public void BivariatePolynomialRegression()
{
// do a set of polynomial regression fits
// make sure not only that the fit parameters are what they should be, but that their variances/covariances are as claimed
Random rng = new Random(271828);
// define logistic parameters
double[] a = new double[] { 0.0, -1.0, 2.0, -3.0 };
// keep track of sample of returned a and b fit parameters
MultivariateSample A = new MultivariateSample(a.Length);
// also keep track of returned covariance estimates
// since these vary slightly from fit to fit, we will average them
SymmetricMatrix C = new SymmetricMatrix(a.Length);
// also keep track of test statistics
Sample F = 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 CauchyDistribution();
Distribution nd = new NormalDistribution(0.0, 4.0);
// generate a synthetic data set
BivariateSample s = new BivariateSample();
for (int j = 0; j < 20; j++) {
double x = xd.GetRandomValue(rng);
double y = nd.GetRandomValue(rng);
for (int i = 0; i < a.Length; i++) {
y += a[i] * MoreMath.Pow(x, i);
}
s.Add(x, y);
}
// do the regression
FitResult r = s.PolynomialRegression(a.Length - 1);
ColumnVector ps = r.Parameters;
//Console.WriteLine("{0} {1} {2}", ps[0], ps[1], ps[2]);
// record best fit parameters
A.Add(ps);
// record estimated covariances
C += r.CovarianceMatrix;
// record the fit statistic
F.Add(r.GoodnessOfFit.Statistic);
//Console.WriteLine("F={0}", r.GoodnessOfFit.Statistic);
}
C = (1.0 / A.Count) * C; // allow matrix division by real numbers
// check that mean parameter estimates are what they should be: the underlying population parameters
for (int i = 0; i < A.Dimension; i++) {
Console.WriteLine("{0} {1}", A.Column(i).PopulationMean, a[i]);
Assert.IsTrue(A.Column(i).PopulationMean.ConfidenceInterval(0.95).ClosedContains(a[i]));
}
// check that parameter covarainces are what they should be: the reported covariance estimates
for (int i = 0; i < A.Dimension; i++) {
for (int j = i; j < A.Dimension; j++) {
Console.WriteLine("{0} {1} {2} {3}", i, j, C[i, j], A.TwoColumns(i, j).PopulationCovariance);
Assert.IsTrue(A.TwoColumns(i, j).PopulationCovariance.ConfidenceInterval(0.95).ClosedContains(C[i, j]));
}
}
// check that F is distributed as it should be
//Console.WriteLine(fs.KolmogorovSmirnovTest(new FisherDistribution(2, 48)).LeftProbability);
}
public void MultivariateLinearRegressionAgreement()
{
Random rng = new Random(1);
MultivariateSample SA = new MultivariateSample(2);
for (int i = 0; i < 10; i++) {
SA.Add(rng.NextDouble(), rng.NextDouble());
}
FitResult RA = SA.LinearRegression(0);
ColumnVector PA = RA.Parameters;
SymmetricMatrix CA = RA.CovarianceMatrix;
MultivariateSample SB = SA.Columns(1, 0);
FitResult RB = SB.LinearRegression(1);
ColumnVector PB = RB.Parameters;
SymmetricMatrix CB = RB.CovarianceMatrix;
Assert.IsTrue(TestUtilities.IsNearlyEqual(PA[0], PB[1])); Assert.IsTrue(TestUtilities.IsNearlyEqual(PA[1], PB[0]));
Assert.IsTrue(TestUtilities.IsNearlyEqual(CA[0, 0], CB[1, 1])); Assert.IsTrue(TestUtilities.IsNearlyEqual(CA[0, 1], CB[1, 0])); Assert.IsTrue(TestUtilities.IsNearlyEqual(CA[1, 1], CB[0, 0]));
Assert.IsTrue(TestUtilities.IsNearlyEqual(RA.GoodnessOfFit.Statistic, RB.GoodnessOfFit.Statistic));
BivariateSample SC = SA.TwoColumns(1, 0);
FitResult RC = SC.LinearRegression();
ColumnVector PC = RC.Parameters;
SymmetricMatrix CC = RC.CovarianceMatrix;
Assert.IsTrue(TestUtilities.IsNearlyEqual(PA, PC));
Assert.IsTrue(TestUtilities.IsNearlyEqual(CA, CC));
Assert.IsTrue(TestUtilities.IsNearlyEqual(RA.GoodnessOfFit.Statistic, RC.GoodnessOfFit.Statistic));
}
public void BivariateLogisticRegression()
{
double[] c = new double[] { -0.1, 1.0 };
Random rng = new Random(1);
UniformDistribution pointDistribution = new UniformDistribution(Interval.FromEndpoints(-4.0, 4.0));
BivariateSample sample1 = new BivariateSample();
MultivariateSample sample2 = new MultivariateSample(2);
for (int k = 0; k < 1000; k++) {
double x = pointDistribution.GetRandomValue(rng);
double z = c[0] * x + c[1];
double ez = Math.Exp(z);
double p = ez / (1.0 + ez);
double y = (rng.NextDouble() < p) ? 1.0 : 0.0;
sample1.Add(x, y);
sample2.Add(x, y);
}
Console.WriteLine(sample1.Covariance / sample1.X.Variance / sample1.Y.Mean / (1.0 - sample1.Y.Mean));
Console.WriteLine(sample1.Covariance / sample1.X.Variance / sample1.Y.Variance);
FitResult result1 = sample1.LinearLogisticRegression();
FitResult result2 = sample2.TwoColumns(0, 1).LinearLogisticRegression();
FitResult result3 = sample2.LogisticLinearRegression(1);
for (int i = 0; i < result1.Dimension; i++) {
Console.WriteLine("{0} {1} {2}", i, result1.Parameter(i), result3.Parameter(i) );
}
}
public void MultivariateMoments()
{
// create a random sample
MultivariateSample M = new MultivariateSample(3);
Distribution d0 = new NormalDistribution();
Distribution d1 = new ExponentialDistribution();
Distribution d2 = new UniformDistribution();
Random rng = new Random(1);
int n = 10;
for (int i = 0; i < n; i++) {
M.Add(d0.GetRandomValue(rng), d1.GetRandomValue(rng), d2.GetRandomValue(rng));
}
// test that moments agree
for (int i = 0; i < 3; i++) {
int[] p = new int[3];
p[i] = 1;
Assert.IsTrue(TestUtilities.IsNearlyEqual(M.Column(i).Mean, M.Moment(p)));
p[i] = 2;
Assert.IsTrue(TestUtilities.IsNearlyEqual(M.Column(i).Variance, M.MomentAboutMean(p)));
for (int j = 0; j < i; j++) {
int[] q = new int[3];
q[i] = 1;
q[j] = 1;
Assert.IsTrue(TestUtilities.IsNearlyEqual(M.TwoColumns(i, j).Covariance, M.MomentAboutMean(q)));
}
}
}
