Venue for CARME in ASSOS Assos View from Turkey of Aegean Sea and island of Lesbos Unifying the geometry of simple and multiple correspondence analysis Michael Greenacre Universitat Pompeu Fabra Barcelona www.econ.upf.edu/~michael www.globalsong.net youtube.com/statisticalsongs youtube.com/youwomangroup youtube.com/arcticfrontiers
SImple correspondence analysis > summary(ca(smoke)) Principal inertias (eigenvalues): dim value % cum% scree plot 1 0.074759 87.8 87.8 ************************* 2 0.010017 11.8 99.5 *** 3 0.000414 0.5 100.0 -------- ----- Total: 0.085190 100.0 First two principal axes of CA Rows: name mass qlt inr k=1 cor ctr k=2 cor ctr 1 SM 57 893 31-66 92 3-194 800 214 2 JM 93 991 139 259 526 84-243 465 551 3 SE 264 1000 450-381 999 512-11 1 3 4 JE 456 1000 308 233 942 331 58 58 152 5 SC 130 999 71-201 865 70 79 133 81 Columns: name mass qlt inr k=1 cor ctr k=2 cor ctr 1 non 316 1000 577-393 994 654-30 6 29 2 lgh 233 984 83 99 327 31 141 657 463 3 mdm 321 983 148 196 982 166 7 1 2 4 hvy 130 995 192 294 684 150-198 310 506 Variance explained = 99.5%
Multiple correspondence analysis (function mjca of ca package in R) N "indicator" "Burt" "adjusted" "JCA" J= q J q J 0100 1000 0001 0010 1000 1000 Z J B = Z T Z * i Q 2 ( 1 ) 2 i ( Q 1) Q 2 X X X X i i same standard coordinates of categories but different principal coordinates same coordinates of supplementary pts
Problem of variance explained > summary(mjca(wg93[,1:4], lambda="indicator")) Principal inertias (eigenvalues): dim value % cum% scree plot 1 0.457379 11.4 11.4 ************************* 2 0.430966 10.8 22.2 *********************** 3 0.321926 8.0 30.3 *************** : : : : > summary(mjca(wg93[,1:4], lambda="burt")) Principal inertias (eigenvalues): dim value % cum% scree plot 1 0.209196 18.6 18.6 ************************* 2 0.185732 16.5 35.0 ********************** 3 0.103636 9.2 44.2 *********** : : : : > summary(mjca(wg93[,1:4], lambda="adjusted")))) #DEFAULT Principal inertias (eigenvalues): dim value % cum% scree plot 1 0.076455 44.9 44.9 ************************* 2 0.058220 34.2 79.1 ******************* 3 0.009197 5.4 84.5 *** : : : : > summary(mjca(wg93[,1:4]), lambda="jca")) Percentage explained by JCA in 2 dimensions: 85.7% (Eigenvalues are not nested) [Iterations in JCA: 44, epsilon = 9.91e-05] increasing inertia explained
Same problem for individual points > summary(mjca(wg93[,1:4], lambda="burt")) Principal inertias (eigenvalues): dim value % cum% scree plot 1 0.209196 18.6 18.6 ************************* 2 0.185732 16.5 35.0 ********************** 3 0.103636 9.2 44.2 *********** : : : : : name mass qlt inr k=1 cor ctr k=2 cor ctr 1 A1 34 445 55-840 391 53-314 54 8 2 A2 92 169 38-250 136 13 123 33 3 3 A3 59 344 47 204 47 5 517 298 36 4 A4 51 350 50 533 258 32-318 92 12 5 A5 14 401 60 913 170 25-1064 231 36 6 B1 20 621 