Package org.hipparchus.stat.inference

Class 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 LARGE_SAMPLE_PRODUCT), 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 LARGE_SAMPLE_PRODUCT, 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 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.

    • Method Summary

      All Methods Instance Methods Concrete Methods 
      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.

    • Field Detail

      • PG_SUM_RELATIVE_ERROR

        protected static final double PG_SUM_RELATIVE_ERROR
        Convergence criterion for the sums in #pelzGood(double, double, int)}
        See Also:
        Constant Field Values

      • 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.
        See Also:
        Constant Field Values

    • Constructor Detail

      • 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 Detail

      • 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
        See Also:
        bootstrap(double[], double[], int, boolean)

      • 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] (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