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 LinearRegressionVariances()
{
// do a set of logistic regression fits
// make sure not only that the fit parameters are what they should be, but that their variances/covariances are as returned
Random rng = new Random(314159);
// define line parameters
double a0 = 2.0; double b0 = -1.0;
// do a lot of fits, recording results of each
FrameTable data = new FrameTable();
data.AddColumns <double>("a", "va", "b", "vb", "abCov", "p", "dp");
for (int k = 0; k < 128; k++)
{
// we should be able to draw x's from any distribution; noise should be drawn from a normal distribution
ContinuousDistribution xd = new LogisticDistribution();
ContinuousDistribution nd = new NormalDistribution(0.0, 2.0);
// generate a synthetic data set
BivariateSample sample = new BivariateSample();
for (int i = 0; i < 12; i++)
{
double x = xd.GetRandomValue(rng);
double y = a0 + b0 * x + nd.GetRandomValue(rng);
sample.Add(x, y);
}
// do the regression
LinearRegressionResult result = sample.LinearRegression();
// record result
UncertainValue p = result.Predict(12.0);
data.AddRow(new Dictionary <string, object>()
{
{ "a", result.Intercept.Value },
{ "va", result.Parameters.VarianceOf("Intercept") },
{ "b", result.Slope.Value },
{ "vb", result.Parameters.VarianceOf("Slope") },
{ "abCov", result.Parameters.CovarianceOf("Slope", "Intercept") },
{ "p", p.Value },
{ "dp", p.Uncertainty }
});
}
// variances of parameters should agree with predictions
Assert.IsTrue(data["a"].As <double>().PopulationVariance().ConfidenceInterval(0.99).ClosedContains(data["va"].As <double>().Median()));
Assert.IsTrue(data["b"].As <double>().PopulationVariance().ConfidenceInterval(0.99).ClosedContains(data["vb"].As <double>().Median()));
Assert.IsTrue(data["a"].As <double>().PopulationCovariance(data["b"].As <double>()).ConfidenceInterval(0.99).ClosedContains(data["abCov"].As <double>().Median()));
// variance of prediction should agree with claim
Assert.IsTrue(data["p"].As <double>().PopulationStandardDeviation().ConfidenceInterval(0.99).ClosedContains(data["dp"].As <double>().Median()));
}
public void BivariateLinearRegression()
{
// do a set of logistic regression fits
// make sure not only that the fit parameters are what they should be, but that their variances/covariances are as returned
Random rng = new Random(314159);
// define logistic parameters
double a0 = 2.0; double b0 = -1.0;
// keep track of sample of returned a and b fit parameters
BivariateSample ps = new BivariateSample();
// also keep track of returned covariance estimates
// since these vary slightly from fit to fit, we will average them
double caa = 0.0;
double cbb = 0.0;
double cab = 0.0;
// also keep track of test statistics
Sample fs = new Sample();
// do 100 fits
for (int k = 0; k < 100; k++)
{
// we should be able to draw x's from any distribution; noise should be drawn from a normal distribution
Distribution xd = new LogisticDistribution();
Distribution nd = new NormalDistribution(0.0, 2.0);
// generate a synthetic data set
BivariateSample s = new BivariateSample();
for (int i = 0; i < 25; i++)
{
double x = xd.GetRandomValue(rng);
double y = a0 + b0 * x + nd.GetRandomValue(rng);
s.Add(x, y);
}
// do the regression
FitResult r = s.LinearRegression();
// record best fit parameters
double a = r.Parameter(0).Value;
double b = r.Parameter(1).Value;
ps.Add(a, b);
// record estimated covariances
caa += r.Covariance(0, 0);
cbb += r.Covariance(1, 1);
cab += r.Covariance(0, 1);
// record the fit statistic
fs.Add(r.GoodnessOfFit.Statistic);
Console.WriteLine("F={0}", r.GoodnessOfFit.Statistic);
}
caa /= ps.Count;
cbb /= ps.Count;
cab /= ps.Count;
// check that mean parameter estimates are what they should be: the underlying population parameters
Assert.IsTrue(ps.X.PopulationMean.ConfidenceInterval(0.95).ClosedContains(a0));
Assert.IsTrue(ps.Y.PopulationMean.ConfidenceInterval(0.95).ClosedContains(b0));
Console.WriteLine("{0} {1}", caa, ps.X.PopulationVariance);
Console.WriteLine("{0} {1}", cbb, ps.Y.PopulationVariance);
// check that parameter covarainces are what they should be: the reported covariance estimates
