# Smooth a NPD R matrix to PD using the Alternating Projection Algorithm Smooth a Non positive defnite (NPD) correlation matrix to PD using the Alternating Projection Algorithm with Dykstra's correction via Theory described in Higham 2002. ```r smoothAPA(R, delta = 1e-06, fixR = NULL, Wghts = NULL, maxTries = 1000) ``` ## Arguments - `R`: A p x p indefinite matrix. - `delta`: Desired value of the smallest eigenvalue of smoothed matrix, RAPA. (Default = 1e-06). - `fixR`: User-supplied integer list that instructs the program to constrain elements in RAPA to equal corresponding elements in R. For example if fixR = c(1,2) then smoothed matrix, RAPA[1:2,1:2] = R[1:2,1:2]. Default (fixR = NULL). - `Wghts`: A p-length vector of weights for differential variable weighting. Default (Wghts = NULL). - `maxTries`: Maximum number of iterations in the alternating projections algorithm. Default (maxTries = 1000). ## Returns - **RAPA**: A smoothed matrix. - **delta**: User-supplied delta value. - **Wghts**: User-supplied weight vector. - **fixR**: User-supplied integer list that instructs the program to constrain elements in RAPA to equal corresponding elements in R. - **convergence**: A value of 0 indicates that the algorithm located a feasible solution. A value of 1 indicates that no feasible solution was located within maxTries. ## Examples ```r data(BadRKtB) ################################################################### ## Replicate analyses in Table 2 of Knol and ten Berge (1989). ################################################################### ## n1 = 0,1 out<-smoothAPA(R = BadRKtB, delta = .0, fixR = NULL, Wghts = NULL, maxTries=1e06) S <- out$RAPA round(S - BadRKtB,3) normF(S - BadRKtB) eigen(S)$val ## n1 = 2 out<-smoothAPA(R = BadRKtB, fixR =c(1,2), delta=.0, Wghts = NULL, maxTries=1e06) S <- out$RAPA round(S - BadRKtB,3) normF(S - BadRKtB) eigen(S)$val ## n1 = 4 out<-smoothAPA(R = BadRKtB, fixR = 1:4, delta=.0, Wghts = NULL, maxTries=1e06) S <- out$RAPA round(S - BadRKtB,3) normF(S - BadRKtB) eigen(S)$val ## n1 = 5 out<-smoothAPA(R = BadRKtB, fixR = 1:5, delta=0, Wghts = NULL, maxTries=1e06) S <- out$RAPA round(S - BadRKtB,3) normF(S - BadRKtB) eigen(S)$val ################################################################### ## Replicate analyses in Table 3 of Knol and ten Berge (1989). ################################################################### ## n1 = 0,1 out<-smoothAPA(R = BadRKtB, delta = .05, fixR = NULL, Wghts = NULL, maxTries=1e06) S <- out$RAPA round(S - BadRKtB,3) normF(S - BadRKtB) eigen(S)$val ## n1 = 2 out<-smoothAPA(R = BadRKtB, fixR =c(1,2), delta=.05, Wghts = NULL, maxTries=1e06) S <- out$RAPA round(S - BadRKtB,3) normF(S - BadRKtB) eigen(S)$val ## n1 = 4 out<-smoothAPA(R = BadRKtB, fixR = 1:4, delta=.05, Wghts = NULL, maxTries=1e06) S <- out$RAPA round(S - BadRKtB,3) normF(S - BadRKtB) eigen(S)$val ## n1 = 5 out<-smoothAPA(R = BadRKtB, fixR = 1:5, delta=.05, Wghts = NULL, maxTries=1e06) S <- out$RAPA round(S - BadRKtB,3) normF(S - BadRKtB) eigen(S)$val ################################################################### ## This example illustrates differential variable weighting. ## ## Imagine a scenerio in which variables 1 & 2 were collected with ## 5 times more subjects than variables 4 - 6 then . . . ################################################################### ## n1 = 2 out<-smoothAPA(R = BadRKtB, delta=.0, fixR = NULL, Wghts = c(5, 5, rep(1,4)), maxTries=1e5) S <- out$RAPA round(S - BadRKtB,3) normF(S - BadRKtB) eigen(S)$val ``` ## Author(s) Niels Waller

smoothAPA function