public void FisherFromChiSquared()
{
// we will need a RNG
Random rng = new Random(314159);
int n1 = 1;
int n2 = 2;
// define chi squared distributions
ContinuousDistribution d1 = new ChiSquaredDistribution(n1);
ContinuousDistribution d2 = new ChiSquaredDistribution(n2);
// create a sample of chi-squared variates
Sample s = new Sample();
for (int i = 0; i < 250; i++)
{
double x1 = d1.GetRandomValue(rng);
double x2 = d2.GetRandomValue(rng);
double x = (x1 / n1) / (x2 / n2);
s.Add(x);
}
// it should match a Fisher distribution with the appropriate parameters
ContinuousDistribution f0 = new FisherDistribution(n1, n2);
TestResult t0 = s.KuiperTest(f0);
Assert.IsTrue(t0.Probability > 0.05);
// it should be distinguished from a Fisher distribution with different parameters
ContinuousDistribution f1 = new FisherDistribution(n1 + 1, n2);
TestResult t1 = s.KuiperTest(f1);
Assert.IsTrue(t1.Probability < 0.05);
}
public void AnovaDistribution()
{
Distribution sDistribution = new NormalDistribution();
Random rng = new Random(1);
Sample fSample = new Sample();
// do 100 ANOVAs
for (int t = 0; t < 100; t++) {
// each ANOVA has 4 groups
List<Sample> groups = new List<Sample>();
for (int g = 0; g < 4; g++) {
// each group has 3 data points
Sample group = new Sample();
for (int i = 0; i < 3; i++) {
group.Add(sDistribution.GetRandomValue(rng));
}
groups.Add(group);
}
OneWayAnovaResult result = Sample.OneWayAnovaTest(groups);
fSample.Add(result.Factor.Result.Statistic);
}
// compare the distribution of F statistics to the expected distribution
Distribution fDistribution = new FisherDistribution(3, 8);
Console.WriteLine("m={0} s={1}", fSample.PopulationMean, fSample.PopulationStandardDeviation);
TestResult kResult = fSample.KolmogorovSmirnovTest(fDistribution);
Console.WriteLine(kResult.LeftProbability);
Assert.IsTrue(kResult.LeftProbability < 0.95);
}
public void BivariateLinearRegressionGoodnessOfFitDistribution()
{
// create uncorrelated x and y values
// the distribution of F-test statistics returned by linear fits should follow the expected F-distribution
Random rng = new Random(987654321);
NormalDistribution xd = new NormalDistribution(1.0, 2.0);
NormalDistribution yd = new NormalDistribution(-3.0, 4.0);
Sample fs = new Sample();
for (int i = 0; i < 127; i++)
{
BivariateSample xys = new BivariateSample();
for (int j = 0; j < 7; j++)
{
xys.Add(xd.GetRandomValue(rng), yd.GetRandomValue(rng));
}
double f = xys.LinearRegression().GoodnessOfFit.Statistic;
fs.Add(f);
}
Distribution fd = new FisherDistribution(1, 5);
Console.WriteLine("{0} v. {1}", fs.PopulationMean, fd.Mean);
TestResult t = fs.KolmogorovSmirnovTest(fd);
Console.WriteLine(t.LeftProbability);
Assert.IsTrue(t.LeftProbability < 0.95);
}
public void BivariateLinearRegressionNullDistribution()
{
// create uncorrelated x and y values
// the distribution of F-test statistics returned by linear fits should follow the expected F-distribution
Random rng = new Random(987654321);
NormalDistribution xd = new NormalDistribution(1.0, 2.0);
NormalDistribution yd = new NormalDistribution(-3.0, 4.0);
Sample fs = new Sample();
Sample rSample = new Sample();
ContinuousDistribution rDistribution = null;
Sample fSample = new Sample();
ContinuousDistribution fDistribution = null;
for (int i = 0; i < 127; i++)
{
BivariateSample sample = new BivariateSample();
for (int j = 0; j < 7; j++)
{
sample.Add(xd.GetRandomValue(rng), yd.GetRandomValue(rng));
