在拟合两个参数的上下文中，“nlsLM”的无非缺失参数是什么？
What is no non-missing arguments for `nlsLM` in the context of fitting two parameters?
我的目标是使用 nlsLM 或 nls 来优化我公式中的参数 p 和 U。但是，我正在 运行 解决算法问题，如下所示，它似乎不满足 nlsLM 所以它可以 运行。下面是更多详细信息，但是有人可以帮我弄清楚在下面的示例中我哪里出错了吗？
df1<-structure(list(a = c(0.826386065758906, 0.804841782886437, 0.783325233355932,
0.760569303063664, 0.727963278438188, 0.703970672785107, 0.680589589769403,
0.65474843620048, 0.629508566083339, 0.602417701399354, 0.576536805662077,
0.548851189058063, 0.519947188978781, 0.490433240798225, 0.462086422390147,
0.431909413297801, 0.403590803993467, 0.373435985401716, 0.343943659176956,
0.31380487904441, 0.284871585752978, 0.254086078501208, 0.231342598159761,
0.198724947119965, 0.16612702603107, 0.141555650677377, 0.109001084106243,
0.0831572469086037, 0.0579197953048283, 0.0332801640177642, 0.00921287687980588,
-0.00683461114689798, -0.0308662529136221, -0.0468697128496984,
-0.0634535329550516, -0.0788157766432729, -0.0947585282387919,
-0.112551607962292, -0.121770647911843, -0.131005124862407, -0.13105454830848,
-0.13913263731113, -0.141015879860184, -0.141727067372491, -0.15092263010162,
-0.151527379077436, -0.152155929282014, -0.144792734494917, -0.146630537024706,
-0.139308277987739, -0.142395755120843, -0.135627970868379, -0.130069772180454,
-0.134347052454578, -0.122661184175998, -0.119541926090815, -0.117691480771668,
-0.0980210085445172, -0.0900039693582695, -0.0734011206583904,
-0.0654294710491197, -0.0488666582927202, -0.0334934180561017,
-0.0181602076105936, -0.00403680892262948, 0.00958389551677818,
0.023138063507762, 0.0360154200368275, 0.0470535195180641, 0.0574121363142126,
0.0819938322538736, 0.104326976159514, 0.118669575746475, 0.123156717504759,
0.12529037342488, 0.11004928326779, 0.111288254062499, 0.112616123288566,
0.123770126197535, 0.125825836909568, 0.12600008978006, 0.133135092174578,
0.13150137631796, 0.137318345531007, 0.142490352653999, 0.127578135774472,
0.11881443261832, 0.101364392587429, 0.0930057723054443, 0.0753897222358955,
0.0657226581165791, 0.0559810295585592, 0.0486890523393608, 0.0359839335111634,
0.0239424532478925, 0.0137758789173201, 0.01148034494644, 0.00959075043614099,
0.00965506497270852, 0.0010321021197246, -0.00878784052445702,
-0.0186955496477701, -0.0286825539248294, -0.0387428469291482,
-0.0457835827509943, -0.0533022515989874, -0.067799838787316,
-0.0740879165672261, -0.0704910195102784, -0.0749170995555968,
-0.0892351910852938, -0.0952381116284169, -0.101154870134375,
-0.116713134920168, -0.123043579564404, -0.122229090993963, -0.131574286614478,
-0.139872347069247, -0.150352573559005, -0.160985422263815, -0.183477294368611,
-0.196028087077558), b = c(2.73720919401795e-07, 2.89275866243833e-07,
3.05704744509841e-07, 3.23056147643161e-07, 3.41383503107939e-07,
3.60740020118146e-07, 3.81182894669021e-07, 4.02772468186816e-07,
4.25573817135869e-07, 4.49652809926895e-07, 4.75077190571432e-07,
5.01924516454949e-07, 5.30273542459307e-07, 5.60209325607651e-07,
5.91819764376075e-07, 6.25191565024464e-07, 6.60423815060183e-07,
6.97623624698831e-07, 7.36899824259678e-07, 7.78364716293996e-07,
8.22141727880177e-07, 8.68358553356049e-07, 9.17146942649445e-07,
9.68648499282585e-07, 1.02301254228321e-06, 1.08039295361324e-06,
1.14095522628006e-06, 1.20488164276889e-06, 1.27235309266431e-06,
1.34356428051965e-06, 1.41872017661923e-06, 1.49803720002012e-06,
1.58174837208432e-06, 1.67008400106882e-06, 1.76329669689182e-06,
1.86165861610844e-06, 1.96545125564181e-06, 2.07495771940026e-06,
2.19051033032321e-06, 2.31244676896853e-06, 2.44109344748083e-06,
2.57682364247604e-06, 2.72002332780748e-06, 2.87110867939643e-06,
3.03049986417657e-06, 3.19862819087422e-06, 3.37598849722003e-06,
3.56309366647097e-06, 3.76046212631028e-06, 3.96865098343192e-06,
4.18824726536018e-06, 4.41986951018994e-06, 4.66416943984418e-06,
4.92183372125163e-06, 5.1935858199306e-06, 5.48018795087751e-06,
5.78244313168551e-06, 6.1011973433322e-06, 6.43734180406693e-06,
6.7918153624452e-06, 7.16563024757515e-06, 7.55980712352996e-06,
7.97541858133744e-06, 8.41364335579908e-06, 8.87569896478336e-06,
9.36286778257749e-06, 9.87650045747503e-06, 1.04180531601306e-05,
1.09889579629179e-05, 1.15908264520961e-05, 1.22254359968269e-05,
1.28944314880812e-05, 1.35996142589735e-05, 1.43429180537019e-05,
1.51263316467067e-05, 1.59520940241802e-05, 1.6822522665106e-05,
1.77398981706101e-05, 1.87067840492661e-05, 1.97258854864512e-05,
2.07998638330026e-05, 2.1931710964562e-05, 2.3124583860732e-05,
2.43817436226076e-05, 2.57064566766011e-05, 2.71023147452351e-05,
2.85732719977727e-05, 3.01232366788732e-05, 3.17564077655259e-05,
3.34772050137959e-05, 3.52903930671811e-05, 3.72007656749664e-05,
3.92134811622193e-05, 4.13339600995988e-05, 4.35677774133232e-05,
4.59210633579047e-05, 4.84002939774841e-05, 5.10121441928917e-05,
