PROC CLUSTER: Cluster Analysis of Flying Mileages between 10 American Cities

The CLUSTER Procedure

Example 29.1 Cluster Analysis of Flying Mileages between 10 American Cities

This example clusters 10 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.

title 'Cluster Analysis of Flying Mileages Between 10 American Cities';

data mileages(type=distance);
   input (Atlanta Chicago Denver Houston LosAngeles
         Miami NewYork SanFrancisco Seattle WashingtonDC) (5.)
         @56 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.
;

goptions htext=0.15in htitle=0.15in;

A partial listing from following statements include Output 29.1.1 and Output 29.1.2:

/*---------------------- Average linkage --------------------*/
title2 'Using METHOD=AVERAGE';
proc cluster data=mileages outtree=tree method=average pseudo;
  id City;
run;

proc tree horizontal; id City; run;

Output 29.1.1 Cluster History Using METHOD=AVERAGE

Cluster Analysis of Flying Mileages Between 10 American Cities

Using METHOD=AVERAGE

The CLUSTER Procedure

Average Linkage Cluster Analysis

Cluster History
NCL	Clusters Joined		FREQ	PSF	PST2	Norm RMS Dist	T i e
9	New York	Washington D.C.	2	66.7	.	0.1297
8	Los Angeles	San Francisco	2	39.2	.	0.2196
7	Atlanta	Chicago	2	21.7	.	0.3715
6	CL7	CL9	4	14.5	3.4	0.4149
5	CL8	Seattle	3	12.4	7.3	0.5255
4	Denver	Houston	2	13.9	.	0.5562
3	CL6	Miami	5	15.5	3.8	0.6185
2	CL3	CL4	7	16.0	5.3	0.8005
1	CL2	CL5	10	.	16.0	1.2967

Output 29.1.2 Tree Diagram Using METHOD=AVERAGE

A partial listing from following statements include Output 29.1.3 and Output 29.1.4:

/*---------------------- Centroid method --------------------*/
title2 'Using METHOD=CENTROID';
proc cluster data=mileages method=centroid pseudo;
   id City;
run;

proc tree horizontal; id City; run;

Output 29.1.3 Cluster History Using METHOD=CENTROID

Cluster Analysis of Flying Mileages Between 10 American Cities

Using METHOD=CENTROID

The CLUSTER Procedure

Centroid Hierarchical Cluster Analysis

Cluster History
NCL	Clusters Joined		FREQ	PSF	PST2	Norm Cent Dist	T i e
9	New York	Washington D.C.	2	66.7	.	0.1297
8	Los Angeles	San Francisco	2	39.2	.	0.2196
7	Atlanta	Chicago	2	21.7	.	0.3715
6	CL7	CL9	4	14.5	3.4	0.3652
5	CL8	Seattle	3	12.4	7.3	0.5139
4	Denver	CL5	4	12.4	2.1	0.5337
3	CL6	Miami	5	14.2	3.8	0.5743
2	CL3	Houston	6	22.1	2.6	0.6091
1	CL2	CL4	10	.	22.1	1.173

Output 29.1.4 Tree Diagram Using METHOD=CENTROID

A partial listing from following statements include Output 29.1.5 and Output 29.1.6:

/*-------- Density linkage with 3rd-nearest-neighbor --------*/
title2 'Using METHOD=DENSITY K=3';
proc cluster data=mileages method=density k=3;
   id City;
run;

proc tree horizontal; id City; run;

Output 29.1.5 Cluster History Using METHOD=DENSITY K=3

Cluster Analysis of Flying Mileages Between 10 American Cities

Using METHOD=DENSITY K=3

The CLUSTER Procedure

Density Linkage Cluster Analysis

Cluster History
NCL			FREQ	Normalized Fusion Density		Maximum Density in Each Cluster		T i e
NCL	Clusters Joined		FREQ	Normalized Fusion Density		Lesser	Greater	T i e
9	Atlanta	Washington D.C.	2	96.106		92.5043	100.0
8	CL9	Chicago	3	95.263		90.9548	100.0
7	CL8	New York	4	86.465		76.1571	100.0
6	CL7	Miami	5	74.079		58.8299	100.0	T
5	CL6	Houston	6	74.079		61.7747	100.0
4	Los Angeles	San Francisco	2	71.968		65.3430	80.0885
3	CL4	Seattle	3	66.341		56.6215	80.0885
2	CL3	Denver	4	63.509		61.7747	80.0885
1	CL5	CL2	10	61.775	*	80.0885	100.0

