拟合一条强制通过 0 的线（渐近？）
Fit a line (asymptotic?) forced through 0
数据
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0.852605164, 0.840772986, 0.808174312, 0.788675547, 0.772639871, 0.756165385, 0.741423607, 0.731267095, 0.744977355, 0.770040751, 0.782511771, 0.774730265, 0.756842554, 0.734819889, 0.698052168, 0.77581346, 0.817140758, 0.885996401, 0.934090912, 0.944526315, 0.955664337, 1.011435747, 1.073157072, 1.119287014, 1.119515657, 1.059037447, 1.000632763, 0.962856472, 0.961162984, 0.974968374, 0.970587075, 0.977721334, 0.957530379, 0.910347521, 0.879712403, 0.858478308, 0.843830824, 0.8379758, 0.842489362, 0.884822607, 0.948875129, 0.994918346, 1.009823918, 1.013503909, 1.019781709, 1.006411076, 0.77581346, 0.817140758, 0.886676908, 0.935586452, 0.94628191, 0.958534539, 1.019246221, 1.08957231, 1.144408226, 1.149548173, 1.089345217, 1.02939105, 0.988438129, 0.982776225, 0.995111585, 0.995346427, 1.01788497, 1.013562918, 0.97511065, 0.951002598, 0.932399511, 0.916437924, 0.908735514, 0.915104389, 0.971065819, 1.056129813, 1.113286853, 1.123075843, 1.109285474, 1.091263294, 1.055615306, 0.77581346, 0.817140758, 0.887305021, 0.936918974, 0.947968066, 0.961458325, 1.026293159, 1.103072286, 1.163753867, 1.171400785, 1.109564304, 1.046773672, 1.004206896, 1.001410365, 1.021429181, 1.028966308, 1.05691576, 1.051712751, 1.008408427, 0.980799317, 0.961422026, 0.94676441, 0.941248, 0.949138701, 1.007747769, 1.095256448, 1.151844382, 1.160300612, 1.149290085, 1.139482856, 1.110224843, 0.850993693, 0.878551781, 0.929460883, 0.968631923, 0.979977369, 0.984748244, 1.006746173, 1.022598743, 1.022056341, 0.995788336, 0.941417933, 0.887557328, 0.844630063, 0.82260114, 0.816226721, 0.811409831, 0.822775304, 0.827547729, 0.816761315, 0.808919251, 0.798904061, 0.785326183, 0.770118356, 0.755392075, 0.753594935, 0.759954751, 0.761058331, 0.750226259, 0.732908666, 0.714197397, 0.686983109, 0.850993693, 0.878551781, 0.930243015, 0.970633924, 0.982279301, 0.986959875, 1.011868954, 1.036545277, 1.050327182, 1.038798094, 0.994740367, 0.94990766, 0.916918397, 0.905929267, 0.908401549, 0.906138301, 0.9154374, 0.913555622, 0.89567697, 0.888012469, 0.887200415, 0.889993548, 0.895718455, 0.904391587, 0.934504628, 0.97859478, 1.01128006, 1.02061367, 1.01873982, 1.017302394, 1.004862547, 0.850993693, 0.878551781, 0.930973589, 0.972435594, 0.984444678, 0.989400923, 1.01672256, 1.047491074, 1.070196271, 1.067319155, 1.02880311, 0.988358259, 0.959646761, 0.953317106, 0.960671306, 0.963204622, 0.978041887, 0.976714373, 0.952274144, 0.933524072, 0.920094073, 0.911611795, 0.908339798, 0.909115911, 0.93042165, 0.96522212, 0.991738021, 1.000063777, 1.00124383, 1.005596638, 0.999989033, 0.850993693, 0.878551781, 0.931660473, 0.974092126, 0.986525893, 0.99185437, 1.02062571, 1.054694772, 1.081895232, 1.082667708, 1.045917034, 1.007757068, 0.984359384, 0.986672878, 1.003036141, 1.011242032, 1.030569553, 1.031354904, 1.006945848, 0.989743054, 0.978098989, 0.96971339, 0.965755701, 0.967847824, 0.997593641, 1.045950651, 1.081306815, 1.089472413, 1.084782481, 1.080476761, 1.064157963, 0.567489982, 0.652162194, 0.785000205, 0.873056889, 0.891931236, 0.912524343, 1.003800631, 1.082395673, 1.125870347, 1.094785452, 0.992700994, 0.896223068, 0.808895707, 0.748320878, 0.704849899, 0.664936841, 0.656069219, 0.633515, 0.597315729, 0.579769373, 0.56124866, 0.537654638, 0.510705411, 0.482994974, 0.46803537, 0.458939075, 0.447710872, 0.431626648, 0.416970104, 0.406424105, 0.38950634, 0.567489982, 0.652162194, 0.785214782, 0.873410881, 0.892192543, 0.915579736, 1.02197063, 1.126847506, 1.201263547, 1.186374187, 1.080291033, 0.984983623, 0.910771549, 0.893572509, 0.904725552, 0.904995918, 0.946030796, 0.933481693, 0.878471017, 0.847143769, 0.818725407, 0.797739804, 0.79148525, 0.80905968, 0.909970701, 1.043915868, 1.117636561, 1.119943023, 1.094645143, 1.06159699, 1.003503442, 0.567489982, 0.652162194, 0.785410047, 0.873727441, 0.892530978, 0.918462813, 1.035583615, 1.156719565, 1.249178052, 1.24364233, 1.134532094, 1.035999656, 0.960602522, 0.950968742, 0.973596573, 0.980182827, 1.03298378, 1.021811604, 0.961205125, 0.93057096, 0.903318107, 0.882389605, 0.876688838, 0.898444176, 1.028598189, 1.207716346, 1.306888223, 1.31149435, 1.280924439, 1.245280743, 1.177808642, 0.567489982, 0.652162194, 0.785588086, 0.874014735, 0.892850876, 0.920784175, 1.046159983, 1.178910494, 1.282226086, 1.280955315, 1.168986797, 1.068514109, 0.993000448, 0.987921119, 1.014108896, 1.018215179, 1.066857815, 1.049017906, 0.981723487, 0.946660042, 0.917113125, 0.895081699, 0.888686717, 0.909516275, 1.039963722, 1.219105005, 1.316385865, 1.319118261, 1.289013147, 1.256706476, 1.191284895, 0.747616649, 0.836237073, 0.988689244, 1.054734826, 1.025774002, 1.015855908, 1.134614944, 1.216524243, 1.254345179, 1.173024774, 0.998351097, 0.874513745, 0.78600049, 0.778888166, 0.806788266, 0.816571891, 0.871123493, 0.853065372, 0.791552305, 0.760557473, 0.717245996, 0.670639575, 0.636679769, 0.62634939, 0.67381537, 0.736528993, 0.77349472, 0.769330204, 0.73253262, 0.673349917, 0.596250415, 0.747616649, 0.836237073, 0.989223421, 1.055233955, 1.025119543, 1.016921043, 1.151469111, 1.258143306, 1.321192265, 1.245432615, 1.057826281, 0.928337336, 0.842428386, 0.865764856, 0.929639697, 0.956855834, 1.033756137, 1.004413009, 0.918607771, 0.881685138, 0.843460083, 0.817945898, 0.818849444, 0.856904984, 1.007567525, 1.156849146, 1.213010311, 1.186402917, 1.130393624, 1.074572086, 1.003096223, 0.747616649, 0.836237073, 0.989702821, 1.055676937, 1.024871826, 1.018258095, 1.163146257, 1.283846498, 1.358478546, 1.283459663, 1.088712811, 0.952908397, 0.860999584, 0.883257091, 0.949974537, 0.981948197, 1.07404995, 1.051186562, 0.96545577, 0.933451772, 0.896999121, 0.871107161, 0.87029767, 0.908490062, 1.089059591, 1.285072803, 1.356228113, 1.312512875, 1.22735393, 1.145342112, 1.055003524, 0.747616649, 0.836237073, 0.990136743, 1.056092143, 1.024971724, 1.020100713, 1.172664881, 1.301743984, 1.38206172, 1.306367397, 1.106599331, 0.967083514, 0.872632623, 0.896234035, 0.964761913, 0.995294869, 1.085453749, 1.059120417, 0.970445096, 0.938069701, 0.90287292, 0.878777742, 0.879172385, 0.917467892, 1.098720551, 1.293762326, 1.362068295, 1.316773057, 1.233494401, 1.155817986, 1.067250609, 0.638475657, 0.701241612, 0.780950189, 0.833968759, 0.847179413, 0.855540872, 0.887958705, 0.901707411, 0.895013452, 0.853711843, 0.775103211, 0.701753795, 0.638828635, 0.593984663, 0.556727886, 0.516842306, 0.488357365, 0.453170925, 0.412938416, 0.381577313, 0.353999287, 0.33014816, 0.309804767, 0.292518765, 0.283133805, 0.275905758, 0.264425516, 0.247465536, 0.228905529, 0.211275622, 0.19250603, 0.638475657, 0.701241612, 0.782851875, 