Regression

Regression model

The regression module implements regression algorithms, where the goal is to find relationships between continuous input and output variables. After being fitted to a learning set, Regression algorithms can predict output values of new input data.

A regression algorithm consists of identifying a function \(f: \mathbb{R}^{n_{\textrm{inputs}}} \to \mathbb{R}^{n_{\textrm{outputs}}}\). Given an input point \(x \in \mathbb{R}^{n_{\textrm{inputs}}}\), the predict method of the regression algorithm will return the output point \(y = f(x) \in \mathbb{R}^{n_{\textrm{outputs}}}\). See supervised for more information.

Wherever possible, regression algorithms should also be able to compute the Jacobian matrix of the function it has learned to represent. Given an input point \(x \in \mathbb{R}^{n_{\textrm{inputs}}}\), the Jacobian predict method of the regression algorithm should thus return the matrix

\[\begin{split}J_f(x) = \frac{\partial f}{\partial x} = \begin{pmatrix} \frac{\partial f_1}{\partial x_1} & \cdots & \frac{\partial f_1} {\partial x_{n_{\textrm{inputs}}}}\\ \vdots & \ddots & \vdots\\ \frac{\partial f_{n_{\textrm{outputs}}}}{\partial x_1} & \cdots & \frac{\partial f_{n_{\textrm{outputs}}}} {\partial x_{n_{\textrm{inputs}}}} \end{pmatrix} \in \mathbb{R}^{n_{\textrm{outputs}}\times n_{\textrm{inputs}}}.\end{split}\]

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

Available regression models are:

• Gaussian process regression
• Linear regression
• Mixture of experts
• Polynomial chaos expansion
• Polynomial regression
• Random forest regression
• RBF regression

Examples

Development