import numpy as np

import matplotlib.pyplot as plt

from scipy.optimize import curve_fit

import statsmodels.api as sm

from statsmodels.base.model import GenericLikelihoodModel

from sklearn.preprocessing import PolynomialFeatures

from sklearn.linear_model import LinearRegression

from sklearn.pipeline import make_pipeline

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# Generate synthetic data

np.random.seed(0)

x_data = np.linspace(0, 4, 50)

y_data = 2.5 * np.exp(-1.3 * x_data) + 0.5 + 0.2 * np.random.normal(size=len(x_data))

x_data_reshaped = x_data[:, np.newaxis]  # For sklearn

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# 1. Using scipy.optimize.curve_fit

def model(x, a, b, c):

    return a * np.exp(-b * x) + c

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popt, pcov = curve_fit(model, x_data, y_data)

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image.mefiz.com

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# 2. Using statsmodels

class NonLinearModel(GenericLikelihoodModel):

    def __init__(self, endog, exog, **kwargs):

        super(NonLinearModel, self).__init__(endog, exog, **kwargs)

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    def nloglikeobs(self, params):

        a, b, c = params

        y_pred = a * np.exp(-b * self.exog) + c

        resid = self.endog - y_pred

        return 0.5 * (resid ** 2)

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    def fit(self, start_params=None, maxiter=10000, method='nm', **kwargs):

        if start_params is None:

            start_params = np.array([1.0, 1.0, 1.0])

        return super(NonLinearModel, self).fit(start_params=start_params, maxiter=maxiter, method=method, **kwargs)

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nlm = NonLinearModel(y_data, x_data)

result = nlm.fit()

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# 3. Using sklearn with Polynomial Features

degree = 4

poly_model = make_pipeline(PolynomialFeatures(degree), LinearRegression())

poly_model.fit(x_data_reshaped, y_data)

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# Plot results

plt.figure(figsize=(12, 8))

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# Original Data

plt.scatter(x_data, y_data, label='Data', color='blue')

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# scipy.optimize.curve_fit

plt.plot(x_data, model(x_data, *popt), label='scipy.optimize.curve_fit', color='red')

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# statsmodels

plt.plot(x_data, result.predict(x_data), label='statsmodels', color='green')

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# sklearn Polynomial Regression

plt.plot(x_data, poly_model.predict(x_data_reshaped), label='sklearn Polynomial Regression', color='purple')

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plt.legend()

plt.xlabel('x')

plt.ylabel('y')

plt.title('Non-linear Regression using different methods')

plt.show()

​