Wie wird der Std-Fehler in der Funktion nls berechnet?

Question

Ich versuche, ein theoretisches Modell an meine Daten anzupassen, und die angegebenen Werte sind passend für die Parameter, die ich finden möchte, sind gut, aber der Fehler (Std. Error) ist sehr groß, wodurch die Zuverlässigkeit der Anpassung verloren geht

Ich würde gerne wissen, ob jemand weiß, wie diese Std. Error Fehler berechnet werden?

Mein Skript:

y=c(133129.8,132171.4,131439,130849.8,130359.6,129942.2,129580.6,129263.1,128981.5,128729.6,128498.8,128281.9,128075.8,127878.4,127687.7,127502.7,127322.7,127146.5,126973.2,126802.1,126633.3,126467.2,126303.2,126140.8,125979.4,125810.1,125624.4,125421.6,125201.5,124964.2,124714.1,124455.8,124189.3,123914.4,123631.3,123344.3,123057.8,122772,122486.6,122201.4,121912.2,121614.8,121309,120994.6,120671.7,120342.6,120009.5,119672.4,119331.4,118986.2,118633.9,118271.9,117899.8,117517.8,117125.6,116722.6,116307.9,115881.4,115443.2,114993.2,114532.4,114061.5,113580.5,113089.6,112588.5,112077.1,111554.7,111021.5,110477.4,109922.4,109357.6,108783.8,108201.2,107609.6,107009.2,106400.9,105785.7,105163.6,104534.7,103898.9,103256.2,102606.4,101949.6,101285.7,100614.8,99936.8,99251.7,98559.5,97860.2,97153.8,96441.3,95723.7,95001,94273.3,93540.4,92804.5,92067.2,91328.6,90588.8,89847.8,89106.5,88365.8,87625.7,86886.3,86147.6,85412.2,84682.6,83958.8,83240.9,82528.8,81821,81116,80413.8,79714.3,79017.5,78324.4,77635.7,76951.3,76271.4,75596,74925.8,74261.6,73603.4,72951.2,72305.1,71665.8,71034,70409.8,69793.2,69184,68581,67983,67389.9,66801.6,66218.1,65636.9,65055.4,64473.7,63891.5,63309.1,62727.6,62148.3,61571.3,60996.6,60424,59853.1,59283.2,58714.4,58146.6,57579.9,57014.7,56451.7,55891,55332.4,54776,54222.4,53672.1,53125,52581.2,52040.7,51504,50971.7,50443.6,49919.8,49400.3,48885.7,48376.2,47872.1,47373.2,46879.6,46392.7,45914.1,45443.7,44981.5,44527.5,44081.2,43641.9,43209.7,42784.4,42366.2,41954.4,41548.4,41148.3,40754,40365.4,39982.1,39603.5,39229.4,38860,38495.1,38135.3,37780.6,37431.3,37087.3,36748.5,36415.3,36088,35766.6,35451.1,35141.4,34837,34537.3,34242.3,33952,33666.4,33385.7,33110.1,32839.8,32574.7,32314.7,32059.4,31808,31560.6,31317.2,31077.8,30843.4,30615.1,30392.8,30176.5,29966.4,29762.4,29564.9,29373.9,29189.3,29011.1,28839.1,28673.2,28513.2,28359.2,28211.2,28068.6,27930.7,27797.7,27669.3,27545.7,27425.5,27307.3,27191.1,27077,26964.8,26854.8,26747.1,26641.5,26538.2,26437.2,26339.2,26245.1,26154.8,26068.5,25985.9,25906.6,25829.9,25755.9,25684.4,25615.5,25548.9,25484.4,25421.9,25361.4,25302.9,25246.1,25190.8,25136.8,25084.3,25033.1,24982.7,24932.5,24882.3,24832.3,24782.3,24732.8,24684.1,24636.1,24588.8,24542.3,24496.1,24450.1,24404.1,24358.2,24312.3,24266.1,24219.3,24171.8,24123.6,24074.8,24025.3,23974.9,23923.8,23871.9,23819.2,23765.9,23712.3,23658.3,23603.9,23549.2,23494.1,23438.3,23382,23325.1,23267.7,.1,23152.9,23096,23039.5,22983.3,22926.7,22869.1,22810.3,22750.4,22689.5,22627.5,22564.7,22501,22436.4,22371,22305,22238.9,22172.7,22106.3,22039.8,21973.6,21907.8,21842.6,21777.9,21713.8,21650,21586.3,21522.6,21459.1,21395.6,21332.5,21270,21208.2,21147.1,21086.5,21026.9,20968.5,20911.2,20855,20800,20746.2,20693.4,20641.7,20591.1,20541.6,20492.8,20444.6,20396.8,20349.5,20302.7,20256.2,20210.1,20164.3,20118.8,20073.6,20028.9,19984.8,19941.3,19898.4,19856.2,19814.6,19773.9,19734.1,19695.1,19657,19619.8,19583.6,19548.5,19514.4,19481.3,19449.4,19418.6,19389,19360.6,19333.4,19307.3,19282.6,19259.1,19236.8,19215.8,19195.8,19176.7,19158.5,19141,19124.4,19108.6,19093.7,19079.5,19066.2,19053.7,19042,19031.1,19020.9,19011.4,19002.7,18994.7,18987.4,18980.7,18974.6,18969.2,18964.4,18960,18956,18952.5,18949.4,18946.7,18944.5,18942.7,18941.3,18940.3,18939.8,18939.8,18940.1,18940.9) 

