This is a syntax question and probably has a simple solution but I can't find it covered anywhere on SO for the optimx package.
Minimal working example & Question
I have a function like:
ToOptimise = function(a,b,d,e){
(a-1)^2 + (b-2)^2 + (d-3)^2 +(e-4)^2
}
I can optimise select parameters in mle2 quite easily:
library(bbmle)
Calib2 = mle2(ToOptimise,
start = list(a = 1, d = 10),
fixed = list(b = 2, e = 2))
This is not really the right tool though as it is not a maximum likelihood problem.
I want to find a way to pass fixed and optimisable parameters to Optimx as easily
I plan to do several calibrations holding different parameters fixed at any time and so do not want to have to redo the function hardcoding in some parameters.
My attempt that don't work
library(optimx)
ToOptimiseVector = function(Theta){
a = Theta[1]
b = Theta[2]
d = Theta[3]
e = Theta[4]
(a-1)^2 + (b-2)^2 + (d-3)^2 +(e-4)^2
}
Calib1 = optimx( par = c(1,2,1,2),
fn = ToOptimiseVector,
Theta[2] = 2, Theta[4] = 2)
Another related Question on Stack Overflow
Another optimx syntax question that does not look at fixed and optimisable parameters is:
R- Optimx for exponential function with 2 parameters - cannot evaluate function at initial parameter values
Added a opt parameter to the fn. Just pass this opt vector into optimx(). Any non NA values will become fixed.
ToOptimiseVector <- function(Theta, opt = rep(NA, 4)){
# Check if any of opt is not NA
if (any(!sapply(opt, is.na))) {
i = !sapply(opt, is.na)
# Fix non-NA values
Theta[i] <- opt[i]
}
a <- Theta[1]
b <- Theta[2]
d <- Theta[3]
e <- Theta[4]
return((a-1)^2 + (b-2)^2 + (d-3)^2 +(e-4)^2)
}
Seems to work.
Calib1 = optimx( par = c(1,2,1,2), fn = ToOptimiseVector, opt = c(NA, 2, NA, 2))
Calib2 = optimx( par = c(1,2,1,2), fn = ToOptimiseVector)
> Calib1
p1 p2 p3 p4 value fevals gevals niter convcode kkt1 kkt2 xtimes
Nelder-Mead 0.9998974 5.517528 3.00022 10.83214 4 103 NA NA 0 TRUE FALSE 0.02
BFGS 1.0000000 4.000000 3.00000 8.00000 4 6 3 NA 0 TRUE FALSE 0.00
> Calib2
p1 p2 p3 p4 value fevals gevals niter convcode kkt1 kkt2 xtimes
Nelder-Mead 1.000936 1.999793 3.0006 4.000256 1.344336e-06 227 NA NA 0 FALSE TRUE 0.01
BFGS 1.000000 2.000000 3.0000 4.000000 3.566556e-23 16 3 NA 0 TRUE TRUE 0.00
The trick is that the starting params and the arguments to the function to be optimized have to be aligned. Please see if the following helps you at all.
library(optimx)
ToOptimiseVector <- function(Theta){
a <- Theta[1]
b <- Theta[2]
d <- Theta[3]
e <- Theta[4]
(a-1)^2 + (b-2)^2 + (d-3)^2 +(e-4)^2
}
start <- c(1,0,1,1)
start <- c(1,0,0,0)
start <- c(1,2,1,2)
Calib1 <- optimx( par=start,
fn = ToOptimiseVector)
This gives warning messages, but you do get a result.
> Calib1
p1 p2 p3 p4 value fevals gevals niter convcode kkt1 kkt2
Nelder-Mead 1 2 3 4 0 4 4 NA 0 TRUE TRUE
BFGS 1 2 3 4 0 4 4 NA 0 TRUE TRUE
xtimes
Nelder-Mead 0
BFGS 0
Hope you can start with this and proceed.
