# Optimality Criteria These are some default design criteria to be minimized. There is a table in the details section that gives the formula for each design criterion and describes their usage. Note that the inputs for these functions come in 3 syntax flavors, namely Type-X, Type-D and Type-K. Users can define and use their owm design criteria as long as it has the Type-X syntax as shown with the examples. ```r AOPT(Train, Test, P, lambda = 1e-05, C=NULL) CDMAX(Train, Test, P, lambda = 1e-05, C=NULL) CDMAX0(Train, Test, P, lambda = 1e-05, C=NULL) CDMAX2(Train, Test, P, lambda = 1e-05, C=NULL) CDMEAN(Train, Test, P, lambda = 1e-05, C=NULL) CDMEAN0(Train, Test, P, lambda = 1e-05, C=NULL) CDMEAN2(Train, Test, P, lambda = 1e-05, C=NULL) CDMEANMM(Train, Test, Kinv,K, lambda = 1e-05, C=NULL, Vg=NULL, Ve=NULL) DOPT(Train, Test, P, lambda = 1e-05, C=NULL) EOPT(Train, Test, P, lambda = 1e-05, C=NULL) GAUSSMEANMM(Train, Test, Kinv, K, lambda = 1e-05, C=NULL, Vg=NULL, Ve=NULL) GOPTPEV(Train, Test, P, lambda = 1e-05, C=NULL) GOPTPEV2(Train, Test, P, lambda = 1e-05, C=NULL) PEVMAX(Train, Test, P, lambda = 1e-05, C=NULL) PEVMAX0(Train, Test, P, lambda = 1e-05, C=NULL) PEVMAX2(Train, Test, P, lambda = 1e-05, C=NULL) PEVMEAN(Train, Test, P, lambda = 1e-05, C=NULL) PEVMEAN0(Train, Test, P, lambda = 1e-05, C=NULL) PEVMEAN2(Train, Test, P, lambda = 1e-05, C=NULL) PEVMEANMM(Train, Test, Kinv,K, lambda = 1e-05, C=NULL, Vg=NULL, Ve=NULL) dist_to_test(Train, Test, Dst, lambda, C) dist_to_test2(Train, Test, Dst, lambda, C) neg_dist_in_train(Train, Test, Dst, lambda, C) neg_dist_in_train2(Train, Test, Dst, lambda, C) ``` ## Arguments - `Train`: vector of identifiers for individuals in the training set - `Test`: vector of identifiers for individuals in the test set - `P`: (Only for Type-X) $n \times k$ matrix of the first PCs of the predictor variables. The matrix needs to have union of the identifiers of the candidate and test individuals as rownames. - `Dst`: (Only for Type-D) $n \times n$ symmetric distance matrix with row and column names. - `Kinv`: (Only for Type-K) $n \times n$ symmetric matrix (inverse of the relationship matrix K between n individuals) with row and column names. - `K`: (Only for Type-K) $n \times n$ symmetric matrix (the relationship matrix K between n individuals). - `lambda`: scalar shrinkage parameter ($\lambda>0$). - `C`: Contrast Matrix. - `Vg`: (Only for PEVMEANMM) covariance matrix between traits generated by the relationship K (multi-trait version). - `Ve`: (Only for PEVMEANMM) residual covariance matrix for the traits (multi-trait version). ## Returns value of the criterion. ## Details |||||||||||||||||||||| |:--|:--|:--|:--|:--|:--|:--|:--|:--|:--|:--|:--|:--|:--|:--|:--|:--|:--|:--|:--|:--| |criterion name|formula|Type| |AOPT|$trace[C(P'_{Train}P_{Train}+lambda*I)^{-1}C']$|X| |||| |CDMAX|$max[diag(CP_{Test}(P'_{Train}P_{Train}+lambda*I)^{-1}P'_{Test}C')/$|X| ||$diag(CP_{Test}P'_{Test}C')]$|| |CDMAX0|$max[diag(CP_{Train}(P'_{Train}P_{Train}+lambda*I)^{-1}P'_{Train}C')/$|X| ||$diag(CP_{Train}P'_{Train}C')]$|| |CDMAX2|$max[diag(CP_{Test}(P'_{Train}P_{Train}+lambda*I)^{-1}P'_{Train}P_{Train}$|X| ||$(P'_{Train}P_{Train}+lambda*I)^{-1}P'_{Test}C')/diag(CP_{Test}P'_{Test}C')]$|| |CDMEAN|$mean[diag(CP_{Test}(P'_{Train}P_{Train}+lambda*I)^{-1}P'_{Test}C