public void WilcoxonNullDistribution()
{
// Pick a very non-normal distribution
ContinuousDistribution d = new ExponentialDistribution();
Random rng = new Random(271828);
foreach (int n in TestUtilities.GenerateIntegerValues(4, 64, 4))
{
Sample wSample = new Sample();
ContinuousDistribution wDistribution = null;
for (int i = 0; i < 128; i++)
{
BivariateSample sample = new BivariateSample();
for (int j = 0; j < n; j++)
{
double x = d.GetRandomValue(rng);
double y = d.GetRandomValue(rng);
sample.Add(x, y);
}
TestResult wilcoxon = sample.WilcoxonSignedRankTest();
wSample.Add(wilcoxon.Statistic);
wDistribution = wilcoxon.Distribution;
}
TestResult ks = wSample.KolmogorovSmirnovTest(wDistribution);
Assert.IsTrue(ks.Probability > 0.05);
Assert.IsTrue(wSample.PopulationMean.ConfidenceInterval(0.99).ClosedContains(wDistribution.Mean));
Assert.IsTrue(wSample.PopulationStandardDeviation.ConfidenceInterval(0.99).ClosedContains(wDistribution.StandardDeviation));
}
}
public void WilcoxonNullDistribution()
{
// Pick a very non-normal distribution
ContinuousDistribution d = new ExponentialDistribution();
Random rng = new Random(271828);
foreach (int n in TestUtilities.GenerateIntegerValues(4, 64, 4))
{
Sample wContinuousSample = new Sample();
ContinuousDistribution wContinuousDistribution = null;
List <int> wDiscreteSample = new List <int>();
DiscreteDistribution wDiscreteDistribution = null;
for (int i = 0; i < 256; i++)
{
BivariateSample sample = new BivariateSample();
for (int j = 0; j < n; j++)
{
double x = d.GetRandomValue(rng);
double y = d.GetRandomValue(rng);
sample.Add(x, y);
}
TestResult wilcoxon = sample.WilcoxonSignedRankTest();
if (wilcoxon.UnderlyingStatistic != null)
{
wDiscreteSample.Add(wilcoxon.UnderlyingStatistic.Value);
wDiscreteDistribution = wilcoxon.UnderlyingStatistic.Distribution;
}
else
{
wContinuousSample.Add(wilcoxon.Statistic.Value);
wContinuousDistribution = wilcoxon.Statistic.Distribution;
}
}
if (wDiscreteDistribution != null)
{
TestResult chi2 = wDiscreteSample.ChiSquaredTest(wDiscreteDistribution);
Assert.IsTrue(chi2.Probability > 0.01);
}
else
{
TestResult ks = wContinuousSample.KolmogorovSmirnovTest(wContinuousDistribution);
Assert.IsTrue(ks.Probability > 0.01);
Assert.IsTrue(wContinuousSample.PopulationMean.ConfidenceInterval(0.99).ClosedContains(wContinuousDistribution.Mean));
Assert.IsTrue(wContinuousSample.PopulationStandardDeviation.ConfidenceInterval(0.99).ClosedContains(wContinuousDistribution.StandardDeviation));
}
}
}
public void GammaFromExponential()
{
// test that x_1 + x_2 + ... + x_n ~ Gamma(n) when z ~ Exponential()
Random rng = new Random(1);
ExponentialDistribution eDistribution = new ExponentialDistribution();
// pick some low values of n so distribution is not simply normal
foreach (int n in new int[] { 2, 3, 4, 5 })
{
Sample gSample = new Sample();
for (int i = 0; i < 100; i++)
{
double sum = 0.0;
for (int j = 0; j < n; j++)
{
sum += eDistribution.GetRandomValue(rng);
}
gSample.Add(sum);
}
GammaDistribution gDistribution = new GammaDistribution(n);
TestResult result = gSample.KolmogorovSmirnovTest(gDistribution);
Assert.IsTrue(result.Probability > 0.05);
}
}
public void MultivariateMoments()
{
// create a random sample
MultivariateSample M = new MultivariateSample(3);
ContinuousDistribution d0 = new NormalDistribution();
ContinuousDistribution d1 = new ExponentialDistribution();
ContinuousDistribution d2 = new UniformDistribution();
Random rng = new Random(1);
int n = 10;
for (int i = 0; i < n; i++)
{
M.Add(d0.GetRandomValue(rng), d1.GetRandomValue(rng), d2.GetRandomValue(rng));
}
// test that moments agree
for (int i = 0; i < 3; i++)
{
int[] p = new int[3];
p[i] = 1;
Assert.IsTrue(TestUtilities.IsNearlyEqual(M.Column(i).Mean, M.RawMoment(p)));
p[i] = 2;
Assert.IsTrue(TestUtilities.IsNearlyEqual(M.Column(i).Variance, M.CentralMoment(p)));
for (int j = 0; j < i; j++)
{
int[] q = new int[3];
q[i] = 1;
q[j] = 1;
Assert.IsTrue(TestUtilities.IsNearlyEqual(M.TwoColumns(i, j).Covariance, M.CentralMoment(q)));
}
}
}
public void KendallNullDistributionTest()
{
// Pick independent distributions for x and y, which needn't be normal and needn't be related.
