Duarte Silva, A. P.

German

Repositório Institucional da Universidade Católica Portuguesa

DISCARDING VARIABLES IN PRINCIPAL COMPONENT ANALYSIS:  ALGORITHMS FOR ALL-SUBSETS COMPARISONS BASED ON THE RV COEFFICIENTANTÓNIO PEDRO DUARTE SILVAFACULDADE DE CIÊNCIAS ECONÓMICAS E EMPRESARIAISUNIVERSIDADE CATÓLICA PORTUGUESACENTRO REGIONAL DO PORTODiscarding and/or Selecting Variables in PCA WHY DISCARD VARIABLES ?•  PARSIMONITY•  REDUCING COSTS OF DATA COLLECTION•  HELPING THE INTREPETATION OF A CLASSICAL PCA- THE TRADITIONAL APPROACH OF IGNORING VARIABLES WITH SMALL LOADINGS, IMPLICITLY DISCARDS THEM- ASSESSING VARIABLE IMPORTANTE ONLY BASED ON LOADINGS CAN BE MISLEADING (CADIMA AND JOLLIFFE 1995)•  AS A  DIRECT APPROACH TO THE PROBLEM OF DIMENSIONALITY REDUCTION  (McCabe 1984)CRITERIA FOR COMPARING VARIABLE SUBSETSMcCABE APROACH (PRINCIPAL VARIABLES 1984):Y  =   A    X  ;       Z =    B  Y(n*k) (n*p)     (n*p)     (n*k)(A = [I1 | 0 ]  ; X = [X1 | X2 ]   Y = X1)(S22|1 = S22 - S21 S11-1 S12)(1)  MIN  |Z - X|          MIN  |S22|1|      =  i(2)  MIN  Tr (Z - X)      MIN Tr  S22|1 =  i(3)  MIN ||Z - X||2  MIN ||S22|1||2   =  i2(4) MIN  (Z - X)T SX-1 (Z - X)    MIN Tr ( S22-1 S22|1) = (p-K) -  i2YANAI’ S GENERALIZED CORRELATION COEFFICIENTCRITERIA BASED ON CORRELATIONAL MEASURES :r1(A,B) = Tr(AT B) / [Tr (ATA) Tr (BTB)]1/2A = PA A QAT B = PB B QBTESCOUFIER’S RV COEFFICIENTGCD(A,B) = r1(PA PAT, PB PBT)MAX   GCD(X1,PCq(X))                      (Cadima and Jolliffe 1996) RV(A,B) = r1(PA A2 PAT, PB B2 PBT)MAX   RV(X,M X1)                              (Robert and Escoufier 1976) yyXyyXXX'´''ALL-SUBSETS COMPARISONS IN LINEAR REGRESSIONy = X’    + eˆCOMMENTS:• MATRIX SIMMETRY IS ALWAYS PRESERVED • ONLY THE RIGHT-LOWER CORNER NEEDS TO BE UPDATED AT EACH STEP  • BY SEQUECING THE SUBSETS EVALUATIONS PROPERLY:1/2 OF THE SWEEPS UPDATE (1*1) SUBMATRICES 1/4 OF THE SWEEPS UPDATE (2*2) SUBMATRICES …ONLY ONE SWEEP UPDATES THE FULL (P*P) MATRIX eeXX''ˆˆ)'( 1FURNIVAL’S ALGORITHM (1971):ELEMENTARY MATRIX OPERATIONS (SYMMETRIC SWEEPS)TOTAL NUMBER OF FLOATINGPOINT OPERATIONS: 6 * 2P( 6 (2p) – p(p+7)/2 – 6 )FURNIVAL AND WILSON’S “LEAPS AND BOUNDS” (1974):SA  SB  (eTe)A  (eTe)BDOUBLE TREE SEARCH X1, X2, … , XP           ADDING VARIABLES          REMOVING VARIABLES                         CREATE BOUNDS eTe   >  BOUND                                  PRUNE BRANCH ALL-SUBSETS COMPARISONS BASED ON THE RV COEFFICIENTMAX RV(X, M X1)    MAX Tr { [ (S11)-1 (S2)11]2 } == i,j { [ (S11)-1 (S2)11](i,j) * [ (S11)-1 (S2)11](j,i) }SYMMETRIC SWEEPING OF COVARIANCE MATRICES:1|221112112111111S)S(SS)S()S(A'1|'2'21'1'1'1'2'2'11'1'11'1'1S)S(SS)S()S(BX1’ = X1  {Xu}  ;   X2’ = X2 \ {Xu}B(u,u) = - 1 / A(u,u)B(i,u) =  A(i,u) * B(u,u)                (i  u)B(i,j) =  A(i,,j)  + A(i,u) * B(j,u)     (i  u, i  u )UPDATING THE ELEMENTS OF [(S11)-1 (S2)11]       )u,u(S)j,a(S*)u,a(S)S()j,u(S)j,u()S()S(1|22XX:a2121112'121'1'11a      *)u,u(S)u,i(S)S()j,i()S()S()j,i()S()S(1|221211112111'121'1'1      1a XX:a2121112)j,a(S*)u,a(S)S()j,u(S* (i  u)NUMBER OF FLOATING POINT OPERATIONSPER SWEEP:(5/2) K12 + (11/2) K1 + 2TOTAL:(5/2) K12 + (11/2) K1 + 2OPERATORS FOR REMOVING VARIABLES      )u,j)(S(*)u,i()S()j,i()S()S()j,i()S()S( 21'1'1'121'1'112111  1a XX:a21'1'11'1'11'1'1 )j,a(S*)u,a()S(*)u,u()S()u,i()S('1|'2'21'1'1'1'2'2'11'1'11'1'1S)S(SS)S()S(A1|221112112111111S)S(SS)S()S(BX1’ = X1  {Xu}  ;   X2’ = X2 \ {Xu}B(u,u) = - 1 / A(u,u)B(i,u) =  A(i,u) * B(u,u)                (i  u)B(i,j) =  A(i,,j)  + A(i,u) * B(j,u)     (i  u, i  u )CONCLUSIONSALL-SUBSETS COMPARISONS ARE POSSIBLE:(b) IF A “FEW” SUBSETS ARE CLEARLYPREFERABLE TO THE OTHERSOR(a) IF THE NUMBER OF ORIGINAL VARIABLES ISNOT TOO LARGE (SAY P  25)SOFTWARE IMPLEMENTATIONLOOK AT:http://www.porto.ucp.pt/psilvaWHEN ALL-SUBSETS COMPARISONS ARE NOT POSSIBLE CAN HEURISTIC METHODS BE ALMOST OPTIMAL ???FURTHER RESEARCH

Discarding variables in principal component analysis:  algorithms for all-subsets comparisons based on the RV coefficient

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Discarding variables in principal component analysis: algorithms for all-subsets comparisons based on the RV coefficient

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Repositório Institucional da Universidade Católica Portuguesa