public void DistributionFunctionTest()
{
double[] ranks = { 1, 2, 3, 4, 5 };
int n1 = 2;
int n2 = 3;
int N = n1 + n2;
Assert.AreEqual(N, ranks.Length);
var target = new MannWhitneyDistribution(ranks, n1, n2);
// Number of possible combinations is 5!/(3!2!) = 10.
int nc = (int)Special.Binomial(5, 3);
Assert.AreEqual(10, nc);
double[] expected = { 0.0, 0.1, 0.1, 0.2, 0.2, 0.2, 0.1, 0.1, 0.0, 0.0 };
expected = expected.CumulativeSum();
for (int i = 0; i < expected.Length; i++)
{
// P(U<=i)
double actual = target.DistributionFunction(i);
Assert.AreEqual(expected[i], actual,1e-10);
}
}
public void ConstructorTest()
{
double[] ranks = { 1, 2, 3, 4, 5 };
var mannWhitney = new MannWhitneyDistribution(ranks, n1: 2, n2: 3);
double mean = mannWhitney.Mean; // 2.7870954605658511
double median = mannWhitney.Median; // 1.5219615583481305
double var = mannWhitney.Variance; // 18.28163603621158
double cdf = mannWhitney.DistributionFunction(x: 4); // 0.6
double pdf = mannWhitney.ProbabilityDensityFunction(x: 4); // 0.2
double lpdf = mannWhitney.LogProbabilityDensityFunction(x: 4); // -1.6094379124341005
double ccdf = mannWhitney.ComplementaryDistributionFunction(x: 4); // 0.4
double icdf = mannWhitney.InverseDistributionFunction(p: cdf); // 3.6666666666666661
double hf = mannWhitney.HazardFunction(x: 4); // 0.5
double chf = mannWhitney.CumulativeHazardFunction(x: 4); // 0.916290731874155
string str = mannWhitney.ToString(); // MannWhitney(u; n1 = 2, n2 = 3)
Assert.AreEqual(3.0, mean);
Assert.AreEqual(3.0000006357828775, median);
Assert.AreEqual(3.0, var);
Assert.AreEqual(0.916290731874155, chf);
Assert.AreEqual(0.6, cdf);
Assert.AreEqual(0.2, pdf);
Assert.AreEqual(-1.6094379124341005, lpdf);
Assert.AreEqual(0.5, hf);
Assert.AreEqual(0.4, ccdf);
Assert.AreEqual(3.6666666666666661, icdf);
Assert.AreEqual("MannWhitney(u; n1 = 2, n2 = 3)", str);
}
public void ProbabilityDensityFunctionTest()
{
double[] ranks = { 1, 2, 3, 4, 5 };
int n1 = 2;
int n2 = 3;
int N = n1 + n2;
Assert.AreEqual(N, ranks.Length);
var target = new MannWhitneyDistribution(ranks, n1, n2);
// Number of possible combinations is 5!/(3!2!) = 10.
int nc = (int)Special.Binomial(5, 3);
Assert.AreEqual(10, nc);
double[] expected = { 0.1, 0.1, 0.2, 0.2, 0.2, 0.1, 0.1, 0.0, 0.0 };
double sum = 0;
for (int i = 0; i < expected.Length; i++)
{
// P(U=i)
double actual = target.ProbabilityDensityFunction(i);
Assert.AreEqual(expected[i], actual);
sum += actual;
}
Assert.AreEqual(1, sum);
}
public void icdf()
{
var dist = new MannWhitneyDistribution(n1: 1, n2: 1);
Assert.AreEqual(Double.NegativeInfinity, dist.Support.Min);
Assert.AreEqual(Double.PositiveInfinity, dist.Support.Max);
{
foreach (var x in new[] { 0.0, 1.0 })
{
double a = dist.InverseDistributionFunction(x);
double ab = dist.DistributionFunction(a);
double abc = dist.InverseDistributionFunction(ab);
double abcd = dist.DistributionFunction(abc);
Assert.AreEqual(x, ab, 1e-5);
Assert.AreEqual(x, abcd, 1e-5);
Assert.AreEqual(a, abc, 1e-5);
}
}
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);
Assert.AreEqual(x, cdf, 1e-5);
}
}
public void MedianTest()
{
double[] ranks = { 1, 1, 2, 3, 4, 7, 5 };
int n1 = 4;
int n2 = 3;
Assert.AreEqual(n1 + n2, ranks.Length);
var target = new MannWhitneyDistribution(ranks, n1, n2);
Assert.AreEqual(target.Median, target.InverseDistributionFunction(0.5));
}
public void RCompatibilityTest1()
{
// Example from https://onlinecourses.science.psu.edu/stat464/node/36
double[] a = { 37, 49, 55, 57 };
double[] b = { 23, 31, 46 };
double[] c = a.Concatenate(b);
double[] rc = c.Rank();
double[] ra = rc.Get(0, a.Length);
double[] rb = rc.Get(a.Length, 0);
Assert.IsTrue(rc.IsEqual(new[] { 3, 5, 6, 7, 1, 2, 4 }));
