Cadima, Jorge

Cerdeira, Jorge Orestes

Duarte Silva, A. P.

Minhoto, Manuel

English

Repositório Institucional da Universidade Católica Portuguesa

Subselect 0.9-99:  Selecting variable subsets in multivariate linear models(1) FEG/CEGE – UNIV. CATÓLICA PORTUGUESA – C.R. PORTO(2) I.S. AGRONOMIA – UNIV. TÉCNICA DE LISBOA(3) DEP. MATEMÁTICA – UNIVERSIDADE DE ÉVORAA. PEDRO DUARTE SILVA(1)(*)JORGE CADIMA(2)MANUEL MINHOTO(3)JORGE ORESTES CERDEIRA(2)(*) Supported by: FEDER / POCI 2010Subselect 0.9-99THE PROBLEM: Finding a k-variable subset that is a good  surrogate for a full p-variable data setCONTEXT:•  Exploratory data analysis•  Multivariate Linear Models(Cadima, Cerdeira, Duarte Silva and Minhoto -- useR! 2004)– Subselect 0.1-- 0.9– Subselect 0.9-99 A LINEAR HYPOTHESIS FRAMEWORKX  =  A  Ψ +  U•  SELECT COLUMNS OF X IN ORDER TO EXPLAIN H1PARTICULAR CASES:H0:  C Ψ =  0CANONICAL CORRELATION ANALYSISLINEAR DISCRIMINANT ANALYSISMULTI-WAY MANOVA/MANCOVA EFFECTSA = [1 | Y] C = [0 | I ]  A = [1g ] Ψ = [µg] ⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−=1...01............0...11CSubselect 0.9-99Subselect 0.9-99Comparison Criteria: Multivariate Indices21ccr1/rr1i2i2 )ccr(11 ⎟⎟⎠⎞⎜⎜⎝⎛ −−= ∏=τ( )∑=−−−= ri 112112iccrrςrccrr1i2i2∑==ξ)H(TEigvalccr 1i2i−= T =  X’ (I - Pω) X H = X’ (PΩ - Pω) XΩ = R(A) ω = R(A)  ∩ N(C) r = dim(Ω) - dim(ω) ( max ccr12 ⇔ max Roy λ1 ) ( max ς2 ⇔ max Lawley-Hotelling trace )( max τ2 ⇔ min Wilks Λ ) ( max ξ2 ⇔ max Bartlei-Pillai trace )The Subselect PackageSearch routines for (combinatorial) criteria optimizationannealExact Algorithm:leaps - based on Furnival and Wilson´s leaps andbounds algorithm for linear regression- viable with up to 30 - 35 original variablesHeuristics:geneticimprove- simulated annealing- genetic algorithm- restricted local improvementSubselect 0.9-99Subselect in Multivariate Linear ModelsPrincipal arguments of search routines :rmat - Total SSCP data matrix (T)criterionkmin, kmax - minimum and maximum subsetdimensionalities soughtH - Effect SSCP data matrix - Expected rank of the H matrix- “ccr12”, “tau2”, “xi2” or “zeta2”Subselect 0.9-99Subselect in Multivariate Linear ModelsOther arguments :- Tuning parameters for heuristics- Maximum time allowed for exact search- Variables forcibly included or excluded in theselected subsets- Number of solutions by subset dimensionality- Numerical tolerance for detecting singular ornon-symmetrical matricesSubselect 0.9-99Subselect in Multivariate Linear ModelsAuxiliary functions:lmHmat - creates H and mat matrices for linearregression/canonical correlation analysisldaHmat - creates H and mat matrices for lineardiscriminant analysisglhHmat - creates H and mat matrices for ananalysis based on a linear hypothesis specified by the user Subselect 0.9-99Subselect in Multivariate Linear ModelsAuxiliary functions :trim.matrix - deletes rows and columns of singularor ill-conditioned matricesccr12.coef, tau2.coefzeta2.coef, xi2.coef- computes a comparisoncriterion for a subsetsupplied by the user- until all linear dependencies (perfector almost perfect) are