62-1338 519 80-590 101 16 7 B2 50 158 47-293 80 9 287 77 10 8 B3 59 227 45-158 29 3 415 198 24 9 B4 81 210 41 327 185 19 121 25 3 10 B5 40 722 60 619 229 34-908 493 77 11 C1 44 732 60-987 632 93-392 100 16 12 C2 91 164 38-113 27 3 255 137 14 13 C3 57 296 48 283 84 10 450 212 27 14 C4 44 345 52 617 289 37-274 57 8 15 C5 15 471 60 671 99 15-1300 372 59 16 D1 17 251 56-551 83 11-785 168 25 17 D2 67 14 42 101 14 1 3 0 0 18 D3 58 303 48 176 33 4 499 269 34 19 D4 65 25 43 101 14 1 91 11 1 20 D5 43 272 50-324 81 10-496 191 25
Same problem for individual points > summary(mjca(wg93[,1:4], lambda="burt")) Principal inertias (eigenvalues): dim value % cum% scree plot 1 0.209196 18.6 18.6 ************************* 2 0.185732 16.5 35.0 ********************** 3 0.103636 9.2 44.2 *********** > mjca(wg93[,1:4])$burt A1 A2 A3 A4 A5 B1 B2 B3 B4 B5 C1 C2 C3 C4 C5 D1 D2 D3 D4 D5 A1 119 0 0 0 0 27 28 30 22 12 49 40 18 7 5 15 25 17 34 28 A2 0 322 0 0 0 38 74 84 96 30 67 142 60 41 12 22 102 76 68 54 A3 0 0 204 0 0 3 48 63 73 17 18 75 70 34 7 10 44 68 58 24 A4 0 0 0 178 0 3 21 23 79 52 16 50 40 56 16 9 52 28 54 35 A5 0 0 0 0 48 0 3 5 11 29 2 9 9 16 12 4 9 13 12 10 B1 27 38 3 3 0 71 0 0 0 0 43 19 4 3 2 9 17 10 10 25 B2 28 74 48 21 3 0 174 0 0 0 36 88 34 15 1 16 51 42 45 20 B3 30 84 63 23 5 0 0 205 0 0 37 90 57 19 2 10 53 63 51 28 B4 22 96 73 79 11 0 0 0 281 0 27 88 75 74 17 6 66 70 92 47 B5 12 30 17 52 29 0 0 0 0 140 9 31 27 43 30 19 45 17 28 31 C1 49 67 18 16 2 43 36 37 27 9 152 0 0 0 0 25 24 15 38 50 C2 40 142 75 50 9 19 88 90 88 31 0 316 0 0 0 15 97 67 89 48 C3 18 60 70 40 9 4 34 57 75 27 0 0 197 0 0 5 51 83 41 17 C4 7 41 34 56 16 3 15 19 74 43 0 0 0 154 0 6 44 30 51 23 C5 5 12 7 16 12 2 1 2 17 30 0 0 0 0 52 9 16 7 7 13 D1 15 22 10 9 4 9 16 10 6 19 25 15 5 6 9 60 0 0 0 0 D2 25 102 44 52 9 17 51 53 66 45 24 97 51 44 16 0 232 0 0 0 D3 17 76 68 28 13 10 42 63 70 17 15 67 83 30 7 0 0 202 0 0 D4 34 68 58 54 12 10 45 51 92 28 38 89 41 51 7 0 0 0 226 0 D5 28 54 24 35 10 25 20 28 47 31 50 48 17 23 13 0 0 0 0 151
Joint correspondence analysis: updated Burt matrix > mjca(wg93[,1:4], lambda="jca")$burt.upd A1 A2 A3 A4 A5 B1 B2 B3 B4 B5 C1 C2 C3 C4 C5 D1 D2 D3 D4 D5 A1 31 53 19 14 3 27 28 30 22 12 49 40 18 7 5 15 25 17 34 28 A2 53 131 77 52 10 38 74 84 96 30 67 142 60 41 12 22 102 76 68 54 A3 19 77 63 39 7 3 48 63 73 17 18 75 70 34 7 10 44 68 58 24 A4 14 52 39 54 20 3 21 23 79 52 16 50 40 56 16 9 52 28 54 35 A5 3 10 7 20 9 0 3 5 11 29 2 9 9 16 12 4 9 13 12 10 B1 27 38 3 3 0 21 20 18 8 3 43 19 4 3 2 9 17 10 10 25 B2 28 74 48 21 3 20 46 54 50 4 36 88 34 15 1 16 51 42 45 20 B3 30 84 63 23 5 18 54 65 64 4 37 90 57 19 2 10 53 63 51 28 B4 22 96 73 79 11 8 50 64 104 55 27 88 75 74 17 6 66 70 92 47 B5 12 30 17 52 29 3 4 4 55 74 9 31 27 43 30 19 45 17 28 31 C1 49 67 18 16 2 43 36 37 27 9 82 55 4 3 7 25 24 15 38 50 C2 40 142 75 50 9 19 88 