public void FitDataToPolynomialTest()
{
Interval r = Interval.FromEndpoints(-10.0, 10.0);
double[] c = new double[] { 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 };
Func <double, double> fv = delegate(double x) {
double f = 0.0;
for (int i = c.Length - 1; i >= 0; i--)
{
f = f * x + c[i];
}
return(f);
};
Func <double, double> fu = delegate(double x) {
return(1.0 + 0.5 * Math.Cos(x));
};
UncertainMeasurementSample set = CreateDataSet(r, fv, fu, 50);
Assert.IsTrue(set.Count == 50);
// fit to an appropriate polynomial
FitResult poly = set.FitToPolynomial(5);
// the coefficients should match
for (int i = 0; i < poly.Dimension; i++)
{
Assert.IsTrue(poly.Parameter(i).ConfidenceInterval(0.95).ClosedContains(c[i]));
}
// the fit should be good
Console.WriteLine(poly.GoodnessOfFit.LeftProbability);
Assert.IsTrue(poly.GoodnessOfFit.LeftProbability < 0.95);
// fit to a lower order polynomial
FitResult low = set.FitToPolynomial(4);
// the fit should be bad
Assert.IsTrue(low.GoodnessOfFit.Statistic > poly.GoodnessOfFit.Statistic);
Assert.IsTrue(low.GoodnessOfFit.LeftProbability > 0.95);
// fit to a higher order polynomial
FitResult high = set.FitToPolynomial(6);
// the higher order coefficients should be compatible with zero
Assert.IsTrue(high.Parameter(6).ConfidenceInterval(0.95).ClosedContains(0.0));
// the fit should be better, but not too much better
Assert.IsTrue(high.GoodnessOfFit.Statistic < poly.GoodnessOfFit.Statistic);
}
public void FitDataToPolynomialChiSquaredTest()
{
// we want to make sure that the chi^2 values we are producing from polynomial fits are distributed as expected
// create a sample to hold chi^2 values
Sample chis = new Sample();
// define a model
Interval r = Interval.FromEndpoints(-5.0, 15.0);
Func <double, double> fv = delegate(double x) {
return(1.0 * x - 2.0 * x * x);
};
Func <double, double> fu = delegate(double x) {
return(1.0 + 0.5 * Math.Sin(x));
};
// draw 50 data sets from the model and fit year
// store the resulting chi^2 value in the chi^2 set
for (int i = 0; i < 50; i++)
{
UncertainMeasurementSample xs = CreateDataSet(r, fv, fu, 10, i);
FitResult fit = xs.FitToPolynomial(2);
double chi = fit.GoodnessOfFit.Statistic;
chis.Add(chi);
}
// sanity check the sample
Assert.IsTrue(chis.Count == 50);
// test whether the chi^2 values are distributed as expected
ContinuousDistribution chiDistribution = new ChiSquaredDistribution(7);
TestResult ks = chis.KolmogorovSmirnovTest(chiDistribution);
Assert.IsTrue(ks.LeftProbability < 0.95);
}
public void Bug6162()
{
// When UncertianMeasurementSample.FitToPolynomial used Cholesky inversion of (A^T A), this inversion
// would fail when roundoff errors would made the matrix non-positive-definite. We have now changed
// to QR decomposition, which is more robust.
