Polynomial regression

Polynomial regression class is a particular case of the linear regression, where the input data is transformed before the regression is applied. This transform consists of creating a matrix of monomials (Vandermonde) by raising the input data to different powers up to a certain degree \(D\). In the case where there is only one input variable, the input data \((x_i)_{i=1, \dots, n}\in\mathbb{R}^n\) is transformed into the Vandermonde matrix

\[\begin{split}\begin{pmatrix} x_1^1 & x_1^2 & \cdots & x_1^D\\ x_2^1 & x_2^2 & \cdots & x_2^D\\ \vdots & \vdots & \ddots & \vdots\\ x_n^1 & x_n^2 & \cdots & x_n^D\\ \end{pmatrix} = (x_i^d)_{i=1, \dots, n;\ d=1, \dots, D}.\end{split}\]

The output is expressed as a weighted sum of monomials:

\[y = w_0 + w_1 x^1 + w_2 x^2 + ... + w_D x^D,\]

where the coefficients \((w_1, w_2, ..., w_d)\) and the intercept \(w_0\) are estimated by least square regression.

In the case of a multidimensional input, i.e. \(X = (x_{ij})_{i=1,\dots,n; j=1,\dots,m}\), where \(n\) is the number of samples and \(m\) is the number of input variables, the Vandermonde matrix is expressed through different combinations of monomials of degree \(d, (1 \leq d \leq D)\); e.g. for three variables \((x, y, z)\) and degree \(D=3\), the different terms are \(x\), \(y\), \(z\), \(x^2\), \(xy\), \(xz\), \(y^2\), \(yz\), \(z^2\), \(x^3\), \(x^2y\) etc. More generally, for m input variables, the total number of monomials of degree \(1 \leq d \leq D\) is given by \(P = \binom{m+D}{m} = \frac{(m+D)!}{m!D!}\). In the case of 3 input variables given above, the total number of monomial combinations of degree lesser than or equal to three is thus \(P = \binom{6}{3} = 20\). The linear regression has to identify the coefficients \((w_1, \dots, w_P)\), in addition to the intercept \(w_0\).

This concept is implemented through the PolynomialRegression class which inherits from the MLRegressionAlgo class.


The polynomial regression model relies on the LinearRegression class of the LinearRegression and PolynomialFeatures classes of the scikit-learn library.

class gemseo.mlearning.regression.polyreg.PolynomialRegression(data, degree, transformer=None, input_names=None, output_names=None, fit_intercept=True, penalty_level=0.0, l2_penalty_ratio=1.0, **parameters)[source]

Polynomial regression.


  • data (Dataset) – learning dataset.

  • degree (int) – Degree of polynomial. Default: 2.

  • transformer (dict(str)) – transformation strategy for data groups. If None, do not transform data. Default: None.

  • input_names (list(str)) – names of the input variables.

  • output_names (list(str)) – names of the output variables.

  • fit_intercept (bool) – if True, fit intercept. Default: True.

  • penalty_level – penalty level greater or equal to 0. If 0, there is no penalty. Default: 0.

  • l2_penalty_ratio (float) – penalty ratio related to the l2 regularization. If 1, the penalty is the Ridge penalty. If 0, this is the Lasso penalty. Between 0 and 1, the penalty is the ElasticNet penalty. Default: None.