The Power STATIS-ACT method Jacques B enass eni, Mohammed Bennani - PowerPoint PPT Presentation

Correspondence Analysis and Related Methods - CARME 2011 The Power STATIS-ACT method Jacques B´ enass´ eni, Mohammed Bennani Dosse Universit´ e Rennes 2, UMR 6620 February 2011, Rennes, France

Contents 1. Data 2. STATIS-ACT method 3. A s -power based Criterion 4. The special case s = 1 5. The general case s > 1 6. Low rank compromises 7. Comparisons

Data ◮ K data matrices X 1 , . . . , X K ◮ Each X k is a n × p k matrix : measurements of the same units on p k different variables. ◮ D = diag( π 1 , . . . , π n ) : diagonal matrix of weights. ◮ Q k : positive definite matrices (metrics). ◮ We assume that D = I n , Q k = I p k and X k centered.

STATIS-ACT method ◮ The STATIS-ACT method is a generalization of principal component analysis used to study several data tables measured on the same observation units (or variables). ◮ The goal of this method is to analyze the relationship between these data tables (Interstructure step) and to combine them into a common structure, called a compromise . ◮ The principal components derived from the compromise solution are analyzed together with the original variables (intrastructure step).

STATIS-ACT method ◮ In STATIS, the individual association matrices W k = X k X ′ k , k = 1 , . . . , K play a central role. ◮ W k contains all the information about the multidimensional structure in the data matrix X k . ◮ The use of association matrices W k instead of X k leads to simplification of computations as it obviates the determination of rotations (GPA, Gower 1975).

STATIS-ACT method ◮ Basic idea of STATIS : derive an optimal set of weights α k for computing a compromise solution K � � W = α k W k k =1 ◮ where the α k , k = 1 , . . . , K maximize the criterion � � 2 K � trace( � WW k ) k =1 ◮ subject to the constraints α k � 0 and K � α 2 k = 1 . k =1

STATIS-ACT method ◮ Define the matrix C = ( C k ℓ ) where p k p ℓ � � 2 � � C k ℓ = trace( W k W ℓ ) = cov( x ki , x ℓ j ) i =1 j =1 ◮ Solution of STATIS : the vector a = ( α 1 , . . . , α K ) ′ is the the eigenvector of C corresponding to the largest eigenvalue of this matrix. ◮ Since C � 0 , the vecteur a can be choosen with all its elements nonnegative (Perron-Frobenius theorem).

A s -power Criterion ◮ Since trace( � WW k ) � 0 there is no specific reason for considering in STATIS-ACT the criterion � � 2 K � trace( � WW k ) k =1 rather than any other power s based criterion ( s � 1) : � � s K � trace( � WW k ) k =1 ◮ We investigate the effect of varying the power s on the optimal weights in the compromise solution.

The special case s = 1 ◮ The most simple choice for s . ◮ Simple solution and straighforward interpretation. ◮ It is possible to find a solution to ”the problem of the low rank compromise” when considering a criterion based on s = 1. ◮ Close analogy between the compromise obtained from the s = 1 criterion and the first principal component derived a PCA.

The special case s = 1 ◮ Maximize the criterion � � K � trace( � WW k ) k =1 subject to the constraints α k � 0 and K � α 2 k = 1 . k =1 ◮ The solution is given by Ce � Ce � where e = (1 , . . . , 1) ′ . a = ◮ Clearly a � 0 .

The special case s = 1 ◮ What happens if the constraint on α k is changed to � K k =1 α k = 1 ? ◮ Standard linear program where the constraint set is a polyhedron (Simplex) ◮ The optimal compromise solution is one of the initial matrices W k !

The general case s > 1 ◮ Motivation : One of the most interesting feature of the classical STATIS-ACT method based on power s = 2 is that the corresponding weights α k represent principal agreement between the given tables. ◮ A data table which is not in agreement with the others has a low weight. ◮ What happens if s > 1 ?

