public void WScore()
{
int[] signs = { -1, -1, +1, -1, +1, +1, +1, +1, +1, +1, +1, +1 };
double[] ranks = { 1, 2, 3, 4, 5.5, 5.5, 7, 8, 9, 10, 11, 12 };
double[] diffs = { 0.1, 0.2, 0.3, 0.6, 1.5, 1.5, 1.8, 2.0, 2.1, 2.3, 2.6, 12.4 };
double wm, wp, u;
{
double expected = 7;
wm = WilcoxonDistribution.WNegative(signs, ranks);
Assert.AreEqual(expected, wm);
}
{
double expected = 71;
wp = WilcoxonDistribution.WPositive(signs, ranks);
Assert.AreEqual(expected, wp);
}
{
double expected = 7;
u = WilcoxonDistribution.WMinimum(signs, ranks);
Assert.AreEqual(expected, u);
}
{
double n = signs.Length;
double total = wm + wp;
double expected = (n * (n + 1)) / 2;
Assert.AreEqual(expected, total);
}
}
public void CumulativeExactTest()
{
// example from https://onlinecourses.science.psu.edu/stat414/node/319
double[] ranks =
{
22, 2, 13, 24, 16, 15, 25, 10, 9, 11, 5,
17, 12, 20, 14, 30, 8, 6, 26, 19, 29, 27,3, 28,
7, 21, 23, 1, 18, 4
};
WilcoxonDistribution target = new WilcoxonDistribution(ranks);
Assert.AreEqual(232.5, target.Mean);
Assert.AreEqual(2363.75, target.Variance);
Assert.AreEqual(Math.Sqrt(2363.75), target.StandardDeviation);
double actual = target.DistributionFunction(200);
double expected = 0.2546;
Assert.AreEqual(expected, actual, 1e-2);
double inv = target.InverseDistributionFunction(actual);
Assert.AreEqual(200, inv);
}
public void ConstructorTest2()
{
double[] ranks = { 1, 2, 3, 4, 5.5, 5.5, 7, 8, 9, 10, 11, 12 };
var W = new WilcoxonDistribution(ranks, forceExact: true);
double mean = W.Mean; // 39
double median = W.Median; // 39
double var = W.Variance; // 162.5
double cdf = W.DistributionFunction(w: 42); // 0.582763671875
double pdf = W.ProbabilityDensityFunction(w: 42); // 0.014404296875
double lpdf = W.LogProbabilityDensityFunction(w: 42); // -4.2402287228136233
double ccdf = W.ComplementaryDistributionFunction(x: 42); // 0.417236328125
double icdf = W.InverseDistributionFunction(p: cdf); // 41.965447500067114
double icdf2 = W.InverseDistributionFunction(p: 0.5); // 39.000000487005138
double hf = W.HazardFunction(x: 42); // 0.03452311293153891
double chf = W.CumulativeHazardFunction(x: 42); // 0.87410248360375287
string str = W.ToString(); // "W+(x; R)"
Assert.AreEqual(39.0, mean);
Assert.AreEqual(39.0, median, 1e-6);
Assert.AreEqual(162.5, var);
Assert.AreEqual(0.87410248360375287, chf);
Assert.AreEqual(0.582763671875, cdf);
Assert.AreEqual(0.014404296875, pdf);
Assert.AreEqual(-4.2402287228136233, lpdf);
Assert.AreEqual(0.03452311293153891, hf);
Assert.AreEqual(0.417236328125, ccdf);
Assert.AreEqual(42, icdf, 0.05);
Assert.AreEqual("W+(x; R)", str);
}
public void ConstructorTest()
{
double[] ranks = { 1, 2, 3, 4, 5.5, 5.5, 7, 8, 9, 10, 11, 12 };
var W = new WilcoxonDistribution(ranks);
double mean = W.Mean; // 39.0
double median = W.Median; // 38.5
double var = W.Variance; // 162.5
double cdf = W.DistributionFunction(w: 42); // 0.60817384423279575
double pdf = W.ProbabilityDensityFunction(w: 42); // 0.38418508862319295
double lpdf = W.LogProbabilityDensityFunction(w: 42); // 0.38418508862319295
double ccdf = W.ComplementaryDistributionFunction(x: 42); // 0.39182615576720425
double icdf = W.InverseDistributionFunction(p: cdf); // 42
double icdf2 = W.InverseDistributionFunction(p: 0.5); // 42
double hf = W.HazardFunction(x: 42); // 0.98049883339449373
double chf = W.CumulativeHazardFunction(x: 42); // 0.936937017743799
