#!/usr/bin/perl -w package Regression; \$VERSION = 0.1; use strict; ################################################################ use constant TINY => 1e-8; use constant DEBUGGING => 0; ################################################################ =pod =head1 NAME Regression.pm - weighted linear regression package (line+plane fitting) =head1 DESCRIPTION Regression.pm is a multivariate linear regression package. That is, it estimates the c coefficients for a line-fit of the type y= b(0)*x(0) + b(1)*x1 + b(2)*x2 + ... + b(k)*xk given a data set of N observations, each with k independent x variables and one y variable. Naturally, N must be greater than k---and preferably considerably greater. Any reasonable undergraduate statistics book will explain what a regression is. Most of the time, the user will provide a constant ('1') as x(0) for each observation in order to allow the regression package to fit an intercept. =head1 USAGE If the sample data for (x1, x2, y) includes (\$x1[\$i], \$x2[\$i], \$y[\$i]) for 0<=\$i<=5, type \$reg = Regression->new( 3, "y", [ "const", "x1", "x2" ] ); for(\$i=0; \$i<6; \$i++){ \$reg->include( \$y[\$i], [ 1.0, \$x1[\$i], \$x2[\$i] ] ); } @coeff= \$reg->theta(); \$b0 = \$coeff[0][0]; \$b1 = \$coeff[0][1]; \$b2 = \$coeff[0][2]; =head1 ALGORITHM =head2 Original Algorithm (ALGOL-60): W. M. Gentleman, University of Waterloo, "Basic Description For Large, Sparse Or Weighted Linear Least Squares Problems (Algorithm AS 75)," Applied Statistics (1974) Vol 23; No. 3 =head2 INTERNALS R=Rbar is an upperright triangular matrix, kept in normalized form with implicit 1's on the diagonal. D is a diagonal scaling matrix. These correspond to "standard Regression usage" as X' X = R' D R A backsubsitution routine (in thetacov) allows to invert the R matrix (the inverse is upper-right triangular, too!). Call this matrix H, that is H=R^(-1). (X' X)^(-1) = [(R' D^(1/2)') (D^(1/2) R)]^(-1) = [ R^-1 D^(-1/2) ] [ R^-1 D^(-1/2) ]' =head2 Remarks This algorithm is the statistical "standard." Insertion of a new observation can be done one obs at any time (WITH A WEIGHT!), and still only takes a low quadratic time. The storage space requirement is of quadratic order (in the indep variables). A practically infinite number of observations can easily be processed! =head1 AUTHOR Naturally, Gentleman invented this algorithm. Adaptation by ivo welch. Alan Miller (alan@dmsmelb.mel.dms.CSIRO.AU) pointed out nicer ways to compute the R^2. =head1 Subroutines =cut ################################################################ #### let's start with handling of missing data ("nan" or "NaN") my \$nan= "NaN"; sub isNaN { if (\$_[0] !~ /[0-9nan]/) { die "definitely not a number in NaN: '\$_[0]'"; } return (\$_[0]=~ /NaN/i) || (\$_[0] != \$_[0]); } ################################################################ =pod =head2 new receives the number of variables on each observations (i.e., an integer) and returns the blessed data structure. Also takes an optional name for this regression to remember, as well as a reference to a k*1 array of names for the X coefficients. =cut ################################################################ sub new { my \$classname= shift(@_); my \$K= shift(@_); # the number of variables my \$regname= shift(@_) || "with no name"; if (!defined(\$K)) { die "Regression->new needs at least one argument for the number of variables"; } if (\$K<=1) { die "Cannot run a regression without at least two variables."; } sub zerovec { my @rv; for (my \$i=0; \$i<=\$_[0]; ++\$i) { \$rv[\$i]=0; } return \@rv; } bless { k => \$K, regname => \$regname, xnames => shift(@_), # constantly updated n => 0, sse => 0, syy => 0, sy => 0, wghtn => 0, d => zerovec(\$K), thetabar => zerovec(\$K), rbarsize => (\$K+1)*\$K/2+1, rbar => zerovec((\$K+1)*\$K/2+1), # other constants neverabort => 0, # computed on demand theta => undef, sigmasq => undef, rsq => undef, adjrsq => undef }, \$classname; } ################################################################ =pod =head2 dump is used for debugging. =cut ################################################################ sub dump { my \$this= \$_[0]; print "****************************************************************\n"; print "Regression '\$this->{regname}'\n"; print "****************************************************************\n"; sub print1val { no strict; print "\$_[1](\$_[2])=\t". ((defined(\$_[0]->{ \$_[2] }) ? \$_[0]->{ \$_[2] } : "intentionally undef")); my \$ref=\$_[0]->{ \$_[2] }; if (ref(\$ref) eq 'ARRAY') { my \$arrayref= \$ref; print " \$#\$arrayref+1 elements:\n"; if (\$#\$arrayref>30) { print "\t"; for(my \$i=0; \$i<\$#\$arrayref+1; ++\$i) { print "\$i='\$arrayref->[\$i]';"; } print "\n"; } else { for(my \$i=0; \$i<\$#\$arrayref+1; ++\$i) { print "\t\$i=\t'\$arrayref->[\$i]'\n"; } } } elsif (ref(\$ref) eq 'HASH') { my \$hashref= \$ref; print " ".scalar(keys(%\$hashref))." elements\n"; while (my (\$key, \$val) = each(%\$hashref)) { print "\t'\$key'=>'\$val';\n"; } } else { print " [was scalar]\n"; } } while (my (\$key, \$val) = each(%\$this)) { \$this->print1val(\$key, \$key); } print "****************************************************************\n"; } ################################################################ =pod =head2 print prints the estimated coefficients, and R^2 and N. =cut ################################################################ sub print { my \$this= \$_[0]; print "****************************************************************\n"; print "Regression '\$this->{regname}'\n"; print "****************************************************************\n"; my \$theta= \$this->theta(); for (my \$i=0; \$i< \$this->k(); ++\$i) { print "Theta[\$i".(defined(\$this->{xnames}->[\$i]) ? "='\$this->{xnames}->[\$i]'":"")."]= ".sprintf("%12.4f", \$theta->[\$i])."\n"; } print "R^2= ".sprintf("%.3f", \$this->rsq()).", N= ".\$this->n()."\n"; print "****************************************************************\n"; } ################################################################ =pod =head2 include receives one new observation. Call is \$blessedregr->include( \$yvariable, [ \$x1, \$x2, \$x3 ... \$xk ], 1.0 ); where 1.0 is an (optional) weight. Note that inclusion with a weight of -1 can be used to delete an observation. The function returns the number of observations so far included. =cut ################################################################ sub include { my \$this = shift(); my \$yelement= shift(); my \$xrow= shift(); my \$weight= shift() || 1.0; # omit observations with missing observations; if (!defined(\$yelement)) { die "Internal Error: yelement is undef"; } if (isNaN(\$yelement)) { return \$this->{n}; } my @xcopy; for (my \$i=1; \$i<=\$this->{k}; ++\$i) { if (!defined(\$xrow->[\$i-1])) { die "Internal Error: xrow [ \$i-1 ] is undef"; } if (isNaN(\$xrow->[\$i-1])) { return \$this->{n}; } \$xcopy[\$i]= \$xrow->[\$i-1]; } \$this->{syy}+= (\$weight*(\$yelement*\$yelement)); \$this->{sy}+= (\$weight*(\$yelement)); if (\$weight>=0.0) { ++\$this->{n}; } else { --\$this->{n}; } \$this->{wghtn}+= \$weight; for (my \$i=1; \$i<=\$this->{k};++\$i) { if (\$weight==0.0) { return \$this->{n}; } if (abs(\$xcopy[\$i])>(TINY)) { my \$xi=\$xcopy[\$i]; my \$di=\$this->{d}->[\$i]; my \$dprimei=\$di+\$weight*(\$xi*\$xi); my \$cbar= \$di/\$dprimei; my \$sbar= \$weight*\$xi/\$dprimei; \$weight*=(\$cbar); \$this->{d}->[\$i]=\$dprimei; my \$nextr=int( ((\$i-1)*( (2.0*\$this->{k}-\$i))/2.0+1) ); if (!(\$nextr<=\$this->{rbarsize}) ) { die "Internal Error 2"; } my \$xk; for (my \$kc=\$i+1;\$kc<=\$this->{k};++\$kc) { \$xk=\$xcopy[\$kc]; \$xcopy[\$kc]=\$xk-\$xi*\$this->{rbar}->[\$nextr]; \$this->{rbar}->[\$nextr]= \$cbar * \$this->{rbar}->[\$nextr]+\$sbar*\$xk; ++\$nextr; } \$xk=\$yelement; \$yelement-= \$xi*\$this->{thetabar}->[\$i]; \$this->{thetabar}->[\$i]= \$cbar*\$this->{thetabar}->[\$i]+\$sbar*\$xk; } } \$this->{sse}+=\$weight*(\$yelement*\$yelement); # indicate that Theta is garbage now \$this->{theta}= undef; \$this->{sigmasq}= undef; \$this->{rsq}= undef; \$this->{adjrsq}= undef; return \$this->{n}; } ################################################################ =pod =head2 theta estimates and returns the vector of coefficients. =cut ################################################################ sub theta { my \$this= shift(); if (defined(\$this->{theta})) { return \$this->{theta}; } if (\$this->{n} < \$this->{k}) { return undef; } for (my \$i=(\$this->{k}); \$i>=1; --\$i) { \$this->{theta}->[\$i]= \$this->{thetabar}->[\$i]; my \$nextr= int ((\$i-1)*((2.0*\$this->{k}-\$i))/2.0+1); if (!