PROC MODECLUS: Cluster Analysis of Flying Mileages between Ten American Cities

Example 59.2 Cluster Analysis of Flying Mileages between Ten American Cities

This example uses distance data and illustrates the use of the TRANSPOSE procedure and the DATA step to fill in the upper triangle of the distance matrix. A data set containing a table of flying mileages between 10 U.S. cities is available in the Sashelp library. The results are displayed in Output 59.2.1 through Output 59.2.3.

The following statements produce Output 59.2.1:

title 'Modeclus Analysis of 10 American Cities';
title2 'Based on Flying Mileages';

*-----Fill in Upper Triangle of Distance Matrix---------------;
proc transpose data=sashelp.mileages out=tran;
   copy city;
run;

data mileages(type=distance drop=col: _: i);
   merge sashelp.mileages tran;
   array var[10] atlanta--washingtondc;
   array col[10];
   do i = 1 to 10;
      var[i] = sum(var[i], col[i]);
   end;
run;

*-----Clustering with K-Nearest-Neighbor Density Estimates-----;
proc modeclus data=mileages all m=1 k=3;
   id CITY;
run;

Output 59.2.1 Clustering with K-Nearest-Neighbor Density Estimates

Modeclus Analysis of 10 American Cities

Based on Flying Mileages

The MODECLUS Procedure

Nearest Neighbor List
City	Neighbor	Distance
Atlanta	Washington D.C.	543.0000000
	Chicago	587.0000000
Chicago	Atlanta	587.0000000
	Washington D.C.	597.0000000
Denver	Los Angeles	831.0000000
	Houston	879.0000000
Houston	Atlanta	701.0000000
	Denver	879.0000000
Los Angeles	San Francisco	347.0000000
	Denver	831.0000000
Miami	Atlanta	604.0000000
	Washington D.C.	923.0000000
New York	Washington D.C.	205.0000000
	Chicago	713.0000000
San Francisco	Los Angeles	347.0000000
	Seattle	678.0000000
Seattle	San Francisco	678.0000000
	Los Angeles	959.0000000
Washington D.C.	New York	205.0000000
	Atlanta	543.0000000

Modeclus Analysis of 10 American Cities

Based on Flying Mileages

The MODECLUS Procedure

K=3 METHOD=1

Sums of Density Estimates Within Neighborhood
Cluster	City	Estimated Density	Same Cluster	Other Clusters	Total	Cluster Proportion Same/Total
1	Atlanta	0.00025554	0.0005275	0	0.0005275	1.000
	Chicago	0.00025126	0.00053178	0	0.00053178	1.000
	Houston	0.00017065	0.00025554	0.00017065	0.00042619	0.600
	Miami	0.00016251	0.00053178	0	0.00053178	1.000
	New York	0.00021038	0.0005275	0	0.0005275	1.000
	Washington D.C.	0.00027624	0.00046592	0	0.00046592	1.000
2	Denver	0.00017065	0.00018051	0.00017065	0.00035115	0.514
	Los Angeles	0.00018051	0.00039189	0	0.00039189	1.000
	San Francisco	0.00022124	0.00033692	0	0.00033692	1.000
	Seattle	0.00015641	0.00040174	0	0.00040174	1.000

Boundary Objects -Cluster Proportions-
City	Density	Cluster	1	2
Denver	0.0001706485	2	0.486	0.514
Houston	0.0001706485	1	0.600	0.400

Cluster Statistics
Cluster	Frequency	Maximum Estimated Density	Boundary Frequency	Estimated Saddle Density
1	6	0.00027624	1	0.00017065
2	4	0.00022124	1	0.00017065

Modeclus Analysis of 10 American Cities

Based on Flying Mileages

The MODECLUS Procedure

Cluster Summary
K	Number of Clusters	Frequency of Unclassified Objects
3	2	0

The following statements produce Output 59.2.2:

*------Clustering with Uniform-Kernel Density Estimates--------;
proc modeclus data=mileages all m=1 r=600 800;
   id CITY;
run;

Output 59.2.2 Clustering with Uniform-Kernel Density Estimates

Modeclus Analysis of 10 American Cities

Based on Flying Mileages

The MODECLUS Procedure

Nearest Neighbor List
City	Neighbor	Distance
Atlanta	Washington D.C.	543.0000000
	Chicago	587.0000000
	Miami	604.0000000
	Houston	701.0000000
	New York	748.0000000
Chicago	Atlanta	587.0000000
	Washington D.C.	597.0000000
	New York	713.0000000
Houston	Atlanta	701.0000000
Los Angeles	San Francisco	347.0000000
Miami	Atlanta	604.0000000
New York	Washington D.C.	205.0000000
	Chicago	713.0000000
	Atlanta	748.0000000
San Francisco	Los Angeles	347.0000000
	Seattle	678.0000000
Seattle	San Francisco	678.0000000
Washington D.C.	New York	205.0000000
	Atlanta	543.0000000
	Chicago	597.0000000

Modeclus Analysis of 10 American Cities

Based on Flying Mileages

The MODECLUS Procedure

R=600 METHOD=1

Sums of Density Estimates Within Neighborhood
Cluster	City	Estimated Density	Same Cluster	Other Clusters	Total	Cluster Proportion Same/Total
1	Atlanta	0.00025	0.00058333	0	0.00058333	1.000
	Chicago	0.00025	0.00058333	0	0.00058333	1.000
	New York	0.00016667	0.00033333	0	0.00033333	1.000
	Washington D.C.	0.00033333	0.00066667	0	0.00066667	1.000
2	Los Angeles	0.00016667	0.00016667	0	0.00016667	1.000
	San Francisco	0.00016667	0.00016667	0	0.00016667	1.000
3	Denver	0.00008333	0	0	0	.
4	Houston	0.00008333	0	0	0	.
5	Miami	0.00008333	0	0	0	.
6	Seattle	0.00008333	0	0	0	.