ContinuousDistribution xDistrubtion = new LogisticDistribution();
ContinuousDistribution yDistribution = new ExponentialDistribution();
Random rng = new Random(314159265);
// generate bivariate samples of various sizes
foreach (int n in TestUtilities.GenerateIntegerValues(8, 64, 4))
{
Sample testStatistics = new Sample();
ContinuousDistribution testDistribution = null;
for (int i = 0; i < 128; i++)
{
BivariateSample sample = new BivariateSample();
for (int j = 0; j < n; j++)
{
sample.Add(xDistrubtion.GetRandomValue(rng), yDistribution.GetRandomValue(rng));
}
TestResult result = sample.KendallTauTest();
testStatistics.Add(result.Statistic);
testDistribution = result.Distribution;
}
TestResult r2 = testStatistics.KolmogorovSmirnovTest(testDistribution);
Assert.IsTrue(r2.RightProbability > 0.05);
Assert.IsTrue(testStatistics.PopulationMean.ConfidenceInterval(0.99).ClosedContains(testDistribution.Mean));
Assert.IsTrue(testStatistics.PopulationVariance.ConfidenceInterval(0.99).ClosedContains(testDistribution.Variance));
}
}
public void BivariateNonlinearFitVariances()
{
// Verify that we can fit a non-linear function,
// that the estimated parameters do cluster around the true values,
// and that the estimated parameter covariances do reflect the actually observed covariances
double a = 2.7;
double b = 3.1;
ContinuousDistribution xDistribution = new ExponentialDistribution(2.0);
ContinuousDistribution eDistribution = new NormalDistribution(0.0, 4.0);
FrameTable parameters = new FrameTable();
parameters.AddColumns <double>("a", "b");
MultivariateSample covariances = new MultivariateSample(3);
for (int i = 0; i < 64; i++)
{
BivariateSample sample = new BivariateSample();
Random rng = new Random(i);
for (int j = 0; j < 8; j++)
{
double x = xDistribution.GetRandomValue(rng);
double y = a * Math.Pow(x, b) + eDistribution.GetRandomValue(rng);
sample.Add(x, y);
}
NonlinearRegressionResult fit = sample.NonlinearRegression(
(IReadOnlyList <double> p, double x) => p[0] * Math.Pow(x, p[1]),
new double[] { 1.0, 1.0 }
);
parameters.AddRow(fit.Parameters.ValuesVector);
covariances.Add(fit.Parameters.CovarianceMatrix[0, 0], fit.Parameters.CovarianceMatrix[1, 1], fit.Parameters.CovarianceMatrix[0, 1]);
}
Assert.IsTrue(parameters["a"].As <double>().PopulationMean().ConfidenceInterval(0.99).ClosedContains(a));
Assert.IsTrue(parameters["b"].As <double>().PopulationMean().ConfidenceInterval(0.99).ClosedContains(b));
Assert.IsTrue(parameters["a"].As <double>().PopulationVariance().ConfidenceInterval(0.99).ClosedContains(covariances.Column(0).Mean));
Assert.IsTrue(parameters["b"].As <double>().PopulationVariance().ConfidenceInterval(0.99).ClosedContains(covariances.Column(1).Mean));
Assert.IsTrue(parameters["a"].As <double>().PopulationCovariance(parameters["b"].As <double>()).ConfidenceInterval(0.99).ClosedContains(covariances.Column(2).Mean));
Assert.IsTrue(Bivariate.PopulationCovariance(parameters["a"].As <double>(), parameters["b"].As <double>()).ConfidenceInterval(0.99).ClosedContains(covariances.Column(2).Mean));
}
public void MultivariateLinearLogisticRegressionVariances()
{
// define model y = a + b0 * x0 + b1 * x1 + noise
double a = -3.0;
double b0 = 2.0;
double b1 = 1.0;
ContinuousDistribution x0distribution = new ExponentialDistribution();
ContinuousDistribution x1distribution = new LognormalDistribution();
FrameTable data = new FrameTable();
data.AddColumns <double>("a", "da", "b0", "db0", "b1", "db1", "p", "dp");
// draw a sample from the model
Random rng = new Random(2);
for (int j = 0; j < 32; j++)
{
List <double> x0s = new List <double>();
List <double> x1s = new List <double>();
List <bool> ys = new List <bool>();
FrameTable table = new FrameTable();
table.AddColumn <double>("x0");
table.AddColumn <double>("x1");
table.AddColumn <bool>("y");
for (int i = 0; i < 32; i++)
{
double x0 = x0distribution.GetRandomValue(rng);
double x1 = x1distribution.GetRandomValue(rng);
double t = a + b0 * x0 + b1 * x1;
double p = 1.0 / (1.0 + Math.Exp(-t));
bool y = (rng.NextDouble() < p);
x0s.Add(x0);
x1s.Add(x1);
ys.Add(y);
}
// do a linear regression fit on the model
MultiLinearLogisticRegressionResult result = ys.MultiLinearLogisticRegression(
new Dictionary <string, IReadOnlyList <double> > {
{ "x0", x0s }, { "x1", x1s }
}
);
UncertainValue pp = result.Predict(0.0, 1.0);
data.AddRow(
result.Intercept.Value, result.Intercept.Uncertainty,
result.CoefficientOf("x0").Value, result.CoefficientOf("x0").Uncertainty,
result.CoefficientOf("x1").Value, result.CoefficientOf("x1").Uncertainty,
pp.Value, pp.Uncertainty
);
}
// The estimated parameters should agree with the model that generated the data.
// The variances of the estimates should agree with the claimed variances
Assert.IsTrue(data["a"].As <double>().PopulationStandardDeviation().ConfidenceInterval(0.99).ClosedContains(data["da"].As <double>().Mean()));
Assert.IsTrue(data["b0"].As <double>().PopulationStandardDeviation().ConfidenceInterval(0.99).ClosedContains(data["db0"].As <double>().Mean()));
Assert.IsTrue(data["b1"].As <double>().PopulationStandardDeviation().ConfidenceInterval(0.99).ClosedContains(data["db1"].As <double>().Mean()));
Assert.IsTrue(data["p"].As <double>().PopulationStandardDeviation().ConfidenceInterval(0.99).ClosedContains(data["dp"].As <double>().Mean()));
}