Multidimensional Scaling
and
The Minimum Spanning Tree

SAS Program

***********************************************************************;
* Towns.sas -- Multidimensional scaling of imaginary map of towns     *;
***********************************************************************;
Options PS=60 LS=65 PageNo=1 NoDate
        FORMCHAR='|----|+|---+=|-/\<>*';

/*
 * Include the SAS plotit macro for properly
 * scaling the MDS plot axes and labelling
 * the points.
 */
%Include "plotit7.sas" / NoSource;

Title1 "Multidimensional Scaling of Towns";
Data Towns;
 Length Town $1;
 Input Town A B C D;
DataLines;
A 0 120 160  97
B .   0 200  85
C .   .   0 115
D .   .   .   0
;

Proc Print Data=Towns;
Run;

Title2 "Absolute Distances";
Proc MDS Data=Towns Level=Absolute Out=MDSout
         Dimension=2 PData PConfig PFinal;
 Var A--D;
 Id Town;
Run;

%Plotit(Data=MDSout,Datatype=mds,Monochro=Black);


Title2 "Non-metric Scaling";
Proc MDS Data=Towns Level=Ordinal Out=MDSout
         Dimension=2 PConfig PFinal;
 Var A--D;
 Id Town;
Run;

%Plotit(Data=MDSout,Datatype=mds,Monochro=Black);

/*
 * Now connect the towns with a minimum spanning tree.
 * First, include the MST macro.
 * Second, reproduce the point configuration so that we
 * have the metric solution.
 * Third, subset MDSOut data set so that only the point
 * configuration is kept.
 * Fourth, compute the minimum spanning tree -- it will
 * output an annotate data set that you can use with GPLOT.
 * Fifth, use GPLOT to plot the graph with MST. Note that
 * the axes may not be properly scaled. See later section
 * to see how to use the MST macro and the PLOTIT macro together.
 */

%Include "mst.sas" / NoSource;

Title2 "Absolute Distances";
Proc MDS Data=Towns Level=Absolute Out=MDSout
         Dimension=2 PData PConfig PFinal;
 Var A--D;
 Id Town;
Run;

Data MDSOut2;
 Set MDSOut;
 Where _TYPE_="CONFIG";
Run;

%MST(data=MDSOut2,Out=MSTAnno,Var=Dim1 Dim2,Color=Black);

Proc GPlot Data=MDSOut2 Annotate=MSTAnno;
 Plot Dim2*Dim1=1;
 Symbol1 C=Black V=Dot I=None;
Run;
Quit;

/*
 * Now combine the PLOTIT and MST macros.
 *
 * First, use the PLOTIT macro to generate the plot information,
 * but specify the METHOD=NONE option so that the plot is not
 * actually drawn.
 * Second, dig into the annotate data set produced by PLOTIT
 * to extract the coordinates corresponding to our points.
 * Third, put the MDS data set and the PLOTIT data set together.
 * Fourth, construct the MST annotate data set, then merge
 * it in with the ANNO data set from the PLOTIT macro.
 * Fifth, display the graph with ganno.
 */ 


%Plotit(Data=MDSout,Datatype=mds,Monochro=Black,Method=None);

Data Ann2;
 Set Anno;
 Where OBSTYPE="CONFIG";
 Retain XC YC;
 If Function="LABEL" Then
  Do;
   Town=Text;
   Output;
  End;
 XC=X; YC=Y;
 Keep XC YC Town;
Run;

Proc Sort Data=Ann2;
 By Town;
Run;

Data MDSOut2;
 Merge MDSOut(FirstObs=2) Ann2;
Run;

%MST(data=MDSOut2,Out=MSTAnno,Var=Dim1 Dim2,Var2=XC YC,Color=Black);

Data Anno2;
 Set Anno MSTAnno;
 Drop HSys XSys YSys ZSys;
Run;

Proc GANNO Annotate=Anno2;
Run;

SAS Output

 