Assert.IsTrue(ra.IsEqual(new[] { 3, 5, 6, 7 }));
Assert.IsTrue(rb.IsEqual(new[] { 1, 2, 4 }));
double w1 = MannWhitneyDistribution.MannWhitneyU(ra);
double w2 = MannWhitneyDistribution.MannWhitneyU(rb);
var target = new MannWhitneyDistribution(rc, a.Length, b.Length);
Assert.IsTrue(target.Exact);
double p1 = target.DistributionFunction(w1);
double p2 = target.DistributionFunction(w2);
Assert.AreEqual(0.057142857142857141, p1, 1e-5);
Assert.AreEqual(0.97142857142857142, p2, 1e-5);
target = new MannWhitneyDistribution(rc, b.Length, a.Length);
Assert.IsTrue(target.Exact);
p1 = target.DistributionFunction(w1);
p2 = target.DistributionFunction(w2);
Assert.AreEqual(0.97142857142857142, p1, 1e-5);
Assert.AreEqual(0.057142857142857141, p2, 1e-5);
target = new MannWhitneyDistribution(rc, a.Length, b.Length, exact: false);
Assert.IsFalse(target.Exact);
p1 = target.DistributionFunction(w1);
p2 = target.DistributionFunction(w2);
Assert.AreEqual(0.038549935871770913, p1, 1e-5);
Assert.AreEqual(0.96145006412822909, p2, 1e-5);
target = new MannWhitneyDistribution(rc, b.Length, a.Length, exact: false);
Assert.IsFalse(target.Exact);
p1 = target.DistributionFunction(w1);
p2 = target.DistributionFunction(w2);
Assert.AreEqual(0.96145006412822909, p1, 1e-5);
Assert.AreEqual(0.038549935871770913, p2, 1e-5);
}
public void ConstructorTest()
{
#region doc_create
double[] ranks = { 1, 2, 3, 4, 5 };
var mannWhitney = new MannWhitneyDistribution(ranks, n1: 2, n2: 3);
double mean = mannWhitney.Mean; // 2.7870954605658511
double median = mannWhitney.Median; // 1.5219615583481305
double var = mannWhitney.Variance; // 18.28163603621158
double mode = mannWhitney.Mode;
double cdf = mannWhitney.DistributionFunction(x: 4); // 0.6
double pdf = mannWhitney.ProbabilityDensityFunction(x: 4); // 0.2
double lpdf = mannWhitney.LogProbabilityDensityFunction(x: 4); // -1.6094379124341005
double ccdf = mannWhitney.ComplementaryDistributionFunction(x: 4); // 0.4
double icdf = mannWhitney.InverseDistributionFunction(p: cdf); // 3.6666666666666661
double hf = mannWhitney.HazardFunction(x: 4); // 0.5
double chf = mannWhitney.CumulativeHazardFunction(x: 4); // 0.916290731874155
string str = mannWhitney.ToString(); // MannWhitney(u; n1 = 2, n2 = 3)
#endregion
Assert.AreEqual(3.0, mean);
Assert.AreEqual(3.0, mode);
Assert.AreEqual(3.0000006357828775, median, 1e-5);
Assert.AreEqual(3.0, var, 1e-5);
Assert.AreEqual(0.916290731874155, chf, 1e-8);
Assert.AreEqual(0.8, cdf, 1e-8);
Assert.AreEqual(0.2, pdf, 1e-8);
Assert.AreEqual(-1.6094379124341005, lpdf, 1e-8);
Assert.AreEqual(0.5, hf, 1e-8);
Assert.AreEqual(0.4, ccdf, 1e-8);
Assert.AreEqual(4, icdf, 1e-8);
Assert.AreEqual("MannWhitney(u; n1 = 2, n2 = 3)", str);
var range1 = mannWhitney.GetRange(0.95);
var range2 = mannWhitney.GetRange(0.99);
var range3 = mannWhitney.GetRange(0.01);
Assert.AreEqual(0, range1.Min, 1e-5);
Assert.AreEqual(6, range1.Max, 1e-5);
Assert.AreEqual(-8.5830688476562492E-07, range2.Min, 1e-5);
Assert.AreEqual(6.0000005561746477, range2.Max, 1e-4);
Assert.AreEqual(-8.58306884765625E-07, range3.Min, 1e-5);
Assert.AreEqual(6.0000005561746477, range3.Max, 1e-5);
}
public void cdf_n2_greater_than_n1()
{
var dist = new MannWhitneyDistribution(n1: 1, n2: 10);
Assert.AreEqual(0, dist.DistributionFunction(Double.NegativeInfinity));
Assert.AreEqual(0, dist.DistributionFunction(-1e100));
Assert.AreEqual(1.050717977957305E-06d, dist.DistributionFunction(-10));
Assert.AreEqual(0.040995160500191474d, dist.DistributionFunction(-0.5));
Assert.AreEqual(0.056923149003329065d, dist.DistributionFunction(0.0));
Assert.AreEqual(0.077364461742689294d, dist.DistributionFunction(+0.5));
Assert.AreEqual(0.94307685099667093d, dist.DistributionFunction(+10));