0.838087022, 0.852667689, 0.868328154, 0.926839471, 0.98254782, 1.025614381, 1.021376729, 0.953225791, 0.892099738, 0.851322412, 0.851359308, 0.868489683, 0.869845033, 0.892966986, 0.877989113, 0.830597103, 0.802697659, 0.78116709, 0.766595185, 0.761856914, 0.770022631, 0.826521039, 0.90397501, 0.951342702, 0.957924068, 0.952868819, 0.952133417, 0.928703249, 0.638475657, 0.701241612, 0.784563363, 0.841383219, 0.856479287, 0.877265215, 0.954805672, 1.039082289, 1.109156251, 1.117263913, 1.04548502, 0.980756104, 0.940173388, 0.946755886, 0.96761173, 0.962748587, 0.977708697, 0.952758789, 0.895089209, 0.858695507, 0.828825116, 0.807197034, 0.797473609, 0.802371204, 0.857669353, 0.936871529, 0.989703417, 1.003184795, 1.006631613, 1.016924024, 1.00193429, 0.638475657, 0.701241612, 0.786107659, 0.844073474, 0.859399199, 0.88437283, 0.976726115, 1.081594229, 1.167513251, 1.18007791, 1.103111386, 1.03039217, 0.979927897, 0.977664709, 0.995771885, 0.997566521, 1.03454566, 1.025245547, 0.971404195, 0.940074384, 0.913797617, 0.893222034, 0.885627687, 0.899616897, 0.992524326, 1.126092672, 1.206882238, 1.213850737, 1.190800905, 1.164836526, 1.112275243, 0.795258403, 0.816856802, 0.857468605, 0.891532302, 0.906502187, 0.92630136, 0.979601145, 1.028863072, 1.063205004, 1.066814184, 1.032617211, 0.994250894, 0.955908895, 0.928678632, 0.908084571, 0.883677125, 0.873843551, 0.855833828, 0.826784849, 0.806078315, 0.786206603, 0.767146707, 0.752018094, 0.742449701, 0.751986265, 0.771394849, 0.784393907, 0.784185886, 0.774534941, 0.758701324, 0.732473671, 0.795258403, 0.816856802, 0.857970893, 0.892682314, 0.90784204, 0.929064393, 0.990200877, 1.053232908, 1.103422642, 1.118505836, 1.08778131, 1.05342114, 1.022762179, 1.012263894, 1.011744976, 1.00061059, 1.002970934, 0.988465309, 0.955877185, 0.931310594, 0.907226205, 0.884828508, 0.869311213, 0.863197386, 0.888297558, 0.932593346, 0.967936397, 0.985253811, 0.996055782, 1.008288145, 1.005891085, 0.795258403, 0.816856802, 0.858424306, 0.893659353, 0.908989906, 0.931387544, 0.99817735, 1.070488811, 1.13034749, 1.151631355, 1.122590423, 1.088484406, 1.056558609, 1.043916106, 1.042092919, 1.032291174, 1.043409228, 1.037230253, 1.008979201, 0.989836633, 0.970193386, 0.95017159, 0.936686635, 0.93526113, 0.982087791, 1.064218163, 1.129002571, 1.158894658, 1.168904543, 1.169321179, 1.149312615, 0.795258403, 0.816856802, 0.85883683, 0.894518554, 0.910082877, 0.933618307, 1.004875779, 1.084069133, 1.150643349, 1.17569387, 1.146507144, 1.110903978, 1.077234268, 1.064907432, 1.065898061, 1.05904305, 1.074055791, 1.068949819, 1.039388657, 1.019022465, 0.998272002, 0.976956725, 0.962277412, 0.959765851, 1.007760406, 1.092259288, 1.156586885, 1.184763908, 1.19553268, 1.201265335, 1.185963511, 0.648370028, 0.688576519, 0.742491126, 0.783990383, 0.80298996, 0.821607351, 0.862388551, 0.894268453, 0.911370516, 0.89651376, 0.842166662, 0.783710659, 0.728130281, 0.684661746, 0.648905814, 0.614272654, 0.594889522, 0.571380973, 0.542023718, 0.521143556, 0.499938488, 0.475976497, 0.449459493, 0.421074629, 0.397123814, 0.374796838, 0.350898117, 0.324967325, 0.300079197, 0.278220892, 0.256880879, 0.648370028, 0.688576519, 0.743724585, 0.787064314, 0.807689011, 0.831625164, 0.888596475, 0.946057677, 0.993717194, 1.00470984, 0.962817013, 0.918186665, 0.884074688, 0.878249466, 0.889110208, 0.893722475, 0.920192778, 0.919718087, 0.890581846, 0.875235379, 0.864329875, 0.856818676, 0.855373979, 0.862291932, 0.910496414, 0.980866313, 1.027168989, 1.036630273, 1.033012152, 1.031717777, 1.011372924, 0.648370028, 0.688576519, 0.744858205, 0.789697766, 0.811312854, 0.838736176, 0.90691787, 0.981273532, 1.046675324, 1.06989634, 1.030948877, 0.989238918, 0.961261451, 0.966397285, 0.985641062, 0.989913285, 1.016537428, 1.012879372, 0.976154804, 0.955336571, 0.939947248, 0.927718401, 0.922940135, 0.930412292, 0.994285345, 1.093555927, 1.160755515, 1.176150799, 1.170561314, 1.16328609, 1.130534887, 0.648370028, 0.688576519, 0.74590379, 0.791982412, 0.814170182, 0.843955517, 0.920372605, 1.006811261, 1.083702803, 1.11277163, 1.07243216, 1.026477456, 0.99338913, 0.995457411, 1.015423536, 1.022041798, 1.05273211, 1.048963904, 1.008393526, 0.983748913, 0.964948297, 0.949395537, 0.941659451, 0.946655393, 1.010100961, 1.110544324, 1.177545547, 1.191351295, 1.184355021, 1.17684567, 1.143859267
我想要这样的东西：
我假设这种关系是非线性和渐近的；我知道它可能不完全符合我画的渐近线。我希望代码根据数据的关系绘制这条线，但强制它通过 (0,0) [我知道围绕强制一条线通过 0 的统计问题，我对它的道德不感兴趣] A以 R 为基础的解决方案将是首选。
假设问题是要求一个参数函数，它给出的曲线类似于图像中的曲线，我们提供了下面 (1) 和 (2) 中 nls 调用中显示的两个备选方案。 （输入 x 和 y 必须存在，并在末尾的注释中显示为 R 代码。）
# add zero
x0 <- c(0, x)
y0 <- c(0, y)
# sort
o <- order(x0)
x0 <- x0[o]
y0 <- y0[o]
(1) y0 ~ a * x0/(b + x0)
fm <- nls(y0 ~ a*x0/(b+x0), start = list(a = 1, b = 1)); fm
## Nonlinear regression model
## model: y0 ~ a * x0/(b + x0)
## data: parent.frame()
## a b
## 1.1888 0.1959
## residual sum-of-squares: 54.08
##
## Number of iterations to convergence: 6
## Achieved convergence tolerance: 6.172e-07
见剧情接近尾声。
(2) y0 ~ a * (1 - exp(-b*x0))
a 的拟合值非常接近 1，您可以将其进一步简化为单参数公式 (1 - exp(-b * x0)) -- 未显示。
fm2 <- nls(y0 ~ a * (1 - exp(-b*x0)), start = list(a = 1, b = 1))
## Nonlinear regression model
## model: y0 ~ a * (1 - exp(-b * x0))
## data: parent.frame()
## a b
## 0.9904 4.3019
## residual sum-of-squares: 54.1
##
## Number of iterations to convergence: 7
## Achieved convergence tolerance: 3.276e-07
plot(y0 ~ x0)
lines(fitted(fm) ~ x0, col = "red", lty = 2)
lines(fitted(fm2) ~ x0, col = "blue")
legend("bottomright", c("fm", "fm2"), col = c("red", "blue"), lty = 2:1)
备注
x <- c(0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25,
0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25,
0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75,
0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75,
0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25,
0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25,
0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75,
0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75,
0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25,
0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25,
0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75,
0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75,
0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25,
0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25,
0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75,
0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75,
0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25,
0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25,
0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75,
0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75,
