# Class KolmogorovSmirnovTest

java.lang.Object
org.hipparchus.stat.inference.KolmogorovSmirnovTest

public class KolmogorovSmirnovTest extends Object
Implementation of the Kolmogorov-Smirnov (K-S) test for equality of continuous distributions.

The K-S test uses a statistic based on the maximum deviation of the empirical distribution of sample data points from the distribution expected under the null hypothesis. For one-sample tests evaluating the null hypothesis that a set of sample data points follow a given distribution, the test statistic is $$D_n=\sup_x |F_n(x)-F(x)|$$, where $$F$$ is the expected distribution and $$F_n$$ is the empirical distribution of the $$n$$ sample data points. The distribution of $$D_n$$ is estimated using a method based on [1] with certain quick decisions for extreme values given in [2].

Two-sample tests are also supported, evaluating the null hypothesis that the two samples x and y come from the same underlying distribution. In this case, the test statistic is $$D_{n,m}=\sup_t | F_n(t)-F_m(t)|$$ where $$n$$ is the length of x, $$m$$ is the length of y, $$F_n$$ is the empirical distribution that puts mass $$1/n$$ at each of the values in x and $$F_m$$ is the empirical distribution of the y values. The default 2-sample test method, kolmogorovSmirnovTest(double[], double[]) works as follows:

If the product of the sample sizes is less than LARGE_SAMPLE_PRODUCT and the sample data contains ties, random jitter is added to the sample data to break ties before applying the algorithm above. Alternatively, the bootstrap(double[], double[], int, boolean) method, modeled after ks.boot in the R Matching package [3], can be used if ties are known to be present in the data.

In the two-sample case, $$D_{n,m}$$ has a discrete distribution. This makes the p-value associated with the null hypothesis $$H_0 : D_{n,m} \ge d$$ differ from $$H_0 : D_{n,m} > d$$ by the mass of the observed value $$d$$. To distinguish these, the two-sample tests use a boolean strict parameter. This parameter is ignored for large samples.

The methods used by the 2-sample default implementation are also exposed directly:

References:

Note that [1] contains an error in computing h, refer to MATH-437 for details.

• ## Field Summary

Fields
Modifier and Type
Field
Description
protected static final double
KS_SUM_CAUCHY_CRITERION
protected static final int
LARGE_SAMPLE_PRODUCT
When product of sample sizes exceeds this value, 2-sample K-S test uses asymptotic distribution to compute the p-value.
protected static final int
MAXIMUM_PARTIAL_SUM_COUNT
Bound on the number of partial sums in ksSum(double, double, int)
protected static final double
PG_SUM_RELATIVE_ERROR
Convergence criterion for the sums in #pelzGood(double, double, int)}
• ## Constructor Summary

Constructors
Constructor
Description
KolmogorovSmirnovTest()
Construct a KolmogorovSmirnovTest instance.
KolmogorovSmirnovTest(long seed)
Construct a KolmogorovSmirnovTest instance providing a seed for the PRNG used by the bootstrap(double[], double[], int) method.
• ## Method Summary