')/$|X| ||$diag(CP_{Test}P'_{Test}C')]$|| |CDMEAN0|$mean[diag(CP_{Train}(P'_{Train}P_{Train}+lambda*I)^{-1}P'_{Train}C')/$|X| ||$diag(CP_{Train}P'_{Train}C')]$|| |CDMEAN2|$mean[diag(CP_{Test}(P'_{Train}P_{Train}+lambda*I)^{-1}P'_{Train}P_{Train}$|X| ||$(P'_{Train}P_{Train}+lambda*I)^{-1}P'_{Test}C')/diag(CP_{Test}P'_{Test}C')] $|| |CDMEANMM|$-mean[diag(CZ_{Test}(K-lambda*(Z_{Train}'MZ_{Train}+\lambda*Kinv)^{-1}Z_{Test}'C')/$|K| ||$(diag(CZ_{Test}KZ_{Test}'C'))]$|| |DOPT|$logdet(C(P'_{Train}P_{Train}+lambda*I))^{-1}C'$|X| |||| |EOPT|$max(eigenval(C(P'_{Train}P_{Train}+lambda*I))^{-1}C'))$|X| |||| |GAUSSMEANMM|$-mean(diag(Z_{Test}KZ_{Test}'-$|K| ||$Z_{Test}KZ_{Train}'(Z_{Train}KZ_{Train}'+\lambda*I_{ntrain})^{-1}Z_{Train}KZ_{Test}')$|| |GOPTPEV|$max(eigenval(CP_{Test}(P_{Train}'P_{Train}+\lambda*I_{ntrain})^{-1}P_{Test}'C'))$|X| |||| |GOPTPEV2|$mean(eigenval(CP_{Test}(P_{Train}'P_{Train}+\lambda*I_{ntrain})^{-1}P_{Test}'C'))$|X| |||| |PEVMAX|$max(diag(CP_{Test}(P'_{Train}P_{Train}+lambda*I)^{-1}P'_{Test}C'))$|X| |||| |PEVMAX0|$max(diag(CP_{Train}(P'_{Train}P_{Train}+lambda*I)^{-1}P'_{Train}C'))$|X| |||| |PEVMAX2|$max[diag(CP_{Test}(P'_{Train}P_{Train}+lambda*I)^{-1}P'_{Train}P_{Train}$|X| ||$(P'_{Train}P_{Train}+lambda*I)^{-1}P'_{Test}C']$|| |PEVMEAN|$mean(diag(CP_{Test}(P'_{Train}P_{Train}+lambda*I)^{-1}P'_{Test}C'))$|X| |||| |PEVMEAN0|$mean(diag(CP_{Train}(P'_{Train}P_{Train}+lambda*I)^{-1}P'_{Train}C'))$|X| |||| |PEVMEAN2|$mean[diag(CP_{Test}(P'_{Train}P_{Train}+lambda*I)^{-1}$|X| ||$P'_{Train}P_{Train}(P'_{Train}P_{Train}+lambda*I)^{-1}P'_{Test}C']$|| |PEVMEANMM|$mean(diag(CZ_{test}(Ztrain'MZtrain+lambda*Kinv)^{-1}Ztest'C')))$|K| |dist_to_test|maximum distance from training set to test set based on Dst|D| |dist_to_test2|mean distance from training set to test set based on Dst|D| |neg_dist_in_train|negative of minimum distance between pairs in the training set based on Dst|D| |neg_dist_in_train2|negative of mean distance between distinct pairs in the training set based on Dst|D| ## Examples ```r ## Not run: #Examples to new criterion: #1- PEVmax STPGAUSERFUNC<-function(Train,Test, P, lambda=1e-6, C=NULL){ PTrain<-P[rownames(P)%in%Train,] PTest<-P[rownames(P)%in%Test,] if (length(Test)==1){PTest=matrix(PTest, nrow=1)} if (!is.null(C)){ PTest<-C%*%PTest} PEV<-PTest%*%solve(crossprod(PTrain)+lambda*diag(ncol(PTrain)),t(PTrain)) PEVmax<-max(diag(tcrossprod(PEV))) return(PEVmax) } ######Here is an example of usage data(iris) #We will try to estimate petal width from #variables sepal length and width and petal length. X<-as.matrix(iris[,1:4]) distX<-as.matrix(dist(X)) rownames(distX)<-colnames(distX)<-rownames(X)<-paste(iris[,5],rep(1:50,3),sep="_" ) #test data 25 iris plants selected at random from the virginica family, #candidates are the plants in the setosa and versicolor families. candidates<-rownames(X)[1:100] test<-sample(setdiff(rownames(X),candidates), 25) #want to select 25 examples using the criterion defined in STPGAUSERFUNC #Increase niterations and npop substantially for better convergence. ListTrain<-GenAlgForSubsetSelection(P=distX,Candidates=candidates, Test=test,ntoselect=25,npop=50, nelite=5, mutprob=.8, niterations=30, lambda=1e-5, errorstat="STPGAUSERFUNC", plotiters=TRUE) ## End(Not run) ``` ## Author(s) Deniz Akdemir

CRITERIA function