Using elimination to describe Maxwell curves

Cartesian ovals are curves in the plane that have been studied for hundreds of years. A Cartesian oval is the set of points whose distances from two fixed points called foci satisfy the property that a linear combination of these distances is a fixed constant. These ovals are a special case of what we call Maxwell curves. A Maxwell curve is the set of points with the property that a specific weighted sum of the distances to n foci is constant. We shall describe these curves geometrically. We will then examine Maxwell curves with two foci and a special case with three foci by deriving a system of equations that describe each of them. Since their solution spaces have too many dimensions, we will eliminate all but two variables from these systems in order to study the curves in xy-space. We will show how to do this first by hand. Then, after some background from algebraic geometry, we will discuss two other methods of eliminating variables, Groebner bases and resultants. Finally we will find the same elimination polynomials with these two methods and study them

Development of a Model-Based Noninvasive Glucose Monitoring Device for Non-Insulin Dependent People

Continuous-time glucose monitoring (CGM) effectively improves glucose control, as oppose to infrequent glucose measurements (i.e. using Lancet Meters), by providing frequent blood glucose concentration (BGC) to better associate this variation with changes in behavior. Currently, the most widely used CGM devices rely on a sensor that is inserted invasively under the skin. Because of the invasive nature and also the replacement cost of sensors, the primary users of current CGM devices are insulin dependent people (type 1 and some type 2 diabetics). Most non-insulin dependent diabetics use only lancet glucose measurements. The ultimate goal of this research is the development of CGM technology that overcomes these limitations (i.e. invasive sensors and their cost) in an effort to increase CGM applications among non-insulin dependent people. To meet this objective, this preliminary work has developed a methodology to mathematically infer BGC from measurements of non-invasive input variables which can be thought of as a “virtual” or “soft” sensor approach. In this work virtual sensors are developed and evaluated on 20 subjects using four BGC measurements per day and eight input variables representing meals, activity, stress, and clock time. Up to four weeks of data are collected for each subject. One evaluation consists of 3 days of training and up to 25 days of testing data. The second one consists of one week of training, one week of validation, and 2 weeks of testing data. The third one consists two weeks of training, one week of validation and one week of testing data. Model acceptability is determined on an individual basis based on the fitted correlation to CGM testing data. For 3 day, 1 week, and 2 weeks training studies, 35%, 55% and 65% of the subjects, respectively, met the Acceptability Criteria that we established based on the concept of usefulness

The Use of Advanced Statistical Concepts and Analysis to Improve Nonlinear Dynamic Glucose Modeling

Type 2 diabetes is one of the greatest burdens on the health care industry today. This condition is characterized by poor control of blood glucose concentration (BGC). In order to help those afflicted with type 2 diabetes better control their BGC, the goal of this research is to develop a device that can noninvasively measure BGC. There are several statistical issues that must be addressed before such a device can be developed. The first is to identify inputs that appear to infer BGC and choose a model that can use these inputs to accurately predict BGC. Due to its ability to assign unique dynamics to each input, the Wiener network model is used to predict BGC for each subject. However, there are several challenges to fitting a Wiener network model that can accurately predict BGC, including estimating a large number of parameters, the nonlinearity of the parameters, the stiffness of the least squares objective function for fitting this model, and possible overfitting. Thus an algorithm is designed to fit a Wiener network model where the correlation between predicted and observed BGC is maximized under supervised learning. However, such models were fit with frequent BGC measurements every five minutes. For a non-insulin dependent person, there may only be four BGC measurements per day, which for a week of data or less, implies that there are fewer observations than parameters. Thus, in order to calibrate a noninvasive device, some parameters were held fixed to reduce parameterization, and a novel scheme was devised to estimate the remaining model parameters. Finally, a method of predicting future BGC should be devised that could be used to warn the user if their BGC is going to be too low or too high in the near future. Time series models that use only outputs, such as autoregressive models, to predict BGC into the future performed well in the very near future, but performance degraded quickly as time increased. By utilizing the Wiener network model and previous measurements of BGC, a k-steps ahead prediction model is devised that predicts BGC 5k minutes into the future. This is used to calculate approximate (1-α) 100% forecast intervals for BGC up to one hour into the future.</p

