/**
* Calculates beta by GLS.
*
* <pre>
* b=(X' Omega^-1 X)^-1X'Omega^-1 y
* </pre>
*
* @return beta
*/
@Override
protected RealVector calculateBeta() {
RealMatrix OI = getOmegaInverse();
RealMatrix XT = X.transpose();
RealMatrix XTOIX = XT.multiply(OI).multiply(X);
RealMatrix inverse = new LUDecompositionImpl(XTOIX).getSolver().getInverse();
return inverse.multiply(XT).multiply(OI).operate(Y);
}
/**
* Calculates the variance-covariance matrix of the regression parameters.
*
* <p>Var(b) = (X<sup>T</sup>X)<sup>-1</sup>
*
* <p>Uses QR decomposition to reduce (X<sup>T</sup>X)<sup>-1</sup> to
* (R<sup>T</sup>R)<sup>-1</sup>, with only the top p rows of R included, where p = the length of
* the beta vector.
*
* @return The beta variance-covariance matrix
*/
@Override
protected RealMatrix calculateBetaVariance() {
int p = X.getColumnDimension();
RealMatrix Raug = qr.getR().getSubMatrix(0, p - 1, 0, p - 1);
RealMatrix Rinv = new LUDecompositionImpl(Raug).getSolver().getInverse();
return Rinv.multiply(Rinv.transpose());
}
/**
* Compute the "hat" matrix.
*
* <p>The hat matrix is defined in terms of the design matrix X by
* X(X<sup>T</sup>X)<sup>-1</sup>X<sup>T</sup>
*
* <p>The implementation here uses the QR decomposition to compute the hat matrix as Q
* I<sub>p</sub>Q<sup>T</sup> where I<sub>p</sub> is the p-dimensional identity matrix augmented
* by 0's. This computational formula is from "The Hat Matrix in Regression and ANOVA", David C.
* Hoaglin and Roy E. Welsch, <i>The American Statistician</i>, Vol. 32, No. 1 (Feb., 1978), pp.
* 17-22.
*
* @return the hat matrix
*/
public RealMatrix calculateHat() {
// Create augmented identity matrix
RealMatrix Q = qr.getQ();
final int p = qr.getR().getColumnDimension();
final int n = Q.getColumnDimension();
Array2DRowRealMatrix augI = new Array2DRowRealMatrix(n, n);
double[][] augIData = augI.getDataRef();
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
if (i == j && i < p) {
augIData[i][j] = 1d;
} else {
augIData[i][j] = 0d;
}
}
}
// Compute and return Hat matrix
return Q.multiply(augI).multiply(Q.transpose());
}
/**
* Calculates the residuals of multiple linear regression in matrix notation.
*
* <pre>
* u = y - X * b
* </pre>
*
* @return The residuals [n,1] matrix
*/
protected RealMatrix calculateResiduals() {
RealMatrix b = calculateBeta();
return Y.subtract(X.multiply(b));
}
/** {@inheritDoc} */
public double[] estimateResiduals() {
RealMatrix b = calculateBeta();
RealMatrix e = Y.subtract(X.multiply(b));
return e.getColumn(0);
}