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##' @title estimate hypothesis c_st
##'
##' @description estimates parameters from hypothesis lambda = c_st * gamma
##'
##' @details There are 2*S*T 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
estCst <- function(Xdst, Ydst, J, I, EM, em_maxiter, BALANCED) {
## some numbers
S <- ncol(Ydst); s <- seq_len(S)
T <- nrow(Ydst); t <- seq_len(T)
ST <- S*T; st <- seq_len(ST)
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
## cHat <- cHat_old <- matrix(0.5, ncol=S, nrow=T) # matrix(runif(ST), nrow=T)
init <- est1(Xdst, Ydst, J, I, EM, em_maxiter, BALANCED)
gammaHat <- gammaHat_old <- init$gamma
cHat <- cHat_old <- Xdst/(J*gammaHat) + pen
## iterate EM
while ( TRUE ) {
## update lambda
lambda <- cHat*gammaHat
elambda <- exp(-lambda)
EX <- lambda/(1-elambda)
## convenience
ZEX <- Xdst*EX + pen # pull ZEX away from zero
## iterate only once simultaneous eqs
gammaHat <- (ZEX + Ydst)/(J*cHat + I)
cHat <- ZEX/(J*gammaHat)
## check convergence
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("estCst: max EM iterations, %d, reached. Please adjust accordingly.", em_maxiter))
}
}
## since estimated 0s don't follow from calculations above
## lambdaHat[which(Xdst == 0, arr.ind=T)] <- 0
## calc standard error with est params
## SE <- seEM(lambdaHat, gammaHat, NULL, Xdst, Ydst, J, I)
Info <- diag(2*ST) # initialize information matrix
g2 <- gammaHat^2 # gamma^2
l <- cHat*gammaHat
expl <- exp(l)
tmp <- J*expl/(expl-1)^2 # a common term
## fill in second derivatives
diag(Info)[-st] <- unlist(Ydst/g2 + cHat^2*tmp) # gamma
diag(Info)[st] <- unlist(g2*tmp) # c
Info[st,-st] <- unlist(-J/(expl-1) + l*tmp) # fill off diags; upper tri
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 loglik wit est params
cv <- unlist(cHat)
loglik <- llEM(Xdst, Ydst, NA, gammaHat, J, I, cv)
list('gamma' = as.matrix(gammaHat), 'c' = cv,
'em_iters' = em_iter, 'll' = loglik, 'var' = var)
} else {
## some numbers
XYdst <- Xdst + Ydst
iter <- 1; maxiter <- 500
## not sure this is the right spot for these checks
T <- nrow(Xdst)
if ( length(J) != T ) {
stop("J indexed oddly says estGen")
}
if ( length(I) != T ) {
stop("I indexed oddly says estGen")
}
## initialize some values
## cHat <- cHat_old <- matrix(0.5, ncol=S, nrow=T) # matrix(runif(ST), nrow=T)
## gammaHat <- gammaHat_old <- XYdst / (J*cHat + I)
gammaHat <- gammaHat_old <- XYdst / (J + I)
## iteratively update; relies on concavity of log-lik
while ( TRUE ) {
## update parameters
cHat <- Xdst / (J*gammaHat)
gammaHat <- XYdst / (J*cHat + I)
## 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
## limit iterations
if ( iter > maxiter ) {
stop(sprintf("estCst: %d not sufficient iterations for simultaneous equations.", maxiter))
}
}
## calc standard error with est params
## SE <- se(lambdaHat, gammaHat, NULL, Xdst, Ydst, J, I)
Info <- diag(2*ST) # initialize information matrix
## fill Info with second derivatives
diag(Info)[-st] <- unlist(XYdst/gammaHat^2) # gamma
diag(Info)[st] <- unlist(Xdst/cHat) # c
for ( i in t ) {
Info[st,-st][,seq(0, (S-1)*T, by=T)+i] <- J[i]
}
lowmat <- lower.tri(Info)
Info[lowmat] <- t(Info)[lowmat] # make symmetric from upper tri
## calc loglik with est params
cv <- unlist(cHat)
loglik <- ll(Xdst, Ydst, NA, gammaHat, J, I, cv)
list(gamma=as.matrix(gammaHat), c=cv, iters=iter,
ll=loglik, var=solve(Info))
}
}
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