--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/PGAP-1.2.1/Statistics/Distributions.pm	Thu Jun 24 13:51:52 2021 +0000
@@ -0,0 +1,529 @@
+package Statistics::Distributions;
+
+use strict;
+use vars qw($VERSION @ISA @EXPORT @EXPORT_OK);
+use constant PI => 3.1415926536;
+use constant SIGNIFICANT => 5; # number of significant digits to be returned
+
+require Exporter;
+
+@ISA = qw(Exporter AutoLoader);
+# Items to export into callers namespace by default. Note: do not export
+# names by default without a very good reason. Use EXPORT_OK instead.
+# Do not simply export all your public functions/methods/constants.
+@EXPORT_OK = qw(chisqrdistr tdistr fdistr udistr uprob chisqrprob tprob fprob);
+$VERSION = '1.02';
+
+# Preloaded methods go here.
+   
+sub chisqrdistr { # Percentage points  X^2(x^2,n)
+	my ($n, $p) = @_;
+	if ($n <= 0 || abs($n) - abs(int($n)) != 0) {
+		die "Invalid n: $n\n"; # degree of freedom
+	}
+	if ($p <= 0 || $p > 1) {
+		die "Invalid p: $p\n"; 
+	}
+	return precision_string(_subchisqr($n, $p));
+}
+
+sub udistr { # Percentage points   N(0,1^2)
+	my ($p) = (@_);
+	if ($p > 1 || $p <= 0) {
+		die "Invalid p: $p\n";
+	}
+	return precision_string(_subu($p));
+}
+
+sub tdistr { # Percentage points   t(x,n)
+	my ($n, $p) = @_;
+	if ($n <= 0 || abs($n) - abs(int($n)) != 0) {
+		die "Invalid n: $n\n";
+	}
+	if ($p <= 0 || $p >= 1) {
+		die "Invalid p: $p\n";
+	}
+	return precision_string(_subt($n, $p));
+}
+
+sub fdistr { # Percentage points  F(x,n1,n2)
+	my ($n, $m, $p) = @_;
+	if (($n<=0) || ((abs($n)-(abs(int($n))))!=0)) {
+		die "Invalid n: $n\n"; # first degree of freedom
+	}
+	if (($m<=0) || ((abs($m)-(abs(int($m))))!=0)) {
+		die "Invalid m: $m\n"; # second degree of freedom
+	}
+	if (($p<=0) || ($p>1)) {
+		die "Invalid p: $p\n";
+	}
+	return precision_string(_subf($n, $m, $p));
+}
+
+sub uprob { # Upper probability   N(0,1^2)
+	my ($x) = @_;
+	return precision_string(_subuprob($x));
+}
+
+sub chisqrprob { # Upper probability   X^2(x^2,n)
+	my ($n,$x) = @_;
+	if (($n <= 0) || ((abs($n) - (abs(int($n)))) != 0)) {
+		die "Invalid n: $n\n"; # degree of freedom
+	}
+	return precision_string(_subchisqrprob($n, $x));
+}
+
+sub tprob { # Upper probability   t(x,n)
+	my ($n, $x) = @_;
+	if (($n <= 0) || ((abs($n) - abs(int($n))) !=0)) {
+		die "Invalid n: $n\n"; # degree of freedom
+	}
+	return precision_string(_subtprob($n, $x));
+}
+
+sub fprob { # Upper probability   F(x,n1,n2)
+	my ($n, $m, $x) = @_;