5.37634740647825e-05, 5.66616536161489e-05, 5.97142485508226e-05,
6.29295921258743e-05, 6.63164729686435e-05, 6.98835178901844e-05,
7.36402014328656e-05, 7.75967281240777e-05, 8.17636166483606e-05,
8.61519334015006e-05, 9.07733206985704e-05, 9.56403286306054e-05,
0.000100765247971781, 0.000106161936108279, 0.000111844981550703,
0.000117828340858234, 0.00012412884956377, 0.000130763552542832,
0.000137749179512345, 0.00014510363839137, 0.000152846660963375,
0.000160998542157569, 0.000169580127136327, 0.000178614349528655
), x = c(7.61206469829168e-10, 8.04880837026198e-10, 8.5102453143301e-10,
8.99793134300951e-10, 9.51328437405998e-10, 1.00578580312419e-09,
1.06332922814733e-09, 1.12413181831763e-09, 1.18838421310043e-09,
1.25626386645407e-09, 1.32797298152365e-09, 1.40374357927037e-09,
1.48378414368651e-09, 1.56836357338967e-09, 1.65770477079087e-09,
1.75207245412028e-09, 1.85175816475582e-09, 1.95707152824522e-09,
2.06831403251254e-09, 2.18581608289076e-09, 2.30994161115441e-09,
2.44104487535132e-09, 2.57951412571437e-09, 2.72575899152997e-09,
2.88021164601164e-09, 3.04330792286779e-09, 3.21554668918285e-09,
3.39745762416176e-09, 3.58953223142937e-09, 3.7923812298687e-09,
4.00655143334275e-09, 4.2327224845197e-09, 4.47153210287025e-09,
4.72364822571277e-09, 4.9898357506258e-09, 5.27087200511913e-09,
5.56757681151882e-09, 5.88073754993049e-09, 6.21145699001951e-09,
6.56054428940019e-09, 6.92905211001358e-09, 7.31804978053444e-09,
7.72866489630656e-09, 8.16213989148934e-09, 8.6196245901883e-09,
9.1024320192891e-09, 9.61207173210311e-09, 1.0149956868384e-08,
1.07176367759663e-08, 1.1316744903592e-08, 1.19490033177631e-08,
1.26162274595822e-08, 1.33203311541876e-08, 1.4063331886057e-08,
1.48473563541273e-08, 1.5674646321422e-08, 1.65475647746164e-08,
1.74686024097974e-08, 1.84403844614874e-08, 1.94656778929232e-08,
2.05475322857365e-08, 2.16887618647287e-08, 2.28927322096252e-08,
2.41628570299852e-08, 2.55027338320421e-08, 2.69161537094373e-08,
2.84071116498747e-08, 2.99800111976379e-08, 3.16387068089972e-08,
3.33886696287756e-08, 3.52346661692384e-08, 3.71814450102107e-08,
3.92346911195274e-08, 4.13999001075981e-08, 4.36830992974598e-08,
4.60912297232476e-08, 4.86307601828189e-08, 5.13084564066871e-08,
5.41324352378923e-08, 5.71102577697511e-08, 6.02498314193268e-08,
6.35606456850457e-08, 6.70515362160507e-08, 7.07326494651599e-08,
7.46128341939538e-08, 7.87042035557664e-08, 8.30176288376039e-08,
8.75650370123332e-08, 9.23589889309948e-08, 9.74127126516464e-08,
1.02740794822333e-07, 1.08356627800041e-07, 1.14277010988554e-07,
1.20516791595619e-07, 1.27093789588104e-07, 1.34026039967841e-07,
1.41333432178137e-07, 1.49035156991607e-07, 1.57152272804214e-07,
1.65706950725174e-07, 1.74721422340645e-07, 1.84222422717041e-07,
1.9423476847285e-07, 2.04784394461598e-07, 2.15901129942397e-07,
2.2761514326591e-07, 2.39958190684945e-07, 2.52963699037024e-07,
2.66666852685233e-07, 2.81106463170876e-07, 2.96316174150964e-07,
3.12344318635138e-07, 3.29228453259451e-07, 3.47013702100588e-07,
3.65754459931466e-07, 3.85496913679509e-07, 4.06294045238492e-07,
4.28198911519565e-07, 4.51275434571536e-07, 4.75579797828766e-07,
5.01179638398094e-07, 5.28143286489079e-07), J = c(314.96456,
315.0479, 315.13122, 315.21455, 315.29788, 315.38121, 315.46454,
315.54787, 315.63121, 315.71454, 315.79786, 315.88119, 315.96451,
316.04785, 316.13118, 316.2145, 316.29782, 316.38115, 316.46448,
316.54781, 316.63115, 316.71449, 316.79783, 316.88117, 316.96451,
317.04784, 317.13117, 317.21451, 317.29784, 317.38118, 317.46451,
317.54785, 317.63119, 317.71452, 317.79785, 317.88118, 317.96451,
318.04782, 318.13116, 318.21449, 318.29782, 318.38115, 318.46448,
318.54782, 318.63115, 318.71447, 318.7978, 318.88113, 318.96446,
319.04779, 319.13112, 319.21445, 319.29778, 319.38111, 319.46444,
319.54777, 319.6311, 319.71443, 319.79776, 319.88109, 319.96443,
320.04776, 320.13109, 320.21442, 320.29775, 320.38108, 320.46441,
320.54775, 320.63107, 320.71441, 320.79776, 320.8811, 320.96444,
321.04777, 321.13109, 321.21442, 321.29775, 321.38107, 321.4644,
321.54773, 321.63105, 321.71438, 321.79771, 321.88105, 321.96437,
322.0477, 322.13103, 322.21436, 322.29769, 322.38102, 322.46436,
322.54769, 322.63103, 322.71436, 322.79769, 322.88102, 322.96436,
323.0477, 323.13104, 323.21438, 323.29771, 323.38105, 323.46439,
323.54772, 323.63105, 323.71438, 323.79771, 323.88104, 323.96437,
324.04771, 324.13103, 324.21437, 324.2977, 324.38102, 324.46436,
324.5477, 324.63104, 324.71437, 324.79771, 324.88104, 324.96437,
325.0477)), class = "data.frame", row.names = c(NA, 122L))
# package(s):
library("minpack.lm")# used for nlsLM()
# objects:
U <-550
K <- 273.15
p<-77+K
R <- 0.00831
# used to fix diff() complications in formula:
a1<-df1$a[-1]
b1<-df1$b[-1]
J2<-df1$J[-1]
J1<-df1$J[-nrow(df1)]
