unstable results of nlxb fit
John/Petr, I think there is an issue between a global optimum and local optima. I added a multistart loop around the code to see if I could find different solutions. Here is the code I added (I am not a great coder so please excuse any inefficiencies in this code segment):
# Multistart approach
NT <- 100
Results <- matrix(data=NA, nrow = NT, ncol=5, dimnames=list(NULL,c("SS", "A", "B", "a", "b")))
A1 <- runif(NT,0,100)
B1 <- runif(NT,0,100)
a1 <- runif(NT,0.0,0.1)
b1 <- runif(NT,0.0,0.1)
for (I in 1:NT) {
if (A1[I] > B1[I]) { # Ensure that the A'a are always the lower so that nlxb() always converge to the same values
A0 <- A1[I]
a0 <- a1[I]
A1[I] <- B1[I]
a1[I] <- b1[I]
B1[I] <- A0
b1[I] <- a0
}
fit <- nlxb(tsmes ~ A*exp(a*plast) + B* exp(b*plast), data=temp,
start=list(A=A1[I], B=B1[I], a=a1[I], b=b1[I]))
ccc <- coef(fit)
Results[I,1] <- fit$ssquares
Results[I,2] <- ccc[1]
Results[I,3] <- ccc[2]
Results[I,4] <- ccc[3]
Results[I,5] <- ccc[4]
}
Results
What I found is that the minimum SS generated at each trial had two distinct values, 417.8 and 3359.2. The A,B,a, and b values when the SS was 417.8 were all the same but I got different values for the case where the minimal SS was 3359.2. This indicates that the SS=417.8 may be the global minimum solution whereas the others are local optima. Here are the iteration results for a 100 trial multistart:
Results
SS A B a b
[1,] 3359.2 8.3546e+03 6.8321e+00 -1.988226 2.6139e-02
[2,] 3359.2 8.2865e+03 6.8321e+00 -5.201735 2.6139e-02
[3,] 417.8 3.9452e-13 9.7727e+00 0.280227 2.1798e-02
[4,] 3359.2 6.8321e+00 7.7888e+02 0.026139 -7.2812e-01
[5,] 3359.2 -3.9020e+01 4.5852e+01 0.026139 2.6139e-02
[6,] 3359.2 6.8321e+00 2.6310e+02 0.026139 -1.8116e+00
[7,] 3359.2 -2.1509e+01 2.8341e+01 0.026139 2.6139e-02
[8,] 3359.2 -3.8075e+01 4.4908e+01 0.026139 2.6139e-02
[9,] 417.8 3.9452e-13 9.7727e+00 0.280227 2.1798e-02
[10,] 3359.2 1.2466e+04 6.8321e+00 -4.196000 2.6139e-02
[11,] 417.8 9.7727e+00 3.9452e-13 0.021798 2.8023e-01
[12,] 417.8 3.9452e-13 9.7727e+00 0.280227 2.1798e-02
[13,] 417.8 3.9452e-13 9.7727e+00 0.280227 2.1798e-02
[14,] 3359.2 3.8018e+02 6.8321e+00 -0.806414 2.6139e-02
[15,] 3359.2 -3.1921e+00 1.0024e+01 0.026139 2.6139e-02
[16,] 417.8 3.9452e-13 9.7727e+00 0.280227 2.1798e-02
[17,] 3359.2 -1.5938e+01 2.2770e+01 0.026139 2.6139e-02
