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```Statistics::LineFit(3)User Contributed Perl DocumentatioStatistics::LineFit(3)

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

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.

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.

perl v5.24.1			  2004-09-02		Statistics::LineFit(3)
```

NAME | SYNOPSIS | DESCRIPTION | ALGORITHM | LIMITATIONS | EXAMPLES | METHODS | SEE ALSO | AUTHOR | LICENSE | DISCLAIMER

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