90 88 31 55 126 79 46 9 15 97 67 89 48 C3 18 60 70 40 9 4 34 57 75 27 4 79 66 41 6 5 51 83 41 17 C4 7 41 34 56 16 3 15 19 74 43 3 46 41 45 18 6 44 30 51 23 C5 5 12 7 16 12 2 1 2 17 30 7 9 6 18 11 9 16 7 7 13 D1 15 22 10 9 4 9 16 10 6 19 25 15 5 6 9 9 15 5 13 18 D2 25 102 44 52 9 17 51 53 66 45 24 97 51 44 16 15 62 56 61 38 D3 17 76 68 28 13 10 42 63 70 17 15 67 83 30 7 5 56 64 56 21 D4 34 68 58 54 12 10 45 51 92 28 38 89 41 51 7 13 61 56 60 36 D5 28 54 24 35 10 25 20 28 47 31 50 48 17 23 13 18 38 21 36 38 default: two-dimensional solution at convergence the diagonal blocks are perfectly fitted
Joint correspondence analysis: discounting inertia on diagonal blocks > summary(mjca(wg93[,1:4], lambda="jca")) Principal inertias (eigenvalues): dim value 1 0.099091 2 0.065033 : : -------- Total: 0.182425 Diagonal inertia discounted from eigenvalues: 0.0547405 Percentage explained by JCA in 2 dimensions: 85.7% (Eigenvalues are not nested) [Iterations in JCA: 44, epsilon = 9.91e-05] (0.099091 0.065033) 0.0547405 0.182425 0.0547405 0.857
JCA: discounting inertia for each point > summary(mjca(wg93[,1:4], lambda="jca")) : : : : : : : : : : Columns: name mass inr k=1 k=2 cor ctr 1 A1 34 55-458 238 955 67 2 A2 92 38-169 -40 720 19 3 A3 59 47 48-281 867 37 4 A4 51 50 364 112 854 51 5 A5 14 60 711 458 871 68 6 B1 20 62-784 431 942 110 7 B2 50 47-267 -156 754 23 8 B3 59 45-201 -240 917 29 9 B4 81 41 194-91 634 25 10 B5 40 60 634 509 958 130 11 C1 44 60-662 417 970 138 12 C2 91 38-103 -130 599 17 13 C3 57 48 170-316 755 39 14 C4 44 52 450 42 865 54 15 C5 15 60 587 640 927 77 16 D1 17 56-212 515 813 37 17 D2 67 42 52-23 132 2 18 D3 58 48 57-297 717 37 19 D4 65 43 37-61 144 2 20 D5 43 50-126 320 910 36 As for the overall inertia, individual contributions of axes to points can only be computed for the solution space, not separately for each axis. Similiarly, contributions of points to the solution space (here, 2-dimensional) are not separated for each axis.
-1.0-0.5 0.0 0.5-0.8-0.6-0.4-0.2 0.0 0.2 0.4 Burt: 1, 2, 35% explained Adjusted MCA D3 A3 C3 B3 Adjusted: 1 *, 2 *, 79% explained B2 A2 C2 D4 D2 B4 B3 B2 C2 A2 D3 A3 C3 D4 B4 D2 B1 C1 A1 D5 C4 A4 B1 A1 C1 D1 D5 C4 A4 B5 A5 D1 C5 B5 A5-0.8-0.6-0.4-0.2 0.0 0.2 0.4 0.6-1.5-1.0-0.5 0.0 0.5 1.0 C5 * i Q 2 ( 1 2 i ( Q 1) Q ) 2 expressed relative to the average inertia in off-diagonal blocks, NOT relative to sum of * i (Benzécri)
Adjusted MCA nullifying the Burt matrix A1 A2 A3 A4 A5 B1 B2 B3 B4 B5 C1 C2 C3 C4 C5 D1 D2 D3 D4 D5 A1 119 0 0 0 0 27 28 30 22 12 49 40 18 7 5 15 25 17 34 28 A2 0 322 0 0 0 38 74 84 96 30 67 142 60 41 12 22 102 76 68 54 A3 0 0 204 0 0 3 48 63 73 17 18 75 70 34 7 10 44 68 58 24 A4 0 0 0 178 0 3 21 23 79 52 16 50 40 56 16 9 52 28 54 35 A5 0 0 0 0 48 0 3 5 11 29 2 9 9 16 12 4 9 13 12 10 B1 27 38 3 3 0 71 0 0 0 0 43 19 4 3 2 9 17 10 10 25 B2 28 74 48 