Statistics::Descriptive - Module of basic descriptive statistical functions.
use Statistics::Descriptive; $stat = Statistics::Descriptive::Full->new(); $stat->add_data(1,2,3,4); $mean = $stat->mean(); $var = $stat->variance(); $tm = $stat->trimmed_mean(.25); $Statistics::Descriptive::Tolerance = 1e-10;
This module provides basic functions used in descriptive statistics. It has an object oriented design and supports two different types of data storage and calculation objects: sparse and full. With the sparse method, none of the data is stored and only a few statistical measures are available. Using the full method, the entire data set is retained and additional functions are available.
Whenever a division by zero may occur, the denominator is checked to be
greater than the value $Statistics::Descriptive::Tolerance
, which
defaults to 0.0. You may want to change this value to some small
positive value such as 1e-24 in order to obtain error messages in case
of very small denominators.
Many of the methods (both Sparse and Full) cache values so that subsequent calls with the same arguments are faster.
Similar to the Sparse Methods above, any Full Method that is called caches the current result so that it doesn't have to be recalculated. In some cases, several values can be cached at the same time.
Adds data to the statistics variable. All of the sparse statistical values are updated and cached. Cached values from Full methods are deleted since they are no longer valid.
Note: Calling add_data with an empty array will delete all of your Full method cached values! Cached values for the sparse methods are not changed
Sorts the data and returns the value that corresponds to the percentile as defined in RFC2330:
For example, given the 6 measurements:
-2, 7, 7, 4, 18, -5
Then F(-8) = 0, F(-5) = 1/6, F(-5.0001) = 0, F(-4.999) = 1/6, F(7) = 5/6, F(18) = 1, F(239) = 1.
Note that we can recover the different measured values and how many times each occurred from F(x) -- no information regarding the range in values is lost. Summarizing measurements using histograms, on the other hand, in general loses information about the different values observed, so the EDF is preferred.
Using either the EDF or a histogram, however, we do lose information regarding the order in which the values were observed. Whether this loss is potentially significant will depend on the metric being measured.
We will use the term "percentile" to refer to the smallest value of x for which F(x) >= a given percentage. So the 50th percentile of the example above is 4, since F(4) = 3/6 = 50%; the 25th percentile is -2, since F(-5) = 1/6 < 25%, and F(-2) = 2/6 >= 25%; the 100th percentile is 18; and the 0th percentile is -infinity, as is the 15th percentile.
Care must be taken when using percentiles to summarize a sample, because they can lend an unwarranted appearance of more precision than is really available. Any such summary must include the sample size N, because any percentile difference finer than 1/N is below the resolution of the sample.
(Taken from: RFC2330 - Framework for IP Performance Metrics, Section 11.3. Defining Statistical Distributions. RFC2330 is available from: http://www.cis.ohio-state.edu/htbin/rfc/rfc2330.html.)
If the percentile method is called in a list context then it will also return the index of the percentile.
trimmed_mean(ltrim)
returns the mean with a fraction ltrim
of entries at each end dropped. trimmed_mean(ltrim,utrim)
returns the mean after a fraction ltrim
has been removed from the
lower end of the data and a fraction utrim
has been removed from the
upper end of the data. This method sorts the data before beginning
to analyze it.
All calls to trimmed_mean() are cached so that they don't have to be calculated a second time.
frequency_distribution($partitions)
slices the data into
$partition
sets (where $partition is greater than 1) and counts the
number of items that fall into each partition. It returns an
associative array where the keys are the numerical values of the
partitions used. The minimum value of the data set is not a key and the
maximum value of the data set is always a key. The number of entries
for a particular partition key are the number of items which are
greater than the previous partition key and less then or equal to the
current partition key. As an example,
$stat->add_data(1,1.5,2,2.5,3,3.5,4); %f = $stat->frequency_distribution(2); for (sort {$a <=> $b} keys %f) { print "key = $_, count = $f{$_}\n"; }
prints
key = 2.5, count = 4 key = 4, count = 3
since there are four items less than or equal to 2.5, and 3 items greater than 2.5 and less than 4.
frequency_distribution(\@bins)
provides the bins that are to be used
for the distribution. This allows for non-uniform distributions as
well as trimmed or sample distributions to be found. @bins
must
be monotonic and contain at least one element. Note that unless the
set of bins contains the range that the total counts returned will
be less than the sample size.
Calling frequency_distribution()
with no arguments returns the last
distribution calculated, if such exists.
least_squares_fit()
performs a least squares fit on the data,
assuming a domain of @x
or a default of 1..$stat->count(). It
returns an array of four elements ($q, $m, $r, $rms)
where
$q and $m
$r
$rms
If case of error or division by zero, the empty list is returned.
The array that is returned can be "coerced" into a hash structure by doing the following:
my %hash = (); @hash{'q', 'm', 'r', 'err'} = $stat->least_squares_fit();
Because calling least_squares_fit()
with no arguments defaults
to using the current range, there is no caching of the results.
I read my email frequently, but since adopting this module I've added 2 children and 1 dog to my family, so please be patient about my response times. When reporting errors, please include the following to help me out:
-v
at
the command line.
Colin Kuskie
My email address can be found at http://www.perl.com under Who's Who or at: http://search.cpan.org/author/COLINK/.
RFC2330, Framework for IP Performance Metrics
The Art of Computer Programming, Volume 2, Donald Knuth.
Handbook of Mathematica Functions, Milton Abramowitz and Irene Stegun.
Probability and Statistics for Engineering and the Sciences, Jay Devore.
Copyright (c) 1997,1998 Colin Kuskie. All rights reserved. This program is free software; you can redistribute it and/or modify it under the same terms as Perl itself.
Copyright (c) 1998 Andrea Spinelli. All rights reserved. This program is free software; you can redistribute it and/or modify it under the same terms as Perl itself.
Copyright (c) 1994,1995 Jason Kastner. All rights reserved. This program is free software; you can redistribute it and/or modify it under the same terms as Perl itself.
Rolled into November 1998
Code provided by Andrea Spinelli to prevent division by zero and to make consistent return values for undefined behavior. Andrea also provided a test bench for the module.
A bug fix for the calculation of frequency distributions. Thanks to Nick Tolli for alerting this to me.
Added 4 lines of code to Makefile.PL to make it easier for the ActiveState installation tool to use. Changes work fine in perl5.004_04, haven't tested them under perl5.005xx yet.
Rolled into March 1998.
Fixed problem with sending 0's and -1's as data. The old 0 : true ? false thing. Use defined to fix.
Provided a fix for AUTOLOAD/DESTROY/Carp bug. Very strange.
August 1997
Fixed errors in statistics algorithms caused by changing the interface.
August 1997
Fixed errors in removing cached values (they weren't being removed!) and added sort_data and presorted methods.
June 1997
Transferred ownership of the module from Jason to Colin.
Rewrote OO interface, modified function distribution, added mindex, maxdex.
April 1995
Added LeastSquaresFit and FrequencyDistribution.
March 1995
Released to comp.lang.perl and placed on archive sites.
December 1994
Complete rewrite after extensive and invaluable e-mail correspondence with Anno Siegel.
December 1994
Initital concept, released to perl5-porters list.