+	if (($n<=0) || ((abs($n)-(abs(int($n))))!=0)) {
+		die "Invalid n: $n\n"; # first degree of freedom
+	}
+	if (($m<=0) || ((abs($m)-(abs(int($m))))!=0)) {
+		die "Invalid m: $m\n"; # second degree of freedom
+	} 
+	return precision_string(_subfprob($n, $m, $x));
+}
+
+
+sub _subfprob {
+	my ($n, $m, $x) = @_;
+	my $p;
+
+	if ($x<=0) {
+		$p=1;
+	} elsif ($m % 2 == 0) {
+		my $z = $m / ($m + $n * $x);
+		my $a = 1;
+		for (my $i = $m - 2; $i >= 2; $i -= 2) {
+			$a = 1 + ($n + $i - 2) / $i * $z * $a;
+		}
+		$p = 1 - ((1 - $z) ** ($n / 2) * $a);
+	} elsif ($n % 2 == 0) {
+		my $z = $n * $x / ($m + $n * $x);
+		my $a = 1;
+		for (my $i = $n - 2; $i >= 2; $i -= 2) {
+			$a = 1 + ($m + $i - 2) / $i * $z * $a;
+		}
+		$p = (1 - $z) ** ($m / 2) * $a;
+	} else {
+		my $y = atan2(sqrt($n * $x / $m), 1);
+		my $z = sin($y) ** 2;
+		my $a = ($n == 1) ? 0 : 1;
+		for (my $i = $n - 2; $i >= 3; $i -= 2) {
+			$a = 1 + ($m + $i - 2) / $i * $z * $a;
+		} 
+		my $b = PI;
+		for (my $i = 2; $i <= $m - 1; $i += 2) {
+			$b *= ($i - 1) / $i;
+		}
+		my $p1 = 2 / $b * sin($y) * cos($y) ** $m * $a;
+
+		$z = cos($y) ** 2;
+		$a = ($m == 1) ? 0 : 1;
+		for (my $i = $m-2; $i >= 3; $i -= 2) {
+			$a = 1 + ($i - 1) / $i * $z * $a;
+		}
+		$p = max(0, $p1 + 1 - 2 * $y / PI
+			- 2 / PI * sin($y) * cos($y) * $a);
+	}
+	return $p;
+}
+
+
+sub _subchisqrprob {
+	my ($n,$x) = @_;
+	my $p;
+
+	if ($x <= 0) {
+		$p = 1;
+	} elsif ($n > 100) {
+		$p = _subuprob((($x / $n) ** (1/3)
+				- (1 - 2/9/$n)) / sqrt(2/9/$n));
+	} elsif ($x > 400) {
+		$p = 0;
+	} else {   
+		my ($a, $i, $i1);
+		if (($n % 2) != 0) {
+			$p = 2 * _subuprob(sqrt($x));
+			$a = sqrt(2/PI) * exp(-$x/2) / sqrt($x);
+			$i1 = 1;
+		} else {
+			$p = $a = exp(-$x/2);
+			$i1 = 2;
+		}
+
+		for ($i = $i1; $i <= ($n-2); $i += 2) {
+			$a *= $x / $i;
+			$p += $a;
+		}
+	}
+	return $p;
+}
+
+sub _subu {
+	my ($p) = @_;
+	my $y = -log(4 * $p * (1 - $p));
+	my $x = sqrt(
+		$y * (1.570796288
+		  + $y * (.03706987906
+		  	+ $y * (-.8364353589E-3
+			  + $y *(-.2250947176E-3
+			  	+ $y * (.6841218299E-5
+				  + $y * (0.5824238515E-5
+					+ $y * (-.104527497E-5
+					  + $y * (.8360937017E-7
+						+ $y * (-.3231081277E-8
+						  + $y * (.3657763036E-10
+							+ $y *.6936233982E-12)))))))))));
+	$x = -$x if ($p>.5);
+	return $x;
+}
+
+sub _subuprob {
+	my ($x) = @_;
+	my $p = 0; # if ($absx > 100)
+	my $absx = abs($x);
+
+	if ($absx < 1.9) {
+		$p = (1 +
+			$absx * (.049867347
+			  + $absx * (.0211410061
