Regression.pm - weighted linear regression package (line+plane fitting)
use Statistics::Regression;
# Create regression object
my $reg = Statistics::Regression->new(
3, "sample regression",
[ "const", "someX", "someY" ]
);
# Add data points
$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 ] );
Please note that you must provide the constant.
# Print the result $reg->print();
This prints the following:
**************************************************************** Regression 'sample regression' **************************************************************** Name Theta StdErr T-stat [0='const'] 0.2950 6.0512 0.05 [1='someX'] 0.6723 0.3278 2.05 [2='someY'] 1.0688 2.7954 0.38 R^2= 0.808, N= 4 ****************************************************************
Or, use the subroutines to do the work yourself:
my @theta = $reg->theta(); my @se = $reg->standarderrors(); my $rsq = $reg->rsq(); my $adjrsq = $reg->adjrsq(); my $ybar = $reg->ybar(); ## the average of the y vector my $sst = $reg->sst(); ## the sum-squares-total my $sigmasq= $reg->sigmasq(); ## the variance of the residual my $k = $reg->k(); ## the number of variables my $n = $reg->n(); ## the number of observations
In addition, there are some other helper routines, and a subroutine linearcombination_variance(). If you don't know what this is, don't use it.
You should have an understanding of OLS regressions if you want to use this package. You can get this from an introductory college econometrics class and/or from most intermediate college statistics classes. If you do not have this background knowledge, then this package will remain a mystery to you. There is no support for this package--please don't expect any.
Regression.pm is a multivariate linear regression package. That is, it estimates the c coefficients for a line-fit of the type
y= c(0)*x(0) + c(1)*x1 + c(2)*x2 + ... + c(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.
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
Gentleman's algorithm is the statistical standard. Insertion of a new observation can be done one observation 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!
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) ]'
None known.
Unfortunately, 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).
0.50, 2007/04/04
Naturally, Gentleman invented this algorithm. It was adaptated by Ivo Welch. Alan Miller (alan\@dmsmelb.mel.dms.CSIRO.AU) pointed out nicer ways to compute the R^2. Ivan Tubert-Brohman helped wrap the module as as a standard CPAN distribution.
This module is released for free public use under a GPL license.
(C) Ivo Welch, 2001,2004, 2007.
2007/04/04: Added Coefficient Standard Errors
