/// <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 NonlinearRegressionResult NonlinearRegression(Func <IReadOnlyList <double>, double, double> f, IReadOnlyList <double> start)
{
if (start == null)
{
throw new ArgumentNullException(nameof(start));
}
return(Bivariate.NonlinearRegression(yData, xData, f, start));
}
/// <summary>
/// Computes the best-fit linear logistic regression from the data.
/// </summary>
/// <returns>The fit result.</returns>
/// <remarks>
/// <para>Linear logistic regression is a way to fit binary outcome data to a linear model.</para>
/// <para>The method assumes that binary outcomes are encoded as 0 and 1. If any y-values other than
/// 0 and 1 are encountered, it throws an <see cref="InvalidOperationException"/>.</para>
/// <para>The fit result is two-dimensional. The first parameter is a, the second b.</para>
/// </remarks>
/// <exception cref="InsufficientDataException">There are fewer than three data points.</exception>
/// <exception cref="InvalidOperationException">There is a y-value other than 0 or 1.</exception>
public LinearLogisticRegressionResult LinearLogisticRegression()
{
List <bool> y = yData.Select(v => { if (v == 0.0)
{
return(false);
}
else if (v == 1.0)
{
return(true);
}
else
{
throw new InvalidOperationException();
} }).ToList();
return(Bivariate.LinearLogisticRegression(y, xData));
}
internal LinearRegressionResult(IReadOnlyList <double> x, IReadOnlyList <double> y) :
base()
{
double yMean, xxSum, xySum, yySum;
Bivariate.ComputeBivariateMomentsUpToTwo(x, y, out n, out xMean, out yMean, out xxSum, out yySum, out xySum);
b = xySum / xxSum;
a = yMean - b * xMean;
residuals = new List <double>(n);
SSR = 0.0;
SSF = 0.0;
for (int i = 0; i < n; i++)
{
double yi = y[i];
double ypi = a + b * x[i];
double zi = yi - ypi;
residuals.Add(zi);
SSR += zi * zi;
SSF += MoreMath.Sqr(ypi - yMean);
}
SST = yySum;
xVariance = xxSum / n;
sigma2 = SSR / (n - 2);
sigma = Math.Sqrt(sigma2);
cbb = sigma2 / xVariance / n;
cab = -xMean * cbb;
caa = (xVariance + xMean * xMean) * cbb;
rTest = new Lazy <TestResult>(() => {
double r = xySum / Math.Sqrt(xxSum * yySum);
TestResult rTest = new TestResult("r", r, new PearsonRDistribution(n), TestType.TwoTailed);
return(rTest);
});
}
/// <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)
{
return(Bivariate.PolynomialRegression(yData, xData, m));
}
/// <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()
{
return(Bivariate.LinearRegression(yData, xData));
}
/// <summary>
/// Performs a Wilcoxon signed rank test.
/// </summary>
/// <returns>The result of the test.</returns>
/// <remarks>
/// <para>The Wilcoxon signed rank test is a non-parametric alternative to the
/// paired t-test (<see cref="PairedStudentTTest"/>). Given two measurements on
/// the same subjects, this method tests for changes in the distribution between
/// the two measurements. It is sensitive primarily to shifts in the median.
/// Note that the distributions of the individual measurements
/// may be far from normal, and may be different for each subject.</para>
/// </remarks>
/// <seealso href="https://en.wikipedia.org/wiki/Wilcoxon_signed-rank_test"/>
public TestResult WilcoxonSignedRankTest()
{
return(Bivariate.WilcoxonSignedRankTest(xData, yData));
}
/// <summary>
/// Performs a paired Student t-test.
/// </summary>
/// <returns>The result of the test.</returns>
/// <remarks>
/// <para>Like a two-sample, unpaired t-test (<see cref="Sample.StudentTTest(Sample,Sample)" />),
/// a paired t-test compares two samples to detect a difference in means.
/// Unlike the unpaired version, the paired version assumes that each </para>
/// </remarks>
/// <exception cref="InsufficientDataException">There are fewer than two data points.</exception>
public TestResult PairedStudentTTest()
{
return(Bivariate.PairedStudentTTest(xData, yData));
}
/// <summary>
/// Performs a Kendall concordance test for association.
