Orthogonal rotation in PCAMIX

Orthogonal rotation in PCAMIX ∗ Marie Cha v en t 1 , 2 † , V anessa Kuen tz 3 and J ´ erˆ ome Saracco 2 , 4 1 Univ ersit´ e de Bordeaux, IMB, CNRS, UMR 5251, F rance 2 INRIA Bordeaux Sud-Ouest, CQFD team, F rance 3 CEMA GREF, UR ADBX, F rance 4 Institut P olytechnique de Bordeaux, F rance Abstract Kiers (1991) considered the orthogonal rotation in PCAMIX, a principal comp o- nen t metho d for a mixture of qualitativ e and quantitativ e v ariables. PCAMIX includes the ordinary principal comp onen t analysis (PCA) and multiple corresp ondence anal- ysis (MCA) as sp ecial cases. In this pap er, we give a new presen tation of PCAMIX where the principal comp onen ts and the squared loadings are obtained from a Singu- lar V alue Decomp osition. The loadings of the quantitativ e v ariables and the principal co ordinates of the categories of the qualitativ e v ariables are also obtained directly . In this context, we prop ose a computationaly efficient pro cedure for v arimax rotation in PCAMIX and a direct solution for the optimal angle of rotation. A sim ulation study sho ws the goo d computational b eha vior of the prop osed algorithm. An application on a real data set illustrates the in terest of using rotation in MCA. All source co des are a v ailable in the R pack age “PCAmixdata”. Keyw ords: mixture of qualitativ e and quan titative data, principal comp onen t analy- sis, m ultiple corresp ondence analysis, rotation. 1 In tro duction Kaiser (1958) in tro duced the v arimax criterion for the attainmen t of simple structures by orthogonal rotation in Principal Comp onen t Analysis (PCA) . This criterion aims at maxi- mizing the sum ov er the columns of the squared elements of the loading matrix. The loading matrix plays a significan t part in the in terpretation of the results since it contains the corre- lations b et w een the v ariables and the principal comp onen ts. The idea is to get comp onen ts so that the interpretation is easier, that is to rotate the loading matrix and the standardized principal comp onen ts so that the groups of v ariables app ear: ha ving high loadings on the same comp onen t, mo derate ones on a few comp onen ts and negligible ones on the remaining comp onen ts. Because the Singular V alue Decomp osition (SVD) approach in PCA giv es one the freedom for orthogonal rotation, the p ercen tage of v ariance explained is redistributed ∗ Submitted pap er, August 2011 † marie.c hav en t@u-b ordeaux2.fr 1 along the newly rotated axes, while still conserving the v ariance explained b y the solution as a whole. Kiers (1991) extended the v arimax criterion for the attainment of simple structures in PCAMIX, a principal comp onen t metho d for the mixture of qualitative and quan titative v ariables. F or qualitativ e v ariables, the co efficien t used to express the link b et w een a v ariable and a comp onen t is the correlation ratio; this correlation ratio plays the role of a squared loading. The v arimax criterion is then expressed with squared loadings defined as correlation ratios for qualitativ e v ariables and squared correlations for quan titativ e v ariables. Algorithms devised for the determination of an optimal orthogonal rotation in the context of PCA, as prop osed for example b y Kaiser’s (1958), Neudeck er (1981) or Jennric h (2001) did not apply to this extended v arimax criterion. So Kiers (1991) proposes a matrix reform ulation of this new v arimax criterion in order to replace the optimization problem with a problem of sim ultaneous diagonalization of a set of symmetric matrices (ten Berge, 1984), and suggests the use of the algorithm of de Leeu w and Pruzansky (1978) to solv e the latter. T o the b est of our knowledge, the resulting algorithm has never b een presented in a single pap er, so w e hav e recalled for comparison purp ose the main steps of the matrix reform ulation and the simultaneous diagonalization. W e shall refer to this algorithm as Kiers’ (1991) original approac h to PCAMIX. In this pap er we will first present a new formulation of PCAMIX. It is similar to that of Escofier (1979) and Pag ` es (2004) in the wa y quantitativ e and qualitative v ariables are transformed, but it is presen ted via a SVD. This presents a direct