    x=c(0.003,0.004,0.005,0.006,0.007,0.008,0.009,0.01,0.011,0.012,0.013,0.014,0.015,0.016,0.017,0.018,0.019,0.02,0.021,0.022,0.023,0.024,0.025,0.026,0.027,0.028,0.029,0.03,0.031,0.032,0.033,0.034,0.035,0.036,0.037,0.038,0.039,0.04,0.041,0.042,0.043,0.044,0.045,0.046,0.047,0.048,0.049,0.05,0.051,0.052,0.053,0.054,0.055,0.056,0.057,0.058,0.059,0.06,0.061,0.062,0.063,0.064,0.065,0.066,0.067,0.068,0.069,0.07,0.071,0.072,0.073,0.074,0.075,0.076,0.077,0.078,0.079,0.08,0.081,0.082,0.083,0.084,0.085,0.086,0.087,0.088,0.089,0.09,0.091,0.092,0.093,0.094,0.095,0.096,0.097,0.098,0.099,0.1,0.101,0.102,0.103,0.104,0.105,0.106,0.107,0.108,0.109,0.11,0.111,0.112,0.113,0.114,0.115,0.116,0.117,0.118,0.119,0.12,0.121,0.122,0.123,0.124,0.125,0.126,0.127,0.128,0.129,0.13,0.131,0.132,0.133,0.134,0.135,0.136,0.137,0.138,0.139,0.14,0.141,0.142,0.143,0.144,0.145,0.146,0.147,0.148,0.149,0.15,0.151,0.152,0.153,0.154,0.155,0.156,0.157,0.158,0.159,0.16,0.161,0.162,0.163,0.164,0.165,0.166,0.167,0.168,0.169,0.17,0.171,0.172,0.173,0.174,0.175,0.176,0.177,0.178,0.179,0.18,0.181,0.182,0.183,0.184,0.185,0.186,0.187,0.188,0.189,0.19,0.191,0.192,0.193,0.194,0.195,0.196,0.197,0.198,0.199,0.2,0.201,0.202,0.203,0.204,0.205,0.206,0.207,0.208,0.209,0.21,0.211,0.212,0.213,0.214,0.215,0.216,0.217,0.218,0.219,0.22,0.221,0.222,0.223,0.224,0.225,0.226,0.227,0.228,0.229,0.23,0.231,0.232,0.233,0.234,0.235,0.236,0.237,0.238,0.239,0.24,0.241,0.242,0.243,0.244,0.245,0.246,0.247,0.248,0.249,0.25,0.251,0.252,0.253,0.254,0.255,0.256,0.257,0.258,0.259,0.26,0.261,0.262,0.263,0.264,0.265,0.266,0.267,0.268,0.269,0.27,0.271,0.272,0.273,0.274,0.275,0.276,0.277,0.278,0.279,0.28,0.281,0.282,0.283,0.284,0.285,0.286,0.287,0.288,0.289,0.29,0.291,0.292,0.293,0.294,0.295,0.296,0.297,0.298,0.299,0.3,0.301,0.302,0.303,0.304,0.305,0.306,0.307,0.308,0.309,0.31,0.311,0.312,0.313,0.314,0.315,0.316,0.317,0.318,0.319,0.32,0.321,0.322,0.323,0.324,0.325,0.326,0.327,0.328,0.329,0.33,0.331,0.332,0.333,0.334,0.335,0.336,0.337,0.338,0.339,0.34,0.341,0.342,0.343,0.344,0.345,0.346,0.347,0.348,0.349,0.35,0.351,0.352,0.353,0.354,0.355,0.356,0.357,0.358,0.359,0.36,0.361,0.362,0.363,0.364,0.365,0.366,0.367,0.368,0.369,0.37,0.371,0.372,0.373,0.374,0.375,0.376,0.377,0.378,0.379,0.38,0.381,0.382,0.383,0.384,0.385,0.386,0.387,0.388,0.389,0.39,0.391,0.392,0.393,0.394,0.395,0.396,0.397,0.398,0.399,0.4,0.401,0.402,0.403,0.404,0.405,0.406,0.407,0.408,0.409,0.41,0.411,0.412,0.413,0.414,0.415,0.416) 

Fitting:

start_ini=data.frame(t=c(0.1,1),o=c(10,1000),k=c(2,3)) 

py=nls2(y ~ o/(x*(x+t)^k), start=start_ini, algorithm="brute-force") 

summary(py) 

Ergebnisse:

> Formula: y ~ o/(x * (x + t)^k) 

Parameters: 
    Estimate Std. Error t value Pr(>|t|) 
t  0.70  17.90 0.039 0.969 
o 670.00 23162.61 0.029 0.977 
k  2.00  46.48 0.043 0.966 

Residual standard error: 55100 on 411 degrees of freedom 

Number of iterations to convergence: 64 
Achieved convergence tolerance: NA 

Dank fortschrittlicher

Answer 1

Die Standardfehler groß sind, weil Ihr Modell nicht passt überhaupt. Werfen Sie einen Blick auf die angepassten Werte (auf einer log-log Plot):

plot(x,y,log="xy",ylim=c(1000,1000000)) 
points(x,predict(py),col="red") 
legend("bottomleft",legend = c("Data","Model"),col=c("black","red"),pch=1) 

Entweder Ihr Modell überhaupt nicht geeignet ist, oder Sie sind nicht in der Lage gute Parameterwerte, da das Netz zu finden in Ihrer Grid-Suche ist zu restriktiv.