21 3 0 174 0 0 0 36 88 34 15 1 16 51 42 45 20 B3 30 84 63 23 5 0 0 205 0 0 37 90 57 19 2 10 53 63 51 28 B4 22 96 73 79 11 0 0 0 281 0 27 88 75 74 17 6 66 70 92 47 B5 12 30 17 52 29 0 0 0 0 140 9 31 27 43 30 19 45 17 28 31 C1 49 67 18 16 2 43 36 37 27 9 152 0 0 0 0 25 24 15 38 50 C2 40 142 75 50 9 19 88 90 88 31 0 316 0 0 0 15 97 67 89 48 C3 18 60 70 40 9 4 34 57 75 27 0 0 197 0 0 5 51 83 41 17 C4 7 41 34 56 16 3 15 19 74 43 0 0 0 154 0 6 44 30 51 23 C5 5 12 7 16 12 2 1 2 17 30 0 0 0 0 52 9 16 7 7 13 D1 15 22 10 9 4 9 16 10 6 19 25 15 5 6 9 60 0 0 0 0 D2 25 102 44 52 9 17 51 53 66 45 24 97 51 44 16 0 232 0 0 0 D3 17 76 68 28 13 10 42 63 70 17 15 67 83 30 7 0 0 202 0 0 D4 34 68 58 54 12 10 45 51 92 28 38 89 41 51 7 0 0 0 226 0 D5 28 54 24 35 10 25 20 28 47 31 50 48 17 23 13 0 0 0 0 151
Adjusted MCA nullified Burt matrix B 0 = A1 A2 A3 A4 A5 B1 B2 B3 B4 B5 C1 C2 C3 C4 C5 D1 D2 D3 D4 D5 A1 119 0 0 0 0 27 28 30 22 12 49 40 18 7 5 15 25 17 34 28 A2 0 322 0 0 0 38 74 84 96 30 67 142 60 41 12 22 102 76 68 54 A3 0 0 204 0 0 0 3 48 63 73 17 18 75 70 34 7 10 44 68 58 24 A4 0 0 0 178 0 3 21 23 79 52 16 50 40 56 16 9 52 28 54 35 A5 0 0 0 0 48 0 3 5 11 29 2 9 9 16 12 4 9 13 12 10 B1 27 38 3 3 0 71 0 0 0 0 43 19 4 3 2 9 17 10 10 25 B2 28 74 48 21 3 0 174 0 0 36 88 34 15 1 16 51 42 45 20 B3 30 84 63 23 5 0 0 205 0 0 0 37 90 57 19 2 10 53 63 51 28 B4 22 96 73 79 11 0 0 0 281 0 27 88 75 74 17 6 66 70 92 47 B5 12 30 17 52 29 0 0 0 0 140 9 31 27 43 30 19 45 17 28 31 C1 49 67 18 16 2 43 36 37 27 9 152 0 0 0 0 25 24 15 38 50 C2 40 142 75 50 9 19 88 90 88 31 0 316 0 0 15 97 67 89 48 C3 18 60 70 40 9 4 34 57 75 27 0 0 197 0 0 0 5 51 83 41 17 C4 7 41 34 56 16 3 15 19 74 43 0 0 0 154 0 6 44 30 51 23 C5 5 12 7 16 12 2 1 2 17 30 0 0 0 0 52 9 16 7 7 13 D1 15 22 10 9 4 9 16 10 6 19 25 15 5 6 9 60 0 0 0 0 D2 25 102 44 52 9 17 51 53 66 45 24 97 51 44 16 0 232 0 0 D3 17 76 68 28 13 10 42 63 70 17 15 67 83 30 7 0 0 202 0 0 0 D4 34 68 58 54 12 10 45 51 92 28 38 89 41 51 7 0 0 0 226 0 D5 28 54 24 35 10 25 20 28 47 31 50 48 17 23 13 0 0 0 0 151 Perform eigendecomposition on B 0 (suitably centred & normalized, as in MCA) note: not SVD. The POSITIVE eigenvalues are exactly the adjusted inertias Adjustments for each category obtained in same way (a little miracle!)
Adjusted MCA nullified Burt matrix This neat result is a special case of the Polygone Convexe d un Tableau Symmétrique in Benzécri s 1973 book (summarized in my 1984 book): For P a square symmetric correspondence matrix with margins r and inertias λ k P (, ) P D (1 ) rr has same standard coordinates as P square roots of inertias are unexplained inertia Special case: r K ( k K* 1 1/ 2 k k T 1/ 2 2 ) where / k Q/( Q 1) 1/( Q 1) k k is eigenvalue s parity gives nullified matrix, inertias are exactly the adjusted inertias, and unexplained inertia is that of the adjusted analysis.
