How to extract equation from a polynomial fit?

Question

My goal is to fit some data to a polynomial function and obtain the actual equation including the fitted parameter values.

I adapted this example to my data and the outcome is as expected.

Here is my code:

import numpy as np
import matplotlib.pyplot as plt

from sklearn.linear_model import Ridge
from sklearn.preprocessing import PolynomialFeatures
from sklearn.pipeline import make_pipeline


x = np.array([0., 4., 9., 12., 16., 20., 24., 27.])
y = np.array([2.9,4.3,66.7,91.4,109.2,114.8,135.5,134.2])

x_plot = np.linspace(0, max(x), 100)
# create matrix versions of these arrays
X = x[:, np.newaxis]
X_plot = x_plot[:, np.newaxis]

plt.scatter(x, y, label="training points")

for degree in np.arange(3, 6, 1):
    model = make_pipeline(PolynomialFeatures(degree), Ridge())
    model.fit(X, y)
    y_plot = model.predict(X_plot)
    plt.plot(x_plot, y_plot, label="degree %d" % degree)

plt.legend(loc='lower left')

plt.show()

However, I now don't know where to extract the actual equation and fitted parameter values for the respective fits. Where do I access the actual fitted equation?

EDIT:

The variable model has the following attributes:

model.decision_function  model.fit_transform      model.inverse_transform  model.predict            model.predict_proba      model.set_params         model.transform          
model.fit                model.get_params         model.named_steps        model.predict_log_proba  model.score              model.steps

model.get_params does not store the desired parameters.

Answer 1

The coefficients of the linear model are stored in the intercept_ and coeff_ attributes of the model.

You can see this more clearly by turning-down the regularization and feeding-in a known model; e.g.

import numpy as np
from sklearn.linear_model import Ridge
from sklearn.pipeline import make_pipeline
from sklearn.preprocessing import PolynomialFeatures

x = 10 * np.random.random(100)
y = -4 + 2 * x - 3 * x ** 2

model = make_pipeline(PolynomialFeatures(2), Ridge(alpha=1E-8, fit_intercept=False))
model.fit(x[:, None], y)
ridge = model.named_steps['ridge']
print(ridge.coef_)
# array([-4.,  2., -3.])

Also note that the PolynomialFeatures by default includes a bias term, so fitting the intercept in Ridge will be redundant for small alpha.