# try , apparently succeeded if GGrothendieck's answer uses it.
df2 <- data.frame(a1,b1,J2,J1)
# optimizing parameters U & p to fit a1:
summary(nlsLM(formula=a1 ~U *(
(1/ (1 + exp((U/ R)*((1/J2) - (1/p)))))
-(1/ (1 + exp((U/ R)*((1/ J1) - (1/p)))))
/(J2-J1)),
data = df2,
start = list(U=550),p=list(p),
lower=c(300,50),upper=c(600,85),
control= nls.lm.control(maxiter=1000),
trace = F))
错误：
Error in model.frame.default(formula = ~a1 + J2 + J1, data = df1, p = list(p)) :
invalid type (NULL) for variable 'J2'
In addition: Warning messages:
1: In min(x) : no non-missing arguments to min; returning Inf
2: In max(x) : no non-missing arguments to max; returning -Inf
试图通过 upper 和 lower 限制解决此问题 algorithm="port":
summary(nlsLM(formula=a1 ~U *(
(1/ (1 + exp((U/ R)*((1/J2) - (1/p)))))
-(1/ (1 + exp((U/ R)*((1/ J1) - (1/p)))))
/(J2-J1)),
data = df1,
start = list(U=550),p=list(p),
algorithm = "port",
lower=c(300,365),upper=c(600,324),
control= nls.lm.control(maxiter=1000),
trace = F))
但是，我实现了这个错误：
Error in upper(600, 324) : could not find function "upper"
In addition: Warning messages:
1: In min(x) : no non-missing arguments to min; returning Inf
2: In max(x) : no non-missing arguments to max; returning -Inf
然而，在另一个不等同于上面的公式的替代公式中，我 nlsLM 运行 没有 algorithm= 就可以了：
summary(nls(formula= a1~U*(diff(x)/diff(J-K)),
data=d, start=list(U=552),
control=nls.control(maxiter=1000),trace=F))
修复语法错误，注意只有nls algorithm = "port"支持upper和lower，而且问题对p的值不敏感。如果我们 运行 nls 重复将 p 固定为 50 到 85 之间的每个整数值并对 U 进行优化，我们每次都会得到相同的结果（相同的 U，相同的残差平方和）。
res <- lapply(50:85, function(p) {
fo <- a1 ~U *(
(1/ (1 + exp((U/ R)*((1/J2) - (1/p)))))-
(1/ (1 + exp((U/ R)*((1/ J1) - (1/p))))) / (J2-J1))
nls(fo, df2, start = list(U = 550),
lower = 300, upper = 600, algorithm = "port")
})
# all iterations converged
all(sapply(res, function(x) x$convInfo$isConv))
## [1] TRUE
# same U regardless of p
unname(sapply(res, coef))
## [1] 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300
# same residual sum of squares regardless of p
unname(sapply(res, deviance))
## [1] 1317808551 1317808551 1317808551 1317808551 1317808551 1317808551 1317808551 1317808551 1317808551 1317808551 1317808551 1317808551 1317808551 1317808551
## [15] 1317808551 1317808551 1317808551 1317808551 1317808551 1317808551 1317808551 1317808551 1317808551 1317808551 1317808551 1317808551 1317808551 1317808551
## [29] 1317808551 1317808551 1317808551 1317808551 1317808551 1317808551 1317808551 1317808551
因此我们可以将 p 固定为我们希望的极限之间的任何值并优化 U，即我们可以选择 res 的任何组件或根据大残差和使用其他模型平方和 J1 几乎等于 J2
with(df2, var(J2 - J1))
## [1] 4.279614e-11
即使是简单的线性模型也给出了低得多的残差平方和：
fm2 <- lm(a1 ~ b1 + I(J1 - J2) + 0, df2)
deviance(fm2)
## [1] 6.015689
备注
R <- 0.00831
df2 <-
structure(list(a1 = c(0.804841782886437, 0.783325233355932, 0.760569303063664,
0.727963278438188, 0.703970672785107, 0.680589589769403, 0.65474843620048,
0.629508566083339, 0.602417701399354, 0.576536805662077, 0.548851189058063,
0.519947188978781, 0.490433240798225, 0.462086422390147, 0.431909413297801,
0.403590803993467, 0.373435985401716, 0.343943659176956, 0.31380487904441,
0.284871585752978, 0.254086078501208, 0.231342598159761, 0.198724947119965,
0.16612702603107, 0.141555650677377, 0.109001084106243, 0.0831572469086037,
0.0579197953048283, 0.0332801640177642, 0.00921287687980588,
-0.00683461114689798, -0.0308662529136221, -0.0468697128496984,
-0.0634535329550516, -0.0788157766432729, -0.0947585282387919,
-0.112551607962292, -0.121770647911843, -0.131005124862407, -0.13105454830848,
-0.13913263731113, -0.141015879860184, -0.141727067372491, -0.15092263010162,
-0.151527379077436, -0.152155929282014, -0.144792734494917, -0.146630537024706,
-0.139308277987739, -0.142395755120843, -0.135627970868379, -0.130069772180454,
-0.134347052454578, -0.122661184175998, -0.119541926090815, -0.117691480771668,
-0.0980210085445172, -0.0900039693582695, -0.0734011206583904,
-0.0654294710491197, -0.0488666582927202, -0.0334934180561017,
-0.0181602076105936, -0.00403680892262948, 0.00958389551677818,
0.023138063507762, 0.0360154200368275, 0.0470535195180641, 0.0574121363142126,
0.0819938322538736, 0.104326976159514, 0.118669575746475, 0.123156717504759,
0.12529037342488, 0.11004928326779, 0.111288254062499, 0.112616123288566,
0.123770126197535, 0.125825836909568, 0.12600008978006, 0.133135092174578,
0.13150137631796, 0.137318345531007, 0.142490352653999, 0.127578135774472,
0.11881443261832, 0.101364392587429, 0.0930057723054443, 0.0753897222358955,
0.0657226581165791, 0.0559810295585592, 0.0486890523393608, 0.0359839335111634,
0.0239424532478925, 0.0137758789173201, 0.01148034494644, 0.00959075043614099,
0.00965506497270852, 0.0010321021197246, -0.00878784052445702,
-0.0186955496477701, -0.0286825539248294, -0.0387428469291482,
-0.0457835827509943, -0.0533022515989874, -0.067799838787316,
-0.0740879165672261, -0.0704910195102784, -0.0749170995555968,
-0.0892351910852938, -0.0952381116284169, -0.101154870134375,
-0.116713134920168, -0.123043579564404, -0.122229090993963, -0.131574286614478,