wa y to determine b oth the comp onen t scores and the squared loadings and also the principal co ordinates of the categories of the qualitativ e v ariables as w ell as the loadings of the qualitative v ariables. Then w e will search for an optimal rotation for the PCAMIX v arimax criterion using the iterativ e pro cedure suggested b y Kaiser (1958) for PCA: we will rotate pairs of dimensions according to an optimal angle θ , iterativ ely un til the process con v erges. A new direct, sp ecific to PCAMIX determination of this angle is proposed. W e shall refer to the resulting algorithm as the SVD approac h to PCAMIX. This algorithm leads to the same final rotation as Kiers’ (1991) original approac h, how ev er a sim ulation study shows that it is computationally more efficien t. When all the v ariables are quantitativ e, the new algorithm reduces to the classical Kaiser’s (1958) pro cedure for orthogonal rotation in PCA with a new direct expression of the optimal planar angle θ . Notice that Kaiser’s v arimax rotation pro cedure do es not alwa ys pro duce an optimal rotation in PCA. ten Berge (1995) made suggestions for addressing this p oin t for PCA. This is an op en problem for PCAMIX. 2 This pap er is organized as follows. Section 2 recalls Kiers’ original PCAMIX metho d and proposes an alternative form ulation using SVD. Section 3 deals with v arimax rotation in PCAMIX. The optimization problem is giv en section 3.1. The determination of the optimal angle of rotation with Kiers’ matrix reform ulation a pproach is describ ed section 3.2.1 for purp ose of comparison with the direct solution prop osed section 3.2.2. The complete pro cedure for orthogonal rotation in more than tw o dimensions is giv en section 3.3. A sim ulation study compares section 4.1 the computational time of the prop osed rotation pro cedure with the rotation pro cedure based on Kiers (1991). In section 4.2 a real data application illustrates the in terest of rotation in MCA and shows some of the outputs and graphical represen tations av ailable in the R pac k age “PCAmixdata” w e ha v e dev elop ed. 2 The PCAMIX metho d Let us first in tro duce some notations used in the presen tation of the PCAMIX metho d. • Let n denote the num ber of observ ation units, p 1 the num b er of quantitativ e v ariables, p 2 the n um b er of qualitativ e v ariables and p = p 1 + p 2 the total num ber of v ariables. • Let z j b e the column vector whic h contains the standardized scores of the n ob jects on v ariable j if the j -th v ariable is quan titative. • Let G j b e the indicator matrix for the v ariable j if the j -th v ariable is qualitativ e and let D j b e the diagonal matrix of frequencies of categories of this v ariable. • Let us denote b y m the n umber of categories of the p 2 qualitativ e v ariables. • Let G = ( G 1 | · · · | G j | · · · | G p 2 ) b e the n × m matrix of the indicator v ariables of the m categories of the p 2 qualitativ e v ariables and let D = diag( D 1 , . . . , D j , . . . , D p 2 ) b e the m × m diagonal matrix of frequencies of the m categories. • Let J = I n − 11 0 /n b e the cen tering op erator where I n denotes the n × n identit y matrix and 1 the v ector of order n with unit entries. In the t w o follo wing subsections, w e giv e t w o form ulations of the PCAMIX metho d and highligh t their main differences. 2.1 The original PCAMIX pro cedure Supp ose k is the num b er of comp onen ts required in PCAMIX. In Kiers (1991), the pro ced ure computes the n × k matrix X of the standardized comp onen t scores, the v ariance of each 3 comp onen t and the p × k matrix C of the squared loadings. The squared loadings are de- fined as squared correlation for quan titativ e v ariables and as correlation ratio for qualitativ e v ariables. This pro cedure is carried out according to the follo wing steps: 1. F or j = 1 , . . . , p : calculate the so-called n × n quan tification matrix S j with:  S j = 1 n z j z 0 j if v ariable j is quan titative , S j = JG j D − 1 j G 0 j J if v ariable j is qualitativ e . 2. Calculate the n × n matrix S = P p j =1 S j . 3. Perform an EigenV alue Decomp osition of S . The matrix X of the standardized com- p onen t scores is given by the first k eigen vectors of S normalized to n (such that X 0 X = n I k ). 