//real data
double[] X_axis = new double[] { 40270.65625, 40270.6569444444, 40270.6576388888, 40270.6583333332, 40270.6590277776,
40270.659722222, 40270.6604166669, 40270.6611111113, 40270.6618055557, 40270.6625000001 };
double[] Y_axis = new double[] { 246.824996948242, 246.850006103516, 245.875, 246.225006103516, 246.975006103516,
247.024993896484, 246.949996948242, 246.875, 247.5, 247.100006103516 };
UncertainMeasurementSample DataSet = new UncertainMeasurementSample();
for (int i = 0; i < 10; i++)
{
DataSet.Add(X_axis[i], Y_axis[i], 1);
}
UncertainMeasurementFitResult DataFit = DataSet.FitToPolynomial(3);
BivariateSample bs = new BivariateSample();
for (int i = 0; i < 10; i++)
{
bs.Add(X_axis[i], Y_axis[i]);
}
PolynomialRegressionResult bsFit = bs.PolynomialRegression(3);
foreach (Parameter p in bsFit.Parameters)
{
Console.WriteLine(p);
}
}
public void FitToFunctionPolynomialCompatibilityTest()
{
// specify a cubic function
Interval r = Interval.FromEndpoints(-10.0, 10.0);
Func <double, double> fv = delegate(double x) {
return(0.0 - 1.0 * x + 2.0 * x * x - 3.0 * x * x * x);
};
Func <double, double> fu = delegate(double x) {
return(1.0 + 0.5 * Math.Cos(x));
};
// create a data set from it
UncertainMeasurementSample set = CreateDataSet(r, fv, fu, 60);
// fit it to a cubic polynomial
UncertainMeasurementFitResult pFit = set.FitToPolynomial(3);
// fit it to a cubic polynomial
Func <double[], double, double> ff = delegate(double[] p, double x) {
return(p[0] + p[1] * x + p[2] * x * x + p[3] * x * x * x);
};
UncertainMeasurementFitResult fFit = set.FitToFunction(ff, new double[] { 0, 0, 0, 0 });
// dimension
Assert.IsTrue(pFit.Parameters.Count == fFit.Parameters.Count);
// chi squared
Assert.IsTrue(TestUtilities.IsNearlyEqual(pFit.GoodnessOfFit.Statistic, fFit.GoodnessOfFit.Statistic, Math.Sqrt(TestUtilities.TargetPrecision)));
// don't demand super-high precision agreement of parameters and covariance matrix
// parameters
Assert.IsTrue(TestUtilities.IsNearlyEqual(pFit.Parameters.ValuesVector, fFit.Parameters.ValuesVector, Math.Pow(TestUtilities.TargetPrecision, 0.3)));
// covariance
Assert.IsTrue(TestUtilities.IsNearlyEqual(pFit.Parameters.CovarianceMatrix, fFit.Parameters.CovarianceMatrix, Math.Pow(TestUtilities.TargetPrecision, 0.3)));
}
public void FitDataToPolynomialTest()
{
Interval r = Interval.FromEndpoints(-10.0, 10.0);
Polynomial p = Polynomial.FromCoefficients(1.0, -2.0, 3.0, -4.0, 5.0, -6.0);
Func <double, double> fu = delegate(double x) {
return(1.0 + 0.5 * Math.Cos(x));
};
UncertainMeasurementSample set = CreateDataSet(r, p.Evaluate, fu, 50);
Assert.IsTrue(set.Count == 50);
// fit to an appropriate polynomial
UncertainMeasurementFitResult poly = set.FitToPolynomial(5);
// the coefficients should match
for (int i = 0; i < poly.Parameters.Count; i++)
{
Assert.IsTrue(poly.Parameters[i].Estimate.ConfidenceInterval(0.95).ClosedContains(p.Coefficient(i)));
}
// the fit should be good
Assert.IsTrue(poly.GoodnessOfFit.Probability > 0.05);
// fit to a lower order polynomial
UncertainMeasurementFitResult low = set.FitToPolynomial(4);
// the fit should be bad
Assert.IsTrue(low.GoodnessOfFit.Statistic > poly.GoodnessOfFit.Statistic);
Assert.IsTrue(low.GoodnessOfFit.Probability < 0.05);
// fit to a higher order polynomial
UncertainMeasurementFitResult high = set.FitToPolynomial(6);
// the higher order coefficients should be compatible with zero
Assert.IsTrue(high.Parameters[6].Estimate.ConfidenceInterval(0.95).ClosedContains(0.0));
// the fit should be better, but not too much better
Assert.IsTrue(high.GoodnessOfFit.Statistic < poly.GoodnessOfFit.Statistic);
}
public void FitDataToPolynomialUncertaintiesTest()
{
// make sure the reported uncertainties it fit parameters really represent their standard deviation,
// and that the reported off-diagonal elements really represent their correlations
double[] xs = TestUtilities.GenerateUniformRealValues(-1.0, 2.0, 10);
Func <double, double> fv = delegate(double x) {
return(0.0 + 1.0 * x + 2.0 * x * x);
