Multidimensional Scaling
Airfares and City Distances
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* cities.sas -- Multidimensional Scaling of Airline Flight Costs *;
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Options PS=60 LS=78 PageNo=1 NoDate
FORMCHAR='|----|+|---+=|-/\<>*';
GOptions Device=WIN TargetDevice=IMGJPEG;
Filename PlotIt "plotit.sas";
%Include Plotit / NoSource;
Title1 "Multidimensional Scaling of Airline Fares";
Data CityFare;
Length City $3 CityName $11;
Input City CityName & ATL DFW MSY ORD LGA LAX DEN SEA;
DataLines;
ATL Atlanta . 433 112 410 412 635 520 411
DFW Dallas . . 88 754 725 539 454 663
MSY New Orleans . . . 624 590 298 542 359
ORD Chicago . . . . 471 752 538 765
LGA New York . . . . . 840 737 815
LAX Los Angeles . . . . . . 240 151
DEN Denver . . . . . . . 472
SEA Seattle . . . . . . . .
;
Proc Print Data=CityFare;
Run;
Title2 "Absolute Distances";
Proc MDS Data=CityFare Level=Absolute Out=MDSout
Dimension=2 PData PConfig PFinal;
Var ATL--SEA;
Id City;
Run;
%Plotit(Data=MDSout,Datatype=mds,
Gopplot=Device=WIN Targetdevice=PSEPSF,Monochro=Black);
Proc MDS Data=CityFare Level=Absolute
Dimension=3 PConfig PFinal;
Var ATL--SEA;
Id City;
Run;
Title2 "Non-metric Scaling";
Proc MDS Data=CityFare Level=Ordinal Out=MDSout
Dimension=2 PConfig PFinal;
Var ATL--SEA;
Id City;
Run;
%Plotit(Data=MDSout,Datatype=mds,
Gopplot=Device=WIN Targetdevice=PSEPSF,Monochro=Black);
Title1 "Multidimensional Scaling of Airline Flight Distances";
Data CityDist;
Length City $3 CityName $11;
Input City CityName & ATL DFW MSY ORD LGA LAX DEN SEA;
DataLines;
ATL Atlanta . 732 425 606 761 1946 1208 2182
DFW Dallas . . 448 1076 1389 1235 644 1660
MSY New Orleans . . . 1031 1186 1683 1092 2108
ORD Chicago . . . . 733 1839 1345 1938
LGA New York . . . . . 2475 1969 2549
LAX Los Angeles . . . . . . 971 954
DEN Denver . . . . . . . 1070
SEA Seattle . . . . . . . .
;
Title2 "Absolute Distances";
Proc MDS Data=CityDist Level=Absolute Out=MDSout
Dimension=2 PData PConfig PFinal;
Var ATL--SEA;
Id City;
Run;
%Plotit(Data=MDSout,Datatype=mds,
Gopplot=Device=WIN Targetdevice=PSEPSF,Monochro=Black);
Multidimensional Scaling of Airline Fares 1
OBS CITY CITYNAME ATL DFW MSY ORD LGA LAX DEN SEA
1 ATL Atlanta . 433 112 410 412 635 520 411
2 DFW Dallas . . 88 754 725 539 454 663
3 MSY New Orleans . . . 624 590 298 542 359
4 ORD Chicago . . . . 471 752 538 765
5 LGA New York . . . . . 840 737 815
6 LAX Los Angeles . . . . . . 240 151
7 DEN Denver . . . . . . . 472
8 SEA Seattle . . . . . . . .
Multidimensional Scaling of Airline Fares 2
Absolute Distances
Multidimensional Scaling: Data=WORK.CITYFARE
Data Matrix
1 ATL DFW MSY ORD
ATL . 433 112 410
DFW 433 . 88 754
MSY 112 88 . 624
ORD 410 754 624 .
LGA 412 725 590 471
LAX 635 539 298 752
DEN 520 454 542 538
SEA 411 663 359 765
1 LGA LAX DEN SEA
ATL 412 635 520 411
DFW 725 539 454 663
MSY 590 298 542 359
ORD 471 752 538 765
LGA . 840 737 815
LAX 840 . 240 151
DEN 737 240 . 472
SEA 815 151 472 .
Multidimensional Scaling of Airline Fares 3
Absolute Distances
Multidimensional Scaling: Data=WORK.CITYFARE
Shape=TRIANGLE Condition=MATRIX Level=ABSOLUTE
Coef=IDENTITY Dimension=2 Formula=1 Fit=1
Gconverge=0.01 Maxiter=100 Over=1 Ridge=0.0001
Badness-of-Fit Change in Convergence
Iteration Type Criterion Criterion Measure
-----------------------------------------------------------------------
0 Initial 0.176401 . 0.499552
1 Lev-Mar 0.152450 0.023950 0.194721
2 Gau-New 0.147422 0.005028 0.121794
3 Gau-New 0.146051 0.001371 0.031780
4 Gau-New 0.145961 0.000089528 0.008191
Convergence criterion is satisfied.