Results for adjusted MCA > summary(mjca(wg93[,1:4])) Principal inertias (eigenvalues): dim value % cum% scree plot 1 0.076455 44.9 44.9 ************************* 2 0.058220 34.2 79.1 ******************* 3 0.009197 5.4 84.5 *** : : : : : name mass qlt inr k=1 cor ctr k=2 cor ctr 1 A1 34 963 55 508 860 115-176 103 18 2 A2 92 659 38 151 546 28 69 113 7 3 A3 59 929 47-124 143 12 289 786 84 4 A4 51 798 50-322 612 69-178 186 28 5 A5 14 799 60-552 369 55-596 430 84 6 B1 20 911 62 809 781 174-331 131 38 7 B2 50 631 47 177 346 21 161 285 22 8 B3 59 806 45 96 117 7 233 690 55 9 B4 81 620 41-197 555 41 68 65 6 10 B5 40 810 60-374 285 74-509 526 179 11 C1 44 847 60 597 746 203-219 101 36 12 C2 91 545 38 68 101 6 143 444 32 13 C3 57 691 48-171 218 22 252 473 62 14 C4 44 788 52-373 674 80-153 114 18 15 C5 15 852 60-406 202 32-728 650 136 16 D1 17 782 56 333 285 25-440 497 57 17 D2 67 126 42-61 126 3 2 0 0 18 D3 58 688 48-106 87 9 280 601 78 19 D4 65 174 43-61 103 3 51 71 3 20 D5 43 869 50 196 288 22-278 581 57
MCA with two variables When Q = 2, usual MCA (version indicator or version Burt ) do not have simple CA as a special case. But both adjusted and JCA versions have simple CA as an exact special case. The nullified Burt matrix has exactly the simple CA inertias coinciding with the (squares) of its positive eigenvalues. No iterations necessary for JCA, its initial solution (based on adjusted MCA) perfectly fits the updated Burt matrix at the first iteraion, and once the inertia created in the diagonal blocks is discounted, the simple CA solution is recovered exactly.
Geometry of MCA indicator matrix version "indicator" J 0100 1000 0001 0010 1000 1000 N Z Dealt with already in The Carroll-Green-Schaffer scaling in correspondence analysis: a theoretical and empirical appraisal (Greenacre, Journal of Marketing Research, 1989)
Geometry of MCA two categories of same variable A1 A2 A3 A4 A5 B1 B2 B3 B4 B5 C1 C2 C3 C4 C5 D1 D2 D3 D4 D5 A1 119 0 0 0 0 27 28 30 22 12 49 40 18 7 5 15 25 17 34 28 A2 0 322 0 0 0 38 74 84 96 30 67 142 60 41 12 22 102 76 68 54 A3 0 0 204 0 0 3 48 63 73 17 18 75 70 34 7 10 44 68 58 24 A4 0 0 0 178 0 3 21 23 79 52 16 50 40 56 16 9 52 28 54 35 A5 0 0 0 0 48 0 3 5 11 29 2 9 9 16 12 4 9 13 12 10 B1 27 38 3 3 0 71 0 0 0 0 43 19 4 3 2 9 17 10 10 25 B2 28 74 48 21 3 0 174 0 0 0 36 88 34 15 1 16 51 42 45 20 B3 30 84 63 23 5 0 0 205 0 0 37 90 57 19 2 10 53 63 51 28 B4 22 96 73 79 11 0 0 0 281 0 27 88 75 74 17 6 66 70 92 47 B5 12 30 17 52 29 0 0 0 0 140 9 31 27 43 30 19 45 17 28 31 C1 49 67 18 16 2 43 36 37 27 9 152 0 0 0 0 25 24 15 38 50 C2 40 142 75 50 