# [R] problem with convergence in mle2/optim function

Mon Oct 8 07:39:36 CEST 2012

```Dear R Help,

Thank you to those who responded to my questions about mle2/optim
convergence.  A few responders pointed out that the optim error seems to
arise when either one of the probabilities P1, P2, or P3 become negative
or infinite.  One suggested examining the exponential terms within the
P1 and P2 equations.

I may have made some progress along these lines.  The exponential terms
in the equations for P1 and P2 go to infinity at certain (large) values
of t. The exponential terms can be found in lines 1, 5, and 7 in the P1
and P2 expressions below.  Here is some example code:

###########################################################################
# P1 and P2 equations
P1 <- expression((p1*((-1 + exp(sqrt((mu1 + mu2 + p1 + p2)^2 -
4*(mu2*p1 + mu1*(mu2 + p2)))*t))*((-mu2)*(mu2 - p1 + p2) +
mu1*(mu2 + 2*p2)) - mu2*sqrt((mu1 + mu2 + p1 + p2)^2 -
4*(mu2*p1 + mu1*(mu2 + p2))) -
exp(sqrt((mu1 + mu2 + p1 + p2)^2 - 4*(mu2*p1 + mu1*(mu2 + p2)))*t)*
mu2*sqrt((mu1 + mu2 + p1 + p2)^2 - 4*(mu2*p1 + mu1*(mu2 + p2))) +
2*exp((1/2)*(mu1 + mu2 + p1 + p2 + sqrt((mu1 + mu2 + p1 + p2)^2 -
4*(mu2*p1 + mu1*(mu2 + p2))))*t)*mu2*
sqrt((mu1 + mu2 + p1 + p2)^2 - 4*(mu2*p1 + mu1*(mu2 + p2)))))/
exp((1/2)*(mu1 + mu2 + p1 + p2 + sqrt((mu1 + mu2 + p1 + p2)^2 -
4*(mu2*p1 + mu1*(mu2 + p2))))*t)/(2*(mu2*p1 + mu1*(mu2 + p2))*
sqrt((mu1 + mu2 + p1 + p2)^2 - 4*(mu2*p1 + mu1*(mu2 + p2)))))
P2 <- expression((p2*((-1 + exp(sqrt((mu1 + mu2 + p1 + p2)^2 -
4*(mu2*p1 + mu1*(mu2 + p2)))*t))*(-mu1^2 + 2*mu2*p1 +
mu1*(mu2 - p1 + p2)) - mu1*sqrt((mu1 + mu2 + p1 + p2)^2 -
4*(mu2*p1 + mu1*(mu2 + p2))) -
exp(sqrt((mu1 + mu2 + p1 + p2)^2 - 4*(mu2*p1 + mu1*(mu2 + p2)))*t)*
mu1*sqrt((mu1 + mu2 + p1 + p2)^2 - 4*(mu2*p1 + mu1*(mu2 + p2))) +
2*exp((1/2)*(mu1 + mu2 + p1 + p2 + sqrt((mu1 + mu2 + p1 + p2)^2 -
4*(mu2*p1 + mu1*(mu2 + p2))))*t)*mu1*
sqrt((mu1 + mu2 + p1 + p2)^2 - 4*(mu2*p1 + mu1*(mu2 + p2)))))/
exp((1/2)*(mu1 + mu2 + p1 + p2 + sqrt((mu1 + mu2 + p1 + p2)^2 -
4*(mu2*p1 + mu1*(mu2 + p2))))*t)/(2*(mu2*p1 + mu1*(mu2 + p2))*
sqrt((mu1 + mu2 + p1 + p2)^2 - 4*(mu2*p1 + mu1*(mu2 + p2)))))

# Vector of t values
tv <- c(1:200)

# 'true' parameter values
p1t = 2; p2t = 2; mu1t = 0.001; mu2t = 0.001

# Function to calculate probabilities from 'true' parameter values
psim <- function(x){
params <- list(p1 = p1t, p2 = p2t, mu1 = mu1t, mu2 = mu2t, t = x)
eval.P1 <- eval(P1, params)
eval.P2 <- eval(P2, params)
P3 <- 1 - eval.P1 - eval.P2
c(x, matrix(c(eval.P1, eval.P2, P3), ncol = 3))
}
pdat <- sapply(tv, psim, simplify = TRUE)
Pdat <- as.data.frame(t(pdat))
names(Pdat) <- c("time", "P1", "P2", "P3")

matplot(Pdat[,-1], type = "l", xlab = "time", ylab = "Probability",
col = c("black", "brown", "blue"),
lty = c(1:3), lwd = 2, ylim = c(0,1))
legend("topright", c("P1", "P2", "P3"),
col = c("black", "brown", "blue"),
lty = c(1:3), lwd = 2)
Pdat[160:180,] # psim function begins to return "NaN" at t = 178

# exponential terms in P1 and P2 expressions are problematic
params <- list(p1 = p1t, p2 = p2t, mu1 = mu1t, mu2 = mu2t, t = 178)

exp1 <- expression(exp(sqrt((mu1 + mu2 + p1 + p2)^2 -
4*(mu2*p1 + mu1*(mu2 + p2)))*t))
eval(exp1, params) # returns Inf at t = 178

exp2 <- expression(exp((1/2)*(mu1 + mu2 + p1 + p2 +
sqrt((mu1 + mu2 + p1 + p2)^2 -
4*(mu2*p1 + mu1*(mu2 + p2))))*t))
eval(exp2, params) # also returns Inf at t = 178
##########################################################################

The time step at which the exponential terms go to infinity depends on
the values of the parameters p1, p2, mu1, mu2.  It seems that the
convergence problems may be due to these exponential terms going to
infinity.  Thus my convergence problem appears to be an overflow problem?