[18,] 3359.2 -3.1205e+01 3.8037e+01 0.026139 2.6139e-02
[19,] 417.8 3.9452e-13 9.7727e+00 0.280227 2.1798e-02
[20,] 417.8 3.9452e-13 9.7727e+00 0.280227 2.1798e-02
[21,] 3359.2 8.6627e+03 6.8321e+00 -3.319778 2.6139e-02
[22,] 3359.2 6.8321e+00 1.9318e+01 0.026139 -6.5773e-01
[23,] 3359.2 6.2991e+01 -5.6159e+01 0.026139 2.6139e-02
[24,] 3359.2 2.8865e-03 6.8321e+00 -1.576307 2.6139e-02
[25,] 3359.2 -1.2496e+01 1.9328e+01 0.026139 2.6139e-02
[26,] 3359.2 -5.9432e+00 1.2775e+01 0.026139 2.6139e-02
[27,] 3359.2 1.6884e+02 6.8321e+00 -211.866423 2.6139e-02
[28,] 417.8 3.9452e-13 9.7727e+00 0.280227 2.1798e-02
[29,] 3359.2 5.4972e+03 6.8321e+00 -3.432094 2.6139e-02
[30,] 3359.2 6.8321e+00 1.4427e+03 0.026139 -4.2771e+02
[31,] 417.8 9.7727e+00 3.9452e-13 0.021798 2.8023e-01
[32,] 3359.2 3.5760e+01 -2.8928e+01 0.026139 2.6139e-02
[33,] 3359.2 6.8321e+00 -4.0737e+02 0.026139 -6.7152e-01
[34,] 3359.2 6.8321e+00 1.2638e+04 0.026139 -2.8070e+00
[35,] 3359.2 1.1813e+01 -4.9807e+00 0.026139 2.6139e-02
[36,] 417.8 3.9452e-13 9.7727e+00 0.280227 2.1798e-02
[37,] 3359.2 6.8321e+00 1.2281e+03 0.026139 -3.0702e+02
[38,] 417.8 3.9452e-13 9.7727e+00 0.280227 2.1798e-02
[39,] 3359.2 -2.6850e+01 3.3682e+01 0.026139 2.6139e-02
[40,] 417.8 3.9452e-13 9.7727e+00 0.280227 2.1798e-02
[41,] 417.8 9.7727e+00 3.9452e-13 0.021798 2.8023e-01
[42,] 3359.2 -2.3279e+01 3.0111e+01 0.026139 2.6139e-02
[43,] 417.8 3.9452e-13 9.7727e+00 0.280227 2.1798e-02
[44,] 3359.2 6.8321e+00 1.4550e+03 0.026139 -4.0303e+00
[45,] 3359.2 -1.1386e+01 1.8218e+01 0.026139 2.6139e-02
[46,] 3359.2 8.8026e+02 6.8321e+00 -65.430608 2.6139e-02
[47,] 3359.2 -8.1985e+00 1.5031e+01 0.026139 2.6139e-02
[48,] 3359.2 -6.7767e+00 1.3609e+01 0.026139 2.6139e-02
[49,] 3359.2 -1.1436e+01 1.8268e+01 0.026139 2.6139e-02
[50,] 3359.2 1.0710e+04 6.8321e+00 -2.349659 2.6139e-02
[51,] 417.8 9.7727e+00 3.9452e-13 0.021798 2.8023e-01
[52,] 3359.2 6.8321e+00 7.1837e+02 0.026139 -7.4681e-01
[53,] 417.8 3.9452e-13 9.7727e+00 0.280227 2.1798e-02
[54,] 417.8 9.7727e+00 3.9452e-13 0.021798 2.8023e-01
[55,] 3359.2 -4.8774e+00 6.8321e+00 -16.405584 2.6139e-02
[56,] 3359.2 1.2687e+03 6.8321e+00 -3.775998 2.6139e-02
[57,] 3359.2 1.5529e+01 -8.6967e+00 0.026139 2.6139e-02
[58,] 3359.2 -1.0003e+01 1.6835e+01 0.026139 2.6139e-02
[59,] 3359.2 6.8321e+00 3.9291e+02 0.026139 -4.1974e+02
[60,] 3359.2 -2.1880e+01 2.8712e+01 0.026139 2.6139e-02
