/// <summary>
/// Fits the data to a linear combination of fit functions.
/// </summary>
/// <param name="functions">The component functions.</param>
/// <returns>A fit result containing the best-fit coefficients of the component functions and a χ<sup>2</sup> test
/// of the quality of the fit.</returns>
/// <exception cref="ArgumentNullException"><paramref name="functions"/> is null.</exception>
/// <exception cref="InsufficientDataException">There are fewer data points than fit parameters.</exception>
public UncertainMeasurementFitResult FitToLinearFunction(Func <T, double>[] functions)
{
if (functions == null)
{
throw new ArgumentNullException(nameof(functions));
}
if (functions.Length > data.Count)
{
throw new InsufficientDataException();
}
// construct the design matrix
RectangularMatrix A = new RectangularMatrix(data.Count, functions.Length);
for (int r = 0; r < data.Count; r++)
{
for (int c = 0; c < functions.Length; c++)
{
A[r, c] = functions[c](data[r].X) / data[r].Y.Uncertainty;
}
}
// construct the right-hand-side
ColumnVector b = new ColumnVector(data.Count);
for (int r = 0; r < data.Count; r++)
{
b[r] = data[r].Y.Value / data[r].Y.Uncertainty;
}
// Solve the system via QR
ColumnVector a;
SymmetricMatrix C;
QRDecomposition.SolveLinearSystem(A, b, out a, out C);
// do a chi^2 test
double chi2 = 0.0;
for (int i = 0; i < data.Count; i++)
{
double f = 0.0;
for (int j = 0; j < functions.Length; j++)
{
f += functions[j](data[i].X) * a[j];
}
chi2 += Math.Pow((data[i].Y.Value - f) / data[i].Y.Uncertainty, 2);
}
TestResult test = new TestResult("χ²", chi2, new ChiSquaredDistribution(data.Count - functions.Length), TestType.RightTailed);
// return the results
string[] names = new string[functions.Length];
for (int j = 0; j < names.Length; j++)
{
names[j] = j.ToString();
}
ParameterCollection parameters = new ParameterCollection(names, a, C);
return(new UncertainMeasurementFitResult(parameters, test));
}
/// <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 FitResult PolynomialRegression(int m)
{
if (m < 0)
{
throw new ArgumentOutOfRangeException("m");
}
int n = Count;
if (n < m + 1)
{
throw new InsufficientDataException();
}
// Construct the n X m design matrix A_{ij} = x_{i}^{j}
RectangularMatrix A = new RectangularMatrix(n, m + 1);
ColumnVector y = new ColumnVector(n);
for (int i = 0; i < n; i++)
{
double x = xData[i];
A[i, 0] = 1.0;
for (int j = 1; j <= m; j++)
{
A[i, j] = A[i, j - 1] * x;
}
y[i] = yData[i];
}
ColumnVector a;
SymmetricMatrix C;
QRDecomposition.SolveLinearSystem(A, y, out a, out C);
// Compute the residual vector r and s^2 = r^2 / dof
ColumnVector r = A * a - y;
double V = r.Transpose() * r;
double ss2 = V / (n - (m + 1));
// Scale up the covariance by s^2
for (int i = 0; i <= m; i++)
{
for (int j = i; j <= m; j++)
{
C[i, j] = C[i, j] * ss2;
}
}
// compute F-statistic
// total variance dof = n - 1, explained variance dof = m, unexplained variance dof = n - (m + 1)
double totalVarianceSum = yData.Variance * n;
double unexplainedVarianceSum = V;
double explainedVarianceSum = totalVarianceSum - unexplainedVarianceSum;
double unexplainedVarianceDof = n - (m + 1);
double explainedVarianceDof = m;
double F = (explainedVarianceSum / explainedVarianceDof) / (unexplainedVarianceSum / unexplainedVarianceDof);
TestResult test = new TestResult("F", F, TestType.RightTailed, new FisherDistribution(explainedVarianceDof, unexplainedVarianceDof));
return(new FitResult(a, C, test));
}
/// <summary>
/// Fits the data to a linear combination of fit functions.
