Adaptive parameter choice for one-sided finite difference schemes and its application in diabetes technology

In this paper we discuss the problem of approximation of the first derivative of a function at the endpoint of its definition interval. This problem is motivated by diabetes therapy management, where it is important to provide estimations of the future blood glucose trend from current and past measurements. A natural way to approach the problem is to use one-sided finite difference schemes for numerical differentiation, but, following this way, one should be aware that the values of the function to be differentiated are noisy and available only at given fixed points. Then (as we argue in the paper) the number of used point values is the only parameter to be employed for regularization of the above mentioned ill-posed problem of numerical differentiation. In this paper we present and theoretically justify an adaptive procedure for choosing such a parameter. We also demonstrate some illustrative tests, as well as the results of numerical experiments with simulated clinical data.

[1]  Jens Flemming,et al.  Sharp converse results for the regularization error using distance functions , 2011 .

[2]  H. Woxniakowski.  Information-Based Complexity , 1988 .

[3]  Boris Kovatchev,et al.  Peculiarities of the Continuous Glucose Monitoring Data Stream and Their Impact on Developing Closed-Loop Control Technology , 2008, Journal of diabetes science and technology.

[4]  Rahul Kumar,et al.  One-sided finite-difference approximations suitable for use with Richardson extrapolation , 2006, J. Comput. Phys..

[5]  Robert S. Anderssen,et al.  Finite difference methods for the numerical differentiation of non-exact data , 2005, Computing.

[6]  P. Mathé.  The Lepskii principle revisited , 2006 .

[7]  Stefan Kindermann,et al.  On the convergence of the quasioptimality criterion for (iterated) Tikhonov regularization , 2008 .

[8]  Alexander G. Ramm,et al.  On stable numerical differentiation , 2001, Math. Comput..

[9]  Giovanni Sparacino,et al.  Glucose Concentration can be Predicted Ahead in Time From Continuous Glucose Monitoring Sensor Time-Series , 2007, IEEE Transactions on Biomedical Engineering.

[10]  Srinivasan Rajaraman,et al.  Predictive Monitoring for Improved Management of Glucose Levels , 2007, Journal of diabetes science and technology.

[11]  R. Qu.  A new approach to numerical differentiation and integration , 1996 .

[12]  Anders Reiz.  On quadrature formulae , 1950 .

[13]  Shuai Lu,et al.  Numerical differentiation from a viewpoint of regularization theory , 2006, Math. Comput..

[14]  C. Cobelli,et al.  In Silico Preclinical Trials: A Proof of Concept in Closed-Loop Control of Type 1 Diabetes , 2009, Journal of diabetes science and technology.

[15]  Henryk Wozniakowski,et al.  Information-based complexity , 1987, Nature.

[16]  Giovanni Sparacino,et al.  Glucose Prediction Algorithms from Continuous Monitoring Data: Assessment of Accuracy via Continuous Glucose Error-Grid Analysis. , 2007, Journal of diabetes science and technology.

[17]  Brent D. Cameron,et al.  Development of a Neural Network for Prediction of Glucose Concentration in Type 1 Diabetes Patients , 2008, Journal of diabetes science and technology.

[18]  B. Fornberg.  Generation of finite difference formulas on arbitrarily spaced grids , 1988 .

[19]  Leszek Plaskota,et al.  Noisy information and computational complexity , 1996 .

[20]  L. Quinn,et al.  Estimation of future glucose concentrations with subject-specific recursive linear models. , 2009, Diabetes technology & therapeutics.

[21]  Peter Mathé,et al.  The use of higher order finite difference schemes is not dangerous , 2009, J. Complex..

[22]  A. Tikhonov,et al.  Use of the regularization method in non-linear problems , 1965 .

[23]  F. Hoog,et al.  A stable finite difference ansatz for higher order differentiation of non-exact data , 1998, Bulletin of the Australian Mathematical Society.

[24]  C. Cobelli,et al.  Assessment of blood glucose predictors: the prediction-error grid analysis. , 2011, Diabetes technology & therapeutics.

[25]  B. Wayne Bequette,et al.  Hypoglycemia Detection and Prediction Using Continuous Glucose Monitoring—A Study on Hypoglycemic Clamp Data , 2007, Journal of diabetes science and technology.

[26]  F. Doyle,et al.  Prevention of Nocturnal Hypoglycemia Using Predictive Alarm Algorithms and Insulin Pump Suspension , 2010, Diabetes Care.

[27]  S. Pereverzyev,et al.  Extrapolation in variable RKHSs with application to the blood glucose reading , 2011 .

[28]  Boris Kovatchev,et al.  Evaluating the clinical accuracy of two continuous glucose sensors using continuous glucose-error grid analysis. , 2005, Diabetes care.