public MultivariateSample CreateMultivariateNormalSample(ColumnVector M, SymmetricMatrix C, int n)
{
int d = M.Dimension;
MultivariateSample S = new MultivariateSample(d);
SquareMatrix A = C.CholeskyDecomposition().SquareRootMatrix();
Random rng = new Random(1);
ContinuousDistribution normal = new NormalDistribution();
for (int i = 0; i < n; i++)
{
// create a vector of normal deviates
ColumnVector V = new ColumnVector(d);
for (int j = 0; j < d; j++)
{
double y = rng.NextDouble();
double z = normal.InverseLeftProbability(y);
V[j] = z;
}
// form the multivariate distributed vector
ColumnVector X = M + A * V;
// add it to the sample
S.Add(X);
}
return(S);
}
public static Tuple <Point, Vector> Compute(Point[] points)
{
Contract.Requires(points.Length >= 2);
Contract.Ensures(Contract.Result <Tuple <Point, Vector> >() != null);
if (points.Length == 2)
{
return(Tuple.Create(points[0], (points[1] - points[0]).Normalized()));
}
var avgX = points.Select(p => p.X).Average();
var avgY = points.Select(p => p.Y).Average();
var shifted = points.Select(p => p - new Vector(avgX, avgY));
var mvSample = new MultivariateSample(2);
foreach (var p in shifted)
{
mvSample.Add(p.X, p.Y);
}
var pca = mvSample.PrincipalComponentAnalysis();
var firstComponentVector = pca.Component(0).NormalizedVector();
return(Tuple.Create(
new Point(avgX, avgY),
new Vector(firstComponentVector[0], firstComponentVector[1])));
}
public void MultivariateMoments()
{
// create a random sample
MultivariateSample M = new MultivariateSample(3);
ContinuousDistribution d0 = new NormalDistribution();
ContinuousDistribution d1 = new ExponentialDistribution();
ContinuousDistribution 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.RawMoment(p)));
p[i] = 2;
Assert.IsTrue(TestUtilities.IsNearlyEqual(M.Column(i).Variance, M.CentralMoment(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.CentralMoment(q)));
}
}
}
public void MultivariateLinearRegressionBadInputTest()
{
// create a sample
MultivariateSample sample = new MultivariateSample(3);
sample.Add(1, 2, 3);
sample.Add(2, 3, 4);
// try to predict with too little data
try {
sample.LinearRegression(2);
Assert.IsTrue(false);
} catch (InvalidOperationException) {
Assert.IsTrue(true);
}
// add enough data
sample.Add(3, 4, 5);
sample.Add(4, 5, 6);
// try to predict a non-existent variable
try {
sample.LinearRegression(-1);
Assert.IsTrue(false);
} catch (ArgumentOutOfRangeException) {
Assert.IsTrue(true);
}
try {
sample.LinearRegression(3);
Assert.IsTrue(false);
} catch (ArgumentOutOfRangeException) {
Assert.IsTrue(true);
}
}
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 MultivariateLinearLogisticRegressionSimple()
{
// define model y = a + b0 * x0 + b1 * x1 + noise
double a = 1.0;
double b0 = -1.0 / 2.0;
double b1 = 1.0 / 3.0;
ContinuousDistribution x0distribution = new LaplaceDistribution();
ContinuousDistribution x1distribution = new NormalDistribution();
// draw a sample from the model
Random rng = new Random(1);
MultivariateSample old = new MultivariateSample("y", "x0", "x1");
FrameTable table = new FrameTable();
table.AddColumn <double>("x0");
table.AddColumn <double>("x1");
table.AddColumn <bool>("y");
for (int i = 0; i < 100; i++)
{
double x0 = x0distribution.GetRandomValue(rng);
double x1 = x1distribution.GetRandomValue(rng);
double t = a + b0 * x0 + b1 * x1;
double p = 1.0 / (1.0 + Math.Exp(-t));
bool y = (rng.NextDouble() < p);
old.Add(y ? 1.0 : 0.0, x0, x1);
table.AddRow(x0, x1, y);
}
// do a linear regression fit on the model
MultiLinearLogisticRegressionResult oldResult = old.LogisticLinearRegression(0);
MultiLinearLogisticRegressionResult newResult = table["y"].As <bool>().MultiLinearLogisticRegression(
table["x0"].As <double>(), table["x1"].As <double>()
);
// the result should have the appropriate dimension
Assert.IsTrue(newResult.Parameters.Count == 3);
// The parameters should match the model
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));
// Our predictions should be better than chance.
