f_red =function(pars,peddata){
u1 <- pars[1]
u2 <- pars[2]	
u3 <- pars[3]
u4 <- pars[4]	
u5 <- pars[5]
u6 <- pars[6]	
M <- sum(peddata[16:22])	
lambdas <- c(exp(u6),exp(u5),exp(u5),exp(u5),exp(u5),
	exp(u4),exp(u4+log(2)),exp(u4),exp(u3),exp(u3),
	exp(u2),exp(u2),exp(u2),exp(u2),exp(u1))
lambdamiss <- c(exp(u1)+exp(u2),exp(u2)+exp(u3),
	exp(u2)+exp(u4),exp(u2)+exp(u4+log(2))+exp(u5),
	exp(u3)+exp(u5),exp(u4)+exp(u5),exp(u5)+exp(u6))
sum(log(dpois(peddata[1:15],lambdas)))+   
(M>0)*sum(log(dpois(peddata[16:22],lambdamiss)))
}

## First order derivative ##
df_red = function(pars,peddata){
u1 <- pars[1]
u2 <- pars[2]	
u3 <- pars[3]
u4 <- pars[4]	
u5 <- pars[5]
u6 <- pars[6]
M <- sum(peddata[16:22])
lambdas <- c(exp(u6),exp(u5),exp(u5),exp(u5),exp(u5),
	exp(u4),exp(u4+log(2)),exp(u4),exp(u3),exp(u3),
	exp(u2),exp(u2),exp(u2),exp(u2),exp(u1))
lambdamiss <- c(exp(u1)+exp(u2),exp(u2)+exp(u3),
	exp(u2)+exp(u4),exp(u2)+exp(u4+log(2))+exp(u5),
	exp(u3)+exp(u5),exp(u4)+exp(u5),exp(u5)+exp(u6))
c(
# dl/du1
sum(peddata[15]-lambdas[15])+(M>0)*(peddata[16]*lambdas[15]/lambdamiss[1]-lambdas[15]),
# dl/du2
sum(peddata[11:14]-lambdas[11:14])+(M>0)*sum(peddata[16:19]*lambdas[14:11]/lambdamiss[1:4]-lambdas[14:11]),
# dl/du3
sum(peddata[9:10]-lambdas[9:10])+(M>0)*sum(peddata[c(17,20)]*lambdas[10:9]/lambdamiss[c(2,5)]-lambdas[10:9]),
# dl/du4
sum(peddata[6:8]-lambdas[6:8])+(M>0)*sum(peddata[c(19,18,21)]*lambdas[c(7,8,6)]/lambdamiss[c(4,3,6)]-lambdas[c(7,8,6)]),
# dl/du5
sum(peddata[2:5]-lambdas[2:5])+(M>0)*sum(peddata[19:22]*lambdas[c(5,3,4,2)]/lambdamiss[c(4:7)]-lambdas[c(5,3,4,2)]),
# dl/du6
sum(peddata[1]-lambdas[1])+(M>0)*sum(peddata[22]*lambdas[1]/lambdamiss[7]-lambdas[1])
)
}