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-#
-# Author: Peter J. Acklam
-# Time-stamp: 2000-06-02 23:42:57
-# E-mail: pjacklam@online.no
-# URL: http://home.online.no/~pjacklam
-
-=head1 NAME
-
-Statistics::Distrib::Normal - the normal distribution
-
-=head1 SYNOPSIS
-
- use Statistics::Distrib::Normal;
-
- $dist = new Statistics::Distrib::Normal;
-
- $dist->mu(3); # set the location parameter
- $dist->sigma(5); # set the scale parameter
- @x = $dist->rand(10); # generate random numbers
-
- # or
-
- @x = Statistics::Distrib::Normal->new(Mu => 3, Sigma => 5)->rand(10);
-
-=head1 DESCRIPTION
-
-This module contains miscellaneous functions related to the normal
-distribution.
-
-=cut
-
-package Statistics::Distrib::Normal;
-require 5.000;
-
-use strict;
-use vars qw($VERSION);
-use Carp;
-
-$VERSION = '0.01';
-
-use constant PI => 4 * atan2 1, 1;
-use constant TWOPI => 2 * PI;
-
-# the smallest positive floating-point number such that 1+EPS > 1
-use constant EPS => 2.220446049250313080847263336181640625e-016;
-
-##
-## Constructor
-##
-
-=head1 CONSTRUCTOR
-
-=over 4
-
-=item new ( [ OPTIONS ] )
-
-C<OPTIONS> is a list of options given in the form of key-value
-pairs, just like a hash table. Valid options are
-
-=over 8
-
-=item B<Mu>
-
-Sets the mu parameter (the mean) of the distribution to the specified
-value.
-
-=item B<Sigma>
-
-Sets the sigma parameter (the standard deviation) of the distribution
-to the specified value.
-
-=back
-
-=back
-
-=cut
-
-sub new {
- my $self = shift;
- my $class = ref($self) || $self;
- my %arg = @_;
-
- my %hash = ( mu => 0,
- sigma => 1,
- );
-
- my $me = bless \%hash, $class;
-
- foreach my $key ( keys %arg ) {
- $me->mu($arg{Mu}), next if $key eq 'Mu';
- $me->sigma($arg{Sigma}), next if $key eq 'Sigma';
- carp "Unknown option $arg{$key} ignored";
- }
-
- return $me;
-}
-
-##
-## Methods
-##
-
-=pod
-
-=head1 METHODS
-
-=over 4
-
-=item mean ( [ MEAN ] )
-
-Set the mu parameter (the mean) of the distribution to C<MU>. If
-C<MU> is omitted, the current value of mu is returned.
-
-=cut
-
-sub mu {
- my $me = shift;
- croak 'Too many arguments' if @_ > 1;
- if ( @_ ) {
- $me->{mu} = shift;
- return 1;
- }
- return $me->{mu};
-}
-
-=pod
-
-=item sigma ( [ SDEV ] )
-
-Set the sigma parameter (the standard deviation) of the distribution
-to C<SIGMA>. If C<SIGMA> is omitted, the current value of sigma is
-returned.
-
-=cut
-
-sub sigma {
- my $me = shift;
- croak 'Too many arguments' if @_ > 1;
- if ( @_ ) {
- my $sigma = shift;
- croak 'Standard deviation most be positive' unless $sigma > 0;
- $me->{sigma} = $sigma;
- return 1;
- }
- return $me->{sigma};
-}
-
-=pod
-
-=item pdf ( X1 [, X2 [, X3 ... ] ] )
-
-Evaluate the probability density function at C<X1>, C<X2>, C<X3>, ...
-
-=cut
-
-sub pdf {
- my $me = shift;
- croak 'Not enough arguments' unless @_;
- my $mu = $me->{mu};
- my $sigma = $me->{sigma};
- my $const = log(TWOPI * $sigma * $sigma);
- my @f;
- foreach my $x ( @_ ) {
- my $z = ( $x - $mu ) / $sigma;
- push @f, exp( -0.5 * ( $const + $z*$z ) );
- }
- return @f;
-}
-
-=pod
-
-=item ltp ( X1 [, X2 [, ... ] ] )
-
-Evaluate the lower tail probability function at C<X1>, C<X2>, C<X3>,
-...
-
-=cut
-
-sub ltp {
- my $me = shift;
- croak 'Not enough arguments' unless @_;
- my $mu = $me->{mu};
- my $sigma = $me->{sigma};
-
- require Math::SpecFun::Erf;
- import Math::SpecFun::Erf qw(erfc);
-
- my @p;
- foreach my $x ( @_ ) {
- my $z = ( $x - $mu ) / $sigma;
- push @p, erfc( - $_[0] / sqrt(2) )/2;
- }
- return @p;
-}
-
-=pod
-
-=item utp ( X1 [, X2 [, ... ] ] )
-
-Evaluate the upper tail probability function at C<X1>, C<X2>, C<X3>,
-...