}
LinearRegressionResult result = sample.LinearRegression();
double f = result.F.Statistic;
fs.Add(f);
rSample.Add(result.R.Statistic);
rDistribution = result.R.Distribution;
fSample.Add(result.F.Statistic);
fDistribution = result.F.Distribution;
Assert.IsTrue(result.F.Statistic == result.Anova.Result.Statistic);
Assert.IsTrue(TestUtilities.IsNearlyEqual(
result.R.Probability, result.F.Probability,
new EvaluationSettings()
{
RelativePrecision = 1.0E-14, AbsolutePrecision = 1.0E-16
}
));
}
ContinuousDistribution fd = new FisherDistribution(1, 5);
Console.WriteLine("{0} v. {1}", fs.PopulationMean, fd.Mean);
TestResult t = fs.KolmogorovSmirnovTest(fd);
Console.WriteLine(t.LeftProbability);
Assert.IsTrue(t.LeftProbability < 0.95);
Assert.IsTrue(rSample.KuiperTest(rDistribution).Probability > 0.05);
Assert.IsTrue(fSample.KuiperTest(fDistribution).Probability > 0.05);
}
public void Bug5886()
{
// the inverse CDF of the F-distribution would fail for d2 <= 2
double d1 = 1.0;
double d2 = 0.1;
FisherDistribution F = new FisherDistribution(d1, d2);
double x1 = F.InverseLeftProbability(0.6);
Console.WriteLine(x1);
double P = F.LeftProbability(x1);
Console.WriteLine(P);
}
public void Bug5886()
{
// the inverse CDF of the F-distribution would fail for d2 <= 2
double d1 = 1.0;
double d2 = 0.1;
FisherDistribution F = new FisherDistribution(d1, d2);
double x1 = F.InverseLeftProbability(0.6);
Console.WriteLine(x1);
double P = F.LeftProbability(x1);
Console.WriteLine(P);
}
public void FisherInversion()
{
// x ~ Fisher(a,b) => 1/x ~ Fisher(b,a)
FisherDistribution f = new FisherDistribution(2.3, 5.6);
FisherDistribution fi = new FisherDistribution(f.DenominatorDegreesOfFreedom, f.NumeratorDegreesOfFreedom);
Random rng = new Random(1);
for (int i = 0; i < 10; i++)
{
double x = f.GetRandomValue(rng);
double xi = 1.0 / x;
// LeftProbability <-> RightProbability because as x increases, 1/x decreases
Assert.IsTrue(TestUtilities.IsNearlyEqual(f.LeftProbability(x), fi.RightProbability(xi)));
}
}
public void BivariateLinearRegressionGoodnessOfFitDistribution()
{
// create uncorrelated x and y values
// the distribution of F-test statistics returned by linear fits should follow the expected F-distribution
Random rng = new Random(987654321);
NormalDistribution xd = new NormalDistribution(1.0, 2.0);
NormalDistribution yd = new NormalDistribution(-3.0, 4.0);
Sample fs = new Sample();
for (int i = 0; i < 127; i++) {
BivariateSample xys = new BivariateSample();
for (int j = 0; j < 7; j++) {
xys.Add(xd.GetRandomValue(rng), yd.GetRandomValue(rng));
}
double f = xys.LinearRegression().GoodnessOfFit.Statistic;
fs.Add(f);
}
Distribution fd = new FisherDistribution(1, 5);
Console.WriteLine("{0} v. {1}", fs.PopulationMean, fd.Mean);
TestResult t = fs.KolmogorovSmirnovTest(fd);
Console.WriteLine(t.LeftProbability);
Assert.IsTrue(t.LeftProbability < 0.95);
}
public void FisherTest()
{
// we will need a RNG
Random rng = new Random(314159);
int n1 = 1;
int n2 = 2;
// define chi squared distributions
Distribution d1 = new ChiSquaredDistribution(n1);
Distribution d2 = new ChiSquaredDistribution(n2);
// create a sample of chi-squared variates
Sample s = new Sample();
for (int i = 0; i < 250; i++) {
double x1 = d1.InverseLeftProbability(rng.NextDouble());
double x2 = d2.InverseLeftProbability(rng.NextDouble());
double x = (x1/n1) / (x2/n2);
s.Add(x);
}
// it should match a Fisher distribution with the appropriate parameters
