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 x t into its dynamic counterpart, v t , and then applies the methods and tools of static regression to v t . 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 ( x i ); 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 ( x i ).