Package org.apache.commons.math3.stat.inference

Class KolmogorovSmirnovTest

java.lang.Object
org.apache.commons.math3.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:

  • For small samples (where the product of the sample sizes is less than 10000), the method presented in [4] is used to compute the exact p-value for the 2-sample test.
  • When the product of the sample sizes exceeds 10000, the asymptotic distribution of \(D_{n,m}\) is used. See approximateP(double, int, int) for details on the approximation.

If the product of the sample sizes is less than 10000 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.

Since:
3.3

  • Field Summary

    Fields
    Modifier and Type
    Field
    Description
    protected static final double
    KS_SUM_CAUCHY_CRITERION
    Convergence criterion for ksSum(double, double, int)
    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 int
    MONTE_CARLO_ITERATIONS
    Deprecated.
    protected static final double
    PG_SUM_RELATIVE_ERROR
    Convergence criterion for the sums in #pelzGood(double, double, int)}
    protected static final int
    SMALL_SAMPLE_PRODUCT
    Deprecated.

  • Constructor Summary

    Constructors
    Constructor
    Description
    KolmogorovSmirnovTest()
    Construct a KolmogorovSmirnovTest instance with a default random data generator.
    KolmogorovSmirnovTest(RandomGenerator rng)
    Deprecated.

  • 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 \(P(D_n invalid input: '<' d)\) 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
    monteCarloP(double d, int n, int m, boolean strict, int iterations)
    Uses Monte Carlo simulation to approximate \(P(D_{n,m} > d)\) where \(D_{n,m}\) is the 2-sample Kolmogorov-Smirnov statistic.
    double
    pelzGood(double d, int n)
    Computes the Pelz-Good approximation for \(P(D_n invalid input: '<' d)\) 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

  • Constructor Details

    • KolmogorovSmirnovTest

      public KolmogorovSmirnovTest()
      Construct a KolmogorovSmirnovTest instance with a default random data generator.

    • KolmogorovSmirnovTest

      @Deprecated public KolmogorovSmirnovTest(RandomGenerator rng)
      Deprecated.
      Construct a KolmogorovSmirnovTest with the provided random data generator. The #monteCarloP(double, int, int, boolean, int) that uses the generator supplied to this constructor is deprecated as of version 3.6.
      Parameters:
      rng - random data generator used by monteCarloP(double, int, int, boolean, int)

  • 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:
      InsufficientDataException - 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:
      InsufficientDataException - 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).
      • For small samples (where the product of the sample sizes is less than 10000), the exact p-value is computed using the method presented in [4], implemented in exactP(double, int, int, boolean).
      • When the product of the sample sizes exceeds 10000, the asymptotic distribution of \(D_{n,m}\) is used. See approximateP(double, int, int) for details on the approximation.

      If x.length * y.length invalid input: '<' 10000 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:
      InsufficientDataException - if either x or y does not have length at least 2
      NullArgumentException - if either x or y is null
      See Also:

    • 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:
      InsufficientDataException - 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:
      InsufficientDataException - 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:
      InsufficientDataException - 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:
      InsufficientDataException - 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 MathArithmeticException
      Calculates \(P(D_n invalid input: '<' d)\) 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:
      \(P(D_n invalid input: '<' d)\)
      Throws:
      MathArithmeticException - if algorithm fails to convert h to a BigFraction in expressing d as \((k - h) / m\) for integer k, m and \(0 \le h invalid input: '<' 1\)

    • cdfExact

      public double cdfExact(double d, int n) throws MathArithmeticException
      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:
      \(P(D_n invalid input: '<' d)\)
      Throws:
      MathArithmeticException - if the algorithm fails to convert h to a BigFraction in expressing d as \((k - h) / m\) for integer k, m and \(0 \le h invalid input: '<' 1\)

    • cdf

      public double cdf(double d, int n, boolean exact) throws MathArithmeticException
      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:
      \(P(D_n invalid input: '<' d)\)
      Throws:
      MathArithmeticException - if algorithm fails to convert h to a BigFraction in expressing d as \((k - h) / m\) for integer k, m and \(0 \le h invalid input: '<' 1\).

    • pelzGood

      public double pelzGood(double d, int n)
      Computes the Pelz-Good approximation for \(P(D_n invalid input: '<' d)\) as described in [2] in the class javadoc.
      Parameters:
      d - value of d-statistic (x in [2])
      n - sample size
      Returns:
      \(P(D_n invalid input: '<' d)\)
      Since:
      3.4

    • 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 TooManyIterationsException 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:
      TooManyIterationsException - 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] (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 1.0E-20 as tolerance and 100000 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

    • monteCarloP

      public double monteCarloP(double d, int n, int m, boolean strict, int iterations)
      Uses Monte Carlo simulation 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 \(D_{n,m}\).

      The simulation generates iterations random partitions of m + n into an n set and an m set, computing \(D_{n,m}\) for each partition and returning the proportion of values that are greater than d, or greater than or equal to d if strict is false.

      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
      iterations - number of random partitions to generate
      Returns:
      proportion of randomly generated m-n partitions of m + n that result in \(D_{n,m}\) greater than (resp. greater than or equal to) d