(373c) Improvements Upon Least-Squares Model Identification - An Application to Diabetes Modeling

In many biomedical applications, process noise is known to be neither white nor normally distributed. When identifying models in these cases, it may be more effective to minimize a different penalty function than the standard sum of squared errors (as in a least-squares identification method). This study investigates model identification based on two different penalty functions: the 1-norm of the prediction errors and a Huber-type penalty function. In certain realistic situations, model identification based on these latter two penalty functions is shown to result in more accurate estimates of parameters than the standard least-squares solution, and also more accurate model predictions for independent test data. The effects of model identification based on these three methods are investigated in this paper. In particular, two measures of model accuracy are quantified: 1) the accuracy with which the model parameters are estimated, and 2) the accuracy of model predictions for test data (i.e., data not used for identification). The first measure is applicable when the structure of the model matches the structure of the process, while the second measure is more generally applicable and relevant even when there is process/model mismatch. The case when the identified model structure is simpler than that of the process (a realistic situation, especially for biomedical applications) is investigated in this paper. The application of interest in this study relates to type 1 diabetes. An "artificial β-cell" is a biomedical device that would automatically regulate glucose levels in type 1 diabetes patients by delivering appropriate insulin doses based on previous glucose and insulin infusion information and predictions of future glucose trends obtained from an accurate model of the subject's glucose-insulin dynamics. In such a model predictive control (MPC) scheme, the mathematical model would be identified from previous data collected from the subject. One fundamental property of the glucose-insulin system in type 1 diabetes patients is the presence of carbohydrate meals, which can be viewed as large disturbances from a modeling-and-control perspective. (In addition to meals, there are other such disturbances like bouts of intense exercise or acute physiological stress.) If these disturbances are unmodeled, then they may be characterized as noise spikes, or outliers. Due to these large, frequent noise outliers, it may be more effective to identify models from type 1 diabetes data based on a 1-norm- or Huber-type penalty function rather than the standard 2-norm. A second fundamental property of an artificial β-cell device relates to characteristics of the continuous sensor (or sensors) required for control. The critical measurement is the glucose concentration obtained from a continuous glucose monitor, but other sensors may be used to measure variables like heart rate and/or activity. These sensors are known to occasionally give missing or erroneous data. For example, some such sensors rely on wireless communication between a transducer and a receiver. A missed sample or value of zero may result when these two units are separated by enough distance. These missing or zero-valued measurements can also be viewed as noise outliers, although they are outliers of a different ilk than the meal-type outliers. The models in this study were identified from simulation data obtained from a single-input, single-output (SISO) process. The input u was a realization of a white, Gaussian signal. In addition to the deterministic effects from the input, the output signal y was also corrupted by two kinds of noise: process noise with randomly-spaced spikes (analogous to meal-related effects) and measurement noise with random samples zero-valued (analogous to sensor-related effects). In all cases, models were identified by optimizing the one-step prediction. The optimizations were based on three different penalty functions: the standard 2-norm type (denoted by l₂), a 1-norm type (denoted by l₁), and a Huber type. For data with disturbance characteristics analogous to meal-related effects, the l₁ and Huber identification methods resulted in more accurate estimates of the process parameters than the l₂ method. Specifically, the distributions of the estimated parameters were narrower for the l₁ and Huber methods. For data with simulated measurement noise analogous to sensor-related effects, a bias is introduced into the parameter estimates. The bias is less pronounced with the l₁ and Huber identification methods than with the l₂ method, though. The quality of the identification techniques was assessed not only by the accuracy with which the parameters were identified, but also the quality of the resulting model predictions. The predictions were quantified with FIT values for independent test data. In words, the FIT value is the percentage of variability in the data that the model prediction explains (Ljung, 1999). For infinite-step predictions of test data with disturbance analogous to meals, the l₂, l₁, and Huber identification techniques resulted in FIT values of 44%, 66%, and 70%, respectively. A representative prediction for an l₂-identified model and a Huber-identified model are shown in the figure. For the case in which the process is undermodeled (one fewer parameter is identified than is present in the process), the predictions of the l₁ and Huber models were more accurate than those of the l₂ models. Model identification results in this study indicate that, for process data with certain characteristics, more accurate models may be identified by using penalty functions other than the standard least-squares method. In particular, data with two different noise characteristics?-large process disturbances and measurement noise outliers?-resulted in more accurate identified models when the model parameters were estimated using l₁-type and Huber-type optimization criteria. These noise characteristics are common to some types of real-world data, e.g., data relevant to an artificial β-cell application for treating type 1 diabetes. Financial support from the Danish Council for Strategic Research and Novo Nordisk is gratefully acknowledged.