R: minpack.lm::nls.lm 失败但结果良好-6ren

R: minpack.lm::nls.lm 失败但结果良好

转载作者：行者123 更新时间：2023-12-03 02:09:36

我使用 minpack.lm 包中的 nls.lm 来拟合许多非线性模型。

由于初始参数估计时的奇异梯度矩阵，它经常在 20 次迭代后失败。

问题是当我在失败之前查看迭代(trace = T)时，我可以看到结果是好的。

可重现的示例:

数据:

df <- structure(list(x1 = c(7L, 5L, 10L, 6L, 9L, 10L, 2L, 4L, 9L, 3L, 
11L, 6L, 4L, 0L, 7L, 12L, 9L, 11L, 11L, 0L, 2L, 3L, 5L, 6L, 6L, 
9L, 1L, 7L, 7L, 4L, 3L, 13L, 12L, 13L, 5L, 0L, 5L, 6L, 6L, 7L, 
5L, 10L, 6L, 10L, 0L, 7L, 9L, 12L, 4L, 5L, 6L, 3L, 4L, 5L, 5L, 
0L, 9L, 9L, 1L, 2L, 2L, 13L, 8L, 2L, 5L, 10L, 6L, 11L, 5L, 0L, 
4L, 4L, 8L, 9L, 4L, 2L, 12L, 4L, 10L, 7L, 0L, 4L, 4L, 5L, 8L, 
8L, 12L, 4L, 6L, 13L, 5L, 12L, 1L, 6L, 4L, 9L, 11L, 11L, 6L, 
10L, 10L, 0L, 3L, 1L, 11L, 4L, 3L, 13L, 5L, 4L, 2L, 3L, 11L, 
7L, 0L, 9L, 6L, 11L, 6L, 13L, 1L, 5L, 0L, 6L, 4L, 8L, 2L, 3L, 
7L, 9L, 12L, 11L, 7L, 4L, 10L, 0L, 6L, 1L, 7L, 2L, 6L, 3L, 1L, 
6L, 10L, 12L, 7L, 7L, 6L, 6L, 1L, 7L, 8L, 7L, 7L, 5L, 7L, 10L, 
10L, 11L, 7L, 1L, 8L, 3L, 12L, 0L, 11L, 8L, 5L, 0L, 6L, 3L, 2L, 
2L, 8L, 9L, 2L, 8L, 2L, 13L, 10L, 2L, 12L, 6L, 13L, 2L, 11L, 
1L, 12L, 6L, 7L, 9L, 8L, 10L, 2L, 6L, 0L, 2L, 11L, 2L, 3L, 9L, 
12L, 1L, 11L, 11L, 12L, 4L, 6L, 9L, 1L, 4L, 1L, 8L, 8L, 6L, 1L, 
9L, 8L, 2L, 10L, 10L, 1L, 2L, 0L, 11L, 6L, 6L, 0L, 4L, 13L, 4L, 
8L, 4L, 10L, 9L, 6L, 11L, 8L, 1L, 6L, 5L, 10L, 8L, 10L, 8L, 0L, 
3L, 0L, 6L, 7L, 4L, 3L, 7L, 7L, 8L, 6L, 2L, 9L, 5L, 7L, 7L, 0L, 
7L, 2L, 5L, 5L, 7L, 5L, 7L, 8L, 6L, 1L, 2L, 6L, 0L, 8L, 10L, 
0L, 10L), x2 = c(4L, 6L, 1L, 5L, 4L, 1L, 8L, 9L, 4L, 7L, 2L, 
6L, 9L, 11L, 5L, 1L, 3L, 2L, 2L, 12L, 8L, 9L, 6L, 4L, 4L, 2L, 
9L, 6L, 6L, 6L, 8L, 0L, 0L, 0L, 8L, 10L, 7L, 7L, 4L, 5L, 5L, 
3L, 6L, 3L, 12L, 6L, 1L, 0L, 8L, 6L, 6L, 7L, 8L, 5L, 8L, 11L, 
3L, 2L, 12L, 11L, 10L, 0L, 2L, 8L, 8L, 3L, 7L, 2L, 7L, 10L, 7L, 
8L, 2L, 4L, 7L, 11L, 1L, 8L, 2L, 5L, 11L, 9L, 7L, 5L, 5L, 3L, 
1L, 8L, 4L, 0L, 5L, 0L, 12L, 5L, 9L, 1L, 2L, 0L, 5L, 0L, 2L, 
10L, 9L, 10L, 0L, 8L, 10L, 0L, 6L, 8L, 8L, 7L, 1L, 6L, 10L, 1L, 
