/// <summary>
/// Performs a linear logistic regression analysis.
/// </summary>
/// <param name="outputIndex">The index of the column to predict.</param>
/// <returns></returns>
/// <remarks>Logistic linear regression is suited to situations where multiple input variables, either continuous or binary indicators, are used to predict
/// the value of a binary output variable. Like a linear regression, a logistic linear regression tries to find a model that predicts the output variable using
/// a linear combination of input variables. Unlike a simple linear regression, the model does not assume that this linear
/// function predicts the output directly; instead it assumes that this function value is then fed into a logit link function, which
/// maps the real numbers into the interval (0, 1), and interprets the value of this link function as the probability of obtaining success value
/// for the output variable.</remarks>
/// <exception cref="InvalidOperationException">The column to be predicted contains values other than 0 and 1.</exception>
/// <exception cref="InsufficientDataException">There are not more rows in the sample than columns.</exception>
public FitResult LogisticLinearRegression(int outputIndex)
{
if ((outputIndex < 0) || (outputIndex >= this.Dimension))
{
throw new ArgumentOutOfRangeException("outputIndex");
}
if (this.Count <= this.Dimension)
{
throw new InsufficientDataException();
}
// Define the log likelihood as a function of the parameter set
Func <IList <double>, double> logLikelihood = (IList <double> a) => {
double L = 0.0;
for (int k = 0; k < this.Count; k++)
{
double z = 0.0;
for (int i = 0; i < this.storage.Length; i++)
{
if (i == outputIndex)
{
z += a[i];
}
else
{
z += a[i] * this.storage[i][k];
}
}
double ez = Math.Exp(z);
double y = this.storage[outputIndex][k];
if (y == 0.0)
{
L -= Math.Log(1.0 + ez);
}
else if (y == 1.0)
{
L -= Math.Log(1.0 + 1.0 / ez);
}
else
{
throw new InvalidOperationException();
}
}
return(L);
};
double[] start = new double[this.Dimension];
//for (int i = 0; i < start.Length; i++) {
// if (i != outputIndex) start[i] = this.TwoColumns(i, outputIndex).Covariance / this.Column(i).Variance / this.Column(outputIndex).Variance;
//}
MultiExtremum maximum = MultiFunctionMath.FindLocalMaximum(logLikelihood, start);
FitResult result = new FitResult(maximum.Location, maximum.HessianMatrix.CholeskyDecomposition().Inverse(), null);
return(result);
}
/// <summary>
/// Fits the data to an arbitrary parameterized function.
/// </summary>
/// <param name="function">The fit function.</param>
/// <param name="start">An initial guess at the parameters.</param>
/// <returns>A fit result containing the best-fitting function parameters
/// and a χ<sup>2</sup> test of the quality of the fit.</returns>
/// <exception cref="ArgumentNullException"><paramref name="function"/> or <paramref name="start"/> are <see langword="null"/>.</exception>
/// <exception cref="InsufficientDataException">There are fewer data points than fit parameters.</exception>
/// <exception cref="DivideByZeroException">The curvature matrix is singular, indicating that the data is independent of
/// one or more parameters, or that two or more parameters are linearly dependent.</exception>
public FitResult FitToFunction(Func <double[], T, double> function, double[] start)
{
if (function == null)
{
throw new ArgumentNullException(nameof(function));
}
if (start == null)
{
throw new ArgumentNullException(nameof(start));
}
// you can't do a fit with less data than parameters
if (this.Count < start.Length)
{
throw new InsufficientDataException();
}
/*
* Func<IList<double>, double> function0 = (IList<double> x0) => {
* double[] x = new double[x0.Count];
* x0.CopyTo(x, 0);
* return(function(x));
* };
* MultiExtremum minimum0 = MultiFunctionMath.FindMinimum(function0, start);
*/
// create a chi^2 fit metric and minimize it
FitMetric <T> metric = new FitMetric <T>(this, function);
SpaceExtremum minimum = FunctionMath.FindMinimum(new Func <double[], double>(metric.Evaluate), start);
// compute the covariance (Hessian) matrix by inverting the curvature matrix
SymmetricMatrix A = 0.5 * minimum.Curvature();
CholeskyDecomposition CD = A.CholeskyDecomposition(); // should not return null if we were at a minimum
if (CD == null)
{
throw new DivideByZeroException();
}
SymmetricMatrix C = CD.Inverse();
// package up the results and return them
TestResult test = new TestResult("ChiSquare", minimum.Value, TestType.RightTailed, new ChiSquaredDistribution(this.Count - minimum.Dimension));
FitResult fit = new FitResult(minimum.Location(), C, test);
return(fit);
}
/// <summary>
/// Finds the parameterized function that best fits the data.
