Ubuntu Manpage: Statistics::LineFit - Least squares line fit, weighted or unweighted

Provided by: libstatistics-linefit-perl_0.07-1_all

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();
        (varianceIntercept, $varianceSlope) = $lineFit->varianceOfEstimates();

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 square of the correlation
coefficient (R squared), the Durbin-Watson statistic, the mean squared error, sigma, the t
statistics, the variance of the estimates of the slope and y-intercept, the predicted y
values and the residuals of the y values. (See the METHODS section for a description of
these statistics.)

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 line fit minimizes the weighted sum of the squared errors and the following
statistics are weighted: the correlation coefficient, the Durbin-Watson statistic, the
mean squared error, sigma and the t statistics.

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. After calling setData(), you can modify the input data
or weights without affecting the module's results.

The decision to use or not use weighting could be made using your a priori knowledge of
the data or using supplemental data. If the data is sparse or contains 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, points that are known to have more
noise).

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

       Note that a and b are undefined if all the x values are the same.

       If you use weights, each term in the above sums is multiplied by the value of the weight
       for that index.  The program normalizes the weights (after copying the input values) so
       that the sum of the weights equals the number of points.  This minimizes the differences
       between the weighted and unweighted equations.

       Statistics::LineFit uses equations that are mathematically equivalent to the above
       equations and computationally more efficient.  The module runs in O(N) (linear time).

LIMITATIONS

       The regression fails if the input x values are all equal or the only unequal x values have
       zero weights.  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.

       As the sum of the squared deviations of the x values approaches zero, the module's results
       becomes 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), (1, 7), (2, 3), (2,
       5) returns a slope of zero, an intercept of 4 and an R squared of zero.  This is correct
       behavior because this line is the best least-squares 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 80286 IBM PC.)

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 same object, validate input, suppress error messages:
        use Statistics::LineFit;
        $lineFit = Statistics::LineFit->new(1, 1);
        while (1) {
            @xy = read2Dxy();  # User-supplied subroutine
            $lineFit->setData(\@xy);
            ($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 error messages (default)

   coefficients() - Return the slope and y intercept
        ($intercept, $slope) = $lineFit->coefficients();

       The returned list is 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 list is 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.
       After you call setData(), 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 list is undefined if the regression fails.

   rSquared() - Return the square of the correlation coefficient
        $rSquared = $lineFit->rSquared();

       R squared, also called the square of the Pearson product-moment 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; fitting 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";

       @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 method identifies the difference between
       the first and fourth calling signatures by examining the first argument.

       The optional weights array must be the same length as the data array(s).  The weights must
       be non-negative numbers; at least two of the weights must be nonzero.  Only the relative
       size of the weights is significant: the program normalizes the weights (after copying the
       input values) so that the sum of the weights equals the number of points.  If you want to
       do multiple line fits using the same weights, the weights must be passed to each call to
       setData().

       The method will return zero if the array lengths don't match, there are less than two data
       points, any weights are negative or less than two of the weights are nonzero. If the new()
       method was called with validate = 1, the method will also verify that the data and weights
       are valid numbers.  Once you successfully call setData(), the next call to any method
       other than new() or setData() invokes the regression.  You can modify the input data or
       weights after calling setData() without affecting the module's results.

   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 list is undefined if the regression fails.  If weights are input, the
       returned values are the weighted t statistics.

   varianceOfEstimates() - Return variances of estimates of intercept, slope
        (varianceIntercept, $varianceSlope) = $lineFit->varianceOfEstimates();

       Assuming the data are noisy or inaccurate, the intercept and slope returned by the
       coefficients() method are only estimates of the true intercept and slope.  The
       varianceofEstimate() method returns the variances of the estimates of the intercept and
       slope, respectively.  See Numerical Recipes in C, section 15.2 (Fitting Data to a Straight
       Line), equation 15.2.9.

       The returned list is undefined if the regression fails.  If weights are input, the
       returned values are the weighted variances.

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.