9 19 88 90 88 31 0 316 0 0 0 15 97 67 89 48 C3 18 60 70 40 9 4 34 57 75 27 0 0 197 0 0 5 51 83 41 17 C4 7 41 34 56 16 3 15 19 74 43 0 0 0 154 0 6 44 30 51 23 C5 5 12 7 16 12 2 1 2 17 30 0 0 0 0 52 9 16 7 7 13 D1 15 22 10 9 4 9 16 10 6 19 25 15 5 6 9 60 0 0 0 0 D2 25 102 44 52 9 17 51 53 66 45 24 97 51 44 16 0 232 0 0 0 D3 17 76 68 28 13 10 42 63 70 17 15 67 83 30 7 0 0 202 0 0 D4 34 68 58 54 12 10 45 51 92 28 38 89 41 51 7 0 0 0 226 0 D5 28 54 24 35 10 25 20 28 47 31 50 48 17 23 13 0 0 0 0 151
Geometry of MCA two categories of same variable A1 A2 A3 A4 A5 B1 B2 B3 B4 B5 C1 C2 C3 C4 C5 D1 D2 D3 D4 D5 A1 119 0 0 0 0 27 28 30 22 12 49 40 18 7 5 15 25 17 34 28 A2 0 322 0 0 0 38 74 84 96 30 67 142 60 41 12 22 102 76 68 54 A3 0 0 204 0 0 3 48 63 73 17 18 75 70 34 7 10 44 68 58 24 A4 0 0 0 178 0 3 21 23 79 52 16 50 40 56 16 9 52 28 54 35 A5 0 0 0 0 48 0 3 5 11 29 2 9 9 16 12 4 9 13 12 10 B1 27 38 3 3 0 71 0 0 0 0 43 19 4 3 2 9 17 10 10 25 B2 28 74 48 21 3 0 174 0 0 0 36 88 34 15 1 16 51 42 45 20 B3 30 84 63 23 5 0 0 205 0 0 37 90 57 19 2 10 53 63 51 28 B4 22 96 73 79 11 0 0 0 281 0 27 88 75 74 17 6 66 70 92 47 B5 12 30 17 52 29 0 0 0 0 140 9 31 27 43 30 19 45 17 28 31 C1 49 67 18 16 2 43 36 37 27 9 152 0 0 0 0 25 24 15 38 50 C2 40 142 75 50 9 19 88 90 88 31 0 316 0 0 0 15 97 67 89 48 C3 18 60 70 40 9 4 34 57 75 27 0 0 197 0 0 5 51 83 41 17 C4 7 41 34 56 16 3 15 19 74 43 0 0 0 154 0 6 44 30 51 23 C5 5 12 7 16 12 2 1 2 17 30 0 0 0 0 52 9 16 7 7 13 D1 15 22 10 9 4 9 16 10 6 19 25 15 5 6 9 60 0 0 0 0 D2 25 102 44 52 9 17 51 53 66 45 24 97 51 44 16 0 232 0 0 0 D3 17 76 68 28 13 10 42 63 70 17 15 67 83 30 7 0 0 202 0 0 D4 34 68 58 54 12 10 45 51 92 28 38 89 41 51 7 0 0 0 226 0 D5 28 54 24 35 10 25 20 28 47 31 50 48 17 23 13 0 0 0 0 151
Geometry of MCA two categories of different variables A1 A2 A3 A4 A5 B1 B2 B3 B4 B5 C1 C2 C3 C4 C5 D1 D2 D3 D4 D5 A1 119 0 0 0 0 27 28 30 22 12 49 40 18 7 5 15 25 17 34 28 A2 0 322 0 0 0 38 74 84 96 30 67 142 60 41 12 22 102 76 68 54 A3 0 0 204 0 0 3 48 63 73 17 18 75 70 34 7 10 44 68 58 24 A4 0 0 0 178 0 3 21 23 79 52 16 50 40 56 16 9 52 28 54 35 A5 0 0 0 0 48 0 3 5 11 29 2 9 9 16 12 4 9 13 12 10 B1 27 38 3 3 0 71 0 0 0 0 43 19 4 3 2 9 17 10 10 25 B2 28 74 48 21 3 0 174 0 0 0 36 88 34 15 1 16 51 42 45 20 B3 30 84 63 23 5 0 0 205 0 0 37 90 57 19 2 10 53 63 51 28 B4 22 96 73 79 11 0 0 0 281 0 27 88 75 74 17 6 66 70 92 47 B5 12 30 17 52 29 0 0 0 0 140 9 31 27 43 30 19 45 17 28 31 C1 49 67 18 16 2 43 36 37 27 9 152 0 0 0 0 25 24 15 38 50 C2 40 142 75 50 9 19 88 90 88 31 0 316 0 0 0 15 97 67 89 48 C3 18 60 70 40 9 4 34 57 75 27 0 0 197 0 