[61,] 3359.2 4.1736e+03 6.8321e+00 -10.711457 2.6139e-02
[62,] 3359.2 -3.3185e+01 4.0017e+01 0.026139 2.6139e-02
[63,] 3359.2 7.6732e+02 6.8321e+00 -0.723977 2.6139e-02
[64,] 3359.2 1.5334e+04 6.8321e+00 -52.573620 2.6139e-02
[65,] 3359.2 -2.9556e+01 3.6388e+01 0.026139 2.6139e-02
[66,] 3359.2 -1.0447e+00 7.8767e+00 0.026139 2.6139e-02
[67,] 3359.2 6.8321e+00 2.1471e+02 0.026139 -7.0582e+01
[68,] 417.8 9.7727e+00 3.9452e-13 0.021798 2.8023e-01
[69,] 3359.2 -2.2293e+01 2.9126e+01 0.026139 2.6139e-02
[70,] 3359.2 6.2259e+02 6.8321e+00 -2.782527 2.6139e-02
[71,] 3359.2 -1.4639e+01 2.1471e+01 0.026139 2.6139e-02
[72,] 417.8 3.9452e-13 9.7727e+00 0.280227 2.1798e-02
[73,] 417.8 3.9452e-13 9.7727e+00 0.280227 2.1798e-02
[74,] 417.8 3.9452e-13 9.7727e+00 0.280227 2.1798e-02
[75,] 3359.2 -2.3449e+01 3.0281e+01 0.026139 2.6139e-02
[76,] 3359.2 -2.5926e+01 6.8321e+00 -0.663656 2.6139e-02
[77,] 417.8 9.7727e+00 3.9452e-13 0.021798 2.8023e-01
[78,] 3359.2 6.8321e+00 6.9426e+02 0.026139 -1.9442e+00
[79,] 3359.2 2.8684e+02 6.8321e+00 -0.854394 2.6139e-02
[80,] 417.8 3.9452e-13 9.7727e+00 0.280227 2.1798e-02
[81,] 3359.2 -4.5066e+01 5.1899e+01 0.026139 2.6139e-02
[82,] 3359.2 4.4678e+03 6.8321e+00 -2.109446 2.6139e-02
[83,] 3359.2 3.1376e+03 6.8321e+00 -1.104803 2.6139e-02
[84,] 3359.2 6.8321e+00 1.1167e+02 0.026139 -1.0280e+00
[85,] 417.8 3.9452e-13 9.7727e+00 0.280227 2.1798e-02
[86,] 3359.2 5.3864e+02 6.8321e+00 -0.657971 2.6139e-02
[87,] 3359.2 4.8227e+01 6.8321e+00 -2.304024 2.6139e-02
[88,] 3359.2 -2.2048e+01 2.8880e+01 0.026139 2.6139e-02
[89,] 417.8 3.9452e-13 9.7727e+00 0.280227 2.1798e-02
[90,] 3359.2 6.8321e+00 -4.1689e+01 0.026139 -3.6049e+00
[91,] 417.8 9.7727e+00 3.9452e-13 0.021798 2.8023e-01
[92,] 3359.2 -4.1265e+01 4.8097e+01 0.026139 2.6139e-02
[93,] 3359.2 -1.1565e+01 1.8397e+01 0.026139 2.6139e-02
[94,] 3359.2 2.3698e+01 -1.6866e+01 0.026139 2.6139e-02
[95,] 3359.2 4.4700e+03 6.8321e+00 -12.836180 2.6139e-02
[96,] 3359.2 4.6052e+04 6.8321e+00 -7.158584 2.6139e-02
[97,] 3359.2 2.5464e+03 6.8321e+00 -1.811626 2.6139e-02
[98,] 3359.2 6.8321e+00 1.0338e+03 0.026139 -1.5365e+01
[99,] 3359.2 1.3783e+01 -6.9507e+00 0.026139 2.6139e-02
[100,] 3359.2 6.8321e+00 6.7153e+02 0.026139 -1.5975e+03
Hope this helps,
Bernard McGarvey
Director, Fort Myers Beach Lions Foundation, Inc.
Retired (Lilly Engineering Fellow).
