public void prediction_test()
{
// Example from http://www.real-statistics.com/multiple-regression/confidence-and-prediction-intervals/
var dt = Accord.IO.CsvReader.FromText(Resources.linreg, true).ToTable();
double[] y = dt.Columns["Poverty"].ToArray();
double[][] x = dt.ToArray("Infant Mort", "White", "Crime");
// Use Ordinary Least Squares to learn the regression
OrdinaryLeastSquares ols = new OrdinaryLeastSquares();
// Use OLS to learn the multiple linear regression
MultipleLinearRegression regression = ols.Learn(x, y);
Assert.AreEqual(3, regression.NumberOfInputs);
Assert.AreEqual(1, regression.NumberOfOutputs);
Assert.AreEqual(0.443650703716698, regression.Intercept, 1e-5);
Assert.AreEqual(1.2791842411083394, regression.Weights[0], 1e-5);
Assert.AreEqual(0.036259242392669415, regression.Weights[1], 1e-5);
Assert.AreEqual(0.0014225014835705938, regression.Weights[2], 1e-5);
double rse = regression.GetStandardError(x, y);
Assert.AreEqual(rse, 2.4703520840798507, 1e-5);
double[][] im = ols.GetInformationMatrix();
double mse = regression.GetStandardError(x, y);
double[] se = regression.GetStandardErrors(mse, im);
Assert.AreEqual(0.30063086032754965, se[0], 1e-10);
Assert.AreEqual(0.033603448179240082, se[1], 1e-10);
Assert.AreEqual(0.0022414548866296342, se[2], 1e-10);
Assert.AreEqual(3.9879881671805824, se[3], 1e-10);
double[] x0 = new double[] { 7, 80, 400 };
double y0 = regression.Transform(x0);
Assert.AreEqual(y0, 12.867680376316864, 1e-5);
double actual = regression.GetStandardError(x0, mse, im);
Assert.AreEqual(0.35902764658470271, actual, 1e-10);
DoubleRange ci = regression.GetConfidenceInterval(x0, mse, x.Length, im);
Assert.AreEqual(ci.Min, 12.144995206616116, 1e-5);
Assert.AreEqual(ci.Max, 13.590365546017612, 1e-5);
actual = regression.GetPredictionStandardError(x0, mse, im);
Assert.AreEqual(2.4963053239397244, actual, 1e-10);
DoubleRange pi = regression.GetPredictionInterval(x0, mse, x.Length, im);
Assert.AreEqual(pi.Min, 7.8428783761994554, 1e-5);
Assert.AreEqual(pi.Max, 17.892482376434273, 1e-5);
}
private void compute(double[][] x, double[] y)
{
int n = x.Length;
int p = NumberOfInputs;
SSt = 0;
SSe = 0;
outputMean = 0.0;
NumberOfSamples = x.Length;
// Compute the regression
OrdinaryLeastSquares.Token = Token;
regression = OrdinaryLeastSquares.Learn(x, y);
informationMatrix = OrdinaryLeastSquares.GetInformationMatrix();
// Calculate mean of the expected outputs
outputMean = y.Mean();
// Calculate actual outputs (results)
#pragma warning disable 612, 618
results = regression.Transform(x);
// Calculate SSe and SSt
for (int i = 0; i < x.Length; i++)
{
double d;
d = y[i] - results[i];
SSe += d * d;
d = y[i] - outputMean;
SSt += d * d;
}
// Calculate SSr
SSr = SSt - SSe;
// Calculate R-Squared
rSquared = (SSt != 0) ? 1.0 - (SSe / SSt) : 1.0;
// Calculated Adjusted R-Squared
if (rSquared == 1)
{
rAdjusted = 1;
}
else
{
if (n - p == 1)
{
rAdjusted = double.NaN;
}
else
{
rAdjusted = 1.0 - (1.0 - rSquared) * ((n - 1.0) / (n - p - 1.0));
}
}
// Calculate Degrees of Freedom
DFr = p;
DFe = n - (p + 1);
DFt = DFr + DFe;
// Calculate Sum of Squares Mean
MSe = SSe / DFe;
MSr = SSr / DFr;
MSt = SSt / DFt;
// Calculate the F statistic
ftest = new FTest(MSr / MSe, DFr, DFe);
stdError = Math.Sqrt(MSe);
// Create the ANOVA table
List <AnovaVariationSource> table = new List <AnovaVariationSource>();
table.Add(new AnovaVariationSource(this, "Regression", SSr, DFr, MSr, ftest));
table.Add(new AnovaVariationSource(this, "Error", SSe, DFe, MSe, null));
table.Add(new AnovaVariationSource(this, "Total", SSt, DFt, MSt, null));
this.anovaTable = new AnovaSourceCollection(table);
// Compute coefficient standard errors;
standardErrors = new double[NumberOfInputs + 1];
for (int i = 0; i < informationMatrix.Length; i++)
{
standardErrors[i] = Math.Sqrt(MSe * informationMatrix[i][i]);
}
// Compute coefficient tests
for (int i = 0; i < CoefficientValues.Length; i++)
{
double tStatistic = CoefficientValues[i] / standardErrors[i];
ttests[i] = new TTest(estimatedValue: CoefficientValues[i],
standardError: standardErrors[i], degreesOfFreedom: DFe);
ftests[i] = new FTest(tStatistic * tStatistic, 1, DFe);
confidences[i] = ttests[i].GetConfidenceInterval(confidencePercent);