-0.139872347069247, -0.150352573559005, -0.160985422263815, -0.183477294368611,
-0.196028087077558), b1 = c(2.89275866243833e-07, 3.05704744509841e-07,
3.23056147643161e-07, 3.41383503107939e-07, 3.60740020118146e-07,
3.81182894669021e-07, 4.02772468186816e-07, 4.25573817135869e-07,
4.49652809926895e-07, 4.75077190571432e-07, 5.01924516454949e-07,
5.30273542459307e-07, 5.60209325607651e-07, 5.91819764376075e-07,
6.25191565024464e-07, 6.60423815060183e-07, 6.97623624698831e-07,
7.36899824259678e-07, 7.78364716293996e-07, 8.22141727880177e-07,
8.68358553356049e-07, 9.17146942649445e-07, 9.68648499282585e-07,
1.02301254228321e-06, 1.08039295361324e-06, 1.14095522628006e-06,
1.20488164276889e-06, 1.27235309266431e-06, 1.34356428051965e-06,
1.41872017661923e-06, 1.49803720002012e-06, 1.58174837208432e-06,
1.67008400106882e-06, 1.76329669689182e-06, 1.86165861610844e-06,
1.96545125564181e-06, 2.07495771940026e-06, 2.19051033032321e-06,
2.31244676896853e-06, 2.44109344748083e-06, 2.57682364247604e-06,
2.72002332780748e-06, 2.87110867939643e-06, 3.03049986417657e-06,
3.19862819087422e-06, 3.37598849722003e-06, 3.56309366647097e-06,
3.76046212631028e-06, 3.96865098343192e-06, 4.18824726536018e-06,
4.41986951018994e-06, 4.66416943984418e-06, 4.92183372125163e-06,
5.1935858199306e-06, 5.48018795087751e-06, 5.78244313168551e-06,
6.1011973433322e-06, 6.43734180406693e-06, 6.7918153624452e-06,
7.16563024757515e-06, 7.55980712352996e-06, 7.97541858133744e-06,
8.41364335579908e-06, 8.87569896478336e-06, 9.36286778257749e-06,
9.87650045747503e-06, 1.04180531601306e-05, 1.09889579629179e-05,
1.15908264520961e-05, 1.22254359968269e-05, 1.28944314880812e-05,
1.35996142589735e-05, 1.43429180537019e-05, 1.51263316467067e-05,
1.59520940241802e-05, 1.6822522665106e-05, 1.77398981706101e-05,
1.87067840492661e-05, 1.97258854864512e-05, 2.07998638330026e-05,
2.1931710964562e-05, 2.3124583860732e-05, 2.43817436226076e-05,
2.57064566766011e-05, 2.71023147452351e-05, 2.85732719977727e-05,
3.01232366788732e-05, 3.17564077655259e-05, 3.34772050137959e-05,
3.52903930671811e-05, 3.72007656749664e-05, 3.92134811622193e-05,
4.13339600995988e-05, 4.35677774133232e-05, 4.59210633579047e-05,
4.84002939774841e-05, 5.10121441928917e-05, 5.37634740647825e-05,
5.66616536161489e-05, 5.97142485508226e-05, 6.29295921258743e-05,
6.63164729686435e-05, 6.98835178901844e-05, 7.36402014328656e-05,
7.75967281240777e-05, 8.17636166483606e-05, 8.61519334015006e-05,
9.07733206985704e-05, 9.56403286306054e-05, 0.000100765247971781,
0.000106161936108279, 0.000111844981550703, 0.000117828340858234,
0.00012412884956377, 0.000130763552542832, 0.000137749179512345,
0.00014510363839137, 0.000152846660963375, 0.000160998542157569,
0.000169580127136327, 0.000178614349528655), J2 = c(315.0479,
315.13122, 315.21455, 315.29788, 315.38121, 315.46454, 315.54787,
315.63121, 315.71454, 315.79786, 315.88119, 315.96451, 316.04785,
316.13118, 316.2145, 316.29782, 316.38115, 316.46448, 316.54781,
316.63115, 316.71449, 316.79783, 316.88117, 316.96451, 317.04784,
317.13117, 317.21451, 317.29784, 317.38118, 317.46451, 317.54785,
317.63119, 317.71452, 317.79785, 317.88118, 317.96451, 318.04782,
318.13116, 318.21449, 318.29782, 318.38115, 318.46448, 318.54782,
318.63115, 318.71447, 318.7978, 318.88113, 318.96446, 319.04779,
319.13112, 319.21445, 319.29778, 319.38111, 319.46444, 319.54777,
319.6311, 319.71443, 319.79776, 319.88109, 319.96443, 320.04776,
320.13109, 320.21442, 320.29775, 320.38108, 320.46441, 320.54775,
320.63107, 320.71441, 320.79776, 320.8811, 320.96444, 321.04777,
321.13109, 321.21442, 321.29775, 321.38107, 321.4644, 321.54773,
321.63105, 321.71438, 321.79771, 321.88105, 321.96437, 322.0477,
322.13103, 322.21436, 322.29769, 322.38102, 322.46436, 322.54769,
322.63103, 322.71436, 322.79769, 322.88102, 322.96436, 323.0477,
323.13104, 323.21438, 323.29771, 323.38105, 323.46439, 323.54772,
323.63105, 323.71438, 323.79771, 323.88104, 323.96437, 324.04771,
324.13103, 324.21437, 324.2977, 324.38102, 324.46436, 324.5477,
324.63104, 324.71437, 324.79771, 324.88104, 324.96437, 325.0477
), J1 = c(314.96456, 315.0479, 315.13122, 315.21455, 315.29788,
315.38121, 315.46454, 315.54787, 315.63121, 315.71454, 315.79786,
315.88119, 315.96451, 316.04785, 316.13118, 316.2145, 316.29782,
316.38115, 316.46448, 316.54781, 316.63115, 316.71449, 316.79783,
316.88117, 316.96451, 317.04784, 317.13117, 317.21451, 317.29784,
317.38118, 317.46451, 317.54785, 317.63119, 317.71452, 317.79785,
317.88118, 317.96451, 318.04782, 318.13116, 318.21449, 318.29782,
318.38115, 318.46448, 318.54782, 318.63115, 318.71447, 318.7978,
318.88113, 318.96446, 319.04779, 319.13112, 319.21445, 319.29778,
319.38111, 319.46444, 319.54777, 319.6311, 319.71443, 319.79776,
319.88109, 319.96443, 320.04776, 320.13109, 320.21442, 320.29775,
320.38108, 320.46441, 320.54775, 320.63107, 320.71441, 320.79776,
320.8811, 320.96444, 321.04777, 321.13109, 321.21442, 321.29775,
321.38107, 321.4644, 321.54773, 321.63105, 321.71438, 321.79771,
