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Dataplot Vol 1 Vol 2

CLUSTER

Name:
    K MEANS CLUSTER
    NORMAL MIXTURE CLUSTER
    K MEDOIDS CLUSTER
    FUZZY CLUSTER
    AGNES CLUSTER
    DIANA CLUSTER
Type:
    Analysis Command
Purpose:
    Perform a cluster analysis.
Description:
    The goal of cluster analysis is to find groups in data. There are many approaches to this task. We can divide this into two primary approaches.

    1. Partitioning Methods

      Given p variables each with n observations, we create k clusters and assign each of the n observations to one of these clusters.

      For these methods, the number of clusters typically has to be specified in advance. In Dataplot, to specify the number of clusters, enter the command

        LET NCLUSTER = <value>

      It is typical to run the cluster analysis for several different values of NCLUSTER.

      Dataplot implements the following partition based methods.

      • K-MEANS

        K-means is the workhorse method for clustering. The k-means critierion is to minimize the within cluster sum of squares based on Euclidean distances between the observations. That is, minimize

        \( \sum_{i=1}^{k}{W(C_{k})} = \sum_{i=1}^{k}{\sum_{x_{i} \in C_{k}} {(x_{i} - \mu_{k})^{2}}} \)

        with \( C_{k} \), \( x_{i} \), and \( \mu_{k} \) denoting the k-th cluster, an observation belonging to cluster k, and the mean value of the observations belonging to \( C_{k} \), respectively.

        Dataplot implements k- means using the Hartigan-Wong algorithm. This algorithm finds a local minimum, so different results can be obtained based on the initial assignment to clusters. The first method is to randomly select observations to use as initial cluster centers. The second method is suggested by Hartigan and Wong. First order the observations by their distances to the overall mean. Then for cluster L (L = 1, 2, ... k), use row \( 1 + (L-1) [\frac{p}{k}] \) as the initial cluster center.

        To specify the initialization method, enter

          SET K MEANS INITIAL

        The default is RANDOM.

      • K-MEDOIDS

        The k-medoids method was proposed by Kaufman and Rousseeuw. In k-medoids clustering, each cluster is represented by one observation in the cluster. These observations are called the cluster medoids. The cluster medoids correspond to the most centrally located observations in the cluster. The k-medoids method is more robust to outliers and noise than the k-means method. The mathematical details of the method are given in the Kaufman and Rousseeuw book (see References below).

        The k-medoids method can start either with the original measurement data or a distance matrix (this matrix will have dimension nxn).

        Kaufman and Rousseeuw provided two algorithms for k-medoid clustering.

        Partioning around medoids (PAM) is used when the number of observations is small (up to 100 observations in the original Kaufman and Rousseeuw code). All of the observations are used to determine the clusters.

        When the number of observations is larger, the CLARA algorithm is used. In CLARA, a number of random samples of the full data set are generated and the PAM algorithm is applied to them. The random sample that generates the best clustering is used to assign the unsampled observations to a cluster.

        In Dataplot, you can specify the cut-off between switching from PAM to CLARA with the command

          SET K MEDOID CLUSTER PAM MAXIMUM SIZE

        where <value> is between 100 and 500.

        You can also specify the number of samples drawn and the sample size for each sample with the commands

          SET K MEDOID CLUSTER NUMBER OF SAMPLES <value>
          SET K MEDOID CLUSTER NUMBER OF SAMPLES <value>

        The default is to draw 5 samples with 40 + 2*(number of clusters) observations per sample. For most applications, these defaults should be sufficient.

        The PAM and CLARA algorithms can be based on either Euclidean distances or Manhattan (city block) distances. To specify which to use, enter

          SET K MEDOID CLUSTER DISTANCE <EUCLIDEAN/MANHATTAN>

        The default is to use euclidean distances.

        Dataplot assumes if the number of rows and columns are equal that a distance matrix is being input. In the unlikely case where the they are equal for measurement data, you can enter the command

          SET K MEDOID CLUSTER TYPE MEASUREMENT

        To restore the default, enter

          SET K MEDOID CLUSTER TYPE DISSIMILARITY

        For the random sampling, Dataplot uses its own random number generator routines by default. You can request the genrator used by Kaufman and Rousseeuw by entering the command (this option is intended primarily to allow validating the Dataplot results against the results given by running the Kaufman and Rousseeuw codes directly)

          SET K MEDOID CLUSTER RANDOM NUMBER GENERATOR ROUSSEEUW

        To reset the default, enter

          SET K MEDOID CLUSTER RANDOM NUMBER GENERATOR DATAPLOT

        You can request that only the final results be printed by entering the command

          SET K MEDOID CLUSTER PRINT FINAL

        To rese the default where results for the individual samples are printed, enter

          SET K MEDOID CLUSTER PRINT ALL

      • FUZZY CLUSTERING (FANNY)

        Partioning algorithms typically assign each observation to a single cluster. Fuzzy clustering assigns each observation to every cluster with a probability for being in that cluster. Kaufman and Rousseeuw provide an algorithm, FANNY, to generate a fuzzy clustering. The details for FANNY are given in the Kaufman and Rousseeuw book.

        The following commands can be used with fanny clustering.

          SET FANNY CLUSTER DISTANCE <EUCLIDEAN/MANHATTAN>
          SET FANNY CLUSTER PRINT <ALL/FINAL>
          SET FANNY CLUSTER TYPE <MEASUREMENT/DISSIMILARITY>
          SET FANNY CLUSTER MAXIMUM SIZE <value>

        These options are similar to the options for k-medoids clustering. As with PAM, a maximum of 500 observations can be set.

        The primary advantage of this approach is that it gives some indication of the uncertainty of the cluster assignments. The algorithm does return the "most likely" cluster assignment which can be used in visualizing the results of the cluster analysis. The drawback is that interpretion can become difficult as the number of observations increases.

      • NORMAL MIXTURES

        This implements Hartigan's MIX algorithm. This is similar to FANNY in that it assigns probabilities for the cluster assignments. This method is based on the model that the observations is selected at random from one of k (where k is the number of clusters) multivariate normal populations. The mathematical details are given in Hartigan's book.

    2. Hierchial Clustering

      Hierchial clustering typically starts with a distance or dissimilarity matrix.

      Hierchial clustering can be divided into agglomerative algorithms and divisive algorithms.

      With agglomerative algorithms, we start with each object in a separate cluster. Then at each step, the two "closest" clusters are merged. This process is repeated until all objects are in a single cluster.

      Divisive algorithms work in the opposite direction. That is, it starts with all objects in a single cluster. Then at each step, a cluster is split into two clusters. This is repeated until all objects are in their own cluster.

      • AGGLOMERATIVE NESTING (AGNES)

        Dataplot implements the AGNES algorithm given in the Kaufman and Rousseeuw book.

        The AGNES algorithm can start either with measurement data (i.e., p variables with n observations) or with a previously created dissimilarity matrix. For measurement data, the first step is to create a dissimilarity matrix. You can request that either Euclidean distances or Manhattan distances be used to create the dissimilarity matrix. To specify which distance measure to use, enter the command

          SET AGNES CLUSTER DISTANCE <EUCLIDEAN/MANHATTAN>

        The default is to use the Manhattan distance. If you want to use some other distance metric, see the Note section below which describes the use of the GENERATE MATRIX command with a number of different distance metrics.