string str = W.ToString(); // "W+(x; R)"
Assert.AreEqual(39.0, mean);
Assert.AreEqual(38.5, median, 1e-6);
Assert.AreEqual(162.5, var);
Assert.AreEqual(0.936937017743799, chf);
Assert.AreEqual(0.60817384423279575, cdf);
Assert.AreEqual(0.38418508862319295, pdf);
Assert.AreEqual(-0.95663084089698047, lpdf);
Assert.AreEqual(0.98049883339449373, hf);
Assert.AreEqual(0.39182615576720425, ccdf);
Assert.AreEqual(42, icdf, 1e-6);
Assert.AreEqual("W+(x; R)", str);
}
public void MedianTest()
{
double[] ranks = { 1, 2, 3, 7 };
WilcoxonDistribution target = new WilcoxonDistribution(ranks);
Assert.AreEqual(target.Median, target.InverseDistributionFunction(0.5));
}
public void MedianTest_approximation()
{
double[] ranks = { 1, 2, 3, 7 };
WilcoxonDistribution target = new WilcoxonDistribution(ranks, exact: false);
Assert.IsFalse(target.Exact);
Assert.AreEqual(target.Median, target.InverseDistributionFunction(0.5));
}
/// <summary>
/// Performs a Wilcoxon signed rank test.
/// </summary>
/// <param name="x">The values of the first variable.</param>
/// <param name="y">The values of the second variable.</param>
/// <returns>The result of the test.</returns>
/// <remarks>
/// <para>The Wilcoxon signed rank test is a non-parametric alternative to the
/// paired t-test (<see cref="PairedStudentTTest"/>). Given two measurements on
/// the same subjects, this method tests for changes in the distribution between
/// the two measurements. It is sensitive primarily to shifts in the median.
/// Note that the distributions of the individual measurements
/// may be far from normal, and may be different for each subject.</para>
/// </remarks>
/// <exception cref="ArgumentNullException"><paramref name="x"/> or <paramref name="y"/> is <see langword="null"/>.</exception>
/// <exception cref="DimensionMismatchException"><paramref name="x"/> and <paramref name="y"/> do not contain the same number of entries.</exception>
/// <exception cref="InsufficientDataException">There are fewer than two data points.</exception>
/// <seealso href="https://en.wikipedia.org/wiki/Wilcoxon_signed-rank_test"/>
public static TestResult WilcoxonSignedRankTest(IReadOnlyList <double> x, IReadOnlyList <double> y)
{
if (x == null)
{
throw new ArgumentNullException(nameof(x));
}
if (y == null)
{
throw new ArgumentNullException(nameof(y));
}
if (x.Count != y.Count)
{
throw new DimensionMismatchException();
}
int n = x.Count;
if (n < 2)
{
throw new InsufficientDataException();
}
double[] z = new double[n];
for (int i = 0; i < z.Length; i++)
{
z[i] = x[i] - y[i];
}
Array.Sort(z, (xValue, yValue) => Math.Abs(xValue).CompareTo(Math.Abs(yValue)));
int W = 0;
for (int i = 0; i < z.Length; i++)
{
if (z[i] > 0.0)
{
W += (i + 1);
}
}
if (n < 32)
{
DiscreteDistribution wDistribution = new WilcoxonDistribution(n);
return(new TestResult("W", W, wDistribution, TestType.TwoTailed));
}
else
{
double mu = n * (n + 1.0) / 4.0;
double sigma = Math.Sqrt(mu * (2.0 * n + 1.0) / 6.0);
ContinuousDistribution wDistribution = new NormalDistribution(mu, sigma);
return(new TestResult("W", W, wDistribution, TestType.TwoTailed));
}
}
public void ConstructorTest2()
{
double[] ranks = { 1, 2, 3, 4, 5.5, 5.5, 7, 8, 9, 10, 11, 12 };
var W = new WilcoxonDistribution(ranks, exact: true);
double mean = W.Mean; // 39
double median = W.Median; // 39
double var = W.Variance; // 162.5
