Unifying the geometry of simple and multiple correspondence analysis

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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

1 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 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.

2 SImple correspondence analysis > summary(ca(smoke)) Principal inertias (eigenvalues): dim value % cum% scree plot ************************* *** Total: First two principal axes of CA Rows: name mass qlt inr k=1 cor ctr k=2 cor ctr 1 SM JM SE JE SC Columns: name mass qlt inr k=1 cor ctr k=2 cor ctr 1 non lgh mdm hvy Variance explained = 99.5%

3 Multiple correspondence analysis (function mjca of ca package in R) N "indicator" "Burt" "adjusted" "JCA" J= q J q J 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

4 Problem of variance explained > summary(mjca(wg93[,1:4], lambda="indicator")) Principal inertias (eigenvalues): dim value % cum% scree plot ************************* *********************** *************** : : : : > summary(mjca(wg93[,1:4], lambda="burt")) Principal inertias (eigenvalues): dim value % cum% scree plot ************************* ********************** *********** : : : : > summary(mjca(wg93[,1:4], lambda="adjusted")))) #DEFAULT Principal inertias (eigenvalues): dim value % cum% scree plot ************************* ******************* *** : : : : > 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

5 Same problem for individual points > summary(mjca(wg93[,1:4], lambda="burt")) Principal inertias (eigenvalues): dim value % cum% scree plot ************************* ********************** *********** : : : : : name mass qlt inr k=1 cor ctr k=2 cor ctr 1 A A A A A B B B B B C C C C C D D D D D

6 Same problem for individual points > summary(mjca(wg93[,1:4], lambda="burt")) Principal inertias (eigenvalues): dim value % cum% scree plot ************************* ********************** *********** > 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 A A A A A B B B B B C C C C C D D D D D

7 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 A A A A A B B B B B C C C C C D D D D D default: two-dimensional solution at convergence the diagonal blocks are perfectly fitted

8 Joint correspondence analysis: discounting inertia on diagonal blocks > summary(mjca(wg93[,1:4], lambda="jca")) Principal inertias (eigenvalues): dim value : : Total: Diagonal inertia discounted from eigenvalues: Percentage explained by JCA in 2 dimensions: 85.7% (Eigenvalues are not nested) [Iterations in JCA: 44, epsilon = 9.91e-05] ( )

9 JCA: discounting inertia for each point > summary(mjca(wg93[,1:4], lambda="jca")) : : : : : : : : : : Columns: name mass inr k=1 k=2 cor ctr 1 A A A A A B B B B B C C C C C D D D D D 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.

10 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 A 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)

11 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 A A A A A B B B B B C C C C C D D D D D

12 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 A A A A A B B B B B C C C C C D D D D D 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!)

13 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.

14 Results for adjusted MCA > summary(mjca(wg93[,1:4])) Principal inertias (eigenvalues): dim value % cum% scree plot ************************* ******************* *** : : : : : name mass qlt inr k=1 cor ctr k=2 cor ctr 1 A A A A A B B B B B C C C C C D D D D D

15 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.

16 Geometry of MCA indicator matrix version "indicator" J 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)

17 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 A A A A A B B B B B C C C C C D D D D D

18 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 A A A A A B B B B B C C C C C D D D D D

19 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 A A A A A B B B B B C C C C C D D D D D

20 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 A A A A A B B B B B C C C C C D D D D D

21 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 A A A A A B B B B B C C C C C D D D D D 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.

22 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 A A A A A B B B B B C C C C C D D D D D 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.

23 From asymmetric to contribution biplot Asymmetric map

24 From asymmetric to contribution biplot Asymmetric biplot Contribution biplot

25 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.

26 Thanks for your attention First two principal axes of CA Variance explained = 99.5%

27 subset versions subset CA subset MCA (indicator/burt/adjusted), subset JCA

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