No Boundary Objects

Cluster Statistics
Cluster	Frequency	Maximum Estimated Density	Boundary Frequency	Estimated Saddle Density
1	4	0.00033333	0	.
2	2	0.00016667	0	.
3	1	0.00008333	0	.
4	1	0.00008333	0	.
5	1	0.00008333	0	.
6	1	0.00008333	0	.

Modeclus Analysis of 10 American Cities

Based on Flying Mileages

The MODECLUS Procedure

R=800 METHOD=1

Sums of Density Estimates Within Neighborhood
Cluster	City	Estimated Density	Same Cluster	Other Clusters	Total	Cluster Proportion Same/Total
1	Atlanta	0.000375	0.001	0	0.001	1.000
	Chicago	0.00025	0.000875	0	0.000875	1.000
	Houston	0.000125	0.000375	0	0.000375	1.000
	Miami	0.000125	0.000375	0	0.000375	1.000
	New York	0.00025	0.000875	0	0.000875	1.000
	Washington D.C.	0.00025	0.000875	0	0.000875	1.000
2	Los Angeles	0.000125	0.0001875	0	0.0001875	1.000
	San Francisco	0.0001875	0.00025	0	0.00025	1.000
	Seattle	0.000125	0.0001875	0	0.0001875	1.000
3	Denver	0.0000625	0	0	0	.

No Boundary Objects

Cluster Statistics
Cluster	Frequency	Maximum Estimated Density	Boundary Frequency	Estimated Saddle Density
1	6	0.000375	0	.
2	3	0.0001875	0	.
3	1	0.0000625	0	.

Modeclus Analysis of 10 American Cities

Based on Flying Mileages

The MODECLUS Procedure

Cluster Summary
R	Number of Clusters	Frequency of Unclassified Objects
600	6	0
800	3	0

The following statements produce Output 59.2.3:

*------Clustering Neighborhoods Extended to Nearest Neighbor--------;
proc modeclus data=mileages list m=1 ck=2 r=600 800;
   id CITY;
run;

Output 59.2.3 Uniform-Kernel Density Estimates, Clustering Neighborhoods Extended to Nearest Neighbor

Modeclus Analysis of 10 American Cities

Based on Flying Mileages

The MODECLUS Procedure

CK=2 R=600 METHOD=1

Sums of Density Estimates Within Neighborhood
Cluster	City	Estimated Density	Same Cluster	Other Clusters	Total	Cluster Proportion Same/Total
1	Atlanta	0.00025	0.00058333	0	0.00058333	1.000
	Chicago	0.00025	0.00058333	0	0.00058333	1.000
	Houston	0.00008333	0.00025	0	0.00025	1.000
	Miami	0.00008333	0.00025	0	0.00025	1.000
	New York	0.00016667	0.00033333	0	0.00033333	1.000
	Washington D.C.	0.00033333	0.00066667	0	0.00066667	1.000
2	Denver	0.00008333	0.00016667	0	0.00016667	1.000
	Los Angeles	0.00016667	0.00016667	0	0.00016667	1.000
	San Francisco	0.00016667	0.00016667	0	0.00016667	1.000
	Seattle	0.00008333	0.00016667	0	0.00016667	1.000

Cluster Statistics
Cluster	Frequency	Maximum Estimated Density	Boundary Frequency	Estimated Saddle Density
1	6	0.00033333	0	.
2	4	0.00016667	0	.

Modeclus Analysis of 10 American Cities

Based on Flying Mileages

The MODECLUS Procedure

CK=2 R=800 METHOD=1

Sums of Density Estimates Within Neighborhood
Cluster	City	Estimated Density	Same Cluster	Other Clusters	Total	Cluster Proportion Same/Total
1	Atlanta	0.000375	0.001	0	0.001	1.000
	Chicago	0.00025	0.000875	0	0.000875	1.000
	Houston	0.000125	0.000375	0	0.000375	1.000
	Miami	0.000125	0.000375	0	0.000375	1.000
	New York	0.00025	0.000875	0	0.000875	1.000
	Washington D.C.	0.00025	0.000875	0	0.000875	1.000
2	Denver	0.0000625	0.000125	0	0.000125	1.000
	Los Angeles	0.000125	0.0001875	0	0.0001875	1.000
	San Francisco	0.0001875	0.00025	0	0.00025	1.000
	Seattle	0.000125	0.0001875	0	0.0001875	1.000

Cluster Statistics
Cluster	Frequency	Maximum Estimated Density	Boundary Frequency	Estimated Saddle Density
1	6	0.000375	0	.
2	4	0.0001875	0	.

Modeclus Analysis of 10 American Cities

Based on Flying Mileages

The MODECLUS Procedure

Cluster Summary
R	CK	Number of Clusters	Frequency of Unclassified Objects
600	2	2	0
800	2	2	0