! Discriminant analysis by linear programming;
! There are two categories of observations, type 0 and type 1,
which we wish to distinguish based on explanatory variables;
! Key idea: Find a scoring formula:
score( i) = beta0 + beta1*xdat( k, 1) + beta2*xdat( i, 2) + ...
and try to enforce:
score( i) < 0 if observation i is of type 0,
score( i) > 0 if observation i is of type 1;
! Key words: Binary choice, Categorical variables, Discriminant analysis, Random forest;
SETS:
var: BETA;
vardep( var); ! The dependent variable;
varexp( var); ! The set of explanatory variables;
varpltx( var); ! Variable on X axis in plot/chart/graph;
varplty( var); ! Variable on Y axis in plot/chart/graph;
varwgt( var); ! The weight vector column;
OBS: ERR, SCORE;
oxv( OBS, var): xdat; ! The big data matrix;
! Two subsets, based on category of the observation for plotting;
OBS0( OBS) : xp0, yp0;
OBS1( OBS) : xp1, yp1;
ENDSETS
DATA:
SCALE = 0.5; ! Scale factor for the solution, used to specify
a strictly positive separation between type 1 and type 2 points;
! Various data sets;
!CaCircle var = X1 X2 X1SQ X2SQ type wgtof;
!CaCircle vardep = type;
!CaCircle varwgt = wgtof;
!CaCircle !CaCircle varexp = X1 X2 X1SQ X2SQ; !Include squared terms!CaCircle varpltx = X1;
!CaCircle varplty = X2;
!CaCircle alpha = 0.0001; !CaCircle xdat =
22.98 21.75 528.30 473.06 0 1
28.27 24.90 799.05 620.01 1 1
21.85 22.88 477.42 523.71 0 1
24.90 28.34 620.01 803.19 1 1
20.30 23.30 412.09 542.89 0 1
20.30 29.60 412.09 876.16 1 1
18.75 22.88 351.56 523.71 0 1
15.70 28.34 246.49 803.19 1 1
17.62 21.75 310.30 473.06 0 1
12.33 24.90 152.09 620.01 1 1
17.20 20.20 295.84 408.04 0 1
11.10 20.20 123.21 408.04 1 1
17.62 18.65 310.30 347.82 0 1
12.33 15.50 152.09 240.25 1 1
18.75 17.52 351.56 306.79 0 1
15.70 12.06 246.49 145.43 1 1
20.30 17.10 412.09 292.41 0 1
20.30 10.80 412.09 116.64 1 1
21.85 17.52 477.42 306.79 0 1
24.90 12.06 620.01 145.43 1 1
22.98 18.65 528.30 347.82 0 1
28.27 15.50 799.05 240.25 1 1
23.40 20.20 547.56 408.04 0 1
29.50 20.20 870.25 408.04 1 1;
! Case 1: Unequal class sizes;
!CaUneq01 var = value type wgtof;
!CaUneq01 vardep = type;
!CaUneq01 wgtv = wgtof;
!CaUneq01 varexp = value;
!CaUnEq01 alpha = 0.0001; !CaUneq01 xdat =
1 1 1
2 0 1
3 0 1
7 0 1
8 0 1
9 0 1;
! Case 2 : Unequal class sizes;
!CaUneq02 var = value type wgtof;
!CaUneq02 vardep = type;
!CaUneq02 varwgt = wgtof;
!CaUneq02 varexp = value;
!CaUneq02 alpha = 0.0001; !CaUneq02 xdat =
1 0 1
2 0 1
3 1 1
7 0 1
8 0 1
9 0 1;
! Case 3: Unequal class sizes, use class size dependent weights;
!CaUneq03 var = value type wgtof;
!CaUneq03 vardep = type;
!CaUneq03 varwgt = wgtof;
!CaUneq03 varexp = value;
!CaUnEq03 alpha = 0.0001; !CaUneq03 xdat =
1 0 0.6
2 0 0.6
3 1 3
7 0 0.6
8 0 0.6
9 0 0.6;
! A scaled version of CaUneq03;
!CaUnEqWg var = value type wgtof;
!CaUnEqWg vardep = type;
!CaUnEqWg varwgt = wgtof;
!CaUnEqWg varexp = value;
!CaUnEqWg alpha = 0.0001; !CaUnEqWg xdat =
2 0 0.6
4 0 0.6
6 1 3
14 0 0.6
16 0 0.6
18 0 0.6;
! An inseparable case;
!CaInsep var = value type wgtof;
