问题
I'm trying to use nls to estimate the parameters of a non linear model.
I first use nls2 to find good initial parameters with Random Search and I then use nls to improve the estimation with a Gauss-Newton approach.
The problem is I always get an "singular gradient matrix at initial parameter estimates" error.
I'm not sure I understand, because the input matrix doesn't seem to be a singular gradient matrix.
Moreover even if the fits I'm looking for is not perfect for this data, nls should find a way to improve the parameters estimations. Isn't it ?
Question: Is there a way to improve the parameters estimation?
I've tried NLS.lm but I had the same problem.
Here is a reproductible example:
Data:
structure(list(x1 = c(0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L,
0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 0L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 2L, 2L, 2L,
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L,
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L,
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L,
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L,
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 4L,
4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L,
4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L,
4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L,
4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 4L, 5L,
5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L,
5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L,
5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L,
5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 5L, 6L, 6L,
6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L,
6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L,
6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L,
6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 6L, 7L, 7L, 7L, 7L,
7L, 7L, 7L, 7L, 7L, 7L, 7L, 7L, 7L, 7L, 7L, 7L, 7L, 7L, 7L, 7L,
7L, 7L, 7L, 7L, 7L, 7L, 7L, 7L, 7L, 7L, 7L, 7L, 7L, 7L, 7L, 7L,
7L, 7L, 7L, 7L, 7L, 7L, 7L, 7L, 7L, 7L), x2 = c(1L, 2L, 3L, 4L,
5L, 6L, 7L, 8L, 9L, 10L, 11L, 12L, 13L, 14L, 15L, 16L, 17L, 18L,
19L, 20L, 21L, 22L, 23L, 24L, 25L, 26L, 27L, 28L, 29L, 30L, 31L,
32L, 33L, 34L, 35L, 36L, 37L, 38L, 39L, 40L, 41L, 42L, 43L, 44L,
45L, 46L, 47L, 48L, 49L, 50L, 51L, 52L, 53L, 54L, 55L, 56L, 57L,
58L, 59L, 60L, 61L, 62L, 63L, 64L, 65L, 66L, 67L, 0L, 1L, 2L,
3L, 4L, 5L, 6L, 7L, 8L, 9L, 10L, 11L, 12L, 13L, 14L, 15L, 16L,
17L, 18L, 19L, 20L, 21L, 22L, 23L, 24L, 25L, 26L, 27L, 28L, 29L,
30L, 31L, 32L, 33L, 34L, 35L, 36L, 37L, 38L, 39L, 40L, 41L, 42L,
43L, 44L, 45L, 46L, 47L, 48L, 49L, 50L, 51L, 52L, 53L, 54L, 55L,
56L, 57L, 58L, 59L, 60L, 61L, 62L, 63L, 64L, 65L, 66L, 0L, 1L,
2L, 3L, 4L, 5L, 6L, 7L, 8L, 9L, 10L, 11L, 12L, 13L, 14L, 15L,
16L, 17L, 18L, 19L, 20L, 21L, 22L, 23L, 24L, 25L, 26L, 27L, 28L,
29L, 30L, 31L, 32L, 33L, 34L, 35L, 36L, 37L, 38L, 39L, 40L, 41L,
42L, 43L, 44L, 45L, 46L, 47L, 48L, 49L, 50L, 51L, 52L, 53L, 54L,
55L, 56L, 57L, 58L, 59L, 60L, 61L, 62L, 63L, 64L, 65L, 0L, 1L,
2L, 3L, 4L, 5L, 6L, 7L, 8L, 9L, 10L, 11L, 12L, 13L, 14L, 15L,
16L, 17L, 18L, 19L, 20L, 21L, 22L, 23L, 24L, 25L, 26L, 27L, 28L,
29L, 30L, 31L, 32L, 33L, 34L, 35L, 36L, 37L, 38L, 39L, 40L, 41L,
42L, 43L, 44L, 45L, 46L, 47L, 48L, 49L, 50L, 51L, 52L, 53L, 54L,
55L, 56L, 57L, 58L, 59L, 60L, 61L, 62L, 63L, 64L, 0L, 1L, 2L,
3L, 4L, 5L, 6L, 7L, 8L, 9L, 10L, 11L, 12L, 13L, 14L, 15L, 16L,