double cdf = W.DistributionFunction(x: 42); // 0.582763671875
double pdf = W.ProbabilityDensityFunction(x: 42); // 0.014404296875
double lpdf = W.LogProbabilityDensityFunction(x: 42); // -4.2402287228136233
double ccdf = W.ComplementaryDistributionFunction(x: 42); // 0.417236328125
double icdf = W.InverseDistributionFunction(p: cdf); // 41.965447500067114
double icdf2 = W.InverseDistributionFunction(p: 0.5); // 39.000000487005138
double hf = W.HazardFunction(x: 42); // 0.03452311293153891
double chf = W.CumulativeHazardFunction(x: 42); // 0.87410248360375287
string str = W.ToString(); // "W+(x; R)"
Assert.AreEqual(39.0, mean);
Assert.AreEqual(39.0, median, 1e-6);
Assert.AreEqual(162.5, var);
Assert.AreEqual(0.87410248360375287, chf);
Assert.AreEqual(0.59716796875, cdf, 1e-5);
Assert.AreEqual(0.014404296875, pdf, 1e-8);
Assert.AreEqual(-4.2402287228136233, lpdf);
Assert.AreEqual(0.03452311293153891, hf);
Assert.AreEqual(0.417236328125, ccdf);
Assert.AreEqual(42.5, icdf, 0.05);
Assert.AreEqual("W+(x; R)", str);
var range1 = W.GetRange(0.95);
var range2 = W.GetRange(0.99);
var range3 = W.GetRange(0.01);
Assert.AreEqual(17.999999736111114, range1.Min, 1e-6);
Assert.AreEqual(60.000000315408002, range1.Max, 1e-6);
Assert.AreEqual(10.000000351098127, range2.Min, 1e-6);
Assert.AreEqual(67.99999981945885, range2.Max, 1e-6);
Assert.AreEqual(10.000000351098119, range3.Min, 1e-6);
Assert.AreEqual(67.99999981945885, range3.Max, 1e-6);
}
public void ProbabilityTest()
{
// Example from https://onlinecourses.science.psu.edu/stat414/node/319
double[] ranks = { 1, 2, 3 };
WilcoxonDistribution target = new WilcoxonDistribution(ranks);
double[] expected = { 1 / 8.0, 1 / 8.0, 1 / 8.0, 2 / 8.0, 1 / 8.0, 1 / 8.0, 1 / 8.0 };
for (int i = 0; i < expected.Length; i++)
{
// P(W=i)
double actual = target.ProbabilityDensityFunction(i);
Assert.AreEqual(expected[i], actual);
}
}
public void CumulativeTest()
{
// Example from https://onlinecourses.science.psu.edu/stat414/node/319
double[] ranks = { 1, 2, 3 };
WilcoxonDistribution target = new WilcoxonDistribution(ranks);
double[] probabilities = { 1 / 8.0, 1 / 8.0, 1 / 8.0, 2 / 8.0, 1 / 8.0, 1 / 8.0, 1 / 8.0 };
double[] expected = Accord.Math.Matrix.CumulativeSum(probabilities);
for (int i = 0; i < expected.Length; i++)
{
// P(W<=i)
double actual = target.DistributionFunction(i);
Assert.AreEqual(expected[i], actual);
}
}
public void icdf()
{
var dist = new WilcoxonDistribution(1);
double[] percentiles = Vector.Range(0.0, 1.0, stepSize: 0.1);
for (int i = 0; i < percentiles.Length; i++)
{
double x = percentiles[i];
double icdf = dist.InverseDistributionFunction(x);
double cdf = dist.DistributionFunction(icdf);
double iicdf = dist.InverseDistributionFunction(cdf);
double iiicdf = dist.DistributionFunction(iicdf);
Assert.AreEqual(iicdf, icdf, 1e-5);
Assert.AreEqual(x, cdf, 1e-5);
Assert.AreEqual(iiicdf, cdf, 1e-5);
}
}
public void ApproximationTest()
{
double[] m = Matrix.Magic(5).Reshape().Get(0, 20);
double[] samples = m.Rank();
var exact = new WilcoxonDistribution(samples, exact: true);
var approx = new WilcoxonDistribution(samples, exact: false);
var nd = NormalDistribution.Estimate(exact.Table);
Assert.AreEqual(nd.Mean, exact.Mean, 1e-10);
Assert.AreEqual(nd.Variance, exact.Variance, 1e-3);
foreach (double x in Vector.Range(0, 100))
{
double e = approx.DistributionFunction(x);
