R nls奇异梯度矩阵-积分中的拟合参数'；s的上限

我试图使一个

nls

适合一个稍微复杂的表达式，它包含两个积分，其中两个拟合参数在其上限

我弄错了

nlsModel（公式、mf、start、wts）中的错误：奇异梯度 初始参数估计的矩阵”

我已经在前面的答案中搜索过了，但是没有帮助。参数初始化似乎是好的，我已经尝试改变参数，但没有工作。如果我的函数只有一个积分，那么一切都很好，但是当添加第二个积分项时，就会出现错误。我不认为这个函数参数化过度，因为我已经用更多的参数进行了其他拟合，而且效果很好。下面我列出了一些数据

最简单的例子如下：

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))

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))

被积函数为参考，适用的拟合如下：

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))

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))

integrand数据来说明问题：

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)

dat您可以尝试其他一些优化器：

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