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第258行： + − One approach to generalizing the Kolmogorov–Smirnov statistic to higher dimensions which meets the above concern is to compare the cdfs of the two samples with all possible orderings, and take the largest of the set of resulting K–S statistics. In ''d'' dimensions, there are 2<sup>''d''</sup>−1 such orderings. One such variation is due to Peacock<ref name="Peacock">{{cite journal |author = Peacock J.A. |title = Two-dimensional goodness-of-fit testing in astronomy |journal = [[Monthly Notices of the Royal Astronomical Society]] |volume = 202 |issue = 3 |pages = 615–627 |year = 1983 |bibcode = 1983MNRAS.202..615P |doi=10.1093/mnras/202.3.615|doi-access = free }}</ref> − − (see also Gosset<ref>{{cite journal − − |author = Gosset E. − − |title = A three-dimensional extended Kolmogorov-Smirnov test as a useful tool in astronomy} − − |journal = Astronomy and Astrophysics − − |volume = 188 − − |issue = 1 − − |pages = 258–264 − − |year = 1987 − − |bibcode = 1987A&A...188..258G − − }}</ref> − − for a 3D version) − − | last = Eadie − − | last = Eadie − − and another to Fasano and Franceschini<ref name="Fasano">{{cite journal |authors= Fasano, G., Franceschini, A. |year=1987 |title= A multidimensional version of the Kolmogorov–Smirnov test |journal= Monthly Notices of the Royal Astronomical Society |issn=0035-8711 |volume= 225 |pages= 155–170 |bibcode=1987MNRAS.225..155F |doi=10.1093/mnras/225.1.155|doi-access= free }}</ref> (see Lopes et al. for a comparison and computational details).<ref name="Lopes">{{cite conference |authors= Lopes, R.H.C., Reid, I., Hobson, P.R. |title= The two-dimensional Kolmogorov–Smirnov test |conference= XI International Workshop on Advanced Computing and Analysis Techniques in Physics Research |date= 23–27 April 2007 |location= Amsterdam, the Netherlands |url= http://dspace.brunel.ac.uk/bitstream/2438/1166/1/acat2007.pdf }}</ref> Critical values for the test statistic can be obtained by simulations, but depend on the dependence structure in the joint distribution. − | first = W.T. |author2=D. Drijard |author3=F.E. James |author4=M. Roos |author5=B. Sadoulet+ − 第一个 = w.t。2 = d.3 = f.e.4 = m.5 = b.女名女子名+ − − | title = Statistical Methods in Experimental Physics − − 实验物理中的统计方法 − | publisher = North-Holland+ − − | publisher = North-Holland − − − − | year = 1971 − − 1971年
→The Kolmogorov–Smirnov statistic in more than one dimension
虽然通常使用Kolmogorov–Smirnov检验法来检验给定的F（x）是否为Fn（x）的潜在概率分布，但可以将过程倒过来给出F（x）本身的置信极限。如果选择检验统计量Dα的临界值，使得P（Dn>Dα）=α，则在Fn（x）周围宽度±Dα内将完全包含概率为1-α的F（x）。
虽然通常使用Kolmogorov–Smirnov检验法来检验给定的F（x）是否为Fn（x）的潜在概率分布，但可以将过程倒过来给出F（x）本身的置信极限。如果选择检验统计量Dα的临界值，使得P（Dn>Dα）=α，则在Fn（x）周围宽度±Dα内将完全包含概率为1-α的F（x）。
==The Kolmogorov–Smirnov statistic in more than one dimension==
== The Kolmogorov–Smirnov statistic in more than one dimension 多个维度的Kolmogorov–Smirnov统计 ==
A distribution-free multivariate Kolmogorov–Smirnov goodness of fit test has been proposed by Justel, Peña and Zamar (1997).<ref>{{cite journal |last=Justel |first=A. |last2=Peña |first2=D. |last3=Zamar |first3=R. |year=1997 |title=A multivariate Kolmogorov–Smirnov test of goodness of fit |journal=Statistics & Probability Letters |volume=35 |issue=3 |pages=251–259 |doi=10.1016/S0167-7152(97)00020-5 |citeseerx=10.1.1.498.7631 }}</ref> The test uses a statistic which is built using Rosenblatt's transformation, and an algorithm is developed to compute it in the bivariate case. An approximate test that can be easily computed in any dimension is also presented.
