STAR_DISCREPANCY is a C program which computes bounds on the star discrepancy of an M-dimensional set of N points contained in the unit hypercube.
The star discrepancy is a commonly cited statistic for determining how uniformly a pointset is distributed over a region. For convenience, this region is usually taken as the unit hypercube; STAR_DISCREPANCY will assume that datasets under investigation are meant to fill up the unit hypercube.
If the pointset to be investigated actually lies in some other hypercube, a simply translation and rescaling may be enough to transform the data. This will probably NOT be satisfactory if the original region is rectangular, but has sides of different length, or if the region is not rectangular.
The discrepancy measures the worst error that would be made in estimating the area of a subregion of the hypercube by simply noting the fraction of the pointset contained in the subregion. If arbitrary subregions were allowed, then it would always be possible to make this error equal to 1 (just take the region consisting of the hypercube minus the pointset.) Since any "reasonable" area can be arbitrarily well approximated by rectangles, the star discrepancy calculation uses only rectangular subregions, whose sides are aligned with coordinates directions, and one of whose corners is at the origin.
Formally, the star discrepancy of a pointset of n points is symbolized by D_{n}^{*} and defined as
D_{n}^{*} = supremum ( P in I* ) | ( A(P,x) / n ) - lambda ( P ) |Here, I* is the set of all M-dimensional subintervals of the form [0,p1] x [0,p2] x ... x [0,ps] where every p is between 0 and 1; P is any such subinterval; lambda(P) is the volume of the subinterval, A(P,x) is the number of points of the point set x that occur in P, and n is the number of points in x.
Clearly, the star discrepancy is measuring how badly the pointset estimates the volume of a subinterval. This worst error is somewhere between 0 (absolutely no error ever) and 1 (totally missing the volume of the unit hypercube). A value of 0.25, for instance, means that there is a subinterval in the unit hypercube for which the difference between its true and estimated volumes is 0.25. (It might have a volume of 0.80, and be estimated at 0.55, for instance, or a volume of 0.05 that is estimated at 0.30.)
Two file formats are allowed to specify the point set x={x(1),...,x(n)} in I^m:
1) As real numbers, x(i)=(x(i,1),...,x(i,m)). The file must contain the n+1 following lines:
m n reals x(1,1) x(1,2) ... x(1,m) ... x(n,1) x(n,2) ... x(n,m)
2) As fractions, x(i)=(num(i,1)/den(i),...,num(i,m)/den(i)). The file must contain the n+1 following lines:
m n fractions num(1,1) num(1,2) ... num(1,m) den(1) ... num(n,1) num(n,2) ... num(n,m) den(n)
STAR_DISCREPANCY is available in a C version and a C++ version.
DIAPHONY, a C program which reads a file of N points in M dimensions and computes its diaphony, a measure of point dispersion.
TABLE_LATINIZE, a C++ program which can read a TABLE file and write out a "latinized" version.
TABLE_QUALITY, a C++ program which can read a TABLE file and print out measures of the quality of dispersion of the points.
Eric Thiemard
star_discrepancy 2 0.001 10 halton_reals.txt
star_discrepancy 2 0.001 10 halton_fractions.txt
You can go up one level to the C source codes.