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));
}