非常高的残差平方和答案

【问题标题】：Very high residual Sum-of-Squares非常高的残差平方和
【发布时间】：2016-05-04 14:22:08
【问题描述】：

我对拟合的残差平方和有疑问。残差的平方和太高，说明拟合不是很好。但是，从视觉上看，拥有如此高的剩余价值看起来不错……谁能帮我知道发生了什么？

我的数据：

x=c(0.017359, 0.019206, 0.020619, 0.021022, 0.021793, 0.022366, 0.025691, 0.025780, 0.026355, 0.028858, 0.029766, 0.029967, 0.030241, 0.032216, 0.033657,
 0.036250, 0.039145, 0.040682, 0.042334, 0.043747, 0.044165, 0.044630, 0.046045, 0.048138, 0.050813, 0.050955, 0.051910, 0.053042, 0.054853, 0.056886,
0.058651, 0.059472, 0.063770,0.064567, 0.067415, 0.067802, 0.068995, 0.070742,0.073486, 0.074085 ,0.074452, 0.075224, 0.075853, 0.076192, 0.077002,
 0.078273, 0.079376, 0.083269, 0.085902, 0.087619, 0.089867, 0.092606, 0.095944, 0.096327, 0.097019, 0.098444, 0.098868, 0.098874, 0.102027, 0.103296,
 0.107682, 0.108392, 0.108719, 0.109184, 0.109623, 0.118844, 0.124023, 0.124244, 0.129600, 0.130892, 0.136721, 0.137456, 0.147343, 0.149027, 0.152818,
0.155706,0.157650, 0.161060, 0.162594, 0.162950, 0.165031, 0.165408, 0.166680, 0.167727, 0.172882, 0.173264, 0.174552,0.176073, 0.185649, 0.194492,
 0.196429, 0.200050, 0.208890, 0.209826, 0.213685, 0.219189, 0.221417, 0.222662, 0.230860, 0.234654, 0.235211, 0.241819, 0.247527, 0.251528, 0.253664,
 0.256740, 0.261723, 0.274585, 0.278340, 0.281521, 0.282332, 0.286166, 0.288103, 0.292959, 0.295201, 0.309456, 0.312158, 0.314132, 0.319906, 0.319924,
 0.322073, 0.325427, 0.328132, 0.333029, 0.334915, 0.342098, 0.345899, 0.345936, 0.350355, 0.355015, 0.355123, 0.356335, 0.364257, 0.371180, 0.375171,
0.377743, 0.383944, 0.388606, 0.390111, 0.395080, 0.398209, 0.409784, 0.410324, 0.424782 )


y= c(34843.40, 30362.66, 27991.80 ,28511.38, 28004.74, 27987.13, 22272.41, 23171.71, 23180.03, 20173.79, 19751.84, 20266.26, 20666.72, 18884.42, 17920.78, 15980.99, 14161.08, 13534.40, 12889.18, 12436.11,
12560.56, 12651.65, 12216.11, 11479.18, 10573.22, 10783.99, 10650.71, 10449.87, 10003.68,  9517.94,  9157.04,  9104.01,  8090.20,  8059.60,  7547.20,  7613.51,  7499.47,  7273.46,  6870.20,  6887.01,
6945.55,  6927.43,  6934.73,  6993.73,  6965.39,  6855.37,  6777.16,  6259.28,  5976.27,  5835.58,  5633.88,  5387.19,  5094.94,  5129.89,  5131.42,  5056.08,  5084.47,  5155.40,  4909.01,  4854.71,
4527.62,  4528.10,  4560.14,  4580.10,  4601.70,  3964.90,  3686.20,  3718.46,  3459.13,  3432.05,  3183.09,  3186.18,  2805.15,  2773.65,  2667.73,  2598.55,  2563.02,  2482.63,  2462.49,  2478.10,
2441.70,  2456.16,  2444.00,  2438.47,  2318.64,  2331.75,  2320.43,  2303.10,  2091.95,  1924.55, 1904.91,  1854.07,  1716.52,  1717.12,  1671.00,  1602.70,  1584.89,  1581.34,  1484.16,  1449.26,
1455.06,  1388.60,  1336.71,  1305.60,  1294.58,  1274.36,  1236.51,  1132.67,  1111.35,  1095.21,  1097.71,  1077.05,  1071.04,  1043.99,  1036.22,   950.26,   941.06,   936.37,   909.72,   916.45,
911.01, 898.94,   890.68,   870.99,   867.45,   837.39,   824.93,   830.61,   815.49,   799.77,   804.84,   804.88,   775.53,   751.95,   741.01,   735.86,   717.03,   704.57,   703.74,   690.63,
684.24,   650.30,   652.74,   612.95 )

然后使用 nlsLM 函数（minpack.lm 包）进行拟合：

library(magicaxis)
library(minpack.lm)

sig.backg=3*10^(-3) 

mod <- nlsLM(y ~ a *( 1 + (x/b)^2 )^c+sig.backg,
             start = c(a = 0, b = 1, c = 0),
             trace = TRUE)

## plot data
magplot(x, y, main = "data", log = "xy", pch=16)
## plot fitted values
lines(x, fitted(mod), col = 2, lwd = 4 )

这个值就是残差：

> print(mod)
Nonlinear regression model
  model: y ~ a * (1 + (x/b)^2)^c + sig.backg
   data: parent.frame()
         a          b          c 
68504.2013     0.0122    -0.6324 
 residual sum-of-squares: 12641435

Number of iterations to convergence: 34 
Achieved convergence tolerance: 0.0000000149

平方和残差太高：12641435 ...

是这样还是调整有问题？不好吗？

【问题讨论】：

“太高”没有量化验证是相当误导

标签： r model-fitting adjustment function-fitting

【解决方案1】：

这是有道理的，因为您的响应变量的平方平均值是 38110960。如果您喜欢使用较小的数字，您可以缩放您的数据。

【讨论】：

我如何扩展我的数据以处理较小的数字？有什么建议吗？
只需将其除以一个常数即可。例如。如果这些是以米为单位的测量值，请将它们转换为以公里为单位的测量值。但正如@o_o 指出的那样，平方和本身就是一个毫无意义的数量。

【解决方案2】：

如果不知道总平方和（从中可以计算 R^2），残差平方和没有多大意义。如果您的数据具有较大的值或添加更多数据点，无论您的拟合程度如何，它的价值都会增加。此外，您可能想查看残差与拟合数据的关系图，您的模型应该解释一个清晰的模式，以确保您的错误是正态分布的。

【讨论】：