KS2D is a two-dimensional extension to the Kolmogorov-Smyrnov test for goodness-of-fit. It is used to compare datasets of points to a distributions, or two datasets of points.
In this case, we check if two-dimensional data fits a particular distribution. The extension to higher dimensions is non-trivial and requires on the order of O(n2) operations i.e. slow for large datasets.
Mainly intended to be interacted with using functions ks2d1s and ks2d2s, which take as inputs one 2-column matrix and one 2D function, or two 2-column matrices respectively.
These algorithms compute the relative probabilities of finding data in orthonormal quadrants that surround each point in the data set, then uses those to compute the K-S statistic with its distribution function (Qks). Look around 14.3.7 and 14.7.1 in [3] for detailed arcane mathemagic explanations.
ks2d1s and ks2d2s outputs the KS statistic D, and the significance level prob of that observed value of D, respectively.
Float number representation and rounding: probabilities expected to sum to 1.0 return 0.99999999 instead, etc... No plans to implement any kind of solution to this: it sounds much more trouble than it is worth. This test is theoretically just an approximation, rounding to a couple digits seem reasonable.
ks2d1s: Input: no 2D density matrices. Integration method: only numerical.
Prerequisites: scipy, numpy.
Keywords: Kolmogorov-smirnov test, multi-dimensional, two-dimensional, KS test, hypothesis testing, statistics, probability theory.
Based on the idea by Peacock (1983), an upgrade by Fasano and Franceschin (1987) with much guidance from Numerical recipes in C by Press and Teukolsky (1996).
[1] Peacock, J. A. (1983). Two-dimensional goodness-of-fit testing in astronomy. Monthly Notices of the Royal Astronomical Society, 202(3), 615-627.
[2] Fasano, G., & Franceschini, A. (1987). A multidimensional version of the Kolmogorov–Smirnov test. Monthly Notices of the Royal Astronomical Society, 225(1), 155-170.
[3] Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. (1996). Numerical recipes in C (Vol. 2). Cambridge: Cambridge university press.