# NAME

PDL::Stats::Kmeans -- classic k-means cluster analysis

# DESCRIPTION

Assumes that we have data pdl dim [observation, variable] and the goal is to put observations into clusters based on their values on the variables. The terms "observation" and "variable" are quite arbitrary but serve as a reminder for "that which is being clustered" and "that which is used to cluster".

The terms FUNCTIONS and METHODS are arbitrarily used to refer to methods that are threadable and methods that are non-threadable, respectively.

# SYNOPSIS

Implement a basic k-means procedure,

```
use PDL::LiteF;
use PDL::NiceSlice;
use PDL::Stats;
my ($data, $idv, $ido) = rtable( $file );
my ($cluster, $centroid, $ss_centroid, $cluster_last);
# start out with 8 random clusters
$cluster = random_cluster( $data->dim(0), 8 );
# iterate to minimize total ss
# stop when no more changes in cluster membership
do {
$cluster_last = $cluster;
($centroid, $ss_centroid) = $data->centroid( $cluster );
$cluster = $data->assign( $centroid );
}
while ( sum(abs($cluster - $cluster_last)) > 0 );
```

or, use the **kmeans** function provided here,

```
my %k = $data->kmeans( \%opt );
print "$_\t$k{$_}\n" for (sort keys %k);
```

plot the clusters if there are only 2 vars in $data,

```
use PDL::Graphics::PGPLOT::Window;
my ($win, $c);
$win = pgwin 'xs';
$win->env($data( ,0)->minmax, $data( ,1)->minmax);
$win->points( $data->dice_axis(0,which($k{cluster}->(,$_)))->dog,
{COLOR=>++$c} )
for (0 .. $k{cluster}->dim(1)-1);
```

# FUNCTIONS

## random_cluster

` Signature: (short [o]cluster(o,c); int obs=>o; int clu=>c)`

Creates masks for random mutually exclusive clusters. Accepts two parameters, num_obs and num_cluster. Extra parameter turns into extra dim in mask. May loop a long time if num_cluster approaches num_obs because empty cluster is not allowed.

` my $cluster = random_cluster( $num_obs, $num_cluster );`

## which_cluster

` Signature: (short a(o,c); indx [o]b(o))`

Given cluster mask dim [obs x clu], returns the cluster index to which an obs belong.

Does not support overlapping clusters. If an obs has TRUE value for multiple clusters, the returned index is the first cluster the obs belongs to. If an obs has no TRUE value for any cluster, the return val is set to -1 or BAD if the input mask has badflag set.

Usage:

```
# create a cluster mask dim [obs x clu]
perldl> p $c_mask = iv_cluster [qw(a a b b c c)]
[
[1 1 0 0 0 0]
[0 0 1 1 0 0]
[0 0 0 0 1 1]
]
# get cluster membership list dim [obs]
perldl> p $ic = $c_mask->which_cluster
[0 0 1 1 2 2]
```

which_cluster processes bad values. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

## assign

` Signature: (data(o,v); centroid(c,v); short [o]cluster(o,c))`

Takes data pdl dim [obs x var] and centroid pdl dim [cluster x var] and returns mask dim [obs x cluster] to cluster membership. An obs is assigned to the first cluster with the smallest distance (ie sum squared error) to cluster centroid. With bad value, obs is assigned by smallest mean squared error across variables.

```
perldl> $centroid = ones 2, 3
perldl> $centroid(0,) .= 0
perldl> p $centroid
[
[0 1]
[0 1]
[0 1]
]
perldl> $b = qsort( random 4, 3 )
perldl> p $b
[
[0.022774068 0.032513883 0.13890034 0.30942479]
[ 0.16943853 0.50262636 0.56251531 0.7152271]
[ 0.23964483 0.59932745 0.60967495 0.78452117]
]
# notice that 1st 3 obs in $b are on average closer to 0
# and last obs closer to 1
perldl> p $b->assign( $centroid )
[
[1 1 1 0] # cluster 0 membership
[0 0 0 1] # cluster 1 membership
]
```

assign processes bad values. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