4. F or l = 1 , . . . , k : calculate the v ariance of the l -th comp onen t given by x 0 l Sx l where x l denotes the l -th column of X . 5. Calculate the matrix C of the squared loadings of the p v ariables on the k comp onen ts with c j l = 1 n x 0 l S j x l . F or quantitativ e (resp. qualitative) v ariables, c j l is the squared correlation (resp. correlation ratio) b et w een the v ariable j and the comp onen t l . When all the v ariables are quantitativ e (resp. qualitativ e), this pro cedure is equiv alent to PCA (resp. MCA). But the loadings (the correlations b et w een the v ariables and the com- p onen ts) and the principal co ordinates of the categories (the barycenters of the component scores) are not directly pro vided and must b e calculated afterw ards if desired. F rom a prac- tical point of view this pro cedure requires the construction and the storage of p matrices of dimension n × n which can leads to memory size problems when n and p increase. 2.2 The SVD based PCAMIX pro cedure This procedure is carried out according to the following steps: 1. Determine the n × ( p 1 + m ) matrix of in terest Z = 1 √ n ( Z 1 | Z 2 ) where : • Z 1 = ( z 1 | · · · | z j | · · · | z p 1 ) is the n × p 1 matrix of the standardized scores of the n observ ation units (ob jects) on the p 1 quan titativ e v ariables. • Z 2 is the n × m matrix obtained by reco ding G in the follo wing wa y: Z 2 = JGD − 1 / 2 . 4 2. Perform the SVD of Z : Z = U Λ V 0 , (1) where U 0 U = V 0 V = I r , Λ is the diagonal matrix of singular v alues (in w eakly de- scending order) and r is the rank of Z . 3. Calculate the n × k matrix of the standardized comp onen t scores: X = √ n U k (2) where U k denotes the matrix of the first k columns of U . 4. F or ` = 1 , . . . , k , the standard deviation of the ` -th comp onen t is giv en b y the ` -th singular v alue in Λ . 5. Calculate the matrix: A = V k Λ k , (3) where V k denote the matrix of the first k columns of V and Λ k the diagonal matrix of the k largest singular v alues. 6. W rite A =  A 1 A 2  the concatenation of a p 1 × k matrix A 1 and a m × k matrix A 2 . • The matrix A 1 con tains the loadings of the quantitativ e v ariables (the correlations b et w een the quantitativ e v ariables and the components). • The matrix D A 2 con tains the principal co ordinates of the categories of the qual- itativ e v ariables. • Calculate the matrix C of the squared loadings of the p v ariables on the k com- p onen ts. This matrix is obtained from the matrix A as follo ws:  c j l = a 2 j l if v ariable j is quan titative , c j l = P s ∈ I j a 2 sl if v ariable j is qualitativ e , where I j is the set of row indices of A asso ciated with the categories of the qualitativ e v ariable j . T o simplify the notations, we note hereafter c j l = P s ∈ I j a 2 sl for b oth quantitativ e and qualitative v ariables with I j = { j } in the quan titativ e case. Note that the matrix X of the standardized comp onen t scores is obtained from the SVD of the reco ded data matrix Z whereas it was obtained from the Eigenv alue Decomp osition of the matrix S (the sum of the quantification matrices S j ) in Kiers’ original approach. Also, the matrix C of the squared loadings (squared correlations or correlation ratios b et w een 5 the v ariables and the comp onen ts) is calculated here from the only matrix A obtained with the SVD of Z whereas it was calculated from the tw o matrices X and S j in Kiers’ original approac h. Con trary to the original PCAMIX approac h, this pro cedure simultaneously provides the loadings of the quantitativ e v ariables and the principal co ordinates of the categories of the qualitativ e v ariables. Moreo ver, when the data are mixed (quan titativ e and qualitativ e), the w ell known barycen tric prop ert y in MCA remains true: the co ordinates of the categories are the av erages of the standardized comp onen t scores of the ob jects in those categories. The matrices X , A 1 and D A 2 are then used to plot the observ ation units, the quan titative v ariables and the categories with the same in terpretation rules as in PCA and MCA. Matrix C is used to plot the quantitativ e and qualitativ e v ariables on a same graphic. 