+z2+z3)

(The area of its triangular base is 1/2*det([x1,x2,x3;y1,y2,y3;1,1,1]) and the average height is 1/3*(z1+z2+z3).) Note that if the ordering of the three points is clockwise, this expression will be the negative of the actual volume.

Thus, your job is now to determine the coordinates of all those points in the surface.

(Corrected)

[Roger Stafford]

Note that I corrected the array in the determinant which had an erroneous comma in place of a semicolon.