Theoretically Based Dynamic Regression (TDR)—A New and Novel Regression Framework for Modeling Dynamic Behavior

The theoretical modeling of a dynamic system will have derivatives of the response (y) with respect to time (t). Two common physical attributes (i.e., parameters) of dynamic systems are dead-time (θ) and lag (τ). Theoretical dynamic modeling will contain physically interpretable parameters such as τ and θ with physical constraints. In addition, the number of unknown model-based parameters can be considerably smaller than empirically based (i.e., lagged-based) approaches. This work proposes a Theoretically based Dynamic Regression (TDR) modeling approach that overcomes critical lagged-based modeling limitations as demonstrated in three large, multiple input, highly dynamic, real data sets. Dynamic Regression (DR) is a lagged-based, empirical dynamic modeling approach that appears in the statistics literature. However, like all empirical approaches, the model structures do not contain first-principle interpretable parameters. Additionally, several time lags are typically needed for the output, y, and input, x, to capture significant dynamic behavior. TDR uses a simplistic theoretically based dynamic modeling approach to transform xt into its dynamic counterpart, vt, and then applies the methods and tools of static regression to vt. TDR is demonstrated on the following three modeling problems of freely existing (i.e., not experimentally designed) real data sets: 1. the weight variation in a person (y) with four measured nutrient inputs (xi); 2. the variation in the tray temperature (y) of a distillation column with nine inputs and eight test data sets over a three year period; and 3. eleven extremely large, highly dynamic, subject-specific models of sensor glucose (y) with 12 inputs (xi).