0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25,
0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25,
0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75,
0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75,
0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25,
0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25,
0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75,
0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75,
0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25,
0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25,
0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75,
0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75,
0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25,
0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25,
0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75,
0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75,
0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25,
0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25,
0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75,
0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75,
0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1)
y <- c(y=0.237584546, 0.28671819, 0.336561114, 0.378876835, 0.410064071,
0.42711702, 0.431941271, 0.422502577, 0.401732266, 0.371183008, 0.333479345,
0.294357896, 0.256349087, 0.220946386, 0.18870756, 0.159970611, 0.13514623,
0.114071228, 0.096631728, 0.083271086, 0.074629657, 0.07140743, 0.073767886,
0.080317959, 0.09055645, 0.099722967, 0.103947006, 0.102038115, 0.095550247,
0.08657901, 0.076411799, 0.237584546, 0.28671819, 0.337439418, 0.381827265,
0.417294532, 0.445565343, 0.47127223, 0.489057124, 0.497978568, 0.494098246,
0.475260139, 0.451871842, 0.427158743, 0.405162275, 0.385870874, 0.368627489,
0.357870609, 0.347339779, 0.336065352, 0.3277376, 0.31923309, 0.310445666,
0.30146125, 0.292309433, 0.285917997, 0.28080225, 0.274813414, 0.266959012,
0.258607358, 0.250720114, 0.241381615, 0.237584546, 0.28671819, 0.338249743,
0.384354472, 0.422907919, 0.459495634, 0.502073705, 0.54344523, 0.579701245,
0.602399588, 0.604650199, 0.603404641, 0.604648113, 0.616592407, 0.636100411,
0.656602085, 0.691333354, 0.715613246, 0.726100028, 0.742937744, 0.75896734,
0.772711635, 0.783817351, 0.792477548, 0.815433204, 0.844297409, 0.860775054,
0.859533668, 0.851030469, 0.842182457, 0.824014425, 0.237584546, 0.28671819,
0.339000225, 0.386546582, 0.427356929, 0.470181704, 0.526094198, 0.586508811,
0.644855261, 0.688954055, 0.707522988, 0.72301662, 0.744395614, 0.78133285,
0.821173847, 0.846684277, 0.874156415, 0.8794052, 0.865071774, 0.85576421,
0.847150624, 0.841573656, 0.842631876, 0.853851676, 0.89989382, 0.968196332,
1.021945477, 1.04606986, 1.053631186, 1.051974535, 1.029160142, 0.526703835,
0.580077052, 0.64041841, 0.685227215, 0.708760738, 0.727063358, 0.756454229,
0.773454905, 0.772194505, 0.745049596, 0.690470636, 0.63082099, 0.570992827,
0.517849863, 0.471707493, 0.430477947, 0.401431292, 0.376345813, 0.352742821,
0.337159961, 0.323135644, 0.306610614, 0.2863518, 0.262777597, 0.239513919,
0.216919169, 0.195270076, 0.175353825, 0.159000337, 0.147969663, 0.141399831,
0.526703835, 0.580077052, 0.641411245, 0.688088119, 0.714152455, 0.739854753,
0.788182974, 0.834289014, 0.867789567, 0.872478724, 0.838853657, 0.802308857,
0.771704018, 0.760362864, 0.764012635, 0.768407643, 0.794624805, 0.806138694,
0.793886721, 0.787385046, 0.778422236, 0.767448068, 0.760419846, 0.763194442,
0.800594032, 0.861122489, 0.909678638, 0.929959297, 0.93331182, 0.929158747,
0.906263888, 0.526703835, 0.580077052, 0.642331958, 0.69060123, 0.718490601,
0.749270976, 0.810674727, 0.87634027, 0.932119727, 0.955627441, 0.931819558,
0.905923545, 0.890946984, 0.903621435, 0.930591166, 0.944273829, 0.972908974,
0.975313425, 0.946814418, 0.929125249, 0.915514231, 0.905400157, 0.903990924,
0.915810287, 0.975515068, 1.064027309, 1.126505017, 1.145498753, 1.144020438,
1.139107943, 1.113434553, 0.526703835, 0.580077052, 0.643188179, 0.692829609,
0.722048342, 0.756346047, 0.826838255, 0.905560315, 0.975319922, 1.010055542,
0.991317511, 0.968984783, 0.956542194, 0.97092247, 0.998722494, 1.012706399,
1.046013355, 1.049696207, 1.017329335, 0.997018099, 0.98111558, 0.96725297,
0.96104753, 0.967863142, 1.026332974, 1.117019534, 1.179564476, 1.195799589,
1.191725969, 1.186194897, 1.159237266, 0.655505419, 0.700835288, 0.778770804,
0.833899736, 0.847185075, 0.862788737, 0.92250669, 0.972223103, 0.995378196,
0.971450984, 0.899317265, 0.831511259, 0.777273357, 0.750439405, 0.736706853,
0.716939509, 0.713318467, 0.689852595, 0.64779079, 0.613008142, 0.577883363,
0.544176877, 0.515529096, 0.494873941, 0.494853497, 0.50579989, 0.51072073,
0.502549946, 0.486712486, 0.4666287, 0.436886549, 0.655505419, 0.700835288,
0.779515505, 0.835625052, 0.849386513, 0.867613256, 0.940745175, 1.015935421,
1.071178913, 1.070588827, 1.003308654, 0.939561248, 0.893934131, 0.890404642,
0.906546175, 0.907787442, 0.930118144, 0.915706635, 0.869787991, 0.841509521,
0.818410397, 0.800443888, 0.792261958, 0.799097538, 0.858240128, 0.941043377,
0.992391586, 1.001775026, 0.997865558, 0.997114182, 0.976159096, 0.655505419,
0.700835288, 0.780204177, 0.837109506, 0.851117432, 0.871370792, 0.954050004,
1.045617461, 1.120124578, 1.133210897, 1.068657398, 1.006111622, 0.961171031,
0.959812284, 0.979586005, 0.985468209, 1.022557616, 1.016664505, 0.970005512,
0.943179786, 0.920201898, 0.900165081, 0.890574932, 0.901043296, 0.988548338,
1.120099664, 1.20433116, 1.22200346, 1.21430707, 1.204596162, 1.166260242,
0.655505419, 0.700835288, 0.780844152, 0.838413477, 0.852482915, 0.873996615,
0.96333921, 1.065954447, 1.1517272, 1.170616388, 1.104780555, 1.041245937,
0.998188853, 1.004236221, 1.031573772, 1.038371921, 1.074363112, 1.063259125,
1.009537697, 0.97778815, 0.951786101, 0.92967248, 0.91851604, 0.927403271,
1.015437007, 1.14773798, 1.229890227, 1.244415402, 1.234895468, 1.22572875,
1.187791944, 0.77581346, 0.817140758, 0.885253429, 0.932349861, 0.942467809,
0.95222801, 1.000935197, 1.049176574, 1.079828262, 1.069961071, 1.007853389,
0.946804225, 0.898908794, 0.875712335, 0.865409732, 0.849046052, 0.852605164,
0.840772986, 0.808174312, 0.788675547, 0.772639871, 0.756165385, 0.741423607,
0.731267095, 0.744977355, 0.770040751, 0.782511771, 0.774730265, 0.756842554,
0.734819889, 0.698052168, 0.77581346, 0.817140758, 0.885996401, 0.934090912,
0.944526315, 0.955664337, 1.011435747, 1.073157072, 1.119287014, 1.119515657,
1.059037447, 1.000632763, 0.962856472, 0.961162984, 0.974968374, 0.970587075,
0.977721334, 0.957530379, 0.910347521, 0.879712403, 0.858478308, 0.843830824,
0.8379758, 0.842489362, 0.884822607, 0.948875129, 0.994918346, 1.009823918,
