Cluster analysis is the collective name given to a number of algorithms for grouping similar
objects into distinct categories. It is a form of exploratory data analysis aimed at grouping
observations in a way that minimizes the difference within groups while maximizing the difference
between groups.
In K-Means clustering, the number of clusters is fixed at the beginning.
A number of initial cluster centers is chosen. The observations are assigned to the closest cluster.
Each center is then recalculated as the mean of its members. This changes the distances between cluster
centers and observations, so the observations are once again reassigned. This process is repeated
until no more observations change cluster.
Note that the final partition depends on the initial location of the centers. Different applications
may return different results for the same dataset.
Running a cluster analysis
K-Means clustering is implemented by the
KMeansClusterAnalysis class.
This class has four constructors.
The first constructor takes two parameters. The first is a
Matrix whose columns contain the data to be analyzed.
The second parameter is the number of clusters to find.
The second constructor also takes two parameters: an array of
NumericalVariable objects and the number of clusters to find..
The third and fourth constructors each take three parameters. The third constructor takes a
VariableCollection as its first argument. The second argument is an array of strings
that contains the names of the variables from the collection that should be included in the analysis.
The last parameter is the number of clusters to find.
The fourth constructor takes a System.Data..::.DataTable as its first argument. The second argument
is once again an array of strings that this time contains the names of the columns to be included in the analysis.
The last parameter is once again the number of clusters to find.
The Compute()()() method performs the actual calculations.
Once the computations are complete, a number of properties and methods give access to the results in detail.
The following code sample sets up a K-means cluster analysis with 3 clusters:
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VariableCollection variables = new VariableCollection(data);
KMeansClusterAnalysis kmc = new KMeansClusterAnalysis(variables, 3);
kmc.Standardize = true;
kmc.Compute();
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Dim variables As New VariableCollection(data)
Dim kmc As New KMeansClusterAnalysis(variables, 3)
kmc.Standardize = True
kmc.Compute()
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Results of the analysis
Use the GetClusters()()() method
to get an object of type
KMeansClusterCollection, which - as the name implies -
is a collection of KMeansCluster objects.
In addition to the usual collection properties and methods, this class has two more methods.
The GetMemberships()()() method,
which returns a CategoricalVariable that for each observation indicates the cluster
to which it belongs.
The GetDistancesToCenters()()() method returns a
NumericalVariable that for each observation indicates the distance of the observation
from the center of its cluster.
Each KMeansCluster describes one cluster in detail.
The Center property returns the
center of the cluster as a Vector.
The Size property returns the number of observations in the cluster.
The MemberFilter property returns a
Filter that selects the members of the cluster from the original dataset.
The SumOfSquares returns the within-cluster
sum of squares of the distances of its members to the center. The sample code below prints information about each cluster.
In addition, it prints out for each observation the cluster it belongs to and the distance to its center:
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KMeansClusterCollection clusters = kmc.GetClusters();
foreach (KMeansCluster cluster in clusters)
{
Console.WriteLine("Cluster {0} has {1} members. Sum of squares: {2:F4}",
cluster.Index, cluster.Size, cluster.SumOfSquares);
Console.WriteLine("Center: {0:F4}", cluster.Center);
}
CategoricalVariable memberships = clusters.GetMemberships();
NumericalVariable distances = clusters.GetDistancesToCenters();
for (int i = 18; i < memberships.Length; i++)
Console.WriteLine("Observation {0} belongs to cluster {1}, distance: {2:F4}.",
i, memberships.GetLevelIndex(i), distances[i]);
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Dim clusters As KMeansClusterCollection = kmc.GetClusters()
For Each cluster As KMeansCluster In clusters
Console.WriteLine("Cluster {0} has {1} members. Sum of squares: {2:F4}", _
cluster.Index, cluster.Size, cluster.SumOfSquares)
Console.WriteLine("Center: {0:F4}", cluster.Center)
Next
Dim memberships As CategoricalVariable = clusters.GetMemberships()
Dim distances As NumericalVariable = clusters.GetDistancesToCenters()
For i As Integer = 18 To memberships.Length - 1
Console.WriteLine("Observation {0} belongs to cluster {1}, distance: {2:F4}.", _
i, memberships.GetLevelIndex(i), distances(i))
Next
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