如何强制零截距以拟合二阶函数?(Python) [英] How to force zero interception for fitting a 2nd order function? (Python)
问题描述
我正在尝试拟合零截距的二阶函数.现在，当我绘制它时，我得到一条 y-int > 0 的线.我试图适应函数输出的一组点:
I am trying to fit a second order function with a zero intercept. Right now when I plot it, I am getting a line with a y-int > 0. I am trying to fit to a set of points output by the function:
y**2 = 14.29566 * np.pi * x
或
y = np.sqrt(14.29566 * np.pi * x)
到两个数据集 x 和 y，其中 D = 3.57391553.我的试衣习惯是:
to two data sets x and y, with D = 3.57391553. My fitting routine is:
z = np.polyfit(x,y,2) # Generate curve coefficients
p = np.poly1d(z) # Generate curve function
xp = np.linspace(0, catalog.tlimit, 10) # generate input values
plt.scatter(x,y)
plt.plot(xp, p(xp), '-')
plt.show()
我也尝试过使用 statsmodels.ols:
mod_ols = sm.OLS(y,x)
res_ols = mod_ols.fit()
但我不明白如何为二阶函数生成系数而不是线性函数，也不明白如何将 y-int 设置为 0.我看到另一个类似的帖子处理强制 y-int 为 0用线性拟合，但我不知道如何用二阶函数做到这一点.
but I don't understand how to generate coefficients for a second order function as opposed to a linear function, nor how to set the y-int to 0. I saw another similar post dealing with forcing a y-int of 0 with a linear fit, but I couldn't figure out how to do that with a second order function.
现在的情节:
数据:
x = [0., 0.00325492, 0.00650985, 0.00976477, 0.01301969, 0.01627462, 0.01952954, 0.02278447,
0.02603939, 0.02929431, 0.03254924, 0.03580416, 0.03905908, 0.04231401]
y = [0., 0.38233801, 0.5407076, 0.66222886, 0.76467602, 0.85493378, 0.93653303, 1.01157129,
1.0814152, 1.14701403, 1.20905895, 1.26807172, 1.32445772, 1.3785393]
推荐答案
如果我理解正确，您想用 OLS 拟合多项式回归线，您可以尝试以下操作(如我们所见，随着拟合多项式次数的增加，模型对数据的过拟合越来越多):
If I understood you correctly you want to fit polynomial regression lines with OLS, you can try the following (as we can see, as the degree of the fitted polynomial increases, the model overfits the data more and more):
xp = np.linspace(0, 0.05, 10) # generate input values
pred1 = p(xp) # prediction with np.poly as you have done
import pandas as pd
data = pd.DataFrame(data={'x':x, 'y':y})
# let's first fit 2nd order polynomial with OLS with intercept
olsres2 = sm.ols(formula = 'y ~ x + I(x**2)', data = data).fit()
print olsres2.summary()
OLS Regression Results
==============================================================================
Dep. Variable: y R-squared: 0.978
Model: OLS Adj. R-squared: 0.974
Method: Least Squares F-statistic: 243.1
Date: Sat, 21 Jan 2017 Prob (F-statistic): 7.89e-10
Time: 04:16:22 Log-Likelihood: 20.323
No. Observations: 14 AIC: -34.65
Df Residuals: 11 BIC: -32.73
Df Model: 2
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [95.0% Conf. Int.]
------------------------------------------------------------------------------
Intercept 0.1470 0.045 3.287 0.007 0.049 0.245
x 52.4655 4.907 10.691 0.000 41.664 63.267
I(x ** 2) -580.4730 111.820 -5.191 0.000 -826.588 -334.358
==============================================================================
Omnibus: 4.803 Durbin-Watson: 1.164
Prob(Omnibus): 0.091 Jarque-Bera (JB): 2.101
Skew: -0.854 Prob(JB): 0.350
Kurtosis: 3.826 Cond. No. 6.55e+03
==============================================================================
pred2 = olsres2.predict(data) # predict with the fitted model
# fit 3rd order polynomial with OLS with intercept
olsres3 = sm.ols(formula = 'y ~ x + I(x**2) + I(x**3)', data = data).fit()
pred3 = olsres3.predict(data) # predict
plt.scatter(x,y)
plt.plot(xp, pred1, '-r', label='np.poly')
plt.plot(x, pred2, '-b', label='ols.o2')
plt.plot(x, pred3, '-g', label='ols.o3')
plt.legend(loc='upper left')
plt.show()
# now let's fit the polynomial regression lines this time without intercept
olsres2 = sm.ols(formula = 'y ~ x + I(x**2)-1', data = data).fit()
print olsres2.summary()
OLS Regression Results
==============================================================================
Dep. Variable: y R-squared: 0.993
Model: OLS Adj. R-squared: 0.992
Method: Least Squares F-statistic: 889.6
Date: Sat, 21 Jan 2017 Prob (F-statistic): 9.04e-14
Time: 04:16:24 Log-Likelihood: 15.532
No. Observations: 14 AIC: -27.06
Df Residuals: 12 BIC: -25.79
Df Model: 2
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [95.0% Conf. Int.]
------------------------------------------------------------------------------
x 65.8170 3.714 17.723 0.000 57.726 73.908
I(x ** 2) -833.6787 109.279 -7.629 0.000 -1071.777 -595.580
==============================================================================
Omnibus: 1.716 Durbin-Watson: 0.537
Prob(Omnibus): 0.424 Jarque-Bera (JB): 1.341
Skew: 0.649 Prob(JB): 0.511
Kurtosis: 2.217 Cond. No. 118.
==============================================================================
pred2 = olsres2.predict(data)
# fit 3rd order polynomial with OLS without intercept
olsres3 = sm.ols(formula = 'y ~ x + I(x**2) + I(x**3) -1', data = data).fit()
pred3 = olsres3.predict(data)
plt.scatter(x,y)
plt.plot(xp, pred1, '-r', label='np.poly')
plt.plot(x, pred2, '-b', label='ols.o2')
plt.plot(x, pred3, '-g', label='ols.o3')
plt.legend(loc='upper left')
plt.show()
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