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module Statistics
module StatisticalTest
class KolmogorovSmirnovTest
# Common alpha, and critical D are calculated following formulas from: https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test#Two-sample_Kolmogorov%E2%80%93Smirnov_test
def self.two_samples(group_one:, group_two:, alpha: 0.05)
samples = group_one + group_two # We can use unbalaced group samples
ecdf_one = Distribution::Empirical.new(samples: group_one)
ecdf_two = Distribution::Empirical.new(samples: group_two)
d_max = samples.sort.map do |sample|
d1 = ecdf_one.cumulative_function(x: sample)
d2 = ecdf_two.cumulative_function(x: sample)
(d1 - d2).abs
end.max
# TODO: Validate calculation of Common alpha.
common_alpha = Math.sqrt((-0.5 * Math.log(alpha)))
radicand = (group_one.size + group_two.size) / (group_one.size * group_two.size).to_f
critical_d = common_alpha * Math.sqrt(radicand)
# critical_d = self.critical_d(alpha: alpha, n: samples.size)
# We are unable to calculate the p_value, because we don't have the Kolmogorov distribution
# defined. We reject the null hypotesis if Dmax is > than Dcritical.
{ d_max: d_max,
d_critical: critical_d,
total_samples: samples.size,
alpha: alpha,
null: d_max <= critical_d,
alternative: d_max > critical_d,
confidence_level: 1.0 - alpha }
end
# This is an implementation of the formula presented by Paul Molin and Hervé Abdi in a paper,
# called "New Table and numerical approximations for Kolmogorov-Smirnov / Lilliefors / Van Soest
# normality test".
# In this paper, the authors defines a couple of 6th-degree polynomial functions that allow us
# to find an aproximation of the real critical value. This is based in the conclusions made by
# Dagnelie (1968), where indicates that critical values given by Lilliefors can be approximated
# numerically.
#
# In general, the formula found is:
# C(N, alpha) ^ -2 = A(alpha) * N + B(alpha).
#
# Where A(alpha), B(alpha) are two 6th degree polynomial functions computed using the principle
# of Monte Carlo simulations.
#
# paper can be found here: https://utdallas.edu/~herve/MolinAbdi1998-LillieforsTechReport.pdf
# def self.critical_d(alpha:, n:)
# confidence = 1.0 - alpha
# a_alpha = 6.32207539843126 -17.1398870006148 * confidence +
# 38.42812675101057 * (confidence ** 2) - 45.93241384693391 * (confidence ** 3) +
# 7.88697700041829 * (confidence ** 4) + 29.79317711037858 * (confidence ** 5) -
# 18.48090137098585 * (confidence ** 6)
# b_alpha = 12.940399038404 - 53.458334259532 * confidence +
# 186.923866119699 * (confidence ** 2) - 410.582178349305 * (confidence ** 3) +
# 517.377862566267 * (confidence ** 4) - 343.581476222384 * (confidence ** 5) +
# 92.123451358715 * (confidence ** 6)
# Math.sqrt(1.0 / (a_alpha * n + b_alpha))
# end
end
KSTest = KolmogorovSmirnovTest # Alias
end
end
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