int correct = 0;
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 p = newResult.Predict(x0, x1).Value;
bool y = (bool)row["y"];
if ((y && p > 0.5) || (!y & p < 0.5))
{
correct++;
}
}
Assert.IsTrue(correct > 0.5 * table.Rows.Count);
}
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
NormalFitResult fit = NormalDistribution.FitToSample(s);
// and record the mu and sigma values from the fit into our bivariate sample
parameters.Add(fit.Mean.Value, fit.StandardDeviation.Value);
// also record the claimed covariances among these parameters
covariances.Add(fit.Parameters.CovarianceMatrix[0, 0], fit.Parameters.CovarianceMatrix[1, 1], fit.Parameters.CovarianceMatrix[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 MultivariateLinearRegressionNullDistribution()
{
int d = 4;
Random rng = new Random(1);
NormalDistribution n = new NormalDistribution();
Sample fs = new Sample();
for (int i = 0; i < 64; i++)
{
MultivariateSample ms = new MultivariateSample(d);
for (int j = 0; j < 8; j++)
{
double[] x = new double[d];
for (int k = 0; k < d; k++)
{
x[k] = n.GetRandomValue(rng);
}
ms.Add(x);
}
RegressionResult r = ms.LinearRegression(0);
fs.Add(r.F.Statistic);
}
// conduct a KS test to check that F follows the expected distribution
TestResult ks = fs.KolmogorovSmirnovTest(new FisherDistribution(3, 4));
Assert.IsTrue(ks.LeftProbability < 0.95);
}
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());
}
RegressionResult RA = SA.LinearRegression(0);
ColumnVector PA = RA.Parameters.Best;
SymmetricMatrix CA = RA.Parameters.Covariance;
MultivariateSample SB = SA.Columns(1, 0);
RegressionResult RB = SB.LinearRegression(1);
ColumnVector PB = RB.Parameters.Best;
SymmetricMatrix CB = RB.Parameters.Covariance;
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]));
BivariateSample SC = SA.TwoColumns(1, 0);
RegressionResult RC = SC.LinearRegression();
ColumnVector PC = RC.Parameters.Best;
SymmetricMatrix CC = RC.Parameters.Covariance;
Assert.IsTrue(TestUtilities.IsNearlyEqual(PA, PC));
Assert.IsTrue(TestUtilities.IsNearlyEqual(CA, CC));
}
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 and variances
for (int i = 0; i < 32; i++)
{
Sample s = SampleTest.CreateSample(d, 64, i);
GumbelFitResult r = GumbelDistribution.FitToSample(s);
parameters.Add(r.Location.Value, r.Scale.Value);
variances.Add(r.Parameters.CovarianceMatrix[0, 0], r.Parameters.CovarianceMatrix[1, 1], r.Parameters.CovarianceMatrix[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 MeansClustering()
{
// Re-create the mouse test
double[] x = new double[3];
double[] y = new double[3];
double[] s = new double[3];
x[0] = 0.25;
y[0] = 0.75;
s[0] = 0.1;
x[1] = 0.75;
y[1] = 0.75;
s[1] = 0.1;
x[2] = 0.5;
y[2] = 0.5;
s[2] = 0.2;
MultivariateSample points = new MultivariateSample(2);
Random rng = new Random(1);
NormalDistribution d = new NormalDistribution();
for (int i = 0; i < 100; i++)
{
int k = rng.Next(3);
points.Add(x[k] + s[k] * d.GetRandomValue(rng), y[k] + s[k] * d.GetRandomValue(rng));
}
MeansClusteringResult result = points.MeansClustering(3);
Assert.IsTrue(result.Count == 3);
Assert.IsTrue(result.Dimension == 2);
}
public static PCA2DResult Compute(IEnumerable <Point> points)
{
Contract.Requires(points != null);
Contract.Requires(points.Any() && points.Skip(1).Any()); // at least two points
var xAvg = points.Select(pnt => pnt.X).Average();
var yAvg = points.Select(pnt => pnt.Y).Average();
var shiftedPoints =
from pnt in points
