What is the correct input for the Two-sample Kolmogorov-Smirnov test, when I need to compare two histograms?

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Sim
Sim 2024-7-16
回答： Divyam 2024-9-13
What is the correct input for the Two-sample Kolmogorov-Smirnov test, when I need to compare two histograms?
Example 1. If the input data are quite similar, it looks like there is no difference in the output of the Two-sample Kolmogorov-Smirnov test, either if I use the original data, "X" and "Y", or the bin counts "NX" and "NY" (returned from the histcounts function), as inputs for the Two-sample Kolmogorov-Smirnov test:
% inputs
X = [2 5 7 10 11 12 13 14 16 17 18 19 22 23 24 29];
Y = [2 5 11 12 13 14 16 17 18 19 20 21 22 23 24 29];
[NX,edgesX] = histcounts(X,'NumBins',6);
[NY,edgesY] = histcounts(Y,'NumBins',6);
% plot
hold on
histogram(X,edgesX,'FaceAlpha',0.1,'EdgeAlpha',0.8)
histogram(Y,edgesY,'FaceAlpha',0.1,'EdgeAlpha',0.2)
% Two-sample Kolmogorov-Smirnov test
[h1,p1,ks2stat1] = kstest2(X,Y); % <-- By using the original input data
[h2,p2,ks2stat2] = kstest2(NX,NY); % <-- By using the Bin counts "NX" and "NY", returned from the "histcounts" function
table([[h1,p1,ks2stat1];[h2,p2,ks2stat2]] ,'VariableNames', {'h | p | ks2stat'},'RowNames', {'kstest2(X,Y)', 'kstest2(NX,NY)'})
% Reusult of Example 1
ans =
2×1 table
h | p | ks2stat
___________________________________________
kstest2(X,Y) 0 0.999035232339821 0.125
kstest2(NX,NY) 0 0.999956514899259 0.166666666666667
Example 2. If the input data are a bit different, it looks like there is difference in the output of the Two-sample Kolmogorov-Smirnov test, if I use the original data, "X" and "Y", or the bin counts "NX" and "NY" (returned from the histcounts function), as inputs for the Two-sample Kolmogorov-Smirnov test:
% inputs
X = [2 5 7 10 11 12 13 14 16 17 18 19 22 23 24 29];
Y = [2 5 11 12 13 14 16 17 18 19 20 21 22 23 24 29 29 29 29 29 29 29 29];
[NX,edgesX] = histcounts(X,'NumBins',6);
[NY,edgesY] = histcounts(Y,'NumBins',6);
% plot
hold on
histogram(X,edgesX,'FaceAlpha',0.1,'EdgeAlpha',0.8)
histogram(Y,edgesY,'FaceAlpha',0.1,'EdgeAlpha',0.2)
% Two-sample Kolmogorov-Smirnov test
[h1,p1,ks2stat1] = kstest2(X,Y);
[h2,p2,ks2stat2] = kstest2(NX,NY);
table([[h1,p1,ks2stat1];[h2,p2,ks2stat2]] ,'VariableNames', {'h | p | ks2stat'},'RowNames', {'kstest2(X,Y)', 'kstest2(NX,NY)'})
% Result of Example 2
ans =
2×1 table
h | p | ks2stat
___________________________________________
kstest2(X,Y) 0 0.251817384522441 0.315217391304348
kstest2(NX,NY) 0 0.809557310616653 0.333333333333333
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Sim
Sim 2024-7-16
I have just checked the kstest2(x1,x2) function by opening it in the Command Window:
>> open kstest2
I can see that the Bin counts are already calculated in the part where the empirical CDFs are derived:
% Calculate F1(x) and F2(x), the empirical (i.e., sample) CDFs.
...
binCounts1 = histc (x1 , binEdges, 1);
binCounts2 = histc (x2 , binEdges, 1);
...
Therefore, to the best of my understanding, the correct usage of kstest2(x1,x2) is only with the original input data "X" and "Y":
[h1,p1,ks2stat1] = kstest2(X,Y); % <-- By using the original input data
While the employment of the Bin counts "NX" and "NY" would lead to a wrong result:
[h2,p2,ks2stat2] = kstest2(NX,NY); % <-- By using the Bin counts "NX" and "NY", returned from the "histcounts" function
If anyone wants to confirm my reasoning is very welcome!
(maybe, from @MathWorks Support Team as well?)
Sim
Sim 2024-7-16
编辑：Sim 2024-7-16
Yes @Divyam, you are right! I just need to use original data "X" and "Y" (that you called the "vectors themselves")!! ...and kstest2 will do the rest of the work for me :-)
Many thanks! :-) :-)

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回答（1 个）

Divyam
Divyam 2024-9-13
Hi @Sim,
The Two-sample Kolmogorov-Smirnov test is used to test whether the data from any two vectors is from the same continuous distributions.
To effectively implement the Kolmogorov-Smirnov test, you should use the vectors "X" and "Y" themselves since they are the direct representatives of the distribution which create the histogram. The "bincounts" variables "NX" and "NY" on the other hand, reveal the underlying shape of the distribution and do not represent the distribution itself in all cases (as evident when you choose different input data).
Note: Answering this question to improve the visibility for anyone referring the community with a similar query.

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