3 V arimax rotation in PCAMIX 3.1 The optimization problem Wh y using rotation ? As shown b y Eck art and Y oung (1936), from the SVD in (1) and definitions of matrices X and A giv en in (2) and (3), the matrix XA 0 is a rank k least squares appro ximation of Z . Let us in tro duce T an orthonormal rotation matrix: TT 0 = T 0 T = I k . Let e X = XT and e A = A T . As XA 0 = e X e A 0 , this appro ximation is not unique ov er orthogonal rotations. This non-uniqueness can b e exploited to improv e the in terpretabilit y of the original so- lutions. T o simplify the in terpretations, the matrices X and A are then rotated in suc h a w a y that when considering one v ariable, few squared loadings are large (close to 1) and as man y as p ossible are close to zero. The v arimax problem. In PCA, since e A contains the loadings of the v ariables after rotation, the v arimax rotation problem is formulated as max T f ( T ) , s.t. TT 0 = T 0 T = I k , (4) where f ( T ) = k X l =1 p X j =1 (˜ a 2 j l ) 2 − 1 p k X l =1 p X j =1 ˜ a 2 j l ! 2 (5) is the v arimax function measuring the simplicity of the components after rotation. In the SVD approach of PCAMIX, the v arimax function f is defined by replacing in (5) 6 the terms ˜ a 2 j l b y ˜ c j l , where the ˜ c j l = P s ∈ I j ˜ a 2 sl are the squared loadings after rotation: f ( T ) = k X l =1 p X j =1 (˜ c j l ) 2 − 1 p k X l =1 p X j =1 ˜ c j l ! 2 . (6) Note that the squared loadings after rotation ˜ c j l are squared correlations (resp. correlation ratios) betw een the quantitativ e (resp. qualitative) v ariables and the rotated comp onen ts. F or comparison purp ose, w e recall Kiers’ original expression of the v arimax function in PCAMIX: the squared loadings after rotation ˜ c j l are given by 1 n ˜ x 0 l S j ˜ x l , where ˜ x l denotes the l -th column of e X . Hence the v arimax function (6) b ecomes: f ( T ) = k X l =1 p X j =1  1 n ˜ x 0 l S j ˜ x l  2 − 1 p k X l =1 p X j =1 1 n ˜ x 0 l S j ˜ x l ! 2 . (7) The iterative optimization pro cedure. Because a direct solution for the optimal T is not a v ailable, an iterativ e optimization pro cedure suggested b y Kaiser (1958) for PCA can b e used for PCAMIX. The idea is to consider at eac h iteration a planar rotation for whic h the rotation matrix T only dep ends of an angle θ (see b elo w for details). This pro cedure rotates pairs of dimensions in the following w ay: the single-plane rotations are applied to dimensions 1 and 2, 1 and 3, . . . , 1 and k , 2 and 3, . . . , ( k − 1) and k , iterativ ely until the pro cess con v erges, i.e. un til k ( k − 1) / 2 successive rotations pro viding an angle of rotation equal to zero are obtained. The k ey point of this rotation procedure is the definition of the single-plane rotation step. W e giv e next details on the calculation of the optimal angle for planar rotation. Then we giv e the complete iterative pro cedure for rotation in more than t w o dimensions. 3.2 Planar rotation Single planar rotations are obtained with a rotation matrix T defined b y T =  cos θ − sin θ sin θ cos θ  (8) where θ is the angle of rotation. The v arimax rotation problem (4) is then rewritten as: max θ ∈ R f ( θ ) . F or purp ose of comparison w e recall first the solution based on Kiers’ matrix reform ulation b efore we giv e our direct solution. 7 3.2.1 Planar rotation using the Kiers’ matrix reform ulation Kiers (1991) prop oses to use a pro cedure of simultaneous diagonalization of a set of sym- metric matrices (ten Berge, 1984; de Leeuw and Pruzansky ,1978) to solve the global v arimax optimization problem (4). F or that purp ose he gives the following matrix reformulation of the form ula (7) giving f : f ( T ) = p − 2 p X j =1 T race ( T 0 E j T (Diag T 0 E j T )) (9) where E j = p X 0 S j X − n Γ (10) and Γ is the diagonal matrix with the k first eigenv alues of S on its diagonal. Careful reading of ten Berge (1984) and de Leeuw and Pruzansky (1978) shows that the procedure for sim ultaneous diagonalization of the matrices E j is equiv alent to Kaiser’s iterativ e optimization pro cedure with the optimal angle θ of single plane rotations defined b y the equation: tan(4 θ ) = a b , (11) where a = 4 p X j =1 e j 12 ( e j 11 − e j 22 ) and b = p X j =1 ( e j 11 − e j 22 ) 2 − 4 p X j =1 ( e j 12 ) 2 (12) and E j =  e j 11 e j 12 e j 21 e j 22  is defined in (10). As mentionned by several authors (see for instance Nev els, 1986; ten Berge, 1984; de Leeu w and Pruzansky , 1978 and Kaiser, 1958) equation (11) is only a necessary condition obtained up on setting the first order deriv ativ e of the ob jectiv e function to zero. Both Kaiser (1958) and de Leeuw and Pruzansky (1978) developed a pro cedure for determining the optimal θ from the sign of the second order deriv ativ e of the ob jective function. These t w o pro cedures, expressed in tabular form, give the appropriate solution for every p ossible com bination of signs of a and b . 