}
// Compute model performance tests
ttest = new TTest(results, outputMean);
ztest = new ZTest(results, outputMean);
chiSquareTest = new ChiSquareTest(y, results, n - p - 1);
#pragma warning restore 612, 618
}
public void prediction_test()
{
// Example from http://www.real-statistics.com/multiple-regression/confidence-and-prediction-intervals/
var dt = Accord.IO.CsvReader.FromText(Resources.linreg, true).ToTable();
double[][] y = dt.ToArray("Poverty");
double[][] x = dt.ToArray("Infant Mort", "White", "Crime");
// Use Ordinary Least Squares to learn the regression
OrdinaryLeastSquares ols = new OrdinaryLeastSquares();
// Use OLS to learn the multiple linear regression
MultivariateLinearRegression regression = ols.Learn(x, y);
Assert.AreEqual(3, regression.NumberOfInputs);
Assert.AreEqual(1, regression.NumberOfOutputs);
Assert.AreEqual(0.443650703716698, regression.Intercepts[0], 1e-5);
Assert.AreEqual(1.2791842411083394, regression.Weights[0][0], 1e-5);
Assert.AreEqual(0.036259242392669415, regression.Weights[1][0], 1e-5);
Assert.AreEqual(0.0014225014835705938, regression.Weights[2][0], 1e-5);
double rse = regression.GetStandardError(x, y)[0];
Assert.AreEqual(rse, 2.4703520840798507, 1e-5);
double[][] im = ols.GetInformationMatrix();
double[] mse = regression.GetStandardError(x, y);
double[][] se = regression.GetStandardErrors(mse, im);
Assert.AreEqual(0.30063086032754965, se[0][0], 1e-10);
Assert.AreEqual(0.033603448179240082, se[0][1], 1e-10);
Assert.AreEqual(0.0022414548866296342, se[0][2], 1e-10);
Assert.AreEqual(3.9879881671805824, se[0][3], 1e-10);
double[] x0 = new double[] { 7, 80, 400 };
double y0 = regression.Transform(x0)[0];
Assert.AreEqual(y0, 12.867680376316864, 1e-5);
double actual = regression.GetStandardError(x0, mse, im)[0];
Assert.AreEqual(0.35902764658470271, actual, 1e-10);
DoubleRange ci = regression.GetConfidenceInterval(x0, mse, x.Length, im)[0];
Assert.AreEqual(ci.Min, 12.144995206616116, 1e-5);
Assert.AreEqual(ci.Max, 13.590365546017612, 1e-5);
actual = regression.GetPredictionStandardError(x0, mse, im)[0];
Assert.AreEqual(2.4963053239397244, actual, 1e-10);
DoubleRange pi = regression.GetPredictionInterval(x0, mse, x.Length, im)[0];
Assert.AreEqual(pi.Min, 7.8428783761994554, 1e-5);
Assert.AreEqual(pi.Max, 17.892482376434273, 1e-5);
}
private static RegressionResult PerformOls(double[] yVal, string[] xVars, double[][] xVals)
{
var ols = new OrdinaryLeastSquares()
{
// intercept should represent the return if no impact from inputs. (mean daily return)
// forcing to 0 as we assume that all explanation of price move should be due to independenet variables.
// in reality there is likely an unexplainable drift. For example daily bleed (MER, funding etc).
UseIntercept = false
};
MultipleLinearRegression regression = ols.Learn(xVals, yVal);
double[] predicted = regression.Transform(xVals);
RegressionResult r = new RegressionResult();
r.ModelType = "OLS";
r.StandardError = regression.GetStandardError(xVals, yVal);
// r.RSquared = new RSquaredLoss(xVals.Length, yVal, ).Loss(predicted);
r.RSquared = regression.CoefficientOfDetermination(xVals, yVal, false);
r.AdjRSquared = regression.CoefficientOfDetermination(xVals, yVal, true);
r.Betas = SgtStringUtils.DoubleArrayToRoundedDelimitedString(regression.Weights);
r.xVars = string.Join(";", xVars);
// TODO need to validate the below section. add p-values?
double[] coeffStandardErrors = regression.GetStandardErrors(r.StandardError, ols.GetInformationMatrix());
double[] coeffTScores = DoubleDivide(regression.Weights, coeffStandardErrors);
double[] coeffPVals = new double[coeffTScores.Length];
double df = xVals.Length - 1;
for (int i = 0; i < coeffTScores.Length; i++)
{
TTest tTest = new TTest(coeffTScores[i], df, OneSampleHypothesis.ValueIsDifferentFromHypothesis);
coeffPVals[i] = tTest.PValue;
}
r.CoeffStandardErrors = SgtStringUtils.DoubleArrayToRoundedDelimitedString(coeffStandardErrors);
r.CoeffTScores = SgtStringUtils.DoubleArrayToRoundedDelimitedString(coeffTScores);
r.CoeffPVals = SgtStringUtils.DoubleArrayToRoundedDelimitedString(coeffPVals);
// /TODO
r.xVarCount = regression.NumberOfInputs;
r.SamplesCount = yVal.Length;
r.Mean = regression.Intercept;
return(r);
}