+			  	+ $absx * (.0032776263
+				  + $absx * (.0000380036
+					+ $absx * (.0000488906
+					  + $absx * .000005383)))))) ** -16/2;
+	} elsif ($absx <= 100) {
+		for (my $i = 18; $i >= 1; $i--) {
+			$p = $i / ($absx + $p);
+		}
+		$p = exp(-.5 * $absx * $absx) 
+			/ sqrt(2 * PI) / ($absx + $p);
+	}
+
+	$p = 1 - $p if ($x<0);
+	return $p;
+}
+
+   
+sub _subt {
+	my ($n, $p) = @_;
+
+	if ($p >= 1 || $p <= 0) {
+		die "Invalid p: $p\n";
+	}
+
+	if ($p == 0.5) {
+		return 0;
+	} elsif ($p < 0.5) {
+		return - _subt($n, 1 - $p);
+	}
+
+	my $u = _subu($p);
+	my $u2 = $u ** 2;
+
+	my $a = ($u2 + 1) / 4;
+	my $b = ((5 * $u2 + 16) * $u2 + 3) / 96;
+	my $c = (((3 * $u2 + 19) * $u2 + 17) * $u2 - 15) / 384;
+	my $d = ((((79 * $u2 + 776) * $u2 + 1482) * $u2 - 1920) * $u2 - 945) 
+				/ 92160;
+	my $e = (((((27 * $u2 + 339) * $u2 + 930) * $u2 - 1782) * $u2 - 765) * $u2
+			+ 17955) / 368640;
+
+	my $x = $u * (1 + ($a + ($b + ($c + ($d + $e / $n) / $n) / $n) / $n) / $n);
+
+	if ($n <= log10($p) ** 2 + 3) {
+		my $round;
+		do { 
+			my $p1 = _subtprob($n, $x);
+			my $n1 = $n + 1;
+			my $delta = ($p1 - $p) 
+				/ exp(($n1 * log($n1 / ($n + $x * $x)) 
+					+ log($n/$n1/2/PI) - 1 
+					+ (1/$n1 - 1/$n) / 6) / 2);
+			$x += $delta;
+			$round = sprintf("%.".abs(int(log10(abs $x)-4))."f",$delta);
+		} while (($x) && ($round != 0));
+	}
+	return $x;
+}
+
+sub _subtprob {
+	my ($n, $x) = @_;
+
+	my ($a,$b);
+	my $w = atan2($x / sqrt($n), 1);
+	my $z = cos($w) ** 2;
+	my $y = 1;
+
+	for (my $i = $n-2; $i >= 2; $i -= 2) {
+		$y = 1 + ($i-1) / $i * $z * $y;
+	} 
+
+	if ($n % 2 == 0) {
+		$a = sin($w)/2;
+		$b = .5;
+	} else {
+		$a = ($n == 1) ? 0 : sin($w)*cos($w)/PI;
+		$b= .5 + $w/PI;
+	}
+	return max(0, 1 - $b - $a * $y);
+}
+
+sub _subf {
+	my ($n, $m, $p) = @_;
+	my $x;
+
+	if ($p >= 1 || $p <= 0) {
+		die "Invalid p: $p\n";
+	}
+
+	if ($p == 1) {
+		$x = 0;
+	} elsif ($m == 1) {
+		$x = 1 / (_subt($n, 0.5 - $p / 2) ** 2);
+	} elsif ($n == 1) {
+		$x = _subt($m, $p/2) ** 2;
+	} elsif ($m == 2) {
+		my $u = _subchisqr($m, 1 - $p);
+		my $a = $m - 2;
+		$x = 1 / ($u / $m * (1 +
+			(($u - $a) / 2 +
+				(((4 * $u - 11 * $a) * $u + $a * (7 * $m - 10)) / 24 +
+					(((2 * $u - 10 * $a) * $u + $a * (17 * $m - 26)) * $u
+						- $a * $a * (9 * $m - 6)
+					)/48/$n
+				)/$n
+			)/$n));
+	} elsif ($n > $m) {
+		$x = 1 / _subf2($m, $n, 1 - $p)
+	} else {
+		$x = _subf2($n, $m, $p)
+	}
+	return $x;
+}
+
+sub _subf2 {
+	my ($n, $m, $p) = @_;
