public void MultivariateLinearRegressionAgreement2()
{
// A multivariate linear regression with just one x-column should be the same as a bivariate linear regression.
double intercept = 1.0;
double slope = -2.0;
ContinuousDistribution yErrDist = new NormalDistribution(0.0, 3.0);
UniformDistribution xDist = new UniformDistribution(Interval.FromEndpoints(-2.0, 3.0));
Random rng = new Random(1111111);
MultivariateSample multi = new MultivariateSample("x", "y");
for (int i = 0; i < 10; i++)
{
double x = xDist.GetRandomValue(rng);
double y = intercept + slope * x + yErrDist.GetRandomValue(rng);
multi.Add(x, y);
}
// Old multi linear regression code.
MultiLinearRegressionResult result1 = multi.LinearRegression(1);
// Simple linear regression code.
LinearRegressionResult result2 = multi.TwoColumns(0, 1).LinearRegression();
Assert.IsTrue(TestUtilities.IsNearlyEqual(result1.Parameters["Intercept"].Estimate, result2.Parameters["Intercept"].Estimate));
// New multi linear regression code.
MultiLinearRegressionResult result3 = multi.Column(1).ToList().MultiLinearRegression(multi.Column(0).ToList());
Assert.IsTrue(TestUtilities.IsNearlyEqual(result1.Parameters["Intercept"].Estimate, result3.Parameters["Intercept"].Estimate));
}
public void MultivariateLinearRegressionVariances()
{
// define model y = a + b0 * x0 + b1 * x1 + noise
double a = -3.0;
double b0 = 2.0;
double b1 = -1.0;
ContinuousDistribution x0distribution = new LaplaceDistribution();
ContinuousDistribution x1distribution = new CauchyDistribution();
ContinuousDistribution eDistribution = new NormalDistribution(0.0, 4.0);
FrameTable data = new FrameTable();
data.AddColumns <double>("a", "da", "b0", "db0", "b1", "db1", "ab1Cov", "p", "dp");
// draw a sample from the model
Random rng = new Random(4);
for (int j = 0; j < 64; j++)
{
List <double> x0s = new List <double>();
List <double> x1s = new List <double>();
List <double> ys = new List <double>();
for (int i = 0; i < 16; i++)
{
double x0 = x0distribution.GetRandomValue(rng);
double x1 = x1distribution.GetRandomValue(rng);
double e = eDistribution.GetRandomValue(rng);
double y = a + b0 * x0 + b1 * x1 + e;
x0s.Add(x0);
x1s.Add(x1);
ys.Add(y);
}
// do a linear regression fit on the model
MultiLinearRegressionResult result = ys.MultiLinearRegression(
new Dictionary <string, IReadOnlyList <double> > {
{ "x0", x0s }, { "x1", x1s }
}
);
UncertainValue pp = result.Predict(-5.0, 6.0);
data.AddRow(
result.Intercept.Value, result.Intercept.Uncertainty,
result.CoefficientOf("x0").Value, result.CoefficientOf("x0").Uncertainty,
result.CoefficientOf("x1").Value, result.CoefficientOf("x1").Uncertainty,
result.Parameters.CovarianceOf("Intercept", "x1"),
pp.Value, pp.Uncertainty
);
}
// The estimated parameters should agree with the model that generated the data.
// The variances of the estimates should agree with the claimed variances
Assert.IsTrue(data["a"].As <double>().PopulationStandardDeviation().ConfidenceInterval(0.99).ClosedContains(data["da"].As <double>().Mean()));
Assert.IsTrue(data["b0"].As <double>().PopulationStandardDeviation().ConfidenceInterval(0.99).ClosedContains(data["db0"].As <double>().Mean()));
Assert.IsTrue(data["b1"].As <double>().PopulationStandardDeviation().ConfidenceInterval(0.99).ClosedContains(data["db1"].As <double>().Mean()));
Assert.IsTrue(data["a"].As <double>().PopulationCovariance(data["b1"].As <double>()).ConfidenceInterval(0.99).ClosedContains(data["ab1Cov"].As <double>().Mean()));
Assert.IsTrue(data["p"].As <double>().PopulationStandardDeviation().ConfidenceInterval(0.99).ClosedContains(data["dp"].As <double>().Median()));
}
public void MultivariateLinearRegressionSimple()
{
// define model y = a + b0 * x0 + b1 * x1 + noise
double a = 1.0;
double b0 = -2.0;
double b1 = 3.0;
ContinuousDistribution x0distribution = new CauchyDistribution(10.0, 5.0);
ContinuousDistribution x1distribution = new UniformDistribution(Interval.FromEndpoints(-10.0, 20.0));
ContinuousDistribution noise = new NormalDistribution(0.0, 10.0);
// draw a sample from the model
Random rng = new Random(1);
MultivariateSample sample = new MultivariateSample("x0", "x1", "y");
FrameTable table = new FrameTable();
table.AddColumns <double>("x0", "x1", "y");
for (int i = 0; i < 100; i++)
{
double x0 = x0distribution.GetRandomValue(rng);
double x1 = x1distribution.GetRandomValue(rng);
double eps = noise.GetRandomValue(rng);
