[Python] Polynomial regression with 2D array

I have two 2D datasets Xand Y, which respectively represent experimental and theoretical data. In the two cases, each row corresponds to a sample (one physical configuration), and each column corresponds to a measurement done by a sensor at one particular location (feature).

These two datasets have the same shape (n_samples, n_features)and are linked by an unknown relationship, that I believe to be some kind of polynomial equation.

Y = a + b*X + c*(X**2) + ...

where :

• Y = theoretical values matrix
• X = experimental values matrix

Therefore, I try to obtain the polynomial coefficients a, b, c for degree = 2, with sklearn.

Before trying with real dataset, I am trying with synthetic data but I don't manage to properly obtain the right polynomial coefficients. I have tried with a 1D-array (1 feature) and it's working well but I don't know why it doesn't work with the whole 2D-array.

I don't know if the problem comes from the method or from my script but any help would be greatly appreciated. Please find below my attempt :


import numpy as np
from sklearn.preprocessing import PolynomialFeatures
from sklearn.linear_model import LinearRegression
from sklearn.pipeline import make_pipeline
from sklearn.model_selection import train_test_split


# =============================================================================
# x = 1D ARRAY EXAMPLE
# =============================================================================
pipe = make_pipeline(PolynomialFeatures(2),LinearRegression())

rng = np.random.RandomState(1)
x = 10 * rng.rand(50)
y = 37 + 3*x + 1.5*x**2

# Fit the model
X = x[:, np.newaxis] # Arrange data into a features matrix and target vector.
pipe.fit(X, y)

# Inspect the fit
coef = pipe[1].coef_
print(coef)

# predict
xfit = np.linspace(0, 10, 1000)
Xfit = xfit[:, np.newaxis]
yfit = pipe.predict(Xfit)



# =============================================================================
# x = 2D ARRAY
# =============================================================================

n_subfeatures = 4 # Number of sub-sensors
n_components_per_subf = 3 # Number of components per sub-sensor
n_features = n_subfeatures * n_components_per_subf # Total number of features

n_samples = 100

## GENERATE SYNTHETIC DATA

# Simulated data
np.random.seed(42)
x = np.random.randn(n_samples, n_features)
y = (
0.3
+ 1.7 * x
- 0.5 * x**2
)


# Pipe version
# -------------------------
pipe = make_pipeline(PolynomialFeatures(2),LinearRegression())

# Fit the model
# X = x[:, np.newaxis] # Arrange data into a features matrix and target vector.
X = x
pipe.fit(X, y)

# Inspect the fit
coef = pipe[1].coef_
print(coef)


# Poly / Reg version
# -------------------------
X=x
poly = PolynomialFeatures(degree=2)
poly_features = poly.fit_transform(X)

X_train, X_test, y_train, y_test = train_test_split(poly_features, y, test_size=0.3, random_state=42)

poly_reg_model = LinearRegression()
poly_reg_model.fit(X_train, y_train)
poly_reg_y_predicted = poly_reg_model.predict(X_test)

coef = poly_reg_model.coef_
print(coef)

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