Assert.IsTrue(ps.X.PopulationVariance.ConfidenceInterval(0.95).ClosedContains(caa));
Assert.IsTrue(ps.Y.PopulationVariance.ConfidenceInterval(0.95).ClosedContains(cbb));
Assert.IsTrue(ps.PopulationCovariance.ConfidenceInterval(0.95).ClosedContains(cab));
// check that F is distributed as it should be
Console.WriteLine(fs.KolmogorovSmirnovTest(new FisherDistribution(2, 48)).LeftProbability);
}
public void BivariateLinearRegression()
{
// do a set of logistic regression fits
// make sure not only that the fit parameters are what they should be, but that their variances/covariances are as returned
Random rng = new Random(314159);
// define line parameters
double a0 = 2.0; double b0 = -1.0;
// keep track of sample of returned a and b fit parameters
BivariateSample pSample = new BivariateSample();
// also keep track of returned covariance estimates
// since these vary slightly from fit to fit, we will average them
double caa = 0.0;
double cbb = 0.0;
double cab = 0.0;
// Record predictions for a new point
double x0 = 12.0;
Sample ySample = new Sample();
double ySigma = 0.0;
// do 100 fits
for (int k = 0; k < 128; k++)
{
// we should be able to draw x's from any distribution; noise should be drawn from a normal distribution
ContinuousDistribution xd = new LogisticDistribution();
ContinuousDistribution nd = new NormalDistribution(0.0, 2.0);
// generate a synthetic data set
BivariateSample sample = new BivariateSample();
for (int i = 0; i < 16; i++)
{
double x = xd.GetRandomValue(rng);
double y = a0 + b0 * x + nd.GetRandomValue(rng);
sample.Add(x, y);
}
// do the regression
LinearRegressionResult result = sample.LinearRegression();
// test consistancy
Assert.IsTrue(result.Intercept == result.Parameters[0].Estimate);
Assert.IsTrue(result.Intercept.Value == result.Parameters.Best[0]);
Assert.IsTrue(TestUtilities.IsNearlyEqual(result.Intercept.Uncertainty, Math.Sqrt(result.Parameters.Covariance[0, 0])));
Assert.IsTrue(result.Slope == result.Parameters[1].Estimate);
Assert.IsTrue(result.Slope.Value == result.Parameters.Best[1]);
Assert.IsTrue(TestUtilities.IsNearlyEqual(result.Slope.Uncertainty, Math.Sqrt(result.Parameters.Covariance[1, 1])));
Assert.IsTrue(TestUtilities.IsNearlyEqual(result.R.Statistic, sample.CorrelationCoefficient));
// record best fit parameters
double a = result.Parameters.Best[0];
double b = result.Parameters.Best[1];
pSample.Add(a, b);
// record estimated covariances
caa += result.Parameters.Covariance[0, 0];
cbb += result.Parameters.Covariance[1, 1];
cab += result.Parameters.Covariance[0, 1];
UncertainValue yPredict = result.Predict(x0);
ySample.Add(yPredict.Value);
ySigma += yPredict.Uncertainty;
double SST = 0.0;
foreach (double y in sample.Y)
{
SST += MoreMath.Sqr(y - sample.Y.Mean);
}
Assert.IsTrue(TestUtilities.IsNearlyEqual(SST, result.Anova.Total.SumOfSquares));
double SSR = 0.0;
foreach (double z in result.Residuals)
{
SSR += z * z;
}
Assert.IsTrue(TestUtilities.IsNearlyEqual(SSR, result.Anova.Residual.SumOfSquares));
}
caa /= pSample.Count;
cbb /= pSample.Count;
cab /= pSample.Count;
ySigma /= pSample.Count;
// check that mean parameter estimates are what they should be: the underlying population parameters
Assert.IsTrue(pSample.X.PopulationMean.ConfidenceInterval(0.95).ClosedContains(a0));
Assert.IsTrue(pSample.Y.PopulationMean.ConfidenceInterval(0.95).ClosedContains(b0));
Console.WriteLine("{0} {1}", caa, pSample.X.PopulationVariance);
Console.WriteLine("{0} {1}", cbb, pSample.Y.PopulationVariance);
// check that parameter covarainces are what they should be: the reported covariance estimates
Assert.IsTrue(pSample.X.PopulationVariance.ConfidenceInterval(0.95).ClosedContains(caa));
Assert.IsTrue(pSample.Y.PopulationVariance.ConfidenceInterval(0.95).ClosedContains(cbb));
Assert.IsTrue(pSample.PopulationCovariance.ConfidenceInterval(0.95).ClosedContains(cab));
// Check that the predicted ys conform to the model and the asserted uncertainty.
Assert.IsTrue(ySample.PopulationMean.ConfidenceInterval(0.95).ClosedContains(a0 + x0 * b0));
//Assert.IsTrue(ySample.PopulationStandardDeviation.ConfidenceInterval(0.95).ClosedContains(ySigma));
}