!CaInsep vardep = type;
!CaInsep varwgt = wgtof;
!CaInsep varexp = value;
!CaInsep alpha = 0.0001; !CaInsep xdat =
1 0 1
2 0 1
2 1 1
3 1 1;
! A 2D inseparable case;
!Ca2D var = X1 X2 type wgtof;
!Ca2D vardep = type;
!Ca2D varexp = X1 X2;
!Ca2D varwgt = wgtof;
!Ca2D alpha = 0.0001; !Ca2D xdat =
1 2 0 0.917
1.1 4 0 0.917
2 1 0 0.917
2 3 0 0.917
2 5 0 0.917
2 5.2 1 1.1
3 1 1 1.1
3 4 1 1.1
3.5 3 1 1.1
4 0 0 0.917
5 2 1 1.1;
! Illustrate separability and parsimony;
!CaSePar var = X1 X2 type wgtof;
!CaSePar vardep = type;
!CaSePar varwgt = wgtof;
!CaSePar varexp = X1 X2;
!CaSePar varpltx = X1;
!CaSePar varplty = X2;
!CaSePar alpha = 0.0001; !CaSePar xdat =
1 8 0 1
2 6 0 1
3 9 0 1
4 7 0 1
6 3 1 1
7 1 1 1
8 4 1 1
9 2 1 1;
! The 66 variable set;
! OBS = 1..66;
!Ca66 var =
OBNO TEST1 TEST2 TEST3 TEST4 TEST5 TYPE;
!Ca66 vardep = type;
!Ca66 varexp=
TEST1 TEST2 TEST3 TEST4 TEST5;
!Ca66 alpha = 0.0001; !Ca66 xdat =
1 346.7000122 247.1999969 220.5 364.1000061 311.7000122 0
2 334 313.2999878 306.5 330.8999939 311.1000061 0
3 248.3999939 189.1999969 206.8000031 334.7000122 312.5 1
4 309 291.8999939 281.2000122 346.2000122 311.1000061 1
5 328.8999939 306.2000122 259.3999939 336.3999939 310.8999939 0
6 252.8000031 248.8000031 253.8000031 321 311.7000122 1
7 313 289.7000122 292.6000061 318 311 0
8 304.8999939 115.5 284.2000122 316.5 310.5 1
9 327.8999939 330.7999878 305.7000122 332.6000061 311 1
10 315.3999939 203.8999939 287.1000061 333.7999878 311.5 1
11 333 270.6000061 274.2999878 379.1000061 311.2000122 0
12 242.3999939 145.8999939 292.2999878 318.7000122 311.2999878 0
13 124.9000015 1.1 244.1999969 345.7000122 310.7999878 0
14 323.5 317.2000122 287.3999939 406.1000061 312 1
15 304.2999878 191.6999969 275.7999878 331.7000122 311.5 0
16 382.3999939 124.0999985 30 322.5 316.7000122 1
17 327 275.3999939 290.6000061 345.5 313.3999939 1
18 278.7999878 282.1000061 316.2999878 317 311.2999878 0
19 324.1000061 261.7999878 316.7999878 326.6000061 311.6000061 0
20 249.3999939 260.7999878 292.7999878 317.2000122 310.2999878 1
21 336.2000122 290.7999878 273.2999878 400.3999939 310.7999878 1
22 317 291.8999939 303.5 326.5 310.8999939 0
23 256.8999939 212 282 336.6000061 311.7000122 1
24 292.7999878 181 295.7999878 577.9000244 311.2999878 0
25 342.7000122 306 294.2000122 487.3999939 312.1000061 0
26 336.7000122 301.2999878 273.7000122 342.5 312.7999878 0
27 302.2999878 250.8000031 297.2000122 331.2999878 312.1000061 0
28 328 296.8999939 292.3999939 324.6000061 310.8999939 1
29 312 272 311.6000061 317.7000122 311.2000122 1
30 274.7000122 252.1000061 310.7000122 323.7000122 310.7999878 0
31 315.1000061 301.2000122 300.8999939 410.8999939 310.8999939 0
32 310 245.3000031 306 310.7000122 310.1000061 0
33 335.2000122 298.6000061 314.7999878 317 310.8999939 1
34 345.2000122 353 326.3999939 409.1000061 311.2999878 1
35 348.7999878 357 326 436.5 311.8999939 0
36 324 306.7000122 314 401.7000122 312.7000122 0
37 365.1000061 345 330.7999878 382.2999878 311.8999939 1
38 369.2999878 356.7000122 322.6000061 1034.1 310.8999939 1
