See the file INSTALL for installation instructions.
Contents:
NAME
SYNOPSIS
DESCRIPTION
ALGORITHM
EXAMPLES
METHODS
LIMITATIONS
SEE ALSO
VERSION
AUTHOR
LICENSE
DISCLAIMER
NAME
Statistics::LineFit - Least squares line fit, weighted or unweighted
SYNOPSIS
use Statistics::LineFit;
$lineFit = Statistics::LineFit->new();
$lineFit->setData (\@xValues, \@yValues) or die "Invalid data";
($intercept, $slope) = $lineFit->coefficients();
defined $intercept or die "Can't fit line if x values are all equal";
$rSquared = $lineFit->rSquared();
$meanSquaredError = $lineFit->meanSqError();
$durbinWatson = $lineFit->durbinWatson();
$sigma = $lineFit->sigma();
($tStatIntercept, $tStatSlope) = $lineFit->tStatistics();
@predictedYs = $lineFit->predictedYs();
@residuals = $lineFit->residuals();
DESCRIPTION
The Statistics::LineFit module does weighted or unweighted least-squares
line fitting to two-dimensional data (y = a + b * x). (This is also
called linear regression.) In addition to the slope and y-intercept, the
module can return the Durbin-Watson statistic, mean squared error,
sigma, t statistics, predicted y values and residuals of the y values.
See the METHODS section for a description of these statistics. See the
SEE ALSO section for a comparison of this module to Statistics::OLS.
The module accepts input data in separate x and y arrays or a single 2-D
array (an array of arrayrefs). The optional weights are input in a
separate array. The module can optionally verify that the input data and
weights are valid numbers. If weights are input, the returned statistics
all reflect the effect of the weights. For example, meanSqError()
returns the weighted mean squared error and rSquared() returns the
weighted correlation coefficient.
The module is state-oriented and caches its results. Once you call the
setData() method, you can call the other methods in any order or call a
method several times without invoking redundant calculations.
The regression fails if the x values are all the same. This is an
inherent limit to fitting a line of the form y = a + b * x. In this
case, the module issues an error message and methods that return
statistical values will return undefined values. You can also use the
return value of the regress() method to check the status of the
regression.
The decision to use or not use weighting could be made using your a
priori knowledge of the data or using supplemental data. In the presence
of non-random noise weighting can degrade the solution. Weighting is a
good option if some points are suspect or less relevant (e.g., older
terms in a time series, data from a suspect source).
ALGORITHM
The least-square line is the line that minimizes the sum of the squares
of the y residuals:
Minimize SUM((y[i] - (a + b * x[i])) ** 2)
Setting the parial derivatives of a and b to zero yields a solution that
can be expressed in terms of the means, variances and covariances of x
and y:
b = SUM((x[i] - meanX) * (y[i] - meanY)) / SUM((x[i] - meanX) ** 2)
a = meanY - b * meanX
If you use weights, each term in the sums is multiplied by the value of
the weight for that index. Note that a and b are undefined if all the x
values are the same.
Statistics::LineFit uses equations that are mathematically equivalent to
the above equations and computationally more efficient. The module runs
in O(N) (linear time).
EXAMPLES
Alternate calling sequence:
use Statistics::LineFit;
$lineFit = Statistics::LineFit->new();
$lineFit->setData(\@x, \@y) or die "Invalid regression data\n";
if (defined $lineFit->rSquared()
and $lineFit->rSquared() > $threshold)
{
($intercept, $slope) = $lineFit->coefficients();
print "Slope: $slope Y-intercept: $intercept\n";
}
Multiple calls with the same object, validate input:
use Statistics::LineFit;
$lineFit = Statistics::LineFit->new(1);
while (1) {
@xy = read2Dxy(); # User-supplied subroutine
last unless @xy;
next unless $lineFit->setData(\@xy);
($intercept, $slope) = $lineFit->coefficients();
if (defined $intercept) {
print "Slope: $slope Y-intercept: $intercept\n";
}
}
METHODS
The module is state-oriented and caches its results. Once you call the
setData() method, you can call the other methods in any order or call a
method several times without invoking redundant calculations.