On May 7, 2020 at 9:33 AM J C Nash <profjcnash at gmail.com> wrote: The double exponential is well-known as a disaster to fit. Lanczos in his 1956 book Applied Analysis, p. 276 gives a good example which is worked through. I've included it with scripts using nlxb in my 2014 book on Nonlinear Parameter Optimization Using R Tools (Wiley). The scripts were on Wiley's site for the book, but I've had difficulty getting Wiley to fix things and not checked lately if it is still accessible. Ask off-list if you want the script and I'll dig into my archives. nlxb (preferably from nlsr which you used rather than nlmrt which is now not maintained), will likely do as well as any general purpose code. There may be special approaches that do a bit better, but I suspect the reality is that the underlying problem is such that there are many sets of parameters with widely different values that will get quite similar sums of squares. Best, JN On 2020-05-07 9:12 a.m., PIKAL Petr wrote:
Dear all I started to use nlxb instead of nls to get rid of singular gradient error. I try to fit double exponential function to my data, but results I obtain are strongly dependent on starting values. tsmes ~ A*exp(a*plast) + B* exp(b*plast) Changing b from 0.1 to 0.01 gives me completely different results. I usually check result by a plot but could the result be inspected if it achieved good result without plotting? Or is there any way how to perform such task? Cheers Petr Below is working example.
dput(temp)
temp <- structure(list(tsmes = c(31, 32, 32, 32, 32, 32, 32, 32, 33,
34, 35, 35, 36, 36, 36, 37, 38, 39, 40, 40, 40, 40, 40, 41, 43,
44, 44, 44, 46, 47, 47, 47, 47, 48, 49, 51, 51, 51, 52, 53, 54,
54, 55, 57, 57, 57, 59, 59, 60, 62, 63, 64, 65, 66, 66, 67, 67,
68, 70, 72, 74, 76, 78, 81, 84, 85, 86, 88, 90, 91, 92, 94, 96,
97, 99, 100, 102, 104, 106, 109, 112, 115, 119, 123, 127, 133,
141, 153, 163, 171), plast = c(50, 51, 52, 52, 53, 53, 53, 54,
55, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 64, 64, 65, 65, 66,
66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 75, 76, 76, 77, 77, 78,
78, 79, 80, 81, 82, 83, 84, 85, 85, 86, 86, 87, 88, 88, 89, 90,
91, 91, 93, 93, 94, 95, 96, 96, 97, 98, 98, 99, 100, 100, 101,
102, 103, 103, 104, 105, 106, 107, 107, 108, 109, 110, 111, 112,
112, 113, 113, 114, 115, 116)), row.names = 2411:2500, class = "data.frame")
library(nlsr)
fit <- nlxb(tsmes ~ A*exp(a*plast) + B* exp(b*plast), data=temp,
start=list(A=1, B=15, a=0.025, b=0.01))
coef(fit)
A B a b
3.945167e-13 9.772749e+00 2.802274e-01 2.179781e-02
plot(temp$plast, temp$tsmes, ylim=c(0,200))
lines(temp$plast, predict(fit, newdata=temp), col="pink", lwd=3)
ccc <- coef(fit)
lines(0:120,ccc[1]*exp(ccc[3]*(0:120)))
lines(0:120,ccc[2]*exp(ccc[4]*(0:120)), lty=3, lwd=2)
# wrong fit with slightly different b
fit <- nlxb(tsmes ~ A*exp(a*plast) + B* exp(b*plast), data=temp,
start=list(A=1, B=15, a=0.025, b=0.1))
coef(fit)
A B a b
2911.6448377 6.8320597 -49.1373979 0.0261391
lines(temp$plast, predict(fit, newdata=temp), col="red", lwd=3)
ccc <- coef(fit)
lines(0:120,ccc[1]*exp(ccc[3]*(0:120)), col="red")
lines(0:120,ccc[2]*exp(ccc[4]*(0:120)), lty=3, lwd=2, col="red")
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______________________________________________ R-help at r-project.org mailing list -- To UNSUBSCRIBE and more, see https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.