};
Func <double, double> fu = delegate(double x) {
return(0.5);
};
// keep track of best-fit parameters and claimed parameter covariances
MultivariateSample sample = new MultivariateSample(3);
// generate 50 small data sets and fit each
UncertainMeasurementFitResult[] fits = new UncertainMeasurementFitResult[50];
for (int i = 0; i < fits.Length; i++)
{
UncertainMeasurementSample set = CreateDataSet(xs, fv, fu, 314159 + i);
fits[i] = set.FitToPolynomial(2);
sample.Add(fits[i].Parameters.ValuesVector);
}
// check that parameters agree
for (int i = 0; i < 3; i++)
{
Console.WriteLine(sample.Column(i).PopulationMean);
}
// for each parameter, verify that the standard deviation of the reported values agrees with the (average) reported uncertainty
double[] pMeans = new double[3];
for (int i = 0; i <= 2; i++)
{
Sample values = new Sample();
Sample uncertainties = new Sample();
for (int j = 0; j < fits.Length; j++)
{
UncertainValue p = fits[j].Parameters[i].Estimate;
values.Add(p.Value);
uncertainties.Add(p.Uncertainty);
}
pMeans[i] = values.Mean;
Assert.IsTrue(values.PopulationStandardDeviation.ConfidenceInterval(0.95).ClosedContains(uncertainties.Mean));
}
}
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 FitToFunctionPolynomialCompatibilityTest()
{
// specify a cubic function
Interval r = Interval.FromEndpoints(-10.0, 10.0);
Func <double, double> fv = delegate(double x) {
return(0.0 - 1.0 * x + 2.0 * x * x - 3.0 * x * x * x);
};
Func <double, double> fu = delegate(double x) {
return(1.0 + 0.5 * Math.Cos(x));
};
// create a data set from it
UncertainMeasurementSample set = CreateDataSet(r, fv, fu, 60);
// fit it to a cubic polynomial
FitResult pFit = set.FitToPolynomial(3);
// fit it to a cubic polynomaial
Func <double[], double, double> ff = delegate(double[] p, double x) {
return(p[0] + p[1] * x + p[2] * x * x + p[3] * x * x * x);
};
FitResult fFit = set.FitToFunction(ff, new double[] { 0, 0, 0, 0 });
// the fits should agree
Console.WriteLine("{0} ?= {1}", pFit.GoodnessOfFit.Statistic, fFit.GoodnessOfFit.Statistic);
for (int i = 0; i < pFit.Dimension; i++)
{
Console.WriteLine("{0} ?= {1}", pFit.Parameter(i), fFit.Parameter(i));
Assert.IsTrue(pFit.Parameter(i).ConfidenceInterval(0.01).ClosedContains(fFit.Parameter(i).Value));
}
// dimension
Assert.IsTrue(pFit.Dimension == fFit.Dimension);
// chi squared
Assert.IsTrue(TestUtilities.IsNearlyEqual(pFit.GoodnessOfFit.Statistic, fFit.GoodnessOfFit.Statistic, Math.Sqrt(TestUtilities.TargetPrecision)));
// don't demand super-high precision agreement of parameters and covariance matrix
// parameters
Assert.IsTrue(TestUtilities.IsNearlyEqual(pFit.Parameters, fFit.Parameters, Math.Pow(TestUtilities.TargetPrecision, 0.3)));
// covariance
Assert.IsTrue(TestUtilities.IsNearlyEqual(pFit.CovarianceMatrix, fFit.CovarianceMatrix, Math.Pow(TestUtilities.TargetPrecision, 0.3)));
}
public void FitDataToPolynomialUncertaintiesTest()
{
// make sure the reported uncertainties it fit parameters really represent their standard deviation,
// and that the reported off-diagonal elements really represent their correlations
double[] xs = TestUtilities.GenerateUniformRealValues(-1.0, 2.0, 10);
Func <double, double> fv = delegate(double x) {
return(0.0 + 1.0 * x + 2.0 * x * x);
};
Func <double, double> fu = delegate(double x) {
return(0.5);
};
// keep track of best-fit parameters and claimed parameter covariances
MultivariateSample sample = new MultivariateSample(3);
// generate 50 small data sets and fit each
FitResult[] fits = new FitResult[50];
for (int i = 0; i < fits.Length; i++)
{
//DataSet set = CreateDataSet(Interval.FromEndpoints(-1.0,2.0), fv, fu, 10, i);
UncertainMeasurementSample set = CreateDataSet(xs, fv, fu, 314159 + i);
/*
* foreach (DataPoint point in set) {
* Console.WriteLine(" i={0} x={1} y={2}", i, point.X, point.Y);
* }
*/
fits[i] = set.FitToPolynomial(2);
/*
* for (int j = 0; j < fits[i].Dimension; j++) {
* Console.WriteLine("p[{0}] = {1}", j, fits[i].Parameter(j));
* }
*/