double y = a + b0 * x0 + b1 * x1 + eps;
sample.Add(x0, x1, y);
table.AddRow(x0, x1, y);
}
// do a linear regression fit on the model
ParameterCollection oldResult = sample.LinearRegression(2).Parameters;
MultiLinearRegressionResult newResult = table["y"].As <double>().MultiLinearRegression(
table["x0"].As <double>(), table["x1"].As <double>()
);
// the result should have the appropriate dimension
Assert.IsTrue(oldResult.Count == 3);
Assert.IsTrue(newResult.Parameters.Count == 3);
// The parameters should match the model
Assert.IsTrue(oldResult[0].Estimate.ConfidenceInterval(0.90).ClosedContains(b0));
Assert.IsTrue(oldResult[1].Estimate.ConfidenceInterval(0.90).ClosedContains(b1));
Assert.IsTrue(oldResult[2].Estimate.ConfidenceInterval(0.90).ClosedContains(a));
Assert.IsTrue(newResult.CoefficientOf(0).ConfidenceInterval(0.99).ClosedContains(b0));
Assert.IsTrue(newResult.CoefficientOf("x1").ConfidenceInterval(0.99).ClosedContains(b1));
Assert.IsTrue(newResult.Intercept.ConfidenceInterval(0.99).ClosedContains(a));
// The residuals should be compatible with the model predictions
for (int i = 0; i < table.Rows.Count; i++)
{
FrameRow row = table.Rows[i];
double x0 = (double)row["x0"];
double x1 = (double)row["x1"];
double yp = newResult.Predict(x0, x1).Value;
double y = (double)row["y"];
Assert.IsTrue(TestUtilities.IsNearlyEqual(newResult.Residuals[i], y - yp));
}
}
public void TestMultivariateRegression()
{
double cz = 1.0;
double cx = 0.0;
double cy = 0.0;
Random rng = new Random(1001110000);
ContinuousDistribution xDistribution = new UniformDistribution(Interval.FromEndpoints(-4.0, 8.0));
ContinuousDistribution yDistribution = new UniformDistribution(Interval.FromEndpoints(-8.0, 4.0));
ContinuousDistribution eDistribution = new NormalDistribution();
Sample r2Sample = new Sample();
for (int i = 0; i < 500; i++)
{
MultivariateSample xyzSample = new MultivariateSample(3);
for (int k = 0; k < 12; k++)
{
double x = xDistribution.GetRandomValue(rng);
double y = yDistribution.GetRandomValue(rng);
double z = cx * x + cy * y + cz + eDistribution.GetRandomValue(rng);
xyzSample.Add(x, y, z);
}
MultiLinearRegressionResult fit = xyzSample.LinearRegression(2);
double fcx = fit.Parameters.Best[0];
double fcy = fit.Parameters.Best[1];
double fcz = fit.Parameters.Best[2];
double ss2 = 0.0;
double ss1 = 0.0;
foreach (double[] xyz in xyzSample)
{
ss2 += MoreMath.Sqr(xyz[2] - (fcx * xyz[0] + fcy * xyz[1] + fcz));
ss1 += MoreMath.Sqr(xyz[2] - xyzSample.Column(2).Mean);
}
double r2 = 1.0 - ss2 / ss1;
r2Sample.Add(r2);
}
Console.WriteLine("{0} {1} {2} {3} {4}", r2Sample.Count, r2Sample.PopulationMean, r2Sample.StandardDeviation, r2Sample.Minimum, r2Sample.Maximum);
ContinuousDistribution r2Distribution = new BetaDistribution((3 - 1) / 2.0, (12 - 3) / 2.0);
//Distribution r2Distribution = new BetaDistribution((10 - 2) / 2.0, (2 - 1) / 2.0);
Console.WriteLine("{0} {1}", r2Distribution.Mean, r2Distribution.StandardDeviation);
TestResult ks = r2Sample.KolmogorovSmirnovTest(r2Distribution);
Console.WriteLine(ks.RightProbability);
Console.WriteLine(ks.Probability);
}
public void TestMultivariateRegression()
{
// Collect r^2 values from multivariate linear regressions.
double cz = 1.0;
double cx = 0.0;
double cy = 0.0;
Random rng = new Random(1001110000);
ContinuousDistribution xDistribution = new UniformDistribution(Interval.FromEndpoints(-4.0, 8.0));
ContinuousDistribution yDistribution = new UniformDistribution(Interval.FromEndpoints(-8.0, 4.0));
ContinuousDistribution eDistribution = new NormalDistribution();
List <double> r2Sample = new List <double>();
for (int i = 0; i < 500; i++)
{
MultivariateSample xyzSample = new MultivariateSample(3);
for (int k = 0; k < 12; k++)
{
double x = xDistribution.GetRandomValue(rng);
double y = yDistribution.GetRandomValue(rng);
double z = cx * x + cy * y + cz + eDistribution.GetRandomValue(rng);
xyzSample.Add(x, y, z);
}
MultiLinearRegressionResult fit = xyzSample.LinearRegression(2);
double fcx = fit.Parameters.ValuesVector[0];
double fcy = fit.Parameters.ValuesVector[1];
double fcz = fit.Parameters.ValuesVector[2];
r2Sample.Add(fit.RSquared);
}
// r^2 values should be distributed as expected.
ContinuousDistribution r2Distribution = new BetaDistribution((3 - 1) / 2.0, (12 - 3) / 2.0);
TestResult ks = r2Sample.KolmogorovSmirnovTest(r2Distribution);
Assert.IsTrue(ks.Probability > 0.05);
}
