Proposed CPAN module Statistics::LineFit

From: Richard Anderson (gg.2.starfire_at_spamgourmet.com)
Date: 11/18/03 

Date: 17 Nov 2003 16:03:35 -0800

Here are the docs for a module I am preparing for release to CPAN.
Any comments? (See the SEE ALSO section for a comparison to
Statistics::OLS.)

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, 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.

Relevant Pages

  • Re: Polynomial Fit Program
    ... least squares fit program? ... In the simplest case (polynomials of moderate degree), ... If you have weights, e.g. w_i, then first formally ... RWTH - Aachen University ...
    (sci.math.num-analysis)
  • Re: Polynomial Fit Program
    ... least squares fit program? ... Java applet, it might help me as well. ... If you have weights, e.g. w_i, then first formally ...
    (sci.math.num-analysis)
  • Re: Cost Functions
    ... obtain the weights of my estimator according to the minimum least squares ... frequencies is supposed to be small whereas the error at high frequencies ... Seems as a weighted least squares. ...
    (comp.dsp)
  • Re: large Residual in lsqnonlin
    ... Thanks for your suggestions, John. ... > weighted least squares. ... Simply choose weights ... > same, cat. ...
    (comp.soft-sys.matlab)
  • Re: A multiple regression stumper
    ... > variable in a standard multiple regression model. ... > of optimal weights that minimize the variance of e. ... > I would like to compute R2 for non-optimal sets of weights. ...
    (sci.stat.math)