for (int j = 0; j < fits[i].Dimension; j++)
{
sample.Add(fits[i].Parameters);
}
}
// check that parameters agree
for (int i = 0; i < 3; i++)
{
Console.WriteLine(sample.Column(i).PopulationMean);
}
//Assert.IsTrue(sample.PopulationMean(0).ConfidenceInterval(0.95).ClosedContains(0.0));
//Assert.IsTrue(sample.PopulationMean(1).ConfidenceInterval(0.95).ClosedContains(1.0));
//Assert.IsTrue(sample.PopulationMean(2).ConfidenceInterval(0.95).ClosedContains(2.0));
/*
* // check that the claimed covariances agree with the measured covariances
* for (int i = 0; i < 3; i++) {
* for (int j = 0; j < 3; j++) {
*
* Console.WriteLine("{0},{1} {2} {3}", i, j, sample.PopulationCovariance(i, j), C[i, j]);
*
* Assert.IsTrue(sample.PopulationCovariance(i, j).ConfidenceInterval(0.95).ClosedContains(C[i, j]));
* }
* }
*/
// for each parameter, verify that the standard devition of the reported values agrees with the (average) reported uncertainty
double[] pMeans = new double[3];
for (int i = 0; i <= 2; i++)
{
Console.WriteLine("paramater {0}", i);
Sample values = new Sample();
Sample uncertainties = new Sample();
for (int j = 0; j < fits.Length; j++)
{
UncertainValue p = fits[j].Parameter(i);
values.Add(p.Value);
uncertainties.Add(p.Uncertainty);
}
pMeans[i] = values.Mean;
Console.WriteLine("reported values mean={0}, standard deviation={1}", values.Mean, values.StandardDeviation);
Console.WriteLine("reported uncertainties mean={0}", uncertainties.Mean);
Assert.IsTrue(values.PopulationStandardDeviation.ConfidenceInterval(0.95).ClosedContains(uncertainties.Mean));
}
// for each parameter pair, verify that the covariances of the reported values agrees with the (average) reported covarance
/*
* for (int i = 0; i <= 2; i++) {
* for (int j = 0; j <= 2; j++) {
* // compute cov(i,j)
* double cov = 0.0;
* for (int k = 0; k < fits.Length; k++) {
* cov += (fits[k].Parameter(i).Value - pMeans[i]) * (fits[k].Parameter(j).Value - pMeans[j]);
* }
* cov = cov / fits.Length;
* // collect the reported covarainces
* Sample covs = new Sample();
* for (int k = 0; k < fits.Length; k++) {
* covs.Add(fits[k].Covariance(i, j));
* }
* Console.WriteLine("cov({0},{1}) computed={2} reported={3}", i, j, cov, covs.PopulationMean);
* // the computed covariance should agree with the (average) reported covariance
* // problem: we need to estimate the uncertainty in our covariance, but that isn't ever computed
* // note: covs just depend on x's, so the reported cov is the same for all fits
* // solution: for now, just assume it scales with 1/Sqrt(N); long term, do a multivariate fit
* //Assert.IsTrue((new UncertainValue(cov, Math.Abs(cov)* Math.Sqrt(2.0/fits.Length))).ConfidenceInterval(0.95).ClosedContains(covs.Mean));
* }
* }
*/
}
public void Bug6162()
{
// When UncertianMeasurementSample.FitToPolynomial used Cholesky inversion of (A^T A), this inversion
// would fail when roundoff errors would made the matrix non-positive-definite. We have now changed
// to QR decomposition, which is more robust.
//real data
double[] X_axis = new double[] { 40270.65625, 40270.6569444444, 40270.6576388888, 40270.6583333332, 40270.6590277776,
40270.659722222, 40270.6604166669, 40270.6611111113, 40270.6618055557, 40270.6625000001 };
double[] Y_axis = new double[] { 246.824996948242, 246.850006103516, 245.875, 246.225006103516, 246.975006103516,
247.024993896484, 246.949996948242, 246.875, 247.5, 247.100006103516 };
UncertainMeasurementSample DataSet = new UncertainMeasurementSample();
for (int i = 0; i < 10; i++) DataSet.Add(X_axis[i], Y_axis[i], 1);
//for (int i = 0; i < 10; i++) DataSet.Add(X_axis[i] - 40270.0, Y_axis[i] - 247.0, 1);
FitResult DataFit = DataSet.FitToPolynomial(3);
for (int i = 0; i < DataFit.Dimension; i++)
Console.WriteLine("a" + i.ToString() + " = " + DataFit.Parameter(i).Value);
BivariateSample bs = new BivariateSample();
for (int i = 0; i < 10; i++) bs.Add(X_axis[i], Y_axis[i]);
FitResult bsFit = bs.PolynomialRegression(3);
for (int i = 0; i < bsFit.Dimension; i++) Console.WriteLine(bsFit.Parameter(i));
}