RKHS_REC_DIAG.md
In BGLR: Bayesian Generalized Linear Regression

Recursive models

Example adapted from MTM package. In recursive models covariances are modeled as regressions. For instance, a recursive model for the covariance matrix of the vector x can be x=Bx + delta where B is matrix of regression coefficients (typically lower-triangular) with zeros in the diagonal and delta is a random vector (independent normal random variables in BGLR). The covariance matrix of x is Cov(x,x')=(I-B)^{-1} * PSI * ((I-B)^{-1})', where PSI is the covariance matrix for delta, and we assume it is diagonal. The parameters of the model include the recursive effects and the variance of delta. The first ones are assigned IID normal priors with null mean and variance var=100 and the variances are assigned IID scaled-inverse chi-squares priors.


library(BGLR)
data(wheat)
K<-wheat.A
y<-wheat.Y

M <- matrix(nrow = 4, ncol = 4, FALSE)
M[3, 2] <- M[4, 2] <- TRUE # Adding recursion from trait 2 onto traits 3 and 4
M[4, 3] <- TRUE # Adding recursion from trait 3 on trait 4

ETA<-list(list(K=K,model="RKHS",Cov=list(type="REC",M=M)))

fm<-Multitrait(y=y,ETA=ETA,resCov=list(type="DIAG"), nIter=1000,burnIn=500)

#Residual covariance matrix
fm$resCov

#Genetic covariance matrix
fm$ETA[[1]]$Cov

#Random effects
fm$ETA[[1]]$u

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