public void NormalFitCovariances()
{
NormalDistribution N = new NormalDistribution(-1.0, 2.0);
// Create a bivariate sample to hold our fitted best mu and sigma values
// so we can determine their covariance as well as their means and variances
BivariateSample parameters = new BivariateSample();
MultivariateSample covariances = new MultivariateSample(3);
// A bunch of times, create a normal sample
for (int i = 0; i < 128; i++)
{
// We use small samples so the variation in mu and sigma will be more substantial.
Sample s = TestUtilities.CreateSample(N, 8, i);
// Fit each sample to a normal distribution
FitResult fit = NormalDistribution.FitToSample(s);
// and record the mu and sigma values from the fit into our bivariate sample
parameters.Add(fit.Parameter(0).Value, fit.Parameter(1).Value);
// also record the claimed covariances among these parameters
covariances.Add(fit.Covariance(0, 0), fit.Covariance(1, 1), fit.Covariance(0, 1));
}
// the mean fit values should agree with the population distribution
Assert.IsTrue(parameters.X.PopulationMean.ConfidenceInterval(0.95).ClosedContains(N.Mean));
Assert.IsTrue(parameters.Y.PopulationMean.ConfidenceInterval(0.95).ClosedContains(N.StandardDeviation));
// but also the covariances of those fit values should agree with the claimed covariances
Assert.IsTrue(parameters.X.PopulationVariance.ConfidenceInterval(0.95).ClosedContains(covariances.Column(0).Mean));
Assert.IsTrue(parameters.Y.PopulationVariance.ConfidenceInterval(0.95).ClosedContains(covariances.Column(1).Mean));
Assert.IsTrue(parameters.PopulationCovariance.ConfidenceInterval(0.95).ClosedContains(covariances.Column(2).Mean));
}
public void FitDataToLinearFunctionTest()
{
// create a data set from a linear combination of sine and cosine
Interval r = Interval.FromEndpoints(-4.0, 6.0);
double[] c = new double[] { 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 };
Func <double, double> fv = delegate(double x) {
return(2.0 * Math.Cos(x) + 1.0 * Math.Sin(x));
};
Func <double, double> fu = delegate(double x) {
return(0.1 + 0.1 * Math.Abs(x));
};
UncertainMeasurementSample set = CreateDataSet(r, fv, fu, 20, 2);
// fit the data set to a linear combination of sine and cosine
Func <double, double>[] fs = new Func <double, double>[]
{ delegate(double x) { return(Math.Cos(x)); }, delegate(double x) { return(Math.Sin(x)); } };
FitResult result = set.FitToLinearFunction(fs);
// the fit should be right right dimension
Assert.IsTrue(result.Dimension == 2);
// the coefficients should match
Console.WriteLine(result.Parameter(0));
Console.WriteLine(result.Parameter(1));
Assert.IsTrue(result.Parameter(0).ConfidenceInterval(0.95).ClosedContains(2.0));
Assert.IsTrue(result.Parameter(1).ConfidenceInterval(0.95).ClosedContains(1.0));
// diagonal covarainces should match errors
Assert.IsTrue(TestUtilities.IsNearlyEqual(Math.Sqrt(result.Covariance(0, 0)), result.Parameter(0).Uncertainty));
Assert.IsTrue(TestUtilities.IsNearlyEqual(Math.Sqrt(result.Covariance(1, 1)), result.Parameter(1).Uncertainty));
}
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 GumbelFit()
{
GumbelDistribution d = new GumbelDistribution(-1.0, 2.0);
MultivariateSample parameters = new MultivariateSample(2);
MultivariateSample variances = new MultivariateSample(3);
// Do a bunch of fits, record reported parameters an variances
for (int i = 0; i < 32; i++)
{
Sample s = SampleTest.CreateSample(d, 64, i);
FitResult r = GumbelDistribution.FitToSample(s);
parameters.Add(r.Parameters);
variances.Add(r.Covariance(0, 0), r.Covariance(1, 1), r.Covariance(0, 1));
Assert.IsTrue(r.GoodnessOfFit.Probability > 0.01);
}
// The reported parameters should agree with the underlying parameters
Assert.IsTrue(parameters.Column(0).PopulationMean.ConfidenceInterval(0.99).ClosedContains(d.Location));
Assert.IsTrue(parameters.Column(1).PopulationMean.ConfidenceInterval(0.99).ClosedContains(d.Scale));
// The reported covariances should agree with the observed covariances
Assert.IsTrue(parameters.Column(0).PopulationVariance.ConfidenceInterval(0.99).ClosedContains(variances.Column(0).Mean));
Assert.IsTrue(parameters.Column(1).PopulationVariance.ConfidenceInterval(0.99).ClosedContains(variances.Column(1).Mean));
Assert.IsTrue(parameters.TwoColumns(0, 1).PopulationCovariance.ConfidenceInterval(0.99).ClosedContains(variances.Column(2).Mean));
}
public void BivariateNonlinearFit()
{
// 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);
MultivariateSample parameters = new MultivariateSample("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);
}
FitResult fit = sample.NonlinearRegression(
(IList <double> p, double x) => p[0] * Math.Pow(x, p[1]),
new double[] { 1.0, 1.0 }
);
parameters.Add(fit.Parameters);
covariances.Add(fit.Covariance(0, 0), fit.Covariance(1, 1), fit.Covariance(0, 1));
}
Assert.IsTrue(parameters.Column(0).PopulationMean.ConfidenceInterval(0.99).ClosedContains(a));
Assert.IsTrue(parameters.Column(1).PopulationMean.ConfidenceInterval(0.99).ClosedContains(b));
Assert.IsTrue(parameters.Column(0).PopulationVariance.ConfidenceInterval(0.99).ClosedContains(covariances.Column(0).Mean));
Assert.IsTrue(parameters.Column(1).PopulationVariance.ConfidenceInterval(0.99).ClosedContains(covariances.Column(1).Mean));
Assert.IsTrue(parameters.TwoColumns(0, 1).PopulationCovariance.ConfidenceInterval(0.99).ClosedContains(covariances.Column(2).Mean));
}
public void RayleighFit()
{
RayleighDistribution rayleigh = new RayleighDistribution(3.2);
Sample parameter = new Sample();
Sample variance = new Sample();
for (int i = 0; i < 128; i++)
{
// We pick a quite-small sample, because we have a finite-n unbiased estimator.