3.2.2 Planar rotation using the SVD approach of PCAMIX The v arimax function f ( T ) defined with the SVD approach in (6) is written: f ( θ ) = p X j =1   X s ∈ I j ˜ a 2 s 1   2 + p X j =1   X s ∈ I j ˜ a 2 s 2   2 − 1 p   p X j =1 X s ∈ I j ˜ a 2 s 1   2 − 1 p   p X j =1 X s ∈ I j ˜ a 2 s 2   2 (13) with ˜ a s 1 = a s 1 cos( θ ) + a s 2 sin( θ ) and ˜ a s 2 = − a s 1 sin( θ ) + a s 2 cos( θ ) . (14) 8 This function is equal to (see App endix): f ( θ ) = f (0) + ρ 4 p  cos(4 θ − ψ ) − cos ψ  (15) where ρ and ψ are defined by : ρ = ( a 2 + b 2 ) 1 / 2 , cos ψ = b/ρ , sin ψ = a/ρ (16) with a and b giv en b y : a = 2 p p X j =1 u j v j − 2 p X j =1 u j p X j =1 v j , b = p p X j =1 ( u j 2 − v j 2 ) − p X j =1 u j ! 2 + p X j =1 v j ! 2 , (17) where u j and v j are defined by : u j = X s ∈ I j ( a 2 s 1 − a 2 s 2 ) and v j = 2 X s ∈ I j a s 1 a s 2 . (18) The function f obtained in (15) is maximum for cos(4 θ − Ψ ) = 1 ⇔ 4 θ − Ψ = 2 k π , th us the optimal angles are : θ = Ψ 4 + k π 2 , k ∈ Z . (19) Note that the ab o v e expressions of u j and v j con tain as sp ecial cases (take I j = { j } ) those defined by Kaiser (1958) for the PCA v arimax solution. Note also that the classical necessary condition (11) immediately follo ws b y setting the expression (23) of pf 0 ( θ ) given in the App endix to zero (the co efficien ts b and a giv en b y (12) on one side, and (17)(18) on the other side are prop ortional). 3.3 The iterativ e rotation pro cedure. W e consider no w the case where the num ber k of dimensions in the rotation is greater than tw o. The iterative rotation pro cedure gives the matrix e X of the rotated standardized comp onen t scores and the matrix e A whic h is used to obtain the rotated squared loadings, the rotated loadings (correlations) of the quan titative v ariables and the rotated principal co ordinates of the categories. This pro cedure is carried out according to the follo wing steps: 1. Initialization : e X = X and e A = A where the n × k matrix X and the ( p 1 + m ) × k matrix A are giv en b y the SVD based PCAMIX procedure given section 2.2 . 2. F or l = 1 , . . . , k − 1 and t = ( l + 1) , . . . , k , calculate for the pair of dimensions ( l, t ): - the angle of rotation θ = Ψ / 4 with Ψ defined in (16) . W e c ho ose: Ψ =      arcos( b √ a 2 + b 2 ) if a ≥ 0 , − arcos( b √ a 2 + b 2 ) if a ≤ 0 . (20) where a and b are defined in (17). 9 - the matrix of rotation T =  cos θ − sin θ sin θ cos θ  , - the matrices e X and e A updated by rotation of their l -th and t -th column. 3. Rep eat the previous step un til the k ( k − 1) / 2 angles θ are equal to zero. 4. Calculate: - the matrix e C with ˜ c j l = P s ∈ I j ˜ a 2 sl . - the matrix e A 1 of the p 1 first rows of e A which contains the rotated loadings of the quan titativ e v ariables. - the matrix e A 2 of the m last ro ws of e A and the matrix D e A 2 whic h con tains the rotated principal co ordinates of the categories of the qualitative v ariables. The main differences b et w een this pro cedure and that constructed with Kiers’ matrix reform ulation are the following: • The expressions of a and b in step (2): in this pro cedure they are expressed according to the matrix A of dimension ( p 1 + m ) × n where p 1 is the num ber of quantitativ e v ariables and m is the total n um b er of categories. With Kiers’ matrix reformulation, a and b are expressed according to the p matrices S j of dimension n × n . Then the calculation and the storage of these matrices ma y b e time and space consuming. • The direct determination of the optimal angle in step (2). Ha ving an explicit ex- pression for the solution is of theoretical in terest and is more straightforw ard from a computational point of view. • The outputs: this pro cedure provides directly the rotated loadings of the quan titative v ariables and the rotated principal co ordinates of the categories which are used for graphical represen tations after rotation. 