0 5 51 83 41 17 C4 7 41 34 56 16 3 15 19 74 43 0 0 0 154 0 6 44 30 51 23 C5 5 12 7 16 12 2 1 2 17 30 0 0 0 0 52 9 16 7 7 13 D1 15 22 10 9 4 9 16 10 6 19 25 15 5 6 9 60 0 0 0 0 D2 25 102 44 52 9 17 51 53 66 45 24 97 51 44 16 0 232 0 0 0 D3 17 76 68 28 13 10 42 63 70 17 15 67 83 30 7 0 0 202 0 0 D4 34 68 58 54 12 10 45 51 92 28 38 89 41 51 7 0 0 0 226 0 D5 28 54 24 35 10 25 20 28 47 31 50 48 17 23 13 0 0 0 0 151
Geometry of MCA two categories of different variables A1 A2 A3 A4 A5 B1 B2 B3 B4 B5 C1 C2 C3 C4 C5 D1 D2 D3 D4 D5 A1 119 0 0 0 0 27 28 30 22 12 49 40 18 7 5 15 25 17 34 28 A2 0 322 0 0 0 38 74 84 96 30 67 142 60 41 12 22 102 76 68 54 A3 0 0 204 0 0 3 48 63 73 17 18 75 70 34 7 10 44 68 58 24 A4 0 0 0 178 0 3 21 23 79 52 16 50 40 56 16 9 52 28 54 35 A5 0 0 0 0 48 0 3 5 11 29 2 9 9 16 12 4 9 13 12 10 B1 27 38 3 3 0 71 0 0 0 0 43 19 4 3 2 9 17 10 10 25 B2 28 74 48 21 3 0 174 0 0 0 36 88 34 15 1 16 51 42 45 20 B3 30 84 63 23 5 0 0 205 0 0 37 90 57 19 2 10 53 63 51 28 B4 22 96 73 79 11 0 0 0 281 0 27 88 75 74 17 6 66 70 92 47 B5 12 30 17 52 29 0 0 0 0 140 9 31 27 43 30 19 45 17 28 31 C1 49 67 18 16 2 43 36 37 27 9 152 0 0 0 0 25 24 15 38 50 C2 40 142 75 50 9 19 88 90 88 31 0 316 0 0 0 15 97 67 89 48 C3 18 60 70 40 9 4 34 57 75 27 0 0 197 0 0 5 51 83 41 17 C4 7 41 34 56 16 3 15 19 74 43 0 0 0 154 0 6 44 30 51 23 C5 5 12 7 16 12 2 1 2 17 30 0 0 0 0 52 9 16 7 7 13 D1 15 22 10 9 4 9 16 10 6 19 25 15 5 6 9 60 0 0 0 0 D2 25 102 44 52 9 17 51 53 66 45 24 97 51 44 16 0 232 0 0 0 D3 17 76 68 28 13 10 42 63 70 17 15 67 83 30 7 0 0 202 0 0 D4 34 68 58 54 12 10 45 51 92 28 38 89 41 51 7 0 0 0 226 0 D5 28 54 24 35 10 25 20 28 47 31 50 48 17 23 13 0 0 0 0 151
Geometry of JCA two categories of same variable A1 A2 A3 A4 A5 B1 B2 B3 B4 B5 C1 C2 C3 C4 C5 D1 D2 D3 D4 D5 A1 31 53 19 14 3 27 28 30 22 12 49 40 18 7 5 15 25 17 34 28 A2 53 131 77 52 10 38 74 84 96 30 67 142 60 41 12 22 102 76 68 54 A3 19 77 63 39 7 3 48 63 73 17 18 75 70 34 7 10 44 68 58 24 A4 14 52 39 54 20 3 21 23 79 52 16 50 40 56 16 9 52 28 54 35 A5 3 10 7 20 9 0 3 5 11 29 2 9 9 16 12 4 9 13 12 10 B1 27 38 3 3 0 21 20 18 8 3 43 19 4 3 2 9 17 10 10 25 B2 28 74 48 21 3 20 46 54 50 4 36 88 34 15 1 16 51 42 45 20 B3 30 84 63 23 5 18 54 65 64 4 37 90 57 19 2 10 53 63 51 28 B4 22 96 73 79 11 8 50 64 104 55 27 88 75 74 17 6 66 70 92 47 B5 12 30 17 52 29 3 4 4 55 74 9 31 27 43 30 19 45 17 28 31 C1 49 67 18 16 2 43 36 37 27 9 82 55 4 3 7 25 24 15 38 50 C2 40 142 75 50 9 19 88 90 88 31 55 126 79 46 9 15 97 67 89 48 C3 18 60 70 40 9 4 34 57 75 27 4 79 66 41 6 5 51 83 41 17 C4 7 41 34 56 16 3 15 19 74 43 3 46 41 45 18 