# Generate Fungible Correlation Matrices Generate fungible correlation matrices. For a given vector of standardized regression coefficients, Beta, and a user-define R-squared value, Rsq, find predictor correlation matrices, R, such that Beta' R Beta = Rsq. The size of the smallest eigenvalue (Lp) of R can be defined. ```r fungibleR(R, Beta, Lp = 0, eps = 1e-08, Print.Warnings = TRUE) ``` ## Arguments - `R`: A p x p predictor correlation matrix. - `Beta`: A p x 1 vector of standardized regression coefficients. - `Lp`: Controls the size of the smallest eigenvalue of RstarLp. - `eps`: Convergence criterion. - `Print.Warnings`: Logical, default = TRUE. When TRUE, convergence failures are printed. ## Returns - **R**: Any input correlation matrix that satisfies Beta' R Beta = Rsq. - **Beta**: Input vector of std reg coefficients. - **Rstar**: A random fungible correlation matrix. - **RstarLp**: A fungible correlation matrix with a fixed minimum eigenvalue (RstarLp can be PD, PSD, or ID). - **s**: Scaling constant for Rstar. - **sLp**: Scaling constant for RstarLp. - **Delta**: Vector in the null space of vecp(Beta Beta'). - **Q**: Left null space of Beta. - **FrobNorm**: Frobenius norm ||R - Rstar||_F. - **FrobNormLp**: Frobenius norm ||R - RstarLp||_F given random Delta. - **converged**: An integer code. 0 indicates successful completion. ## Examples ```r library(fungible) ## ===== Example 1 ===== ## Generate 5 random PD fungible R matrices ## that are consistent with a user-defined predictive ## structure: B' Rxx B = .30 set.seed(246) ## Create a 5 x 5 correlation matrix, R, with all r_ij = .25 R.ex1 <- matrix(.25, 5, 5) diag(R.ex1) <- 1 ## create a 5 x 1 vector of standardized regression coefficients, ## Beta.ex1 Beta.ex1 <- c(-.4, -.2, 0, .2, .4) cat("\nModel Rsq = ", t(Beta.ex1) %*% R.ex1 %*% Beta.ex1) ## Generate fungible correlation matrices, Rstar, with smallest ## eigenvalues > 0. Rstar.list <- list(rep(99,5)) i <- 0 while(i <= 5){ out <- fungibleR(R = R.ex1, Beta = Beta.ex1, Lp = 1e-8, eps = 1e-8, Print.Warnings = TRUE) if(out$converged==0){ i <- i + 1 Rstar.list[[i]] <- out$Rstar } } ## Check Results cat("\n *** Check Results ***") for(i in 1:5){ cat("\n\n\n+++++++++++++++++++++++++++++++++++++++++++++++++") cat("\nRstar", i,"\n") print(round(Rstar.list[[i]], 2),) cat("\neigenvalues of Rstar", i,"\n") print(eigen(Rstar.list[[i]])$values) cat("\nBeta' Rstar",i, "Beta = ", t(Beta.ex1) %*% Rstar.list[[i]] %*% Beta.ex1) } ## ===== Example 2 ===== ## Generate a PD fungible R matrix with a fixed smallest ## eigenvalue (Lp). ## Create a 5 x 5 correlation matrix, R, with all r_ij = .5 R <- matrix(.5, 5, 5) diag(R) <- 1 ## create a 5 x 1 vector of standardized regression coefficients, Beta, ## such that Beta_i = .1 for all i Beta <- rep(.1, 5) ## Generate fungible correlation matrices (a) Rstar and (b) RstarLp. ## Set Lp = 0.12345678 so that the smallest eigenvalue (Lp) of RstarLp ## = 0.12345678 out <- fungibleR(R, Beta, Lp = 0.12345678, eps = 1e-10, Print.Warnings = TRUE) ## print R cat("\nR: a user-specified seed matrix") print(round(out$R,3)) ## Rstar cat("\nRstar: A random fungible correlation matrix for R") print(round(out$Rstar,3)) cat("\nCoefficient of determination when using R\n") print( t(Beta) %*% R %*% Beta ) cat("\nCoefficient of determination when using Rstar\n") print( t(Beta) %*% out$Rstar %*% Beta) ## Eigenvalues of R cat("\nEigenvalues of R\n") print(round(eigen(out$R)$values, 9)) ## Eigenvalues of Rstar cat("\nEigenvalues of Rstar\n") print(round(eigen(out$Rstar)$values, 9)) ## What is the Frobenius norm (Euclidean distance) between ## R and Rstar cat("\nFrobenious norm ||R - Rstar||\n") print( out$FrobNorm) ## RstarLp is a random fungible correlation