public void WaldFit()
{
WaldDistribution wald = new WaldDistribution(3.5, 2.5);
FrameTable results = new FrameTable();
results.AddColumns <double>("Mean", "Shape", "MeanVariance", "ShapeVariance", "MeanShapeCovariance");
for (int i = 0; i < 128; i++)
{
Sample sample = SampleTest.CreateSample(wald, 16, i);
WaldFitResult result = WaldDistribution.FitToSample(sample);
Assert.IsTrue(result.Mean.Value == result.Parameters.ValuesVector[result.Parameters.IndexOf("Mean")]);
Assert.IsTrue(result.Shape.Value == result.Parameters.ValuesVector[result.Parameters.IndexOf("Shape")]);
Assert.IsTrue(TestUtilities.IsNearlyEqual(result.Parameters.VarianceOf("Mean"), MoreMath.Sqr(result.Mean.Uncertainty)));
Assert.IsTrue(TestUtilities.IsNearlyEqual(result.Parameters.VarianceOf("Shape"), MoreMath.Sqr(result.Shape.Uncertainty)));
results.AddRow(
result.Mean.Value, result.Shape.Value,
result.Parameters.VarianceOf("Mean"), result.Parameters.VarianceOf("Shape"), result.Parameters.CovarianceOf("Mean", "Shape")
);
}
Assert.IsTrue(results["Mean"].As <double>().PopulationMean().ConfidenceInterval(0.99).ClosedContains(wald.Mean));
Assert.IsTrue(results["Shape"].As <double>().PopulationMean().ConfidenceInterval(0.99).ClosedContains(wald.Shape));
Assert.IsTrue(results["Mean"].As <double>().PopulationVariance().ConfidenceInterval(0.99).ClosedContains(results["MeanVariance"].As <double>().Median()));
Assert.IsTrue(results["Shape"].As <double>().PopulationVariance().ConfidenceInterval(0.99).ClosedContains(results["ShapeVariance"].As <double>().Median()));
Assert.IsTrue(results["Mean"].As <double>().PopulationCovariance(results["Shape"].As <double>()).ConfidenceInterval(0.99).ClosedContains(results["MeanShapeCovariance"].As <double>().Median()));
}
public void WaldFit()
{
WaldDistribution wald = new WaldDistribution(3.5, 2.5);
BivariateSample parameters = new BivariateSample();
MultivariateSample variances = new MultivariateSample(3);
for (int i = 0; i < 128; i++)
{
Sample s = SampleTest.CreateSample(wald, 16, i);
FitResult r = WaldDistribution.FitToSample(s);
parameters.Add(r.Parameters[0], r.Parameters[1]);
variances.Add(r.Covariance(0, 0), r.Covariance(1, 1), r.Covariance(0, 1));
Assert.IsTrue(r.GoodnessOfFit.Probability > 0.01);
}
Assert.IsTrue(parameters.X.PopulationMean.ConfidenceInterval(0.99).ClosedContains(wald.Mean));
Assert.IsTrue(parameters.Y.PopulationMean.ConfidenceInterval(0.99).ClosedContains(wald.Shape));
Assert.IsTrue(parameters.X.PopulationVariance.ConfidenceInterval(0.99).ClosedContains(variances.Column(0).Median));
Assert.IsTrue(parameters.Y.PopulationVariance.ConfidenceInterval(0.99).ClosedContains(variances.Column(1).Median));
Assert.IsTrue(parameters.PopulationCovariance.ConfidenceInterval(0.99).ClosedContains(variances.Column(2).Median));
}
public void InverseGaussianSummation()
{
// X_i ~ IG(\mu,\lambda) \rightarrow \sum_{i=0}^{n} X_i ~ IG(n \mu, n^2 \lambda)
Random rng = new Random(1);
WaldDistribution d0 = new WaldDistribution(1.0, 2.0);
List <double> s = new List <double>();
for (int i = 0; i < 64; i++)
{
s.Add(d0.GetRandomValue(rng) + d0.GetRandomValue(rng) + d0.GetRandomValue(rng));
}
WaldDistribution d1 = new WaldDistribution(3.0 * 1.0, 9.0 * 2.0);
TestResult r = s.KolmogorovSmirnovTest(d1);
Assert.IsTrue(r.Probability > 0.05);
}
public void InverseGaussianSummation()
{
// X_i ~ IG(\mu,\lambda) \rightarrow \sum_{i=0}^{n} X_i ~ IG(n \mu, n^2 \lambda)
Random rng = new Random(0);
WaldDistribution d0 = new WaldDistribution(1.0, 2.0);
Sample s = new Sample();
for (int i = 0; i < 64; i++)
{
s.Add(d0.GetRandomValue(rng) + d0.GetRandomValue(rng) + d0.GetRandomValue(rng));
}
WaldDistribution d1 = new WaldDistribution(3.0 * 1.0, 9.0 * 2.0);
TestResult r = s.KolmogorovSmirnovTest(d1);
Console.WriteLine(r.LeftProbability);
}
/// <summary>
/// Finds the Wald distribution that best fits a sample.