Modifier and Type
Method
Description
double
approximateP(double d, int n, int m)
Uses the Kolmogorov-Smirnov distribution to approximate $$P(D_{n,m} > d)$$ where $$D_{n,m}$$ is the 2-sample Kolmogorov-Smirnov statistic.
double
bootstrap(double[] x, double[] y, int iterations)
Computes bootstrap(x, y, iterations, true).
double
bootstrap(double[] x, double[] y, int iterations, boolean strict)
Estimates the p-value of a two-sample Kolmogorov-Smirnov test evaluating the null hypothesis that x and y are samples drawn from the same probability distribution.
double
cdf(double d, int n)
Calculates P(D_n < d) using the method described in [1] with quick decisions for extreme values given in [2] (see above).
double
cdf(double d, int n, boolean exact)
Calculates P(D_n < d) using method described in [1] with quick decisions for extreme values given in [2] (see above).
double
cdfExact(double d, int n)
Calculates P(D_n < d).
double
exactP(double d, int n, int m, boolean strict)
Computes $$P(D_{n,m} > d)$$ if strict is true; otherwise $$P(D_{n,m} \ge d)$$, where $$D_{n,m}$$ is the 2-sample Kolmogorov-Smirnov statistic.
double
kolmogorovSmirnovStatistic(double[] x, double[] y)
Computes the two-sample Kolmogorov-Smirnov test statistic, $$D_{n,m}=\sup_x |F_n(x)-F_m(x)|$$ where $$n$$ is the length of x, $$m$$ is the length of y, $$F_n$$ is the empirical distribution that puts mass $$1/n$$ at each of the values in x and $$F_m$$ is the empirical distribution of the y values.
double
kolmogorovSmirnovStatistic(RealDistribution distribution, double[] data)
Computes the one-sample Kolmogorov-Smirnov test statistic, $$D_n=\sup_x |F_n(x)-F(x)|$$ where $$F$$ is the distribution (cdf) function associated with distribution, $$n$$ is the length of data and $$F_n$$ is the empirical distribution that puts mass $$1/n$$ at each of the values in data.
double
kolmogorovSmirnovTest(double[] x, double[] y)
Computes the p-value, or observed significance level, of a two-sample Kolmogorov-Smirnov test evaluating the null hypothesis that x and y are samples drawn from the same probability distribution.
double
kolmogorovSmirnovTest(double[] x, double[] y, boolean strict)
Computes the p-value, or observed significance level, of a two-sample Kolmogorov-Smirnov test evaluating the null hypothesis that x and y are samples drawn from the same probability distribution.
double
kolmogorovSmirnovTest(RealDistribution distribution, double[] data)
Computes the p-value, or observed significance level, of a one-sample Kolmogorov-Smirnov test evaluating the null hypothesis that data conforms to distribution.
double
kolmogorovSmirnovTest(RealDistribution distribution, double[] data, boolean exact)
Computes the p-value, or observed significance level, of a one-sample Kolmogorov-Smirnov test evaluating the null hypothesis that data conforms to distribution.
boolean
kolmogorovSmirnovTest(RealDistribution distribution, double[] data, double alpha)
Performs a Kolmogorov-Smirnov test evaluating the null hypothesis that data conforms to distribution.
double
ksSum(double t, double tolerance, int maxIterations)
Computes $$1 + 2 \sum_{i=1}^\infty (-1)^i e^{-2 i^2 t^2}$$ stopping when successive partial sums are within tolerance of one another, or when maxIterations partial sums have been computed.
double
pelzGood(double d, int n)
Computes the Pelz-Good approximation for $$P(D_n < d)$$ as described in [2] in the class javadoc.

### Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
• ## Field Details

• ### MAXIMUM_PARTIAL_SUM_COUNT

protected static final int MAXIMUM_PARTIAL_SUM_COUNT
Bound on the number of partial sums in ksSum(double, double, int)
• ### KS_SUM_CAUCHY_CRITERION

protected static final double KS_SUM_CAUCHY_CRITERION
• ### PG_SUM_RELATIVE_ERROR

protected static final double PG_SUM_RELATIVE_ERROR
Convergence criterion for the sums in #pelzGood(double, double, int)}
• ### LARGE_SAMPLE_PRODUCT

protected static final int LARGE_SAMPLE_PRODUCT
When product of sample sizes exceeds this value, 2-sample K-S test uses asymptotic distribution to compute the p-value.
• ## Constructor Details

• ### KolmogorovSmirnovTest

public KolmogorovSmirnovTest()
Construct a KolmogorovSmirnovTest instance.
• ### KolmogorovSmirnovTest

public KolmogorovSmirnovTest(long seed)
Construct a KolmogorovSmirnovTest instance providing a seed for the PRNG used by the bootstrap(double[], double[], int) method.
Parameters:
seed - the seed for the PRNG
• ## Method Details

• ### kolmogorovSmirnovTest

public double kolmogorovSmirnovTest(RealDistribution distribution, double[] data, boolean exact)
Computes the p-value, or observed significance level, of a one-sample Kolmogorov-Smirnov test evaluating the null hypothesis that data conforms to distribution. If exact is true, the distribution used to compute the p-value is computed using extended precision. See cdfExact(double, int).
Parameters:
distribution - reference distribution
data - sample being being evaluated
exact - whether or not to force exact computation of the p-value
Returns:
the p-value associated with the null hypothesis that data is a sample from distribution
Throws:
MathIllegalArgumentException - if data does not have length at least 2
NullArgumentException - if data is null
• ### kolmogorovSmirnovStatistic

public double kolmogorovSmirnovStatistic(RealDistribution distribution, double[] data)
Computes the one-sample Kolmogorov-Smirnov test statistic, $$D_n=\sup_x |F_n(x)-F(x)|$$ where $$F$$ is the distribution (cdf) function associated with distribution, $$n$$ is the length of data and $$F_n$$ is the empirical distribution that puts mass $$1/n$$ at each of the values in data.
Parameters:
distribution - reference distribution
data - sample being evaluated
Returns:
Kolmogorov-Smirnov statistic $$D_n$$
Throws:
MathIllegalArgumentException - if data does not have length at least 2
NullArgumentException - if data is null
• ### kolmogorovSmirnovTest

public double kolmogorovSmirnovTest(double[] x, double[] y, boolean strict)
Computes the p-value, or observed significance level, of a two-sample Kolmogorov-Smirnov test evaluating the null hypothesis that x and y are samples drawn from the same probability distribution. Specifically, what is returned is an estimate of the probability that the kolmogorovSmirnovStatistic(double[], double[]) associated with a randomly selected partition of the combined sample into subsamples of sizes x.length and y.length will strictly exceed (if strict is true) or be at least as large as strict = false) as kolmogorovSmirnovStatistic(x, y).

