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I have want to estimate the parameters of the function which involves Bessel function and integration. However, when i tried to run it, i got a message that "Error in f(x, ...) : could not find function "BesselI" ". I don't know to fix it and would appreciate any related proposal.

library(Bessel)
library(maxLik)
library(miscTools)


K<-300

f  <- function(theta,lambda,u) {exp(-u*theta)*Vectorize(BesselI(2*sqrt(t*u*theta*lambda),1))/u^0.5}   
F  <- function(theta,lambda){integrate(f,0,K,theta=theta,lambda=lambda)$value}
tt <- function(theta,lambda){(sqrt(lambda)*exp(-t*lambda)/(2*sqrt(t*theta)))*
                                   (theta*(2*t*lambda-1)*F(theta,lambda))}
loglik <- function(param) {
   theta <- param[1]
   lambda <- param[2]
   ll <-sum(log(tt(theta,lambda)))
}
t <- c(24,220,340,620,550,559,689,543)
res <- maxNR(loglik, start=c(0.001,0.0005),print.level=1,tol = 1e-08) 
summary(res)

Newton-Raphson maximisation Number of iterations: 0 Return code: 100 Initial value out of range.

I got "There were 50 or more warnings (use warnings() to see the first 50)" and when I used warnings(), following is the warning.

In t * u : longer object length is not a multiple of shorter object length.

sessionInfo()
R version 2.14.2 (2012-02-29)
Platform: i386-pc-mingw32/i386 (32-bit)
locale:
[1] LC_COLLATE=English_United States.1252 
[2] LC_CTYPE=English_United States.1252  
[3] LC_MONETARY=English_United States.1252
[4] LC_NUMERIC=C                    
[5] LC_TIME=English_United States.1252    

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] maxLik_1.1-2     miscTools_0.6-16   Bessel_0.5-4     Rmpfr_0.5-1  
[5] gmp_0.5-4       
loaded via a namespace (and not attached):
[1] sandwich_2.2-10
Ahmed
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2 Answers2

1

Warning: incomplete answer/exploration.

Let's strip this down a little to try to see where the warnings are coming from, which may give further insight. At least we can eliminate that as a possibility.

K <- 300 ## global variable, maybe a bad idea
t <- c(24,220,340,620,550,559,689,543)  ## global/same name as t(), ditto
library(Bessel)
f  <- function(theta,lambda,u) {
    exp(-u*theta)*BesselI(2*sqrt(t*u*theta*lambda),1)/u^0.5}
## Vectorize() isn't doing any good here anyway, take it out ...
F  <- function(theta,lambda){
       integrate(f,0,K,theta=theta,lambda=lambda)$value}

F() works but still gives the vectorization warnings:

F(theta=1e-3,lambda=5e-4)
## [1] 3.406913
## There were 50 or more warnings (use warnings() to see the first 50)
f(theta=1e-3,lambda=5e-4,u=0:10)
##  [1]         NaN 0.010478182 0.013014566
##  [4] 0.017562239 0.016526010 0.016646497
##  [7] 0.018468755 0.016377872 0.003436664
## [10] 0.010399265 0.012919646
## Warning message:
## In t * u : longer object length is not a multiple of shorter object length

We still get a warning. We can also see that evaluating the integrand at 0 is likely to be a problem (we have a square-root of u in the denominator ...)

It looks like we might need to evaluate f() over the outer product (all combinations of t and u), and then perhaps sum f over the values of t? This really needs to be resolved, because t has a fixed length (presumably it's a data sample of some sort) while the integration variable u is going to have arbitrarily many values ...

Can you provide a link to the original derivation of the log-likelihood function?

Ben Bolker
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  • I have no link of this derivation. The "tt" function is density function and since it involves Bessel function with integration, that's why i just used sum(log(tt)) because likelihood is product of tt and after using log product will be sum. – Ahmed Aug 26 '14 at 20:12
  • you'll need to rearrange things a bit (I don't have time right now) -- you have to integrate over values of `u` for individual values of `t`, *then* summ them up. – Ben Bolker Aug 26 '14 at 23:30
0

Here's what I've got. Unfortunately I wasn't able to figure out how to run it across the t vector but I suppose you could just do a for loop. 

library(base)
install.packages("maxLik")
library(maxLik)

#setting some initial values to test as I go
K <- 300
u <- 1
theta <- 1
T <- c(1,2,3,4)
lambda <- 1

#I got it to work with besselI instead of BesselI
#I also dropped Vectorize and instead do a for loop at the end
f  <- function ( theta, lambda, u ) {
    exp(-u * theta) * besselI(2 * sqrt(t * u * theta * lambda), 1) / u^0.5
}
#testing to see if everything is working
f(1,1,1)

#making a function with only one input so it can be integrated 
F <- function ( u ) {
    f(theta, lambda, u)
}
F(1)

#testing for integration
t <- T[1]
integrate(F, 0, K)

#entered the integration function inside here at the bottom
tt <- function ( theta, lambda ) {
    (sqrt(lambda) * exp(-t * lambda) / (2 * sqrt(t * theta))) *
        (theta * (2 * t * lambda - 1) * integrate(F, 0, K)$value)
}
tt(1,1)

#took the absolute value of theta and lambda just in case they turn out negative
loglik <- function(param) {
    theta <- param[1]
    lambda <- param[2]
    ll <-sum(log(tt(abs(theta),abs(lambda))))
    return(ll)
}

#example for using the for loop
for ( i in 1 : length(T) ) {
    t <- T[i]
    print(loglik(c(1,1)))
}

maxNR(loglik, start = c(1, 5), print.level = 1, tol = 1e-08)

I'm not sure what you want your parameters theta and lamba but my settings worked and with the values of t used. There may be some dependencies on the the magnitude of t and theta/lambda as far as singular matrices go. Not sure if maxNR uses generalized inverses but I'm sure it does.

Rex
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