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, the mean squared error,
   sigma, t statistics, the predicted y values and the 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 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 certain measurements 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.

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 module Statistics::OLS.

   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 will return incorrect results
       (e.g., only two data points). 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 (or at least different) 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 0.01. 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.

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