The general case s > 1 ◮ Maximizing K � � s � trace( � WW k ) k =1 ◮ subject to � K k =1 α 2 k = 1 and α k � 0 ◮ is equivalent to maximizing � � s K � f ( a ) = Ca k k =1 subject to a ′ a = 1. ◮ f is convex and differentiable function on R K + .

The general case s > 1 ◮ Iterative solution. ◮ Algorithm: ◮ Choose a (0) (randomly such that � a (0) � = 1). ν = 0 ◮ Repeat until convergence ◮ ν := ν + 1. � Ca ( ν ) � s − 1 ◮ Calculate z = ( z 1 , . . . , z K ) ′ where z k = . k ◮ set a ( ν +1) = Cz � Cz � ◮ End.

The general case s > 1 ◮ We prove monotone convegence. ◮ When s = 2, this algorithm is simply the power method used in the numerical calculation of the dominant eigenvector of C . ◮ Convergence to a global maximum is not necessarily guaranteed (multistart, ...) ◮ Algebraic solution when s tends to infinity.

Low rank compromises ◮ The configuration of observations given by the compromise solution is derived from principal components which are the eigenvectors of � W . ◮ In practice, interest mainly focuses on graphical representations based on the first R principal components (with R = 2 in most situations). ◮ However if � W corresponds to the maximum of the criterion of interest (with general power s ), this point is no longer true when considering the approximation of rank R of � W . ◮ For the s = 1 case, we can derive an algebraic solution to the low rank compromise of the form � R ℓ =1 u ℓ u ′ ℓ with u ℓ u ′ j = 0.

Applications ◮ Real data sets (from sensory analysis, ecology) ◮ Comparison of the weights EN (1) EN (5) EN ( ∞ ) IN (1) IN (5) IN ( ∞ ) Data set 1 0.002 0.006 0.074 0.002 0.005 0.049 2 0.026 0.053 0.084 0.023 0.049 0.087 3 0.008 0.026 0.133 0.005 0.017 0.081 4 0.015 0.054 0.238 0.012 0.046 0.167 5 0.087 0.170 0.292 0.052 0.079 0.139 6 0.024 0.045 0.182 0.015 0.025 0.111 7 0.212 0.302 0.474 0.185 0.235 0.284 Table: Comparison of a (2) and a ( s ) for s = 1 , 5 , ∞ . where EN ( s ) = � a (2) − a ( s ) � 2 and IN ( s ) = � a (2) − a ( s ) � ∞ .

Applications ◮ Monte-Carlo simulations ◮ Comparison of the weights a ( s ) s = 1 s = 2 s = 5 s = ∞ α ( s ) 0.489 0.386 0.239 0.223 1 (0.031) (0.092) (0.124) (0.108) α ( s ) 0.622 0.654 0.684 0.693 2 (0.023) (0.034) (0.033) (0.043) α ( s ) 0.610 0.643 0.676 0.674 3 (0.024) (0.036) (0.036) (0.046) Table: Comparison of a ( s ) for s = 1 , 2 , 5 , ∞ .

Conclusions ◮ The weights attached to the compromise solution for s = 1 are in general fairly close to those obtained in the usual STATIS-ACT method. ◮ The compromise obtained with the s = 1 approach simply requires elementary operations whereas the usual compromise needs calculation of the dominant eigenvector of C . ◮ The power parameter s is is relation with robustness of the compromise solution. ◮ When there are some ”outlying” matrices W k , increasing the power parameter s in the generalized criterion downloads the influence of these matrices on the compromise, thus enhancing the well known ”majority effect” of the STATIS method.

References ◮ B´ enass´ eni, J. & Bennani Dosse, M., 2010. Analyzing multiset data by the Power STATIS-ACT method, Advances in Data Analysis and Classification, to appear. ◮ Lavit, C., Escoufier, Y., Sabatier, R. & Traissac, P., 1994. The ACT (STATIS method). Computational Statistics & Data Analysis, 18, 97-117. ◮ Lavit, C., 1985. Application de la m´ ethode STATIS. Statistique et Analyse des donn´ ees, 10(1), 103-116. ◮ Gower, J.C., 1975. Generalised Procrustes Analysis. Psychometrika, 40, 33-51.

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