/// </summary>
/// <param name="f">The parameterized function.</param>
/// <param name="start">An initial guess for the parameters.</param>
/// <returns>The fit result.</returns>
/// <remarks>
/// <para>
/// In the returned <see cref="FitResult"/>, the parameters appear in the same order as in
/// the supplied fit function and initial guess vector. No goodness-of-fit test is returned.
/// </para>
/// </remarks>
/// <exception cref="ArgumentNullException"><paramref name="f"/> or <paramref name="start"/> is null.</exception>
/// <exception cref="InsufficientDataException">There are not more data points than fit parameters.</exception>
/// <exception cref="DivideByZeroException">The curvature matrix is singular, indicating that the data is independent of
/// one or more parameters, or that two or more parameters are linearly dependent.</exception>
public FitResult NonlinearRegression(Func <IList <double>, double, double> f, IList <double> start)
{
if (f == null)
{
throw new ArgumentNullException(nameof(f));
}
if (start == null)
{
throw new ArgumentNullException(nameof(start));
}
int n = this.Count;
int d = start.Count;
if (n <= d)
{
throw new InsufficientDataException();
}
MultiExtremum min = MultiFunctionMath.FindLocalMinimum((IList <double> a) => {
double ss = 0.0;
for (int i = 0; i < n; i++)
{
double r = yData[i] - f(a, xData[i]);
ss += r * r;
}
return(ss);
}, start);
CholeskyDecomposition cholesky = min.HessianMatrix.CholeskyDecomposition();
if (cholesky == null)
{
throw new DivideByZeroException();
}
SymmetricMatrix curvature = cholesky.Inverse();
curvature = (2.0 * min.Value / (n - d)) * curvature;
FitResult result = new FitResult(min.Location, curvature, null);
return(result);
}
/// <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);
}
/// <summary>
/// Fits an MA(1) model to the time series.
/// </summary>
/// <returns>The fit with parameters lag-1 coefficient, mean, and standard deviation.</returns>
public FitResult FitToMA1()
{
if (data.Count < 4)
{
throw new InsufficientDataException();
}
// MA(1) model is
// y_t - \mu = u_t + \beta u_{t-1}
// where u_t ~ N(0, \sigma) are IID.
// It's easy to show that
// m = E(y_t) = \mu
// c_0 = V(y_t) = E((y_t - m)^2) = (1 + \beta^2) \sigma^2
// c_1 = V(y_t, y_{t-1}) = \beta \sigma^2
// So the method of moments gives
// \beta = \frac{1 - \sqrt{1 - (2 g_1)^2}}{2}
// \mu = m
// \sigma^2 = \frac{c_0}{1 + \beta^2}
// It turns out these are very poor (high bias, high variance) estimators,
// but they illustrate a basic requirement that g_1 < 1/2.
// The MLE estimator
int n = data.Count;
double m = data.Mean;
Func <double, double> fnc = (double theta) => {
double s = 0.0;
double uPrevious = 0.0;
for (int i = 0; i < data.Count; i++)
{
double u = data[i] - m - theta * uPrevious;
s += u * u;
uPrevious = u;
}
;
return(s);
};
Extremum minimum = FunctionMath.FindMinimum(fnc, Interval.FromEndpoints(-1.0, 1.0));
double beta = minimum.Location;
double sigma2 = minimum.Value / (data.Count - 3);
// While there is significant evidence that the MLE value for \beta is biased
// for small-n, I know of no anlytic correction.
double[] parameters = new double[] { beta, m, Math.Sqrt(sigma2) };
// The calculation of the variance for \mu can be improved over the MLE
// result by plugging the values for \gamma_0 and \gamma_1 into the
// exact formula.
SymmetricMatrix covariances = new SymmetricMatrix(3);
if (minimum.Curvature > 0.0)
{
covariances[0, 0] = sigma2 / minimum.Curvature;
}
else
{
covariances[0, 0] = MoreMath.Sqr(1.0 - beta * beta) / n;
}
covariances[1, 1] = sigma2 * (MoreMath.Sqr(1.0 + beta) - 2.0 * beta / n) / n;
covariances[2, 2] = sigma2 / 2.0 / n;
TimeSeries residuals = new TimeSeries();
double u1 = 0.0;
for (int i = 0; i < data.Count; i++)
{
double u0 = data[i] - m - beta * u1;
residuals.Add(u0);
u1 = u0;
}
;
TestResult test = residuals.LjungBoxTest();
FitResult result = new FitResult(
parameters, covariances, test
);
return(result);
}