A distribution-free multivariate Kolmogorov–Smirnov goodness of fit test has been proposed by Justel, Peña and Zamar (1997). The test uses a statistic which is built using Rosenblatt's transformation, and an algorithm is developed to compute it in the bivariate case. An approximate test that can be easily computed in any dimension is also presented.
Justel，Peña和Zamar（1997）提出了无分布的多元Kolmogorov-Smirnov拟合优度检验。该检验使用通过Rosenblatt变换建立的统计量，并开发了一种算法来计算双变量情况。还介绍了可以在任何维度上轻松计算的近似检测法。
The Kolmogorov–Smirnov test statistic needs to be modified if a similar test is to be applied to [[multivariate statistics|multivariate data]]. This is not straightforward because the maximum difference between two joint [[cumulative distribution function]]s is not generally the same as the maximum difference of any of the complementary distribution functions. Thus the maximum difference will differ depending on which of <math>\Pr(x < X \land y < Y)</math> or <math>\Pr(X < x \land Y > y)</math> or any of the other two possible arrangements is used. One might require that the result of the test used should not depend on which choice is made.
The Kolmogorov–Smirnov test statistic needs to be modified if a similar test is to be applied to [[multivariate statistics|multivariate data]]. This is not straightforward because the maximum difference between two joint [[cumulative distribution function]]s is not generally the same as the maximum difference of any of the complementary distribution functions. Thus the maximum difference will differ depending on which of <math>\Pr(x < X \land y < Y)</math> or <math>\Pr(X < x \land Y > y)</math> or any of the other two possible arrangements is used. One might require that the result of the test used should not depend on which choice is made.
如果要将类似的检验应用于多元数据，则需要修改Kolmogorov–Smirnov检验统计量。过程略显复杂，因为两个联合累积分布函数之间的最大差异通常与任何互补分布函数中的最大差异都不相同。因此，最大差异将取决于使用或中的哪一个，或者使用其他两种可能分布中的任何一种。当然也有可能要求所用的检测结果无关于使用选择。
One approach to generalizing the Kolmogorov–Smirnov statistic to higher dimensions which meets the above concern is to compare the cdfs of the two samples with all possible orderings, and take the largest of the set of resulting K–S statistics. In ''d'' dimensions, there are 2<sup>''d''</sup>−1 such orderings. One such variation is due to Peacock(see also Gosset for a 3D version) and another to Fasano and Franceschini (see Lopes et al. for a comparison and computational details). Critical values for the test statistic can be obtained by simulations, but depend on the dependence structure in the joint distribution.
在满足以上要求的同时，将Kolmogorov-Smirnov统计量泛化为更高维度的一种方法是，比较两个样本的累积分布函数与所有可能的排序，并从所得的K-S统计量集中最大。在d维中，有2d-1个这样的排序。Peacock得出了一种这样的变化量（有关3D版本，另请参见Gosset），另一种由Fasano和Franceschini得出（有关比较和计算细节，请参见Lopes等人）。可以通过模拟获得检测统计量的临界值，但取决于联合分布中的依存关系的结构。
In one dimension, the Kolmogorov–Smirnov statistic is identical to the so-called star discrepancy D, so another native KS extension to higher dimensions would be simply to use D also for higher dimensions. Unfortunately, the star discrepancy is hard to calculate in high dimensions.
In one dimension, the Kolmogorov–Smirnov statistic is identical to the so-called star discrepancy D, so another native KS extension to higher dimensions would be simply to use D also for higher dimensions. Unfortunately, the star discrepancy is hard to calculate in high dimensions.
一维的Kolmogorov-Smirnov统计量与所谓的星差异D相同，因此，另一个对更高维度的本地KS扩展是将D也用于更高维度。可惜的是，很难从高维度上计算出星差异。
==Implementations==
==Implementations==