1.013503909, 1.019781709, 1.006411076, 0.77581346, 0.817140758, 0.886676908,
0.935586452, 0.94628191, 0.958534539, 1.019246221, 1.08957231, 1.144408226,
1.149548173, 1.089345217, 1.02939105, 0.988438129, 0.982776225, 0.995111585,
0.995346427, 1.01788497, 1.013562918, 0.97511065, 0.951002598, 0.932399511,
0.916437924, 0.908735514, 0.915104389, 0.971065819, 1.056129813, 1.113286853,
1.123075843, 1.109285474, 1.091263294, 1.055615306, 0.77581346, 0.817140758,
0.887305021, 0.936918974, 0.947968066, 0.961458325, 1.026293159, 1.103072286,
1.163753867, 1.171400785, 1.109564304, 1.046773672, 1.004206896, 1.001410365,
1.021429181, 1.028966308, 1.05691576, 1.051712751, 1.008408427, 0.980799317,
0.961422026, 0.94676441, 0.941248, 0.949138701, 1.007747769, 1.095256448,
1.151844382, 1.160300612, 1.149290085, 1.139482856, 1.110224843, 0.850993693,
0.878551781, 0.929460883, 0.968631923, 0.979977369, 0.984748244, 1.006746173,
1.022598743, 1.022056341, 0.995788336, 0.941417933, 0.887557328, 0.844630063,
0.82260114, 0.816226721, 0.811409831, 0.822775304, 0.827547729, 0.816761315,
0.808919251, 0.798904061, 0.785326183, 0.770118356, 0.755392075, 0.753594935,
0.759954751, 0.761058331, 0.750226259, 0.732908666, 0.714197397, 0.686983109,
0.850993693, 0.878551781, 0.930243015, 0.970633924, 0.982279301, 0.986959875,
1.011868954, 1.036545277, 1.050327182, 1.038798094, 0.994740367, 0.94990766,
0.916918397, 0.905929267, 0.908401549, 0.906138301, 0.9154374, 0.913555622,
0.89567697, 0.888012469, 0.887200415, 0.889993548, 0.895718455, 0.904391587,
0.934504628, 0.97859478, 1.01128006, 1.02061367, 1.01873982, 1.017302394,
1.004862547, 0.850993693, 0.878551781, 0.930973589, 0.972435594, 0.984444678,
0.989400923, 1.01672256, 1.047491074, 1.070196271, 1.067319155, 1.02880311,
0.988358259, 0.959646761, 0.953317106, 0.960671306, 0.963204622, 0.978041887,
0.976714373, 0.952274144, 0.933524072, 0.920094073, 0.911611795, 0.908339798,
0.909115911, 0.93042165, 0.96522212, 0.991738021, 1.000063777, 1.00124383,
1.005596638, 0.999989033, 0.850993693, 0.878551781, 0.931660473, 0.974092126,
0.986525893, 0.99185437, 1.02062571, 1.054694772, 1.081895232, 1.082667708,
1.045917034, 1.007757068, 0.984359384, 0.986672878, 1.003036141, 1.011242032,
1.030569553, 1.031354904, 1.006945848, 0.989743054, 0.978098989, 0.96971339,
0.965755701, 0.967847824, 0.997593641, 1.045950651, 1.081306815, 1.089472413,
1.084782481, 1.080476761, 1.064157963, 0.567489982, 0.652162194, 0.785000205,
0.873056889, 0.891931236, 0.912524343, 1.003800631, 1.082395673, 1.125870347,
1.094785452, 0.992700994, 0.896223068, 0.808895707, 0.748320878, 0.704849899,
0.664936841, 0.656069219, 0.633515, 0.597315729, 0.579769373, 0.56124866,
0.537654638, 0.510705411, 0.482994974, 0.46803537, 0.458939075, 0.447710872,
0.431626648, 0.416970104, 0.406424105, 0.38950634, 0.567489982, 0.652162194,
0.785214782, 0.873410881, 0.892192543, 0.915579736, 1.02197063, 1.126847506,
1.201263547, 1.186374187, 1.080291033, 0.984983623, 0.910771549, 0.893572509,
0.904725552, 0.904995918, 0.946030796, 0.933481693, 0.878471017, 0.847143769,
0.818725407, 0.797739804, 0.79148525, 0.80905968, 0.909970701, 1.043915868,
1.117636561, 1.119943023, 1.094645143, 1.06159699, 1.003503442, 0.567489982,
0.652162194, 0.785410047, 0.873727441, 0.892530978, 0.918462813, 1.035583615,
1.156719565, 1.249178052, 1.24364233, 1.134532094, 1.035999656, 0.960602522,
0.950968742, 0.973596573, 0.980182827, 1.03298378, 1.021811604, 0.961205125,
0.93057096, 0.903318107, 0.882389605, 0.876688838, 0.898444176, 1.028598189,
1.207716346, 1.306888223, 1.31149435, 1.280924439, 1.245280743, 1.177808642,
0.567489982, 0.652162194, 0.785588086, 0.874014735, 0.892850876, 0.920784175,
1.046159983, 1.178910494, 1.282226086, 1.280955315, 1.168986797, 1.068514109,
0.993000448, 0.987921119, 1.014108896, 1.018215179, 1.066857815, 1.049017906,
0.981723487, 0.946660042, 0.917113125, 0.895081699, 0.888686717, 0.909516275,
1.039963722, 1.219105005, 1.316385865, 1.319118261, 1.289013147, 1.256706476,
1.191284895, 0.747616649, 0.836237073, 0.988689244, 1.054734826, 1.025774002,
1.015855908, 1.134614944, 1.216524243, 1.254345179, 1.173024774, 0.998351097,
0.874513745, 0.78600049, 0.778888166, 0.806788266, 0.816571891, 0.871123493,
0.853065372, 0.791552305, 0.760557473, 0.717245996, 0.670639575, 0.636679769,
0.62634939, 0.67381537, 0.736528993, 0.77349472, 0.769330204, 0.73253262,
0.673349917, 0.596250415, 0.747616649, 0.836237073, 0.989223421, 1.055233955,
1.025119543, 1.016921043, 1.151469111, 1.258143306, 1.321192265, 1.245432615,
1.057826281, 0.928337336, 0.842428386, 0.865764856, 0.929639697, 0.956855834,
1.033756137, 1.004413009, 0.918607771, 0.881685138, 0.843460083, 0.817945898,
0.818849444, 0.856904984, 1.007567525, 1.156849146, 1.213010311, 1.186402917,
1.130393624, 1.074572086, 1.003096223, 0.747616649, 0.836237073, 0.989702821,
1.055676937, 1.024871826, 1.018258095, 1.163146257, 1.283846498, 1.358478546,
1.283459663, 1.088712811, 0.952908397, 0.860999584, 0.883257091, 0.949974537,
0.981948197, 1.07404995, 1.051186562, 0.96545577, 0.933451772, 0.896999121,
0.871107161, 0.87029767, 0.908490062, 1.089059591, 1.285072803, 1.356228113,
1.312512875, 1.22735393, 1.145342112, 1.055003524, 0.747616649, 0.836237073,
0.990136743, 1.056092143, 1.024971724, 1.020100713, 1.172664881, 1.301743984,
1.38206172, 1.306367397, 1.106599331, 0.967083514, 0.872632623, 0.896234035,
0.964761913, 0.995294869, 1.085453749, 1.059120417, 0.970445096, 0.938069701,
0.90287292, 0.878777742, 0.879172385, 0.917467892, 1.098720551, 1.293762326,
1.362068295, 1.316773057, 1.233494401, 1.155817986, 1.067250609, 0.638475657,
0.701241612, 0.780950189, 0.833968759, 0.847179413, 0.855540872, 0.887958705,
0.901707411, 0.895013452, 0.853711843, 0.775103211, 0.701753795, 0.638828635,
0.593984663, 0.556727886, 0.516842306, 0.488357365, 0.453170925, 0.412938416,
0.381577313, 0.353999287, 0.33014816, 0.309804767, 0.292518765, 0.283133805,
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0.926839471, 0.98254782, 1.025614381, 1.021376729, 0.953225791, 0.892099738,
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0.877265215, 0.954805672, 1.039082289, 1.109156251, 1.117263913, 1.04548502,
0.980756104, 0.940173388, 0.946755886, 0.96761173, 0.962748587, 0.977708697,
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0.802371204, 0.857669353, 0.936871529, 0.989703417, 1.003184795, 1.006631613,
1.016924024, 1.00193429, 0.638475657, 0.701241612, 0.786107659, 0.844073474,
0.859399199, 0.88437283, 0.976726115, 1.081594229, 1.167513251, 1.18007791,
1.103111386, 1.03039217, 0.979927897, 0.977664709, 0.995771885, 0.997566521,
1.03454566, 1.025245547, 0.971404195, 0.940074384, 0.913797617, 0.893222034,