+	my $u = _subchisqr($n, $p);
+	my $n2 = $n - 2;
+	my $x = $u / $n * 
+		(1 + 
+			(($u - $n2) / 2 + 
+				(((4 * $u - 11 * $n2) * $u + $n2 * (7 * $n - 10)) / 24 + 
+					(((2 * $u - 10 * $n2) * $u + $n2 * (17 * $n - 26)) * $u 
+						- $n2 * $n2 * (9 * $n - 6)) / 48 / $m) / $m) / $m);
+	my $delta;
+	do {
+		my $z = exp(
+			(($n+$m) * log(($n+$m) / ($n * $x + $m)) 
+				+ ($n - 2) * log($x)
+				+ log($n * $m / ($n+$m))
+				- log(4 * PI)
+				- (1/$n  + 1/$m - 1/($n+$m))/6
+			)/2);
+		$delta = (_subfprob($n, $m, $x) - $p) / $z;
+		$x += $delta;
+	} while (abs($delta)>3e-4);
+	return $x;
+}
+
+sub _subchisqr {
+	my ($n, $p) = @_;
+	my $x;
+
+	if (($p > 1) || ($p <= 0)) {
+		die "Invalid p: $p\n";
+	} elsif ($p == 1){
+		$x = 0;
+	} elsif ($n == 1) {
+		$x = _subu($p / 2) ** 2;
+	} elsif ($n == 2) {
+		$x = -2 * log($p);
+	} else {
+		my $u = _subu($p);
+		my $u2 = $u * $u;
+
+		$x = max(0, $n + sqrt(2 * $n) * $u 
+			+ 2/3 * ($u2 - 1)
+			+ $u * ($u2 - 7) / 9 / sqrt(2 * $n)
+			- 2/405 / $n * ($u2 * (3 *$u2 + 7) - 16));
+
+		if ($n <= 100) {
+			my ($x0, $p1, $z);
+			do {
+				$x0 = $x;
+				if ($x < 0) {
+					$p1 = 1;
+				} elsif ($n>100) {
+					$p1 = _subuprob((($x / $n)**(1/3) - (1 - 2/9/$n))
+						/ sqrt(2/9/$n));
+				} elsif ($x>400) {
+					$p1 = 0;
+				} else {
+					my ($i0, $a);
+					if (($n % 2) != 0) {
+						$p1 = 2 * _subuprob(sqrt($x));
+						$a = sqrt(2/PI) * exp(-$x/2) / sqrt($x);
+						$i0 = 1;
+					} else {
+						$p1 = $a = exp(-$x/2);
+						$i0 = 2;
+					}
+
+					for (my $i = $i0; $i <= $n-2; $i += 2) {
+						$a *= $x / $i;
+						$p1 += $a;
+					}
+				}
+				$z = exp((($n-1) * log($x/$n) - log(4*PI*$x) 
+					+ $n - $x - 1/$n/6) / 2);
+				$x += ($p1 - $p) / $z;
+				$x = sprintf("%.5f", $x);
+			} while (($n < 31) && (abs($x0 - $x) > 1e-4));
+		}
+	}
+	return $x;
+}
+
+sub log10 {
+	my $n = shift;
+	return log($n) / log(10);
+}
+ 
+sub max {
+	my $max = shift;
+	my $next;
+	while (@_) {
+		$next = shift;
+		$max = $next if ($next > $max);
+	}	
+	return $max;
+}
+
+sub min {
+	my $min = shift;
+	my $next;
+	while (@_) {
+		$next = shift;
+		$min = $next if ($next < $min);
+	}	
+	return $min;
+}
+
+sub precision {
+	my ($x) = @_;
+	return abs int(log10(abs $x) - SIGNIFICANT);
+}
+
+sub precision_string {
+	my ($x) = @_;
+	if ($x) {
+		return sprintf "%." . precision($x) . "f", $x;
+	} else {
+		return "0";
+	}
+}
+
+
+# Autoload methods go after =cut, and are processed by the autosplit program.
