com.imsl.stat
Class ClusterHierarchical

java.lang.Object
  com.imsl.stat.ClusterHierarchical

All Implemented Interfaces:

Serializable, Cloneable

public class ClusterHierarchical
extends Object
implements Serializable, Cloneable

Performs a hierarchical cluster analysis from a distance matrix.

Class ClusterHierarchical conducts a hierarchical cluster analysis based upon a distance matrix, or, by appropriate use of the transformation specified in the method setTransformType, based upon a similarity matrix. Only the upper triangular part of the input matrix is used.

Hierarchical clustering in ClusterHierarchical proceeds as follows:

Initially, each data point is considered to be a cluster, numbered 1 to n, where n is the number of rows in the input matrix, dist.

1. If the input matrix contains similarities, the matrix is transformed to a distance matrix using the transform type specified by the method setTransformType. Set k = 1.
2. A search is made of the distance matrix to find the two closest clusters. These clusters are merged to form a new cluster, numbered n + k. The cluster numbers of the two clusters joined at this stage are saved as Right Sons and Left Sons, and the distance measure between the two clusters is stored as Cluster Level.
3. Based upon the method of clustering, updating of the distance measure in the row and column of dist corresponding to the new cluster is performed.
4. Set k = k + 1. If k is less than n, go to Step 2.

The five methods differ primarily in how the distance matrix is updated after two clusters have been joined. The argument method in setMethod specifies how the distance of the cluster just merged with each of the remaining clusters will be updated. Class ClusterHierarchical allows five methods for computing the distances. To understand these measures, suppose in the following discussion that clusters A and B have just been joined to form cluster Z, and interest is in computing the distance of Z with another cluster called C.

method	Description
LINKAGE_SINGLE	Single linkage (minimum distance). The distance from Z to C is the minimum of the distances (A to C, B to C).
LINKAGE_COMPLETE	Complete linkage (maximum distance). The distance from Z to C is the maximum of the distances (A to C, B to C).
LINKAGE_AVG_WITHIN_CLUSTERS	Average-distance-within-clusters method. The distance from Z to C is the average distance of all objects that would be within the cluster formed by merging clusters Z and C. This average may be computed according to formulas given by Anderberg (1973, page 139).
LINKAGE_AVG_BETWEEN_CLUSTERS	Average-distance-between-clusters method. The distance from Z to C is the average distance of objects within cluster Z to objects within cluster C. This average may be computed according to methods given by Anderberg (1973, page 140).
LINKAGE_WARDS	Ward's method: Clusters are formed so as to minimize the increase in the within-cluster sums of squares. The distance between two clusters is the increase in these sums of squares if the two clusters were merged. A method for computing this distance from a squared Euclidean distance matrix is given by Anderberg (1973, pages 142-145).

In general, single linkage will yield long thin clusters while complete linkage will yield clusters that are more spherical. Average linkage and Ward's linkage tend to yield clusters that are similar to those obtained with complete linkage.

Class ClusterHierarchical produces a unique representation of the binary cluster tree via the following three conventions; the fact that the tree is unique should aid in interpreting the clusters. First, when two clusters are joined and each cluster contains two or more data points, the cluster that was initially formed with the smallest level becomes the left son. Second, when a cluster containing more than one data point is joined with a cluster containing a single data point, the cluster with the single data point becomes the right son. Finally, when two clusters containing only one object are joined, the cluster with the smallest cluster number becomes the right son.

Comments

1. The clusters corresponding to the original data points are numbered from 1 to n, where n is the number of rows in dist. The n - 1 clusters formed by merging clusters are numbered n + 1 to n + (n - 1 ).
2. Raw correlations, if used as similarities, should be made positive and transformed to a distance measure. One such transformation can be performed by setting transform = RECIPROCAL_ABS, in the setTransformType method.
3. The user may cluster either variables or observations with ClusterHierarchical since a dissimilarity matrix, not the original data, is used. Class Dissimilarities may be used to compute the matrix dist for either the variables or observations.

See Also:

Example 1, Serialized Form

Field Summary

static int	LINKAGE_AVG_BETWEEN_CLUSTERS Indicates the average distance between (average distance between objects in the two clusters) method.
static int	LINKAGE_AVG_WITHIN_CLUSTERS Indicates the average distance within (average distance between objects within the merged cluster) method.
static int	LINKAGE_COMPLETE Indicates the complete linkage (maximum distance) method.
static int	LINKAGE_SINGLE Indicates the single linkage (minimum distance) method.
static int	LINKAGE_WARDS Indicates the Ward's method.
static int	MULTIPLICATION Indicates transformation by multiplication by -1.0.
static int	NONE Indicates no transformation.
static int	RECIPROCAL_ABS Indicates transformation by taking the reciprocal of the absolute value.