5L, 1L, 6L, 0L, 12L, 7L, 13L, 6L, 9L, 2L, 11L, 10L, 5L, 2L, 0L, 
2L, 5L, 6L, 2L, 10L, 4L, 10L, 4L, 9L, 5L, 9L, 11L, 4L, 3L, 1L, 
6L, 3L, 7L, 7L, 10L, 3L, 3L, 6L, 3L, 7L, 4L, 1L, 0L, 1L, 4L, 
11L, 4L, 10L, 0L, 11L, 0L, 3L, 5L, 11L, 5L, 8L, 10L, 9L, 4L, 
3L, 10L, 4L, 10L, 0L, 3L, 9L, 1L, 7L, 0L, 8L, 1L, 11L, 0L, 5L, 
4L, 2L, 2L, 0L, 11L, 6L, 13L, 9L, 1L, 9L, 7L, 3L, 1L, 12L, 2L, 
2L, 1L, 6L, 4L, 2L, 10L, 6L, 10L, 2L, 3L, 4L, 9L, 2L, 5L, 10L, 
0L, 0L, 10L, 9L, 12L, 0L, 7L, 5L, 10L, 6L, 0L, 9L, 4L, 8L, 1L, 
3L, 5L, 2L, 4L, 12L, 4L, 5L, 2L, 5L, 0L, 2L, 10L, 8L, 10L, 7L, 
3L, 8L, 8L, 6L, 3L, 5L, 6L, 11L, 4L, 5L, 4L, 3L, 10L, 6L, 8L, 
6L, 7L, 4L, 8L, 5L, 3L, 7L, 12L, 8L, 4L, 11L, 2L, 3L, 12L, 1L
), x3 = c(1, 1, 1, 1, 3, 1, 0, 3, 3, 0, 3, 2, 3, 1, 2, 3, 2, 
3, 3, 2, 0, 2, 1, 0, 0, 1, 0, 3, 3, 0, 1, 3, 2, 3, 3, 0, 2, 3, 
0, 2, 0, 3, 2, 3, 2, 3, 0, 2, 2, 1, 2, 0, 2, 0, 3, 1, 2, 1, 3, 
3, 2, 3, 0, 0, 3, 3, 3, 3, 2, 0, 1, 2, 0, 3, 1, 3, 3, 2, 2, 2, 
1, 3, 1, 0, 3, 1, 3, 2, 0, 3, 0, 2, 3, 1, 3, 0, 3, 1, 1, 0, 2, 
0, 2, 1, 1, 2, 3, 3, 1, 2, 0, 0, 2, 3, 0, 0, 1, 2, 2, 3, 3, 2, 
3, 2, 3, 0, 3, 3, 2, 1, 2, 3, 2, 0, 2, 0, 0, 1, 1, 1, 1, 2, 2, 
0, 3, 3, 3, 0, 3, 3, 1, 0, 1, 3, 0, 2, 1, 1, 0, 2, 1, 2, 2, 3, 
2, 1, 1, 1, 0, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 3, 3, 1, 3, 3, 3, 
0, 2, 2, 2, 1, 1, 1, 0, 0, 3, 2, 3, 1, 2, 1, 0, 2, 3, 3, 3, 3, 
3, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 3, 2, 0, 0, 1, 1, 2, 1, 3, 
1, 0, 0, 3, 3, 2, 2, 1, 2, 1, 3, 2, 3, 0, 0, 2, 3, 0, 0, 0, 1, 
0, 3, 0, 2, 1, 3, 0, 3, 2, 3, 3, 0, 1, 0, 0, 3, 0, 1, 2, 1, 3, 
2, 1, 3, 3, 0, 0, 1, 0, 3, 2, 1), y = c(0.03688, 0.09105, 0.16246, 
0, 0.11024, 0.16246, 0.13467, 0, 0.11024, 0.0807, 0.12726, 0.03934, 
0, 0.0826, 0.03688, 0.06931, 0.1378, 0.12726, 0.12726, 0.08815, 
0.13467, 0.01314, 0.09105, 0.12077, 0.12077, 0.02821, 0.15134, 
0.03604, 0.03604, 0.08729, 0.04035, 0.46088, 0.20987, 0.46088, 
0.06672, 0.24121, 0.08948, 0.07867, 0.12077, 0.03688, 0.02276, 
0.04535, 0.03934, 0.04535, 0.08815, 0.03604, 0.50771, 0.20987, 