        The link function defines the critierion that will be used to decide which two clusters are "closest" and will therefore be joined at each step. The supported link functions are

        • Average Linkage - The distance between two clusters, A and B, is defined as the average distance between the elements in cluster A and the elements in cluster B.

          This is the recommended choice of Kaufman and Rousseeuw and is the default used by Dataplot.

        • Complete Linkage - The distance between two clusters, A and B, is defined as the maximum distance of all pairwise distances between the elements in cluster A and the elements in cluster B.

        • Single Linkage - The distance between two clusters, A and B, is defined as the minimum distance of all pairwise distances between the elements in cluster A and the elements in cluster B. Single linkage clustering is also referred to as "nearest neighbor" clustering.

        • Centroid Linkage - The distance between two clusters, A and B, is defined as the distance between the centroid for cluster A and the centroid for cluster B.

          This method should be restricted to the case where the dissimilarity matrix is defined by Euclidean distances. Another drawback is that the dissimilarities between clusters are no longer monotone which makes visualizing the results problematic. For these reasons, average linkage is typically preferred to centroid linkage.

        • Ward's Linkage - This method minimizes the total within-cluster variance. At each step, the pair of clusters with minimum between-cluster distance are merged.

          As with centroid linkage, Ward's linkage is intended for the case where Euclidean distances are used. According to Kaufman and Rousseeuw, this method only performs well if an approximately equal number of objects is drawn from each population and it has problems with clusters of unequal diameter. Also, it may have problems when the clusters are ellipsoidal (i.e., variables are correlated within clusters) rather than spherical.

        • Weighted Average Linkage - This method is a variant of average linkage. This method was proposed by Sokal and Sneath. See Kaufman and Rousseeuw for details.

        • Gower's Linkage - This is a variant of the centroid method and should also be restricted to the case of Euclidean distances. See Kaufman and Rousseeus for details.

        In practice, average linkage, complete linkage, and single linkage are the methods most commonly used. Kaufman and Rousseeuw review the properties of various linkage methods and reference several other studies. In summary, although no method is best in all cases, they find that average linkage typically performs well in practice and is reasonably robust to slight distortions. Studies they cite indicate that single linkage, although easy to implement and understand, typically does not perform as well as average linkage or complete linkage.

        To specify the linkage method to use, enter one of the following commands

          SET AGNES CLUSTER METHOD AVERAGE LINKAGE
          SET AGNES CLUSTER METHOD COMPLETE LINKAGE
          SET AGNES CLUSTER METHOD SINGLE LINKAGE
          SET AGNES CLUSTER METHOD WEIGHTED AVERAGE ...
                      LINKAGE
          SET AGNES CLUSTER METHOD WARD
          SET AGNES CLUSTER METHOD CENTROID
          SET AGNES CLUSTER METHOD GOWER

        You can specify the maximum number of rows/columns in the distance matrix (if you start with measurement data, this is the number of columns) with the command

          SET AGNES CLUSTER MAXIMUM SIZE <value>

        where <value> is between 100 and 500 (the default is 100).

        You can control the amount of output generated by agnes clustering with the command

          SET AGNES CLUSTER PRINT <ALL/FINAL>

        Using FINAL omits the printing of the distance matrix.

        Dataplot assumes if the number of rows and columns are equal that a distance matrix is being input. In the unlikely case where the they are equal for measurement data, you can enter the command

          SET AGNES CLUSTER TYPE MEASUREMENT

        To restore the default, enter

          SET AGNES CLUSTER TYPE DISSIMILARITY

      • DIVISIVE (DIANA)

        Dataplot implements the DIANA algorithm given in the Kaufman and Rousseeuw book. The details of the algorithm are given there. DIANA is currently limited to using average distances between clusters.

        The options used for agnes clustering also apply to diana clustering. The exception is that diana only supports the average linkage method.

    The traditional clustering methods described above are heuristic methods and are intented for small to moderate size data sets. These methods tend to work reasonably well for spherical shaped or convex clusters. If clusters are not compact and well separated, these methods may not be effective. The k-means algorithm is sensitive to noise and outliers (the k-medoids method may work better in these cases).

    Dataplot does not currently support model-based clustering or some of the newer cluster methods such as DBSCAN that can work better for non-spherical shapes in the presence of significant noise.

Syntax 1:
    K MEANS CLUSTER <y1> ... <yk>
                            <SUBSET/EXCEPT/FOR qualification>
    where <y1> ... <yk> is a list of response variables;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    This syntax performs Hartigan's k-means clustering.

Syntax 2:
    NORMAL MIXTURE CLUSTER <y1> ... <yk>
                            <SUBSET/EXCEPT/FOR qualification>
    where <y1> ... <yk> is a list of response variables;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    This syntax performs Hartigan's normal mixture clustering.

Syntax 3:
    K MEDIODS CLUSTER <y1> ... <yk>
                            <SUBSET/EXCEPT/FOR qualification>
    where <y1> ... <yk> is a list of response variables;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    This syntax performs Kaufman and Rousseeuw k-medoids clustering. The use of PAM or CLARA will be determined based on the number of objects to be clustered.

Syntax 4:
    FANNY CLUSTER <y1> ... <yk>
                            <SUBSET/EXCEPT/FOR qualification>
    where <y1> ... <yk> is a list of response variables;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    This syntax performs Kaufman and Rousseeuw fuzzy clustering using the FANNY algorithm.

Syntax 5:
    AGNES CLUSTER <y1> ... <yk>
                            <SUBSET/EXCEPT/FOR qualification>
    where <y1> ... <yk> is a list of response variables;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    This syntax performs Kaufman and Rousseeuw agglomerative nesting clustering using the AGNES algorithm.

    By default, this algoritm uses the average distance linking critierion. However, it can also be used for single linkage (nearest neighbor), complete linkage, Ward's method, the centroid method, and Gower's method. See above for details.

Syntax 6:
    DIANA CLUSTER <y1> ... <yk>
                            <SUBSET/EXCEPT/FOR qualification>
    where <y1> ... <yk> is a list of response variables;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    This syntax performs Kaufman and Rousseeuw divisive clustering using the DIANA algorithm.

Examples:
    K MEANS CLUSTERING Y1 Y2 Y3 Y4 Y5 Y6
    K MEANS CLUSTERING Y1 TO Y6
    K MEDOIDS CLUSTERING Y1 TO Y6
    AGNES CLUSTERING M
Note:
    When starting with measurement data, if the variables being clustered use different measurement scales, it may be desirable to standardize the data before applying the clustering algorithm. Standardization creates unitless variables.

    The desirability of standardization will depend on the specific data set. Kaufman and Rousseeuw (pp. 8-11) discuss some of the issues in deciding whether or not to standardize. By default, Dataplot does not standardize the variables.

    The following commands can be used to specify whether or not you want the variables to be standardized

      SET K MEANS SCALE <ON/OFF>
      SET NORMAL MIXTURE SCALE <ON/OFF>
      SET K MEDOIDS SCALE <ON/OFF>
      SET FANNY SCALE <ON/OFF>
      SET AGNES SCALE <ON/OFF>

    The SET AGNES SCALE command also applies to the DIANA CLUSTER command.