Distribution f0 = new FisherDistribution(n1, n2);
TestResult t0 = s.KuiperTest(f0);
Console.WriteLine(t0.LeftProbability);
Assert.IsTrue(t0.LeftProbability < 0.95);
// it should be distinguished from a Fisher distribution with different parameters
Distribution f1 = new FisherDistribution(n1 + 1, n2);
TestResult t1 = s.KuiperTest(f1);
Console.WriteLine(t1.LeftProbability);
Assert.IsTrue(t1.LeftProbability > 0.95);
}
public void FisherInversion()
{
// x ~ Fisher(a,b) => 1/x ~ Fisher(b,a)
FisherDistribution f = new FisherDistribution(2.3, 5.6);
FisherDistribution fi = new FisherDistribution(f.DenominatorDegreesOfFreedom, f.NumeratorDegreesOfFreedom);
Random rng = new Random(1);
for (int i = 0; i < 10; i++) {
double x = f.GetRandomValue(rng);
double xi = 1.0 / x;
// LeftProbability <-> RightProbability because as x increases, 1/x decreases
Assert.IsTrue(TestUtilities.IsNearlyEqual(f.LeftProbability(x), fi.RightProbability(xi)));
}
}
// the internal linear regression routine, which assumes inputs are entirely valid
private FitResult LinearRegression_Internal(int outputIndex)
{
// to do a fit, we need more data than parameters
if (Count < Dimension)
{
throw new InsufficientDataException();
}
// construct the design matrix
SymmetricMatrix D = new SymmetricMatrix(Dimension);
for (int i = 0; i < Dimension; i++)
{
for (int j = 0; j <= i; j++)
{
if (i == outputIndex)
{
if (j == outputIndex)
{
D[i, j] = Count;
}
else
{
D[i, j] = storage[j].Mean * Count;
}
}
else
{
if (j == outputIndex)
{
D[i, j] = storage[i].Mean * Count;
}
else
{
double Dij = 0.0;
for (int k = 0; k < Count; k++)
{
Dij += storage[i][k] * storage[j][k];
}
D[i, j] = Dij;
}
}
}
}
// construct the right hand side
ColumnVector b = new ColumnVector(Dimension);
for (int i = 0; i < Dimension; i++)
{
if (i == outputIndex)
{
b[i] = storage[i].Mean * Count;
}
else
{
double bi = 0.0;
for (int k = 0; k < Count; k++)
{
bi += storage[outputIndex][k] * storage[i][k];
}
b[i] = bi;
}
}
// solve the system for the linear model parameters
CholeskyDecomposition CD = D.CholeskyDecomposition();
ColumnVector parameters = CD.Solve(b);
// find total sum of squares, with dof = # points - 1 (minus one for the variance-minimizing mean)
double totalSumOfSquares = storage[outputIndex].Variance * Count;
// find remaining unexplained sum of squares, with dof = # points - # parameters
double unexplainedSumOfSquares = 0.0;
for (int r = 0; r < Count; r++)
{
double y = 0.0;
for (int c = 0; c < Dimension; c++)
{
if (c == outputIndex)
{
y += parameters[c];
}
else
{
y += parameters[c] * storage[c][r];
}
}
unexplainedSumOfSquares += MoreMath.Sqr(y - storage[outputIndex][r]);
}
int unexplainedDegreesOfFreedom = Count - Dimension;
double unexplainedVariance = unexplainedSumOfSquares / unexplainedDegreesOfFreedom;
// find explained sum of squares, with dof = # parameters - 1
double explainedSumOfSquares = totalSumOfSquares - unexplainedSumOfSquares;
int explainedDegreesOfFreedom = Dimension - 1;
double explainedVariance = explainedSumOfSquares / explainedDegreesOfFreedom;
// compute F statistic from sums of squares
double F = explainedVariance / unexplainedVariance;
Distribution fDistribution = new FisherDistribution(explainedDegreesOfFreedom, unexplainedDegreesOfFreedom);
SymmetricMatrix covariance = unexplainedVariance * CD.Inverse();
return(new FitResult(parameters, covariance, new TestResult("F", F, TestType.RightTailed, fDistribution)));
}
