Smoothing in an underdetermined linear model with random explanatory variables

In some physical systems, where the goal is to describe behaviour over an entire field using scattered observations, a multiple regression model can be derived from the discretization of a continuous process. These models often have more parameters than observations. We propose a technique for constructing smoothed estimators in this situation. Our method assumes the model has random explanatory and response variables, and imposes a smoothness penalty based on the signal-to-noise ratio of the model. Results are presented using a known value for the ratio, and a method for estimating the ratio is discussed. The procedure is applied to modelling temperature measurements taken in the California Current.