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"""

Polynomial regression is a type of regression analysis that models the relationship

between a predictor x and the response y as an mth-degree polynomial:

y = β₀ + β₁x + β₂x² + ... + βₘxᵐ + ε

By treating x, x², ..., xᵐ as distinct variables, we see that polynomial regression is a

special case of multiple linear regression. Therefore, we can use ordinary least squares

(OLS) estimation to estimate the vector of model parameters β = (β₀, β₁, β₂, ..., βₘ)

for polynomial regression:

β = (XᵀX)⁻¹Xᵀy = X⁺y

where X is the design matrix, y is the response vector, and X⁺ denotes the Moore-Penrose

pseudoinverse of X. In the case of polynomial regression, the design matrix is

|1 x₁ x₁² ⋯ x₁ᵐ|

X = |1 x₂ x₂² ⋯ x₂ᵐ|

|⋮ ⋮ ⋮ ⋱ ⋮ |

|1 xₙ xₙ² ⋯ xₙᵐ|

In OLS estimation, inverting XᵀX to compute X⁺ can be very numerically unstable. This

implementation sidesteps this need to invert XᵀX by computing X⁺ using singular value

decomposition (SVD):

β = VΣ⁺Uᵀy

where UΣVᵀ is an SVD of X.

References:

- https://en.wikipedia.org/wiki/Polynomial_regression

- https://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_inverse

- https://en.wikipedia.org/wiki/Numerical_methods_for_linear_least_squares

- https://en.wikipedia.org/wiki/Singular_value_decomposition

"""

import matplotlib.pyplot as plt

import numpy as np

class PolynomialRegression:

__slots__ = "degree", "params"

def __init__(self, degree: int) -> None:

"""

@raises ValueError: if the polynomial degree is negative

"""

if degree < 0:

raise ValueError("Polynomial degree must be non-negative")

self.degree = degree

self.params = None

@staticmethod

def _design_matrix(data: np.ndarray, degree: int) -> np.ndarray:

"""

Constructs a polynomial regression design matrix for the given input data. For

input data x = (x₁, x₂, ..., xₙ) and polynomial degree m, the design matrix is

the Vandermonde matrix

|1 x₁ x₁² ⋯ x₁ᵐ|

X = |1 x₂ x₂² ⋯ x₂ᵐ|

|⋮ ⋮ ⋮ ⋱ ⋮ |

|1 xₙ xₙ² ⋯ xₙᵐ|

Reference: https://en.wikipedia.org/wiki/Vandermonde_matrix

@param data: the input predictor values x, either for model fitting or for

prediction

@param degree: the polynomial degree m

@returns: the Vandermonde matrix X (see above)

@raises ValueError: if input data is not N x 1

>>> x = np.array([0, 1, 2])

>>> PolynomialRegression._design_matrix(x, degree=0)

array([[1],

[1],

[1]])

>>> PolynomialRegression._design_matrix(x, degree=1)

array([[1, 0],

[1, 1],

[1, 2]])

>>> PolynomialRegression._design_matrix(x, degree=2)

array([[1, 0, 0],

[1, 1, 1],

[1, 2, 4]])

>>> PolynomialRegression._design_matrix(x, degree=3)

array([[1, 0, 0, 0],

[1, 1, 1, 1],

[1, 2, 4, 8]])

>>> PolynomialRegression._design_matrix(np.array([[0, 0], [0 , 0]]), degree=3)

Traceback (most recent call last):

...

ValueError: Data must have dimensions N x 1

"""

_rows, *remaining = data.shape

if remaining:

raise ValueError("Data must have dimensions N x 1")

return np.vander(data, N=degree + 1, increasing=True)

def fit(self, x_train: np.ndarray, y_train: np.ndarray) -> None:

"""

Computes the polynomial regression model parameters using ordinary least squares

(OLS) estimation:

β = (XᵀX)⁻¹Xᵀy = X⁺y

where X⁺ denotes the Moore-Penrose pseudoinverse of the design matrix X. This

function computes X⁺ using singular value decomposition (SVD).

References:

- https://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_inverse

- https://en.wikipedia.org/wiki/Singular_value_decomposition

- https://en.wikipedia.org/wiki/Multicollinearity

@param x_train: the predictor values x for model fitting

@param y_train: the response values y for model fitting

@raises ArithmeticError: if X isn't full rank, then XᵀX is singular and β

doesn't exist

>>> x = np.array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10])

>>> y = x**3 - 2 * x**2 + 3 * x - 5

>>> poly_reg = PolynomialRegression(degree=3)

>>> poly_reg.fit(x, y)

>>> poly_reg.params

array([-5., 3., -2., 1.])

>>> poly_reg = PolynomialRegression(degree=20)

>>> poly_reg.fit(x, y)

Traceback (most recent call last):

...

ArithmeticError: Design matrix is not full rank, can't compute coefficients

Make sure errors don't grow too large:

>>> coefs = np.array([-250, 50, -2, 36, 20, -12, 10, 2, -1, -15, 1])

>>> y = PolynomialRegression._design_matrix(x, len(coefs) - 1) @ coefs

>>> poly_reg = PolynomialRegression(degree=len(coefs) - 1)

>>> poly_reg.fit(x, y)

>>> np.allclose(poly_reg.params, coefs, atol=10e-3)

True

"""

X = PolynomialRegression._design_matrix(x_train, self.degree) # noqa: N806

_, cols = X.shape

if np.linalg.matrix_rank(X) < cols:

raise ArithmeticError(

"Design matrix is not full rank, can't compute coefficients"

)

# np.linalg.pinv() computes the Moore-Penrose pseudoinverse using SVD

self.params = np.linalg.pinv(X) @ y_train

def predict(self, data: np.ndarray) -> np.ndarray:

"""

Computes the predicted response values y for the given input data by

constructing the design matrix X and evaluating y = Xβ.

@param data: the predictor values x for prediction

@returns: the predicted response values y = Xβ

@raises ArithmeticError: if this function is called before the model

parameters are fit

>>> x = np.array([0, 1, 2, 3, 4])

>>> y = x**3 - 2 * x**2 + 3 * x - 5

>>> poly_reg = PolynomialRegression(degree=3)

>>> poly_reg.fit(x, y)

>>> poly_reg.predict(np.array([-1]))

array([-11.])

>>> poly_reg.predict(np.array([-2]))

array([-27.])

>>> poly_reg.predict(np.array([6]))

array([157.])

>>> PolynomialRegression(degree=3).predict(x)

Traceback (most recent call last):

...

ArithmeticError: Predictor hasn't been fit yet

"""

if self.params is None:

raise ArithmeticError("Predictor hasn't been fit yet")

return PolynomialRegression._design_matrix(data, self.degree) @ self.params

def main() -> None:

"""

Fit a polynomial regression model to predict fuel efficiency using seaborn's mpg

dataset

>>> pass # Placeholder, function is only for demo purposes

"""

import seaborn as sns

mpg_data = sns.load_dataset("mpg")

poly_reg = PolynomialRegression(degree=2)

poly_reg.fit(mpg_data.weight, mpg_data.mpg)

weight_sorted = np.sort(mpg_data.weight)

predictions = poly_reg.predict(weight_sorted)

plt.scatter(mpg_data.weight, mpg_data.mpg, color="gray", alpha=0.5)

plt.plot(weight_sorted, predictions, color="red", linewidth=3)

plt.title("Predicting Fuel Efficiency Using Polynomial Regression")

plt.xlabel("Weight (lbs)")

plt.ylabel("Fuel Efficiency (mpg)")

plt.show()

if __name__ == "__main__":

import doctest

doctest.testmod()

main()

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