4 Numerical studies The pro cedure proposed in this pap er for v arimax orthogonal rotation in PCAMIX has b een implemented in R. A pac k age called “PCAmixdata” is already a v ailable on the CRAN w ebsite. In this section, this algorithm is compared on sim ulated data with Kiers’ rotation pro cedure. Then an application on a real data example illustrates the p ossible b enefits of using rotation in MCA as particular case of PCAMIX. 10 4.1 A sim ulation study: comparison of computational times An iterativ e rotation pro cedure based on Kiers’ matrix reform ulation has also b een imple- men ted in R. This pro cedure is that prop osed section 3.3 with the follo wing mo difications: • Kiers’ original PCAMIX procedure is used in the initialization step in place of the SVD based PCAMIX pro cedure. • All the calculations and outputs based on the matrix A are remov ed b ecause this matrix is not part of the original PCAMIX procedure. • The co efficien ts a and b in step 2 are calculated according to their expressions (12) asso ciated to Kiers’ matrix reform ulation. Note that the ratio a b is the same with the t w o approac hes (SVD and matrix reform ulation) so the optimal angle θ is the same. • In step 4 the squared loadings are calculated with their expression in the original PCAMIX approac h. The computation time of the tw o rotation pro cedures (the one based on Kier’s matrix refor- m ulation and the one based on the SVD approac h of PCAMIX) is compared from simulated datasets with v arying parameters: the num b er p of v ariables ( p/ 2 quantitativ e and p/ 2 qual- itativ e) and the num b er n of observ ations. F or each set of parameters ( n , p ), 20 simulations are dra wn. More precisely the datasets are built using the following pro cedure: • A dataset with n observ ations and p v ariables is drawn from a multiv ariate normal distribution with a co v ariance matrix Σ = Q 0 Q where Q is a p × p matrix drawn from a uniform distribution on the interv al [0 . 2; 0 . 4]. • The p/ 2 last v ariables are distributed in three equal-count categories. Eac h dataset is then constituted of p 1 = p/ 2 quantitativ e v ariable, p 2 = p/ 2 qualitative v ariable and the total num ber of categories is m = 3 ∗ p/ 2. Because the tw o rotation pro cedures iterate planar rotations until conv ergence, w e compare their computation time for k = 2. The median computation times (o v er the 20 replications) are given in T able 1 and the ratio b et w een the computation time of the t w o approac hes are giv en in T able 2. T able 1 sho ws that the SVD approach is faster than the matrix reformulation approac h for all configurations. F or configurations where p = 10, T able 2 sho ws that the SVD approac h is from 3 times faster for n = 50 to 214 times faster for n = 800. F or configurations with greater v alues of p , this ratio is less imp ortan t but still increases with n . F or the configuration where 11 p =10 p =50 p =100 p =200 n =50 Matrix reformulation 0.05 0.12 0.22 0.44 n =50 SVD 0.02 0.06 0.12 0.27 n =100 Matrix reform ulation 0.14 0.33 0.56 1.04 n =100 SVD 0.02 0.09 0.17 0.34 n =200 Matrix reform ulation 0.55 1.12 1.86 3.38 n =200 SVD 0.02 0.11 0.26 0.53 n =400 Matrix reform ulation 2.15 4.32 7.1 12.65 n =400 SVD 0.03 0.16 0.37 0.89 n =800 Matrix reform ulation 10.06 19.27 30.54 error n =800 SVD 0.05 0.25 0.58 1.79 T able 1: Median computation time (in seconds) of t w o PCAMIX rotation pro cedures: the one based on Kiers’ matrix reformulation and the one based on the SVD app oac h. p =10 p =50 p =100 p =200 n =50 2.9 2.0 1.8 1.6 n =100 8.7 3.8 3.3 3.0 n =200 23.2 10.3 7.0 6.4 n =400 69.4 27.7 19.0 14.2 n =800 214.1 77.4 52.9 error T able 2: Ratio betw een the median computation time of the t w o rotation pro cedures (Matrix reform ulation/SVD). n and p are great ( n = 800 and p = 200) an error o ccurs with the rotation pro cedure based on Kiers’matrix refromulation. The maxim um capacit y of memory size of the computer was reac hed in that case. This error o ccurs during the calculation of the p matrices S j of size n × n . This confirms the computational efficiency of the prop osed SVD approach. 