Assert.AreEqual(1, dist.DistributionFunction(+1e100));
Assert.AreEqual(1, dist.DistributionFunction(Double.PositiveInfinity));
}
public void cdf()
{
var dist = new MannWhitneyDistribution(n1: 1, n2: 1);
Assert.AreEqual(0, dist.DistributionFunction(Double.NegativeInfinity));
Assert.AreEqual(0, dist.DistributionFunction(-1e100));
Assert.AreEqual(0, dist.DistributionFunction(-10));
Assert.AreEqual(0.022750131948179209d, dist.DistributionFunction(-0.5));
Assert.AreEqual(0.15865525393145707d, dist.DistributionFunction(0.0));
Assert.AreEqual(0.5, dist.DistributionFunction(+0.5));
Assert.AreEqual(1, dist.DistributionFunction(+10));
Assert.AreEqual(1, dist.DistributionFunction(+1e100));
Assert.AreEqual(1, dist.DistributionFunction(Double.PositiveInfinity));
}
public void ApproximationTest()
{
int t = 14;
double[] m = Matrix.Magic(6).Reshape().Get(0, t * 2);
double[] samples = m.Rank();
double[] rank1 = samples.Get(0, t);
double[] rank2 = samples.Get(t, 0);
var exact = new MannWhitneyDistribution(rank1, rank2, exact: true);
var approx = new MannWhitneyDistribution(rank1, rank2, exact: false);
var nd = NormalDistribution.Estimate(exact.Table);
Assert.AreEqual(nd.Mean, exact.Mean, 1e-10);
Assert.AreEqual(nd.Variance, exact.Variance, 2e-5);
foreach (double x in Vector.Range(0, 100))
{
double e = approx.DistributionFunction(x);
double a = exact.DistributionFunction(x);
if (e > 0.23)
{
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 > 0.23)
{
Assert.AreEqual(e, a, 0.1);
}
else
{
Assert.AreEqual(e, a, 0.01);
}
}
}
public void ConstructorTest()
{
double[] ranks = { 1, 2, 3, 4, 5 };
var mannWhitney = new MannWhitneyDistribution(ranks, n1: 2, n2: 3);
double mean = mannWhitney.Mean; // 2.7870954605658511
double median = mannWhitney.Median; // 1.5219615583481305
double var = mannWhitney.Variance; // 18.28163603621158
try { double mode = mannWhitney.Mode; Assert.Fail(); }
catch { }
double cdf = mannWhitney.DistributionFunction(x: 4); // 0.6
double pdf = mannWhitney.ProbabilityDensityFunction(x: 4); // 0.2
double lpdf = mannWhitney.LogProbabilityDensityFunction(x: 4); // -1.6094379124341005
double ccdf = mannWhitney.ComplementaryDistributionFunction(x: 4); // 0.4
double icdf = mannWhitney.InverseDistributionFunction(p: cdf); // 3.6666666666666661
double hf = mannWhitney.HazardFunction(x: 4); // 0.5
double chf = mannWhitney.CumulativeHazardFunction(x: 4); // 0.916290731874155
string str = mannWhitney.ToString(); // MannWhitney(u; n1 = 2, n2 = 3)
Assert.AreEqual(3.0, mean);
Assert.AreEqual(3.0000006357828775, median);
Assert.AreEqual(3.0, var);
Assert.AreEqual(0.916290731874155, chf);
Assert.AreEqual(0.6, cdf);
Assert.AreEqual(0.2, pdf);
Assert.AreEqual(-1.6094379124341005, lpdf);
Assert.AreEqual(0.5, hf);
Assert.AreEqual(0.4, ccdf);
Assert.AreEqual(3.6666666666666661, icdf);
Assert.AreEqual("MannWhitney(u; n1 = 2, n2 = 3)", str);
var range1 = mannWhitney.GetRange(0.95);
var range2 = mannWhitney.GetRange(0.99);
var range3 = mannWhitney.GetRange(0.01);
Assert.AreEqual(0.00000095367431640625085, range1.Min);
Assert.AreEqual(5.9999995430310555, range1.Max);
Assert.AreEqual(0, range2.Min);
Assert.AreEqual(6.000000194140088, range2.Max);
Assert.AreEqual(0, range3.Min);
Assert.AreEqual(6.000000194140088, range3.Max);
}
public void MedianTest_approximation()
{
double[] ranks = { 1, 1, 2, 3, 4, 7, 5 };
foreach (bool exact in new bool[] { false, true })
{
for (int i = 1; i < ranks.Length; i++)
{
int n1 = i;
int n2 = ranks.Length - i;
Assert.AreEqual(n1 + n2, ranks.Length);
var target = new MannWhitneyDistribution(ranks, n1, n2, exact: exact);
Assert.AreEqual(exact, target.Exact);
Assert.AreEqual(target.Median, target.InverseDistributionFunction(0.5));
}
}
}
