如何定量测量SciPy的适合度？

Question

我想找出最合适的数据.我做的是循环遍历各种n值并使用公式((y_fit - y_actual)/ y_actual)x 100计算每个p的残差.然后我计算每个n的平均值,然后找出最小残差均值和相应的n值并使用此值拟合.可重现的代码包括:

import numpy as np
import matplotlib.pyplot as plt
from scipy import optimize   

x = np.array([12.4, 18.2, 20.3, 22.9, 27.7, 35.5, 53.9])
y = np.array([1, 50, 60, 70, 80, 90, 100])
y_residual = np.empty(shape=(1, len(y)))
residual_mean = []

n = np.arange(0.01, 10, 0.01)

def fit(x, a, b):
    return a * x + b
for i in range (len(n)):
    x_fit = 1 / np.log(x) ** n[i]
    y_fit = y
    fit_a, fit_b = optimize.curve_fit(fit, x_fit, y_fit)[0]
    y_fit = (fit_a * x_fit) + fit_b
    y_residual = (abs(y_fit - y) / y) * 100
    residual_mean = np.append(residual_mean, np.mean(y_residual[np.isfinite(y_residual)]))
p = n[np.where(residual_mean == residual_mean.min())]
p = p[0]
print p
x_fit = 1 / np.log(x) ** p
y_fit = y
fit_a, fit_b = optimize.curve_fit(fit, x_fit, y_fit)[0]
y_fit = (fit_a * x_fit) + fit_b
y_residual = (abs(y_fit - y) / y) * 100

fig = plt.figure(1, figsize=(5, 5))
fig.clf()
plot = plt.subplot(111)
plot.plot(x, y, linestyle = '', marker='^')
plot.plot(x, y_fit, linestyle = ':')
plot.set_ylabel('y')
plot.set_xlabel('x')
plt.show()

fig_1 = plt.figure(2, figsize=(5, 5))
fig_1.clf()
plot_1 = plt.subplot(111)
plot_1.plot(1 / np.log(x) ** p, y, linestyle = '-')
plot_1.set_xlabel('pow(x, -p)' )
plot_1.set_ylabel('y' )
plt.show()

fig_2 = plt.figure(2, figsize=(5, 5))
fig_2.clf()
plot_2 = plt.subplot(111)
plot_2.plot(n, residual_mean, linestyle = '-')
plot_2.set_xlabel('n' )
plot_2.set_ylabel('Residual mean')
plt.show()

用n绘制残差均值,这是我得到的:

我需要知道这种方法是否正确以确定最佳拟合.如果可以使用SciPy或任何其他软件包中的其他功能完成.实质上我想要的是定量地知道哪个是最合适的.我已经在SciPy中完成了适合度测试,但它对我没什么帮助.

Answer 1

可能最常用的拟合优度度量是确定系数(也称为R ²值).

公式是:

哪里:

在这里,y _i指的是你的输入y值,f _i指的是你的拟合y值,而̅y指的是平均输入y值.

计算起来非常简单:

# residual sum of squares
ss_res = np.sum((y - y_fit) ** 2)

# total sum of squares
ss_tot = np.sum((y - np.mean(y)) ** 2)

# r-squared
r2 = 1 - (ss_res / ss_tot)