+
+1;
+__END__
+# Below is the stub of documentation for your module. You better edit it!
+
+=head1 NAME
+
+Statistics::Distributions - Perl module for calculating critical values and upper probabilities of common statistical distributions
+
+=head1 SYNOPSIS
+
+  use Statistics::Distributions;
+
+  $chis=Statistics::Distributions::chisqrdistr (2,.05);
+  print "Chi-squared-crit (2 degrees of freedom, 95th percentile "
+       ."= 0.05 level) = $chis\n";
+  
+  $u=Statistics::Distributions::udistr (.05);
+  print "u-crit (95th percentile = 0.05 level) = $u\n";
+  
+  $t=Statistics::Distributions::tdistr (1,.005);
+  print "t-crit (1 degree of freedom, 99.5th percentile = 0.005 level) "
+       ."= $t\n";
+  
+  $f=Statistics::Distributions::fdistr (1,3,.01);
+  print "F-crit (1 degree of freedom in numerator, 3 degrees of freedom "
+       ."in denominator, 99th percentile = 0.01 level) = $f\n";
+  
+  $uprob=Statistics::Distributions::uprob (-0.85);
+  print "upper probability of the u distribution (u = -0.85): Q(u) "
+       ."= 1-G(u) = $uprob\n";
+  
+  $chisprob=Statistics::Distributions::chisqrprob (3,6.25);
+  print "upper probability of the chi-square distribution (3 degrees "
+       ."of freedom, chi-squared = 6.25): Q = 1-G = $chisprob\n";
+  
+  $tprob=Statistics::Distributions::tprob (3,6.251);
+  print "upper probability of the t distribution (3 degrees of "
+       ."freedom, t = 6.251): Q = 1-G = $tprob\n";
+  
+  $fprob=Statistics::Distributions::fprob (3,5,.625);
+  print "upper probability of the F distribution (3 degrees of freedom "
+       ."in numerator, 5 degrees of freedom in denominator, F = 6.25): "
+       ."Q = 1-G = $fprob\n";
+
+=head1 DESCRIPTION
+
+This Perl module calculates percentage points (5 significant digits) of the u (standard normal) distribution, the student's t distribution, the chi-square distribution and the F distribution. It can also calculate the upper probability (5 significant digits) of the u (standard normal), the chi-square, the t and the F distribution.
+These critical values are needed to perform statistical tests, like the u test, the t test, the F test and the chi-squared test, and to calculate confidence intervals.
+
+If you are interested in more precise algorithms you could look at:
+ StatLib: http://lib.stat.cmu.edu/apstat/ ; 
+ Applied Statistics Algorithms by Griffiths, P. and Hill, I.D., Ellis Horwood: Chichester (1985)
+
+=head1 BUGS 
+
+This final version 1.02 has been released after more than one year without a bug report on the previous version 0.07.
+Nevertheless, if you find any bugs or oddities, please do inform the author. 
+
+=head1 INSTALLATION 
+
+See perlmodinstall for information and options on installing Perl modules. 
+
+=head1 AVAILABILITY 
+
+The latest version of this module is available from the Distribution Perl Archive Network (CPAN). Please visit http://www.cpan.org/ to find a CPAN site near you or see http://www.cpan.org/authors/id/M/MI/MIKEK/ .
+
+=head1 AUTHOR
+
+Michael Kospach <mike.perl@gmx.at>
+
+Nice formating, simplification and bug repair by Matthias Trautner Kromann <mtk@id.cbs.dk>
+
+=head1 COPYRIGHT 
+
+Copyright 2003 Michael Kospach. All rights reserved. 
+
+This library is free software; you can redistribute it and/or modify it under the same terms as Perl itself. 
+
+=head1 SEE ALSO
+
+Statistics::ChiSquare, Statistics::Table::t, Statistics::Table::F, perl(1).
+
+=cut
+
+
+
+
+
+
+
+
+