4

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.

erasmortg
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LauraT
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1 Answers1

2

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
Roland
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