public function testInverse()
{
// see http://www.mathwords.com/i/inverse_of_a_matrix.htm
$a = new SquareMatrix(4, 3, 3, 2);
$b = new SquareMatrix(-2, 3, 3, -4);
$aInverse = $a->getInverse();
$this->assertTrue($b->equals($aInverse));
$identity2x2 = new IdentityMatrix(2);
$aaInverse = Matrix::multiply($a, $aInverse);
$this->assertTrue($identity2x2->equals($aaInverse));
$c = new SquareMatrix(1, 2, 3, 0, 4, 5, 1, 0, 6);
$cInverse = $c->getInverse();
$d = Matrix::scalarMultiply(1.0 / 22, new SquareMatrix(24, -12, -2, 5, 3, -5, -4, 2, 4));
$this->assertTrue($d->equals($cInverse));
$identity3x3 = new IdentityMatrix(3);
$ccInverse = Matrix::multiply($c, $cInverse);
$this->assertTrue($identity3x3->equals($ccInverse));
}
public function calculateMatchQuality(GameInfo $gameInfo, array $teams)
{
// We need to create the A matrix which is the player team assigments.
$teamAssignmentsList = $teams;
$skillsMatrix = $this->getPlayerCovarianceMatrix($teamAssignmentsList);
$meanVector = $this->getPlayerMeansVector($teamAssignmentsList);
$meanVectorTranspose = $meanVector->getTranspose();
$playerTeamAssignmentsMatrix = $this->createPlayerTeamAssignmentMatrix($teamAssignmentsList, $meanVector->getRowCount());
$playerTeamAssignmentsMatrixTranspose = $playerTeamAssignmentsMatrix->getTranspose();
$betaSquared = square($gameInfo->getBeta());
$start = Matrix::multiply($meanVectorTranspose, $playerTeamAssignmentsMatrix);
$aTa = Matrix::multiply(Matrix::scalarMultiply($betaSquared, $playerTeamAssignmentsMatrixTranspose), $playerTeamAssignmentsMatrix);
$aTSA = Matrix::multiply(Matrix::multiply($playerTeamAssignmentsMatrixTranspose, $skillsMatrix), $playerTeamAssignmentsMatrix);
$middle = Matrix::add($aTa, $aTSA);
$middleInverse = $middle->getInverse();
$end = Matrix::multiply($playerTeamAssignmentsMatrixTranspose, $meanVector);
$expPartMatrix = Matrix::scalarMultiply(-0.5, Matrix::multiply(Matrix::multiply($start, $middleInverse), $end));
$expPart = $expPartMatrix->getDeterminant();
$sqrtPartNumerator = $aTa->getDeterminant();
$sqrtPartDenominator = $middle->getDeterminant();
$sqrtPart = $sqrtPartNumerator / $sqrtPartDenominator;
$result = exp($expPart) * sqrt($sqrtPart);
return $result;
}