select new Point(pnt.X - xAvg, pnt.Y - yAvg);
var mvSample = new MultivariateSample(2);
foreach (var pnt in shiftedPoints)
{
mvSample.Add(pnt.X, pnt.Y);
}
var pca = mvSample.PrincipalComponentAnalysis();
var first = pca.Component(0).NormalizedVector();
var second = pca.Component(1).NormalizedVector();
return(new PCA2DResult(
new Point(xAvg, yAvg),
new Vector(first[0], first[1]),
new Vector(second[0], second[1])));
}
public void FitDataToLineUncertaintyTest()
{
double[] xs = TestUtilities.GenerateUniformRealValues(0.0, 10.0, 10);
Func <double, double> fv = delegate(double x) {
return(2.0 * x - 1.0);
};
Func <double, double> fu = delegate(double x) {
return(1.0 + x);
};
MultivariateSample sample = new MultivariateSample(2);
SymmetricMatrix covariance = new SymmetricMatrix(2);
// create a bunch of small data sets
for (int i = 0; i < 100; i++)
{
UncertainMeasurementSample data = CreateDataSet(xs, fv, fu, i);
FitResult fit = data.FitToLine();
sample.Add(fit.Parameters);
covariance = fit.CovarianceMatrix;
// because it depends only on the x's and sigmas, the covariance is always the same
Console.WriteLine("cov_00 = {0}", covariance[0, 0]);
}
// the measured covariances should agree with the claimed covariances
//Assert.IsTrue(sample.PopulationCovariance(0,0).ConfidenceInterval(0.95).ClosedContains(covariance[0,0]));
//Assert.IsTrue(sample.PopulationCovariance(0,1).ConfidenceInterval(0.95).ClosedContains(covariance[0,1]));
//Assert.IsTrue(sample.PopulationCovariance(1,0).ConfidenceInterval(0.95).ClosedContains(covariance[1,0]));
//Assert.IsTrue(sample.PopulationCovariance(1,1).ConfidenceInterval(0.95).ClosedContains(covariance[1,1]));
}
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 MultivariateLinearRegressionTest()
{
// define model y = a + b0 * x0 + b1 * x1 + noise
double a = 1.0;
double b0 = -2.0;
double b1 = 3.0;
ContinuousDistribution noise = new NormalDistribution(0.0, 10.0);
// draw a sample from the model
Random rng = new Random(1);
MultivariateSample sample = new MultivariateSample(3);
for (int i = 0; i < 100; i++)
{
double x0 = -10.0 + 20.0 * rng.NextDouble();
double x1 = -10.0 + 20.0 * rng.NextDouble();
double eps = noise.InverseLeftProbability(rng.NextDouble());
double y = a + b0 * x0 + b1 * x1 + eps;
sample.Add(x0, x1, y);
}
// do a linear regression fit on the model
ParameterCollection result = sample.LinearRegression(2).Parameters;
// the result should have the appropriate dimension
Assert.IsTrue(result.Count == 3);
// the parameters should match the model
Assert.IsTrue(result[0].Estimate.ConfidenceInterval(0.90).ClosedContains(b0));
Assert.IsTrue(result[1].Estimate.ConfidenceInterval(0.90).ClosedContains(b1));
Assert.IsTrue(result[2].Estimate.ConfidenceInterval(0.90).ClosedContains(a));
}
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 Bug6391()
{
// this simple PCA caused a NonConvergenceException
var mvSample = new MultivariateSample(2);
mvSample.Add(0, 1);
mvSample.Add(0, -1);
var pca = mvSample.PrincipalComponentAnalysis();
}
private double GetTotalVariance(MultivariateSample sample)
{
double total = 0.0;
for (int i = 0; i < sample.Dimension; i++)
{
total += sample.Column(i).Variance;
}
return(total);
}
public void TimeSeriesFitAR1()
{
double alpha = 0.3;
double mu = 0.2;
double sigma = 0.4;
int n = 20;
// For our fit to AR(1), we have incorporated bias correction (at least
// for the most important parameter alpha), so we can do a small-n test.