0.885627687, 0.899616897, 0.992524326, 1.126092672, 1.206882238, 1.213850737,
1.190800905, 1.164836526, 1.112275243, 0.795258403, 0.816856802, 0.857468605,
0.891532302, 0.906502187, 0.92630136, 0.979601145, 1.028863072, 1.063205004,
1.066814184, 1.032617211, 0.994250894, 0.955908895, 0.928678632, 0.908084571,
0.883677125, 0.873843551, 0.855833828, 0.826784849, 0.806078315, 0.786206603,
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0.857970893, 0.892682314, 0.90784204, 0.929064393, 0.990200877, 1.053232908,
1.103422642, 1.118505836, 1.08778131, 1.05342114, 1.022762179, 1.012263894,
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0.907226205, 0.884828508, 0.869311213, 0.863197386, 0.888297558, 0.932593346,
0.967936397, 0.985253811, 0.996055782, 1.008288145, 1.005891085, 0.795258403,
0.816856802, 0.858424306, 0.893659353, 0.908989906, 0.931387544, 0.99817735,
1.070488811, 1.13034749, 1.151631355, 1.122590423, 1.088484406, 1.056558609,
1.043916106, 1.042092919, 1.032291174, 1.043409228, 1.037230253, 1.008979201,
0.989836633, 0.970193386, 0.95017159, 0.936686635, 0.93526113, 0.982087791,
1.064218163, 1.129002571, 1.158894658, 1.168904543, 1.169321179, 1.149312615,
0.795258403, 0.816856802, 0.85883683, 0.894518554, 0.910082877, 0.933618307,
1.004875779, 1.084069133, 1.150643349, 1.17569387, 1.146507144, 1.110903978,
1.077234268, 1.064907432, 1.065898061, 1.05904305, 1.074055791, 1.068949819,
1.039388657, 1.019022465, 0.998272002, 0.976956725, 0.962277412, 0.959765851,
1.007760406, 1.092259288, 1.156586885, 1.184763908, 1.19553268, 1.201265335,
1.185963511, 0.648370028, 0.688576519, 0.742491126, 0.783990383, 0.80298996,
0.821607351, 0.862388551, 0.894268453, 0.911370516, 0.89651376, 0.842166662,
0.783710659, 0.728130281, 0.684661746, 0.648905814, 0.614272654, 0.594889522,
0.571380973, 0.542023718, 0.521143556, 0.499938488, 0.475976497, 0.449459493,
0.421074629, 0.397123814, 0.374796838, 0.350898117, 0.324967325, 0.300079197,
0.278220892, 0.256880879, 0.648370028, 0.688576519, 0.743724585, 0.787064314,
0.807689011, 0.831625164, 0.888596475, 0.946057677, 0.993717194, 1.00470984,
0.962817013, 0.918186665, 0.884074688, 0.878249466, 0.889110208, 0.893722475,
0.920192778, 0.919718087, 0.890581846, 0.875235379, 0.864329875, 0.856818676,
0.855373979, 0.862291932, 0.910496414, 0.980866313, 1.027168989, 1.036630273,
1.033012152, 1.031717777, 1.011372924, 0.648370028, 0.688576519, 0.744858205,
0.789697766, 0.811312854, 0.838736176, 0.90691787, 0.981273532, 1.046675324,
1.06989634, 1.030948877, 0.989238918, 0.961261451, 0.966397285, 0.985641062,
0.989913285, 1.016537428, 1.012879372, 0.976154804, 0.955336571, 0.939947248,
0.927718401, 0.922940135, 0.930412292, 0.994285345, 1.093555927, 1.160755515,
1.176150799, 1.170561314, 1.16328609, 1.130534887, 0.648370028, 0.688576519,
0.74590379, 0.791982412, 0.814170182, 0.843955517, 0.920372605, 1.006811261,
1.083702803, 1.11277163, 1.07243216, 1.026477456, 0.99338913, 0.995457411,
1.015423536, 1.022041798, 1.05273211, 1.048963904, 1.008393526, 0.983748913,
0.964948297, 0.949395537, 0.941659451, 0.946655393, 1.010100961, 1.110544324,
1.177545547, 1.191351295, 1.184355021, 1.17684567, 1.143859267)
数据
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0.952133417, 0.928703249, 0.638475657, 0.701241612, 0.784563363, 0.841383219, 0.856479287, 0.877265215, 0.954805672, 1.039082289, 1.109156251, 1.117263913, 1.04548502, 0.980756104, 0.940173388, 0.946755886, 0.96761173, 0.962748587, 0.977708697, 0.952758789, 0.895089209, 0.858695507, 0.828825116, 0.807197034, 0.797473609, 0.802371204, 0.857669353, 0.936871529, 0.989703417, 1.003184795, 1.006631613, 1.016924024, 1.00193429, 0.638475657, 0.701241612, 0.786107659, 0.844073474, 0.859399199, 0.88437283, 0.976726115, 1.081594229, 1.167513251, 1.18007791, 1.103111386, 1.03039217, 0.979927897, 0.977664709, 0.995771885, 0.997566521, 1.03454566, 1.025245547, 0.971404195, 0.940074384, 0.913797617, 0.893222034, 0.885627687, 0.899616897, 0.992524326, 1.126092672, 1.206882238, 1.213850737, 1.190800905, 1.164836526, 1.112275243, 0.795258403, 0.816856802, 0.857468605, 0.891532302, 0.906502187, 0.92630136, 0.979601145, 1.028863072, 1.063205004, 1.066814184, 1.032617211, 0.994250894, 0.955908895, 0.928678632, 0.908084571, 0.883677125, 0.873843551, 0.855833828, 0.826784849, 0.806078315, 0.786206603, 0.767146707, 0.752018094, 0.742449701, 0.751986265, 0.771394849, 0.784393907, 0.784185886, 0.774534941, 0.758701324, 0.732473671, 0.795258403, 0.816856802, 0.857970893, 0.892682314, 0.90784204, 0.929064393, 0.990200877, 1.053232908, 1.103422642, 1.118505836, 1.08778131, 1.05342114, 1.022762179, 1.012263894, 1.011744976, 1.00061059, 1.002970934, 0.988465309, 0.955877185, 0.931310594, 0.907226205, 0.884828508, 0.869311213, 0.863197386, 0.888297558, 0.932593346, 0.967936397, 0.985253811, 0.996055782, 1.008288145, 1.005891085, 0.795258403, 0.816856802, 0.858424306, 0.893659353, 0.908989906, 0.931387544, 0.99817735, 1.070488811, 1.13034749, 1.151631355, 1.122590423, 1.088484406, 1.056558609, 1.043916106, 1.042092919, 1.032291174, 1.043409228, 1.037230253, 1.008979201, 0.989836633, 0.970193386, 0.95017159, 0.936686635, 0.93526113, 0.982087791, 1.064218163, 1.129002571, 1.158894658, 1.168904543, 1.169321179, 1.149312615, 0.795258403, 0.816856802, 0.85883683, 0.894518554, 0.910082877, 0.933618307, 1.004875779, 1.084069133, 1.150643349, 1.17569387, 1.146507144, 1.110903978, 1.077234268, 1.064907432, 1.065898061, 1.05904305, 1.074055791, 1.068949819, 1.039388657, 1.019022465, 0.998272002, 0.976956725, 0.962277412, 0.959765851, 1.007760406, 1.092259288, 1.156586885, 1.184763908, 1.19553268, 1.201265335, 1.185963511, 0.648370028, 0.688576519, 0.742491126, 0.783990383, 0.80298996, 0.821607351, 0.862388551, 0.894268453, 0.911370516, 0.89651376, 0.842166662, 0.783710659, 0.728130281, 0.684661746, 0.648905814, 0.614272654, 0.594889522, 0.571380973, 0.542023718, 0.521143556, 0.499938488, 0.475976497, 0.449459493, 0.421074629, 0.397123814, 0.374796838, 0.350898117, 0.324967325, 0.300079197, 0.278220892, 0.256880879, 0.648370028, 0.688576519, 0.743724585, 0.787064314, 0.807689011, 0.831625164, 0.888596475, 0.946057677, 0.993717194, 1.00470984, 0.962817013, 0.918186665, 0.884074688, 0.878249466, 0.889110208, 0.893722475, 0.920192778, 0.919718087, 0.890581846, 0.875235379, 0.864329875, 0.856818676, 0.855373979, 0.862291932, 0.910496414, 0.980866313, 1.027168989, 1.036630273, 1.033012152, 1.031717777, 1.011372924, 0.648370028, 0.688576519, 0.744858205, 0.789697766, 0.811312854, 0.838736176, 0.90691787, 0.981273532, 1.046675324, 1.06989634, 1.030948877, 0.989238918, 0.961261451, 0.966397285, 0.985641062, 0.989913285, 1.016537428, 1.012879372, 0.976154804, 0.955336571, 0.939947248, 0.927718401, 0.922940135, 0.930412292, 0.994285345, 1.093555927, 1.160755515, 1.176150799, 1.170561314, 1.16328609, 1.130534887, 0.648370028, 0.688576519, 0.74590379, 0.791982412, 0.814170182, 0.843955517, 0.920372605, 1.006811261, 1.083702803, 1.11277163, 1.07243216, 1.026477456, 0.99338913, 0.995457411, 1.015423536, 1.022041798, 1.05273211, 1.048963904, 1.008393526, 0.983748913, 0.964948297, 0.949395537, 0.941659451, 0.946655393, 1.010100961, 1.110544324, 1.177545547, 1.191351295, 1.184355021, 1.17684567, 1.143859267