Unfortunately, I'm not sure where to go from here.  Due to the
complexity of the P1 and P2 equations, there's no clear way to rearrange
the equations to eliminate t from the exponential terms.

Does anyone have any suggestions on how to address this problem?
Perhaps there is a way to bound p1, p2, mu1, and mu2 to avoid the
exponential terms going to infinity?  Or bound P1 and P2?

Any suggestions would be greatly appreciated.

On 10/5/2012 3:06 AM, Berend Hasselman wrote:
>
> On 05-10-2012, at 07:12, Adam Zeilinger wrote:
>
>> Hello R Help,
>>
>> I am trying solve an MLE convergence problem: I would like to estimate four parameters, p1, p2, mu1, mu2, which relate to the probabilities, P1, P2, P3, of a multinomial (trinomial) distribution.  I am using the mle2() function and feeding it a time series dataset composed of four columns: time point, number of successes in category 1, number of successes in category 2, and number of success in category 3.  The column headers are: t, n1, n2, and n3.
>>
>> The mle2() function converges occasionally, and I need to improve the rate of convergence when used in a stochastic simulation, with multiple stochastically generated datasets.  When mle2() does not converge, it returns an error: "Error in optim(par = c(2, 2, 0.001, 0.001), fn = function (p) : L-BFGS-B needs finite values of 'fn'."  I am using the L-BFGS-B optimization method with a lower box constraint of zero for all four parameters.  While I do not know any theoretical upper limit(s) to the parameter values, I have not seen any parameter estimates above 2 when using empirical data.  It seems that when I start with certain 'true' parameter values, the rate of convergence is quite high, whereas other "true" parameter values are very difficult to estimate.  For example, the true parameter values p1 = 2, p2 = 2, mu1 = 0.001, mu2 = 0.001 causes convergence problems, but the parameter values p1 = 0.3, p2 = 0.3, mu1 = 0.08, mu2 = 0.08 lead to high convergence rate.  I've chose!
> ! n these two sets of values because they represent the upper and lower estimates of parameter values derived from graphical methods.
>>
>> First, do you have any suggestions on how to improve the rate of convergence and avoid the "finite values of 'fn'" error?  Perhaps it has to do with the true parameter values being so close to the boundary?  If so, any suggestions on how to estimate parameter values that are near zero?