I am attempting to reproduce the solutions of paper by Kostakis. In this paper an abridged mortality table is expanded to a complete life table using de Heligman-Pollard model. The model has 8 parameters which have to be fitted. The author used a modified Gauss-Newton algorithm; this algorithm (E04FDF) is part of the NAG library of computer programs. Should not Levenberg Marquardt yield the same set of parameters? What is wrong with my code or application of the LM algorithm?
library(minpack.lm)
## Heligman-Pollard is used to expand an abridged table.
## nonlinear least squares algorithm is used to fit the parameters on nqx observed over 5 year intervals (5qx)
AGE <- c(0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70)
MORTALITY <- c(0.010384069, 0.001469140, 0.001309318, 0.003814265, 0.005378395, 0.005985625, 0.006741766, 0.009325056, 0.014149626, 0.021601755, 0.034271934, 0.053836246, 0.085287751, 0.136549522, 0.215953304)
## The start parameters for de Heligman-Pollard Formula (Converged set a=0.0005893,b=0.0043836,c=0.0828424,d=0.000706,e=9.927863,f=22.197312,g=0.00004948,h=1.10003)
## I modified a random parameter "a" in order to have a start values. The converged set is listed above.
parStart <- list(a=0.0008893,b=0.0043836,c=0.0828424,d=0.000706,e=9.927863,f=22.197312,g=0.00004948,h=1.10003)
## The Heligman-Pollard Formula (HP8) = qx/px = ...8 parameter equation
HP8 <-function(parS,x)
ifelse(x==0, parS$a^((x+parS$b)^parS$c) + parS$g*parS$h^x,
parS$a^((x+parS$b)^parS$c) + parS$d*exp(-parS$e*(log(x/parS$f))^2) +
parS$g*parS$h^x)
## Define qx = HP8/(1+HP8)
qxPred <- function(parS,x) HP8(parS,x)/(1+HP8(parS,x))
## Calculate nqx predicted by HP8 model (nqxPred(parStart,x))
nqxPred <- function(parS,x)
(1 -(1-qxPred(parS,x)) * (1-qxPred(parS,x+1)) *
(1-qxPred(parS,x+2)) * (1-qxPred(parS,x+3)) *
(1-qxPred(parS,x+4)))
##Define Residual Function, the relative squared distance is minimized
ResidFun <- function(parS, Observed,x) (nqxPred(parS,x)/Observed-1)^2
## Applying the nls.lm algo.
nls.out <- nls.lm(par=parStart, fn = ResidFun, Observed = MORTALITY, x = AGE,
control = nls.lm.control(nprint=1,
ftol = .Machine$double.eps,
ptol = .Machine$double.eps,
maxfev=10000, maxiter = 500))
summary(nls.out)
## The author used a modified Gauss-Newton algorithm, this alogorithm (E04FDF) is part of the NAG library of computer programs
## Should not Levenberg Marquardt yield the same set of parameters
The bottom line here is that #Roland is absolutely right, this is a very ill-posed problem, and you shouldn't necessarily expect to get reliable answers. Below I've
cleaned up the code in a few small ways (this is just aesthetic)
changed the ResidFun to return residuals, not squared residuals. (The former is correct, but this doesn't make very much difference.)
explored results from several different optimizers. It actually looks like the answer you're getting is better than the "converged parameters" you list above, which I'm assuming are the parameters from the original study (can you please provide a reference?).
Load package:
library(minpack.lm)
Data, as a data frame:
d <- data.frame(
AGE = seq(0,70,by=5),
MORTALITY=c(0.010384069, 0.001469140, 0.001309318, 0.003814265,
0.005378395, 0.005985625, 0.006741766, 0.009325056,
0.014149626, 0.021601755, 0.034271934, 0.053836246,
0.085287751, 0.136549522, 0.215953304))
First view of the data:
library(ggplot2)
(g1 <- ggplot(d,aes(AGE,MORTALITY))+geom_point())
g1+geom_smooth() ## with loess fit
Parameter choices:
Presumably these are the parameters from the original paper ...