321.88105, 321.96437, 322.0477, 322.13103, 322.21436, 322.29769,
322.38102, 322.46436, 322.54769, 322.63103, 322.71436, 322.79769,
322.88102, 322.96436, 323.0477, 323.13104, 323.21438, 323.29771,
323.38105, 323.46439, 323.54772, 323.63105, 323.71438, 323.79771,
323.88104, 323.96437, 324.04771, 324.13103, 324.21437, 324.2977,
324.38102, 324.46436, 324.5477, 324.63104, 324.71437, 324.79771,
324.88104, 324.96437)), class = "data.frame", row.names = c(NA,
-121L))
我的目标是使用 nlsLM 或 nls 来优化我公式中的参数 p 和 U。但是，我正在 运行 解决算法问题，如下所示，它似乎不满足 nlsLM 所以它可以 运行。下面是更多详细信息，但是有人可以帮我弄清楚在下面的示例中我哪里出错了吗？
df1<-structure(list(a = c(0.826386065758906, 0.804841782886437, 0.783325233355932,
0.760569303063664, 0.727963278438188, 0.703970672785107, 0.680589589769403,
0.65474843620048, 0.629508566083339, 0.602417701399354, 0.576536805662077,
0.548851189058063, 0.519947188978781, 0.490433240798225, 0.462086422390147,
0.431909413297801, 0.403590803993467, 0.373435985401716, 0.343943659176956,
0.31380487904441, 0.284871585752978, 0.254086078501208, 0.231342598159761,
0.198724947119965, 0.16612702603107, 0.141555650677377, 0.109001084106243,
0.0831572469086037, 0.0579197953048283, 0.0332801640177642, 0.00921287687980588,
-0.00683461114689798, -0.0308662529136221, -0.0468697128496984,
-0.0634535329550516, -0.0788157766432729, -0.0947585282387919,
-0.112551607962292, -0.121770647911843, -0.131005124862407, -0.13105454830848,
-0.13913263731113, -0.141015879860184, -0.141727067372491, -0.15092263010162,
-0.151527379077436, -0.152155929282014, -0.144792734494917, -0.146630537024706,
-0.139308277987739, -0.142395755120843, -0.135627970868379, -0.130069772180454,
-0.134347052454578, -0.122661184175998, -0.119541926090815, -0.117691480771668,
-0.0980210085445172, -0.0900039693582695, -0.0734011206583904,
-0.0654294710491197, -0.0488666582927202, -0.0334934180561017,
-0.0181602076105936, -0.00403680892262948, 0.00958389551677818,
0.023138063507762, 0.0360154200368275, 0.0470535195180641, 0.0574121363142126,
0.0819938322538736, 0.104326976159514, 0.118669575746475, 0.123156717504759,
0.12529037342488, 0.11004928326779, 0.111288254062499, 0.112616123288566,
0.123770126197535, 0.125825836909568, 0.12600008978006, 0.133135092174578,
0.13150137631796, 0.137318345531007, 0.142490352653999, 0.127578135774472,
0.11881443261832, 0.101364392587429, 0.0930057723054443, 0.0753897222358955,
0.0657226581165791, 0.0559810295585592, 0.0486890523393608, 0.0359839335111634,
0.0239424532478925, 0.0137758789173201, 0.01148034494644, 0.00959075043614099,
0.00965506497270852, 0.0010321021197246, -0.00878784052445702,
-0.0186955496477701, -0.0286825539248294, -0.0387428469291482,
-0.0457835827509943, -0.0533022515989874, -0.067799838787316,
-0.0740879165672261, -0.0704910195102784, -0.0749170995555968,
-0.0892351910852938, -0.0952381116284169, -0.101154870134375,
-0.116713134920168, -0.123043579564404, -0.122229090993963, -0.131574286614478,
-0.139872347069247, -0.150352573559005, -0.160985422263815, -0.183477294368611,
-0.196028087077558), b = c(2.73720919401795e-07, 2.89275866243833e-07,
3.05704744509841e-07, 3.23056147643161e-07, 3.41383503107939e-07,
3.60740020118146e-07, 3.81182894669021e-07, 4.02772468186816e-07,
4.25573817135869e-07, 4.49652809926895e-07, 4.75077190571432e-07,
5.01924516454949e-07, 5.30273542459307e-07, 5.60209325607651e-07,
5.91819764376075e-07, 6.25191565024464e-07, 6.60423815060183e-07,
6.97623624698831e-07, 7.36899824259678e-07, 7.78364716293996e-07,
8.22141727880177e-07, 8.68358553356049e-07, 9.17146942649445e-07,
9.68648499282585e-07, 1.02301254228321e-06, 1.08039295361324e-06,
1.14095522628006e-06, 1.20488164276889e-06, 1.27235309266431e-06,
1.34356428051965e-06, 1.41872017661923e-06, 1.49803720002012e-06,
1.58174837208432e-06, 1.67008400106882e-06, 1.76329669689182e-06,
1.86165861610844e-06, 1.96545125564181e-06, 2.07495771940026e-06,
2.19051033032321e-06, 2.31244676896853e-06, 2.44109344748083e-06,
2.57682364247604e-06, 2.72002332780748e-06, 2.87110867939643e-06,
3.03049986417657e-06, 3.19862819087422e-06, 3.37598849722003e-06,
3.56309366647097e-06, 3.76046212631028e-06, 3.96865098343192e-06,
4.18824726536018e-06, 4.41986951018994e-06, 4.66416943984418e-06,
4.92183372125163e-06, 5.1935858199306e-06, 5.48018795087751e-06,
5.78244313168551e-06, 6.1011973433322e-06, 6.43734180406693e-06,
6.7918153624452e-06, 7.16563024757515e-06, 7.55980712352996e-06,
7.97541858133744e-06, 8.41364335579908e-06, 8.87569896478336e-06,
9.36286778257749e-06, 9.87650045747503e-06, 1.04180531601306e-05,
1.09889579629179e-05, 1.15908264520961e-05, 1.22254359968269e-05,
1.28944314880812e-05, 1.35996142589735e-05, 1.43429180537019e-05,
1.51263316467067e-05, 1.59520940241802e-05, 1.6822522665106e-05,
1.77398981706101e-05, 1.87067840492661e-05, 1.97258854864512e-05,
2.07998638330026e-05, 2.1931710964562e-05, 2.3124583860732e-05,
2.43817436226076e-05, 2.57064566766011e-05, 2.71023147452351e-05,
2.85732719977727e-05, 3.01232366788732e-05, 3.17564077655259e-05,
