internal override OneWayAnovaResult CreateAnova()
{
AnovaRow fit = new AnovaRow(SSF, 1);
AnovaRow residual = new AnovaRow(SSR, n - 2);
AnovaRow total = new AnovaRow(SST, n - 1);
OneWayAnovaResult anova = new OneWayAnovaResult(fit, residual, total);
return(anova);
}
internal OneWayAnovaResult(AnovaRow factor, AnovaRow residual, AnovaRow total)
{
Debug.Assert(factor != null);
Debug.Assert(residual != null);
Debug.Assert(total != null);
Debug.Assert(factor.DegreesOfFreedom + residual.DegreesOfFreedom == total.DegreesOfFreedom);
this.Factor = new AnovaTestRow(factor.SumOfSquares, factor.DegreesOfFreedom, this);
this.Residual = residual;
this.Total = total;
}
internal override OneWayAnovaResult CreateAnova()
{
// Use sums-of-squares to do ANOVA
AnovaRow fit = new AnovaRow(SSF, m);
AnovaRow residual = new AnovaRow(SSR, n - (m + 1));
AnovaRow total = new AnovaRow(SST, n - 1);
OneWayAnovaResult anova = new OneWayAnovaResult(fit, residual, total);
return(anova);
}
internal TwoWayAnovaResult(AnovaRow row, AnovaRow column, AnovaRow interaction, AnovaRow residual)
{
Debug.Assert(row != null);
Debug.Assert(column != null);
Debug.Assert(interaction != null);
Debug.Assert(residual != null);
this.row = row;
this.column = column;
this.interaction = interaction;
this.residual = residual;
this.total = new AnovaRow(
row.SumOfSquares + column.SumOfSquares + interaction.SumOfSquares + residual.SumOfSquares,
row.DegreesOfFreedom + column.DegreesOfFreedom + interaction.DegreesOfFreedom + residual.DegreesOfFreedom
);
}
internal OneWayAnovaResult(AnovaRow factor, AnovaRow residual, AnovaRow total)
{
this.Factor = new AnovaTestRow(factor.SumOfSquares, factor.DegreesOfFreedom, this);
this.Residual = residual;
this.Total = total;
}
/// <summary>
/// Computes the polynomial of given degree which best fits the data.
/// </summary>
/// <param name="m">The degree, which must be non-negative.</param>
/// <returns>The fit result.</returns>
/// <exception cref="ArgumentOutOfRangeException"><paramref name="m"/> is negative.</exception>
/// <exception cref="InsufficientDataException">There are fewer data points than coefficients to be fit.</exception>
public PolynomialRegressionResult PolynomialRegression(int m)
{
if (m < 0)
{
throw new ArgumentOutOfRangeException(nameof(m));
}
int n = Count;
if (n < (m + 1))
{
throw new InsufficientDataException();
}
// Construct the n X m design matrix X_{ij} = x_{i}^{j}
RectangularMatrix X = new RectangularMatrix(n, m + 1);
ColumnVector y = new ColumnVector(n);
for (int i = 0; i < n; i++)
{
double x = xData[i];
X[i, 0] = 1.0;
for (int j = 1; j <= m; j++)
{
X[i, j] = X[i, j - 1] * x;
}
y[i] = yData[i];
}
// Use X = QR to solve X b = y and compute C
ColumnVector b;
SymmetricMatrix C;
QRDecomposition.SolveLinearSystem(X, y, out b, out C);
// Compute residuals
double SSR = 0.0;
double SSF = 0.0;
ColumnVector yHat = X * b;
Sample residuals = new Sample();
for (int i = 0; i < n; i++)
{
double z = yData[i] - yHat[i];
residuals.Add(z);
SSR += z * z;
SSF += MoreMath.Sqr(yHat[i] - yData.Mean);
}
double sigma2 = SSR / (n - (m + 1));
// Scale up C by \sigma^2
// (It sure would be great to be able to overload *=.)
for (int i = 0; i <= m; i++)
{
for (int j = i; j <= m; j++)
{
C[i, j] = C[i, j] * sigma2;
}
}
// Compute remaing sums-of-squares
double SST = yData.Variance * n;
// Use sums-of-squares to do ANOVA
AnovaRow fit = new AnovaRow(SSF, m);
AnovaRow residual = new AnovaRow(SSR, n - (m + 1));
AnovaRow total = new AnovaRow(SST, n - 1);
OneWayAnovaResult anova = new OneWayAnovaResult(fit, residual, total);
string[] names = new string[m + 1];
names[0] = "1";
if (m > 0)
{
names[1] = "x";
}
for (int i = 2; i <= m; i++)
{
names[i] = $"x^{i}";
}
ParameterCollection parameters = new ParameterCollection(names, b, C);
return(new PolynomialRegressionResult(parameters, anova, residuals));
}
/// <summary>
/// Computes the best-fit linear regression from the data.
/// </summary>
/// <returns>The result of the fit.</returns>
/// <remarks>
/// <para>Linear regression assumes that the data have been generated by a function y = a + b x + e, where e is
/// normally distributed noise, and determines the values of a and b that best fit the data. It also
/// determines an error matrix on the parameters a and b, and does an F-test to</para>
/// <para>The fit result is two-dimensional. The first parameter is the intercept a, the second is the slope b.