MultivariateSample parameters = new MultivariateSample(3);
MultivariateSample covariances = new MultivariateSample(6);
Sample tests = new Sample();
for (int i = 0; i < 100; i++)
{
TimeSeries series = GenerateAR1TimeSeries(alpha, mu, sigma, n, i + 314159);
FitResult result = series.FitToAR1();
parameters.Add(result.Parameters);
covariances.Add(
result.CovarianceMatrix[0, 0],
result.CovarianceMatrix[1, 1],
result.CovarianceMatrix[2, 2],
result.CovarianceMatrix[0, 1],
result.CovarianceMatrix[0, 2],
result.CovarianceMatrix[1, 2]
);
tests.Add(result.GoodnessOfFit.Probability);
}
// Check that fit parameters agree with inputs
Assert.IsTrue(parameters.Column(0).PopulationMean.ConfidenceInterval(0.99).ClosedContains(alpha));
Assert.IsTrue(parameters.Column(1).PopulationMean.ConfidenceInterval(0.99).ClosedContains(mu));
Assert.IsTrue(parameters.Column(2).PopulationMean.ConfidenceInterval(0.99).ClosedContains(sigma));
// Check that reported variances agree with actual variances
Assert.IsTrue(parameters.Column(0).PopulationVariance.ConfidenceInterval(0.99).ClosedContains(covariances.Column(0).Median));
Assert.IsTrue(parameters.Column(1).PopulationVariance.ConfidenceInterval(0.99).ClosedContains(covariances.Column(1).Median));
Assert.IsTrue(parameters.Column(2).PopulationVariance.ConfidenceInterval(0.99).ClosedContains(covariances.Column(2).Median));
Assert.IsTrue(parameters.TwoColumns(0, 1).PopulationCovariance.ConfidenceInterval(0.99).ClosedContains(covariances.Column(3).Mean));
Assert.IsTrue(parameters.TwoColumns(0, 2).PopulationCovariance.ConfidenceInterval(0.99).ClosedContains(covariances.Column(4).Mean));
Assert.IsTrue(parameters.TwoColumns(1, 2).PopulationCovariance.ConfidenceInterval(0.99).ClosedContains(covariances.Column(5).Mean));
// For small n, the fitted alpha can vary considerably, and the formula for var(m) varies
// quite strongly with alpha, so the computed var(m) have a very long tail. This pushes the
// mean computed var(m) quite a bit higher than a typical value, so we use medians instead
// of means for our best guess for the predicted variance.
TestResult ks = tests.KolmogorovSmirnovTest(new UniformDistribution());
Assert.IsTrue(ks.Probability > 0.05);
}
public void TestMultivariateRegression()
{
double cz = 1.0;
double cx = 0.0;
double cy = 0.0;
Random rng = new Random(1001110000);
Distribution xDistribution = new UniformDistribution(Interval.FromEndpoints(-4.0, 8.0));
Distribution yDistribution = new UniformDistribution(Interval.FromEndpoints(-8.0, 4.0));
Distribution 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);
}
FitResult fit = xyzSample.LinearRegression(2);
double fcx = fit.Parameters[0];
double fcy = fit.Parameters[1];
double fcz = fit.Parameters[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);
Distribution 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 static MultivariateSample ToSample(this IEnumerable <Series> series)
{
var sl = series.ToArray();
var sample = new MultivariateSample(sl.Length);
var row = new double[sl.Length];
var count = series.Select(s => s.Count).Max();
for (int i = 0; i < sl.Length; i++)
{
row [i] = sl[i].As <double>()[i];
sample.Add(row);
}
return(sample);
}
public void TimeSeriesFitToMA1()
{
double beta = -0.2;
double mu = 0.4;
double sigma = 0.6;
int n = 100;
// If we are going to strictly test parameter values and variances,
// we can't pick n too small, because the formulas we use are only
// asymptotically unbiased.
MultivariateSample parameters = new MultivariateSample(3);
MultivariateSample covariances = new MultivariateSample(6);
Sample tests = new Sample("p");
for (int i = 0; i < 64; i++)
{
TimeSeries series = GenerateMA1TimeSeries(beta, mu, sigma, n, n * i + 314159);
Debug.Assert(series.Count == n);
MA1FitResult result = series.FitToMA1();
//Assert.IsTrue(result.Dimension == 3);
parameters.Add(result.Parameters.ValuesVector);