0.08569, 0.09105, 0.03934, 0.0807, 0.08569, 0.02276, 0.06672, 
0.0826, 0.1378, 0.02821, 0.03943, 0.03589, 0.04813, 0.46088, 
0.22346, 0.13467, 0.06672, 0.04535, 0.07867, 0.12726, 0.08948, 
0.24121, 0.06983, 0.08569, 0.22346, 0.11024, 0.06983, 0.03589, 
0.06931, 0.08569, 0.04589, 0.03688, 0.0826, 0, 0.06983, 0.02276, 
0.06238, 0.03192, 0.06931, 0.08569, 0.12077, 0.46088, 0.02276, 
0.20987, 0.03943, 0, 0, 0.50771, 0.12726, 0.1628, 0, 0.41776, 
0.04589, 0.24121, 0.01314, 0.03027, 0.1628, 0.08569, 0, 0.46088, 
0.09105, 0.08569, 0.13467, 0.0807, 0.12912, 0.03604, 0.24121, 
0.50771, 0, 0.12912, 0.03934, 0.46088, 0.03943, 0.08948, 0.07103, 
0.03934, 0, 0.22346, 0.03589, 0, 0.03688, 0.02821, 0.20987, 0.12726, 
0.03688, 0.08729, 0.04589, 0.24121, 0.12077, 0.03027, 0.03688, 
0.03673, 0, 0.01314, 0.02957, 0.12077, 0.04535, 0.06931, 0.03604, 
0.36883, 0.07867, 0.07867, 0.03027, 0.36883, 0.03192, 0.03604, 
0.36883, 0.08948, 0.03688, 0.16246, 0.41776, 0.12912, 0.03688, 
0.02957, 0.1255, 0, 0.20987, 0.0826, 0.1628, 0.03192, 0.02276, 
0.0826, 0, 0.04035, 0.04813, 0.03673, 0.1255, 0.1378, 0.04813, 
0.1255, 0.04813, 0.46088, 0.04535, 0.03673, 0.06931, 0.07867, 
0.46088, 0.13467, 0.12912, 0.02957, 0.20987, 0, 0.03688, 0.02821, 
0.22346, 0.41776, 0.03589, 0.03934, 0.07103, 0.03673, 0.12912, 
0.03673, 0.0807, 0.1378, 0.06931, 0.03943, 0.12726, 0.12726, 
0.06931, 0.08729, 0.12077, 0.02821, 0.03027, 0.08729, 0.03027, 
0.22346, 0.03192, 0.12077, 0.15134, 0.02821, 0.06238, 0.04813, 
0.41776, 0.41776, 0.03027, 0.03673, 0.08815, 0.1628, 0.07867, 
0, 0.24121, 0.08729, 0.46088, 0, 0.1255, 0.08569, 0.16246, 0.1378, 
0, 0.12726, 0.1255, 0.03943, 0.12077, 0.02276, 0.04589, 0.06238, 
0.41776, 0.22346, 0.24121, 0.04035, 0.24121, 0.07867, 0.36883, 
0.08569, 0.04035, 0.03604, 0.36883, 0.06238, 0.03934, 0.03589, 
0.11024, 0.02276, 0.03688, 0.36883, 0.24121, 0.03604, 0.13467, 
0.09105, 0.08948, 0.03688, 0.06672, 0.03688, 0.03192, 0.07867, 
0.03943, 0.13467, 0.12077, 0.0826, 0.22346, 0.04535, 0.08815, 