>>
>> Here is reproducible and relevant code from my stochastic simulation:
>>
>> ########################################################################
>> library(bbmle)
>> library(combinat)
>>
>> # define multinomial distribution
>> dmnom2 <- function(x,prob,log=FALSE) {
>>   r <- lgamma(sum(x) + 1) + sum(x * log(prob) - lgamma(x + 1))
>>   if (log) r else exp(r)
>> }
>>
>> # vector of time points
>> tv <- 1:20
>>
>> # Negative log likelihood function
>> NLL.func <- function(p1, p2, mu1, mu2, y){
>>   t <- y\$tv
>>   n1 <- y\$n1
>>   n2 <- y\$n2
>>   n3 <- y\$n3
>>   P1 <- (p1*((-1 + exp(sqrt((mu1 + mu2 + p1 + p2)^2 -
>>     4*(mu2*p1 + mu1*(mu2 + p2)))*t))*((-mu2)*(mu2 - p1 + p2) +
>>     mu1*(mu2 + 2*p2)) - mu2*sqrt((mu1 + mu2 + p1 + p2)^2 -
>>     4*(mu2*p1 + mu1*(mu2 + p2))) -
>>     exp(sqrt((mu1 + mu2 + p1 + p2)^2 - 4*(mu2*p1 + mu1*(mu2 + p2)))*t)*
>>     mu2*sqrt((mu1 + mu2 + p1 + p2)^2 - 4*(mu2*p1 + mu1*(mu2 + p2))) +
>>     2*exp((1/2)*(mu1 + mu2 + p1 + p2 + sqrt((mu1 + mu2 + p1 + p2)^2 -
>>     4*(mu2*p1 + mu1*(mu2 + p2))))*t)*mu2*
>>     sqrt((mu1 + mu2 + p1 + p2)^2 - 4*(mu2*p1 + mu1*(mu2 + p2)))))/
>>     exp((1/2)*(mu1 + mu2 + p1 + p2 + sqrt((mu1 + mu2 + p1 + p2)^2 -
>>     4*(mu2*p1 + mu1*(mu2 + p2))))*t)/(2*(mu2*p1 + mu1*(mu2 + p2))*
>>     sqrt((mu1 + mu2 + p1 + p2)^2 - 4*(mu2*p1 + mu1*(mu2 + p2))))
>>   P2 <- (p2*((-1 + exp(sqrt((mu1 + mu2 + p1 + p2)^2 -
>>     4*(mu2*p1 + mu1*(mu2 + p2)))*t))*(-mu1^2 + 2*mu2*p1 +
>>     mu1*(mu2 - p1 + p2)) - mu1*sqrt((mu1 + mu2 + p1 + p2)^2 -
>>     4*(mu2*p1 + mu1*(mu2 + p2))) -
>>     exp(sqrt((mu1 + mu2 + p1 + p2)^2 - 4*(mu2*p1 + mu1*(mu2 + p2)))*t)*
>>     mu1*sqrt((mu1 + mu2 + p1 + p2)^2 - 4*(mu2*p1 + mu1*(mu2 + p2))) +
>>     2*exp((1/2)*(mu1 + mu2 + p1 + p2 + sqrt((mu1 + mu2 + p1 + p2)^2 -
>>     4*(mu2*p1 + mu1*(mu2 + p2))))*t)*mu1*
>>     sqrt((mu1 + mu2 + p1 + p2)^2 - 4*(mu2*p1 + mu1*(mu2 + p2)))))/
>>     exp((1/2)*(mu1 + mu2 + p1 + p2 + sqrt((mu1 + mu2 + p1 + p2)^2 -
>>     4*(mu2*p1 + mu1*(mu2 + p2))))*t)/(2*(mu2*p1 + mu1*(mu2 + p2))*
>>     sqrt((mu1 + mu2 + p1 + p2)^2 - 4*(mu2*p1 + mu1*(mu2 + p2))))
>>   P3 <- 1 - P1 - P2
>>   p.all <- c(P1, P2, P3)
>>   -sum(dmnom2(c(n1, n2, n3), prob = p.all, log = TRUE))
>> }
>>
>> ## Generate simulated data
>> # Model equations as expressions,
>> P1 <- expression((p1*((-1 + exp(sqrt((mu1 + mu2 + p1 + p2)^2 -
>>   4*(mu2*p1 + mu1*(mu2 + p2)))*t))*((-mu2)*(mu2 - p1 + p2) +
>>   mu1*(mu2 + 2*p2)) - mu2*sqrt((mu1 + mu2 + p1 + p2)^2 -
>>   4*(mu2*p1 + mu1*(mu2 + p2))) -
>>   exp(sqrt((mu1 + mu2 + p1 + p2)^2 - 4*(mu2*p1 + mu1*(mu2 + p2)))*t)*
>>   mu2*sqrt((mu1 + mu2 + p1 + p2)^2 - 4*(mu2*p1 + mu1*(mu2 + p2))) +
>>   2*exp((1/2)*(mu1 + mu2 + p1 + p2 + sqrt((mu1 + mu2 + p1 + p2)^2 -
>>   4*(mu2*p1 + mu1*(mu2 + p2))))*t)*mu2*
>>   sqrt((mu1 + mu2 + p1 + p2)^2 - 4*(mu2*p1 + mu1*(mu2 + p2)))))/
>>   exp((1/2)*(mu1 + mu2 + p1 + p2 + sqrt((mu1 + mu2 + p1 + p2)^2 -
>>   4*(mu2*p1 + mu1*(mu2 + p2))))*t)/(2*(mu2*p1 + mu1*(mu2 + p2))*
>>   sqrt((mu1 + mu2 + p1 + p2)^2 - 4*(mu2*p1 + mu1*(mu2 + p2)))))
>>
>> P2 <- expression((p2*((-1 + exp(sqrt((mu1 + mu2 + p1 + p2)^2 -
>>   4*(mu2*p1 + mu1*(mu2 + p2)))*t))*(-mu1^2 + 2*mu2*p1 +
