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package org.apache.commons.math3.distribution;

import java.io.Serializable;
import java.math.BigDecimal;

import org.apache.commons.math3.exception.MathArithmeticException;
import org.apache.commons.math3.exception.NotStrictlyPositiveException;
import org.apache.commons.math3.exception.NumberIsTooLargeException;
import org.apache.commons.math3.exception.util.LocalizedFormats;
import org.apache.commons.math3.fraction.BigFraction;
import org.apache.commons.math3.fraction.BigFractionField;
import org.apache.commons.math3.fraction.FractionConversionException;
import org.apache.commons.math3.linear.Array2DRowFieldMatrix;
import org.apache.commons.math3.linear.Array2DRowRealMatrix;
import org.apache.commons.math3.linear.FieldMatrix;
import org.apache.commons.math3.linear.RealMatrix;
import org.apache.commons.math3.util.FastMath;

Implementation of the Kolmogorov-Smirnov distribution.
 Treats the distribution of the two-sided P(D_n < d) where D_n = sup_x |G(x) - G_n (x)| for the theoretical cdf G and the empirical cdf G_n. 

This implementation is based on [1] with certain quick decisions for extreme
values given in [2].

 In short, when wanting to evaluate P(D_n < d), the method in [1] is to write d = (k - h) / n for positive integer k and 0 <= h < 1. Then P(D_n < d) = (n! / n^n) * t_kk, where t_kk is the (k, k)'th entry in the special matrix H^n, i.e. H to the n'th power. 

References:

[1] 
Evaluating Kolmogorov's Distribution by George Marsaglia, Wai
Wan Tsang, and Jingbo Wang
[2] 
Computing the Two-Sided Kolmogorov-Smirnov Distribution by Richard Simard
and Pierre L'Ecuyer

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

See Also:

 * Kolmogorov-Smirnov test (Wikipedia)
Deprecated:
to be removed in version 4.0 - use KolmogorovSmirnovTest/**
 * Implementation of the Kolmogorov-Smirnov distribution.
 *
 * <p>
 * Treats the distribution of the two-sided {@code P(D_n < d)} where
 * {@code D_n = sup_x |G(x) - G_n (x)|} for the theoretical cdf {@code G} and
 * the empirical cdf {@code G_n}.
 * </p>
 * <p>
 * This implementation is based on [1] with certain quick decisions for extreme
 * values given in [2].
 * </p>
 * <p>
 * In short, when wanting to evaluate {@code P(D_n < d)}, the method in [1] is
 * to write {@code d = (k - h) / n} for positive integer {@code k} and
 * {@code 0 <= h < 1}. Then {@code P(D_n < d) = (n! / n^n) * t_kk}, where
 * {@code t_kk} is the {@code (k, k)}'th entry in the special matrix
 * {@code H^n}, i.e. {@code H} to the {@code n}'th power.
 * </p>
 * <p>
 * References:
 * <ul>
 * <li>[1] <a href="http://www.jstatsoft.org/v08/i18/">
 * Evaluating Kolmogorov's Distribution</a> by George Marsaglia, Wai
 * Wan Tsang, and Jingbo Wang</li>
 * <li>[2] <a href="http://www.jstatsoft.org/v39/i11/">
 * Computing the Two-Sided Kolmogorov-Smirnov Distribution</a> by Richard Simard
 * and Pierre L'Ecuyer</li>
 * </ul>
 * Note that [1] contains an error in computing h, refer to
 * <a href="https://issues.apache.org/jira/browse/MATH-437">MATH-437</a> for details.
 * </p>
 *
 * @see <a href="http://en.wikipedia.org/wiki/Kolmogorov-Smirnov_test">
 * Kolmogorov-Smirnov test (Wikipedia)</a>
 * @deprecated to be removed in version 4.0 -
 *  use {@link org.apache.commons.math3.stat.inference.KolmogorovSmirnovTest}
 */
public class KolmogorovSmirnovDistribution implements Serializable {

    
Serializable version identifier. /** Serializable version identifier. */
    private static final long serialVersionUID = -4670676796862967187L;