covariances.Add(
result.Parameters.CovarianceMatrix[0, 0],
result.Parameters.CovarianceMatrix[1, 1],
result.Parameters.CovarianceMatrix[2, 2],
result.Parameters.CovarianceMatrix[0, 1],
result.Parameters.CovarianceMatrix[0, 2],
result.Parameters.CovarianceMatrix[1, 2]
);
tests.Add(result.GoodnessOfFit.Probability);
}
Assert.IsTrue(parameters.Column(0).PopulationMean.ConfidenceInterval(0.99).ClosedContains(mu));;
Assert.IsTrue(parameters.Column(1).PopulationMean.ConfidenceInterval(0.99).ClosedContains(beta));
Assert.IsTrue(parameters.Column(2).PopulationMean.ConfidenceInterval(0.99).ClosedContains(sigma));
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.Column(2).PopulationVariance.ConfidenceInterval(0.99).ClosedContains(covariances.Column(2).Mean));
Assert.IsTrue(parameters.TwoColumns(0, 1).PopulationCovariance.ConfidenceInterval(0.99).ClosedContains(covariances.Column(3).Mean));
Assert.IsTrue(parameters.TwoColumns(0, 2).PopulationCovariance.ConfidenceInterval(0.99).ClosedContains(covariances.Column(4).Mean));
Assert.IsTrue(parameters.TwoColumns(1, 2).PopulationCovariance.ConfidenceInterval(0.99).ClosedContains(covariances.Column(5).Mean));
Assert.IsTrue(tests.KuiperTest(new UniformDistribution()).Probability > 0.01);
}
public List <double> Remove(List <double> Data, ref Stack <ITransformation> Transforms)
{
double yestHigh = 0.0;
double yestLow = 0.0;
double lastHigh = 0.0;
double lastLow = 0.0;
MultivariateSample mvS = new MultivariateSample(3);
for (int i = 0; i < Data.Count; i++)
{
yestHigh = lastHigh;
yestLow = lastLow;
lastHigh = Highs[i];
lastLow = Lows[i];
if (i > 0)
{
mvS.Add(yestHigh, yestLow, Closes[i]);
}
}
List <double> detrendedData = (List <double>)Utilities.DeepClone(Data);
double[] parameters = mvS.LinearRegression(2).Parameters();
for (int i = 0; i < Data.Count; i++)
{
double regression;
if (i > 0)
{
regression =
(parameters[0] * Highs[i - 1])
+ (parameters[1] * Lows[i - 1])
+ parameters[2];
}
else
{
regression = Closes[i];
}
detrendedData[i] -= regression;
}
Transforms.Push(new HighLowTransformation(yestHigh, yestLow, lastHigh, lastLow, parameters));
return(detrendedData);
}
public GeometricClass Classify(PointsSequence p)
{
var points = p.Points;
var xAvg = points.Select(pnt => pnt.X).Average();
var yAvg = points.Select(pnt => pnt.Y).Average();
var shiftedPoints =
from pnt in points
select new Point(pnt.X - xAvg, pnt.Y - yAvg);
var mvSample = new MultivariateSample(2);
foreach (var pnt in shiftedPoints)
{
mvSample.Add(pnt.X, pnt.Y);
}
var pca = mvSample.PrincipalComponentAnalysis();
var firstSize = pca.Component(0).ScaledVector().Norm();
var secondSize = pca.Component(1).ScaledVector().Norm();
var fraction = secondSize / firstSize;
if (fraction < pcaFractionThreshold)
{
return(GeometricClass.Line);
}
else
{
var conicEllipse = EllipseFitter.ConicFit(p.Points);
var parametricEllipse = EllipseFitter.Fit(p.Points);
var deviations =
from pnt in p.Points
select EllipseFitter.ComputeDeviation(conicEllipse, parametricEllipse, pnt);
var avgDeviation = deviations.Average();
if (avgDeviation < ellipseFitThreshold)
{
return(GeometricClass.Ellipse);
}
else
{
return(GeometricClass.Line);
}
}
}
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 BivariateNonlinearFitVariances()
{
// 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);
FrameTable parameters = new FrameTable();
parameters.AddColumns <double>("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);
}
NonlinearRegressionResult fit = sample.NonlinearRegression(
(IReadOnlyList <double> p, double x) => p[0] * Math.Pow(x, p[1]),
new double[] { 1.0, 1.0 }
);
parameters.AddRow(fit.Parameters.ValuesVector);
covariances.Add(fit.Parameters.CovarianceMatrix[0, 0], fit.Parameters.CovarianceMatrix[1, 1], fit.Parameters.CovarianceMatrix[0, 1]);
}
Assert.IsTrue(parameters["a"].As <double>().PopulationMean().ConfidenceInterval(0.99).ClosedContains(a));