Configuration
DIM1 DIM2
-------------------------------
ATL 130.51 96.57
DFW -85.25 345.75
MSY -61.99 184.19
ORD 356.76 -291.46
LGA 512.93 88.15
LAX -340.62 -122.69
DEN -138.15 -284.60
SEA -374.18 -15.91
Multidimensional Scaling of Airline Fares 7
Absolute Distances
Multidimensional Scaling: Data=WORK.CITYFARE
Shape=TRIANGLE Condition=MATRIX Level=ABSOLUTE
Coef=IDENTITY Dimension=3 Formula=1 Fit=1
Gconverge=0.01 Maxiter=100 Over=1 Ridge=0.0001
Badness-of-Fit Change in Convergence
Iteration Type Criterion Criterion Measure
-----------------------------------------------------------------------
0 Initial 0.146307 . 0.694576
1 Lev-Mar 0.110843 0.035464 0.337436
2 Lev-Mar 0.108632 0.002211 0.173064
3 Gau-New 0.108513 0.000118 0.200919
4 Lev-Mar 0.107877 0.000637 0.107072
5 Lev-Mar 0.107813 0.000063676 0.108142
6 Lev-Mar 0.107623 0.000190 0.057716
7 Lev-Mar 0.107614 0.000009523 0.060291
8 Lev-Mar 0.107554 0.000059701 0.031988
9 Lev-Mar 0.107553 0.000001125 0.033958
10 Lev-Mar 0.107534 0.000019059 0.017892
11 Lev-Mar 0.107528 0.000005750 0.009516
Convergence criterion is satisfied.
Configuration
DIM1 DIM2 DIM3
---------------------------------------------
ATL 125.91 116.50 -53.58
DFW -107.35 272.78 210.30
MSY -66.48 184.24 19.74
ORD 353.53 -274.36 97.13
LGA 496.32 73.34 -127.65
LAX -342.36 -97.36 -59.69
DEN -180.56 -230.01 153.89
SEA -279.01 -45.13 -240.15
Multidimensional Scaling of Airline Fares 8
Non-metric Scaling
Multidimensional Scaling: Data=WORK.CITYFARE
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
Convergence Measures
Badness-of-Fit Change in --------------------------
Iteration Type Criterion Criterion Monotone Gradient
-----------------------------------------------------------------------------
0 Initial 0.176401 . . .
1 Monotone 0.137762 0.038638 0.074158 0.495492
2 Gau-New 0.119345 0.018418 . .
3 Monotone 0.107294 0.012050 0.051200 0.405550
4 Gau-New 0.104231 0.003063 . .
5 Monotone 0.085845 0.018386 0.056882 0.235042
6 Gau-New 0.084602 0.001243 . .
7 Monotone 0.079948 0.004653 0.026339 0.128947
8 Gau-New 0.079916 0.000032367 . .
9 Monotone 0.079143 0.000773 0.011026 0.086691
10 Gau-New 0.079124 0.000019113 . .
11 Monotone 0.078905 0.000218 0.005886 0.068250
12 Gau-New 0.078715 0.000190 . 0.004963
Convergence criteria are satisfied.
Configuration
DIM1 DIM2
---------------------------
ATL 0.43 0.14
DFW -0.40 1.17
MSY 0.07 0.59
ORD 1.38 -1.12
LGA 2.07 0.52
LAX -1.44 -0.38
DEN -0.48 -1.04
SEA -1.62 0.13
Multidimensional Scaling of Airline Flight Distances 12
Absolute Distances
Multidimensional Scaling: Data=WORK.CITYDIST
Data Matrix
1 ATL DFW MSY ORD
ATL . 732 425 606
DFW 732 . 448 1076
MSY 425 448 . 1031
ORD 606 1076 1031 .
LGA 761 1389 1186 733
LAX 1946 1235 1683 1839
DEN 1208 644 1092 1345
SEA 2182 1660 2108 1938
1 LGA LAX DEN SEA
ATL 761 1946 1208 2182
DFW 1389 1235 644 1660
MSY 1186 1683 1092 2108
ORD 733 1839 1345 1938
LGA . 2475 1969 2549
LAX 2475 . 971 954
DEN 1969 971 . 1070
SEA 2549 954 1070 .
Multidimensional Scaling of Airline Flight Distances 13
Absolute Distances
Multidimensional Scaling: Data=WORK.CITYDIST
Shape=TRIANGLE Condition=MATRIX Level=ABSOLUTE
Coef=IDENTITY Dimension=2 Formula=1 Fit=1
Gconverge=0.01 Maxiter=100 Over=1 Ridge=0.0001
Badness-of-Fit Change in Convergence
Iteration Type Criterion Criterion Measure
-----------------------------------------------------------------------
0 Initial 0.059587 . 0.586762
1 Lev-Mar 0.047273 0.012313 0.149260
2 Gau-New 0.046372 0.000901 0.106897
3 Gau-New 0.045924 0.000448 0.072792
4 Gau-New 0.045722 0.000201 0.046916
5 Gau-New 0.045640 0.000081884 0.029079
6 Gau-New 0.045609 0.000031054 0.017578
7 Gau-New 0.045598 0.000011262 0.010464
8 Gau-New 0.045594 0.000003974 0.006173
Convergence criterion is satisfied.
Configuration
DIM1 DIM2
---------------------------------
ATL 704.23 137.41
DFW 30.31 391.47
MSY 465.74 517.89
ORD 581.80 -493.81
LGA 1267.20 -342.78
LAX -1214.30 329.78
DEN -519.20 77.60
SEA -1315.77 -617.56