double a = exact.DistributionFunction(x);
if (e > 0.25)
{
Assert.AreEqual(e, a, 0.1);
}
else
{
Assert.AreEqual(e, a, 0.01);
}
}
foreach (double x in Vector.Range(0, 100))
{
double e = approx.ComplementaryDistributionFunction(x);
double a = exact.ComplementaryDistributionFunction(x);
if (e > 0.25)
{
Assert.AreEqual(e, a, 0.1);
}
else
{
Assert.AreEqual(e, a, 0.01);
}
}
}
public void ConstructorTest()
{
double[] ranks = { 1, 2, 3, 4, 5.5, 5.5, 7, 8, 9, 10, 11, 12 };
var W = new WilcoxonDistribution(ranks);
double mean = W.Mean; // 39.0
double median = W.Median; // 38.5
double var = W.Variance; // 162.5
double mode = W.Mode; // 39.0
double cdf = W.DistributionFunction(x: 42); // 0.60817384423279575
double pdf = W.ProbabilityDensityFunction(x: 42); // 0.38418508862319295
double lpdf = W.LogProbabilityDensityFunction(x: 42); // 0.38418508862319295
double ccdf = W.ComplementaryDistributionFunction(x: 42); // 0.39182615576720425
double icdf = W.InverseDistributionFunction(p: cdf); // 42
double icdf2 = W.InverseDistributionFunction(p: 0.5); // 42
double hf = W.HazardFunction(x: 42); // 0.98049883339449373
double chf = W.CumulativeHazardFunction(x: 42); // 0.936937017743799
string str = W.ToString(); // "W+(x; R)"
Assert.AreEqual(39.0, mean);
Assert.AreEqual(39, median, 1e-6);
Assert.AreEqual(39.0, mode, 1e-8);
Assert.AreEqual(162.5, var);
Assert.AreEqual(0.87410248360375287, chf, 1e-8);
Assert.AreEqual(0.59716796875, cdf, 1e-8);
Assert.AreEqual(0.014404296875, pdf, 1e-8);
Assert.AreEqual(-4.2402287228136233, lpdf, 1e-8);
Assert.AreEqual(0.03452311293153891, hf, 1e-8);
Assert.AreEqual(0.417236328125, ccdf, 1e-8);
Assert.AreEqual(42.5, icdf, 0.05);
Assert.AreEqual("W+(x; R)", str);
Assert.AreEqual(0, W.Support.Min);
Assert.AreEqual(double.PositiveInfinity, W.Support.Max);
Assert.AreEqual(W.InverseDistributionFunction(0), W.Support.Min);
Assert.AreEqual(W.InverseDistributionFunction(1), W.Support.Max);
}
/// <summary>
/// Performs a Wilcoxon signed rank test.
/// </summary>
/// <returns>The result of the test.</returns>
/// <remarks>
/// <para>The Wilcoxon signed rank test is a non-parametric alternative to the
/// paired t-test (<see cref="PairedStudentTTest"/>). Given two measurements on
/// the same subjects, this method tests for changes in the distribution between
/// the two measurements. It is sensitive primarily to shifts in the median.
/// Note that the distributions of the individual measurements
/// may be far from normal, and may be different for each subject.</para>
/// </remarks>
/// <seealso href="https://en.wikipedia.org/wiki/Wilcoxon_signed-rank_test"/>
public TestResult WilcoxonSignedRankTest()
{
int n = this.Count;
if (n < 2)
{
throw new InsufficientDataException();
}
double[] z = new double[n];
for (int i = 0; i < z.Length; i++)
{
z[i] = xData[i] - yData[i];
}
Array.Sort(z, (x, y) => Math.Abs(x).CompareTo(Math.Abs(y)));
int W = 0;
for (int i = 0; i < z.Length; i++)
{
if (z[i] > 0.0)
{
W += (i + 1);
}
}
ContinuousDistribution nullDistribution;
if (Count < 32)
{
DiscreteDistribution wilcoxon = new WilcoxonDistribution(n);
nullDistribution = new DiscreteAsContinuousDistribution(wilcoxon);
}
else
{
double mu = n * (n + 1.0) / 4.0;
double sigma = Math.Sqrt(mu * (2.0 * n + 1.0) / 6.0);
nullDistribution = new NormalDistribution(mu, sigma);
}
return(new TestResult("W", W, TestType.TwoTailed, nullDistribution));
}
