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);
});
}
// 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;
}