    If you choose to standardize, the basic formula is

      \( Y_{i} = \frac{X_{i} - loc}{scale} \)

    where loc and scale denote the desired location and scale parameters.

    To specify the location statistic, enter

      SET LOCATION STATISTIC <stat>

    where <stat> is one of: MEAN, MEDIAN, MIDMEAN. HARMONIC MEAN, GEOMETRIC MEAN, BIWEIGHT LOCATION, H10, H12, H15, H17, or H20.

    To specify the scale statistic, enter

      SET SCALE STATISTIC <stat>

    where <stat> is one of: STANDARD DEVIATION, H10, H12, H15, H17, H20, BIWEIGHT SCALE, MEDIAN ABSOLUTE DEVIATION, SCALED MEDIAN ABSOLUTE DEVIATION, AVERAGE ABSOLUTE DEVIATION, INTERQUARTILE RANGE, NORMALIZED INTERQUARTILE RANGE, SN SCALE, or RANGE.

    The default is to use the mean for the location statistic and the standard deviation for the scale statistic. Rousseeuw recommends using the mean for the location statistic and the average absolute deviation for the scale statistic.

Note: Note:
    One issue with clustering is how to visualize the results. The Dataplot clustering commands do not generate any graphics directly. Instead, Dataplot writes information to files that allow several different approaches to visualization. Different graphical approaches are typically used for partitioning and hierchial methods, so we will discuss these separately.

    1. Partion Methods

      • Dataplot writes the cluster id for each observation to the file dpst1f.dat. So one visualization approach is to generate a scatter plot matrix of the variables and use the cluster id to identify the different clusters in the plots. This is demonstrated in the Program 1 and 2 examples below.

      • Another approach is to plot the first two principal components. Again, the cluster id can be used to identify the clusters. This is demonstrated in the Program 1 and 2 examples below.

      • Rousseeuw advocated the silhouette plot. For each observation, compute

          \( s_{i} = \frac{b_{i} - a_{i}} {\max(a_{i},b_{i})} \)

        where

          \( a_{i} \) = the average dissimilarity of the i-th point with all other points in the cluster to which it belongs
          \( b_{i} \) = the lowest average dissimilarity of the i-th point with all other clusters. The \( b_{i} \) value can be considered the second best choice for observation i.

        The \( s_{i} \) values will be between -1 and 1. A value near 0 indicates that \( a_{i} \) and \( b_{i} \) are nearly equal and thus indicating the choice between assigning object i to A or B is ambiguous. On the other hand, when \( s_{i} \) is close to 1, this indicates that the within dissimilarity is much smaller than the smallest between dissimilarity. This indicates good clustering. Negative values of \( s_{i} \) indicate that B may in fact be a better choice than A, so the observation may be misclassified.

        The average \( s_{i} \) for each cluster and the average \( s_{i} \) for all observations can be computed. These provide a measure of the quality of the clustering. In particular, they can be used to pick appropriate number of clusters to use (i.e., the number of clusters which results in the highest average for all the \( s_{i} \) values).

        Dataplot writes the \( s_{i} \) values (with the cluser id values) to dpst4f.dat. This is demonstrated in the Program 1, 2 and 3 examples below.

    2. Hierchial Methods

      • Kaufman and Rousseeuw provide a line printer "banner" plot for their AGNES and DIANA algorithms. To include this plot in the clustering output, enter the command

        SET AGNES CLUSTERING BANNER PLOT ON

        The default is OFF.

      • The most commonly used visualization technique for hierchial clustering is the dendogram. Dataplot writes the plot coordinates for the dendogram to dpst3f.dat. The ordering of the clusters is written to dpst1f.dat. Program examples 4 and 5 demonstrate how to plot the dendogram from the information in these files. The Program 4 example generates a horizontal dendogram and the Program 5 example generates a vertical dendogram.

        The dendogram is basically a variant of a tree diagram. It shows the order the clusters were joined as well as the distance between clusters. One axis contains the objects to be clustered in a sorted order while the other axis is distance. The dendogram shows which clusters were connected at each step and shows the distance between these clusters.

      • Another popular technique is the icicle plot introduced by Kruskal and Landwehr (1983). Although the original article introduced this as a line printer graphic, it can be adapted for modern graphical displays. The Program 4 and Program 5 examples demonstrate how to generate an icicle plot from the information written to files dpst1f.dat and dpst2f.dat.

        Many different variants of the icicle plot are shown in the literature. But the basic idea is that one axis contains the number of clusters while the other axis shows the objects being clustered. For each object, two rows (or columns) are drawn (the last object only has a single row). The coordinate for one row (or column) shows where the object joined the cluster from one direction (i.e., top or left) while the coordinate for the second row (or column) shows where the object joined the cluster from the other direction. If you scan down the "number of cluster" axis, contiguous rows (or columns) indicate objects that belong to the same cluster. Note that the icicle plot does not give any indication of the distance. The banner plot of Kaufman and Rousseeuw is similar to the icicle plot although they do show distances (on a 0 to 100 percentage scale rather than in raw distance units).

        The Program examples below show the icicle plots as simple bar graphs that are read from left to right (or bottom to top). Variants of the icicle plot often show these as rows (or columns) of asterisks or use a right to left (or top to bottom) orientation. These variants are a matter of taste and can be generated from the information written to files dpst1f.dat and dpst2f.dat.

Default:
    None
Synonyms:
    K MEANS is a synonym for K MEANS CLUSTER
    K MEDOIDS is a synonym for K MEDOIDS CLUSTER
    FANNY is a synonym for FANNY CLUSTER
    AGNES is a synonym for AGNES CLUSTER
    DIANA is a synonym for DIANA CLUSTER
Related Commands: References:
    Hartigan and Wong (1979), "Algorithm AS 136: A K-Means Clustering Algorithm", Applied Statistics, Vol. 28, No. 1.

    Hartigan (1975), "Clustering Algorithms", Wiley.

    Kaufman and Rousseeuw (1990), "Finding Groups in Data: An Introduction To Cluster Analysis", Wiley.

    Rousseeuw (1987), "Silhouettes: A Graphical Aid to the Interpretation and Validation of Cluster Analysis", Journal of Computational and Applied Mathematics, North Holland, Vol. 20, pp. 53-65.

    Kruskal and Landwehr (1983), "Icicle Plots: Better Displays for Hierarchial Clustering", The American Statistician, Vol. 37, No. 2, pp. 168.