39 343.6000061 330.7999878 322.5 462.7999878 312.3999939 1
40 362.7999878 343 333.6000061 785.9000244 311.5 1
41 355.6000061 336.1000061 320.3999939 597.9000244 312.1000061 0
42 357.3999939 378.6000061 323.7999878 891.2999878 311.6000061 1
43 350 347.2999878 343.3999939 538.7999878 313.5 1
44 379 369 333.1000061 716 315.5 1
45 344.2000122 359.6000061 333.7999878 436.6000061 311.8999939 1
46 357 322.5 317 363.3999939 311.7999878 0
47 325.3999939 347.2999878 344.1000061 880.0999756 311.5 1
48 366.8999939 345.2999878 314.2000122 550.2999878 310.8999939 0
49 353.7999878 359.5 335.1000061 425 312.6000061 0
50 330.7000122 328.1000061 323.5 373.1000061 314 1
51 343.7999878 341.3999939 325.7000122 454.7999878 311.8999939 0
52 345.2999878 331.5 295.6000061 400 311 1
53 334.3999939 318.5 315.7999878 459.1000061 311.5 0
54 358.8999939 350.6000061 315.7999878 392 311.7999878 0
55 359.8999939 344.6000061 336.3999939 620 311.7999878 0
56 364.7999878 329.8999939 336.7000122 549.9000244 312.2999878 1
57 349 327.3999939 322.6000061 370.5 311.2999878 1
58 363 364.7000122 324.6000061 1081.7 311.7000122 1
59 330.1000061 363.5 330.6000061 617.5 311.1000061 1
60 363.7000122 345.8999939 336.3999939 599.5 312 0
61 356.1000061 349.3999939 340.5 1010 311.8999939 1
62 358.2999878 363.1000061 317.1000061 474.3999939 311.8999939 0
63 356.7000122 349.7999878 323.7999878 539.0999756 311.2000122 0
64 370.2999878 369.5 317 536.5999756 312 1
65 327.8999939 326.2999878 330.3999939 415.6000061 311 0
66 334.7000122 331.7000122 302.2000122 428.6000061 311.6000061 0;
! Genuine and counterfeit banknotes (100 Swiss Francs),
various measurements.
Banknotes BN1 to BN100 are genuine (Good=1),
all others are counterfeit (Good=0).
Dataset courtesy of H. Riedwyl, Bern, Switzerland;
!CaSwiss; var =
Length Left Right Bottom Top Diagonal Good wgtof ;
!CaSwiss; vardep = Good!CaSwiss; varexp = Length Left Right Bottom Top Diagonal!CaSwiss; varpltx = Right!CaSwiss; varplty = Diagonal!CaSwiss; alpha = 0.0001!CaSwiss; varwgt = wgtof
!CaSwiss; OBS, xdat =
BN1 214.8 131.0 131.1 9.0 9.7 141.0 1 1
BN2 214.6 129.7 129.7 8.1 9.5 141.7 1 1
BN3 214.8 129.7 129.7 8.7 9.6 142.2 1 1
BN4 214.8 129.7 129.6 7.5 10.4 142.0 1 1
BN5 215.0 129.6 129.7 10.4 7.7 141.8 1 1
BN6 215.7 130.8 130.5 9.0 10.1 141.4 1 1
BN7 215.5 129.5 129.7 7.9 9.6 141.6 1 1
BN8 214.5 129.6 129.2 7.2 10.7 141.7 1 1
BN9 214.9 129.4 129.7 8.2 11.0 141.9 1 1
BN10 215.2 130.4 130.3 9.2 10.0 140.7 1 1
BN11 215.3 130.4 130.3 7.9 11.7 141.8 1 1
BN12 215.1 129.5 129.6 7.7 10.5 142.2 1 1
BN13 215.2 130.8 129.6 7.9 10.8 141.4 1 1
BN14 214.7 129.7 129.7 7.7 10.9 141.7 1 1
BN15 215.1 129.9 129.7 7.7 10.8 141.8 1 1
BN16 214.5 129.8 129.8 9.3 8.5 141.6 1 1
BN17 214.6 129.9 130.1 8.2 9.8 141.7 1 1
BN18 215.0 129.9 129.7 9.0 9.0 141.9 1 1
BN19 215.2 129.6 129.6 7.4 11.5 141.5 1 1
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BN184 214.6 130.5 130.4 10.1 11.4 139.3 0 1
BN185 214.7 130.2 130.1 10.7 11.1 139.5 0 1
BN186 214.7 130.4 130.0 11.5 10.7 139.4 0 1
BN187 214.5 130.4 130.0 8.0 12.2 138.5 0 1
BN188 214.8 130.0 129.7 11.4 10.6 139.2 0 1
