Multidimensional Scaling of Airline Fares

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
Absolute Distances
Two-dimensional Solution

Multidimensional Scaling: Data=WORK.CITYFARE.DATA

Data Matrix
1 ATL DFW MSY ORD LGA LAX DEN SEA
ATL . 433 112 410 412 635 520 411
DFW 433 . 88 754 725 539 454 663
MSY 112 88 . 624 590 298 542 359
ORD 410 754 624 . 471 752 538 765
LGA 412 725 590 471 . 840 737 815
LAX 635 539 298 752 840 . 240 151
DEN 520 454 542 538 737 240 . 472
SEA 411 663 359 765 815 151 472 .

Multidimensional Scaling of Airline Fares
Absolute Distances
Two-dimensional Solution

Multidimensional Scaling: Data=WORK.CITYFARE.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 0.1764 . 0.4996
1 Lev-Mar 0.1525 0.0240 0.1947
2 Gau-New 0.1474 0.005028 0.1218
3 Gau-New 0.1461 0.001371 0.0318
4 Gau-New 0.1460 0.0000895 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
Absolute Distances
Three-dimensional Solution

Multidimensional Scaling: Data=WORK.CITYFARE.DATA
Shape=TRIANGLE Condition=MATRIX Level=ABSOLUTE
Coef=IDENTITY Dimension=3 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 0.1463 . 0.6946
1 Lev-Mar 0.1108 0.0355 0.3374
2 Lev-Mar 0.1086 0.002211 0.1731
3 Gau-New 0.1085 0.000118 0.2009
4 Lev-Mar 0.1079 0.000637 0.1071
5 Lev-Mar 0.1078 0.0000637 0.1081
6 Lev-Mar 0.1076 0.000190 0.0577
7 Lev-Mar 0.1076 9.523E-6 0.0603
8 Lev-Mar 0.1076 0.0000597 0.0320
9 Lev-Mar 0.1076 1.1255E-6 0.0340
10 Lev-Mar 0.1075 0.0000191 0.0179
11 Lev-Mar 0.1075 5.7505E-6 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
Non-metric Scaling
Two-dimensional Solution

Multidimensional Scaling: Data=WORK.CITYFARE.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 0.1764 . . .
1 Monotone 0.1378 0.0386 0.0742 0.4955
2 Gau-New 0.1193 0.0184 . .
3 Monotone 0.1073 0.0121 0.0512 0.4056
4 Gau-New 0.1042 0.003063 . .
5 Monotone 0.0858 0.0184 0.0569 0.2350
6 Gau-New 0.0846 0.001243 . .
7 Monotone 0.0799 0.004653 0.0263 0.1289
8 Gau-New 0.0799 0.0000324 . .
9 Monotone 0.0791 0.000773 0.0110 0.0867
10 Gau-New 0.0791 0.0000191 . .
11 Monotone 0.0789 0.000218 0.005886 0.0682
12 Gau-New 0.0787 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 Fares
Non-metric Scaling
Three-dimensional Solution

Multidimensional Scaling: Data=WORK.CITYFARE.DATA
Shape=TRIANGLE Condition=MATRIX Level=ORDINAL
Coef=IDENTITY Dimension=3 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 0.1463 . . .
1 Monotone 0.0699 0.0764 0.1180 0.6889
2 Gau-New 0.0492 0.0207 . .
3 Monotone 0.0424 0.006827 0.0219 0.4845
4 Gau-New 0.0399 0.002410 . .
5 Monotone 0.0286 0.0114 0.0273 0.4268
6 Gau-New 0.0275 0.001081 . .
7 Monotone 0.0221 0.005359 0.0158 0.3715
8 Gau-New 0.0215 0.000595 . .
9 Monotone 0.0182 0.003353 0.0115 0.3455
10 Gau-New 0.0177 0.000428 . .
11 Monotone 0.0154 0.002337 0.008721 0.3193
12 Gau-New 0.0146 0.000812 . 0.0102
13 Gau-New 0.0146 8.0534E-7 . 0.000780

Convergence criteria are satisfied.

Configuration
  Dim1 Dim2 Dim3
ATL 0.55 0.10 0.85
DFW -0.31 1.40 -0.77
MSY -0.11 0.90 0.29
ORD 1.39 -1.47 -0.16
LGA 2.45 0.46 -0.00
LAX -1.83 -0.21 -0.35
DEN -0.59 -0.84 -0.94
SEA -1.53 -0.34 1.08

Multidimensional Scaling of Airline Flight Distances
Absolute Distances
Two-dimensional Solution

Multidimensional Scaling: Data=WORK.CITYDIST.DATA

Data Matrix
1 ATL DFW MSY ORD LGA LAX DEN SEA
ATL . 732 425 606 761 1946 1208 2182
DFW 732 . 448 1076 1389 1235 644 1660
MSY 425 448 . 1031 1186 1683 1092 2108
ORD 606 1076 1031 . 733 1839 1345 1938
LGA 761 1389 1186 733 . 2475 1969 2549
LAX 1946 1235 1683 1839 2475 . 971 954
DEN 1208 644 1092 1345 1969 971 . 1070
SEA 2182 1660 2108 1938 2549 954 1070 .

Multidimensional Scaling of Airline Flight Distances
Absolute Distances
Two-dimensional Solution

Multidimensional Scaling: Data=WORK.CITYDIST.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 0.0596 . 0.5868
1 Lev-Mar 0.0473 0.0123 0.1493
2 Gau-New 0.0464 0.000901 0.1069
3 Gau-New 0.0459 0.000448 0.0728
4 Gau-New 0.0457 0.000201 0.0469
5 Gau-New 0.0456 0.0000819 0.0291
6 Gau-New 0.0456 0.0000311 0.0176
7 Gau-New 0.0456 0.0000113 0.0105
8 Gau-New 0.0456 3.9736E-6 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

TEMPLATE GRAPH PRODUCED BY GREPLAY