Characterization of Ridge Trace Behavior

The main objective of this article is to provide some new findings on the algebraic properties of a ridge trace (i.e., the graph of ridge regression coefficients vs. the scalar ridge parameter). These properties evolve by first expressing ridge regression coefficients as rational functions, i.e., the ratio of polynomials in the ridge parameter, k. Specific properties can then be derived assuming the correlation matrix for the independent variables has a specific structure. Some results are possible without imposing this later condition. In particular, we focus on sign changes, criss-crossing, and rate-of-change of the ridge coefficients. This work provides explicit quantitative insights into the behavior depicted by a ridge trace.