/// The goodness-of-fit test is a F-test comparing the variance accounted for by the model to the remaining,
/// unexplained variance.</para>
/// </remarks>
/// <exception cref="InsufficientDataException">There are fewer than three data points.</exception>
public LinearRegressionResult LinearRegression()
{
int n = this.Count;
if (n < 3)
{
throw new InsufficientDataException();
}
// The means and covariances are the inputs to most of the regression formulas.
double mx = xData.Mean;
double my = yData.Mean;
double cxx = xData.Variance;
double cyy = yData.Variance;
double cxy = this.Covariance;
Debug.Assert(cxx >= 0.0);
Debug.Assert(cyy >= 0.0);
// Compute the best-fit parameters
double b = cxy / cxx;
double a = my - b * mx;
// Since cov(x,y) = (n S_xy - S_x S_y)/n^2 and var(x) = (n S_xx - S_x^2) / n^2,
// these formulas are equivilent to the
// to the usual formulas for a and b involving sums, but it is more stable against round-off
ColumnVector v = new ColumnVector(a, b);
v.IsReadOnly = true;
// Compute Pearson r value
double r = cxy / Math.Sqrt(cxx * cyy);
TestResult rTest = new TestResult("r", r, TestType.TwoTailed, new Distributions.PearsonRDistribution(n));
// Compute residuals and other sum-of-squares
double SSR = 0.0;
double SSF = 0.0;
Sample residuals = new Sample();
foreach (XY point in this)
{
double y = a + b * point.X;
double z = point.Y - y;
SSR += z * z;
residuals.Add(z);
SSF += MoreMath.Sqr(y - my);
}
double SST = cyy * n;
// Note SST = SSF + SSR because \sum_{i} ( y_i - \bar{y})^2 = \sum_i (y_i - f_i)^2 + \sum_i (f_i - \bar{y})^2
// Use sums-of-squares to do ANOVA
AnovaRow fit = new AnovaRow(SSF, 1);
AnovaRow residual = new AnovaRow(SSR, n - 2);
AnovaRow total = new AnovaRow(SST, n - 1);
OneWayAnovaResult anova = new OneWayAnovaResult(fit, residual, total);
// Compute covariance of parameters matrix
double s2 = SSR / (n - 2);
double cbb = s2 / cxx / n;
double cab = -mx * cbb;
double caa = (cxx + mx * mx) * cbb;
SymmetricMatrix C = new SymmetricMatrix(2);
C[0, 0] = caa;
C[1, 1] = cbb;
C[0, 1] = cab;
C.IsReadOnly = true;
// Package the parameters
ParameterCollection parameters = new ParameterCollection(
new string[] { "Intercept", "Slope" }, v, C
);
// Prepare the prediction function
Func <double, UncertainValue> predict = (double x) => {
double y = a + b * x;
return(new UncertainValue(y, Math.Sqrt(s2 * (1.0 + (1.0 + MoreMath.Sqr(x - mx) / cxx) / n))));
};
return(new LinearRegressionResult(parameters, rTest, anova, residuals, predict));
}
// the internal linear regression routine, which assumes inputs are entirely valid
private MultiLinearRegressionResult LinearRegression_Internal(int outputIndex)
{
// To do a fit, we need more data than parameters.
if (Count < Dimension)
{
throw new InsufficientDataException();
}
// Compute the design matrix X.
int n = Count;
int m = Dimension;
RectangularMatrix X = new RectangularMatrix(n, m);
ColumnVector y = new ColumnVector(n);
for (int i = 0; i < n; i++)
{
for (int j = 0; j < m; j++)
{
if (j == outputIndex)
{
X[i, j] = 1.0;
}
else
{
X[i, j] = storage[j][i];
}
}
y[i] = storage[outputIndex][i];
}
// Use X = QR to solve X b = y and compute C.
ColumnVector b;
SymmetricMatrix C;
QRDecomposition.SolveLinearSystem(X, y, out b, out C);
// Compute residuals
double SSR = 0.0;
double SSF = 0.0;
ColumnVector yHat = X * b;
Sample residuals = new Sample();
for (int i = 0; i < n; i++)
{
double z = storage[outputIndex][i] - yHat[i];
residuals.Add(z);
SSR += z * z;
SSF += MoreMath.Sqr(yHat[i] - storage[outputIndex].Mean);
}
double sigma2 = SSR / (n - m);
// Scale up C by \sigma^2
// (It sure would be great to be able to overload *=.)
for (int i = 0; i < m; i++)
{
for (int j = i; j < m; j++)
{
C[i, j] = C[i, j] * sigma2;
}
}
// Compute remaing sums-of-squares
double SST = storage[outputIndex].Variance * n;
// Use sums-of-squares to do ANOVA
AnovaRow fit = new AnovaRow(SSF, m - 1);
AnovaRow residual = new AnovaRow(SSR, n - m);
AnovaRow total = new AnovaRow(SST, n - 1);
OneWayAnovaResult anova = new OneWayAnovaResult(fit, residual, total);
string[] names = new string[m];
for (int j = 0; j < m; j++)
{
if (j == outputIndex)
{
names[j] = "Intercept";
}
else
{
names[j] = $"[{j}]";
}
}
ParameterCollection parameters = new ParameterCollection(names, b, C);
return(new MultiLinearRegressionResult(parameters, anova, residuals));
}
internal OneWayAnovaResult(AnovaRow factor, AnovaRow residual, AnovaRow total)
{
this.Factor = new AnovaTestRow(factor.SumOfSquares, factor.DegreesOfFreedom, this);
this.Residual = residual;
this.Total = total;
}