4.2 A real data application This real data application illustrates the in terest of rotation in MCA. A fo od habits survey 1 w as carried out in 1999 on studen ts living in the region “Aquitaine” in south of w est F rance. W e focus on the answers of 2885 studen ts to 12 binary questions concerning their consump- tion at breakfeast (coffe, cereals, eggs...). The PCAMIX metho d (equiv alent here to MCA) has been applied to this dataset and the first 4 comp onen ts hav e b een rotated. In Figure 1 the asso ciation of the v ariables with the first tw o comp onen ts is ob viously easier after rotation. This rotation of the first four comp onen ts leads in T able 3 to clear asso ciations b et w een the binary v ariables: coffe is asso ciated with milk, eggs with cheese and deli, bread with jam and cereals with pure milk. The effect of the rotation on the ob jects’ scores and on the categories’ co ordinates can also b e visualized in Figures 2 and 3. 1 This survey was realized by the Bordeaux School of Public Health (Institut de San t´ e Publique, d’Epid ´ emiologie et de D ´ ev elopp emen t - ISPED) 12 The interpretation rule asso ciated with the barycentric prop ert y remains true after rotation. Befor e r otation After r otation 1 2 3 4 1 2 3 4 coffe 0.23 0.22 0.06 0.05 0.49 0.00 0.00 0.07 tea 0.05 0.01 0.06 0.18 0.05 0.02 0.06 0.17 milk 0.15 0.16 0.08 0.00 0.37 0.00 0.01 0.01 milk chocolate 0.43 0.18 0.01 0.06 0.62 0.00 0.01 0.05 pure milk 0.02 0.00 0.05 0.40 0.01 0.00 0.02 0.44 cheese 0.18 0.23 0.01 0.00 0.00 0.42 0.00 0.00 deli 0.20 0.27 0.00 0.05 0.00 0.51 0.00 0.01 eggs 0.20 0.37 0.00 0.01 0.00 0.58 0.00 0.00 jam 0.06 0.02 0.49 0.02 0.00 0.00 0.59 0.00 honey 0.00 0.05 0.14 0.20 0.00 0.02 0.20 0.16 bread 0.11 0.01 0.45 0.00 0.01 0.01 0.53 0.03 cereals 0.01 0.01 0.12 0.22 0.05 0.01 0.04 0.27 T able 3: Correlation ratio (squared loadings) b et ween the v ariables and the first 4 comp o- nen ts b efore and after rotation 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 Correlation ratios before r otation Dimension 1 Dimension 2 cafe tea milk milkchoc puremilk cheese deli eggs jam honey bread cereals 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 Correlation ratios after rotation Dimension 1 after rotation Dimension 2 after rotation cafe tea milk milkchoc puremilk cheese deli eggs jam honey bread cereals Figure 1: Plots of the correlation ratios b et w een the v ariables and the tw o first comp onen ts b efore rotation and after rotation. Note that for binary v ariables MCA and PCA lead to equiv alent ob ject scores and squared loadings (correlations are equal to correlation ratio). Then considering the data as quanti- tativ e in PCAMIX (equiv alen t to PCA in that case) gives the same results except for the plots of the categories whic h are not defined in that case. 5 Conclusion W e ha ve giv en in this pap er a SVD based formulation of the PCAMIX metho d. This new form ulation leads to an efficient pro cedure for v arimax rotation in PCAMIX where a direct solution for the optimal angle of rotation θ has b een obtained. The n umerical results ha v e shown on simulations that this pro cedure is computationally more efficien t than the 13 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 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● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −2 0 2 4 6 8 −2 0 2 4 6 8 Scores before r otation Dimension 1 Dimension 2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 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● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −2 0 2 4 6 8 −2 0 2 4 6 8 Scores after rotation Dimension 1 after rotation Dimension 2 after rotation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 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2792 2793 2794 2795 2796 2797 2798 2799 2800 2801 2802 2803 2804 2805 2806 2807 2808 2809 2810 2811 2812 2813 2814 2815 2816 2817 2818 2819 2820 2821 2822 2823 2824 2825 2826 2827 2828 2829 2830 2831 2832 2833 2834 2835 2836 2837 2838 2839 2840 2841 2842 2843 2844 2845 2846 2847 2848 2849 2850 2851 