nls singular gradient matrix - fit parameters in integral's upper limits - r

Question

I am trying to make a nls fit for a little bit complicated expression that includes two integrals with two of the fit parameters in their upper limits.
I got the error
"Error in nlsModel(formula, mf, start, wts) : singular gradient
matrix at initial parameter estimates".
I have searched already in the previous answers, but didn't help. The parameters initialization seem to be ok, I have tried to change the parameters but none work. If my function has just one integral everything works very nicely, but when adding a second integral term just got the error. I don't believe the function is over-parametrized, as I have performed other fits with much more parameters and they worked. Below I have wrote a list with some data.
The minimal example is the following:
integrand <- function(X) {
return(X^4/(2*sinh(X/2))^2)
}
fitting = function(T1, T2, N, D, x){
int1 = integrate(integrand, lower=0, upper = T1)$value
int2 = integrate(integrand, lower=0, upper = T2)$value
return(N*(D/x)^2*(exp(D/x)/(1+exp(D/x))^2
)+(448.956*(x/T1)^3*int1)+(299.304*(x/T2)^3*int2))
}
fit = nls(y ~ fitting(T1, T2, N, D, x),
start=list(T1=400,T2=200,N=0.01,D=2))
------>For reference, the fit that worked is the following:
integrand <- function(X) {
return(X^4/(2*sinh(X/2))^2)
}
fitting = function(T1, N, D, x){
int = integrate(integrand, lower=0, upper = T1)$value
return(N*(D/x)^2*(exp(D/x)/(1+exp(D/x))^2 )+(748.26)*(x/T1)^3*int)
}
fit = nls(y ~ fitting(T1 , N, D, x), start=list(T1=400,N=0.01,D=2))
------->Data to illustrate the problem:
dat<- read.table(text="x y
0.38813 0.0198
0.79465 0.02206
1.40744 0.01676
1.81532 0.01538
2.23105 0.01513
2.64864 0.01547
3.05933 0.01706
3.47302 0.01852
3.88791 0.02074
4.26301 0.0256
4.67607 0.03028
5.08172 0.03507
5.48327 0.04283
5.88947 0.05017
6.2988 0.05953
6.7022 0.07185
7.10933 0.08598
7.51924 0.0998
7.92674 0.12022
8.3354 0.1423
8.7384 0.16382
9.14656 0.19114
9.55062 0.22218
9.95591 0.25542", header=TRUE)
I cannot figure out what happen. I need to perform this fit for three integral components, but even for two I have this problem. I appreciate so much your help. Thank you.

Answer 1

You could try some other optimizers:
fitting1 <- function(par, x, y) {
sum((fitting(par[1], par[2], par[3], par[4], x) - y)^2)
}
library(optimx)
res <- optimx(c(400, 200, 0.01, 2),
fitting1,
x = DF$x, y = DF$y,
control = list(all.methods = TRUE))
print(res)
# p1 p2 p3 p4 value fevals gevals niter convcode kkt1 kkt2 xtimes
#BFGS 409.7992 288.6416 -0.7594461 39.00871 1.947484e-03 101 100 NA 1 NA NA 0.22
#CG 401.1281 210.9087 -0.9026459 20.80900 3.892929e-01 215 101 NA 1 NA NA 0.25
#Nelder-Mead 414.6402 446.5080 -1.1298606 -227.81280 2.064842e-03 89 NA NA 0 NA NA 0.02
#L-BFGS-B 412.4477 333.1338 -0.3650530 37.74779 1.581643e-03 34 34 NA 0 NA NA 0.06
#nlm 411.8639 333.4776 -0.3652356 37.74855 1.581644e-03 NA NA 45 0 NA NA 0.04
#nlminb 411.9678 333.4449 -0.3650271 37.74753 1.581643e-03 50 268 48 0 NA NA 0.07
#spg 422.0394 300.5336 -0.5776862 38.48655 1.693119e-03 1197 NA 619 0 NA NA 1.06
#ucminf 412.7390 332.9228 -0.3652029 37.74829 1.581644e-03 45 45 NA 0 NA NA 0.05
#Rcgmin NA NA NA NA 8.988466e+307 NA NA NA 9999 NA NA 0.00
#Rvmmin NA NA NA NA 8.988466e+307 NA NA NA 9999 NA NA 0.00
#newuoa 396.3071 345.1165 -0.3650286 37.74754 1.581643e-03 3877 NA NA 0 NA NA 1.02
#bobyqa 410.0392 334.7074 -0.3650289 37.74753 1.581643e-03 7866 NA NA 0 NA NA 2.07
#nmkb 569.0139 346.0856 282.6526588 -335.32320 2.064859e-03 75 NA NA 0 NA NA 0.01
#hjkb 400.0000 200.0000 0.0100000 2.00000 3.200269e+00 1 NA 0 9999 NA NA 0.01
Levenberg-Marquardt converges too, but nlsLM fails when it tries to create an nls model object from the result because the gradient matrix is singular:
library(minpack.lm)
fit <- nlsLM(y ~ fitting(T1, T2, N, D, x),
start=list(T1=412,T2=333,N=-0.36,D=38), data = DF, trace = TRUE)
#It. 0, RSS = 0.00165827, Par. = 412 333 -0.36 38
#It. 1, RSS = 0.00158186, Par. = 417.352 329.978 -0.3652 37.746
#It. 2, RSS = 0.00158164, Par. = 416.397 330.694 -0.365025 37.7475
#It. 3, RSS = 0.00158164, Par. = 416.618 330.568 -0.365027 37.7475
#It. 4, RSS = 0.00158164, Par. = 416.618 330.568 -0.365027 37.7475
#Error in nlsModel(formula, mf, start, wts) :
# singular gradient matrix at initial parameter estimates