removedSubselect 0.9-99Example:  Hubbard Brook Forest soil dataDescription:Al  - Aluminum58 pits were analyzed before (1983) and after (1986)harvesting (83-84) trees larger than a minimum diameterContinuous variables: gr/m2 of exchangeable cationsCa  - CalciumMg  - MagnesiumK  - PotassiumNa - SodiumSource:  Morrison (1990)Subselect 0.9-99Example:  Hubbard Brook Forest soil dataF  - Forest TypeFactors:1  - Spruce- firSource:  Morrison (1990)Factor levels:2  - High elevation hardwood3  - Low elevation hardwoodD  - Logging Disturbance0  - Uncut forest1  - Cut, undisturbed by machinery2  - Cut, disturbed by machineryYear 1983  or  1986Subselect 0.9-99Example:  Hubbard Brook Forest soil dataReading and preparing the data:Source:  Morrison (1990)> library(subselect)> HubForest <- read.table("Hubbard Brook.txt“ ,header=T,col.names=c("Pit","F","D","Al","Ca","Mg","K","Na","Year"),colClasses=c("factor","factor","factor","numeric","numeric","numeric","numeric","numeric","factor") )Analysis #1: Explaining the levels of calcium> Hmat <- lmHmat(Ca ~ F*D + Al + Mg+ K + Na ,HubForest)> colnames(Hmat$mat)> leaps(Hmat$mat,H=Hmat$H,r=1,nsol=3)Subselect 0.9-99Example:  Hubbard Brook Forest soil dataSource:  Morrison (1990)Analysis #2: Looking for combinations of Forest type and Disturbance that best explain the nutrient levels> Hmat <- lmHmat(cbind(Al,Ca,Mg,K,Na) ~ F*D,HubForest)> colnames(Hmat$mat)> leaps(Hmat$mat,H=Hmat$H,r=5,criterion=“tau2”,nsol=3)Analysis #3: Finding which subsets of nutrients were most affected by the harvesting in 1983-84> Hmat <- ldaHmat(Year ~  Al + Ca + Mg + K + Na , HubForest)> leaps(Hmat$mat,H=Hmat$H,r=1,nsol=3)Subselect 0.9-99Example:  Hubbard Brook Forest soil dataSource:  Morrison (1990)Analysis #4: Finding which subsets of nutrients are most affected by interactions between harvesting and logging disturbances, after controlling for the effect of forest type>  C <- matrix(0.,2,8)>  C[1,7] = C[2,8] = 1.>   Hmat <- glhHmat(cbind(Al,Ca,Mg,K,Na) ~ D*Year + F, C,HubForest)>   leaps(Hmat$mat,H=Hmat$H,r=2, criterion="tau2",nsol=3,tolsym=1E-10)Subselect 0.9-99ReferencesCadima J, Cerdeira JO and  Minhoto M (2004). Computational Aspects of Algorithms for Variable Selection in the Context of PrincipalComponents. Computational Statistics and Data Analysis 47: 225-236.Cadima J, Cerdeira JO, Duarte Silva AP and Minhoto M (2004). The Subselect Package; Selecting Variable Subsets in an Exploratory Data Analysis. useR! 2004. 1rst Internatinal R User Conference. Vienna, Austria. Duarte Silva, A.P. (2001). Efficient Variable Screening for Multivariate Analysis. Journal of Multivariate Analysis 76, 35-62.Furnival, G.M. & Wilson, R.W. (1974). Regressions by Leaps and Bounds. Technometrics 16: 499-511.Morrison D.F. (1990). Multivariate Statistical Methods, 3rd ed., McGraw-Hill. New York, NY. 

Subselect 0.9-99: Selecting variable subsets in multivariate linear models

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Subselect 0.9-99: Selecting variable subsets in multivariate linear models

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