3.34772050137959e-05, 3.52903930671811e-05, 3.72007656749664e-05,
3.92134811622193e-05, 4.13339600995988e-05, 4.35677774133232e-05,
4.59210633579047e-05, 4.84002939774841e-05, 5.10121441928917e-05,
5.37634740647825e-05, 5.66616536161489e-05, 5.97142485508226e-05,
6.29295921258743e-05, 6.63164729686435e-05, 6.98835178901844e-05,
7.36402014328656e-05, 7.75967281240777e-05, 8.17636166483606e-05,
8.61519334015006e-05, 9.07733206985704e-05, 9.56403286306054e-05,
0.000100765247971781, 0.000106161936108279, 0.000111844981550703,
0.000117828340858234, 0.00012412884956377, 0.000130763552542832,
0.000137749179512345, 0.00014510363839137, 0.000152846660963375,
0.000160998542157569, 0.000169580127136327, 0.000178614349528655
), x = c(7.61206469829168e-10, 8.04880837026198e-10, 8.5102453143301e-10,
8.99793134300951e-10, 9.51328437405998e-10, 1.00578580312419e-09,
1.06332922814733e-09, 1.12413181831763e-09, 1.18838421310043e-09,
1.25626386645407e-09, 1.32797298152365e-09, 1.40374357927037e-09,
1.48378414368651e-09, 1.56836357338967e-09, 1.65770477079087e-09,
1.75207245412028e-09, 1.85175816475582e-09, 1.95707152824522e-09,
2.06831403251254e-09, 2.18581608289076e-09, 2.30994161115441e-09,
2.44104487535132e-09, 2.57951412571437e-09, 2.72575899152997e-09,
2.88021164601164e-09, 3.04330792286779e-09, 3.21554668918285e-09,
3.39745762416176e-09, 3.58953223142937e-09, 3.7923812298687e-09,
4.00655143334275e-09, 4.2327224845197e-09, 4.47153210287025e-09,
4.72364822571277e-09, 4.9898357506258e-09, 5.27087200511913e-09,
5.56757681151882e-09, 5.88073754993049e-09, 6.21145699001951e-09,
6.56054428940019e-09, 6.92905211001358e-09, 7.31804978053444e-09,
7.72866489630656e-09, 8.16213989148934e-09, 8.6196245901883e-09,
9.1024320192891e-09, 9.61207173210311e-09, 1.0149956868384e-08,
1.07176367759663e-08, 1.1316744903592e-08, 1.19490033177631e-08,
1.26162274595822e-08, 1.33203311541876e-08, 1.4063331886057e-08,
1.48473563541273e-08, 1.5674646321422e-08, 1.65475647746164e-08,
1.74686024097974e-08, 1.84403844614874e-08, 1.94656778929232e-08,
2.05475322857365e-08, 2.16887618647287e-08, 2.28927322096252e-08,
2.41628570299852e-08, 2.55027338320421e-08, 2.69161537094373e-08,
2.84071116498747e-08, 2.99800111976379e-08, 3.16387068089972e-08,
3.33886696287756e-08, 3.52346661692384e-08, 3.71814450102107e-08,
3.92346911195274e-08, 4.13999001075981e-08, 4.36830992974598e-08,
4.60912297232476e-08, 4.86307601828189e-08, 5.13084564066871e-08,
5.41324352378923e-08, 5.71102577697511e-08, 6.02498314193268e-08,
6.35606456850457e-08, 6.70515362160507e-08, 7.07326494651599e-08,
7.46128341939538e-08, 7.87042035557664e-08, 8.30176288376039e-08,
8.75650370123332e-08, 9.23589889309948e-08, 9.74127126516464e-08,
1.02740794822333e-07, 1.08356627800041e-07, 1.14277010988554e-07,
1.20516791595619e-07, 1.27093789588104e-07, 1.34026039967841e-07,
1.41333432178137e-07, 1.49035156991607e-07, 1.57152272804214e-07,
1.65706950725174e-07, 1.74721422340645e-07, 1.84222422717041e-07,
1.9423476847285e-07, 2.04784394461598e-07, 2.15901129942397e-07,
2.2761514326591e-07, 2.39958190684945e-07, 2.52963699037024e-07,
2.66666852685233e-07, 2.81106463170876e-07, 2.96316174150964e-07,
3.12344318635138e-07, 3.29228453259451e-07, 3.47013702100588e-07,
3.65754459931466e-07, 3.85496913679509e-07, 4.06294045238492e-07,
4.28198911519565e-07, 4.51275434571536e-07, 4.75579797828766e-07,
5.01179638398094e-07, 5.28143286489079e-07), J = c(314.96456,
315.0479, 315.13122, 315.21455, 315.29788, 315.38121, 315.46454,
315.54787, 315.63121, 315.71454, 315.79786, 315.88119, 315.96451,
316.04785, 316.13118, 316.2145, 316.29782, 316.38115, 316.46448,
316.54781, 316.63115, 316.71449, 316.79783, 316.88117, 316.96451,
317.04784, 317.13117, 317.21451, 317.29784, 317.38118, 317.46451,
317.54785, 317.63119, 317.71452, 317.79785, 317.88118, 317.96451,
318.04782, 318.13116, 318.21449, 318.29782, 318.38115, 318.46448,
318.54782, 318.63115, 318.71447, 318.7978, 318.88113, 318.96446,
319.04779, 319.13112, 319.21445, 319.29778, 319.38111, 319.46444,
319.54777, 319.6311, 319.71443, 319.79776, 319.88109, 319.96443,
320.04776, 320.13109, 320.21442, 320.29775, 320.38108, 320.46441,
320.54775, 320.63107, 320.71441, 320.79776, 320.8811, 320.96444,
321.04777, 321.13109, 321.21442, 321.29775, 321.38107, 321.4644,
321.54773, 321.63105, 321.71438, 321.79771, 321.88105, 321.96437,
322.0477, 322.13103, 322.21436, 322.29769, 322.38102, 322.46436,
322.54769, 322.63103, 322.71436, 322.79769, 322.88102, 322.96436,
323.0477, 323.13104, 323.21438, 323.29771, 323.38105, 323.46439,
323.54772, 323.63105, 323.71438, 323.79771, 323.88104, 323.96437,
324.04771, 324.13103, 324.21437, 324.2977, 324.38102, 324.46436,
324.5477, 324.63104, 324.71437, 324.79771, 324.88104, 324.96437,
325.0477)), class = "data.frame", row.names = c(NA, 122L))
# package(s):
library("minpack.lm")# used for nlsLM()
# objects:
U <-550
K <- 273.15
p<-77+K
R <- 0.00831
# used to fix diff() complications in formula:
a1<-df1$a[-1]
b1<-df1$b[-1]
J2<-df1$J[-1]
J1<-df1$J[-nrow(df1)]