If x.length * y.length < LARGE_SAMPLE_PRODUCT and the combined set of values in x and y contains ties, random jitter is added to x and y to break ties before computing $$D_{n,m}$$ and the p-value. The jitter is uniformly distributed on (-minDelta / 2, minDelta / 2) where minDelta is the smallest pairwise difference between values in the combined sample.

If ties are known to be present in the data, bootstrap(double[], double[], int, boolean) may be used as an alternative method for estimating the p-value.

Parameters:
x - first sample dataset
y - second sample dataset
strict - whether or not the probability to compute is expressed as a strict inequality (ignored for large samples)
Returns:
p-value associated with the null hypothesis that x and y represent samples from the same distribution
Throws:
MathIllegalArgumentException - if either x or y does not have length at least 2
NullArgumentException - if either x or y is null
• ### kolmogorovSmirnovTest

public double kolmogorovSmirnovTest(double[] x, double[] y)
Computes the p-value, or observed significance level, of a two-sample Kolmogorov-Smirnov test evaluating the null hypothesis that x and y are samples drawn from the same probability distribution. Assumes the strict form of the inequality used to compute the p-value. See kolmogorovSmirnovTest(RealDistribution, double[], boolean).
Parameters:
x - first sample dataset
y - second sample dataset
Returns:
p-value associated with the null hypothesis that x and y represent samples from the same distribution
Throws:
MathIllegalArgumentException - if either x or y does not have length at least 2
NullArgumentException - if either x or y is null
• ### kolmogorovSmirnovStatistic

public double kolmogorovSmirnovStatistic(double[] x, double[] y)
Computes the two-sample Kolmogorov-Smirnov test statistic, $$D_{n,m}=\sup_x |F_n(x)-F_m(x)|$$ where $$n$$ is the length of x, $$m$$ is the length of y, $$F_n$$ is the empirical distribution that puts mass $$1/n$$ at each of the values in x and $$F_m$$ is the empirical distribution of the y values.
Parameters:
x - first sample
y - second sample
Returns:
test statistic $$D_{n,m}$$ used to evaluate the null hypothesis that x and y represent samples from the same underlying distribution
Throws:
MathIllegalArgumentException - if either x or y does not have length at least 2
NullArgumentException - if either x or y is null
• ### kolmogorovSmirnovTest

public double kolmogorovSmirnovTest(RealDistribution distribution, double[] data)
Computes the p-value, or observed significance level, of a one-sample Kolmogorov-Smirnov test evaluating the null hypothesis that data conforms to distribution.
Parameters:
distribution - reference distribution
data - sample being being evaluated
Returns:
the p-value associated with the null hypothesis that data is a sample from distribution
Throws:
MathIllegalArgumentException - if data does not have length at least 2
NullArgumentException - if data is null
• ### kolmogorovSmirnovTest

public boolean kolmogorovSmirnovTest(RealDistribution distribution, double[] data, double alpha)
Performs a Kolmogorov-Smirnov test evaluating the null hypothesis that data conforms to distribution.
Parameters:
distribution - reference distribution
data - sample being being evaluated
alpha - significance level of the test
Returns:
true iff the null hypothesis that data is a sample from distribution can be rejected with confidence 1 - alpha
Throws:
MathIllegalArgumentException - if data does not have length at least 2
NullArgumentException - if data is null
• ### bootstrap

public double bootstrap(double[] x, double[] y, int iterations, boolean strict)
Estimates the p-value of a two-sample Kolmogorov-Smirnov test evaluating the null hypothesis that x and y are samples drawn from the same probability distribution. This method estimates the p-value by repeatedly sampling sets of size x.length and y.length from the empirical distribution of the combined sample. When strict is true, this is equivalent to the algorithm implemented in the R function ks.boot, described in
 Jasjeet S. Sekhon. 2011. 'Multivariate and Propensity Score Matching
Software with Automated Balance Optimization: The Matching package for R.'
Journal of Statistical Software, 42(7): 1-52.