Multidimensional Scaling of Towns

Obs	Town	A	B	C	D
1	A	0	120	160	97
2	B	.	0	200	85
3	C	.	.	0	115
4	D	.	.	.	0

 

---

 

Multidimensional Scaling of Towns

Absolute Distances

Multidimensional Scaling: Data=WORK.TOWNS.DATA

Data Matrix
1	A	B	C	D
A	0	120	160	97
B	120	0	200	85
C	160	200	0	115
D	97	85	115	0

 

---

 

Multidimensional Scaling of Towns

Absolute Distances

Multidimensional Scaling: Data=WORK.TOWNS.DATA

Shape=TRIANGLE Condition=MATRIX Level=ABSOLUTE

Coef=IDENTITY Dimension=2 Formula=1 Fit=1

Gconverge=0.01 Maxiter=100 Over=1 Ridge=0.0001

Iteration	Type	Badness- of-Fit Criterion	Change in Criterion	Convergence Measure
0	Initial	3.6864E-7	.	0.6568

 

Convergence assumed because the badness-of-fit criterion is less than or equal to MINCRIT=1E-6.

 

Configuration
	Dim1	Dim2
A	25.69	69.47
B	85.77	-34.41
C	-112.81	-10.63
D	1.36	-24.43

 

---

 

 

---

 

Multidimensional Scaling: Data=WORK.TOWNS.DATA

Shape=TRIANGLE Condition=MATRIX Level=ORDINAL

Coef=IDENTITY Dimension=2 Formula=1 Fit=1

Mconverge=0.01 Gconverge=0.01 Maxiter=100 Over=2 Ridge=0.0001

Iteration	Type	Badness- of-Fit Criterion	Change in Criterion	Convergence Measures
				Monotone	Gradient
0	Initial	3.6864E-7	.	.	.

 

Convergence assumed because the badness-of-fit criterion is less than or equal to MINCRIT=1E-6.

 

Configuration
	Dim1	Dim2
A	0.44	1.19
B	1.46	-0.59
C	-1.93	-0.18
D	0.02	-0.42

 

---

 

 

---

 

Multidimensional Scaling of Towns

Absolute Distances

Multidimensional Scaling: Data=WORK.TOWNS.DATA

Data Matrix
1	A	B	C	D
A	0	120	160	97
B	120	0	200	85
C	160	200	0	115
D	97	85	115	0

 

---

 

Multidimensional Scaling of Towns

Absolute Distances

Multidimensional Scaling: Data=WORK.TOWNS.DATA

Shape=TRIANGLE Condition=MATRIX Level=ABSOLUTE

Coef=IDENTITY Dimension=2 Formula=1 Fit=1

Gconverge=0.01 Maxiter=100 Over=1 Ridge=0.0001

Iteration	Type	Badness- of-Fit Criterion	Change in Criterion	Convergence Measure
0	Initial	3.6864E-7	.	0.6568

 

Convergence assumed because the badness-of-fit criterion is less than or equal to MINCRIT=1E-6.

 

Configuration
	Dim1	Dim2
A	25.69	69.47
B	85.77	-34.41
C	-112.81	-10.63
D	1.36	-24.43

 

---

 

Multidimensional Scaling of Towns

Absolute Distances

X Contains the From Coordinates,

 

Y Contains the To Coordinates

 

OBSNUM	LIST2
1	1
2	4
3	4
4	1

 

COORD		Y
25.690672	69.469791	25.690672	69.469791
85.767694	-34.40875	1.35564	-24.42806
-112.814	-10.63298	1.35564	-24.42806
1.35564	-24.42806	25.690672	69.469791

 

---

 

 

---

 

Multidimensional Scaling of Towns

Absolute Distances

X Contains the From Coordinates,

 

Y Contains the To Coordinates

 

OBSNUM	LIST2
1	1
2	4
3	4
4	1

 

COORD		Y
25.690672	69.469791	25.690672	69.469791
85.767694	-34.40875	1.35564	-24.42806
-112.814	-10.63298	1.35564	-24.42806
1.35564	-24.42806	25.690672	69.469791

 

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