# try , apparently succeeded if GGrothendieck's answer uses it.
df2 <- data.frame(a1,b1,J2,J1)
# optimizing parameters U & p to fit a1:
summary(nlsLM(formula=a1 ~U *(
(1/ (1 + exp((U/ R)*((1/J2) - (1/p)))))
-(1/ (1 + exp((U/ R)*((1/ J1) - (1/p)))))
/(J2-J1)),
data = df2,
start = list(U=550),p=list(p),
lower=c(300,50),upper=c(600,85),
control= nls.lm.control(maxiter=1000),
trace = F))
错误：
Error in model.frame.default(formula = ~a1 + J2 + J1, data = df1, p = list(p)) :
invalid type (NULL) for variable 'J2'
In addition: Warning messages:
1: In min(x) : no non-missing arguments to min; returning Inf
2: In max(x) : no non-missing arguments to max; returning -Inf
试图通过 upper 和 lower 限制解决此问题 algorithm="port":
summary(nlsLM(formula=a1 ~U *(
(1/ (1 + exp((U/ R)*((1/J2) - (1/p)))))
-(1/ (1 + exp((U/ R)*((1/ J1) - (1/p)))))
/(J2-J1)),
data = df1,
start = list(U=550),p=list(p),
algorithm = "port",
lower=c(300,365),upper=c(600,324),
control= nls.lm.control(maxiter=1000),
trace = F))
但是，我实现了这个错误：
Error in upper(600, 324) : could not find function "upper"
In addition: Warning messages:
1: In min(x) : no non-missing arguments to min; returning Inf
2: In max(x) : no non-missing arguments to max; returning -Inf
然而，在另一个不等同于上面的公式的替代公式中，我 nlsLM 运行 没有 algorithm= 就可以了：
summary(nls(formula= a1~U*(diff(x)/diff(J-K)),
data=d, start=list(U=552),
control=nls.control(maxiter=1000),trace=F))
修复语法错误，注意只有nls algorithm = "port"支持upper和lower，而且问题对p的值不敏感。如果我们 运行 nls 重复将 p 固定为 50 到 85 之间的每个整数值并对 U 进行优化，我们每次都会得到相同的结果（相同的 U，相同的残差平方和）。
res <- lapply(50:85, function(p) {
fo <- a1 ~U *(
(1/ (1 + exp((U/ R)*((1/J2) - (1/p)))))-
(1/ (1 + exp((U/ R)*((1/ J1) - (1/p))))) / (J2-J1))
nls(fo, df2, start = list(U = 550),
lower = 300, upper = 600, algorithm = "port")
})
# all iterations converged
all(sapply(res, function(x) x$convInfo$isConv))
## [1] TRUE
# same U regardless of p
unname(sapply(res, coef))
## [1] 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300
# same residual sum of squares regardless of p
unname(sapply(res, deviance))
## [1] 1317808551 1317808551 1317808551 1317808551 1317808551 1317808551 1317808551 1317808551 1317808551 1317808551 1317808551 1317808551 1317808551 1317808551
## [15] 1317808551 1317808551 1317808551 1317808551 1317808551 1317808551 1317808551 1317808551 1317808551 1317808551 1317808551 1317808551 1317808551 1317808551
## [29] 1317808551 1317808551 1317808551 1317808551 1317808551 1317808551 1317808551 1317808551
因此我们可以将 p 固定为我们希望的极限之间的任何值并优化 U，即我们可以选择 res 的任何组件或根据大残差和使用其他模型平方和 J1 几乎等于 J2
with(df2, var(J2 - J1))
## [1] 4.279614e-11
即使是简单的线性模型也给出了低得多的残差平方和：
fm2 <- lm(a1 ~ b1 + I(J1 - J2) + 0, df2)
deviance(fm2)
## [1] 6.015689
备注
R <- 0.00831
df2 <-
structure(list(a1 = c(0.804841782886437, 0.783325233355932, 0.760569303063664,
0.727963278438188, 0.703970672785107, 0.680589589769403, 0.65474843620048,
0.629508566083339, 0.602417701399354, 0.576536805662077, 0.548851189058063,
0.519947188978781, 0.490433240798225, 0.462086422390147, 0.431909413297801,
0.403590803993467, 0.373435985401716, 0.343943659176956, 0.31380487904441,
0.284871585752978, 0.254086078501208, 0.231342598159761, 0.198724947119965,
0.16612702603107, 0.141555650677377, 0.109001084106243, 0.0831572469086037,
0.0579197953048283, 0.0332801640177642, 0.00921287687980588,
-0.00683461114689798, -0.0308662529136221, -0.0468697128496984,
-0.0634535329550516, -0.0788157766432729, -0.0947585282387919,
-0.112551607962292, -0.121770647911843, -0.131005124862407, -0.13105454830848,
-0.13913263731113, -0.141015879860184, -0.141727067372491, -0.15092263010162,
-0.151527379077436, -0.152155929282014, -0.144792734494917, -0.146630537024706,
-0.139308277987739, -0.142395755120843, -0.135627970868379, -0.130069772180454,
-0.134347052454578, -0.122661184175998, -0.119541926090815, -0.117691480771668,
-0.0980210085445172, -0.0900039693582695, -0.0734011206583904,
-0.0654294710491197, -0.0488666582927202, -0.0334934180561017,
-0.0181602076105936, -0.00403680892262948, 0.00958389551677818,
0.023138063507762, 0.0360154200368275, 0.0470535195180641, 0.0574121363142126,
0.0819938322538736, 0.104326976159514, 0.118669575746475, 0.123156717504759,
0.12529037342488, 0.11004928326779, 0.111288254062499, 0.112616123288566,
0.123770126197535, 0.125825836909568, 0.12600008978006, 0.133135092174578,
0.13150137631796, 0.137318345531007, 0.142490352653999, 0.127578135774472,
0.11881443261832, 0.101364392587429, 0.0930057723054443, 0.0753897222358955,
0.0657226581165791, 0.0559810295585592, 0.0486890523393608, 0.0359839335111634,
0.0239424532478925, 0.0137758789173201, 0.01148034494644, 0.00959075043614099,
0.00965506497270852, 0.0010321021197246, -0.00878784052445702,
-0.0186955496477701, -0.0286825539248294, -0.0387428469291482,
-0.0457835827509943, -0.0533022515989874, -0.067799838787316,
-0.0740879165672261, -0.0704910195102784, -0.0749170995555968,
-0.0892351910852938, -0.0952381116284169, -0.101154870134375,
-0.116713134920168, -0.123043579564404, -0.122229090993963, -0.131574286614478,
-0.139872347069247, -0.150352573559005, -0.160985422263815, -0.183477294368611,
-0.196028087077558), b1 = c(2.89275866243833e-07, 3.05704744509841e-07,
3.23056147643161e-07, 3.41383503107939e-07, 3.60740020118146e-07,
