Penalized angular regression for personalized predictions

TL;DR

This paper introduces Penalized Angular (PAN) regression, a novel personalized regression method that penalizes angles in a hyperspherical parameterization, improving prediction accuracy especially in medical applications.

Contribution

The paper proposes PAN regression, a new personalized regression approach that inherently incorporates individual-specific coefficients through angle penalization, with theoretical and empirical validation.

Findings

PAN regression outperforms OLS and ridge in prediction error.

Combining PAN with L2 penalty yields smaller mean squared prediction error.

Method demonstrated effectively in a medical application.

Abstract

Personalization is becoming an important feature in many predictive applications. We introduce a penalized regression method implementing personalization inherently in the penalty. Personalized angle (PAN) regression constructs regression coefficients that are specific to the covariate vector for which one is producing a prediction, thus personalizing the regression model itself. This is achieved by penalizing the angles in a hyperspherical parametrization of the regression coefficients. For an orthogonal design matrix, it is shown that the PAN estimate is the solution to a low-dimensional eigenvector equation. Using a parametric bootstrap procedure to select the tuning parameter, simulations show that PAN regression can outperform ordinary least squares and ridge regression in terms of prediction error. We further prove that by combining the PAN penalty with an $L_{2}$ penalty the resulting method will have uniformly smaller mean squared prediction error than ridge regression, asymptotically. Finally, we demonstrate the method in a medical application.