17L, 18L, 19L, 20L, 21L, 22L, 23L, 24L, 25L, 26L, 27L, 28L, 29L,
30L, 31L, 32L, 33L, 34L, 35L, 36L, 37L, 38L, 39L, 40L, 41L, 42L,
43L, 44L, 45L, 46L, 47L, 48L, 49L, 50L, 51L, 52L, 53L, 54L, 55L,
56L, 57L, 58L, 59L, 60L, 61L, 62L, 63L, 0L, 1L, 2L, 3L, 4L, 5L,
6L, 7L, 8L, 9L, 10L, 11L, 12L, 13L, 14L, 15L, 16L, 17L, 18L,
19L, 20L, 21L, 22L, 23L, 24L, 25L, 26L, 27L, 28L, 29L, 30L, 31L,
32L, 33L, 34L, 35L, 36L, 37L, 38L, 39L, 40L, 41L, 42L, 43L, 44L,
45L, 46L, 47L, 48L, 49L, 50L, 51L, 52L, 53L, 54L, 55L, 56L, 57L,
58L, 59L, 60L, 61L, 62L, 0L, 1L, 2L, 3L, 4L, 5L, 6L, 7L, 8L,
9L, 10L, 11L, 12L, 13L, 14L, 15L, 16L, 17L, 18L, 19L, 20L, 21L,
22L, 23L, 24L, 25L, 26L, 27L, 28L, 29L, 30L, 31L, 32L, 33L, 34L,
35L, 36L, 37L, 38L, 39L, 40L, 41L, 42L, 43L, 44L, 45L, 46L, 47L,
48L, 49L, 50L, 51L, 52L, 53L, 54L, 55L, 56L, 57L, 58L, 59L, 60L,
61L, 0L, 1L, 2L, 3L, 4L, 5L, 6L, 7L, 8L, 9L, 10L, 11L, 12L, 13L,
14L, 15L, 16L, 17L, 18L, 19L, 20L, 21L, 22L, 23L, 24L, 25L, 26L,
27L, 28L, 29L, 30L, 31L, 32L, 33L, 34L, 35L, 36L, 37L, 38L, 39L,
40L, 41L, 42L, 43L, 44L, 45L), y = c(0.0689464583349188, 0.0358227182166929,
0.0187034836294036, 0.0227081421239796, 0.0146603483536504, 0.00562771204350896,
0.00411351161052011, 0.00356917888321555, 0.0028017552960605,
0.0024750328652541, 0.00243175013170564, 0.00242654283706898,
0.00235224917236107, 0.00176144220485858, 0.00138071934398105,
0.000696375069179013, 0.00106282865382483, 0.00114735219137874,
0.00277256441625284, 0.00214359572321392, 0.00144935953386591,
0.00249732559162499, 0.00225859018399108, 0.00201642941663214,
0.00232438586834105, 0.0016083751355862, 0.00143118376291818,
0.00158323933266031, 0.00157585431454131, 0.00169206800399143,
0.00158514119474578, 0.00134506293557103, 0.00119442163345335,
0.00101284069499962, 0.0012621113004254, 0.00128964367655383,
0.00102819258807122, 0.00125345601171754, 0.00116155619985178,
0.00142466624262548, 0.00141075318725309, 0.00106556656123991,
0.0010976347045814, 0.0012442089226047, 0.0010627617251863, 0.00125322168410487,
0.00112108560656369, 0.0012459199320756, 0.00135773322693401,
0.0013997982284804, 0.00155012485145915, 0.00151108062240688,
0.00149570655260348, 0.00152598641103596, 0.00108261570337346,
0.000992225418429453, 0.000769588971038765, 0.000700496873143604,
0.000688378351958078, 0.000595007407260441, 0.000557615594951187,
0.00040476923690092, 0.000492276455560289, 0.000447248723966691,
0.000388694992851599, 0.000346087542525691, 0.000189803623801549,
0.0709302325562937, 0.0424623423412875, 0.019085896698975, 0.0190650552541205,
0.014276898897581, 0.00593407290200902, 0.00445528598343583,
0.00371231334350143, 0.00253909496678967, 0.00263487912423124,
0.00248012072619926, 0.00263786771266913, 0.00219351150766708,
0.00179271674850348, 0.00139646119589996, 0.000911560061336614,
0.000989537441246412, 0.001046390000492, 0.00223993432619926,
0.00164189356162362, 0.00106041866437064, 0.00194151698794588,
0.0014213192200082, 0.00165239495268553, 0.00196583929282493,
0.00120501090643706, 0.001141403899631, 0.00122398595424354,
0.00124538223829438, 0.00123370121853218, 0.00136883147552275,
0.00110907318146781, 0.000965843164247642, 0.000859986264862649,
0.00104695561918819, 0.00103985460139401, 0.000455832014104141,
0.000704296760639607, 0.000870145383845838, 0.000919870911357114,
0.00101396309667897, 0.000781894087412874, 0.000909712365723658,
0.000889897365477655, 0.000933063039278393, 0.000779395399425994,
