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