    
Number of observations. /** Number of observations. */
    private int n;

    
Params:
n – Number of observations
Throws:
NotStrictlyPositiveException – if n <= 0/**
     * @param n Number of observations
     * @throws NotStrictlyPositiveException if {@code n <= 0}
     */
    public KolmogorovSmirnovDistribution(int n)
        throws NotStrictlyPositiveException {
        if (n <= 0) {
            throw new NotStrictlyPositiveException(LocalizedFormats.NOT_POSITIVE_NUMBER_OF_SAMPLES, n);
        }

        this.n = n;
    }

    
Calculates P(D_n < d) using method described in [1] with quick decisions for extreme values given in [2] (see above). The result is not exact as with cdfExact(double) because calculations are based on double rather than BigFraction. 
Params:
d – statistic
Throws:
MathArithmeticException – if algorithm fails to convert h to a BigFraction in expressing d as (k - h) / m for integer k, m and 0 <= h < 1.
Returns:
the two-sided probability of P(D_n < d)/**
     * Calculates {@code P(D_n < d)} using method described in [1] with quick
     * decisions for extreme values given in [2] (see above). The result is not
     * exact as with
     * {@link KolmogorovSmirnovDistribution#cdfExact(double)} because
     * calculations are based on {@code double} rather than
     * {@link org.apache.commons.math3.fraction.BigFraction}.
     *
     * @param d statistic
     * @return the two-sided probability of {@code P(D_n < d)}
     * @throws MathArithmeticException if algorithm fails to convert {@code h}
     * to a {@link org.apache.commons.math3.fraction.BigFraction} in expressing
     * {@code d} as {@code (k - h) / m} for integer {@code k, m} and
     * {@code 0 <= h < 1}.
     */
    public double cdf(double d) throws MathArithmeticException {
        return this.cdf(d, false);
    }

    
Calculates P(D_n < d) using method described in [1] with quick decisions for extreme values given in [2] (see above). 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) 
Params:
d – statistic
Throws:
MathArithmeticException – if algorithm fails to convert h to a BigFraction in expressing d as (k - h) / m for integer k, m and 0 <= h < 1.
Returns:
the two-sided probability of P(D_n < d)/**
     * Calculates {@code P(D_n < d)} using method described in [1] with quick
     * decisions for extreme values given in [2] (see above). 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
     * {@link KolmogorovSmirnovDistribution#cdf(double)}
     *
     * @param d statistic
     * @return the two-sided probability of {@code P(D_n < d)}
     * @throws MathArithmeticException if algorithm fails to convert {@code h}
     * to a {@link org.apache.commons.math3.fraction.BigFraction} in expressing
     * {@code d} as {@code (k - h) / m} for integer {@code k, m} and
     * {@code 0 <= h < 1}.
     */
    public double cdfExact(double d) throws MathArithmeticException {
        return this.cdf(d, true);
    }

    
Calculates P(D_n < d) using method described in [1] with quick decisions for extreme values given in [2] (see above). 
Params:
d – statistic
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.
Throws:
MathArithmeticException – if algorithm fails to convert h to a BigFraction in expressing d as (k - h) / m for integer k, m and 0 <= h < 1.
Returns:
the two-sided probability of P(D_n < d)/**
     * Calculates {@code P(D_n < d)} using method described in [1] with quick
     * decisions for extreme values given in [2] (see above).
     *
     * @param d statistic
     * @param exact whether the probability should be calculated exact using
     * {@link org.apache.commons.math3.fraction.BigFraction} everywhere at the
     * expense of very slow execution time, or if {@code double} should be used
     * convenient places to gain speed. Almost never choose {@code true} in real
     * applications unless you are very sure; {@code true} is almost solely for
     * verification purposes.
     * @return the two-sided probability of {@code P(D_n < d)}
     * @throws MathArithmeticException if algorithm fails to convert {@code h}
     * to a {@link org.apache.commons.math3.fraction.BigFraction} in expressing
     * {@code d} as {@code (k - h) / m} for integer {@code k, m} and
     * {@code 0 <= h < 1}.
     */
    public double cdf(double d, boolean exact) throws MathArithmeticException {

        final double ninv = 1 / ((double) n);
        final double ninvhalf = 0.5 * ninv;

        if (d <= ninvhalf) {

            return 0;