Assert.IsTrue(parameters["b"].As <double>().PopulationMean().ConfidenceInterval(0.99).ClosedContains(b));
Assert.IsTrue(parameters["a"].As <double>().PopulationVariance().ConfidenceInterval(0.99).ClosedContains(covariances.Column(0).Mean));
Assert.IsTrue(parameters["b"].As <double>().PopulationVariance().ConfidenceInterval(0.99).ClosedContains(covariances.Column(1).Mean));
Assert.IsTrue(parameters["a"].As <double>().PopulationCovariance(parameters["b"].As <double>()).ConfidenceInterval(0.99).ClosedContains(covariances.Column(2).Mean));
Assert.IsTrue(Bivariate.PopulationCovariance(parameters["a"].As <double>(), parameters["b"].As <double>()).ConfidenceInterval(0.99).ClosedContains(covariances.Column(2).Mean));
}
public void OldMultivariateLinearRegressionTest()
{
MultivariateSample sample = new MultivariateSample(3);
sample.Add(98322, 81449, 269465);
sample.Add(65060, 31749, 121900);
sample.Add(36052, 14631, 37004);
sample.Add(31829, 27732, 91400);
sample.Add(7101, 9693, 54900);
sample.Add(41294, 4268, 16160);
sample.Add(16614, 4697, 21500);
sample.Add(3449, 4233, 9306);
sample.Add(3386, 5293, 38300);
sample.Add(6242, 2039, 13369);
sample.Add(14036, 7893, 29901);
sample.Add(2636, 3345, 10930);
sample.Add(869, 1135, 5100);
sample.Add(452, 727, 7653);
/*
* sample.Add(41.9, 29.1, 251.3);
* sample.Add(43.4, 29.3, 251.3);
* sample.Add(43.9, 29.5, 248.3);
* sample.Add(44.5, 29.7, 267.5);
* sample.Add(47.3, 29.9, 273.0);
* sample.Add(47.5, 30.3, 276.5);
* sample.Add(47.9, 30.5, 270.3);
* sample.Add(50.2, 30.7, 274.9);
* sample.Add(52.8, 30.8, 285.0);
* sample.Add(53.2, 30.9, 290.0);
* sample.Add(56.7, 31.5, 297.0);
* sample.Add(57.0, 31.7, 302.5);
* sample.Add(63.5, 31.9, 304.5);
* sample.Add(65.3, 32.0, 309.3);
* sample.Add(71.1, 32.1, 321.7);
* sample.Add(77.0, 32.5, 330.7);
* sample.Add(77.8, 32.9, 349.0);
*/
Console.WriteLine(sample.Count);
//sample.LinearRegression(0);
sample.LinearRegression(0);
}
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);
}
public void MultivariateLinearRegressionTest()
{
// define model y = a + b0 * x0 + b1 * x1 + noise
double a = 1.0;
double b0 = -2.0;
double b1 = 3.0;
Distribution noise = new NormalDistribution(0.0, 10.0);
// draw a sample from the model
Random rng = new Random(1);
MultivariateSample sample = new MultivariateSample(3);
for (int i = 0; i < 100; i++)
{
double x0 = -10.0 + 20.0 * rng.NextDouble();
double x1 = -10.0 + 20.0 * rng.NextDouble();
double eps = noise.InverseLeftProbability(rng.NextDouble());
double y = a + b0 * x0 + b1 * x1 + eps;
sample.Add(x0, x1, y);
}
// do a linear regression fit on the model
FitResult result = sample.LinearRegression(2);
// the result should have the appropriate dimension
Assert.IsTrue(result.Dimension == 3);
// the result should be significant
Console.WriteLine("{0} {1}", result.GoodnessOfFit.Statistic, result.GoodnessOfFit.LeftProbability);
Assert.IsTrue(result.GoodnessOfFit.LeftProbability > 0.95);
// the parameters should match the model
Console.WriteLine(result.Parameter(0));
Assert.IsTrue(result.Parameter(0).ConfidenceInterval(0.90).ClosedContains(b0));
Console.WriteLine(result.Parameter(1));
Assert.IsTrue(result.Parameter(1).ConfidenceInterval(0.90).ClosedContains(b1));
Console.WriteLine(result.Parameter(2));
Assert.IsTrue(result.Parameter(2).ConfidenceInterval(0.90).ClosedContains(a));
}
public void MultivariateManipulations()
{
MultivariateSample S = new MultivariateSample(3);
Assert.IsTrue(S.Dimension == 3);
Assert.IsTrue(S.Count == 0);
S.Add(1.1, 1.2, 1.3);
S.Add(2.1, 2.2, 2.3);
Assert.IsTrue(S.Count == 2);
// check that an entry is there, remove it, check that it is not there
Assert.IsTrue(S.Contains(1.1, 1.2, 1.3));
Assert.IsTrue(S.Remove(1.1, 1.2, 1.3));
Assert.IsFalse(S.Contains(1.1, 1.2, 1.3));
// clear it and check that the count went to zero
S.Clear();
Assert.IsTrue(S.Count == 0);
}
