Low-rank scale-invariant tensor product smooths for generalized additive mixed models

Area of Science:

• Statistics
• Statistical Modeling

Background:

• Generalized additive models (GAMs) and generalized additive mixed models (GAMMs) are powerful tools for flexible regression analysis.
• Existing methods for tensor product smooths can lack interpretability or computational efficiency.

Purpose of the Study:

• To present a general method for constructing low-rank tensor product smooths for GAMs and GAMMs.
• To offer smooths with desirable properties such as covariate rescaling invariance and tunable smoothness.

Main Methods:

• A penalized regression approach is used to build tensor product smooths from marginal smooths.
• Low-rank bases with quadratic wiggliness penalties are employed for computational efficiency.
• The method allows automatic generation from various marginal bases and penalties.

Main Results:

• The proposed tensor product smooths offer invariance to linear covariate rescaling.
• They provide a tunable range of smoothness, unlike scale-invariant single-penalty smooths.
• The method is computationally efficient due to the low rank of the smooths.

Conclusions:

• The developed method provides flexible, computationally efficient, and interpretable tensor product smooths for GAMs and GAMMs.
• These smooths can be readily integrated into standard linear and generalized linear mixed models.
• Simulation studies indicate favorable comparisons with smoothing spline ANOVA methods.