An algorithm for optimally fitting a wiener model

The purpose of this work is to present a new methodology for fitting Wiener networks to datasets with a large number of variables. Wiener networks have the ability to model a wide range of data types, and their structures can yield parameters with phenomenological meaning. There are several challenges to fitting such a model: model stiffness, the nonlinear nature of a Wiener network, possible overfitting, and the large number of parameters inherent with large input sets. This work describes a methodology to overcome these challenges by using several iterative algorithms under supervised learning and fitting subsets of the parameters at a time. This methodology is applied to Wiener networks that are used to predict blood glucose concentrations. The predictions of validation sets from models fit to four subjects using this methodology yielded a higher correlation between observed and predicted observations than other algorithms, including the Gauss-Newton and Levenberg-Marquardt algorithms.This article is from Mathematical Problems in Engineering 2011 (2011): article no.570509, doi: 10.1155/2011/570509.</p

An Algorithm for Optimally Fitting a Wiener Model

The purpose of this work is to present a new methodology for fitting Wiener networks to datasets with a large number of variables. Wiener networks have the ability to model a wide range of data types, and their structures can yield parameters with phenomenological meaning. There are several challenges to fitting such a model: model stiffness, the nonlinear nature of a Wiener network, possible overfitting, and the large number of parameters inherent with large input sets. This work describes a methodology to overcome these challenges by using several iterative algorithms under supervised learning and fitting subsets of the parameters at a time. This methodology is applied to Wiener networks that are used to predict blood glucose concentrations. The predictions of validation sets from models fit to four subjects using this methodology yielded a higher correlation between observed and predicted observations than other algorithms, including the Gauss-Newton and Levenberg-Marquardt algorithms

Development of a Model-Based Noninvasive Glucose Monitoring Device for Non-Insulin Dependent People

Continuous-time glucose monitoring (CGM) effectively improves glucose control, as oppose to infrequent glucose measurements (i.e. using Lancet Meters), by providing frequent blood glucose concentration (BGC) to better associate this variation with changes in behavior. Currently, the most widely used CGM devices rely on a sensor that is inserted invasively under the skin. Because of the invasive nature and also the replacement cost of sensors, the primary users of current CGM devices are insulin dependent people (type 1 and some type 2 diabetics). Most non-insulin dependent diabetics use only lancet glucose measurements. The ultimate goal of this research is the development of CGM technology that overcomes these limitations (i.e. invasive sensors and their cost) in an effort to increase CGM applications among non-insulin dependent people. To meet this objective, this preliminary work has developed a methodology to mathematically infer BGC from measurements of non-invasive input variables which can be thought of as a “virtual” or “soft” sensor approach. In this work virtual sensors are developed and evaluated on 20 subjects using four BGC measurements per day and eight input variables representing meals, activity, stress, and clock time. Up to four weeks of data are collected for each subject. One evaluation consists of 3 days of training and up to 25 days of testing data. The second one consists of one week of training, one week of validation, and 2 weeks of testing data. The third one consists two weeks of training, one week of validation and one week of testing data. Model acceptability is determined on an individual basis based on the fitted correlation to CGM testing data. For 3 day, 1 week, and 2 weeks training studies, 35%, 55% and 65% of the subjects, respectively, met the Acceptability Criteria that we established based on the concept of usefulness.This article is published as Rollins DK, Beverlin L, Mei Y, Kotz K, Andre D, Vyas N, Welk G, Franke WD. The development of a virtual sensor in glucose monitoring for non-insulin dependent people. Bioinformatics and Diabetes. 2014:1, 19; doi:10.14302;issn.2374-9431.jbd-13-283. Posted with permission.</p