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Example 23.1: Cluster Analysis of Flying Mileages between Ten American Cities

This first example clusters ten American cities based on the flying mileages between them. Six clustering methods are shown with corresponding tree diagrams produced by the TREE procedure. The EML method cannot be used because it requires coordinate data. The other omitted methods produce the same clusters, although not the same distances between clusters, as one of the illustrated methods: complete linkage and the flexible-beta method yield the same clusters as Ward's method, McQuitty's similarity analysis produces the same clusters as average linkage, and the median method corresponds to the centroid method.

All of the methods suggest a division of the cities into two clusters along the east-west dimension. There is disagreement, however, about which cluster Denver should belong to. Some of the methods indicate a possible third cluster containing Denver and Houston. The following statements produce Output 23.1.1:

   title 'Cluster Analysis of Flying Mileages Between 10 American Cities';
   data mileages(type=distance);
      input (atlanta chicago denver houston losangeles
            miami newyork sanfran seattle washdc) (5.)
            @55 city $15.;
      datalines;
       0                                                 ATLANTA
     587    0                                            CHICAGO
    1212  920    0                                       DENVER
     701  940  879    0                                  HOUSTON
    1936 1745  831 1374    0                             LOS ANGELES
     604 1188 1726  968 2339    0                        MIAMI
     748  713 1631 1420 2451 1092    0                   NEW YORK
    2139 1858  949 1645  347 2594 2571    0              SAN FRANCISCO
    2182 1737 1021 1891  959 2734 2408  678    0         SEATTLE
     543  597 1494 1220 2300  923  205 2442 2329    0    WASHINGTON D.C.
   ;


   /*---------------------- Average linkage --------------------*/
    proc cluster data=mileages method=average pseudo;
      id city;
   run;

   proc tree horizontal spaces=2;
      id city;
   run;

   /*---------------------- Centroid method --------------------*/
   proc cluster data=mileages method=centroid pseudo;
      id city;
   run;

   proc tree horizontal spaces=2;
      id city;
   run;

   /*-------- Density linkage with 3rd-nearest-neighbor --------*/
   proc cluster data=mileages method=density k=3;
      id city;
   run;

   proc tree horizontal spaces=2;
      id city;
   run;

   /*--------------------- Single linkage ----------------------*/
   proc cluster data=mileages method=single;
      id city;
   run;

   proc tree horizontal spaces=2;
      id city;
   run;

   /*--- Two-stage density linkage with 3rd-nearest-neighbor ---*/
   proc cluster data=mileages method=twostage k=3;
      id city;
   run;

   proc tree horizontal spaces=2;
      id city;
   run;

   /* Ward's minimum variance with pseudo $F$ and $t^2$ statistics */
   proc cluster data=mileages method=ward pseudo;
      id city;
   run;

   proc tree horizontal spaces=2;
      id city;
   run;

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