我想要这样的东西：
我假设这种关系是非线性和渐近的；我知道它可能不完全符合我画的渐近线。我希望代码根据数据的关系绘制这条线，但强制它通过 (0,0) [我知道围绕强制一条线通过 0 的统计问题，我对它的道德不感兴趣] A以 R 为基础的解决方案将是首选。
假设问题是要求一个参数函数，它给出的曲线类似于图像中的曲线，我们提供了下面 (1) 和 (2) 中 nls 调用中显示的两个备选方案。 （输入 x 和 y 必须存在，并在末尾的注释中显示为 R 代码。）
# add zero
x0 <- c(0, x)
y0 <- c(0, y)
# sort
o <- order(x0)
x0 <- x0[o]
y0 <- y0[o]
(1) y0 ~ a * x0/(b + x0)
fm <- nls(y0 ~ a*x0/(b+x0), start = list(a = 1, b = 1)); fm
## Nonlinear regression model
## model: y0 ~ a * x0/(b + x0)
## data: parent.frame()
## a b
## 1.1888 0.1959
## residual sum-of-squares: 54.08
##
## Number of iterations to convergence: 6
## Achieved convergence tolerance: 6.172e-07
见剧情接近尾声。
(2) y0 ~ a * (1 - exp(-b*x0))
a 的拟合值非常接近 1，您可以将其进一步简化为单参数公式 (1 - exp(-b * x0)) -- 未显示。
fm2 <- nls(y0 ~ a * (1 - exp(-b*x0)), start = list(a = 1, b = 1))
## Nonlinear regression model
## model: y0 ~ a * (1 - exp(-b * x0))
## data: parent.frame()
## a b
## 0.9904 4.3019
## residual sum-of-squares: 54.1
##
## Number of iterations to convergence: 7
## Achieved convergence tolerance: 3.276e-07
plot(y0 ~ x0)
lines(fitted(fm) ~ x0, col = "red", lty = 2)
lines(fitted(fm2) ~ x0, col = "blue")
legend("bottomright", c("fm", "fm2"), col = c("red", "blue"), lty = 2:1)
备注
x <- c(0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25,
0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25,
0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75,
0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75,
0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25,
0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25,
0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75,
0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75,
0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25,
0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25,
0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75,
0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75,
0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25,
0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25,
0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75,
0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75,
0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25,
0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25,
0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75,
0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75,
0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25,
0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25,
0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75,
0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75,
0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25,
0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25,
0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75,
0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75,
0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25,
0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25,
0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75,
0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75,
0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25,
0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25,
0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75,
0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75,
0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25,
0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25,
0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5,
0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75,
0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75,
0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 0.75, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1)
y <- c(y=0.237584546, 0.28671819, 0.336561114, 0.378876835, 0.410064071,
0.42711702, 0.431941271, 0.422502577, 0.401732266, 0.371183008, 0.333479345,
0.294357896, 0.256349087, 0.220946386, 0.18870756, 0.159970611, 0.13514623,
0.114071228, 0.096631728, 0.083271086, 0.074629657, 0.07140743, 0.073767886,
0.080317959, 0.09055645, 0.099722967, 0.103947006, 0.102038115, 0.095550247,
0.08657901, 0.076411799, 0.237584546, 0.28671819, 0.337439418, 0.381827265,
0.417294532, 0.445565343, 0.47127223, 0.489057124, 0.497978568, 0.494098246,
0.475260139, 0.451871842, 0.427158743, 0.405162275, 0.385870874, 0.368627489,
0.357870609, 0.347339779, 0.336065352, 0.3277376, 0.31923309, 0.310445666,
0.30146125, 0.292309433, 0.285917997, 0.28080225, 0.274813414, 0.266959012,
0.258607358, 0.250720114, 0.241381615, 0.237584546, 0.28671819, 0.338249743,
0.384354472, 0.422907919, 0.459495634, 0.502073705, 0.54344523, 0.579701245,
0.602399588, 0.604650199, 0.603404641, 0.604648113, 0.616592407, 0.636100411,
0.656602085, 0.691333354, 0.715613246, 0.726100028, 0.742937744, 0.75896734,
0.772711635, 0.783817351, 0.792477548, 0.815433204, 0.844297409, 0.860775054,
0.859533668, 0.851030469, 0.842182457, 0.824014425, 0.237584546, 0.28671819,
0.339000225, 0.386546582, 0.427356929, 0.470181704, 0.526094198, 0.586508811,
0.644855261, 0.688954055, 0.707522988, 0.72301662, 0.744395614, 0.78133285,
0.821173847, 0.846684277, 0.874156415, 0.8794052, 0.865071774, 0.85576421,
0.847150624, 0.841573656, 0.842631876, 0.853851676, 0.89989382, 0.968196332,
1.021945477, 1.04606986, 1.053631186, 1.051974535, 1.029160142, 0.526703835,
0.580077052, 0.64041841, 0.685227215, 0.708760738, 0.727063358, 0.756454229,
0.773454905, 0.772194505, 0.745049596, 0.690470636, 0.63082099, 0.570992827,
0.517849863, 0.471707493, 0.430477947, 0.401431292, 0.376345813, 0.352742821,
0.337159961, 0.323135644, 0.306610614, 0.2863518, 0.262777597, 0.239513919,
0.216919169, 0.195270076, 0.175353825, 0.159000337, 0.147969663, 0.141399831,
0.526703835, 0.580077052, 0.641411245, 0.688088119, 0.714152455, 0.739854753,
0.788182974, 0.834289014, 0.867789567, 0.872478724, 0.838853657, 0.802308857,
0.771704018, 0.760362864, 0.764012635, 0.768407643, 0.794624805, 0.806138694,
0.793886721, 0.787385046, 0.778422236, 0.767448068, 0.760419846, 0.763194442,
0.800594032, 0.861122489, 0.909678638, 0.929959297, 0.93331182, 0.929158747,
0.906263888, 0.526703835, 0.580077052, 0.642331958, 0.69060123, 0.718490601,