3.81182894669021e-07, 4.02772468186816e-07, 4.25573817135869e-07,
4.49652809926895e-07, 4.75077190571432e-07, 5.01924516454949e-07,
5.30273542459307e-07, 5.60209325607651e-07, 5.91819764376075e-07,
6.25191565024464e-07, 6.60423815060183e-07, 6.97623624698831e-07,
7.36899824259678e-07, 7.78364716293996e-07, 8.22141727880177e-07,
8.68358553356049e-07, 9.17146942649445e-07, 9.68648499282585e-07,
1.02301254228321e-06, 1.08039295361324e-06, 1.14095522628006e-06,
1.20488164276889e-06, 1.27235309266431e-06, 1.34356428051965e-06,
1.41872017661923e-06, 1.49803720002012e-06, 1.58174837208432e-06,
1.67008400106882e-06, 1.76329669689182e-06, 1.86165861610844e-06,
1.96545125564181e-06, 2.07495771940026e-06, 2.19051033032321e-06,
2.31244676896853e-06, 2.44109344748083e-06, 2.57682364247604e-06,
2.72002332780748e-06, 2.87110867939643e-06, 3.03049986417657e-06,
3.19862819087422e-06, 3.37598849722003e-06, 3.56309366647097e-06,
3.76046212631028e-06, 3.96865098343192e-06, 4.18824726536018e-06,
4.41986951018994e-06, 4.66416943984418e-06, 4.92183372125163e-06,
5.1935858199306e-06, 5.48018795087751e-06, 5.78244313168551e-06,
6.1011973433322e-06, 6.43734180406693e-06, 6.7918153624452e-06,
7.16563024757515e-06, 7.55980712352996e-06, 7.97541858133744e-06,
8.41364335579908e-06, 8.87569896478336e-06, 9.36286778257749e-06,
9.87650045747503e-06, 1.04180531601306e-05, 1.09889579629179e-05,
1.15908264520961e-05, 1.22254359968269e-05, 1.28944314880812e-05,
1.35996142589735e-05, 1.43429180537019e-05, 1.51263316467067e-05,
1.59520940241802e-05, 1.6822522665106e-05, 1.77398981706101e-05,
1.87067840492661e-05, 1.97258854864512e-05, 2.07998638330026e-05,
2.1931710964562e-05, 2.3124583860732e-05, 2.43817436226076e-05,
2.57064566766011e-05, 2.71023147452351e-05, 2.85732719977727e-05,
3.01232366788732e-05, 3.17564077655259e-05, 3.34772050137959e-05,
3.52903930671811e-05, 3.72007656749664e-05, 3.92134811622193e-05,
4.13339600995988e-05, 4.35677774133232e-05, 4.59210633579047e-05,
4.84002939774841e-05, 5.10121441928917e-05, 5.37634740647825e-05,
5.66616536161489e-05, 5.97142485508226e-05, 6.29295921258743e-05,
6.63164729686435e-05, 6.98835178901844e-05, 7.36402014328656e-05,
7.75967281240777e-05, 8.17636166483606e-05, 8.61519334015006e-05,
9.07733206985704e-05, 9.56403286306054e-05, 0.000100765247971781,
0.000106161936108279, 0.000111844981550703, 0.000117828340858234,
0.00012412884956377, 0.000130763552542832, 0.000137749179512345,
0.00014510363839137, 0.000152846660963375, 0.000160998542157569,
0.000169580127136327, 0.000178614349528655), J2 = c(315.0479,
315.13122, 315.21455, 315.29788, 315.38121, 315.46454, 315.54787,
315.63121, 315.71454, 315.79786, 315.88119, 315.96451, 316.04785,
316.13118, 316.2145, 316.29782, 316.38115, 316.46448, 316.54781,
316.63115, 316.71449, 316.79783, 316.88117, 316.96451, 317.04784,
317.13117, 317.21451, 317.29784, 317.38118, 317.46451, 317.54785,
317.63119, 317.71452, 317.79785, 317.88118, 317.96451, 318.04782,
318.13116, 318.21449, 318.29782, 318.38115, 318.46448, 318.54782,
318.63115, 318.71447, 318.7978, 318.88113, 318.96446, 319.04779,
319.13112, 319.21445, 319.29778, 319.38111, 319.46444, 319.54777,
319.6311, 319.71443, 319.79776, 319.88109, 319.96443, 320.04776,
320.13109, 320.21442, 320.29775, 320.38108, 320.46441, 320.54775,
320.63107, 320.71441, 320.79776, 320.8811, 320.96444, 321.04777,
321.13109, 321.21442, 321.29775, 321.38107, 321.4644, 321.54773,
321.63105, 321.71438, 321.79771, 321.88105, 321.96437, 322.0477,
322.13103, 322.21436, 322.29769, 322.38102, 322.46436, 322.54769,
322.63103, 322.71436, 322.79769, 322.88102, 322.96436, 323.0477,
323.13104, 323.21438, 323.29771, 323.38105, 323.46439, 323.54772,
323.63105, 323.71438, 323.79771, 323.88104, 323.96437, 324.04771,
324.13103, 324.21437, 324.2977, 324.38102, 324.46436, 324.5477,
324.63104, 324.71437, 324.79771, 324.88104, 324.96437, 325.0477
), J1 = c(314.96456, 315.0479, 315.13122, 315.21455, 315.29788,
315.38121, 315.46454, 315.54787, 315.63121, 315.71454, 315.79786,
315.88119, 315.96451, 316.04785, 316.13118, 316.2145, 316.29782,
316.38115, 316.46448, 316.54781, 316.63115, 316.71449, 316.79783,
316.88117, 316.96451, 317.04784, 317.13117, 317.21451, 317.29784,
317.38118, 317.46451, 317.54785, 317.63119, 317.71452, 317.79785,
317.88118, 317.96451, 318.04782, 318.13116, 318.21449, 318.29782,
318.38115, 318.46448, 318.54782, 318.63115, 318.71447, 318.7978,
318.88113, 318.96446, 319.04779, 319.13112, 319.21445, 319.29778,
319.38111, 319.46444, 319.54777, 319.6311, 319.71443, 319.79776,
319.88109, 319.96443, 320.04776, 320.13109, 320.21442, 320.29775,
320.38108, 320.46441, 320.54775, 320.63107, 320.71441, 320.79776,
320.8811, 320.96444, 321.04777, 321.13109, 321.21442, 321.29775,
321.38107, 321.4644, 321.54773, 321.63105, 321.71438, 321.79771,
321.88105, 321.96437, 322.0477, 322.13103, 322.21436, 322.29769,
322.38102, 322.46436, 322.54769, 322.63103, 322.71436, 322.79769,
322.88102, 322.96436, 323.0477, 323.13104, 323.21438, 323.29771,
323.38105, 323.46439, 323.54772, 323.63105, 323.71438, 323.79771,
323.88104, 323.96437, 324.04771, 324.13103, 324.21437, 324.2977,
324.38102, 324.46436, 324.5477, 324.63104, 324.71437, 324.79771,
324.88104, 324.96437)), class = "data.frame", row.names = c(NA,
-121L))