>>   mu1*(mu2 - p1 + p2)) - mu1*sqrt((mu1 + mu2 + p1 + p2)^2 -
>>   4*(mu2*p1 + mu1*(mu2 + p2))) -
>>   exp(sqrt((mu1 + mu2 + p1 + p2)^2 - 4*(mu2*p1 + mu1*(mu2 + p2)))*t)*
>>   mu1*sqrt((mu1 + mu2 + p1 + p2)^2 - 4*(mu2*p1 + mu1*(mu2 + p2))) +
>>   2*exp((1/2)*(mu1 + mu2 + p1 + p2 + sqrt((mu1 + mu2 + p1 + p2)^2 -
>>   4*(mu2*p1 + mu1*(mu2 + p2))))*t)*mu1*
>>   sqrt((mu1 + mu2 + p1 + p2)^2 - 4*(mu2*p1 + mu1*(mu2 + p2)))))/
>>   exp((1/2)*(mu1 + mu2 + p1 + p2 + sqrt((mu1 + mu2 + p1 + p2)^2 -
>>   4*(mu2*p1 + mu1*(mu2 + p2))))*t)/(2*(mu2*p1 + mu1*(mu2 + p2))*
>>   sqrt((mu1 + mu2 + p1 + p2)^2 - 4*(mu2*p1 + mu1*(mu2 + p2)))))
>>
>> # True parameter values
>> p1t = 2; p2t = 2; mu1t = 0.001; mu2t = 0.001
>>
>> # Function to calculate probabilities from 'true' parameter values
>> psim <- function(x){
>>   params <- list(p1 = p1t, p2 = p2t, mu1 = mu1t, mu2 = mu2t, t = x)
>>   eval.P1 <- eval(P1, params)
>>   eval.P2 <- eval(P2, params)
>>   P3 <- 1 - eval.P1 - eval.P2
>>   c(x, matrix(c(eval.P1, eval.P2, P3), ncol = 3))
>> }
>> pdat <- sapply(tv, psim, simplify = TRUE)
>> Pdat <- as.data.frame(t(pdat))
>> names(Pdat) <- c("time", "P1", "P2", "P3")
>>
>> # Generate simulated data set from probabilities
>> n = rep(20, length(tv))
>> p = as.matrix(Pdat[,2:4])
>> y <- as.data.frame(rmultinomial(n,p))
>> yt <- cbind(tv, y)
>> names(yt) <- c("tv", "n1", "n2", "n3")
>>
>> # mle2 call
>> mle.fit <- mle2(NLL.func, data = list(y = yt),
>>                 start = list(p1 = p1t, p2 = p2t, mu1 = mu1t, mu2 = mu2t),
>>                 control = list(maxit = 5000, factr = 1e-10, lmm = 17),
>>                 method = "L-BFGS-B", skip.hessian = TRUE,
>>                 lower = list(p1 = 0, p2 = 0, mu1 = 0, mu2 = 0))
>>
>> ###########################################################################
>>
>> I interpret the error as having to do with the finite difference approximation failing.  If so, perhaps a gradient function would help? If you agree, I've described my unsuccessful attempt at writing a gradient function below.  If a gradient function is unnecessary, ignore the remainder of this message.
>>
>
> After playing with your function, I can't agree with your interpretation of what could be wrong.
> During optim iterations your function is dmnom2 is getting negative values for prob and that leads to the error messages.
> I checked this by inserting the following lines in NLL.func after the assignment to p.all:
>
>   cat("NLL.func p.all {P1,P2,P3}\n")
>   print(matrix(p.all, ncol=3))
>
> At some stage entries for P1, P2, P3 become negative (which ones and how many depends on the random number generator).
> Try set.seed(1), set.seed(11) and set.seed(413) to see what happens.
>
> The expressions are too complicated for further analysis.
> Assuming your expressions are correct, you will need to restrict P1,P2,P3 to take on valid values.
>
> Berend
>
> ______________________________________________
> R-help at r-project.org mailing list
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> and provide commented, minimal, self-contained, reproducible code.
>

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