/// </summary>
/// <returns>The result of the test.</returns>
/// <remarks>
/// <para>Kendall's τ is a non-parametric and robust test of association
/// between two variables. It simply measures the number of cases where an increase
/// in one variable is associated with an increase in the other (concordant pairs),
/// compared with the number of cases where an increase in one variable is associated
/// with a decrease in the other (discordant pairs).</para>
/// <para>Because τ depends only on the sign
/// of a change and not its magnitude, it is not skewed by outliers exhibiting very large
/// changes, nor by cases where the degree of change in one variable associated with
/// a given change in the other changes over the range of the variables. Of course, it may
/// still miss an association whose sign changes over the range of the variables. For example,
/// if data points lie along a semi-circle in the plane, an increase in the first variable
/// is associated with an increase in the second variable along the rising arc and and decrease in
/// the second variable along the falling arc. No test that looks for single-signed correlation
/// will catch this association.
/// </para>
/// <para>Because it examine all pairs of data points, the Kendall test requires
/// O(N<sup>2</sup>) operations. It is thus impractical for very large data sets. While
/// not quite as robust as the Kendall test, the Spearman test is a good fall-back in such cases.</para>
/// </remarks>
/// <exception cref="InsufficientDataException"><see cref="Count"/> is less than two.</exception>
/// <seealso cref="PearsonRTest"/>
/// <seealso cref="SpearmanRhoTest"/>
/// <seealso href="http://en.wikipedia.org/wiki/Kendall_tau_test" />
public TestResult KendallTauTest()
{
return(Bivariate.KendallTauTest(xData, yData));
}
/// <summary>
/// Performs a Spearman rank-order test of association between the two variables.
/// </summary>
/// <returns>The result of the test.</returns>
/// <remarks>
/// <para>The Spearman rank-order test of association is a non-parametric test for association between
/// two variables. The test statistic rho is the correlation coefficient of the <em>rank</em> of
/// each entry in the sample. It is thus invariant over monotonic re-parameterizations of the data,
/// and will, for example, detect a quadratic or exponential association just as well as a linear
/// association.</para>
/// <para>The Spearman rank-order test requires O(N log N) operations.</para>
/// </remarks>
/// <exception cref="InsufficientDataException">There are fewer than three data points.</exception>
/// <seealso cref="PearsonRTest"/>
/// <seealso cref="KendallTauTest"/>
/// <seealso href="http://en.wikipedia.org/wiki/Spearman%27s_rank_correlation_coefficient"/>
public TestResult SpearmanRhoTest()
{
return(Bivariate.SpearmanRhoTest(xData, yData));
}
/// <summary>
/// Performs a Pearson correlation test for association.
/// </summary>
/// <returns>The result of the test.</returns>
/// <remarks>
/// <para>This test measures the strength of the linear correlation between two variables. The
/// test statistic r is simply the covariance of the two variables, scaled by their respective
/// standard deviations so as to obtain a number between -1 (perfect linear anti-correlation)
/// and +1 (perfect linear correlation).</para>
/// <para>The Pearson test cannot reliably detect or rule out non-linear associations. For example,
/// variables with a perfect quadratic association may have only a weak linear correlation. If
/// you wish to test for associations that may not be linear, consider using the Spearman or
/// Kendall tests instead.</para>
/// <para>The Pearson correlation test requires O(N) operations.</para>
/// <para>The Pearson test requires at least three bivariate values.</para>
/// </remarks>
