Kolmogorov-Smirnov Statistic

Background

The Kolmogorov-Smirnov (KS) statistic detects drift in a continuous feature without assuming a parametric form. It compares two empirical CDFs and reports the maximum gap between them - a single scalar that is easy to monitor and threshold.

Problem statement

Implement ks_statistic(sample_a, sample_b):

$$\text{KS} \;=\; \sup_x \bigl|\,F_A(x) - F_B(x)\,\bigr|$$

where $F_A, F_B$ are the empirical CDFs of the two samples. Returns a float in $[0, 1]$.

Input

• sample_a, sample_b - array-like of floats.

Output

• float in $[0, 1]$ - maximum absolute CDF difference.

Examples

Input: sample_a=[1,2,3,4,5], sample_b=[1,2,3,4,5]
Output: 0.0

Input: sample_a=[1,2,3], sample_b=[10,11,12]
Output: 1.0      # the two CDFs are 100% apart at any point in between

Constraints

• Empty input on either side raises ValueError.
• The standard algorithm: pool and sort all observations, walk through computing CDFs, take max abs diff.

Notes

• vs PSI. PSI is a sum (KL-flavored) over binned probabilities; KS is a sup over CDFs. KS is bin-free; PSI is bin-dependent but more interpretable in industry.
• Two-sided. This is the standard two-sided KS; the one-sided variant only checks one direction.