parConv <- c(a=0.0005893,b=0.0043836,c=0.0828424,
d=0.000706,e=9.927863,f=22.197312,g=0.00004948,h=1.10003)
Perturbed parameters:
parStart <- parConv
parStart["a"] <- parStart["a"]+3e-4
The formulae:
HP8 <-function(parS,x)
with(as.list(parS),
ifelse(x==0, a^((x+b)^c) + g*h^x,
a^((x+b)^c) + d*exp(-e*(log(x/f))^2) + g*h^x))
## Define qx = HP8/(1+HP8)
qxPred <- function(parS,x) {
h <- HP8(parS,x)
h/(1+h)
}
## Calculate nqx predicted by HP8 model (nqxPred(parStart,x))
nqxPred <- function(parS,x)
(1 -(1-qxPred(parS,x)) * (1-qxPred(parS,x+1)) *
(1-qxPred(parS,x+2)) * (1-qxPred(parS,x+3)) *
(1-qxPred(parS,x+4)))
##Define Residual Function, the relative squared distance is minimized
ResidFun <- function(parS, Observed,x) (nqxPred(parS,x)/Observed-1)
n.b. this is changed slightly from the OP's version; nls.lm wants residuals, not squared residuals.
A sum-of-squares function for use with other optimizers:
ssqfun <- function(parS, Observed, x) {
sum(ResidFun(parS, Observed, x)^2)
}
Applying nls.lm. (Not sure why ftol and ptol were lowered
from sqrt(.Machine$double.eps) to .Machine$double.eps -- the
former is generally a practical limit to precision ...
nls.out <- nls.lm(par=parStart, fn = ResidFun,
Observed = d$MORTALITY, x = d$AGE,
control = nls.lm.control(nprint=0,
ftol = .Machine$double.eps,
ptol = .Machine$double.eps,
maxfev=10000, maxiter = 1000))
parNLS <- coef(nls.out)
pred0 <- nqxPred(as.list(parConv),d$AGE)
pred1 <- nqxPred(as.list(parNLS),d$AGE)
dPred <- with(d,rbind(data.frame(AGE,MORTALITY=pred0,w="conv"),
data.frame(AGE,MORTALITY=pred1,w="nls")))
g1 + geom_line(data=dPred,aes(colour=w))
The lines are indistinguishable, but the parameters have some big
differences:
round(cbind(parNLS,parConv),5)
## parNLS parConv
## a 1.00000 0.00059
## b 50.46708 0.00438
## c 3.56799 0.08284
## d 0.00072 0.00071
## e 6.05200 9.92786
## f 21.82347 22.19731
## g 0.00005 0.00005
## h 1.10026 1.10003
d,f,g,h are close, but a,b,c are orders of magnitude different and e is 50% different.
Looking at the original equations, what's happening here is that a^((x+b)^c) is getting set to a constant, because a is approaching 1: once a is approximately 1, b and c are essentially irrelevant.
Let's check the correlation (we need a generalized inverse because
the matrix is so strongly correlated):
obj <- nls.out
vcov <- with(obj,deviance/(length(fvec) - length(par)) *
MASS::ginv(hessian))
cmat <- round(cov2cor(vcov),1)
dimnames(cmat) <- list(letters[1:8],letters[1:8])
## a b c d e f g h
## a 1.0 0.0 0.0 0.0 0.0 0.0 -0.1 0.0
## b 0.0 1.0 -1.0 1.0 -1.0 -1.0 -0.4 -1.0
## c 0.0 -1.0 1.0 -1.0 1.0 1.0 0.4 1.0
## d 0.0 1.0 -1.0 1.0 -1.0 -1.0 -0.4 -1.0
## e 0.0 -1.0 1.0 -1.0 1.0 1.0 0.4 1.0
## f 0.0 -1.0 1.0 -1.0 1.0 1.0 0.4 1.0
## g -0.1 -0.4 0.4 -0.4 0.4 0.4 1.0 0.4
## h 0.0 -1.0 1.0 -1.0 1.0 1.0 0.4 1.0
This is not actually so useful -- it really just confirms that lots
of the variables are strongly correlated ...