        } else if (ninvhalf < d && d <= ninv) {

            double res = 1;
            double f = 2 * d - ninv;

            // n! f^n = n*f * (n-1)*f * ... * 1*x
            for (int i = 1; i <= n; ++i) {
                res *= i * f;
            }

            return res;

        } else if (1 - ninv <= d && d < 1) {

            return 1 - 2 * FastMath.pow(1 - d, n);

        } else if (1 <= d) {

            return 1;
        }

        return exact ? exactK(d) : roundedK(d);
    }

    
Calculates the exact value of P(D_n < d) using method described in [1] and BigFraction (see above). 
Params:
d – statistic
Throws:
MathArithmeticException – if algorithm fails to convert h to a BigFraction in expressing d as (k - h) / m for integer k, m and 0 <= h < 1.
Returns:
the two-sided probability of P(D_n < d)/**
     * Calculates the exact value of {@code P(D_n < d)} using method described
     * in [1] and {@link org.apache.commons.math3.fraction.BigFraction} (see
     * above).
     *
     * @param d statistic
     * @return the two-sided probability of {@code P(D_n < d)}
     * @throws MathArithmeticException if algorithm fails to convert {@code h}
     * to a {@link org.apache.commons.math3.fraction.BigFraction} in expressing
     * {@code d} as {@code (k - h) / m} for integer {@code k, m} and
     * {@code 0 <= h < 1}.
     */
    private double exactK(double d) throws MathArithmeticException {

        final int k = (int) FastMath.ceil(n * d);

        final FieldMatrix<BigFraction> H = this.createH(d);
        final FieldMatrix<BigFraction> Hpower = H.power(n);

        BigFraction pFrac = Hpower.getEntry(k - 1, k - 1);

        for (int i = 1; i <= n; ++i) {
            pFrac = pFrac.multiply(i).divide(n);
        }

        /*
         * BigFraction.doubleValue converts numerator to double and the
         * denominator to double and divides afterwards. That gives NaN quite
         * easy. This does not (scale is the number of digits):
         */
        return pFrac.bigDecimalValue(20, BigDecimal.ROUND_HALF_UP).doubleValue();
    }

    
Calculates P(D_n < d) using method described in [1] and doubles (see above). 
Params:
d – statistic
Throws:
MathArithmeticException – if algorithm fails to convert h to a BigFraction in expressing d as (k - h) / m for integer k, m and 0 <= h < 1.
Returns:
the two-sided probability of P(D_n < d)/**
     * Calculates {@code P(D_n < d)} using method described in [1] and doubles
     * (see above).
     *
     * @param d statistic
     * @return the two-sided probability of {@code P(D_n < d)}
     * @throws MathArithmeticException if algorithm fails to convert {@code h}
     * to a {@link org.apache.commons.math3.fraction.BigFraction} in expressing
     * {@code d} as {@code (k - h) / m} for integer {@code k, m} and
     * {@code 0 <= h < 1}.
     */
    private double roundedK(double d) throws MathArithmeticException {

        final int k = (int) FastMath.ceil(n * d);
        final FieldMatrix<BigFraction> HBigFraction = this.createH(d);
        final int m = HBigFraction.getRowDimension();

        /*
         * Here the rounding part comes into play: use
         * RealMatrix instead of FieldMatrix<BigFraction>
         */
        final RealMatrix H = new Array2DRowRealMatrix(m, m);

        for (int i = 0; i < m; ++i) {
            for (int j = 0; j < m; ++j) {
                H.setEntry(i, j, HBigFraction.getEntry(i, j).doubleValue());
            }
        }

        final RealMatrix Hpower = H.power(n);

        double pFrac = Hpower.getEntry(k - 1, k - 1);

        for (int i = 1; i <= n; ++i) {
            pFrac *= (double) i / (double) n;
        }