Parameters:
x - first sample
y - second sample
iterations - number of bootstrap resampling iterations
strict - whether or not the null hypothesis is expressed as a strict inequality
Returns:
estimated p-value
• ### bootstrap

public double bootstrap(double[] x, double[] y, int iterations)
Computes bootstrap(x, y, iterations, true). This is equivalent to ks.boot(x,y, nboots=iterations) using the R Matching package function. See #bootstrap(double[], double[], int, boolean).
Parameters:
x - first sample
y - second sample
iterations - number of bootstrap resampling iterations
Returns:
estimated p-value
• ### cdf

public double cdf(double d, int n) throws MathRuntimeException
Calculates P(D_n < d) using the method described in [1] with quick decisions for extreme values given in [2] (see above). The result is not exact as with cdfExact(double, int) because calculations are based on double rather than BigFraction.
Parameters:
d - statistic
n - sample size
Returns:
$$P(D_n < d)$$
Throws:
MathRuntimeException - if algorithm fails to convert h to a BigFraction in expressing d as $$(k - h) / m$$ for integer k, m and $$0 <= h < 1$$
• ### cdfExact

public double cdfExact(double d, int n) throws MathRuntimeException
Calculates P(D_n < d). The result is exact in the sense that BigFraction/BigReal is used everywhere at the expense of very slow execution time. Almost never choose this in real applications unless you are very sure; this is almost solely for verification purposes. Normally, you would choose cdf(double, int). See the class javadoc for definitions and algorithm description.
Parameters:
d - statistic
n - sample size
Returns:
$$P(D_n < d)$$
Throws:
MathRuntimeException - if the algorithm fails to convert h to a BigFraction in expressing d as $$(k - h) / m$$ for integer k, m and $$0 <= h < 1$$
• ### cdf

public double cdf(double d, int n, boolean exact) throws MathRuntimeException
Calculates P(D_n < d) using method described in [1] with quick decisions for extreme values given in [2] (see above).
Parameters:
d - statistic
n - sample size
exact - whether the probability should be calculated exact using BigFraction everywhere at the expense of very slow execution time, or if double should be used convenient places to gain speed. Almost never choose true in real applications unless you are very sure; true is almost solely for verification purposes.
Returns:
$$P(D_n < d)$$
Throws:
MathRuntimeException - if algorithm fails to convert h to a BigFraction in expressing d as $$(k - h) / m$$ for integer k, m and $$0 \lt;= h < 1$$.
• ### pelzGood

public double pelzGood(double d, int n)
Computes the Pelz-Good approximation for $$P(D_n < d)$$ as described in [2] in the class javadoc.
Parameters:
d - value of d-statistic (x in [2])
n - sample size
Returns:
$$P(D_n < d)$$
• ### ksSum

public double ksSum(double t, double tolerance, int maxIterations)
Computes $$1 + 2 \sum_{i=1}^\infty (-1)^i e^{-2 i^2 t^2}$$ stopping when successive partial sums are within tolerance of one another, or when maxIterations partial sums have been computed. If the sum does not converge before maxIterations iterations a MathIllegalStateException is thrown.
Parameters:
t - argument
tolerance - Cauchy criterion for partial sums
maxIterations - maximum number of partial sums to compute
Returns:
Kolmogorov sum evaluated at t
Throws:
MathIllegalStateException - if the series does not converge
• ### exactP

public double exactP(double d, int n, int m, boolean strict)
Computes $$P(D_{n,m} > d)$$ if strict is true; otherwise $$P(D_{n,m} \ge d)$$, where $$D_{n,m}$$ is the 2-sample Kolmogorov-Smirnov statistic. See kolmogorovSmirnovStatistic(double[], double[]) for the definition of $$D_{n,m}$$.

The returned probability is exact, implemented by unwinding the recursive function definitions presented in [4] from the class javadoc.

Parameters:
d - D-statistic value
n - first sample size
m - second sample size
strict - whether or not the probability to compute is expressed as a strict inequality
Returns:
probability that a randomly selected m-n partition of m + n generates $$D_{n,m}$$ greater than (resp. greater than or equal to) d
• ### approximateP

public double approximateP(double d, int n, int m)
Uses the Kolmogorov-Smirnov distribution to approximate $$P(D_{n,m} > d)$$ where $$D_{n,m}$$ is the 2-sample Kolmogorov-Smirnov statistic. See kolmogorovSmirnovStatistic(double[], double[]) for the definition of $$D_{n,m}$$.

Specifically, what is returned is $$1 - k(d \sqrt{mn / (m + n)})$$ where $$k(t) = 1 + 2 \sum_{i=1}^\infty (-1)^i e^{-2 i^2 t^2}$$. See ksSum(double, double, int) for details on how convergence of the sum is determined. This implementation passes ksSum KS_SUM_CAUCHY_CRITERION as tolerance and MAXIMUM_PARTIAL_SUM_COUNT as maxIterations.

Parameters:
d - D-statistic value
n - first sample size
m - second sample size
Returns:
approximate probability that a randomly selected m-n partition of m + n generates $$D_{n,m}$$ greater than d