## centroid

` Signature: (data(o,v); cluster(o,c); float+ [o]m(c,v); float+ [o]ss(c,v))`

Takes data dim [obs x var] and mask dim [obs x cluster], returns mean and ss (ms when data contains bad values) dim [cluster x var], using data where mask == 1. Multiple cluster membership for an obs is okay. If a cluster is empty all means and ss are set to zero for that cluster.

```
# data is 10 obs x 3 var
perldl> p $d = sequence 10, 3
[
[ 0 1 2 3 4 5 6 7 8 9]
[10 11 12 13 14 15 16 17 18 19]
[20 21 22 23 24 25 26 27 28 29]
]
# create two clusters by value on 1st var
perldl> p $a = $d( ,(0)) <= 5
[1 1 1 1 1 1 0 0 0 0]
perldl> p $b = $d( ,(0)) > 5
[0 0 0 0 0 0 1 1 1 1]
perldl> p $c = cat $a, $b
[
[1 1 1 1 1 1 0 0 0 0]
[0 0 0 0 0 0 1 1 1 1]
]
perldl> p $d->centroid($c)
# mean for 2 cluster x 3 var
[
[ 2.5 7.5]
[12.5 17.5]
[22.5 27.5]
]
# ss for 2 cluster x 3 var
[
[17.5 5]
[17.5 5]
[17.5 5]
]
```

centroid processes bad values. It will set the bad-value flag of all output piddles if the flag is set for any of the input piddles.

## kmeans

Implements classic k-means cluster analysis. Given a number of observations with values on a set of variables, kmeans puts the observations into clusters that maximizes within-cluster similarity with respect to the variables. Tries several different random seeding and clustering in parallel. Stops when cluster assignment of the observations no longer changes. Returns the best result in terms of R2 from the random-seeding trials.

Instead of random seeding, kmeans also accepts manual seeding. This is done by providing a centroid to the function, in which case clustering will proceed from the centroid and there is no multiple tries.

There are two distinct advantages from seeding with a centroid compared to seeding with predefined cluster membership of a subset of the observations ie "seeds",

(1) a centroid could come from a previous study with a different set of observations;

(2) a centroid could even be "fictional", or in more proper parlance, an idealized prototype with respect to the actual data. For example, if there are 10 person's ratings of 1 to 5 on 4 movies, ie a ratings pdl of dim [10 obs x 4 var], providing a centroid like

```
[
[5 0 0 0]
[0 5 0 0]
[0 0 5 0]
[0 0 0 5]
]
```

will produce 4 clusters of people with each cluster favoring a different one of the 4 movies. Clusters from an idealized centroid may not give the best result in terms of R2, but they sure are a lot more interpretable.

If clustering has to be done from predefined clusters of seeds, simply calculate the centroid using the **centroid** function and feed it to kmeans,

```
my ($centroid, $ss) = $rating($iseeds, )->centroid( $seeds_cluster );
my %k = $rating->kmeans( { CNTRD=>$centroid } );
```

kmeans supports bad value*.

Default options (case insensitive):

```
V => 1, # prints simple status
FULL => 0, # returns results for all seeding trials
CNTRD => PDL->null, # optional. pdl [clu x var]. disables next 3 opts
NTRY => 5, # num of random seeding trials
NSEED => 1000, # num of initial seeds, use NSEED up to max obs
NCLUS => 3, # num of clusters
```

Usage:

```
# suppose we have 4 person's ratings on 5 movies
perldl> p $rating = ceil( random(4, 5) * 5 )
[
[3 2 2 3]
[2 4 5 4]
[5 3 2 3]
[3 3 1 5]
[4 3 3 2]
]
# we want to put the 4 persons into 2 groups
perldl> %k = $rating->kmeans( {NCLUS=>2} )
# by default prints back options used
# as well as info for all tries and iterations
CNTRD => Null
FULL => 0
NCLUS => 2
NSEED => 4
NTRY => 5
V => 1
ss total: 20.5
iter 0 R2 [0.024390244 0.024390244 0.26829268 0.4796748 0.4796748]
iter 1 R2 [0.46341463 0.46341463 0.4796748 0.4796748 0.4796748]
perldl> p "$_\t$k{$_}\n" for (sort keys %k)
R2 0.479674796747968
centroid # mean ratings for 2 group x 5 movies
[
[ 3 2.3333333]
[ 2 4.3333333]
[ 5 2.6666667]
[ 3 3]
[ 4 2.6666667]
]
cluster # 4 persons' membership in two groups
[
[1 0 0 0]
[0 1 1 1]
]
n [1 3] # cluster size
ss
[
[ 0 0.66666667]
[ 0 0.66666667]
[ 0 0.66666667]
[ 0 8]
[ 0 0.66666667]
]
```