Applications:
    Multivariate Analysis, Exploratory Data Analysis
Implementation Date:
    2017/09
Program 1:
     
    case asis
    label case asis
    title case asis
    title offset 2
    .
    . Step 1:   Read the data
    .
    dimension 100 columns
    skip 25
    read iris.dat y1 y2 y3 y4 x
    skip 0
    set write decimals 3
    .
    . Step 2:   Perform the k-means cluster analysis with 3 clusters
    .
    set random number generator fibbonacci congruential
    seed 45617
    let ncluster = 3
    set k means initial distance
    set k means silhouette on
    feedback off
    k-means y1 y2 y3 y4
        
    The following output is generated
                Summary of K-Means Cluster Analysis
     
    ---------------------------------------------
                         Number            Within
                      of Points           Cluster
         Cluster     in Cluster    Sum of Squares
    ---------------------------------------------
               1             53            64.496
               2             49            39.774
               3             48            53.736
        
    read dpst4f.dat clustid si
    .
    . Step 3:   Scatter plot matrix with clusters identified
    .
    line blank all
    char 1 2 3
    char color blue red green
    frame corner coordinates 5 5 95 95
    multiplot scale factor 4
    tic offset units screen
    tic offset 5 5
    .
    set scatter plot matrix tag on
    scatter plot matrix y1 y2 y3 y4 clustid
    .
    justification center
    move 50 97
    text K-Means Clusters for IRIS.DAT
        
    plot generated by sample program
    .
    . Step 4:   Silhouette Plot
    .
    .           For better resolution, show the results for
    .           each cluster separately
    .
    let ntemp = size clustid
    let indx = sequence 1 1 ntemp
    let clustid = sortc clustid si indx
    let x = sequence 1 1 ntemp
    loop for k = 1 1 ntemp
        let itemp = indx(k)
        let string t^k = ^itemp
    end of loop
    .
    orientation portrait
    device 2 color on
    frame corner coordinates 15 20 85 90
    tic offset units data
    horizontal switch on
    .
    spike on
    char blank all
    line blank all
    .
    label size 1.7
    xlimits 0 1
    xtic mark offset 0 0
    x1label S(i)
    x1tic mark label size 1.7
    y1tic mark offset 0.8 0.8
    minor y1tic mark number 0
    y1tic mark label format group label
    y1tic mark label size 1.2
    y1tic mark size 0.8
    y1label Sequence Number
    .
    let simean  = mean si
    let simean  = round(simean,2)
    x3label Mean of All s(i) values: ^simean
    .
    loop for k = 1 1 ncluster
        let sit = si
        let xt  = x
        retain sit xt subset clustid = k
        let ntemp2 = size sit
        let y1min = minimum xt
        let y1max = maximum xt
        y1limits y1min y1max
        major y1tic mark number ntemp2
        let ig = group label t^y1min to t^y1max
        y1tic mark label content ig
        title Silhouette Plot for Cluster ^k Based on K-Means Clustering
        .
        let simean^k = mean si subset clustid = k
        let simean^k = round(simean^k,2)
        x2label Mean of s(i) values for cluster ^k: ^simean^k
        .
        plot si x subset clustid = k
    end of loop
    .
    label
    ylimits
    major y1tic mark number
    minor y1tic mark number
    y1tic mark label format numeric
    y1tic mark label content
    y1tic mark label size
        
    plot generated by sample program

    plot generated by sample program

    plot generated by sample program

    .
    . Step 5:   Display clusters in terms of first 2 principal components
    .
    orientation landscape
    .
    let ym = create matrix y1 y2 y3 y4
    let pc = principal components ym
    read dpst1f.dat clustid
    spike blank all
    character 1 2 3
    character color red blue green
    horizontal switch off
    tic mark offset 0 0
    limits
    title Clusters for First Two Principal Components
    y1label First Principal Component
    x1label Second Principal Component
    x2label
    .
    plot pc1 pc2 clustid
        
    plot generated by sample program
Program 2:
     
    case asis
    label case asis
    title case asis
    title offset 2
    .
    . Step 1:   Read the data
    .
    dimension 100 columns
    skip 25
    read iris.dat y1 y2 y3 y4 x
    skip 0
    set write decimals 3
    .
    . Step 2:   Perform the k-medoids cluster analysis with 3 clusters
    .
    set random number generator fibbonacci congruential
    seed 45617
    let ncluster = 3
    set k medoids cluster distance manhattan
    k medoids y1 y2 y3 y4
        
    The following output is generated
               **********************************************
               *                                            *
               *  ROUSSEEUW/KAUFFMAN K-MEDOID CLUSTERING    *
               *  (USING THE CLARA ROUTINE).                *
               *                                            *
               **********************************************
      
      
     **********************************************
     *                                            *
     *  NUMBER OF REPRESENTATIVE OBJECTS     3    *
     *                                            *
     **********************************************
      
        5 SAMPLES OF    46 OBJECTS WILL NOW BE DRAWN.
      
     SAMPLE NUMBER    1
     ******************
      
     RANDOM SAMPLE =
                2      4      8      9     14     16     19     23     26     27
               30     32     37     38     39     40     43     44     45     46
               49     50     52     53     54     57     62     64     72     87
               89     94     97    102    104    106    109    117    127    130
              135    141    142    143    147    148
      
     RESULT OF BUILD FOR THIS SAMPLE
       AVERAGE DISTANCE =       1.00870
      
     FINAL RESULT FOR THIS SAMPLE
       AVERAGE DISTANCE  =          0.978
      
     RESULTS FOR THE ENTIRE DATA SET
       TOTAL DISTANCE    =         174.900
       AVERAGE DISTANCE  =           1.166
      
       CLUSTER SIZE MEDOID    COORDINATES OF MEDOID
      
             1   50      8         5.00       3.40       0.50       0.20
      
             2   51     62         5.90       3.00       4.20       0.50
      
             3   49    117         6.50       3.00       5.50       1.80
      
       AVERAGE DISTANCE TO EACH MEDOID
              0.75       1.34
      
       MAXIMUM DISTANCE TO EACH MEDOID
              1.90       3.10
       MAXIMUM DISTANCE TO A MEDOID DIVIDED BY MINIMUM
       DISTANCE OF THE MEDOID TO ANOTHER MEDOID
              0.36       0.97
      
     SAMPLE NUMBER    2
     ******************
      
     RANDOM SAMPLE =
                2      8     20     22     24     27     30     32     34     35
               36     37     39     40     43     49     50     52     56     61
               62     63     65     66     71     72     73     74     83     86
               95     97     98    101    117    118    121    126    132    133
              140    141    143    144    146    150
      
     RESULT OF BUILD FOR THIS SAMPLE
       AVERAGE DISTANCE =       0.97174
      
     FINAL RESULT FOR THIS SAMPLE
       AVERAGE DISTANCE  =          0.970
      
     RESULTS FOR THE ENTIRE DATA SET
       TOTAL DISTANCE    =         181.100
       AVERAGE DISTANCE  =           1.207
      
       CLUSTER SIZE MEDOID    COORDINATES OF MEDOID
      
             1   50      8         5.00       3.40       0.50       0.20
      
             2   55     97         5.70       2.90       4.20       0.30
      
             3   45    121         6.90       3.20       5.70       2.30
      
       AVERAGE DISTANCE TO EACH MEDOID
              0.75       1.38
      
       MAXIMUM DISTANCE TO EACH MEDOID
              1.90       3.00
       MAXIMUM DISTANCE TO A MEDOID DIVIDED BY MINIMUM
       DISTANCE OF THE MEDOID TO ANOTHER MEDOID
              0.38       0.60
      