0.16246)), .Names = c("x1", "x2", "x3", "y"), row.names = c(995L, 
1416L, 281L, 1192L, 1075L, 294L, 1812L, 2235L, 1097L, 1583L, 
670L, 1485L, 2199L, 2495L, 1259L, 436L, 803L, 631L, 617L, 2654L, 
1813L, 2180L, 1403L, 911L, 927L, 533L, 2024L, 1517L, 1522L, 1356L, 
1850L, 222L, 115L, 204L, 1974L, 2292L, 1695L, 1746L, 915L, 1283L, 
1128L, 880L, 1467L, 887L, 2665L, 1532L, 267L, 155L, 1933L, 1447L, 
1488L, 1609L, 1922L, 1168L, 1965L, 2479L, 813L, 550L, 2707L, 
2590L, 2373L, 190L, 504L, 1810L, 2007L, 843L, 1770L, 659L, 1730L, 
2246L, 1668L, 1923L, 465L, 1108L, 1663L, 2616L, 409L, 1946L, 
589L, 1277L, 2493L, 2210L, 1662L, 1142L, 1331L, 735L, 430L, 1916L, 
922L, 208L, 1134L, 127L, 2693L, 1213L, 2236L, 240L, 623L, 108L, 
1190L, 9L, 575L, 2268L, 2171L, 2308L, 103L, 1953L, 2409L, 184L, 
1437L, 1947L, 1847L, 1570L, 365L, 1550L, 2278L, 270L, 1204L, 
384L, 1472L, 205L, 2694L, 1727L, 2800L, 1476L, 2229L, 453L, 2630L, 
2426L, 1275L, 523L, 163L, 635L, 1287L, 1349L, 561L, 2261L, 931L, 
2339L, 973L, 2113L, 1229L, 2155L, 2554L, 936L, 892L, 433L, 1560L, 
697L, 1791L, 1755L, 2351L, 720L, 740L, 1558L, 674L, 1736L, 988L, 
321L, 18L, 375L, 959L, 2560L, 1047L, 2429L, 119L, 2468L, 98L, 
773L, 1158L, 2520L, 1216L, 1872L, 2364L, 2094L, 1035L, 826L, 
2374L, 1028L, 2368L, 176L, 895L, 2090L, 399L, 1789L, 179L, 1800L, 
369L, 2568L, 140L, 1207L, 1001L, 518L, 481L, 12L, 2597L, 1474L, 
2749L, 2097L, 379L, 2110L, 1615L, 800L, 423L, 2733L, 626L, 662L, 
421L, 1363L, 898L, 530L, 2315L, 1365L, 2331L, 468L, 768L, 900L, 
2027L, 544L, 1337L, 2376L, 53L, 44L, 2338L, 2075L, 2655L, 78L, 
1782L, 1231L, 2291L, 1379L, 212L, 2212L, 1032L, 1929L, 331L, 
790L, 1226L, 664L, 1018L, 2735L, 916L, 1157L, 590L, 1343L, 7L, 
490L, 2257L, 1853L, 2251L, 1748L, 719L, 1941L, 1885L, 1544L, 
725L, 1294L, 1494L, 2601L, 1077L, 1169L, 979L, 709L, 2282L, 1526L, 
1797L, 1424L, 1690L, 993L, 1979L, 1268L, 730L, 1739L, 2697L, 
1842L, 952L, 2483L, 479L, 864L, 2677L, 283L), class = "data.frame")

