Package 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_SINGLESingle linkage (minimum distance). The distance from Z to C is the minimum of the distances (A to C, B to C).
LINKAGE_COMPLETEComplete linkage (maximum distance). The distance from Z to C is the maximum of the distances (A to C, B to C).
LINKAGE_AVG_WITHIN_CLUSTERSAverage-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_CLUSTERSAverage-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_WARDSWard'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:

  • Field Details

    • LINKAGE_SINGLE

      public static final int LINKAGE_SINGLE
      Indicates the single linkage (minimum distance) method.
      See Also:

    • LINKAGE_COMPLETE

      public static final int LINKAGE_COMPLETE
      Indicates the complete linkage (maximum distance) method.
      See Also:

    • 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:

    • 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:

    • LINKAGE_WARDS

      public static final int LINKAGE_WARDS
      Indicates the Ward's method.
      See Also:

    • NONE

      public static final int NONE
      Indicates no transformation.
      See Also:

    • MULTIPLICATION

      public static final int MULTIPLICATION
      Indicates transformation by multiplication by -1.0.
      See Also:

    • RECIPROCAL_ABS

      public static final int RECIPROCAL_ABS
      Indicates transformation by taking the reciprocal of the absolute value.
      See Also:

  • Constructor Details

    • ClusterHierarchical

      public ClusterHierarchical(double[][] dist, int method, int transform)
      Deprecated.
      Use ClusterHierarchical(double[][]) instead.
      Constructor for ClusterHierarchical.
      Parameters:
      dist - A double symmetric matrix containing the distance (or similarity) matrix. On input, only the upper triangular part needs to be present. ClusterHierarchical saves the upper triangular part of dist in the lower triangle. On return, the upper triangular part of dist is restored, and the matrix is made symmetric.
      method - An int identifying the clustering method to be used.

      method Description
      0Single linkage (minimum distance).
      1Complete linkage (maximum distance).
      2Average distance within (average distance between objects within the merged cluster).
      3Average distance between (average distance between objects in the two clusters).
      4Ward's method (minimize the within-cluster sums of squares). For Ward's method, the elements of dist are assumed to be Euclidean distances.

      transform - An int identifying the type of transformation applied to the measures in dist.

      transform Description
      0No transformation is required. The elements of dist are distances.
      1Convert similarities to distances by multiplication by -1.0.
      2Convert similarities (usually correlations) to distances by taking the reciprocal of the absolute value.

      Throws:
      IllegalArgumentException - is thrown when the row lengths of input matrix a are not equal (i.e. the matrix edges are "jagged")

    • 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 Details

    • 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_SINGLESingle linkage (minimum distance).
      LINKAGE_COMPLETEComplete linkage (maximum distance).
      LINKAGE_AVG_WITHIN_CLUSTERSAverage distance within (average distance between objects within the merged cluster).
      LINKAGE_AVG_BETWEEN_CLUSTERSAverage distance between (average distance between objects in the two clusters).
      LINKAGE_WARDSWard's method (minimize the within-cluster sums of squares). For Ward's method, the elements of dist are assumed to be Euclidean distances.

    • getMethod

      public int getMethod()
      Returns the clustering method used.

    • 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
      NONENo transformation is required. The elements of dist are distances.
      MULTIPLICATIONConvert similarities to distances by multiplication by -1.0.
      RECIPROCAL_ABSConvert similarities (usually correlations) to distances by taking the reciprocal of the absolute value.

    • getTransformType

      public int getTransformType()
      Returns the type of transformation.

    • compute

      public void compute()
      Performs a hierarchical cluster analysis.

    • 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.

    • 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.

    • 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.

    • 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.

    • 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.