haichaozhang / 2dks

2 Dimensional Kolmogorov-Smirnov test for goodness-of-fit

KS2D

2 Dimensional Kolmogorov-Smirnov test for goodness-of-fit.

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.

Issues

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.

MISC

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).

References

[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.