起始值

starting_value <- structure(c(0.177698291502873, 0.6, 0.0761564106440883, 0.05, 
1.9, 1.1, 0.877181493020499, 1.9), .Names = c("F_initial_x2", 
"F_decay_x2", "S_initial_x2", "S_decay_x2", "initial_x1", "decay_x1", 
"initial_x3", "decay_x3"))

NLSLM 失败

coef(nlsLM( 
  formula   = y ~ (F_initial_x2   * exp(- F_decay_x2  * x2) + S_initial_x2 * exp(- S_decay_x2 * x2)) *
    (1 + initial_x1      * exp(- decay_x1      * x1)) *
    (1 + initial_x3      * exp(- decay_x3      * x3 )),
  data     = df,
  start    = coef(brute_force),
  lower    = c(0, 0, 0, 0, 0, 0, 0, 0),
  control  = nls.lm.control(maxiter = 200),
  trace    = T))

It.    0, RSS =    1.36145, Par. =   0.177698        0.6  0.0761564       0.05        1.9        1.1   0.877181        1.9
It.    1, RSS =    1.25401, Par. =   0.207931   0.581039  0.0769047  0.0577244    2.01947    1.22911   0.772957    5.67978
It.    2, RSS =    1.19703, Par. =   0.188978   0.604515  0.0722749  0.0792141    2.44179     1.1258    0.96305    8.67253
It.    3, RSS =     1.1969, Par. =   0.160885   0.640958  0.0990201   0.145187     3.5853   0.847158   0.961844    13.2183
It.    4, RSS =    1.19057, Par. =   0.142138   0.685678    0.11792   0.167417    4.27977   0.936981   0.959606    13.2644
It.    5, RSS =    1.19008, Par. =   0.124264   0.757088   0.136277   0.188896    4.76578    0.91274   0.955142    21.0167
It.    6, RSS =    1.18989, Par. =   0.118904   0.798296   0.141951   0.194167    4.93099    0.91529   0.952972     38.563
It.    7, RSS =    1.18987, Par. =   0.115771   0.821874   0.145398   0.197773    5.02251   0.914204   0.949906     38.563
It.    8, RSS =    1.18986, Par. =   0.113793   0.837804   0.147573   0.199943    5.07456   0.914192   0.948289     38.563
It.    9, RSS =    1.18986, Par. =   0.112458   0.848666   0.149033   0.201406    5.11024   0.914099   0.947232     38.563
It.   10, RSS =    1.18986, Par. =   0.111538   0.856282   0.150035   0.202411    5.13491   0.914051   0.946546     38.563
It.   11, RSS =    1.18986, Par. =   0.110889   0.861702    0.15074   0.203118    5.15244   0.914013   0.946076     38.563
It.   12, RSS =    1.18986, Par. =   0.110426   0.865606   0.151243   0.203623    5.16501   0.913986   0.945747     38.563
It.   13, RSS =    1.18986, Par. =   0.110092   0.868441   0.151605   0.203986    5.17412   0.913966   0.945512     38.563
It.   14, RSS =    1.18986, Par. =   0.109849    0.87051   0.151868    0.20425    5.18075   0.913952   0.945343     38.563
It.   15, RSS =    1.18985, Par. =   0.109672   0.872029    0.15206   0.204443    5.18561   0.913941    0.94522     38.563
It.   16, RSS =    1.18985, Par. =   0.109542   0.873147   0.152201   0.204585    5.18918   0.913933   0.945131     38.563
It.   17, RSS =    1.18985, Par. =   0.109446   0.873971   0.152305   0.204689    5.19181   0.913927   0.945065     38.563
Error in nlsModel(formula, mf, start, wts) : 
  singular gradient matrix at initial parameter estimates

问题:

使用奇异梯度矩阵问题之前找到的最佳参数(即迭代 = 17 时找到的参数)是否有意义？
如果是，有办法获取它们吗？发生错误时，我未能成功保存结果。
我注意到，如果我将 maxiter 的数量减少到 17 以下，我仍然会出现在新的最后一次迭代中出现的相同错误，这对我来说没有意义