2852 2853 2854 2855 2856 2857 2858 2859 2860 2861 2862 2863 2864 2865 2866 2867 2868 2869 2870 2871 2872 2873 2874 2875 2876 2877 2878 2879 2880 2881 2882 2883 2884 2885 Figure 2: Plots of the (standardized) scores of the 2885 students on the first t w o components b efore and after rotation. −1 0 1 2 3 4 5 −1 0 1 2 3 4 5 Categories before r otation Dimension 1 Dimension 2 cafe=no cafe=y es tea=no tea=yes milk=no milk=yes milkchoc=no milkchoc=yes puremilk=no puremilk=yes cheese=no cheese=yes deli=no deli=yes eggs=no eggs=yes jam=no jam=yes honey=no honey=yes bread=no bread=yes cereals=no cereals=yes −1 0 1 2 3 4 5 −1 0 1 2 3 4 5 Categories after rotation Dimension 1 after rotation Dimension 2 after rotation cafe=no cafe=y es tea=no tea=yes milk=no milk=yes milkchoc=no milkchoc=yes puremilk=no puremilk=yes cheese=no cheese=yes deli=no deli=yes eggs=no eggs=yes jam=no jam=yes honey=no honey=yes bread=no bread=yes cereals=no cereals=yes Figure 3: Plots of the category co ordinates on the first t w o comp onen ts b efore and after rotation. pro cedure based on Kiers’ matrix reformulation. The n umerical results hav e also sho wn on a real data application the in terest of this algorithm in the con text of MCA with graphical represen tations of b oth v ariables and categories after rotation. The PCAMIX pro cedure as w ell as the rotation pro cedure hav e b een implemented in the R pack age “PCAmixdata”. 14 App endix Define the complex n umbers: a s def = a s, 1 + ia s, 2 , ˜ a s def = e − iθ a s = ˜ a s, 1 + i ˜ a s, 2 , t j def = P s ∈ I j a 2 s = u j + iv j , ˜ t j def = P s ∈ I j ˜ a 2 s = e − 2 iθ t j = ˜ u j + i ˜ v j , where ˜ a s, 1 , ˜ a s, 2 ha v e b een defined in (14), u j , v j in (18), and where ˜ u j , ˜ v j are giv en b y the same form ula as u j , v j , but with a tilde ov er a s, 1 , a s, 2 . W e introduce no w a complex-v alued v arimax function F ( θ ) of the rotation angle θ b y: F ( θ ) def = p p X j =1 ˜ t 2 j − ( p X j =1 ˜ t j ) 2 = e − 4 iθ F (0) , where F (0) is simply obtained b y suppressing the tilde in F ( θ ). Developmen t of F ( θ ) gives : F ( θ ) = p p X j =1 ( ˜ u 2 j − ˜ v 2 j ) − ( p X j =1 ˜ u j ) 2 + ( p X j =1 ˜ v j ) 2 | {z } g ( θ ) + 2 i  p p X j =1 ˜ u j ˜ v j − p X j =1 ˜ u j p X j =1 ˜ v j } | {z } i h ( θ ) (21) Comparison with the form ula (16), (17), (18) defining b, a, ρ, ψ shows that : F (0) = g (0) + ih (0) = b + ia = ρ e iψ . Hence : F ( θ ) = ρ e i ( ψ − 4 θ ) = ρ  cos(4 θ − ψ ) − i sin(4 θ − ψ )  . But deriv ation of the v arimax function f ( θ ) defined in (13) giv es, using the fact that a 0 s, 1 ( θ ) = a s, 2 ( θ ) and a 0 s, 2 ( θ ) = − a s, 1 ( θ ) : pf 0 ( θ ) = 2  p p X j =1 ˜ u j ˜ v j − p X j =1 ˜ u j p X j =1 ˜ v j  = h ( θ ) = − ρ sin(4 θ − ψ ) , (22) = a cos 4 θ − b sin 4 θ , (23) and (22) prov es (15) b y in tegration. References de Leeu w, J., and Pruzansky , S., (1978), A new computational metho d to fit the weigh ted Euclidean distance mo del, Psychometrika , 43 , 479-490. Escofier, B., (1979), T raitemen t sim ultan´ e de v ariables qualitatives et quantitativ es en anal- yse factorielle [Sim ultaneous treatment of qualitativ e and quantitativ e v ariables in factor analysis], Cahiers de l’A nalyse des Donn ´ ees , 4 , 137-146. 15 Jennric h, R.I., (2001), A simple general pro cedure for orthogonal rotation, Psychometrika , 66 (2), 289-306. Kaiser, H.F., (1958), The v arimax criterion for analytic rotation in factor analysis, Psy- chometrika , 23 (3), 187-200. Kiers, H.A.L., (1991), Simple structure in Comp onen t Analysis T ec hniques for mixtures of qualitativ e and quantitativ e v ariables, Psychometrika , 56 , 197-212. Neudec k er, H., (1981), On the matrix formulation of Kaiser’s v arimax criterion, Psychome- trika , 46 , 343-345. P ag ` es, J., (2004), Analyse F actorielle de donn ´ ees mixtes [F actor Analysis for Mixed Data], R evue de Statistique Appliqu ´ ee , 52 (4), 93-11. ten Berge, J.M.F., (1984), A join t treatmen t of v arimax rotation and the problem of diag- onalizing symmetric matrices sim ultaneously in the least-squares sense, Psychometrika , 49 , 347-358. ten Berge, J.M.F., (1995), Suppressing p erm utations or rigid planar rotations: a remedy against nonoptimal v arimax rotations, Psychometrika , 46 60, 437-446. 16