0.000789546295038951, 0.000773432990897909, 0.00125614787798278,
0.00123172652693727, 0.00078936677195572, 0.000952107503075031,
0.00105449131480115, 0.00123128091742517, 0.000889501370397704,
0.00085648642099221, 0.000830097733497335, 0.000653482256334563,
0.000521696831160312, 0.000612702433456335, 0.000513576588109881,
0.000475289330709307, 0.00041141913800738, 0.000328157997211972,
0.00031336264403444, 0.000328784093808938, 0.000237448446412464,
0.0520691145678866, 0.0281929482152033, 0.0219024230330532, 0.0141074098760277,
0.00691341703402584, 0.00445785262213699, 0.0034569415664917,
0.00234406584844369, 0.00257369504707459, 0.00234047371531346,
0.00227286083862502, 0.00248544382019894, 0.00180810413760828,
0.00138986347039715, 0.000911936124008956, 0.000932783218782117,
0.00108887529088974, 0.0017855660833578, 0.00159768589505946,
0.00124091041330201, 0.00203036436876009, 0.00154489107876964,
0.00111687975012847, 0.00163256939968433, 0.00143626193198502,
0.000996683818914256, 0.0010781399542101, 0.00122575793431581,
0.00115671467616723, 0.001069532453476, 0.0010106869893371, 0.000978618104445015,
0.000894478048836441, 0.000842874700392747, 0.000819009288742475,
0.000843003919670386, 0.000964158733115548, 0.000877802228013507,
0.00087592051873807, 0.000935810596369843, 0.000879047729316546,
0.000829181439950081, 0.0010295792954412, 0.000765620227389517,
0.00102511256239906, 0.000823109180461753, 0.00111669534392894,
0.000802757620485245, 0.00103231207284173, 0.000884354083467919,
0.00109278942886507, 0.000969283099489796, 0.000827480664091176,
0.000798564447676552, 0.000909248326695786, 0.000682209033640434,
0.000780593294853913, 0.000485172195712818, 0.000467514093470122,
0.000295219649739392, 0.000460636351123183, 0.00045060371687344,
0.000492590160218764, 0.000402536549331963, 0.000271941766535751,
0.000171012123770371, 0.0267385565244063, 0.0275426278720772,
0.0154589149018475, 0.00729065000152096, 0.00513675524527996,
0.00378848397112206, 0.00305965140790087, 0.00240428827949139,
0.00233604733730811, 0.00199601458903693, 0.00198302547453915,
0.00137121122011316, 0.00126241982975401, 0.0012413298189045,
0.00103044327584109, 0.00106759120581615, 0.00190957422380402,
0.00124400301656831, 0.000989035353673623, 0.00160702520431547,
0.0011515826661394, 0.00153203681379408, 0.00134897491229138,
0.000916492937174261, 0.00072393419977287, 0.00115124473393361,
0.00104241370079698, 0.000953324905193568, 0.00121656899373365,
0.000891420608484922, 0.000671666092758208, 0.000659860761797571,
0.000586145968952161, 0.00072735268499929, 0.000658407622538582,
0.000498831767252743, 0.000658345030520574, 0.000542106922897528,
0.000874560054044737, 0.000543320226217274, 0.000751139509440084,
0.000668632963233356, 0.000656903021131188, 0.000574965903652329,
0.0006661524076778, 0.000605171890653201, 0.000527045917239561,
0.000985791370586684, 0.000899420142057553, 0.000933015548254953,
0.00082137283567561, 0.000870124781995904, 0.000498046123582973,
0.000540181050881142, 0.000596948101336416, 0.000405622486362069,
0.000631594016548032, 0.000468749313033603, 0.000389576698910993,
0.000335624642574679, 0.000286763668856847, 0.000439039581432135,
0.000244767908276044, 0.000303911794528604, 0.000160988671898765,
0.0365772382134747, 0.0255898183301035, 0.010327803963121, 0.00714710822108354,
0.00506253612461807, 0.00447056668291465, 0.00322822676102386,