The regression fails if the x values are all the same. In this case, the
module issues an error message and methods that return statistical
values will return undefined values. You can also use the return value
of the regress() method to check the status of the regression.
new() - create a new Statistics::LineFit object
$lineFit = Statistics::LineFit->new();
$lineFit = Statistics::LineFit->new($validate);
$lineFit = Statistics::LineFit->new($validate, $hush);
$validate = 1 -> Verify input data is numeric (slower execution)
0 -> Don't verify input data (default, faster execution)
$hush = 1 -> Suppress error messages
= 0 -> Enable warning messages (default)
coefficients() - Return the slope and y intercept
($intercept, $slope) = $lineFit->coefficients();
The returned values are undefined if the regression fails.
durbinWatson() - Return the Durbin-Watson statistic
$durbinWatson = $lineFit->durbinWatson();
The Durbin-Watson test is a test for first-order autocorrelation in the
residuals of a time series regression. The Durbin-Watson statistic has a
range of 0 to 4; a value of 2 indicates there is no autocorrelation.
The return value is undefined if the regression fails. If weights are
input, the return value is the weighted Durbin-Watson statistic.
meanSqError() - Return the mean squared error
$meanSquaredError = $lineFit->meanSqError();
The return value is undefined if the regression fails. If weights are
input, the return value is the weighted mean squared error.
predictedYs() - Return the predicted y values
@predictedYs = $lineFit->predictedYs();
The returned values are undefined if the regression fails.
regress() - Do the least squares line fit (if not already done)
$lineFit->regress() or die "Regression failed"
You don't need to call this method because it is invoked by the other
methods as needed. You can call regress() at any time to get the status
of the regression for the current data.
residuals() - Return predicted y values minus input y values
@residuals = $lineFit->residuals();
The returned values are undefined if the regression fails.
rSquared() - Return the correlation coefficient
$rSquared = $lineFit->rSquared();
R squared, also called the correlation coefficient, is a measure of
goodness-of-fit. It is the fraction of the variation in Y that can be
attributed to the variation in X. A perfect fit will have an R squared
of 1; an attempt to fit a line to the vertices of a regular polygon will
yield an R squared of zero. Graphical displays of data with an R squared
of less than about 0.1 do not show a visible linear trend.
The return value is undefined if the regression fails. If weights are
input, the return value is the weighted correlation coefficient.
setData() - Initialize (x,y) values and optional weights
$lineFit->setData(\@x, \@y) or die "Invalid regression data";
$lineFit->setData(\@x, \@y, \@weights) or die "Invalid regression data";
$lineFit->setData(\@xy) or die "Invalid regression data";
$lineFit->setData(\@xy, \@weights) or die "Invalid regression data";
If the new() method was called with validate = 1, setData() will verify
that the data and weights are valid numbers. @xy is an array of
arrayrefs; x values are $xy[$i][0], y values are $xy[$i][1]. The module
does not access any indices greater than $xy[$i][1], so the arrayrefs
can point to arrays that are longer than two elements.
The optional weights array must be the same length as the data arrays.
The weights must be non-negative numbers. Only the relative size of the
weights is significant: the results are not affected if the weights are
all multiplied by a constant. If you want to do multiple line fits using
the same weights, the weights must be passed to each call to setData().
Once you successfully call setData(), the next call to any other method
invokes the regression.
sigma() - Return the standard error of the estimate
$sigma = $lineFit->sigma();
Sigma is an estimate of the homoscedastic standard deviation of the
error. Sigma is also known as the standard error of the estimate.