(\$nextr<=\$this->{rbarsize})) { die "Internal Error 3"; } for (my \$kc=\$i+1;\$kc<=\$this->{k};++\$kc) { \$this->{theta}->[\$i]-=(\$this->{rbar}->[\$nextr]*\$this->{theta}->[\$kc]); ++\$nextr; } } my \$ref= \$this->{theta}; shift(@\$ref); # we are counting from 0 # if in a scalar context, otherwise please return the array directly return \$this->{theta}; } ################################################################ =pod =head2 rsq, adjrsq, sigmasq, ybar, sst, k, n These functions provide common auxiliary information. rsq, adjrsq, sigmasq, sst, and ybar have not been checked but are likely correct. The results are stored for later usage, although this is somewhat unnecessary because the computation is so simple anyway. =cut ################################################################ sub rsq { my \$this= shift(); return \$this->{rsq}= 1.0- \$this->{sse} / \$this->sst(); } sub adjrsq { my \$this= shift(); return \$this->{adjrsq}= 1.0- (1.0- \$this->rsq())*(\$this->{n}-1)/(\$this->{n} - \$this->{k}); } sub sigmasq { my \$this= shift(); return \$this->{sigmasq}= (\$this->{n}<=\$this->{k}) ? "Inf" : (\$this->{sse}/(\$this->{n} - \$this->{k})); } sub ybar { my \$this= shift(); return \$this->{ybar}= \$this->{sy}/\$this->{wghtn}; } sub sst { my \$this= shift(); return \$this->{sst}= (\$this->{syy} - \$this->{wghtn}*(\$this->ybar())**2); } sub k { my \$this= shift(); return \$this->{k}; } sub n { my \$this= shift(); return \$this->{n}; } ################################################################ =pod =head1 DEBUGGING = SAMPLE USAGE CODE The sample code included with this package demonstrates regression usage. To execute it, just set the constant DEBUGGING at the script head to 1, and do perl Regression.pm The printout should be **************************************************************** Regression 'sample regression' **************************************************************** Theta[0='const']= 0.2950 Theta[1='someX']= 0.6723 Theta[2='someY']= 1.0688 R^2= 0.808, N= 4 **************************************************************** =cut ################################################################ if (DEBUGGING) { package main; my \$reg= Statistics::Regression->new( 3, "sample regression", [ "const", "someX", "someY" ] ); \$reg->include( 2.0, [ 1.0, 3.0, -1.0 ] ); \$reg->include( 1.0, [ 1.0, 5.0, 2.0 ] ); \$reg->include( 20.0, [ 1.0, 31.0, 0.0 ] ); \$reg->include( 15.0, [ 1.0, 11.0, 2.0 ] ); # \$reg->print(); or: my \$coefs= \$reg->theta(); print @coefs; print \$reg->rsq; # my \$coefs= \$reg->theta(); print \$coeff[0]; } ################################################################ =pod =head1 BUGS/PROBLEMS =over 4 =item Missing This package lacks routines to compute the standard errors of the coefficients. This requires access to a matrix inversion package, and I do not have one at my disposal. If you want to add one, please let me know. =item Perl Problem perl is unaware of IEEE number representations. This makes it a pain to test whether an observation contains any missing variables (coded as 'NaN' in Regression.pm). =item Others I am a novice perl programmer, so this is probably ugly code. However, it does seem to work, and I could not find anything equivalent on cpan. =back =head1 INSTALLATION and DOCUMENTATION Installation consists of moving the file 'Regression.pm' into a subdirectory Statistics of your modules path (e.g., /usr/lib/perl5/site_perl/5.6.0/). The documentation was produced from the module: pod2html -noindex -title "perl weighted least squares regression package" Regression.pm > Regression.html The documentation was slightly modified by Maria Voloshina, University of Rochester. =head1 LICENSE This module is released for free public use under a GPL license. (C) Ivo Welch, 2001. =cut ################################################################ 1;