Re: Making sense of data: Clustering OR A coding challenge

[jdporter]

So here's a cluster finder which uses a Genetic Algorithm based approach. I've tested it with trivial data point sets that have very clear (to human eyes) clusters. On these, it finds perfect clustering almost every time. I'd be interested to see how well it works on real-world data.

The parameters are not well isolated in this code. They are:

• Maximum number of clusters (the algorithm is free to find fewer)
• Number of generations to run
• Number of individuals to kill, clone, and mutate in each generation
• In a mutation, how many datapoints to re-cluster

use Carp;
use List::Util qw( shuffle );
use strict;
use warnings;

# an individual represents a distribution of points among clusters.
# that is, it is a specific allocation of points to clusters.
# in the initial population, in each individual, the points are random
+ly assigned to clusters.

# each individual is an array. 
# each element represents a point in the data set, and its value
# is the number of the cluster to which it has been assigned.

my @datapoints;

# The subs in Point:: need to be customized for the type/representatio
+n of a "point".

sub Point::set_metric; # "distance" or "area" or something like that. 
+small values mean "close"
sub Point::as_string;

sub Point::ScalarNumber::set_metric
{
    my $set = shift;
    my @set = @datapoints[@$set];
    @set == 0 and return 1;
    @set == 1 and return 2;
    # RMS
    my $total = 0;
    my $n = 0;
    for my $i ( 1 .. $#set )
    {
        for my $j ( $i .. $#set )
        {
            my $dist = abs( $set[$i-1] - $set[$j] );
            $total += $dist ** 2;
            $n++;
        }
    }
    sqrt( $total / $n )
}
sub Point::ScalarNumber::as_string
{
    $_[0]
}

sub Point::NumberPair::set_metric
{
    my $set = shift;
    my @set = @datapoints[@$set];
    @set == 0 and return 1;
    @set == 1 and return 2;
    # RMS
    my $total = 0;
    my $n = 0;
    for my $i ( 1 .. $#set )
    {
        for my $j ( $i .. $#set )
        {
            my $dist2 = ( ( $set[$i-1][0] - $set[$j][0] ) ** 2 )
                      + ( ( $set[$i-1][1] - $set[$j][1] ) ** 2 );
            $total += $dist2;
            $n++;
        }
    }
    sqrt( $total / $n )
}
sub Point::NumberPair::as_string
{
    "[$_[0][0],$_[0][1]]"
}

######################################################################
+#

my @clusters;

sub Ind::new_randomized
{
    #@datapoints <= 0 and croak "No datapoints defined!\n";
    #@datapoints <  1 and croak "Only one cluster defined!\n";
    #@clusters   <= 0 and croak "No clusters defined!\n";
    #@clusters   <  1 and croak "Only one cluster defined!\n";
    [ map { int( rand @clusters ) } @datapoints ]
}

sub Ind::clone
{
    my $ind = shift;
    [ @$ind ]
}

# optional arg: number of points to move
sub Ind::mutate
{
    my( $ind, $n ) = @_;
    for my $i ( 0 .. ($n||1) )
    {
        my $j = int( rand @datapoints );
        $ind->[$j] = int( rand @clusters );
    }
    $ind
}

    sub Ind::_crossover_points
    {
        my $l = @datapoints;
        my $seglen = 1 + int rand( $l - 1 );
        my $start = int rand( $l - $seglen );
        ( $start .. ($start+$seglen-1) )
    }

sub Ind::crossover
{
    my( $ind1, $ind2 ) = @_;
    my @xo = Ind::_crossover_points();
    for my $i ( @xo )
    {
        ( $ind1->[$i], $ind2->[$i] ) =
        ( $ind2->[$i], $ind1->[$i] )
    }
}

sub Ind::fitness
{
    my $ind = shift;
    my @cluster_points = map {
        my $cl = $_;
        [ grep { $ind->[$_] eq $cl } 0 .. $#{$ind} ]
    } 0 .. $#clusters;

    my $total_metric = 0;
    for my $ci ( 0 .. $#cluster_points )
    {
        my $val = Point::set_metric( $cluster_points[$ci] );
        $total_metric += $val;
    }