     SAMPLE NUMBER    3
     ******************
      
     RANDOM SAMPLE =
                8     12     13     15     22     23     24     25     26     27
               32     33     35     39     40     43     44     46     47     49
               52     58     59     62     63     67     72     75     80     86
               97     99    100    110    113    115    117    119    123    125
              137    139    143    145    148    149
      
     RESULT OF BUILD FOR THIS SAMPLE
       AVERAGE DISTANCE =       1.01522
      
     FINAL RESULT FOR THIS SAMPLE
       AVERAGE DISTANCE  =          1.015
      
     RESULTS FOR THE ENTIRE DATA SET
       TOTAL DISTANCE    =         171.100
       AVERAGE DISTANCE  =           1.141
      
       CLUSTER SIZE MEDOID    COORDINATES OF MEDOID
      
             1   50      8         5.00       3.40       0.50       0.20
      
             2   50     97         5.70       2.90       4.20       0.30
      
             3   50    113         6.80       3.00       5.50       2.10
      
       AVERAGE DISTANCE TO EACH MEDOID
              0.75       1.25
      
       MAXIMUM DISTANCE TO EACH MEDOID
              1.90       2.90
       MAXIMUM DISTANCE TO A MEDOID DIVIDED BY MINIMUM
       DISTANCE OF THE MEDOID TO ANOTHER MEDOID
              0.38       0.67
      
     SAMPLE NUMBER    4
     ******************
      
     RANDOM SAMPLE =
                4      5      6      8     11     12     15     20     23     26
               37     40     42     43     45     47     53     56     61     63
               68     72     73     90     93     97    103    104    105    108
              113    117    120    122    126    127    129    130    134    135
              138    140    143    144    149    150
      
     RESULT OF BUILD FOR THIS SAMPLE
       AVERAGE DISTANCE =       1.00435
      
     FINAL RESULT FOR THIS SAMPLE
       AVERAGE DISTANCE  =          0.983
      
     RESULTS FOR THE ENTIRE DATA SET
       TOTAL DISTANCE    =         177.100
       AVERAGE DISTANCE  =           1.181
      
       CLUSTER SIZE MEDOID    COORDINATES OF MEDOID
      
             1   50     40         5.10       3.40       0.50       0.20
      
             2   49     93         5.80       2.60       4.00       0.20
      
             3   51    117         6.50       3.00       5.50       1.80
      
       AVERAGE DISTANCE TO EACH MEDOID
              0.76       1.34
      
       MAXIMUM DISTANCE TO EACH MEDOID
              2.00       3.00
       MAXIMUM DISTANCE TO A MEDOID DIVIDED BY MINIMUM
       DISTANCE OF THE MEDOID TO ANOTHER MEDOID
              0.40       0.71
      
     SAMPLE NUMBER    5
     ******************
      
     RANDOM SAMPLE =
                8     12     16     17     18     23     24     26     29     41
               44     48     49     51     52     54     55     56     57     59
               62     66     67     71     73     77     79     81     97    100
              101    102    106    108    111    113    114    117    118    120
              121    123    127    134    137    146
      
     RESULT OF BUILD FOR THIS SAMPLE
       AVERAGE DISTANCE =       1.09130
      
     FINAL RESULT FOR THIS SAMPLE
       AVERAGE DISTANCE  =          1.091
      
     RESULTS FOR THE ENTIRE DATA SET
       TOTAL DISTANCE    =         172.800
       AVERAGE DISTANCE  =           1.152
      
       CLUSTER SIZE MEDOID    COORDINATES OF MEDOID
      
             1   50      8         5.00       3.40       0.50       0.20
      
             2   53     79         6.00       2.90       4.50       0.50
      
             3   47    113         6.80       3.00       5.50       2.10
      
       AVERAGE DISTANCE TO EACH MEDOID
              0.75       1.33
      
       MAXIMUM DISTANCE TO EACH MEDOID
              1.90       3.40
       MAXIMUM DISTANCE TO A MEDOID DIVIDED BY MINIMUM
       DISTANCE OF THE MEDOID TO ANOTHER MEDOID
              0.33       0.97
      
      
      
     FINAL RESULTS
     *************
      
     SAMPLE NUMBER   3 WAS SELECTED, WITH OBJECTS =
           8     12     13     15     22     23     24     25     26     27
          32     33     35     39     40     43     44     46     47     49
          52     58     59     62     63     67     72     75     80     86
          97     99    100    110    113    115    117    119    123    125
         137    139    143    145    148    149
      
        AVERAGE DISTANCE FOR THE ENTIRE DATA SET =        1.141
      
      
        CLUSTERING VECTOR
        *****************
      
           1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
           1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
           2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2
           2  2  3  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2
           3  3  3  3  3  3  2  3  3  3  3  3  3  3  3  3  3  3  3  3  3  3  3  3  3
           3  3  3  3  3  3  3  3  3  3  3  3  3  3  3  3  3  3  3  3  3  3  3  3  3
      
      
         CLUSTER SIZE MEDOID OBJECTS
      
               1   50      8
                                 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   30
                                31   32   33   34   35   36   37   38   39   40
                                41   42   43   44   45   46   47   48   49   50
      
               2   50     97
                                51   52   53   54   55   56   57   58   59   60
                                61   62   63   64   65   66   67   68   69   70
                                71   72   73   74   75   76   77   79   80   81
                                82   83   84   85   86   87   88   89   90   91
                                92   93   94   95   96   97   98   99  100  107
      
               3   50    113
                                78  101  102  103  104  105  106  108  109  110
                               111  112  113  114  115  116  117  118  119  120
                               121  122  123  124  125  126  127  128  129  130
                               131  132  133  134  135  136  137  138  139  140
                               141  142  143  144  145  146  147  148  149  150
      
      
         AVERAGE DISTANCE TO EACH MEDOID
              0.750       1.248       1.424
      
         MAXIMUM DISTANCE TO EACH MEDOID
              1.900       2.900       3.000
      
         MAXIMUM DISTANCE TO A MEDOID DIVIDED BY MINIMUM
              0.380       0.674       0.698
        
    skip 1
    read dpst4f.dat clustid si
    skip 0
    .
    . Step 3:   Scatter plot matrix with clusters identified
    .
    line blank all
    char 1 2 3
    char color blue red green
    frame corner coordinates 5 5 95 95
    multiplot scale factor 4
    tic offset units screen
    tic offset 5 5
    .
    set scatter plot matrix tag on
    scatter plot matrix y1 y2 y3 y4 clustid
    .
    justification center
    move 50 97
    text K Medoids Clusters for IRIS.DAT
        