0.00328154620569948, 0.0028470908747756, 0.00253477302081723,
0.00187837758253778, 0.00116416512964702, 0.00119557763663167,
0.000993575112051645, 0.00136274483135782, 0.00204131052512691,
0.00157953945941769, 0.00116523253183218, 0.00190793844827791,
0.00144595416523011, 0.00157423646879793, 0.00126996001866537,
0.00115283860342634, 0.00116894693507543, 0.000930041619012519,
0.00106545753272384, 0.00123507493015348, 0.00130865599847824,
0.000940647984853709, 0.000836521897923032, 0.000778436697656724,
0.00100773629284415, 0.000956581999215341, 0.000808036977042788,
0.000597930101173421, 0.000776453419209873, 0.000630241947142534,
0.000649832426616575, 0.000782188275296327, 0.00102823806308181,
0.000830656989407107, 0.00051915559901561, 0.000537114715917872,
0.000872430107712244, 0.000549284113632851, 0.000738257038745497,
0.00097442578198376, 0.000879724260815807, 0.000884543540237537,
0.00100038027474944, 0.00103543285342337, 0.000875585441608313,
0.000829083410412184, 0.000760316116414823, 0.000712211369823927,
0.000386744815307978, 0.000428331410721292, 0.000397681982571065,
0.000213938551710199, 0.000370800615243779, 0.000281234314553042,
0.000267359921177464, 0.000358376119030352, 0.000337361541022196,
0.0310029062887812, 0.0154963087949333, 0.00959302943445506,
0.00645674376405936, 0.00525321947702945, 0.00386084394749159,
0.00374364242039947, 0.00351047952579374, 0.00298556939927835,
0.00199158625919048, 0.00206559575086432, 0.00169077836254661,
0.00139156751815451, 0.00170363478493893, 0.00250481301085496,
0.00182474837251083, 0.00116804333227652, 0.00155778636185214,
0.00183778204100427, 0.00135012918459471, 0.00166904872503284,
0.00120137403943415, 0.00108307957787943, 0.00146041465872549,
0.0014437889563235, 0.000975926161359965, 0.00102580511345623,
0.00112145083941, 0.000921884915530595, 0.00082253191796126,
0.000634876416504371, 0.00108601324863747, 0.000830573067167897,
0.000965052460105379, 0.000922667052402736, 0.000863193817654785,
0.000982111173513293, 0.000763009170856168, 0.000921755812461313,
0.000771609983091022, 0.000669047474976222, 0.000773869648383834,
0.00072022523061129, 0.000742426347056781, 0.000718728249316847,
0.000761437280522971, 0.000833112611531319, 0.000794451658438637,
0.000907360341651947, 0.00112083735676435, 0.00102996529205731,
0.000651843453054939, 0.000640968179416338, 0.000549646466476441,
0.000778958256714525, 0.000627413038784969, 0.000523658918731223,
0.000418571973368359, 0.000643352520494588, 0.000351378727146459,
0.000504093577607682, 0.000333827596358531, 0.000339505558071773,
0.0181836504450303, 0.0135527124187004, 0.00780738765319868,
0.00643260738080874, 0.00476881905655232, 0.00406986745617877,
0.00400325917456592, 0.00277499160186111, 0.00198311377238581,
0.00241837807740304, 0.00141018451525995, 0.00166798657140732,
0.0013970042073337, 0.00237332662413329, 0.00146721126831566,
0.000990562316636778, 0.00186106889002752, 0.00186322276224556,
0.00140391140302307, 0.00139027556176293, 0.00125730361478641,
0.00127044200804939, 0.00126655503830484, 0.00133956330669488,
0.00128219844136096, 0.00109531452608613, 0.00112195611926977,
0.00101411381866565, 0.00104786051750783, 0.000798711632769435,
0.000852432172756047, 0.000852720107765923, 0.00110385307389073,
0.00081385514739304, 0.00102898862672826, 0.000710330768658628,
0.000803425598538879, 0.000723455383750816, 0.00075034248654992,
0.000864917906994041, 0.000799733114881449, 0.000608518601191706,