Now, for the valiant, kmeans is threadable. Say you gathered 10 persons' ratings on 5 movies from 2 countries, so the data is dim [10,5,2], and you want to put the 10 persons from each country into 3 clusters, just specify NCLUS => [3,1], and there you have it. The key is for NCLUS to include $data->ndims - 1 numbers. The 1 in [3,1] turns into a dummy dim, so the 3-cluster operation is repeated on both countries. Similarly, when seeding, CNTRD needs to have ndims that at least match the data ndims. Extra dims in CNTRD will lead to threading (convenient if you want to try out different centroid locations, for example, but you will have to hand pick the best result). See stats_kmeans.t for examples w 3D and 4D data.

*With bad value, R2 is based on average of variances instead of sum squared error.

# METHODS

## iv_cluster

Turns an independent variable into a cluster pdl. Returns cluster pdl and level-to-pdl_index mapping in list context and cluster pdl only in scalar context.

This is the method used for mean and var in anova. The difference between iv_cluster and dummy_code is that iv_cluster returns pdl dim [obs x level] whereas dummy_code returns pdl dim [obs x (level - 1)].

Usage:

```
perldl> @bake = qw( y y y n n n )
# accepts @ ref or 1d pdl
perldl> p $bake = iv_cluster( \@bake )
[
[1 1 1 0 0 0]
[0 0 0 1 1 1]
]
perldl> p $rating = sequence 6
[0 1 2 3 4 5]
perldl> p $rating->centroid( $bake )
# mean for each iv level
[
[1 4]
]
# ss
[
[2 2]
]
```

## pca_cluster

Assgin variables to components ie clusters based on pca loadings or scores. One way to seed kmeans (see Ding & He, 2004, and Su & Dy, 2004 for other ways of using pca with kmeans). Variables are assigned to their most associated component. Note that some components may not have any variable that is most associated with them, so the returned number of clusters may be smaller than NCOMP.

Default options (case insensitive):

```
V => 1,
ABS => 1, # high pos and neg loadings on a comp in same cluster
NCOMP => undef, # max number of components to consider. determined by
# scree plot black magic if not specified
PLOT => 1, # pca scree plot with cutoff at NCOMP
```

Usage:

```
# say we need to cluster a group of documents
# $data is pdl dim [word x doc]
($data, $idd, $idw) = get_data 'doc_word_info.txt';
perldl> %p = $data->pca;
# $cluster is pdl mask dim [doc x ncomp]
perldl> $cluster = $p{loading}->pca_cluster;
# pca clusters var while kmeans clusters obs. hence transpose
perldl> ($m, $ss) = $data->transpose->centroid( $cluster );
perldl> %k = $data->transpose->kmeans( { cntrd=>$m } );
# take a look at cluster 0 doc ids
perldl> p join("\n", @$idd[ list which $k{cluster}->( ,0) ]);
```

# REFERENCES

Ding, C., & He, X. (2004). K-means clustering via principal component analysis. Proceedings of the 21st International Conference on Machine Learning, 69, 29.

Su, T., & Dy, J. (2004). A deterministic method for initializing K-means clustering. 16th IEEE International Conference on Tools with Artificial Intelligence, 784-786.

Romesburg, H.C. (1984). Cluster Analysis for Researchers. NC: Lulu Press.

Wikipedia (retrieved June, 2009). K-means clustering. http://en.wikipedia.org/wiki/K-means_algorithm

# AUTHOR

Copyright (C) 2009 Maggie J. Xiong <maggiexyz users.sourceforge.net>

All rights reserved. There is no warranty. You are allowed to redistribute this software / documentation as described in the file COPYING in the PDL distribution.