mcmodels.regressors.nonnegative_ridge_regression¶

mcmodels.regressors.nonnegative_ridge_regression(X, y, alpha, sample_weight=None, solver='SLSQP', **solver_kwargs)[source]¶

Solve the nonnegative least squares estimate ridge regression problem.

Solves

\[\underset{x}{\text{argmin}} \| Ax - b \|_2^2 + \alpha^2 \| x \|_2^2 \quad \text{s.t.} \quad x \geq 0\]

We can write this as the quadratic programming (QP) problem:

\[\underset{x}{\text{argmin}} x^TQx - c^Tx \quad \text{s.t.} \quad x \geq 0\]

where

\[Q = A^TA + \alpha I \quad \text{and} \quad c = -2A^Ty\]

Parameters:

X : array, shape = (n_samples, n_features): Training data.
y : array, shape = (n_samples,) or (n_samples, n_targets): Target values.
alpha : float or array with shape = (n_features,): Regularization strength; must be a positive float. Improves the conditioning of the problem and reduces the variance of the estimates. Larger values specify stronger regularization.
sample_weight : float or array-like, shape (n_samples,), optional (default = None): Individual weights for each sample.
solver : string, optional (default = ‘SLSQP’): Solver with which to solve the QP. Must be one that supports bounds (i.e. ‘L-BFGS-B’, ‘TNC’, ‘SLSQP’).
**solver_kwargs: See scipy.optimize.minimize for valid keyword arguments

Returns:

coef : array, shape = (n_features,) or (n_features, n_targets): Weight vector(s).
res : float: The residual, \(\| Qx - c \|_2\)