/// <exception cref="InsufficientDataException"><see cref="Count"/> is less than three.</exception>
/// <seealso cref="SpearmanRhoTest"/>
/// <seealso cref="KendallTauTest"/>
/// <seealso href="http://en.wikipedia.org/wiki/Pearson_correlation_coefficient" />
public TestResult PearsonRTest()
{
return(Bivariate.PearsonRTest(xData, yData));
}
// We need a goodness-of-fit measurement
internal LinearLogisticRegressionResult(IReadOnlyList <double> x, IReadOnlyList <bool> y)
{
Debug.Assert(x != null);
Debug.Assert(y != null);
Debug.Assert(x.Count == y.Count);
// check size of data set
int n = x.Count;
if (n < 3)
{
throw new InsufficientDataException();
}
// The linear logistic model is:
// p_i = \sigma(t_i) \quad t_i = a + b x_i
// So the log likelihood of the data set under the model is:
// \ln L = \sum_{{\rm true} i} \ln p_i + \sum_{{\rm false} i} \ln (1 - p_i)
// = \sum_{{\rm true} i} \ln \sigma(t_i) + \sum_{{\rm false} i} \ln (1 - \sigma(t_i))
// Taking derivatives:
// \frac{\partial L}{\partial a} = \sum_{{\rm true} i} \frac{\sigma'(t_i)}{\sigma(t_i)}
// + \sum_{{\rm false} i} \frac{-\sigma'(t_i)}{1 - \sigma(t_i)}
// \frac{\partial L}{\partial b} = \sum_{{\rm true} i} \frac{\sigma'(t_i)}{\sigma(t_i)} x_i
// + \sum_{{\rm false} i} \frac{-\sigma'(t_i)}{1 - \sigma(t_i)} x_i
// Using \sigma(t) = \frac{1}{1 + e^{-t}}, we can derive:
// \frac{\sigma'(t)}{\sigma(t)} = \sigma(-t)
// \frac{\sigma'(t)}{1 - \sigma(t)} = \sigma(t)
// So this becomes
// \frac{\partial L}{\partial a} = \sum_i \pm \sigma(\mp t_i)
// \frac{\partial L}{\partial b} = \sum_i \pm \sigma(\mp t_i) x_i
// where the upper sign is for true values and the lower sign is for false values.
// Find the simultaneous zeros of these equations to obtain the likelihood-maximizing a, b.
// To get the curvature matrix, we need the second derivatives.
// \frac{\partial^2 L}{\partial a^2} = - \sum_i \sigma'(\mp t_i)
// \frac{\partial^2 L}{\partial a \partial b} = - \sum_i \sigma'(\mp t_i) x_i
// \frac{\partial^2 L}{\partial b^2} = - \sum_i \sigma'(\mp t_i) x_i^2
// We need an initial guess at the parameters. Begin with the Ansatz of the logistic model:
// \frac{p}{1-p} = e^{\alpha + \beta x}
// Differentiate and do some algebra to get:
// \frac{\partial p}{\partial x} = \beta p ( 1 - p)
// Evaluating at means, and noting that p (1 - p) = var(y) and that, in a development around the means,
// cov(p, x) = \frac{\partial p}{\partial x} var(x)
// we get
// \beta = \frac{cov(y, x)}{var(x) var(y)}
// This approximation gets the sign right, but it looks like it usually gets the magnitude quite wrong.
// The problem with the approach is that var(y) = p (1 - p) assumes y are chosen with fixed p, but they aren't.
// We need to re-visit this analysis.
double xMean, yMean, xxSum, yySum, xySum;
Bivariate.ComputeBivariateMomentsUpToTwo(x, y.Select(z => z ? 1.0 : 0.0), out n, out xMean, out yMean, out xxSum, out yySum, out xySum);
double p = yMean;
double b0 = xySum / xxSum / yySum * n;
double a0 = Math.Log(p / (1.0 - p)) - b0 * xMean;
Func <IReadOnlyList <double>, IReadOnlyList <double> > J = (IReadOnlyList <double> a) => {
double dLda = 0.0;
double dLdb = 0.0;
for (int i = 0; i < n; i++)
{
double t = a[0] + a[1] * x[i];
if (y[i])
{
double s = Sigma(-t);
dLda += s;
dLdb += s * x[i];
}
else
{
double s = Sigma(t);
dLda -= s;
dLdb -= s * x[i];
}
}
return(new double[] { dLda, dLdb });
};
ColumnVector b = MultiFunctionMath.FindZero(J, new double[] { a0, b0 });
SymmetricMatrix C = new SymmetricMatrix(2);
for (int i = 0; i < n; i++)
{
double t = b[0] + b[1] * x[i];
if (y[i])
{
t = -t;
}
double e = Math.Exp(-t);
double sp = e / MoreMath.Sqr(1.0 + e);
C[0, 0] += sp;
C[0, 1] += sp * x[i];
C[1, 1] += sp * x[i] * x[i];
}
CholeskyDecomposition CD = C.CholeskyDecomposition();
if (CD == null)
{
throw new DivideByZeroException();
}
C = CD.Inverse();
best = b;
covariance = C;
}