0.000855476747683942, 0.000988548021123443, 0.00104800683206201,
0.000997051779707941, 0.000796235203259423, 0.000910577791459715,
0.000869997383535945, 0.000557402535474327, 0.000757813148434336,
0.000480807445269952, 0.000553425518375578, 0.000633029237291637,
0.00050222863978579, 0.000390945889771328, 0.000430333228928208,
0.000425167676834459, 0.000239604519722651, 0.000357021364759551,
0.000292330910803864, 0.000288851701197491, 0.0198837196044917,
0.0142208140311702, 0.00733039271103269, 0.00609158853724431,
0.00487605866828399, 0.00382636157210858, 0.00411545257392807,
0.00235906433257981, 0.00228491326937568, 0.00109255715480326,
0.00158036861847788, 0.00122011020381908, 0.00223761733564904,
0.00173284341769128, 0.00117538923471357, 0.00219622963095698,
0.00214263916211795, 0.0013198229549172, 0.00172951959530242,
0.00128074705482347, 0.00124062569884766, 0.00144218669111025,
0.00148407512819099, 0.00100716026446858, 0.0010842890711437,
0.000800686408079248, 0.000890454658065465, 0.000887152794471706,
0.00105780722647994, 0.000874948318354744, 0.000569126715186268,
0.000924642167943982, 0.000857013884141074, 0.000823122890591976,
0.00073038777177409, 0.000522615873628494, 0.00070936497950782,
0.000823074755104667, 0.000720588701733105, 0.000722724038337836,
0.00063458965098969, 0.000620049346639466, 0.000842327487089008,
0.000617708212493797, 0.000783953750160813, 0.00112567150392384
)), .Names = c("x1", "x2", "y"), class = c("tbl_df", "data.frame"
), row.names = c(NA, -500L))
Initial parameters: initial_par
structure(list(A1 = 0.0529486559121727, alpha1 = 0.00888818269595504,
B1 = 0.250994319084551, beta1 = 0.471984946168959, A2 = 0.281956987357551,
alpha2 = 0.325086771510541, B2 = 0.0562204262765557, beta2 = 0.725645614322275), class = "data.frame", row.names = c(NA,
-1L), .Names = c("A1", "alpha1", "B1", "beta1", "A2", "alpha2",
"B2", "beta2"))
Formula:
formula = y ~
(A1*exp(-alpha1*x1) + B1*exp(-beta1*x1)) *
(A2*exp(-alpha2*x2) + B2*exp(-beta2*x2))
Nls and the error message
final = nls(formula,
data=df,
start = as.list(as.vector(initial_par)))
Error in nlsModel(formula, mf, start, wts) :
singular gradient matrix at initial parameter estimates
回答1:
The problem is that there is not a one to one relationship between your model and parameters. To see this write A1 = exp(a1+d), A2 = exp(a2-d), B1 = exp(b1+d), B2 = exp(b2-d) in which case we have:
y ~ exp(-alpha1 * x1 + a1 + d) * exp(-alpha2 * x2 + a2 - d) +
exp(-alpha1 * x1 + a1 + d) * exp(-beta2 * x2 + b2 - d) +
exp(-beta1 * x1 + b1 + d) * exp(-alpha2 * x2 + a2 - d) +
exp(-beta1 * x1 + b1 + d) * exp(-beta2 * x2 + b2 - d)
But d cancels in each of the 4 terms and so cancels entirely from the RHS. That is, the RHS is the same for any value of d thus the model is overparameterized and so will give a singular gradient.
Fix one of A1, A2, B1, B2 and then you should be able to get a solution:
A1 <- 1
nls(formula, df, start = initial_par[-1])
giving:
Nonlinear regression model
model: y ~ (A1 * exp(-alpha1 * x1) + B1 * exp(-beta1 * x1)) * (A2 * exp(-alpha2 * x2) + B2 * exp(-beta2 * x2))
data: df
alpha1 B1 beta1 A2 alpha2 B2 beta2
0.11902 1.21030 0.79076 0.04604 0.51697 0.00183 0.02317
residual sum-of-squares: 0.000685
Number of iterations to convergence: 11
Achieved convergence tolerance: 6.686e-06
来源：https://stackoverflow.com/questions/34201377/r-and-nls-singular-gradient-matrix-at-initial-parameter