-
-=cut
-
-sub utp {
- my $me = shift;
- croak 'Not enough arguments' unless @_;
- my $mu = $me->{mu};
- my $sigma = $me->{sigma};
-
- require Math::SpecFun::Erf;
- import Math::SpecFun::Erf qw(erfc);
-
- my @p;
- foreach my $x ( @_ ) {
- my $z = ( $x - $mu ) / $sigma;
- push @p, erfc( $_[0] / sqrt(2) )/2;
- }
- return @p;
-}
-
-=pod
-
-=item ltq ( P1 [, P2 [, ... ] ] )
-
-Returns the lower tail quantile for the probabilities C<P1>, C<P2>,
-C<P3>, ...
-
-=cut
-
-sub ltq {
- croak 'Method not implemented yet';
-}
-
-=pod
-
-=item utq ( P1 [, P2 [, P3 ... ] ] )
-
-Returns the upper tail quantile for the probabilities C<P1>, C<P2>,
-C<P3>, ...
-
-=cut
-
-sub utq {
- croak 'Method not implemented yet';
-}
-
-=pod
-
-=item intprob( XLOW, XHIGH )
-
-Interval probability. Returns the probability of an outcome in the
-interval from XLOW to XHIGH.
-
-=cut
-
-sub intprob {
- my $me = shift;
- croak 'Bad number of arguments' unless @_ == 2;
- my ($xlow, $xhigh) = @_;
- return 0 unless $xlow < $xhigh;
- my $mu = $me->{mu};
- my $sigma = $me->{sigma};
-
- if ( $mu < $xlow ) {
- return $me->utp($xlow) - $me->utp($xhigh);
- } else {
- return $me->ltp($xhigh) - $me->ltp($xlow);
- }
-}
-
-=pod
-
-=item rand( [ NUM ] )
-
-Generate random variables. If C<NUM> is omitted, a single variable is
-returned.
-
-=cut
-
-sub rand {
- my $me = shift;
- my $num;
- if ( @_ ) {
- $num = shift;
- croak 'Too many arguments' if @_;
- croak 'Argument must be positive integer'
- unless ($num == int $num) && ($num > 0);
- } else {
- $num = 1;
- }
-
- # Generate the random variables by the Box-Muller method.
- my @z;
- my $mu = $me->{mu};
- my $sigma = $me->{sigma};
- my $const = -2 * $sigma * $sigma;
- my $i;
- for ( $i = 0 ; $i < $num ; $i += 2 ) {
- my $r = sqrt $const * log rand;
- my $t = TWOPI * rand;
- push @z, $mu + $r * sin $t, $mu + $r * cos $t;
- }
- pop @z if $i > $num;
- return @z;
-}
-
-=pod
-
-=item expectation ()
-
-Return the expectation of the distribution.
-
-=cut
-
-sub expectation {
- my $me = shift;
- croak 'Too many arguments' if @_;
- return $me->{mu};
-}
-
-=pod
-
-=item variance ()
-
-Return the variance of the distribution.
-
-=cut
-
-sub variance {
- my $me = shift;
- croak 'Too many arguments' if @_;
- return $me->{sigma}**2;
-}
-
-=pod
-
-=item skewness ()
-
-Return the skewness of the distribution.
-
-=cut
-
-sub skewness {
- my $me = shift;
- croak 'Too many arguments' if @_;
- return 0;
-}
-
-=pod
-
-=item kurtosis ()
-
-Return the kurtosis (normalized) of the distribution.
-
-=cut
-
-sub kurtosis {
- my $me = shift;
- croak 'Too many arguments' if @_;
- return 0;
-}
-
-=item dmo
-
-Direct moments for the distribution.
-
-Not implemented yet.
-
-=cut
-
-sub dmo {
- croak 'Method not implemented yet';
-}
-
-=pod
-
-=item cmo
-
-Central moments for the distribution.
-
-=cut
-
-sub cmo {
- croak 'Method not implemented yet';
-}
-
-=pod
-
-=item mode ()
-
-Returns the mode of the distribution.
-
-=cut
-
-sub mode {
- my $me = shift;
- croak 'Too many input arguments' if @_;
- return $me->{mu};
-}
-
-=back
-
-=head1 BUGS
-
-None known.
-
-=head1 LIMITATIONS
-
-Degenerate cases are not allowed for most methods; e.g., a
-distribution with zero variance.
-
-=head1 HISTORY
-
-=over 4
-
-=item Version 0.02
-
-Code formatting changes.
-
-=item Version 0.01
-
-First release.
-
-=back
-
-=head1 AUTHOR
-
-Peter J. Acklam E<lt>pjacklam@online.noE<gt>.
-
-=head1 COPYRIGHT/LICENSE
-
-Copyright (c) 1999-2000 Peter J. Acklam. All rights reserved.
-This program is free software; you can redistribute it and/or
-modify it under the same terms as Perl itself.
-
-=cut
-
-1; # Modules must return true.
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