matrix with ## a fixed smallest eigenvalue of 0.12345678 cat("\nRstarLp: a random fungible correlation matrix with a user-defined smallest eigenvalue\n") print(round(out$RstarLp, 3)) ## Eigenvalues of RstarLp cat("\nEigenvalues of RstarLp") print(eigen(out$RstarLp)$values, digits = 9) cat("\nCoefficient of determination when using RstarLp\n") print( t(Beta) %*% out$RstarLp %*% Beta) ## Check function convergence if(out$converged) print("Falied to converge") ## ===== Example 3 ===== ## This examples demonstrates how fungibleR can be used ## to generate improper correlation matrices (i.e., pseudo ## correlation matrices with negative eigenvalues). library(fungible) ## We desire an improper correlation matrix that ## is close to a user-supplied seed matrix. Create an ## interesting seed matrix that reflects a Big Five ## factor structure. set.seed(123) minCrossLoading <- -.2 maxCrossLoading <- .2 F1 <- c(rep(.6,5),runif(20,minCrossLoading, maxCrossLoading)) F2 <- c(runif(5,minCrossLoading, maxCrossLoading), rep(.6,5), runif(15,minCrossLoading, maxCrossLoading)) F3 <- c(runif(10,minCrossLoading,maxCrossLoading), rep(.6,5), runif(10,minCrossLoading,maxCrossLoading) ) F4 <- c(runif(15,minCrossLoading,maxCrossLoading), rep(.6,5), runif(5,minCrossLoading,maxCrossLoading)) F5 <- c(runif(20,minCrossLoading,maxCrossLoading), rep(.6,5)) FacMat <- cbind(F1,F2,F3,F4,F5) R.bfi <- FacMat %*% t(FacMat) diag(R.bfi) <- 1 ## Set Beta to a null vector to inform fungibleR that we are ## not interested in placing constraints on the predictive structure ## of the fungible R matrices. Beta <- rep(0, 25) ## We seek a NPD fungible R matrix that is close to the bfi seed matrix. ## To find a suitable matrix we generate a large number (e.g., 50000) ## fungible R matrices. For illustration purposes I will set Nmatrices ## to a smaller number: 10. Nmatrices<-10 ## Initialize a list to contain the Nmatrices fungible R objects RstarLp.list <- as.list( rep(0, Nmatrices ) ) ## Initialize a vector for the Nmatrices Frobeius norms ||R - RstarLp|| FrobLp.vec <- rep(0, Nmatrices) ## Constraint the smallest eigenvalue of RStarLp by setting ## Lp = -.1 (or any suitably chosen user-defined value). ## Generate Nmatrices fungibleR matrices and identify the NPD correlation ## matrix that is "closest" (has the smallest Frobenious norm) to the bfi ## seed matrix. BestR.i <- 0 BestFrob <- 99 i <- 0 set.seed(1) while(i < Nmatrices){ out<-fungibleR(R = R.bfi, Beta, Lp = -.1, eps=1e-10) ## retain solution if algorithm converged if(out$converged == 0) { i<- i + 1 ## print progress cat("\nGenerating matrix ", i, " Current minimum ||R - RstarLp|| = ",BestFrob) tmp <- FrobLp.vec[i] <- out$FrobNormLp #Frobenious Norm ||R - RstarLp|| RstarLp.list[[i]]<-out$RstarLp if( tmp < BestFrob ) { BestR.i <- i # matrix with lowest ||R - RstarLp|| BestFrob <- tmp # value of lowest ||R - RstarLp|| } } } # CloseR is an improper correlation matrix that is close to the seed matrix. CloseR<-RstarLp.list[[BestR.i]] plot(1:25, eigen(R.bfi)$values, type = "b", lwd = 2, main = "Scree Plots for R and RstarLp", cex.main = 1.5, ylim = c(-.2,6), ylab = "Eigenvalues", xlab = "Dimensions") points(1:25,eigen(CloseR)$values, type = "b", lty = 2, lwd = 2, col = "red") abline(h = 0, col = "grey") legend(legend=c(expression(paste(lambda[i]~" of R",sep = "")), expression(paste(lambda[i]~" of RstarLp",sep = ""))), lty=c(1,2), x = 17,y = 5.75, cex = 1.5, col=c("black","red"), text.width = 5.5, lwd = 2) ``` ## References Waller, N. (2016). Fungible Correlation Matrices: A method for generating nonsingular, singular, and improper correlation matrices for Monte Carlo research. Multivariate Behavioral Research. ## Author(s) Niels Waller

fungibleR function