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##' @title estimate hypothesis c_st = c_s
##'
##' @description estimates parameters from hypothesis lambda_s = c_s * gamma_s
##'
##' @details There are S*T + S free parameters under this hypothesis.
##'
##' @param Xdst matrix of sums of number of eaten prey species s during occurrence t; rows indexed by time, and cols indexed by prey species, TxS
##' @param Ydst matrix sum of number of caught prey species s during occurrence t; rows indexed by time, and cols indexed by prey species, TxS
##' @param J vector of predators caught in each time period
##' @param I vector of number of days all traps were left out in a given time period
##' @param EM boolean; whether or not EM algorithm is used
##' @param em_maxiter integer specifying max number of EM iterations
##' @param BALANCED boolean; whether or not data are BALANCED
estCs <- function(Xdst, Ydst, J, I, EM, em_maxiter, BALANCED) {
## some numbers
S <- ncol(Xdst); s <- seq_len(S)
T <- nrow(Xdst); t <- seq_len(T)
ST <- S*T
if (EM) {
## avoiding estimated zeros
pen <- matrix(0, T, S)
zos <- which(Xdst==0, arr.ind=TRUE)
if (length(zos)>0) {
pen[zos] <- 1
pen[zos] <- pen[zos]/(J[zos[,1]]+1) # add imaginary observation
pen <- 0.001*pen # weight it
}
## initialize some values
em_iter <- 1
init <- est1(Xdst, Ydst, J, I, EM, em_maxiter, BALANCED)
gammaHat <- gammaHat_old <- init$gamma
cHat <- cHat_old <- sumT(Xdst) / sumT(J*gammaHat) + sumT(pen)
## iterate EM
while ( TRUE ) {
## expected value of Xjst
lambda <- sapply(s, function(j) cHat[j]*gammaHat[,j]) # col-wise `*`
elambda <- exp(-lambda)
EX <- lambda/(1-elambda)
## convenience
ZEX <- Xdst*EX + pen # pull ZEX away from zero
## iterate only once simultaneous eqs
ZEXY <- ZEX + Ydst
gammaHat <- sapply(s, function(j) ZEXY[,j] / (cHat[j]*J + I)) # col-wise `/`
cHat <- sumT(ZEX) / sumT(J*gammaHat)
## check convergence of EM
if ( converged(cHat, cHat_old) &&
converged(gammaHat, gammaHat_old) ) {
break
}
## if not converged, store updated estimates
cHat_old <- cHat
gammaHat_old <- gammaHat
## print(sprintf('EM iteration %d found values: c = %f', em_iter, cHat))
em_iter <- em_iter+1
## limit iterations
if ( em_iter > em_maxiter ) {
stop(sprintf("estCs: max EM iterations, %d, reached. Please adjust accordingly.", em_maxiter))
}
}
## calc standard error with est params
## SE <- seEM(NULL, gammaHat, cHat, Xdst, Ydst, J, I)
Info <- diag(ST+S) # initialize information matrix
g2 <- gammaHat^2 # gamma^2
l <- sapply(s, function(j) gammaHat[,j]*cHat[j]) # col-wise `*`
expl <- exp(l); tmp <- J*expl/(expl-1)^2 # a common term
## fill Info with second derivatives
d2g2 <- sapply(s, function(j) tmp[,j]*cHat[j]^2 + Ydst[,j]/g2[,j]) # col-wise `*`
diag(Info)[-s] <- unlist(d2g2) # gamma
diag(Info)[s] <- sumT(tmp*g2) # c
for ( i in s ) { # fill in off diags; upper tri only
Info[i,-s][t+T*(i-1)] <- unlist(-J/(expl[,i] -1) + l[,i]*tmp[,i])
}
lowmat <- lower.tri(Info)
Info[lowmat] <- t(Info)[lowmat] # make symmetric from upper tri
tryCatch(var <- solve(Info),
error=function(e) {
print("Variances not calculated; system is singular.")
var <- NULL
})
## calc log-lik with est params
loglik <- llEM(Xdst, Ydst, NA, gammaHat, J, I, cHat)
list(c=cHat, gamma=as.matrix(gammaHat), em_iters=em_iter,
ll=loglik, var=var)
} else {
XYdst <- Xdst + Ydst
stXdst <- sumST(Xdst)
iter <- 1; maxiter <- 500
## not sure this is the right spot for these checks
## ensure J & I have dimension T or 1
if ( length(J) != T ) stop("J indexed oddly says est0.")
if ( length(I) != T ) stop("I indexed oddly says est0.")
## initialize some values
## cHat <- cHat_old <- rep(0.5, S) # runif(S)
## gammaHat <- gammaHat_old <- sapply(s, function(j) XYdst[,j] / (J*cHat[j] + I))
gammaHat <- gammaHat_old <- sapply(s, function(j) XYdst[,j] / (J + I))
## iteratively update; relies on concavity of log-lik
while ( TRUE ) {
## update parameters
cHat <- sumT(Xdst) / (sumT(J*gammaHat))
gammaHat <- sapply(s, function(j) XYdst[,j] / (J*cHat[j] + I)) # col-wise `/`
## check convergence
if ( converged(gammaHat, gammaHat_old) &&
converged(cHat, cHat_old) ) {
break
}
## if not converged, update estimates for next iteration
gammaHat_old <- gammaHat
cHat_old <- cHat
iter <- iter+1
if ( iter > maxiter ) {
stop(sprintf("estCt: %d not sufficient iterations for simultaneous equations.", maxiter))
}
}
## calc standard error with est params
## SE <- se(NULL, gammaHat, cHat, Xdst, Ydst, J, I)
Info <- diag(ST+S) # initialize information matrix
## fill Info with second derivatives
diag(Info)[-s] <- unlist(XYdst/gammaHat^2) # gamma
diag(Info)[s] <- sumT(Xdst)/cHat^2 # c
for ( i in s ) {
Info[i,-s][t+T*(i-1)] <- J # fill off diags; upper tri only
}
lowmat <- lower.tri(Info)
Info[lowmat] <- t(Info)[lowmat] # make symmetric from upper tri
## calc log-lik with est params
loglik <- ll(Xdst, Ydst, NA, gammaHat, J, I, cHat)
list(gamma=as.matrix(gammaHat), c=cHat, iters=iter,
ll=loglik, var=solve(Info))
}
}
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