    plot generated by sample program
    .
    . Step 4:   Silhouette Plot
    .
    .           For better resolution, show the results for
    .           each cluster separately
    .
    let ntemp = size clustid
    let indx = sequence 1 1 ntemp
    let clustid = sortc clustid si indx
    let x = sequence 1 1 ntemp
    loop for k = 1 1 ntemp
        let itemp = indx(k)
        let string t^k = ^itemp
    end of loop
    .
    orientation portrait
    device 2 color on
    frame corner coordinates 15 20 85 90
    tic offset units data
    horizontal switch on
    .
    spike on
    char blank all
    line blank all
    .
    label size 1.7
    xlimits 0 1
    xtic mark offset 0 0
    x1label S(i)
    x1tic mark label size 1.7
    y1tic mark offset 0.8 0.8
    minor y1tic mark number 0
    y1tic mark label format group label
    y1tic mark label size 1.2
    y1tic mark size 0.8
    y1label Sequence Number
    .
    let simean  = mean si
    let simean  = round(simean,2)
    x3label Mean of All s(i) values: ^simean
    .
    orientation portait
    device 2 color on
    loop for k = 1 1 ncluster
        .
        .
        let sit = si
        let xt  = x
        retain sit xt subset clustid = k
        let ntemp2 = size sit
        let y1min = minimum xt
        let y1max = maximum xt
        y1limits y1min y1max
        major y1tic mark number ntemp2
        let ig = group label t^y1min to t^y1max
        y1tic mark label content ig
        title Silhouette Plot for Cluster ^k Based on K-Medoids Clustering
        .
        let simean^k = mean si subset clustid = k
        let simean^k = round(simean^k,2)
        x2label Mean of s(i) values for cluster ^k: ^simean^k
        .
        plot si x subset clustid = k
    end of loop
    .
    label
    ylimits
    major y1tic mark number
    minor y1tic mark number
    y1tic mark label format numeric
    y1tic mark label content
    y1tic mark label size
        
    plot generated by sample program

    plot generated by sample program

    plot generated by sample program

    .
    . Step 5:   Display clusters in terms of first 2 principal components
    .
    orientation landscape
    device 2 color on
    .
    let ym = create matrix y1 y2 y3 y4
    let pc = principal components ym
    read dpst1f.dat clustid
    spike blank all
    character 1 2 3
    character color red blue green
    horizontal switch off
    tic mark offset 0 0
    limits
    title Clusters for First Two Principal Components
    y1label First Principal Component
    x1label Second Principal Component
    x2label
    .
    plot pc1 pc2 clustid
        
    plot generated by sample program
Program 3:
     
    orientation portait
    .
    case asis
    label case asis
    title case asis
    title offset 2
    .
    . Step 1:   Read the data
    .
    set write decimals 3
    dimension 100 columns
    .
    skip 25
    read matrix rouss1.dat y
    skip 0
    .
    let string s1  = Belgium
    let string s2  = Brazil
    let string s3  = China
    let string s4  = Cuba
    let string s5  = Egypt
    let string s6  = France
    let string s7  = India
    let string s8  = Israel
    let string s9  = USA
    let string s10 = USSR
    let string s11 = Yugoslavia
    let string s12 = Zaire
    .
    . Step 2:   Perform the k-mediods cluster analysis with 3 clusters
    .
    let ncluster = 3
    .
    capture screen on
    capture CLUST4A.OUT
    k medioids y
    end of capture
    skip 1
    read dpst4f.dat indx clustid si neighbor
    skip 0
    .
    . Step 3:   Silhouette Plot
    .
    .           Create axis label
    .
    .           First sort by cluster and then sort by
    .           silhouette within cluster (this second step
    .           is a bit convoluted)
    .
    let simean = mean si
    let simean = round(simean,2)
    .
    let ntemp = size indx
    let clustid = sortc clustid si indx neighbor
    .
    loop for k = 1 1 ncluster
        .
        let simean^k = mean si subset clustid = ^k
        let simean^k = round(simean^k,2)
        .
        let clustidt = clustid
        let sit = si
        let indxt = indx
        let neight = neighbor
        retain clustidt sit indxt neight subset clustid = k
        .
        let sit = sortc sit clustidt indxt neight
        if k = 1
           let clustid2 = clustidt
           let si2 = sit
           let indx2 = indxt
           let neigh2 = neight
        else
           let clustid2 = combine clustid2 clustidt
           let si2 = combine si2 sit
           let indx2 = combine indx2 indxt
           let neigh2 = combine neigh2 neight
        end of if
    end of loop
    let clustid = clustid2
    let si = si2
    let indx = indx2
    let neighbor = neigh2
    .
    loop for k = 1 1 ntemp
        let itemp = indx(k)
        let string t^k = ^s^itemp
    end of loop
    let ig = group label t1 to t^ntemp
    .
    let x = sequence 1 1 ntemp
    .
    frame corner coordinates 15 20 85 90
    tic offset units data
    horizontal switch on
    .
    spike on all
    spike color red blue green
    char blank all
    line blank all
    .
    xlimits 0 1
    xtic mark offset 0 0
    major xtic mark number 6
    x1tic mark decimal 1
    y1limits 1 ntemp
    y1tic mark offset 1 1
    major y1tic mark number ntemp
    minor y1tic mark number 0
    y1tic mark label format group label
    y1tic mark label content ig
    y1tic mark label size 1.1
    y1tic mark size 0.1
    x1label S(i)
    x3label Mean of All s(i) values: ^simean
    title Silhouette Plot Based on K-Medoids Clustering
    .
    plot si x clustid
    .
    height 1.0
    justification left
    movesd 87 3
    text Mean s(i): ^simean1
    movesd 87 7
    text Mean s(i): ^simean2
    movesd 87 10.5
    text Mean s(i): ^simean3
    height 2
    .
    print indx clustid neighbor si
        
    The following output is generated
     
               **********************************************
               *                                            *
               *  ROUSSEEUW/KAUFFMAN K-MEDOID CLUSTERING    *
               *  (USING THE PAM ROUTINE).                  *
               *                                            *
               **********************************************
      
      
      
     DISSIMILARITY MATRIX
     --------------------
        1
        2       5.58
        3       7.00     6.50
        4       7.08     7.00     3.83
        5       4.83     5.08     8.17     5.83
        6       2.17     5.75     6.67     6.92     4.92
        7       6.42     5.00     5.58     6.00     4.67     6.42
        8       3.42     5.50     6.42     6.42     5.00     3.92     6.17
        9       2.50     4.92     6.25     7.33     4.50     2.25     6.33     2.75
       10       6.08     6.67     4.25     2.67     6.00     6.17     6.17     6.92
                6.17
       11       5.25     6.83     4.50     3.75     5.75     5.42     6.08     5.83
                6.67     3.67
       12       4.75     3.00     6.08     6.67     5.00     5.58     4.83     6.17
                5.67     6.50     6.92
      
      
     **********************************************
     *                                            *
     *  NUMBER OF REPRESENTATIVE OBJECTS     3    *
     *                                            *
     **********************************************
      
     RESULT OF BUILD
       AVERAGE DISSIMILARITY =       2.58333
      
     FINAL RESULTS
      
       AVERAGE DISSIMILARITY =        2.507
      
     CLUSTERS
        NUMBER  MEDOID   SIZE      OBJECTS
      
         1        9       5       1   5   6   8   9
      
         2       12       3       2   7  12
      
         3        4       4       3   4  10  11
      
     CLUSTERING VECTOR
     *****************
      
                  1  2  3  3  1  1  2  1  1  3  3  2
      
      
     CLUSTERING CHARACTERISTICS
     **************************
     CLUSTER    3 IS ISOLATED
              WITH DIAMETER  =       4.50 AND SEPARATION =       5.25
              THEREFORE IT IS AN L*-CLUSTER.
      