BN189 214.8 129.9 130.2 9.6 11.9 139.4 0 1
BN190 214.6 130.3 130.2 12.7 9.1 139.2 0 1
BN191 215.1 130.2 129.8 10.2 12.0 139.4 0 1
BN192 215.4 130.5 130.6 8.8 11.0 138.6 0 1
BN193 214.7 130.3 130.2 10.8 11.1 139.2 0 1
BN194 215.0 130.5 130.3 9.6 11.0 138.5 0 1
BN195 214.9 130.3 130.5 11.6 10.6 139.8 0 1
BN196 215.0 130.4 130.3 9.9 12.1 139.6 0 1
BN197 215.1 130.3 129.9 10.3 11.5 139.7 0 1
BN198 214.8 130.3 130.4 10.6 11.1 140.0 0 1
BN199 214.7 130.7 130.8 11.2 11.2 139.4 0 1
BN200 214.3 129.9 129.9 10.2 11.5 139.6 0 1
;
ENDDATA
SUBMODEL discrim:
! Minimize sum of weighted errors;
MIN = errsum + alpha * errmax;
errsum = @SUM( OBS( I): xdat( i, wgtndx) * ERR( I) );
@FREE( BETA0);
@FOR( var( J): @FREE( BETA( J)););
! For bad observations we want SCORE( i) <= - SCALE;
@FOR( OBS(I)| ( xdat( I, depndx) #LT# 1) :
SCORE( I) =
BETA0 + @SUM( varexp( J): BETA( J)* xdat(I,J));
@FREE( SCORE( I));
ERR( I) >= SCORE( I) + SCALE;
ERRMAX >= ERR( i)
);
! For good observations we want SCORE( i) >= SCALE;
@FOR( OBS( I)| (xdat( I, depndx) #GT# 0) :
SCORE( I) =
BETA0 + @SUM( varexp( J): BETA( J)* xdat(I,J));
@FREE( SCORE(I));
ERR( I) >= SCALE - SCORE( i);
ERRMAX >= ERR( i);
);
ENDSUBMODEL
CALC:
@SET( 'OROUTE',1); ! Output to window after this many lines in buffer;
@SET( 'WNLINE',10000); ! (27) Max command window lines (Windows only);
! Get the index number of the dependent var;
@for( vardep( j): depndx = j);! Count observations of each type;
nobs0 = @SUM( obs( i) | xdat( i, depndx) #EQ# 0: 1);
nobs1 = @SUM( obs( i) | xdat( i, depndx) #EQ# 1: 1);
nobs = nobs0 + nobs1; ! Number of observations;
! Get the index number of the wgt var;
@FOR( varwgt( j): wgtndx = j);
! @gen( discrim);
@solve( discrim); ! Get best single number estimate;
objsing = obj; ! Store error measure for single number estimator case;
@WRITE(' ExpVar Beta ', @NEWLINE(1));
@WRITE(' Beta0 ', @FORMAT( beta0, '12.4f'), @NEWLINE( 1));
@FOR( varexp( k):
@WRITE( ' ', @FORMAT( varexp( k),'9s'),' ', @FORMAT( beta( k), '12.4f'), @NEWLINE( 1));
);
! Write score of each observation;
@WRITE( @NEWLINE(1),' Obs Type Score Error', @NEWLINE( 1));
@FOR( obs( i):
@WRITE( @FORMAT( i, '4.0f'), ' ', xdat( i, depndx), ' ',
@FORMAT( score( i),'9.2f'), ' ', @FORMAT( ERR( i), '9.2f'),@NEWLINE( 1));
);
! Prepare data for a chart of the dependent variable vs. one other variable;
! Get the index of the explanatory variables x and y axis;
@FOR( varpltx( j): pltxndx = j);
@FOR( varplty( j): pltyndx = j);
! Create two sets, based on the category of the dependent variable;
@FOR( obs( i) | xdat( i, depndx) #EQ# 0:
@INSERT( OBS0, i);
xp0(i) = xdat( i, pltxndx);
yp0(i) = xdat( i, pltyndx);
);
@FOR( obs( i) | xdat( i, depndx) #EQ# 1:
@INSERT( OBS1, i);
xp1(i) = xdat( i, pltxndx);
yp1(i) = xdat( i, pltyndx);
);
! Beware when interpreting scatter plots, a red point may hide a green point;
@CHARTSCATTER( 'Discriminant Analysis in 2-space', var( pltxndx), var( pltyndx),
'Type 0 points', xp0, yp0, 'Type 1 points', xp1, yp1) ;
ENDCALC
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