/// </summary>
/// <param name="functions">The component functions.</param>
/// <returns>A fit result containing the best-fit coefficients of the component functions and a χ<sup>2</sup> test
/// of the quality of the fit.</returns>
/// <exception cref="ArgumentNullException"><paramref name="functions"/> is null.</exception>
/// <exception cref="InsufficientDataException">There are fewer data points than fit parameters.</exception>
public FitResult FitToLinearFunction(Func <T, double>[] functions)
{
if (functions == null)
{
throw new ArgumentNullException(nameof(functions));
}
if (functions.Length > data.Count)
{
throw new InsufficientDataException();
}
// construct the design matrix
RectangularMatrix A = new RectangularMatrix(data.Count, functions.Length);
for (int r = 0; r < data.Count; r++)
{
for (int c = 0; c < functions.Length; c++)
{
A[r, c] = functions[c](data[r].X) / data[r].Y.Uncertainty;
}
}
// construct the right-hand-side
ColumnVector b = new ColumnVector(data.Count);
for (int r = 0; r < data.Count; r++)
{
b[r] = data[r].Y.Value / data[r].Y.Uncertainty;
}
// Solve the system via QR
ColumnVector a;
SymmetricMatrix C;
QRDecomposition.SolveLinearSystem(A, b, out a, out C);
/*
* // construct the data matrix
* SymmetricMatrix A = new SymmetricMatrix(functions.Length);
* for (int i = 0; i < A.Dimension; i++) {
* for (int j = 0; j <= i; j++) {
* double Aij = 0.0;
* for (int k = 0; k < data.Count; k++) {
* Aij += functions[i](data[k].X) * functions[j](data[k].X) / Math.Pow(data[k].Y.Uncertainty, 2);
* }
* A[i, j] = Aij;
* }
* }
*
* // construct the rhs
* double[] b = new double[functions.Length];
* for (int i = 0; i < b.Length; i++) {
* b[i] = 0.0;
* for (int j = 0; j < data.Count; j++) {
* b[i] += data[j].Y.Value * functions[i](data[j].X) / Math.Pow(data[j].Y.Uncertainty, 2);
* }
* }
*
* // solve the system
* CholeskyDecomposition CD = A.CholeskyDecomposition();
* if (CD == null) throw new InvalidOperationException();
* Debug.Assert(CD != null);
* SymmetricMatrix C = CD.Inverse();
* ColumnVector a = CD.Solve(b);
*/
// do a chi^2 test
double chi2 = 0.0;
for (int i = 0; i < data.Count; i++)
{
double f = 0.0;
for (int j = 0; j < functions.Length; j++)
{
f += functions[j](data[i].X) * a[j];
}
chi2 += Math.Pow((data[i].Y.Value - f) / data[i].Y.Uncertainty, 2);
}
TestResult test = new TestResult("ChiSquare", chi2, TestType.RightTailed, new ChiSquaredDistribution(data.Count - functions.Length));
// return the results
FitResult fit = new FitResult(a, C, test);
return(fit);
}
internal MultiLinearRegressionResult(IReadOnlyList <double> yColumn, IReadOnlyList <IReadOnlyList <double> > xColumns, IReadOnlyList <string> xNames) : base()
{
Debug.Assert(yColumn != null);
Debug.Assert(xColumns != null);
Debug.Assert(xColumns.Count > 0);
Debug.Assert(xNames.Count == xColumns.Count);
n = yColumn.Count;
m = xColumns.Count;
if (n <= m)
{
throw new InsufficientDataException();
}
// Compute the design matrix X.
interceptIndex = -1;
RectangularMatrix X = new RectangularMatrix(n, m);
for (int c = 0; c < m; c++)
{
IReadOnlyList <double> xColumn = xColumns[c];
if (xColumn == null)
{
Debug.Assert(xNames[c] == "Intercept");
Debug.Assert(interceptIndex < 0);
for (int r = 0; r < n; r++)
{
X[r, c] = 1.0;
}
interceptIndex = c;
}
else
{
Debug.Assert(xNames[c] != null);
if (xColumn.Count != n)
{
throw new DimensionMismatchException();
}
for (int r = 0; r < n; r++)
{
X[r, c] = xColumn[r];
}
}
}
Debug.Assert(interceptIndex >= 0);
ColumnVector v = new ColumnVector(yColumn);
// Use X = QR to solve X b = y and compute C.
QRDecomposition.SolveLinearSystem(X, v, out b, out C);
// For ANOVA, we will need mean and variance of y
int yn;
double ym;
Univariate.ComputeMomentsUpToSecond(yColumn, out yn, out ym, out SST);
// Compute residuals
SSR = 0.0;
SSF = 0.0;
ColumnVector yHat = X * b;
residuals = new List <double>(n);
for (int i = 0; i < n; i++)
{
double z = yColumn[i] - yHat[i];
residuals.Add(z);
SSR += z * z;
SSF += MoreMath.Sqr(yHat[i] - ym);
}
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;
}
}
names = xNames;
}
internal PolynomialRegressionResult(IReadOnlyList <double> x, IReadOnlyList <double> y, int degree) : base()
{
Debug.Assert(x != null);
Debug.Assert(y != null);
Debug.Assert(x.Count == y.Count);
Debug.Assert(degree >= 0);
m = degree;
n = x.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_i = x[i];
X[i, 0] = 1.0;
for (int j = 1; j <= m; j++)
{
X[i, j] = X[i, j - 1] * x_i;
}
double y_i = y[i];
Y[i] = y_i;
}
// Use X = QR to solve X b = y and compute C
QRDecomposition.SolveLinearSystem(X, Y, out b, out C);
// Compute mean and total sum of squares.
// This could be done inside loop above, but this way we get to re-use code from Univariate.
double yMean;
Univariate.ComputeMomentsUpToSecond(y, out n, out yMean, out SST);
// Compute residuals
SSR = 0.0;
SSF = 0.0;
ColumnVector yHat = X * b;
residuals = new List <double>(n);
for (int i = 0; i < n; i++)
{
double z = y[i] - yHat[i];
residuals.Add(z);
SSR += z * z;
SSF += MoreMath.Sqr(yHat[i] - yMean);
}
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;
}
}
}
/// <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));
}
// 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));
}
