Calculate confidence band of least-square fit

Question

I got a question that I fight around for days with now.

How do I calculate the (95%) confidence band of a fit?

Fitting curves to data is the every day job of every physicist -- so I think this should be implemented somewhere -- but I can't find an implementation for this neither do I know how to do this mathematically.

The only thing I found is seaborn that does a nice job for linear least-square.

import numpy as np                                                                                                                                                                                                                         
from matplotlib import pyplot as plt
import seaborn as sns
import pandas as pd

x = np.linspace(0,10)
y = 3*np.random.randn(50) + x

data = {'x':x, 'y':y}
frame = pd.DataFrame(data, columns=['x', 'y'])
sns.lmplot('x', 'y', frame, ci=95)

plt.savefig("confidence_band.pdf")

But this is just linear least-square. When I want to fit e.g. a saturation curve like , I'm screwed.

Sure, I can calculate the t-distribution from the std-error of a least-square method like scipy.optimize.curve_fit but that is not what I'm searching for.

Thanks for any help!!

Answer 1

You can achieve this easily using StatsModels module.

Also see this example and this answer.

Here is an answer for your question:

import numpy as np
from matplotlib import pyplot as plt

import statsmodels.api as sm
from statsmodels.stats.outliers_influence import summary_table

x = np.linspace(0,10)
y = 3*np.random.randn(50) + x
X = sm.add_constant(x)
res = sm.OLS(y, X).fit()

st, data, ss2 = summary_table(res, alpha=0.05)
fittedvalues = data[:,2]
predict_mean_se  = data[:,3]
predict_mean_ci_low, predict_mean_ci_upp = data[:,4:6].T
predict_ci_low, predict_ci_upp = data[:,6:8].T

fig, ax = plt.subplots(figsize=(8,6))
ax.plot(x, y, 'o', label="data")
ax.plot(X, fittedvalues, 'r-', label='OLS')
ax.plot(X, predict_ci_low, 'b--')
ax.plot(X, predict_ci_upp, 'b--')
ax.plot(X, predict_mean_ci_low, 'g--')
ax.plot(X, predict_mean_ci_upp, 'g--')
ax.legend(loc='best');
plt.show()