library(optimx)
mvec <- c('Nelder-Mead','BFGS','CG','L-BFGS-B',
'nlm','nlminb','spg','ucminf')
opt1 <- optimx(par=parStart, fn = ssqfun,
Observed = d$MORTALITY, x = d$AGE,
itnmax=5000,
method=mvec,control=list(kkt=TRUE))
## control=list(all.methods=TRUE,kkt=TRUE)) ## Boom!
## fvalues method fns grs itns conv KKT1 KKT2 xtimes
## 2 8.988466e+307 BFGS NA NULL NULL 9999 NA NA 0
## 3 8.988466e+307 CG NA NULL NULL 9999 NA NA 0
## 4 8.988466e+307 L-BFGS-B NA NULL NULL 9999 NA NA 0
## 5 8.988466e+307 nlm NA NA NA 9999 NA NA 0
## 7 0.3400858 spg 1 NA 1 3 NA NA 0.064
## 8 0.3400858 ucminf 1 1 NULL 0 NA NA 0.032
## 1 0.06099295 Nelder-Mead 501 NA NULL 1 NA NA 0.252
## 6 0.009275733 nlminb 200 1204 145 1 NA NA 0.708
This warns about bad scaling, and also finds a variety of different
answers: only ucminf claims to have converged, but nlminb gets a
better answer -- and the itnmax parameter seems to be ignored ...
opt2 <- nlminb(start=parStart, objective = ssqfun,
Observed = d$MORTALITY, x = d$AGE,
control= list(eval.max=5000,iter.max=5000))
parNLM <- opt2$par
Finishes, but with a false convergence warning ...
round(cbind(parNLS,parConv,parNLM),5)
## parNLS parConv parNLM
## a 1.00000 0.00059 1.00000
## b 50.46708 0.00438 55.37270
## c 3.56799 0.08284 3.89162
## d 0.00072 0.00071 0.00072
## e 6.05200 9.92786 6.04416
## f 21.82347 22.19731 21.82292
## g 0.00005 0.00005 0.00005
## h 1.10026 1.10003 1.10026
sapply(list(parNLS,parConv,parNLM),
ssqfun,Observed=d$MORTALITY,x=d$AGE)
## [1] 0.006346250 0.049972367 0.006315034
It looks like nlminb and minpack.lm are getting similar answers, and are actually doing better than the originally stated parameters (by quite a bit):
pred2 <- nqxPred(as.list(parNLM),d$AGE)
dPred <- with(d,rbind(dPred,
data.frame(AGE,MORTALITY=pred2,w="nlminb")))
g1 + geom_line(data=dPred,aes(colour=w))
ggsave("cmpplot.png")
ggplot(data=dPred,aes(x=AGE,y=MORTALITY-d$MORTALITY,colour=w))+
geom_line()+geom_point(aes(shape=w),alpha=0.3)
ggsave("residplot.png")
Other things one could try would be:
appropriate scaling -- although a quick test of this doesn't seem to help that much
provide analytical gradients
use AD Model Builder
use the slice function from bbmle to explore whether the old and new parameters seem to represent distinct minima, or whether the old parameters are just a false convergence ...
get the KKT (Karsh-Kuhn-Tucker) criterion calculators from optimx or related packages working for similar checks
PS: the largest deviations (by far) are for the oldest age classes, which probably also have small samples. From a statistical point of view it would probably be worth doing a fit that weighted by the precision of the individual points ...
