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)); }