/// </summary>
/// <param name="sample">The sample to fit.</param>
/// <returns>The fit.</returns>
/// <exception cref="ArgumentNullException"><paramref name="sample"/> is null.</exception>
/// <exception cref="InvalidOperationException"><paramref name="sample"/> contains non-positive values.</exception>
/// <exception cref="InsufficientDataException"><paramref name="sample"/> contains fewer than three values.</exception>
public static WaldFitResult FitToWald(this IReadOnlyList <double> sample)
{
if (sample == null)
{
throw new ArgumentNullException(nameof(sample));
}
if (sample.Count < 3)
{
throw new InsufficientDataException();
}
// For maximum likelihood estimation, take logs of pdfs and sum:
// \ln p = \frac{1}{2} \ln \lambda - \frac{1}{2} \ln (2\pi) - \frac{3}{2} \ln x
// - \frac{\lambda x}{2\mu^2} + \frac{\lambda}{\mu} - \frac{\lambda}{2x}
// \ln L = \sum_i p_i
// Take derivative wrt \mu
// \frac{\partial \ln L}{\partial \mu} = \sum_i \left[ \frac{\lambda x_i}{\mu^3} - \frac{\lambda}{\mu^2} \right]
// and set equal to zero to obtain
// \mu = \frac{1}{n} \sum_i x_i = <x>
// which agrees with method of moments.
// Take derivative wrt \lambda
// \frac{\partial \ln L}{\partial \lambda} = \sum_i \left[ \frac{1}{2 \lambda} -\frac{x_i}{2\mu^2} + \frac{1}{\mu} - \frac{1}{2 x_i} \right]
// Set equal to zero, plug in our expression for \mu, and solve for \lambda to get
// \frac{n}{\lambda} = \sum_i \left( \frac{1}{x_i} - \frac{1}{\mu} \right)
// i.e. \lambda^{-1} = <(x^{-1} - \mu^{-1})>
int n;
double mu;
ComputeMomentsUpToFirst(sample, out n, out mu);
double mui = 1.0 / mu;
double lambda = 0.0;
foreach (double value in sample)
{
if (value <= 0.0)
{
throw new InvalidOperationException();
}
lambda += (1.0 / value - mui);
}
lambda = (n - 3) / lambda;
// If x ~ IG(\mu, \lambda), then \sum_i x_i ~ IG(n \mu, n^2 \lambda), so \hat{\mu} ~ IG (\mu, n \lambda). This gives us
// not just the exact mean and variance of \hat{\mu}, but its entire distribution. Since its mean is \mu, \hat{\mu} is an
// unbiased estimator. And by the variance formula for IG, the variance of \hat{\mu} is \frac{\mu^3}{n \lambda}.
// Tweedie, "Statistical Properties of Inverse Gaussian Distributions" (http://projecteuclid.org/download/pdf_1/euclid.aoms/1177706964)
// showed that \frac{n \lambda}{\hat{\lambda}} ~ \chi^2_{n-1}. Since the mean of \chi^2_{k} is k, the MLE estimator of
// \frac{1}{\lambda} can be made unbiased by replacing n by (n-1). However, we are estimating \lambda, not \frac{1}{\lambda}.
// By the relation between chi squared and inverse chi squared distributions, \frac{\hat{\lambda}}{n \lambda} ~ I\chi^2_{n-1}.
// The mean of I\chi^2_{k} is \frac{1}{n-2}, so to get an unbiased estimator of \lambda, we need to replace n by (n-3). This is
// what we have done above. Furthermore, the variance of I\chi^2_{k} is \frac{2}{(k-2)^2 (k-4)}, so the variance of \hat{\lambda}
// is \frac{2 \lambda^2}{(n-5)}.
// We can also get covariances from the MLE approach. To get a curvature matrix, take additional derivatives
// \frac{\partial^2 \ln L}{\partial \mu^2} = \sum_i \left[ -\frac{3 \lambda x_i}{\mu^4} + \frac{2 \lambda}{\mu^3} \right]
// \frac{\partial^2 \ln L}{\partial \mu \partial \lambda} = \sum_i \left[ \frac{x_i}{\mu^3} - \frac{1}{\mu^2} \right]
// \frac{\partial^2 \ln L}{\partial \lambda^2} =\sum_i \left[ - \frac{1}{2 \lambda^2} \right]
// and substitutue in best-fit values of \mu and \lambda
// \frac{\partial^2 \ln L}{\partial \mu^2} = - \frac{n \lambda}{\mu^3}
// \frac{\partial^2 \ln L}{\partial \mu \partial \lambda} = 0
// \frac{\partial^2 \ln L}{\partial \lambda^2} = - \frac{n}{2 \lambda^2}
// Mixed derivative vanishes, so matrix is trivially invertible to obtain covariances. These results agree with the
// results from exact distributions in the asymptotic regime.
double v_mu_mu = mu * mu * mu / lambda / n;
double v_lambda_lambda = 2.0 * lambda * lambda / (n - 5);
//double v_mu_lambda = 0.0;
WaldDistribution dist = new WaldDistribution(mu, lambda);
TestResult test = sample.KolmogorovSmirnovTest(dist);
return(new WaldFitResult(mu, lambda, v_mu_mu, v_lambda_lambda, dist, test));
}
