(404e) Forecast Intervals In K-Steps-Ahead Prediction Modeling Under Continuous-Time Monitoring with Application to Blood Glucose Inference
The ability to forecast the onset of hypo- or hyperglycemia is of critical importance in the management of glucose levels for insulin dependent diabetics. To achieve this objective, the individual must be under glucose monitoring. Recent years have seen the development of continuous glucose monitoring systems (CGMS) that report glucose measurements every five minutes. It is the purpose of this research to exploit this capability under modeling to forecast sufficiently early and with high certainty undesirable glucose levels so that corrective action can prevent these levels from occurring.
In the statistical literature a prediction interval gives a set of bounds that contain the value of the response given a value of the explanatory (i.e., input) variable (or vector for multiple explanatory variables), x*, in the future, with a given level of confidence. In the context of this research, which is dynamic modeling, prediction intervals must address future input levels, x*, as well as future time, kDt, where k is an integer and Dt is the sampling time. To meet this requirement, forecast terminology from the time series literature is used. Thus, the goal of this work is to obtain a 100(1 - a)% forecast interval k-steps-ahead (KSA) (i.e., a distance of time kDt into the future) for glucose response (y| kDt, x*) at a given input level of x*.
This forecast interval is developed using classical linear regression as a basis. The expectation function is a static linear regression function in a Wiener block-oriented structure. In this structure each of the p inputs, xi, i = 1, . . ., p, is passed through a linear dynamic block producing outputs from these blocks vi(qi), where qi is a vector of dynamic parameters corresponding to the transfer of xi to vi. (In this work p = 11, consisting of 3 food variables, 7 activity variables from the BodyMedia® armband and time of day on a 24 hour clock.) The KSA forecast intervals are obtained under serially correlated noise from a normal distribution with mean 0 and a variance of s2 for all time steps. In modeling real systems, one of the challenges is obtaining an estimator that maintains its accuracy in the presence of unmeasured disturbances. To insure this feature, this work uses a novel model development method based on the work of Rollins, et al. (2010), that maintains fitted correlation performance in training, validation and test sets. In addition, since the amount of data in training is very large, the vi(qi) can be assumed to be fixed, i.e., with standard errors of zero, making an errors-in-variable approach unnecessary. Moreover, for this same reason, the standard errors in the static and noise parameters can also be assumed negligible giving an approximate 100(1 - a)% forecast interval for (y| kDt, x*) as: , where is the 100(1 - a/2)th percentile of the standard normal distribution and is the estimate of s. Note that this interval does not depend on x*. The lack of dependence on x* comes from the use of a large training set of data which occurs due to frequent sampling of CGM systems, that results in nearly zero estimation error for the parameters. Thus, essentially all the forecast errors come from the s that includes modeling lack of fit which is inversely related to rfit, the correlation between the measured and estimated response.
The results of this work demonstrate high reliability of the proposed forecast interval using about 20 diabetic subjects over a validation period of two weeks or approximately 4,000 time intervals. For small values of k the forecast bands are narrow but grow as k increases since the forecast error, i.e., s, increases as k increases. The implications are that to obtain precise intervals far into the future, this will be primarily achieved by obtaining accurate fits for large values of k.
Rollins, D. K., N. Bhandari, J. Kleinedler, K. Kotz, A. Strohbehn, L. Boland, M. Murphy, D. Andre, N. Vyas, G.Welk and W. Franke, "Free-living inferential modeling of blood glucose level using only noninvasive inputs," Journal of Process Control 20 95-107 (2010).