例如 maxiter = 10

It.    0, RSS =    1.36145, Par. =   0.177698        0.6  0.0761564       0.05        1.9        1.1   0.877181        1.9
It.    1, RSS =    1.25401, Par. =   0.207931   0.581039  0.0769047  0.0577244    2.01947    1.22911   0.772957    5.67978
It.    2, RSS =    1.19703, Par. =   0.188978   0.604515  0.0722749  0.0792141    2.44179     1.1258    0.96305    8.67253
It.    3, RSS =     1.1969, Par. =   0.160885   0.640958  0.0990201   0.145187     3.5853   0.847158   0.961844    13.2183
It.    4, RSS =    1.19057, Par. =   0.142138   0.685678    0.11792   0.167417    4.27977   0.936981   0.959606    13.2644
It.    5, RSS =    1.19008, Par. =   0.124264   0.757088   0.136277   0.188896    4.76578    0.91274   0.955142    21.0167
It.    6, RSS =    1.18989, Par. =   0.118904   0.798296   0.141951   0.194167    4.93099    0.91529   0.952972     38.563
It.    7, RSS =    1.18987, Par. =   0.115771   0.821874   0.145398   0.197773    5.02251   0.914204   0.949906     38.563
It.    8, RSS =    1.18986, Par. =   0.113793   0.837804   0.147573   0.199943    5.07456   0.914192   0.948289     38.563
It.    9, RSS =    1.18986, Par. =   0.112458   0.848666   0.149033   0.201406    5.11024   0.914099   0.947232     38.563
It.   10, RSS =    0.12289, Par. =   0.112458   0.848666   0.149033   0.201406    5.11024   0.914099   0.947232     38.563
Error in nlsModel(formula, mf, start, wts) : 
  singular gradient matrix at initial parameter estimates
In addition: Warning message:
In nls.lm(par = start, fn = FCT, jac = jac, control = control, lower = lower,  :
  lmdif: info = -1. Number of iterations has reached `maxiter' == 10.

你看到任何解释了吗？

最佳答案

这个问题的根本问题是没有实现收敛。这可以通过使用 Y = log(X+1) 转换衰减参数，然后使用 X = exp(Y)-1 将它们转换回来来解决。这种变换可以有益地修改雅可比矩阵。不幸的是，这种转换的应用往往主要是反复试验。 (另请参阅注释 1。)

ix <- grep("decay", names(starting_value))
fm <- nlsLM( 
   formula   = y ~ (F_initial_x2   * exp(- log(F_decay_x2+1)  * x2) + 
                    S_initial_x2 * exp(- log(S_decay_x2+1) * x2)) *
                    (1 + initial_x1 * exp(- log(decay_x1+1) * x1)) *
                    (1 + initial_x3 * exp(- log(decay_x3+1) * x3 )),
   data     = df,
   start    = replace(starting_value, ix, exp(starting_value[ix]) - 1),
   lower    = c(0, 0, 0, 0, 0, 0, 0, 0),
   control  = nls.lm.control(maxiter = 200),
   trace    = TRUE)

给出相似的残差平方和但实现收敛:

> fm
Nonlinear regression model
  model: y ~ (F_initial_x2 * exp(-log(F_decay_x2 + 1) * x2) + S_initial_x2 *     exp(-log(S_decay_x2 + 1) * x2)) * (1 + initial_x1 * exp(-log(decay_x1 +     1) * x1)) * (1 + initial_x3 * exp(-log(decay_x3 + 1) * x3))
   data: df
F_initial_x2   F_decay_x2 S_initial_x2   S_decay_x2   initial_x1     decay_x1 
   1.092e-01    1.402e+00    1.526e-01    2.275e-01    5.199e+00    1.494e+00 
  initial_x3     decay_x3 
   9.449e-01    1.375e+07 
 residual sum-of-squares: 1.19

Number of iterations to convergence: 38 
Achieved convergence tolerance: 1.49e-08

> replace(coef(fm), ix, log(coef(fm)[ix]+1))
F_initial_x2   F_decay_x2 S_initial_x2   S_decay_x2   initial_x1     decay_x1 
   0.1091735    0.8763253    0.1525997    0.2049852    5.1993194    0.9139096 
  initial_x3     decay_x3 
   0.9448779   16.4368001

