/// <summary>
/// Determines the parameters of the Wald distribution that best fits a sample.
/// </summary>
/// <param name="sample">The sample to fit.</param>
/// <returns>The fit.</returns>
/// <remarks>
/// <para>The returned fit parameters are the <see cref="Mean"/> and <see cref="ShapeParameter"/>, in that order.
/// These are the same parameters, in the same order, that are required by the <see cref="WaldDistribution(double,double)"/> constructor to
/// specify a new Wald distribution.</para>
/// </remarks>
/// <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 FitResult FitToSample(Sample sample)
{
if (sample == null)
{
throw new ArgumentNullException("sample");
}
if (sample.Count < 3)
{
throw new InsufficientDataException();
}
// best fit mu = <x>
double mu = sample.Mean;
// best fit lambda is 1/lambda = <1/x - 1/mu>
double lambda = 0.0;
foreach (double value in sample)
{
if (value <= 0.0)
{
throw new InvalidOperationException();
}
lambda += (1.0 / value - 1.0 / mu);
}
lambda = sample.Count / lambda;
// correct lambda estimate for its bias in the non-asymptotic regime
lambda = lambda * (1.0 - 3.0 / sample.Count);
// variances are expressible in closed form, covariance is zero
double v_mu_mu = MoreMath.Pow(mu, 3) / lambda / sample.Count;
double v_lambda_lambda = 2.0 * lambda * lambda / sample.Count;
double v_mu_lambda = 0.0;
Distribution dist = new WaldDistribution(mu, lambda);
TestResult test = sample.KolmogorovSmirnovTest(dist);
return(new FitResult(mu, Math.Sqrt(v_mu_mu), lambda, Math.Sqrt(v_lambda_lambda), v_mu_lambda, test));
}
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);
}
public void WaldFitUncertainties()
{
WaldDistribution wald = new WaldDistribution(3.5, 2.5);
Random rng = new Random(314159);
BivariateSample P = new BivariateSample();
double cmm = 0.0;
double css = 0.0;
double cms = 0.0;
for (int i = 0; i < 50; i++) {
Sample s = new Sample();
for (int j = 0; j < 50; j++) {
s.Add(wald.GetRandomValue(rng));
}
FitResult r = WaldDistribution.FitToSample(s);
P.Add(r.Parameter(0).Value, r.Parameter(1).Value);
cmm += r.Covariance(0, 0);
css += r.Covariance(1, 1);
cms += r.Covariance(0, 1);
}
cmm /= P.Count;
css /= P.Count;
cms /= P.Count;
Console.WriteLine("{0} {1}", P.X.PopulationMean, P.Y.PopulationMean);
Assert.IsTrue(P.X.PopulationMean.ConfidenceInterval(0.95).ClosedContains(wald.Mean));
Assert.IsTrue(P.Y.PopulationMean.ConfidenceInterval(0.95).ClosedContains(wald.ShapeParameter));
// the ML shape parameter estimate appears to be asymptoticly unbiased, as it must be according to ML fit theory,
// but detectably upward biased for small n. we now correct for this.
Console.WriteLine("{0} {1} {2}", P.X.PopulationVariance, P.Y.PopulationVariance, P.PopulationCovariance);
Console.WriteLine("{0} {1} {2}", cmm, css, cms);
Assert.IsTrue(P.X.PopulationVariance.ConfidenceInterval(0.95).ClosedContains(cmm));
Assert.IsTrue(P.Y.PopulationVariance.ConfidenceInterval(0.95).ClosedContains(css));
Assert.IsTrue(P.PopulationCovariance.ConfidenceInterval(0.95).ClosedContains(cms));
}
/// <summary>
/// Determines the parameters of the Wald distribution that best fits a sample.
/// </summary>
/// <param name="sample">The sample to fit.</param>
/// <returns>The fit.</returns>
/// <remarks>
/// <para>The returned fit parameters are the <see cref="Mean"/> and <see cref="Shape"/>, in that order.
/// These are the same parameters, in the same order, that are required by the <see cref="WaldDistribution(double,double)"/> constructor to
/// specify a new Wald distribution.</para>
/// </remarks>
/// <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 FitResult FitToSample(Sample 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})>
double mu = sample.Mean;
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 = (sample.Count - 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 / sample.Count;
double v_lambda_lambda = 2.0 * lambda * lambda / (sample.Count - 5);
double v_mu_lambda = 0.0;
ContinuousDistribution dist = new WaldDistribution(mu, lambda);
TestResult test = sample.KolmogorovSmirnovTest(dist);
return(new FitResult(mu, Math.Sqrt(v_mu_mu), lambda, Math.Sqrt(v_lambda_lambda), v_mu_lambda, test));
}
/// <summary>
/// Determines the parameters of the Wald distribution that best fits a sample.
/// </summary>
/// <param name="sample">The sample to fit.</param>
/// <returns>The fit.</returns>
/// <remarks>
/// <para>The returned fit parameters are the <see cref="Mean"/> and <see cref="ShapeParameter"/>, in that order.
/// These are the same parameters, in the same order, that are required by the <see cref="WaldDistribution(double,double)"/> constructor to
/// specify a new Wald distribution.</para>
/// </remarks>
/// <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 FitResult FitToSample(Sample sample)
{
if (sample == null) throw new ArgumentNullException("sample");
if (sample.Count < 3) throw new InsufficientDataException();
// best fit mu = <x>
double mu = sample.Mean;
// best fit lambda is 1/lambda = <1/x - 1/mu>
double lambda = 0.0;
foreach (double value in sample) {
if (value <= 0.0) throw new InvalidOperationException();
lambda += (1.0 / value - 1.0 / mu);
}
lambda = sample.Count / lambda;
// correct lambda estimate for its bias in the non-asymptotic regime
lambda = lambda * (1.0 - 3.0 / sample.Count);
// variances are expressible in closed form, covariance is zero
double v_mu_mu = MoreMath.Pow(mu, 3) / lambda / sample.Count;
double v_lambda_lambda = 2.0 * lambda * lambda / sample.Count;
double v_mu_lambda = 0.0;
Distribution dist = new WaldDistribution(mu, lambda);
TestResult test = sample.KolmogorovSmirnovTest(dist);
return (new FitResult(mu, Math.Sqrt(v_mu_mu), lambda, Math.Sqrt(v_lambda_lambda), v_mu_lambda, test));
}