#BenBolker, fitting the parameters with the entire dataset (underlying qx) values. Still not able to reproduce parameters
library(minpack.lm)
library(ggplot2)
library(optimx)
getwd()
d <- data.frame(AGE = seq(0,74), MORTALITY=c(869,58,40,37,36,35,32,28,29,23,24,22,24,28,
33,52,57,77,93,103,103,109,105,114,108,112,119,
125,117,127,125,134,134,131,152,179,173,182,199,
203,232,245,296,315,335,356,405,438,445,535,594,
623,693,749,816,915,994,1128,1172,1294,1473,
1544,1721,1967,2129,2331,2559,2901,3203,3470,
3782,4348,4714,5245,5646))
d$MORTALITY <- d$MORTALITY/100000
ggplot(d,aes(AGE,MORTALITY))+geom_point()
##Not allowed to post Images
g1 <- ggplot(d,aes(AGE,MORTALITY))+geom_point()
g1+geom_smooth()## with loess fit
Reported Parameters:
parConv <- c(a=0.0005893,b=0.0043836,c=0.0828424,d=0.000706,e=9.927863,f=22.197312,
g=0.00004948,h=1.10003)
parStart <- parConv
parStart["a"] <- parStart["a"]+3e-4
## Define qx = HP8/(1+HP8)
HP8 <-function(parS,x)
with(as.list(parS),
ifelse(x==0, a^((x+b)^c) + g*h^x, a^((x+b)^c) + d*exp(-e*(log(x/f))^2) + g*h^x))
qxPred <- function(parS,x) {
h <- HP8(parS,x)
h/(1+h)
}
##Define Residual Function, the relative squared distance is minimized,
ResidFun <- function(parS, Observed,x) (qxPred(parS,x)/Observed-1)
ssqfun <- function(parS, Observed, x) {
sum(ResidFun(parS, Observed, x)^2)
}
nls.out <- nls.lm(par=parStart, fn = ResidFun, Observed = d$MORTALITY, x = d$AGE,
control = nls.lm.control(nprint=1, ftol = sqrt(.Machine$double.eps),
ptol = sqrt(.Machine$double.eps), maxfev=1000, maxiter=1000))
parNLS <- coef(nls.out)
pred0 <- qxPred(as.list(parConv),d$AGE)
pred1 <- qxPred(as.list(parNLS),d$AGE)
#Binds Row wise the dataframes from pred0 and pred1
dPred <- with(d,rbind(data.frame(AGE,MORTALITY=pred0,w="conv"),
data.frame(AGE,MORTALITY=pred1,w="nls")))
g1 + geom_line(data=dPred,aes(colour=w))
round(cbind(parNLS,parConv),7)
mvec <- c('Nelder-Mead','BFGS','CG','L-BFGS-B','nlm','nlminb','spg','ucminf')
opt1 <- optimx(par=parStart, fn = ssqfun,
Observed = d$MORTALITY, x = d$AGE,
itnmax=5000,
method=mvec, control=list(all.methods=TRUE,kkt=TRUE,)
## control=list(all.methods=TRUE,kkt=TRUE)) ## Boom
get.result(opt1, attribute= c("fvalues","method", "grs", "itns",
"conv", "KKT1", "KKT2", "xtimes"))
## method fvalues grs itns conv KKT1 KKT2 xtimes
##5 nlm 8.988466e+307 NA NA 9999 NA NA 0
##4 L-BFGS-B 8.988466e+307 NULL NULL 9999 NA NA 0
##2 CG 8.988466e+307 NULL NULL 9999 NA NA 0.02
##1 BFGS 8.988466e+307 NULL NULL 9999 NA NA 0
##3 Nelder-Mead 0.5673864 NA NULL 0 NA NA 0.42
##6 nlminb 0.4127198 546 62 0 NA NA 0.17
opt2 <- nlminb(start=parStart, objective = ssqfun,
Observed = d$MORTALITY, x = d$AGE,
control= list(eval.max=5000,iter.max=5000))
parNLM <- opt2$par
Check on parameters:
round(cbind(parNLS,parConv,parNLM),5)
## parNLS parConv parNLM
##a 0.00058 0.00059 0.00058
##b 0.00369 0.00438 0.00369
##c 0.08065 0.08284 0.08065
##d 0.00070 0.00071 0.00070
##e 9.30948 9.92786 9.30970
##f 22.30769 22.19731 22.30769
##g 0.00005 0.00005 0.00005
##h 1.10084 1.10003 1.10084
SSE Review:
sapply(list(parNLS,parConv,parNLM),
ssqfun,Observed=d$MORTALITY,x=d$AGE)
##[1] 0.4127198 0.4169513 0.4127198
Not able to upload graphs but the code is here. Still appears that the parameters found in the article are not the best fit when the complete mortality data (not abridged or subset) is used
##pred2 <- qxPred(as.list(parNLM),d$AGE)
##dPred <- with(d,rbind(dPred,
data.frame(AGE,MORTALITY=pred2,w="nlminb")))
##g1 + geom_line(data=dPred,aes(colour=w))
ggplot(data=dPred,aes(x=AGE,y=MORTALITY-d$MORTALITY,colour=w))
+ geom_line()+geom_point(aes(shape=w),alpha=0.3)