Output 29.1.6 Tree Diagram Using METHOD=DENSITY K=3

A partial listing from following statements include Output 29.1.7 and Output 29.1.8:

/*--------------------- Single linkage ----------------------*/
title2 'Using METHOD=SINGLE';
proc cluster data=mileages method=single;
   id City;
run;

proc tree horizontal; id City; run;

Output 29.1.7 Cluster History Using METHOD=SINGLE

Cluster Analysis of Flying Mileages Between 10 American Cities

Using METHOD=SINGLE

The CLUSTER Procedure

Single Linkage Cluster Analysis

Cluster History
NCL	Clusters Joined		FREQ	Norm Min Dist	T i e
9	New York	Washington D.C.	2	0.1447
8	Los Angeles	San Francisco	2	0.2449
7	Atlanta	CL9	3	0.3832
6	CL7	Chicago	4	0.4142
5	CL6	Miami	5	0.4262
4	CL8	Seattle	3	0.4784
3	CL5	Houston	6	0.4947
2	Denver	CL4	4	0.5864
1	CL3	CL2	10	0.6203

Output 29.1.8 Tree Diagram Using METHOD=SINGLE

A partial listing from following statements include Output 29.1.9 and Output 29.1.10:

/*--- Two-stage density linkage with 3rd-nearest-neighbor ---*/
title2 'Using METHOD=TWOSTAGE K=3';
proc cluster data=mileages method=twostage k=3;
   id City;
run;

proc tree horizontal; id City; run;

Output 29.1.9 Cluster History Using METHOD=TWOSTAGE K=3

Cluster Analysis of Flying Mileages Between 10 American Cities

Using METHOD=TWOSTAGE K=3

The CLUSTER Procedure

Two-Stage Density Linkage Clustering

Cluster History
NCL			FREQ	Normalized Fusion Density	Maximum Density in Each Cluster		T i e
NCL	Clusters Joined		FREQ	Normalized Fusion Density	Lesser	Greater	T i e
9	Atlanta	Washington D.C.	2	96.106	92.5043	100.0
8	CL9	Chicago	3	95.263	90.9548	100.0
7	CL8	New York	4	86.465	76.1571	100.0
6	CL7	Miami	5	74.079	58.8299	100.0	T
5	CL6	Houston	6	74.079	61.7747	100.0
4	Los Angeles	San Francisco	2	71.968	65.3430	80.0885
3	CL4	Seattle	3	66.341	56.6215	80.0885
2	CL3	Denver	4	63.509	61.7747	80.0885
1	CL5	CL2	10	61.775	80.0885	100.0

Output 29.1.10 Tree Diagram Using METHOD=TWOSTAGE K=3

A partial listing from following statements include Output 29.1.11 and Output 29.1.12:

/*------------- Ward's minimum variance method --------------*/
title2 'Using METHOD=WARD';
proc cluster data=mileages method=ward pseudo;
   id City;
run;

proc tree horizontal; id City; run;

Output 29.1.11 Cluster History Using METHOD=WARD

Cluster Analysis of Flying Mileages Between 10 American Cities

Using METHOD=WARD

The CLUSTER Procedure

Ward's Minimum Variance Cluster Analysis

Cluster History
NCL	Clusters Joined		FREQ	SPRSQ	RSQ	PSF	PST2	T i e
9	New York	Washington D.C.	2	0.0019	.998	66.7	.
8	Los Angeles	San Francisco	2	0.0054	.993	39.2	.
7	Atlanta	Chicago	2	0.0153	.977	21.7	.
6	CL7	CL9	4	0.0296	.948	14.5	3.4
5	Denver	Houston	2	0.0344	.913	13.2	.
4	CL8	Seattle	3	0.0391	.874	13.9	7.3
3	CL6	Miami	5	0.0586	.816	15.5	3.8
2	CL3	CL5	7	0.1488	.667	16.0	5.3
1	CL2	CL4	10	0.6669	.000	.	16.0

Output 29.1.12 Tree Diagram Using METHOD=WARD

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