        return pFrac;
    }

    
Creates H of size m x m as described in [1] (see above). 
Params:
d – statistic
Throws:
NumberIsTooLargeException – if fractional part is greater than 1
FractionConversionException – if algorithm fails to convert h to a BigFraction in expressing d as (k - h) / m for integer k, m and 0 <= h < 1.
Returns:
H matrix/***
     * Creates {@code H} of size {@code m x m} as described in [1] (see above).
     *
     * @param d statistic
     * @return H matrix
     * @throws NumberIsTooLargeException if fractional part is greater than 1
     * @throws FractionConversionException if algorithm fails to convert
     * {@code h} to a {@link org.apache.commons.math3.fraction.BigFraction} in
     * expressing {@code d} as {@code (k - h) / m} for integer {@code k, m} and
     * {@code 0 <= h < 1}.
     */
    private FieldMatrix<BigFraction> createH(double d)
            throws NumberIsTooLargeException, FractionConversionException {

        int k = (int) FastMath.ceil(n * d);

        int m = 2 * k - 1;
        double hDouble = k - n * d;

        if (hDouble >= 1) {
            throw new NumberIsTooLargeException(hDouble, 1.0, false);
        }

        BigFraction h = null;

        try {
            h = new BigFraction(hDouble, 1.0e-20, 10000);
        } catch (FractionConversionException e1) {
            try {
                h = new BigFraction(hDouble, 1.0e-10, 10000);
            } catch (FractionConversionException e2) {
                h = new BigFraction(hDouble, 1.0e-5, 10000);
            }
        }

        final BigFraction[][] Hdata = new BigFraction[m][m];

        /*
         * Start by filling everything with either 0 or 1.
         */
        for (int i = 0; i < m; ++i) {
            for (int j = 0; j < m; ++j) {
                if (i - j + 1 < 0) {
                    Hdata[i][j] = BigFraction.ZERO;
                } else {
                    Hdata[i][j] = BigFraction.ONE;
                }
            }
        }

        /*
         * Setting up power-array to avoid calculating the same value twice:
         * hPowers[0] = h^1 ... hPowers[m-1] = h^m
         */
        final BigFraction[] hPowers = new BigFraction[m];
        hPowers[0] = h;
        for (int i = 1; i < m; ++i) {
            hPowers[i] = h.multiply(hPowers[i - 1]);
        }

        /*
         * First column and last row has special values (each other reversed).
         */
        for (int i = 0; i < m; ++i) {
            Hdata[i][0] = Hdata[i][0].subtract(hPowers[i]);
            Hdata[m - 1][i] = Hdata[m - 1][i].subtract(hPowers[m - i - 1]);
        }

        /*
         * [1] states: "For 1/2 < h < 1 the bottom left element of the matrix
         * should be (1 - 2*h^m + (2h - 1)^m )/m!" Since 0 <= h < 1, then if h >
         * 1/2 is sufficient to check:
         */
        if (h.compareTo(BigFraction.ONE_HALF) == 1) {
            Hdata[m - 1][0] = Hdata[m - 1][0].add(h.multiply(2).subtract(1).pow(m));
        }

        /*
         * Aside from the first column and last row, the (i, j)-th element is
         * 1/(i - j + 1)! if i - j + 1 >= 0, else 0. 1's and 0's are already
         * put, so only division with (i - j + 1)! is needed in the elements
         * that have 1's. There is no need to calculate (i - j + 1)! and then
         * divide - small steps avoid overflows.
         *
         * Note that i - j + 1 > 0 <=> i + 1 > j instead of j'ing all the way to
         * m. Also note that it is started at g = 2 because dividing by 1 isn't
         * really necessary.
         */
        for (int i = 0; i < m; ++i) {
            for (int j = 0; j < i + 1; ++j) {
                if (i - j + 1 > 0) {
                    for (int g = 2; g <= i - j + 1; ++g) {
                        Hdata[i][j] = Hdata[i][j].divide(g);
                    }
                }
            }
        }

        return new Array2DRowFieldMatrix<BigFraction>(BigFractionField.getInstance(), Hdata);
    }
}