    1000/$total_metric # convert it to "large = good"
}

sub Ind::display
{
    my $ind = shift;
    my @cluster_points = map {
        my $cl = $_;
        [ grep { $ind->[$_] eq $cl } 0 .. $#{$ind} ]
    } 0 .. $#clusters;

    my $total_metric = 0;
    for my $ci ( 0 .. $#cluster_points )
    {
        my $val = Point::set_metric( $cluster_points[$ci] );
        $total_metric += $val;
        printf "$ci: Cluster $clusters[$ci]: %5.2f ( ", $val;
        print join ' ', map { Point::as_string($_) }
            @datapoints[@{$cluster_points[$ci]}];
        print " )\n";
    }
    printf "Total metric: %.2f\n", $total_metric;

    $ind
}

######################################################################
+#

if(0)
{
@datapoints = shuffle( 11..14, 21..24, 31..34, 41..44 );
*Point::set_metric = \&Point::ScalarNumber::set_metric;
*Point::as_string = \&Point::ScalarNumber::as_string;
}
else
{
@datapoints = shuffle(
    [ 1, 2], [ 2, 1], [ 2, 3], [ 3, 2],
    [ 1,12], [ 2,11], [ 2,13], [ 3,12],
    [11, 2], [12, 1], [12, 3], [13, 2],
    [11,12], [12,11], [12,13], [13,12],
);
*Point::set_metric = \&Point::NumberPair::set_metric;
*Point::as_string = \&Point::NumberPair::as_string;
}

@clusters = ( 1 .. 4 );

my @pop =
    sort { $b->[0] <=> $a->[0] }
    map { [ Ind::fitness($_), $_ ] }
    map { Ind::new_randomized }
    1 .. 100;

#print "Before:"; printf " %.1f", $_->[0] for @pop; print "\n";

    # this clones an element of @pop
    sub clone { [ $_[0]->[0], Ind::clone($_[0]->[1]) ] }

for my $iter ( 1 .. 200 )
{
    # kill the bottom 30:
    splice @pop, @pop-30, 30;

    # make 10 new ones:
    push @pop,
    map { [ Ind::fitness($_), $_ ] }
    map { Ind::new_randomized }
    1 .. 10;

    # clone the top 20:
    push @pop, map clone($_), @pop[0 .. 19];

    # mutate the top 20:
    for my $e ( @pop[0 .. 19] )
    {
        my $n = 1;
        unless ( int(rand 2) )
        {
            $n++;
            unless ( int(rand 3) )
            {
                $n++;
                unless ( int(rand 4) )
                {
                    $n++;
                }
            }
        }
        #warn "mut $n\n";
        Ind::mutate( $e->[1], $n );
        $e->[0] = Ind::fitness( $e->[1] );
    }

    # sort by fitness again:
    @pop = sort { $b->[0] <=> $a->[0] } @pop;
    # print "Iter $iter: $pop[0][0]\n";
}

# print "\nAfter:"; printf " %.1f", $_->[0] for @pop; print "\n";

Ind::display( $pop[0][1] );

Note that, as it stands, it's not doing any crossover, only mutation, so probably it isn't a GA, technically.
I'm sure improvements could be made in this area.

[belg4mit]

With 5 clusters and the dataset in Re^2: Making sense of data: Clustering OR A coding challenge I get some nasty results like:
Read more... (794 Bytes)
With 5000 generations, 50 cullings and 20 spawn it yields slightly more reasonable, though still not too helpful results:
Read more... (803 Bytes)
Given another order of magnitude or more runtime it might reach palatable results ;-)

[jdporter]

Indeed, there were a number of other parameters that could be tweaked, and by doing so, I was able to get better results than that.

However, in the end, it turns out there are some special properties of your problem that allow much simpler and more effective solutions. Namely, the fact that your data points are one-dimensional. (I'm assuming they are.)

It means, for example, that (1 3),(2 4) is never an optimal clustering. Neither is (1 2),(1 2).

From these, we can define the following constraints on clusters:

1. clusters are simply segments within the ordered data set.
2. all repetitions of a number must be kept together.

In the following solution, I'm using variance as a measure of the "coherence" or "binding strength" or whatever you want to call it within each cluster. You could use other measures; I wouldn't be surprised if variance doesn't necessarily give the best results. I've seen discussions of clustering that talk about maximizing variance between clusters, in addition to minimizing it within clusters. I have my doubts that that would help in a simple one-dimensional problem like this one.

Read more... (5 kB)