      THE NUMBER OF ISOLATED CLUSTERS =    1
      
       DIAMETER OF EACH CLUSTER
            5.00     5.00     4.50
      
       SEPARATION OF EACH CLUSTER
            5.00     4.50     5.25
      
       AVERAGE DISSIMILARITY TO EACH MEDOID
            2.40     2.61     2.56
      
       MAXIMUM DISSIMILARITY TO EACH MEDOID
            4.50     4.83     3.83
      
      
     ------------------------------------------------------------
                INDX        CLUSTID       NEIGHBOR             SI
     ------------------------------------------------------------
               5.000          1.000          2.000          0.021
               8.000          1.000          2.000          0.366
               1.000          1.000          2.000          0.421
               6.000          1.000          2.000          0.440
               9.000          1.000          2.000          0.468
               7.000          2.000          3.000          0.175
               2.000          2.000          1.000          0.255
              12.000          2.000          1.000          0.280
               3.000          3.000          2.000          0.307
              11.000          3.000          1.000          0.313
              10.000          3.000          1.000          0.437
               4.000          3.000          2.000          0.479
      
        
    plot generated by sample program
Program 4:
     
    . Step 1:   Read the data - a dissimilarity matrix
    .
    dimension 100 columns
    set write decimals 3
    .
    skip 25
    read matrix rouss1.dat y
    skip 0
    .
    let string s1  = Belgium
    let string s2  = Brazil
    let string s3  = China
    let string s4  = Cuba
    let string s5  = Egypt
    let string s6  = France
    let string s7  = India
    let string s8  = Israel
    let string s9  = USA
    let string s10 = USSR
    let string s11 = Yugoslavia
    let string s12 = Zaire
    .
    . Step 2:   Perform the agnes cluster analysis
    .
    set agnes cluster banner plot on
    agnes y
        
    The following output is generated
              **********************************************
              *                                            *
              *  ROUSSEEUW/KAUFFMAN AGGLOMERATIVE NESTING  *
              *  CLUSTERING (USING THE AGNES ROUTINE).     *
              *                                            *
              *  DATA IS A DISSIMILARITY MATRIX.           *
              *                                            *
              *  USE AVERAGE LINKAGE METHOD.               *
              *                                            *
              **********************************************
     
     
     
    DISSIMILARITY MATRIX
    -------------------------
     
    001
    002       5.58
    003       7.00     6.50
    004       7.08     7.00     3.83
    005       4.83     5.08     8.17     5.83
    006       2.17     5.75     6.67     6.92     4.92
    007       6.42     5.00     5.58     6.00     4.67     6.42
    008       3.42     5.50     6.42     6.42     5.00     3.92     6.17
    009       2.50     4.92     6.25     7.33     4.50     2.25     6.33     2.75
    010       6.08     6.67     4.25     2.67     6.00     6.17     6.17     6.92
              6.17
    011       5.25     6.83     4.50     3.75     5.75     5.42     6.08     5.83
              6.67     3.67
    012       4.75     3.00     6.08     6.67     5.00     5.58     4.83     6.17
              5.67     6.50     6.92
     
     
     
     
     
    CLUSTER RESULTS
    ---------------
     
     
    THE FINAL ORDERING OF THE OBJECTS IS
     
            1              6              9              8              2
           12              5              7              3              4
           10             11
     
     
    THE DISSIMILARITIES BETWEEN CLUSTERS ARE
     
                 2.170          2.375          3.363          5.532          3.000
                 4.978          4.670          6.417          4.193          2.670
                 3.710
     
     
     
                                      ************
                                      *          *
                                      *  BANNER  *
                                      *          *
                                      ************
     
     
    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1
    .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .
    0  0  0  1  1  2  2  2  3  3  4  4  4  5  5  6  6  6  7  7  8  8  8  9  9  0
    0  4  8  2  6  0  4  8  2  6  0  4  8  2  6  0  4  8  2  6  0  4  8  2  6  0
     
     
                              001+001+001+001+001+001+001+001+001+001+001+001+001+0
                              *****************************************************
                              006+006+006+006+006+006+006+006+006+006+006+006+006+0
                                ***************************************************
                                009+009+009+009+009+009+009+009+009+009+009+009+009
                                            ***************************************
                                            008+008+008+008+008+008+008+008+008+008
                                                                     **************
                                        002+002+002+002+002+002+002+002+002+002+002
                                        *******************************************
                                        012+012+012+012+012+012+012+012+012+012+012
                                                               ********************
                                                           005+005+005+005+005+005+
                                                           ************************
                                                           007+007+007+007+007+007+
                                                                                ***
                                                      003+003+003+003+003+003+003+0
                                                      *****************************
                                    004+004+004+004+004+004+004+004+004+004+004+004
                                    ***********************************************
                                    010+010+010+010+010+010+010+010+010+010+010+010
                                                ***********************************
                                                011+011+011+011+011+011+011+011+011
    0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1
    .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .
    0  0  0  1  1  2  2  2  3  3  4  4  4  5  5  6  6  6  7  7  8  8  8  9  9  0
    0  4  8  2  6  0  4  8  2  6  0  4  8  2  6  0  4  8  2  6  0  4  8  2  6  0
     
     
     THE ACTUAL HIGHEST LEVEL IS                6.4171875000
     
     
     THE AGGLOMERATIVE COEFFICIENT OF THIS DATA SET IS   0.50
        
    .
    . Step 3:   Generate dendogram from dpst3f.dat file
    .
    skip 0
    read dpst1f.dat indx
    read dpst3f.dat xd yd tag
    .
    orientation portrait
    case asis
    label case asis
    title case asis
    title offset 2
    label size 1.5
    tic mark label size 1.5
    title size 1.5
    tic mark offset units data
    .
    let ntemp = size indx
    loop for k = 1 1 ntemp
        let itemp = indx(k)
        let string t^k = ^s^itemp
    end of loop
    let ig = group label t1 to t^ntemp
    .
    x1label Distance
    ylimits 1 12
    major ytic mark number 12
    minor ytic mark number 0
    y1tic mark label format group label
    y1tic mark label content ig
    ytic mark offset 0.9 0.9
    frame corner coordinates 15 20 95 90
    .
    pre-sort off
    horizontal switch on
    title Dendogram of Kauffman and Rousseeuw Data Set (Average Linkage)
    plot yd xd tag
        
    plot generated by sample program
    .
    . Step 4:   Generate icicle plot from dpst2f.dat file
    .
    delete xd yd tag
    skip 0
    read dpst1f.dat indx
    read dpst2f.dat xd yd tag
    .
    set string space ignore
    let ntemp = size indx
    let ntic = 2*ntemp - 1
    let string tcr = sp()cr()
    loop for k = 1 1 ntemp
        let itemp = indx(k)
        let ktemp1 = (k-1)*2 + 1
        let ktemp2 = ktemp1 + 1
        let string t^ktemp1 = ^s^itemp
        if k < ntemp
           let string t^ktemp2 = sp()
        end of if
    end of loop
    let ig = group label t1 to t^ntic
    .
    ylimits 1 ntic
    major ytic mark number ntic
    minor ytic mark number 0
    y1tic mark label format group label
    y1tic mark label content ig
    ytic mark offset 0.9 0.9
    frame corner coordinates 15 20 95 90
    .
    xlimits 0 12
    major x1tic mark number 13
    minor x1tic mark number 0
    .
    line blank all
    character blank all
    bar on all
    bar fill on all
    bar fill color blue all
    .
    x1label Number of Clusters
    title Icicle Plot of Kauffman and Rousseeuw Data Set (Average Linkage)
    plot yd xd tag
        
