/// The "V" function where the team performance difference is greater than the draw margin.
///
/// In the reference F# implementation, this is referred to as "the additive
/// correction of a single-sided truncated Gaussian with unit variance."
///
/// In the paper, it's referred to as just "ε".
///
public static function vExceedsMarginScaled($teamPerformanceDifference, $drawMargin, $c)
{
return self::vExceedsMargin($teamPerformanceDifference/$c, $drawMargin/$c);
}
public static function vExceedsMargin($teamPerformanceDifference, $drawMargin)
{
$denominator = GaussianDistribution::cumulativeTo($teamPerformanceDifference - $drawMargin);
if ($denominator < 2.222758749e-162)
{
return -$teamPerformanceDifference + $drawMargin;
}
return GaussianDistribution::at($teamPerformanceDifference - $drawMargin)/$denominator;
}
///
/// The "W" function where the team performance difference is greater than the draw margin.
///
/// In the reference F# implementation, this is referred to as "the multiplicative
/// correction of a single-sided truncated Gaussian with unit variance."
///
///
///
///
public static function wExceedsMarginScaled($teamPerformanceDifference, $drawMargin, $c)
{
return self::wExceedsMargin($teamPerformanceDifference/$c, $drawMargin/$c);
}
public static function wExceedsMargin($teamPerformanceDifference, $drawMargin)
{
$denominator = GaussianDistribution::cumulativeTo($teamPerformanceDifference - $drawMargin);
if ($denominator < 2.222758749e-162)
{
if ($teamPerformanceDifference < 0.0)
{
return 1.0;
}
return 0.0;
}
$vWin = self::vExceedsMargin($teamPerformanceDifference, $drawMargin);
return $vWin*($vWin + $teamPerformanceDifference - $drawMargin);
}
// the additive correction of a double-sided truncated Gaussian with unit variance
public static function vWithinMarginScaled($teamPerformanceDifference, $drawMargin, $c)
{
return self::vWithinMargin($teamPerformanceDifference/$c, $drawMargin/$c);
}
// from F#:
public static function vWithinMargin($teamPerformanceDifference, $drawMargin)
{
$teamPerformanceDifferenceAbsoluteValue = abs($teamPerformanceDifference);
$denominator =
GaussianDistribution::cumulativeTo($drawMargin - $teamPerformanceDifferenceAbsoluteValue) -
GaussianDistribution::cumulativeTo(-$drawMargin - $teamPerformanceDifferenceAbsoluteValue);
if ($denominator < 2.222758749e-162)
{
if ($teamPerformanceDifference < 0.0)
{
return -$teamPerformanceDifference - $drawMargin;
}
return -$teamPerformanceDifference + $drawMargin;
}
$numerator = GaussianDistribution::at(-$drawMargin - $teamPerformanceDifferenceAbsoluteValue) -
GaussianDistribution::at($drawMargin - $teamPerformanceDifferenceAbsoluteValue);
if ($teamPerformanceDifference < 0.0)
{
return -$numerator/$denominator;
}
return $numerator/$denominator;
}
// the multiplicative correction of a double-sided truncated Gaussian with unit variance
public static function wWithinMarginScaled($teamPerformanceDifference, $drawMargin, $c)
{
return self::wWithinMargin($teamPerformanceDifference/$c, $drawMargin/$c);
}
// From F#:
public static function wWithinMargin($teamPerformanceDifference, $drawMargin)
{
$teamPerformanceDifferenceAbsoluteValue = abs($teamPerformanceDifference);
$denominator = GaussianDistribution::cumulativeTo($drawMargin - $teamPerformanceDifferenceAbsoluteValue)
-
GaussianDistribution::cumulativeTo(-$drawMargin - $teamPerformanceDifferenceAbsoluteValue);
if ($denominator < 2.222758749e-162)
{
return 1.0;
}
$vt = self::vWithinMargin($teamPerformanceDifferenceAbsoluteValue, $drawMargin);
return $vt*$vt +
(
($drawMargin - $teamPerformanceDifferenceAbsoluteValue)
*
GaussianDistribution::at(
$drawMargin - $teamPerformanceDifferenceAbsoluteValue)
- (-$drawMargin - $teamPerformanceDifferenceAbsoluteValue)
*
GaussianDistribution::at(-$drawMargin - $teamPerformanceDifferenceAbsoluteValue))/$denominator;
}
}
?>