The return value is undefined if the regression fails. If weights are
input, the return value is the weighted standard error.
tStatistics() - Return the t statistics
(tStatIntercept, $tStatSlope) = $lineFit->tStatistics();
The t statistic, also called the t ratio or Wald statistic, is used to
accept or reject a hypothesis using a table of cutoff values computed
from the t distribution. The t-statistic suggests that the estimated
value is (reasonable, too small, too large) when the t-statistic is
(close to zero, large and positive, large and negative).
The returned values are undefined if the regression fails. If weights
are input, the returned values are the weighted t statistics.
LIMITATIONS
The module cannot fit a line to a set of points that have the same x
values. This is an inherent limit to fitting a line of the form y = a +
b * x. As the sum of the squared deviations of the x values approaches
zero, the module's results becomes unstable and sensitive to the
precision of floating point operations on the host system.
If the x values are not all the same and the apparent "best fit" line is
vertical, the module will fit a horizontal line. For example, an input
of (1, 1), (2, 3), (2, 5), (1, 7) returns a slope of zero, an intercept
of 4 and an R squared of zero. This is correct behavior because this is
the best least-squares line fit to the data for the given
parameterization (y = a + b * x).
On a 32-bit system the results are accurate to about 11 significant
digits, depending on the input data. Many of the installation tests will
fail on a system with word lengths of 16 bits or fewer. (You might want
to upgrade your old Intel 80286 IBM PC.)
SEE ALSO
Mendenhall, W., and Sincich, T.L., 2003, A Second Course in Statistics:
Regression Analysis, 6th ed., Prentice Hall.
The man page for perl(1).
The CPAN modules Statistics::OLS, Statistics::GaussHelmert and
Statistics::Regression.
Statistics::LineFit is simpler to use than Statistics::GaussHelmert or
Statistics::Regression. Statistics::LineFit was inspired by and borrows
some ideas from the venerable Statistics::OLS module. The significant
differences between Statistics::LineFit and Statistics::OLS are:
Statistics::LineFit is more robust.
For certain datasets Statistics::OLS returns incorrect results.
Statistics::OLS does not deep copy its input arrays, which can lead
to subtle bugs. The Statistics::OLS installation test has only one
test and does not verify that the regression returned correct
results. In contrast, Statistics::LineFit has over 200 installation
tests that use various datasets / calling sequences and it verifies
the accuracy of the regression to within 1.0e-10.
Statistics::LineFit is faster.
For a sequence of calls to new(), setData(\@x, \@y) and regress(),
Statistics::LineFit is faster than Statistics::OLS by factors of
2.0, 1.6 and 2.4 for array lengths of 5, 100 and 10000,
respectively.
Statistics::LineFit can do weighted or unweighted regression.
Statistics::OLS lacks this option.
Statistics::LineFit has a better interface.
Once you call the Statistics::LineFit::setData() method, you can
call the other methods in any order and call methods multiple times
without invoking redundant calculations. Statistics::LineFit lets
you enable or disable data verification or error messages.
Statistics::LineFit has better code and documentation.
The code in Statistics::LineFit is more readable, more object
oriented and more compliant with Perl coding standards than the code
in Statistics::OLS. The documentation for Statistics::LineFit is
more detailed and complete.
VERSION
This document describes Statistics::LineFit version 0.03. The comments
about Statistics::OLS refer to version 0.07 of that module.
AUTHOR
Richard Anderson, cpan(AT)richardanderson(DOT)org,
http://www.richardanderson.org
LICENSE
This program is free software; you can redistribute it and/or modify it
under the same terms as Perl itself.
The full text of the license can be found in the LICENSE file included
in the distribution and available in the CPAN listing for
Statistics::LineFit (see www.cpan.org or search.cpan.org).
DISCLAIMER
To the maximum extent permitted by applicable law, the author of this
module disclaims all warranties, either express or implied, including
but not limited to implied warranties of merchantability and fitness for
a particular purpose, with regard to the software and the accompanying
documentation.