    plot generated by sample program
Program 5:
    case asis
    label case asis
    title case asis
    title offset 2
    .
    . Step 1:   Read the data - a dissimilarity matrix
    .
    dimension 100 columns
    set write decimals 3
    .
    skip 25
    read matrix rouss1.dat y
    skip 0
    .
    let string s1  = Belgium
    let string s2  = Brazil
    let string s3  = China
    let string s4  = Cuba
    let string s5  = Egypt
    let string s6  = France
    let string s7  = India
    let string s8  = Israel
    let string s9  = USA
    let string s10 = USSR
    let string s11 = Yugoslavia
    let string s12 = Zaire
    .
    . Step 2:   Perform the agnes cluster analysis
    .
    set agnes cluster banner plot on
    set agnes cluster method average linkage
    agnes y
        
    The following output is generated
               **********************************************
               *                                            *
               *  ROUSSEEUW/KAUFFMAN AGGLOMERATIVE NESTING  *
               *  CLUSTERING (USING THE AGNES ROUTINE).     *
               *                                            *
               *  DATA IS A DISSIMILARITY MATRIX.           *
               *                                            *
               *  USE AVERAGE LINKAGE METHOD.               *
               *                                            *
               **********************************************
      
      
      
     DISSIMILARITY MATRIX
     -------------------------
      
     001
     002       5.58
     003       7.00     6.50
     004       7.08     7.00     3.83
     005       4.83     5.08     8.17     5.83
     006       2.17     5.75     6.67     6.92     4.92
     007       6.42     5.00     5.58     6.00     4.67     6.42
     008       3.42     5.50     6.42     6.42     5.00     3.92     6.17
     009       2.50     4.92     6.25     7.33     4.50     2.25     6.33     2.75
     010       6.08     6.67     4.25     2.67     6.00     6.17     6.17     6.92
               6.17
     011       5.25     6.83     4.50     3.75     5.75     5.42     6.08     5.83
               6.67     3.67
     012       4.75     3.00     6.08     6.67     5.00     5.58     4.83     6.17
               5.67     6.50     6.92
      
      
      
      
      
     CLUSTER RESULTS
     ---------------
      
      
     THE FINAL ORDERING OF THE OBJECTS IS
      
             1              6              9              8              2
            12              5              7              3              4
            10             11
      
      
     THE DISSIMILARITIES BETWEEN CLUSTERS ARE
      
                  2.170          2.375          3.363          5.532          3.000
                  4.978          4.670          6.417          4.193          2.670
                  3.710
      
      
      
                                       ************
                                       *          *
                                       *  BANNER  *
                                       *          *
                                       ************
      
      
     0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1
     .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .
     0  0  0  1  1  2  2  2  3  3  4  4  4  5  5  6  6  6  7  7  8  8  8  9  9  0
     0  4  8  2  6  0  4  8  2  6  0  4  8  2  6  0  4  8  2  6  0  4  8  2  6  0
      
      
                               001+001+001+001+001+001+001+001+001+001+001+001+001+0
                               *****************************************************
                               006+006+006+006+006+006+006+006+006+006+006+006+006+0
                                 ***************************************************
                                 009+009+009+009+009+009+009+009+009+009+009+009+009
                                             ***************************************
                                             008+008+008+008+008+008+008+008+008+008
                                                                      **************
                                         002+002+002+002+002+002+002+002+002+002+002
                                         *******************************************
                                         012+012+012+012+012+012+012+012+012+012+012
                                                                ********************
                                                            005+005+005+005+005+005+
                                                            ************************
                                                            007+007+007+007+007+007+
                                                                                 ***
                                                       003+003+003+003+003+003+003+0
                                                       *****************************
                                     004+004+004+004+004+004+004+004+004+004+004+004
                                     ***********************************************
                                     010+010+010+010+010+010+010+010+010+010+010+010
                                                 ***********************************
                                                 011+011+011+011+011+011+011+011+011
     0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1
     .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .
     0  0  0  1  1  2  2  2  3  3  4  4  4  5  5  6  6  6  7  7  8  8  8  9  9  0
     0  4  8  2  6  0  4  8  2  6  0  4  8  2  6  0  4  8  2  6  0  4  8  2  6  0
      
      
      THE ACTUAL HIGHEST LEVEL IS                6.4171875000
      
      
      THE AGGLOMERATIVE COEFFICIENT OF THIS DATA SET IS   0.50
        
    .
    . Step 3:   Generate dendogram from dpst3f.dat file
    .
    skip 0
    read dpst1f.dat indx
    read dpst3f.dat xd yd tag
    .
    let ntemp = size indx
    let string tcr = sp()cr()
    loop for k = 1 1 ntemp
        let itemp = indx(k)
        let string t^k = ^s^itemp
        let ival1 = mod(k,2)
        if ival1 = 0
           let t^k = string concatenate tcr t^k
        end of if
    end of loop
    let ig = group label t1 to t^ntemp
    .
    xlimits 1 12
    major xtic mark number 12
    minor xtic mark number 0
    x1tic mark label format group label
    x1tic mark label content ig
    xtic mark offset 0.9 0.9
    frame corner coordinates 15 20 95 90
    .
    y1label Distance
    title Dendogram of Kauffman and Rousseeuw Data Set (Average Linkage)
    plot yd xd tag
        
    plot generated by sample program
    .
    . Step 4:   Generate icicle plot from dpst2f.dat file
    .
    delete xd yd tag
    skip 0
    read dpst1f.dat indx
    read dpst2f.dat xd yd tag
    .
    set string space ignore
    let ntemp = size indx
    let ntic = 2*ntemp - 1
    let string tcr = sp()cr()
    loop for k = 1 1 ntemp
        let itemp = indx(k)
        let ktemp1 = (k-1)*2 + 1
        let ktemp2 = ktemp1 + 1
        let string t^ktemp1 = ^s^itemp
        if k < ntemp
           let string t^ktemp2 = sp()
        end of if
        let ival1 = mod(k,2)
        if ival1 = 0
           let t^ktemp1 = string concatenate tcr t^ktemp1
        end of if
    end of loop
    let ig = group label t1 to t^ntic
    .
    xlimits 1 ntic
    major xtic mark number ntic
    minor xtic mark number 0
    x1tic mark label format group label
    x1tic mark label content ig
    xtic mark offset 0.9 0.9
    frame corner coordinates 15 20 95 90
    .
    ylimits 0 12
    major y1tic mark number 13
    minor y1tic mark number 0
    .
    line blank all
    character blank all
    bar on all
    bar fill on all
    bar fill color blue all
    .
    y1label Number of Clusters
    title Icicle Plot of Kauffman and Rousseeuw Data Set (Average Linkage)
    plot yd xd tag
        
    plot generated by sample program

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