Statistics/Chi-squared distribution: Difference between revisions

=={{header|Java}}==

<syntaxhighlight lang="java">

import java.util.List;

public final class StatisticsChiSquaredDistribution {

public static void main(String[] aArgs) {

System.out.println(" Values of the Chi-squared probability distribution function");

System.out.println(" x/k 1 2 3 4 5");

for ( int x = 0; x <= 10; x++ ) {

System.out.print(String.format("%2d", x));

for ( int k = 1; k <= 5; k++ ) {

System.out.print(String.format("%10.6f", pdf(x, k)));

}

System.out.println();

}

System.out.println();

System.out.println(" Values for the Chi-squared distribution with 3 degrees of freedom");

System.out.println("Chi-squared cumulative pdf p-value");

for ( int x : List.of( 1, 2, 4, 8, 16, 32 ) ) {

final double cdf = cdf(x, 3);

System.out.println(String.format("%5d%20.6f%20.6f", x, cdf, ( 1.0 - cdf )));

}

final int[][] observed = { { 77, 23 }, { 88, 12 }, { 79, 21 }, { 81, 19 } };

final double[][] expected = { { 81.25, 18.75 }, { 81.25, 18.75 }, { 81.25, 18.75 }, { 81.25, 18.75 } };

double testStatistic = 0.0;

for ( int i = 0; i < observed.length; i++ ) {

for ( int j = 0; j < observed[0].length; j++ ) {

testStatistic += Math.pow(observed[i][j] - expected[i][j], 2.0) / expected[i][j];

}

}

final int degreesFreedom = ( observed.length - 1 ) / ( observed[0].length - 1 );

System.out.println();

System.out.println("For the airport data:");

System.out.println(" test statistic : " + String.format("%.6f", testStatistic));

System.out.println(" degrees of freedom : " + degreesFreedom);

System.out.println(" Chi-squared : " + String.format("%.6f", pdf(testStatistic, degreesFreedom)));

System.out.println(" p-value : " + String.format("%.6f", cdf(testStatistic, degreesFreedom)));

}

// The gamma function.

private static double gamma(double aX) {

if ( aX < 0.5 ) {

return Math.PI / ( Math.sin(Math.PI * aX ) * gamma(1.0 - aX) );

}

final double[] probabilities = new double[] {

0.99999999999980993, 676.5203681218851, -1259.1392167224028, 771.32342877765313, -176.61502916214059,

12.507343278686905, -0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7 };

aX -= 1.0;

double t = probabilities[0];

for ( int i = 1; i < 9; i++ ) {

t += probabilities[i] / ( aX + i );

}

final double w = aX + 7.5;

return Math.sqrt(2.0 * Math.PI) * Math.pow(w, aX + 0.5) * Math.exp(-w) * t;

}

// The probability density function of the Chi-squared distribution.

private static double pdf(double aX, double aK) {

return ( aX > 0.0 ) ?

Math.pow(aX, aK / 2 - 1) * Math.exp(-aX / 2) / ( Math.pow(2, aK / 2) * gamma(aK / 2) ) : 0.0;

}

// The cumulative probability function of the Chi-squared distribution.

private static double cdf(double aX, double aK) {

return ( aX > 0.0 && aK > 0.0 ) ? gammaCDF(aX / 2, aK / 2) : 0.0;

}

// The normalised lower incomplete gamma function.

private static double gammaCDF(double aX, double aK) {

double result = 0.0;

for ( int m = 0; m <= 99; m++ ) {

result += Math.pow(aX, m) / gamma(aK + m + 1);

}

result *= Math.pow(aX, aK) * Math.exp(-aX);

return result;

}

}

</syntaxhighlight>

{{ out }}

<pre>

Values of the Chi-squared probability distribution function

x/k 1 2 3 4 5

0 0.000000 0.000000 0.000000 0.000000 0.000000

1 0.241971 0.303265 0.241971 0.151633 0.080657

2 0.103777 0.183940 0.207554 0.183940 0.138369

3 0.051393 0.111565 0.154180 0.167348 0.154180

4 0.026995 0.067668 0.107982 0.135335 0.143976

5 0.014645 0.041042 0.073225 0.102606 0.122042

6 0.008109 0.024894 0.048652 0.074681 0.097304

7 0.004553 0.015099 0.031873 0.052845 0.074371

8 0.002583 0.009158 0.020667 0.036631 0.055112

9 0.001477 0.005554 0.013296 0.024995 0.039887

10 0.000850 0.003369 0.008500 0.016845 0.028335

Values for the Chi-squared distribution with 3 degrees of freedom

Chi-squared cumulative pdf p-value

1 0.198748 0.801252

2 0.427593 0.572407

4 0.738536 0.261464

8 0.953988 0.046012

16 0.998866 0.001134

32 0.999999 0.000001

For the airport data:

test statistic : 4.512821

degrees of freedom : 3

Chi-squared : 0.088754

p-value : 0.788850

</pre>