0.749270976, 0.810674727, 0.87634027, 0.932119727, 0.955627441, 0.931819558,
0.905923545, 0.890946984, 0.903621435, 0.930591166, 0.944273829, 0.972908974,
0.975313425, 0.946814418, 0.929125249, 0.915514231, 0.905400157, 0.903990924,
0.915810287, 0.975515068, 1.064027309, 1.126505017, 1.145498753, 1.144020438,
1.139107943, 1.113434553, 0.526703835, 0.580077052, 0.643188179, 0.692829609,
0.722048342, 0.756346047, 0.826838255, 0.905560315, 0.975319922, 1.010055542,
0.991317511, 0.968984783, 0.956542194, 0.97092247, 0.998722494, 1.012706399,
1.046013355, 1.049696207, 1.017329335, 0.997018099, 0.98111558, 0.96725297,
0.96104753, 0.967863142, 1.026332974, 1.117019534, 1.179564476, 1.195799589,
1.191725969, 1.186194897, 1.159237266, 0.655505419, 0.700835288, 0.778770804,
0.833899736, 0.847185075, 0.862788737, 0.92250669, 0.972223103, 0.995378196,
0.971450984, 0.899317265, 0.831511259, 0.777273357, 0.750439405, 0.736706853,
0.716939509, 0.713318467, 0.689852595, 0.64779079, 0.613008142, 0.577883363,
0.544176877, 0.515529096, 0.494873941, 0.494853497, 0.50579989, 0.51072073,
0.502549946, 0.486712486, 0.4666287, 0.436886549, 0.655505419, 0.700835288,
0.779515505, 0.835625052, 0.849386513, 0.867613256, 0.940745175, 1.015935421,
1.071178913, 1.070588827, 1.003308654, 0.939561248, 0.893934131, 0.890404642,
0.906546175, 0.907787442, 0.930118144, 0.915706635, 0.869787991, 0.841509521,
0.818410397, 0.800443888, 0.792261958, 0.799097538, 0.858240128, 0.941043377,
0.992391586, 1.001775026, 0.997865558, 0.997114182, 0.976159096, 0.655505419,
0.700835288, 0.780204177, 0.837109506, 0.851117432, 0.871370792, 0.954050004,
1.045617461, 1.120124578, 1.133210897, 1.068657398, 1.006111622, 0.961171031,
0.959812284, 0.979586005, 0.985468209, 1.022557616, 1.016664505, 0.970005512,
0.943179786, 0.920201898, 0.900165081, 0.890574932, 0.901043296, 0.988548338,
1.120099664, 1.20433116, 1.22200346, 1.21430707, 1.204596162, 1.166260242,
0.655505419, 0.700835288, 0.780844152, 0.838413477, 0.852482915, 0.873996615,
0.96333921, 1.065954447, 1.1517272, 1.170616388, 1.104780555, 1.041245937,
0.998188853, 1.004236221, 1.031573772, 1.038371921, 1.074363112, 1.063259125,
1.009537697, 0.97778815, 0.951786101, 0.92967248, 0.91851604, 0.927403271,
1.015437007, 1.14773798, 1.229890227, 1.244415402, 1.234895468, 1.22572875,
1.187791944, 0.77581346, 0.817140758, 0.885253429, 0.932349861, 0.942467809,
0.95222801, 1.000935197, 1.049176574, 1.079828262, 1.069961071, 1.007853389,
0.946804225, 0.898908794, 0.875712335, 0.865409732, 0.849046052, 0.852605164,
0.840772986, 0.808174312, 0.788675547, 0.772639871, 0.756165385, 0.741423607,
0.731267095, 0.744977355, 0.770040751, 0.782511771, 0.774730265, 0.756842554,
0.734819889, 0.698052168, 0.77581346, 0.817140758, 0.885996401, 0.934090912,
0.944526315, 0.955664337, 1.011435747, 1.073157072, 1.119287014, 1.119515657,
1.059037447, 1.000632763, 0.962856472, 0.961162984, 0.974968374, 0.970587075,
0.977721334, 0.957530379, 0.910347521, 0.879712403, 0.858478308, 0.843830824,
0.8379758, 0.842489362, 0.884822607, 0.948875129, 0.994918346, 1.009823918,
1.013503909, 1.019781709, 1.006411076, 0.77581346, 0.817140758, 0.886676908,
0.935586452, 0.94628191, 0.958534539, 1.019246221, 1.08957231, 1.144408226,
1.149548173, 1.089345217, 1.02939105, 0.988438129, 0.982776225, 0.995111585,
0.995346427, 1.01788497, 1.013562918, 0.97511065, 0.951002598, 0.932399511,
0.916437924, 0.908735514, 0.915104389, 0.971065819, 1.056129813, 1.113286853,
1.123075843, 1.109285474, 1.091263294, 1.055615306, 0.77581346, 0.817140758,
0.887305021, 0.936918974, 0.947968066, 0.961458325, 1.026293159, 1.103072286,
1.163753867, 1.171400785, 1.109564304, 1.046773672, 1.004206896, 1.001410365,
1.021429181, 1.028966308, 1.05691576, 1.051712751, 1.008408427, 0.980799317,
0.961422026, 0.94676441, 0.941248, 0.949138701, 1.007747769, 1.095256448,
1.151844382, 1.160300612, 1.149290085, 1.139482856, 1.110224843, 0.850993693,
0.878551781, 0.929460883, 0.968631923, 0.979977369, 0.984748244, 1.006746173,
1.022598743, 1.022056341, 0.995788336, 0.941417933, 0.887557328, 0.844630063,
0.82260114, 0.816226721, 0.811409831, 0.822775304, 0.827547729, 0.816761315,
0.808919251, 0.798904061, 0.785326183, 0.770118356, 0.755392075, 0.753594935,
0.759954751, 0.761058331, 0.750226259, 0.732908666, 0.714197397, 0.686983109,
0.850993693, 0.878551781, 0.930243015, 0.970633924, 0.982279301, 0.986959875,
1.011868954, 1.036545277, 1.050327182, 1.038798094, 0.994740367, 0.94990766,
0.916918397, 0.905929267, 0.908401549, 0.906138301, 0.9154374, 0.913555622,
0.89567697, 0.888012469, 0.887200415, 0.889993548, 0.895718455, 0.904391587,
0.934504628, 0.97859478, 1.01128006, 1.02061367, 1.01873982, 1.017302394,
1.004862547, 0.850993693, 0.878551781, 0.930973589, 0.972435594, 0.984444678,
0.989400923, 1.01672256, 1.047491074, 1.070196271, 1.067319155, 1.02880311,
0.988358259, 0.959646761, 0.953317106, 0.960671306, 0.963204622, 0.978041887,
0.976714373, 0.952274144, 0.933524072, 0.920094073, 0.911611795, 0.908339798,
0.909115911, 0.93042165, 0.96522212, 0.991738021, 1.000063777, 1.00124383,
1.005596638, 0.999989033, 0.850993693, 0.878551781, 0.931660473, 0.974092126,
0.986525893, 0.99185437, 1.02062571, 1.054694772, 1.081895232, 1.082667708,
1.045917034, 1.007757068, 0.984359384, 0.986672878, 1.003036141, 1.011242032,
1.030569553, 1.031354904, 1.006945848, 0.989743054, 0.978098989, 0.96971339,
0.965755701, 0.967847824, 0.997593641, 1.045950651, 1.081306815, 1.089472413,
1.084782481, 1.080476761, 1.064157963, 0.567489982, 0.652162194, 0.785000205,
0.873056889, 0.891931236, 0.912524343, 1.003800631, 1.082395673, 1.125870347,
1.094785452, 0.992700994, 0.896223068, 0.808895707, 0.748320878, 0.704849899,
0.664936841, 0.656069219, 0.633515, 0.597315729, 0.579769373, 0.56124866,
0.537654638, 0.510705411, 0.482994974, 0.46803537, 0.458939075, 0.447710872,
0.431626648, 0.416970104, 0.406424105, 0.38950634, 0.567489982, 0.652162194,
0.785214782, 0.873410881, 0.892192543, 0.915579736, 1.02197063, 1.126847506,
1.201263547, 1.186374187, 1.080291033, 0.984983623, 0.910771549, 0.893572509,
0.904725552, 0.904995918, 0.946030796, 0.933481693, 0.878471017, 0.847143769,
0.818725407, 0.797739804, 0.79148525, 0.80905968, 0.909970701, 1.043915868,
1.117636561, 1.119943023, 1.094645143, 1.06159699, 1.003503442, 0.567489982,
0.652162194, 0.785410047, 0.873727441, 0.892530978, 0.918462813, 1.035583615,
1.156719565, 1.249178052, 1.24364233, 1.134532094, 1.035999656, 0.960602522,
0.950968742, 0.973596573, 0.980182827, 1.03298378, 1.021811604, 0.961205125,
0.93057096, 0.903318107, 0.882389605, 0.876688838, 0.898444176, 1.028598189,
1.207716346, 1.306888223, 1.31149435, 1.280924439, 1.245280743, 1.177808642,
0.567489982, 0.652162194, 0.785588086, 0.874014735, 0.892850876, 0.920784175,