6 44 30 51 23 C5 5 12 7 16 12 2 1 2 17 30 7 9 6 18 11 9 16 7 7 13 D1 15 22 10 9 4 9 16 10 6 19 25 15 5 6 9 9 15 5 13 18 D2 25 102 44 52 9 17 51 53 66 45 24 97 51 44 16 15 62 56 61 38 D3 17 76 68 28 13 10 42 63 70 17 15 67 83 30 7 5 56 64 56 21 D4 34 68 58 54 12 10 45 51 92 28 38 89 41 51 7 13 61 56 60 36 D5 28 54 24 35 10 25 20 28 47 31 50 48 17 23 13 18 38 21 36 38 2 -distance between A2 & A4 = average of 2 -distances between profiles of A2 and A4 across the other three variables; For two variables, just the usual 2 -distance in simple CA.
Geometry of JCA two categories of different variables A1 A2 A3 A4 A5 B1 B2 B3 B4 B5 C1 C2 C3 C4 C5 D1 D2 D3 D4 D5 A1 31 53 19 14 3 27 28 30 22 12 49 40 18 7 5 15 25 17 34 28 A2 53 131 77 52 10 38 74 84 96 30 67 142 60 41 12 22 102 76 68 54 A3 19 77 63 39 7 3 48 63 73 17 18 75 70 34 7 10 44 68 58 24 A4 14 52 39 54 20 3 21 23 79 52 16 50 40 56 16 9 52 28 54 35 A5 3 10 7 20 9 0 3 5 11 29 2 9 9 16 12 4 9 13 12 10 B1 27 38 3 3 0 21 20 18 8 3 43 19 4 3 2 9 17 10 10 25 B2 28 74 48 21 3 20 46 54 50 4 36 88 34 15 1 16 51 42 45 20 B3 30 84 63 23 5 18 54 65 64 4 37 90 57 19 2 10 53 63 51 28 B4 22 96 73 79 11 8 50 64 104 55 27 88 75 74 17 6 66 70 92 47 B5 12 30 17 52 29 3 4 4 55 74 9 31 27 43 30 19 45 17 28 31 C1 49 67 18 16 2 43 36 37 27 9 82 55 4 3 7 25 24 15 38 50 C2 40 142 75 50 9 19 88 90 88 31 55 126 79 46 9 15 97 67 89 48 C3 18 60 70 40 9 4 34 57 75 27 4 79 66 41 6 5 51 83 41 17 C4 7 41 34 56 16 3 15 19 74 43 3 46 41 45 18 6 44 30 51 23 C5 5 12 7 16 12 2 1 2 17 30 7 9 6 18 11 9 16 7 7 13 D1 15 22 10 9 4 9 16 10 6 19 25 15 5 6 9 9 15 5 13 18 D2 25 102 44 52 9 17 51 53 66 45 24 97 51 44 16 15 62 56 61 38 D3 17 76 68 28 13 10 42 63 70 17 15 67 83 30 7 5 56 64 56 21 D4 34 68 58 54 12 10 45 51 92 28 38 89 41 51 7 13 61 56 60 36 D5 28 54 24 35 10 25 20 28 47 31 50 48 17 23 13 18 38 21 36 38 2 -distances between-sets still problematic ( you don t get something for nothing!), especially for two-variables. The biplot interpretation, however, is perfectly valid, and optimal for all the off-diagonal elements of the Burt matrix.
From asymmetric to contribution biplot Asymmetric map
From asymmetric to contribution biplot Asymmetric biplot Contribution biplot
From asymmetric to contribution biplot Contribution biplot profile values on A1 project categories of all OTHER variables (B, C and D) on A1, NOT the categories of A; again, this agrees perfectly with the two-variable case.
Thanks for your attention First two principal axes of CA Variance explained = 99.5%
subset versions subset CA subset MCA (indicator/burt/adjusted), subset JCA