Sample s = SampleTest.CreateSample(rayleigh, 8, i);
FitResult r = RayleighDistribution.FitToSample(s);
parameter.Add(r.Parameters[0]);
variance.Add(r.Covariance(0, 0));
Assert.IsTrue(r.GoodnessOfFit.Probability > 0.01);
}
Assert.IsTrue(parameter.PopulationMean.ConfidenceInterval(0.99).ClosedContains(rayleigh.Scale));
Assert.IsTrue(parameter.PopulationVariance.ConfidenceInterval(0.99).ClosedContains(variance.Median));
}
public void FitDataToLineTest()
{
Interval r = Interval.FromEndpoints(0.0, 10.0);
Func <double, double> fv = delegate(double x) {
return(2.0 * x - 1.0);
};
Func <double, double> fu = delegate(double x) {
return(1.0 + x);
};
UncertainMeasurementSample data = CreateDataSet(r, fv, fu, 20);
// sanity check the data set
Assert.IsTrue(data.Count == 20);
// fit to a line
FitResult line = data.FitToLine();
Assert.IsTrue(line.Dimension == 2);
Assert.IsTrue(line.Parameter(0).ConfidenceInterval(0.95).ClosedContains(-1.0));
Assert.IsTrue(line.Parameter(1).ConfidenceInterval(0.95).ClosedContains(2.0));
Assert.IsTrue(line.GoodnessOfFit.LeftProbability < 0.95);
// correlation coefficient should be related to covariance as expected
Assert.IsTrue(TestUtilities.IsNearlyEqual(line.CorrelationCoefficient(0, 1), line.Covariance(0, 1) / line.Parameter(0).Uncertainty / line.Parameter(1).Uncertainty));
// fit to a 1st order polynomial and make sure it agrees
FitResult poly = data.FitToPolynomial(1);
Assert.IsTrue(poly.Dimension == 2);
Assert.IsTrue(TestUtilities.IsNearlyEqual(poly.Parameters, line.Parameters));
Assert.IsTrue(TestUtilities.IsNearlyEqual(poly.CovarianceMatrix, line.CovarianceMatrix));
Assert.IsTrue(TestUtilities.IsNearlyEqual(poly.GoodnessOfFit.Statistic, line.GoodnessOfFit.Statistic));
Assert.IsTrue(TestUtilities.IsNearlyEqual(poly.GoodnessOfFit.LeftProbability, line.GoodnessOfFit.LeftProbability));
Assert.IsTrue(TestUtilities.IsNearlyEqual(poly.GoodnessOfFit.RightProbability, line.GoodnessOfFit.RightProbability));
// fit to a constant; the result should be poor
FitResult constant = data.FitToConstant();
Assert.IsTrue(constant.GoodnessOfFit.LeftProbability > 0.95);
}
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 LinearLogisticRegression()
{
// 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 = 1.0; double b0 = -1.0 / 2.0;
//double a0 = -0.5; double b0 = 2.0;
// 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;
// do 50 fits
for (int k = 0; k < 50; k++)
{
Console.WriteLine("k={0}", k);
// generate a synthetic data set
BivariateSample s = new BivariateSample();
for (int i = 0; i < 50; i++)
{
double x = 2.0 * rng.NextDouble() - 1.0;
double ez = Math.Exp(a0 + b0 * x);
double P = ez / (1.0 + ez);
if (rng.NextDouble() < P)
{
s.Add(x, 1.0);
}
else
{
s.Add(x, 0.0);
}
}
//if (k != 27) continue;
// do the regression
FitResult r = s.LinearLogisticRegression();
// record best fit parameters
double a = r.Parameter(0).Value;
double b = r.Parameter(1).Value;
ps.Add(a, b);
Console.WriteLine("{0}, {1}", a, b);
// record estimated covariances
caa += r.Covariance(0, 0);
cbb += r.Covariance(1, 1);
cab += r.Covariance(0, 1);
}
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));
// 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));
}