1.046159983, 1.178910494, 1.282226086, 1.280955315, 1.168986797, 1.068514109,
0.993000448, 0.987921119, 1.014108896, 1.018215179, 1.066857815, 1.049017906,
0.981723487, 0.946660042, 0.917113125, 0.895081699, 0.888686717, 0.909516275,
1.039963722, 1.219105005, 1.316385865, 1.319118261, 1.289013147, 1.256706476,
1.191284895, 0.747616649, 0.836237073, 0.988689244, 1.054734826, 1.025774002,
1.015855908, 1.134614944, 1.216524243, 1.254345179, 1.173024774, 0.998351097,
0.874513745, 0.78600049, 0.778888166, 0.806788266, 0.816571891, 0.871123493,
0.853065372, 0.791552305, 0.760557473, 0.717245996, 0.670639575, 0.636679769,
0.62634939, 0.67381537, 0.736528993, 0.77349472, 0.769330204, 0.73253262,
0.673349917, 0.596250415, 0.747616649, 0.836237073, 0.989223421, 1.055233955,
1.025119543, 1.016921043, 1.151469111, 1.258143306, 1.321192265, 1.245432615,
1.057826281, 0.928337336, 0.842428386, 0.865764856, 0.929639697, 0.956855834,
1.033756137, 1.004413009, 0.918607771, 0.881685138, 0.843460083, 0.817945898,
0.818849444, 0.856904984, 1.007567525, 1.156849146, 1.213010311, 1.186402917,
1.130393624, 1.074572086, 1.003096223, 0.747616649, 0.836237073, 0.989702821,
1.055676937, 1.024871826, 1.018258095, 1.163146257, 1.283846498, 1.358478546,
1.283459663, 1.088712811, 0.952908397, 0.860999584, 0.883257091, 0.949974537,
0.981948197, 1.07404995, 1.051186562, 0.96545577, 0.933451772, 0.896999121,
0.871107161, 0.87029767, 0.908490062, 1.089059591, 1.285072803, 1.356228113,
1.312512875, 1.22735393, 1.145342112, 1.055003524, 0.747616649, 0.836237073,
0.990136743, 1.056092143, 1.024971724, 1.020100713, 1.172664881, 1.301743984,
1.38206172, 1.306367397, 1.106599331, 0.967083514, 0.872632623, 0.896234035,
0.964761913, 0.995294869, 1.085453749, 1.059120417, 0.970445096, 0.938069701,
0.90287292, 0.878777742, 0.879172385, 0.917467892, 1.098720551, 1.293762326,
1.362068295, 1.316773057, 1.233494401, 1.155817986, 1.067250609, 0.638475657,
0.701241612, 0.780950189, 0.833968759, 0.847179413, 0.855540872, 0.887958705,
0.901707411, 0.895013452, 0.853711843, 0.775103211, 0.701753795, 0.638828635,
0.593984663, 0.556727886, 0.516842306, 0.488357365, 0.453170925, 0.412938416,
0.381577313, 0.353999287, 0.33014816, 0.309804767, 0.292518765, 0.283133805,
0.275905758, 0.264425516, 0.247465536, 0.228905529, 0.211275622, 0.19250603,
0.638475657, 0.701241612, 0.782851875, 0.838087022, 0.852667689, 0.868328154,
0.926839471, 0.98254782, 1.025614381, 1.021376729, 0.953225791, 0.892099738,
0.851322412, 0.851359308, 0.868489683, 0.869845033, 0.892966986, 0.877989113,
0.830597103, 0.802697659, 0.78116709, 0.766595185, 0.761856914, 0.770022631,
0.826521039, 0.90397501, 0.951342702, 0.957924068, 0.952868819, 0.952133417,
0.928703249, 0.638475657, 0.701241612, 0.784563363, 0.841383219, 0.856479287,
0.877265215, 0.954805672, 1.039082289, 1.109156251, 1.117263913, 1.04548502,
0.980756104, 0.940173388, 0.946755886, 0.96761173, 0.962748587, 0.977708697,
0.952758789, 0.895089209, 0.858695507, 0.828825116, 0.807197034, 0.797473609,
0.802371204, 0.857669353, 0.936871529, 0.989703417, 1.003184795, 1.006631613,
1.016924024, 1.00193429, 0.638475657, 0.701241612, 0.786107659, 0.844073474,
0.859399199, 0.88437283, 0.976726115, 1.081594229, 1.167513251, 1.18007791,
1.103111386, 1.03039217, 0.979927897, 0.977664709, 0.995771885, 0.997566521,
1.03454566, 1.025245547, 0.971404195, 0.940074384, 0.913797617, 0.893222034,
0.885627687, 0.899616897, 0.992524326, 1.126092672, 1.206882238, 1.213850737,
1.190800905, 1.164836526, 1.112275243, 0.795258403, 0.816856802, 0.857468605,
0.891532302, 0.906502187, 0.92630136, 0.979601145, 1.028863072, 1.063205004,
1.066814184, 1.032617211, 0.994250894, 0.955908895, 0.928678632, 0.908084571,
0.883677125, 0.873843551, 0.855833828, 0.826784849, 0.806078315, 0.786206603,
0.767146707, 0.752018094, 0.742449701, 0.751986265, 0.771394849, 0.784393907,
0.784185886, 0.774534941, 0.758701324, 0.732473671, 0.795258403, 0.816856802,
0.857970893, 0.892682314, 0.90784204, 0.929064393, 0.990200877, 1.053232908,
1.103422642, 1.118505836, 1.08778131, 1.05342114, 1.022762179, 1.012263894,
1.011744976, 1.00061059, 1.002970934, 0.988465309, 0.955877185, 0.931310594,
0.907226205, 0.884828508, 0.869311213, 0.863197386, 0.888297558, 0.932593346,
0.967936397, 0.985253811, 0.996055782, 1.008288145, 1.005891085, 0.795258403,
0.816856802, 0.858424306, 0.893659353, 0.908989906, 0.931387544, 0.99817735,
1.070488811, 1.13034749, 1.151631355, 1.122590423, 1.088484406, 1.056558609,
1.043916106, 1.042092919, 1.032291174, 1.043409228, 1.037230253, 1.008979201,
0.989836633, 0.970193386, 0.95017159, 0.936686635, 0.93526113, 0.982087791,
1.064218163, 1.129002571, 1.158894658, 1.168904543, 1.169321179, 1.149312615,
0.795258403, 0.816856802, 0.85883683, 0.894518554, 0.910082877, 0.933618307,
1.004875779, 1.084069133, 1.150643349, 1.17569387, 1.146507144, 1.110903978,
1.077234268, 1.064907432, 1.065898061, 1.05904305, 1.074055791, 1.068949819,
1.039388657, 1.019022465, 0.998272002, 0.976956725, 0.962277412, 0.959765851,
1.007760406, 1.092259288, 1.156586885, 1.184763908, 1.19553268, 1.201265335,
1.185963511, 0.648370028, 0.688576519, 0.742491126, 0.783990383, 0.80298996,
0.821607351, 0.862388551, 0.894268453, 0.911370516, 0.89651376, 0.842166662,
0.783710659, 0.728130281, 0.684661746, 0.648905814, 0.614272654, 0.594889522,
0.571380973, 0.542023718, 0.521143556, 0.499938488, 0.475976497, 0.449459493,
0.421074629, 0.397123814, 0.374796838, 0.350898117, 0.324967325, 0.300079197,
0.278220892, 0.256880879, 0.648370028, 0.688576519, 0.743724585, 0.787064314,
0.807689011, 0.831625164, 0.888596475, 0.946057677, 0.993717194, 1.00470984,
0.962817013, 0.918186665, 0.884074688, 0.878249466, 0.889110208, 0.893722475,
0.920192778, 0.919718087, 0.890581846, 0.875235379, 0.864329875, 0.856818676,
0.855373979, 0.862291932, 0.910496414, 0.980866313, 1.027168989, 1.036630273,
1.033012152, 1.031717777, 1.011372924, 0.648370028, 0.688576519, 0.744858205,
0.789697766, 0.811312854, 0.838736176, 0.90691787, 0.981273532, 1.046675324,
1.06989634, 1.030948877, 0.989238918, 0.961261451, 0.966397285, 0.985641062,
0.989913285, 1.016537428, 1.012879372, 0.976154804, 0.955336571, 0.939947248,
0.927718401, 0.922940135, 0.930412292, 0.994285345, 1.093555927, 1.160755515,
1.176150799, 1.170561314, 1.16328609, 1.130534887, 0.648370028, 0.688576519,
0.74590379, 0.791982412, 0.814170182, 0.843955517, 0.920372605, 1.006811261,
1.083702803, 1.11277163, 1.07243216, 1.026477456, 0.99338913, 0.995457411,
1.015423536, 1.022041798, 1.05273211, 1.048963904, 1.008393526, 0.983748913,
0.964948297, 0.949395537, 0.941659451, 0.946655393, 1.010100961, 1.110544324,
1.177545547, 1.191351295, 1.184355021, 1.17684567, 1.143859267)