注释 1:经过一些实验，我发现只需在 decay_x3 上应用转换就足够了。

注释 2:关于您想要自动注释的评论，请注意与 lm 拟合的三次多项式将更一致地不会遇到麻烦并且具有较低的残差和平方数 - 1.14 与 1.19 - 但以更多参数为代价 - 10 与 8。

# lm poly fit
fm.poly <- lm(y ~ poly(x1, x2, degree = 3), df)
deviance(fm.poly) # residual sum of squares
## [1] 1.141398
length(coef(fm.poly)) # no. of coefficients
## [1] 10

# nlsLM fit transforming decay parameters
deviance(fm)
## [1] 1.189855
length(coef(fm))
## [1] 8

注释 3: 这是另一个模型，通过用二次多项式替换 x3 部分并在它变得多余时删除 F_initial_x2 来形成。它还有 8 个参数，它收敛并且比问题中的模型更好地拟合数据(即具有较低的残差平方和)。

fm3 <- nlsLM(formula   = y ~ (exp(- F_decay_x2  * x2) + 
              S_initial_x2 * exp(- S_decay_x2 * x2)) *
              (1 + initial_x1 * exp(- decay_x1      * x1)) *
              cbind(1, poly(x3, degree = 2)) %*% c(p1,p2,p3),
          data     = df,
          start    = c(starting_value[-c(1, 7:8)], p1=0, p2=0, p3=0),
          lower = c(0, 0, 0, 0, 0, 0, NA, NA),
          control  = nls.lm.control(maxiter = 200),
          trace    = TRUE)

给予:

> fm3
Nonlinear regression model
  model: y ~ (exp(-F_decay_x2 * x2) + S_initial_x2 * exp(-S_decay_x2 *     x2)) * (1 + initial_x1 * exp(-decay_x1 * x1)) * cbind(1,     poly(x3, degree = 2)) %*% c(p1, p2, p3)
   data: df
  F_decay_x2 S_initial_x2   S_decay_x2   initial_x1     decay_x1           p1 
     3.51614      2.60886      0.26304      8.26244      0.81232      0.09031 
          p2           p3 
    -0.16968      0.53324 
 residual sum-of-squares: 1.019

Number of iterations to convergence: 20 
Achieved convergence tolerance: 1.49e-08

注释 4: nlmrt 包中的 nlxb 无需执行任何特殊操作即可收敛。

library(nlmrt)
nlxb( 
  formula   = y ~ (F_initial_x2   * exp(- F_decay_x2  * x2) + S_initial_x2 * exp(- S_decay_x2 * x2)) *
    (1 + initial_x1      * exp(- decay_x1      * x1)) *
    (1 + initial_x3      * exp(- decay_x3      * x3 )),
  data     = df,
  start    = starting_value,
  lower    = c(0, 0, 0, 0, 0, 0, 0, 0),
  control  = nls.lm.control(maxiter = 200),
  trace    = TRUE)

给予:

residual sumsquares =  1.1899  on  280 observations
    after  31    Jacobian and  33 function evaluations
  name            coeff          SE       tstat      pval      gradient    JSingval   
F_initial_x2      0.109175            NA         NA         NA   3.372e-11        15.1  
F_decay_x2      0.876313            NA         NA         NA   -5.94e-12       8.083  
S_initial_x2      0.152598            NA         NA         NA    6.55e-11       2.163  
S_decay_x2      0.204984            NA         NA         NA   4.206e-11      0.6181  
initial_x1       5.19928            NA         NA         NA  -1.191e-12      0.3601  
decay_x1         0.91391            NA         NA         NA   6.662e-13      0.1315  
initial_x3      0.944879            NA         NA         NA   2.736e-12     0.02247  
decay_x3         33.9921            NA         NA         NA  -1.056e-15   2.928e-15

关于R: minpack.lm::nls.lm 失败但结果良好，我们在Stack Overflow上找到一个类似的问题： https://stackoverflow.com/questions/35409099/

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R: minpack.lm::nls.lm 失败但结果良好