NAME
Statistics::Gap - Perl extension for the "Gap Statistic"

SYNOPSIS
use Statistics::Gap;
&gap("GapPrefix", "InputFile", "squared", "agglo", 5, 100, "unif", 90);

use Statistics::Gap;
&gap("GapPrefix", "InputFile", "squared", "agglo", 5, 100, "prop", 90);

Input file is expected in the "dense" format -
Sample Input file:

6 5
1 1 0 0 1
1 0 0 0 0
1 1 0 0 1
1 1 0 0 1
1 0 0 0 1
1 1 0 0 1

DESCRIPTION
Given a dataset how does one automatically find the optimal number
of clusters that the dataset should be grouped into? - is one of the
prevailing problems. Statisticians Robert Tibshirani, Guenther Walther
and Trevor Hastie propose a solution for this problem in a Techinal
Report named - "Estimating the number of clusters in a dataset via
the Gap Statistic". This perl module implements the approach proposed
in the above paper.

If one tries to cluster a dataset (i.e. numerous observations described
in terms of a feature space) into n groups/clusters and if we plot the
graph of Within Cluster (Dis)Similarity along Y-axis and Number of clusters
along X-axis then this graph generally takes a form of a elbow/knee depending
upon the measure on the Y-axis. The Gap Statistic seeks to locate this
elbow/knee because the value on the X-axis at this elbow is the optimal
number of clusters for the data.

NOTE:
Gap Statistic uses reference distribution in the process of estimating
the number of clusters. The appropriate methodology for generation of this
reference distribution is dependent on the data to be clustered.
This module was implemented for data with following characteristics:
1. highly sparse - very few features occur in any given observation.
2. high multivariate dimensionality (i.e. large feature space)
3. binary feature frequency - feature either occurs or does not occur
in an observation but rarely occurs multiple times in the same observation.

EXPORT
"gap" function by default.

INPUT
Prefix
The string that should be used to as a prefix while naming the
intermediate files and the .png files (graph files).

InputFile
The input dataset is expected in "dense" matrix format.
The input dense matrix is expected in a plain text file where the first
line in the file gives the dimensions of the dataset and then the
dataset in a matrix format should follow. The contexts / observations
should be along the rows and the features should be along the column.

eg:
6 5
1 1 0 0 1
1 0 0 0 0
1 1 0 0 1
1 1 0 0 1
1 0 0 0 1
1 1 0 0 1

The first line (6 5) gives the number of rows (observations) and the
number of columns (features) present in the following matrix.
Following each line records the frequency of occurrence of the feature
at the column in the given observation. Thus features1 (1st column) occurs
once in the observation1 and infact once in all the other observations too
while the feature3 does not occur in observation1.

DistanceMeasure
The Distance Measure that should be used.
Currrently this module supports the following distance measure:
1. Squared Euclidean (string that should be used as an argument: "squared")
2. Manhattan (string that should be used as an argument: "manhattan")
3. Euclidean (string that should be used as an argument: "euclidean")

ClusteringAlgorithm
The Clustering Measures that can be used are:
1. rb - Repeated Bisections [Default]
2. rbr - Repeated Bisections for by k-way refinement
3. direct - Direct k-way clustering
4. agglo - Agglomerative clustering
5. graph - Graph partitioning-based clustering
6. bagglo - Partitional biased Agglomerative clustering

K value
This is an approximate upper bound for the number of clusters that may be
present in the dataset. Thus for a dataset that you expect to be seperated
into 3 clusters this value should be set some integer value greater than 3.

B value
Specifies the number of time the reference distribution should be generated.

ReferenceGenerationMethod
1. Uniform - While generating the reference distribution, all the features
in the feature set have equal probability of being selected for the observation
under consideration.

2. Proportional - Each feature is assigned a probability of being selected
depending upon its frequency of occurrence in the observed data. Thus feature
distribution is taken into consideration while selecting the features for the
reference distribution generation.

Percentage Confidence Interval
This parameter specifies the percentage confidence to be reported in the log file.
Since Statistics::Gap uses parametric bootstrap method for reference distribution
generation, it is critical to understand the interval around the sample mean that
could contain the population ("true") mean and with what certainty.

OUTPUT
1. The PREFIX.out file contains a single integer number which is the Gap Statistic's
estimate of number of clusters present in the input dataset.
2. The PREFIX.log file contains the log of various values at different K values.
The first table in the file gives values like Gap(k), log(W(k)) etc. for every K value
experimented with.
3. The PREFIX.fig*.png files are the graphical representations of different values which
help locate the knee/elbow.

PRE-REQUISITES
1. This module uses suite of C programs called CLUTO for clustering purposes.
Thus CLUTO needs to be installed for this module to be functional.
CLUTO can be downloaded from http://www-users.cs.umn.edu/~karypis/cluto/

2. Following Perl Modules
1. GD (http://search.cpan.org/~lds/GD-2.19/GD.pm)
2. GD::Text (http://search.cpan.org/~mverb/GDTextUtil-0.86/Text.pm)
3. GD::Graph::lines (http://search.cpan.org/~mverb/GDGraph-1.43/)
4. GD::Graph::colour (http://search.cpan.org/~mverb/GDGraph-1.43/Graph/colour.pm)

SEE ALSO
http://citeseer.ist.psu.edu/tibshirani00estimating.html
http://www-users.cs.umn.edu/~karypis/cluto/

AUTHOR
Anagha Kulkarni, University of Minnesota Duluth
kulka020 <at> d.umn.edu

Guergana Savova, Mayo Clinic
savova.guergana <at> mayo.edu

This program is free software; you can redistribute it and/or
modify it under the terms of the GNU General Public License
as published by the Free Software Foundation; either version 2
of the License, or (at your option) any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.

You should have received a copy of the GNU General Public License
along with this program; if not, write to the Free Software
Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.