Constructor Summary

ClusterHierarchical(double[][] dist)
Constructor for ClusterHierarchical.

Method Summary

void	compute() Performs a hierarchical cluster analysis.
int[]	getClusterLeftSons() Returns the left sons of each merged cluster.
double[]	getClusterLevel() Returns the level at which the clusters are joined.
int[]	getClusterMembership(int nClusters) Returns the cluster membership of each observation.
int[]	getClusterRightSons() Returns the right sons of each merged cluster.
int	getMethod() Returns the clustering method used.
int[]	getObsPerCluster(int nClusters) Returns the number of observations in each cluster.
int	getTransformType() Returns the type of transformation.
void	setMethod(int method) Sets the clustering method to be used.
void	setTransformType(int transform) Sets the type of transformation.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

LINKAGE_AVG_BETWEEN_CLUSTERS

public static final int LINKAGE_AVG_BETWEEN_CLUSTERS

Indicates the average distance between (average distance between objects in the two clusters) method.

See Also:

Constant Field Values

LINKAGE_AVG_WITHIN_CLUSTERS

public static final int LINKAGE_AVG_WITHIN_CLUSTERS

Indicates the average distance within (average distance between objects within the merged cluster) method.

See Also:

Constant Field Values

LINKAGE_COMPLETE

public static final int LINKAGE_COMPLETE

Indicates the complete linkage (maximum distance) method.

See Also:

Constant Field Values

LINKAGE_SINGLE

public static final int LINKAGE_SINGLE

Indicates the single linkage (minimum distance) method.

See Also:

Constant Field Values

LINKAGE_WARDS

public static final int LINKAGE_WARDS

Indicates the Ward's method.

See Also:

Constant Field Values

MULTIPLICATION

public static final int MULTIPLICATION

Indicates transformation by multiplication by -1.0.

See Also:

Constant Field Values

NONE

public static final int NONE

Indicates no transformation.

See Also:

Constant Field Values

RECIPROCAL_ABS

public static final int RECIPROCAL_ABS

Indicates transformation by taking the reciprocal of the absolute value.

See Also:

Constant Field Values

Constructor Detail

ClusterHierarchical

public ClusterHierarchical(double[][] dist)

Constructor for ClusterHierarchical.

Parameters:

dist - A double symmetric matrix containing the distance (or similarity) matrix. Only the upper triangular part is used.

Method Detail

compute

public void compute()

Performs a hierarchical cluster analysis.

getClusterLeftSons

public final int[] getClusterLeftSons()

Returns the left sons of each merged cluster.

Returns:

An int array containing the left sons of each merged cluster.

getClusterLevel

public final double[] getClusterLevel()

Returns the level at which the clusters are joined.

Returns:

A double array containing the level at which the clusters are joined. Element [k-1] contains the distance (or similarity) level at which cluster n + k was formed.

getClusterMembership

public final int[] getClusterMembership(int nClusters)

Returns the cluster membership of each observation.

Parameters:

nClusters - An int which specifies the desired number of clusters.

Returns:

An int array containing the cluster membership of each observation.

getClusterRightSons

public final int[] getClusterRightSons()

Returns the right sons of each merged cluster.

Returns:

An int array containing the right sons of each merged cluster.

getMethod

public int getMethod()

Returns the clustering method used.

getObsPerCluster

public final int[] getObsPerCluster(int nClusters)

Returns the number of observations in each cluster.

Parameters:

nClusters - An int which specifies the desired number of clusters.

Returns:

An int array containing the number of observations in each cluster.

getTransformType

public int getTransformType()

Returns the type of transformation.

setMethod

public void setMethod(int method)

Sets the clustering method to be used.

Parameters:

method - An int identifying the clustering method to be used. By default, method = LINKAGE_SINGLE.

method	Description
LINKAGE_SINGLE	Single linkage (minimum distance).
LINKAGE_COMPLETE	Complete linkage (maximum distance).
LINKAGE_AVG_WITHIN_CLUSTERS	Average distance within (average distance between objects within the merged cluster).
LINKAGE_AVG_BETWEEN_CLUSTERS	Average distance between (average distance between objects in the two clusters).
LINKAGE_WARDS	Ward's method (minimize the within-cluster sums of squares). For Ward's method, the elements of dist are assumed to be Euclidean distances.

setTransformType

public void setTransformType(int transform)

Sets the type of transformation.

Parameters:

transform - An int identifying the type of transformation applied to the measures in dist. By default, transform = NONE.

transform	Description
NONE	No transformation is required. The elements of dist are distances.
MULTIPLICATION	Convert similarities to distances by multiplication by -1.0.
RECIPROCAL_ABS	Convert similarities (usually correlations) to distances by taking the reciprocal of the absolute value.