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], y values are \$xy[\$i].  (The
module does not access any indices greater than \$xy[\$i], 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.

```

#### SEEALSO

```        Mendenhall, W., and Sincich, T.L., 2003, A Second Course in Statistics:
Regression Analysis, 6th ed., Prentice Hall.
Press, W. H., Flannery, B. P., Teukolsky, S. A., Vetterling, W. T., 1992,
Numerical Recipes in C : The Art of Scientific Computing, 2nd ed.,
Cambridge University Press.
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 (version 0.07)
are:

Statistics::LineFit is more robust.
Statistics::OLS returns incorrect results for certain input datasets.  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 returns correct results.  In contrast, Statistics::LineFit has over 200
installation tests that use various datasets/calling sequences to verify 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.

```

#### AUTHOR

```       Richard Anderson, cpan(AT)richardanderson(DOT)org, http://www.richardanderson.org

```

```       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.
```