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));
}
}
