Taylor Series Approach Research Articles

Structural identifiability and indistinguishability analyses of the Minimal Model and a Euglycemic Hyperinsulinemic Clamp model for glucose–insulin dynamics

Many mathematical models have been developed to describe glucose–insulin kinetics as a means of analysing the effective control of diabetes. This paper concentrates on the structural identifiability analysis of certain well-established mathematical models that have been developed to characterise glucose–insulin kinetics under different experimental scenarios. Such analysis is a pre-requisite to experiment design and parameter estimation and is applied for the first time to these models with the specific structures considered. The analysis is applied to a basic (original) form of the Minimal Model (MM) using the Taylor Series approach and a now well-accepted extended form of the MM by application of the Taylor Series approach and a form of the Similarity Transformation approach. Due to the established inappropriate nature of the MM with regard to glucose clamping experiments an alternative model describing the glucose–insulin dynamics during a Euglycemic Hyperinsulinemic Clamp (EIC) experiment was considered. Structural identifiability analysis of the EIC model is also performed using the Taylor Series approach and shows that, with glucose infusion as input alone, the model is structurally globally identifiable. Additional analysis demonstrates that the two different model forms are structurally distinguishable for observation of both glucose and insulin.

A Taylor series approach to the numerical analysis of the M/D/1/N queue

This paper presents a functional approximation of the M/D/1/N built on a Taylor series approximation. Numerical examples are carried out to illustrate the performance of our approach.

Polynomial Fuzzy Models for Nonlinear Control: A Taylor Series Approach

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Classical Takagi–Sugeno (T–S) fuzzy models are formed by <emphasis emphasistype="italic"> convex</emphasis> combinations of <emphasis emphasistype="italic">linear</emphasis> consequent local models. Such fuzzy models can be obtained from nonlinear first-principle equations by the well-known sector-nonlinearity modeling technique. This paper extends the sector-nonlinearity approach to the polynomial case. This way, generalized polynomial fuzzy models are obtained. The new class of models is polynomial, both in the membership functions and in the consequent models. Importantly, T–S models become a particular case of the proposed technique. Recent possibilities for stability analysis and controller synthesis are also discussed. A set of examples shows that polynomial modeling is able to reduce conservativeness with respect to standard T–S approaches as the degrees of the involved polynomials increase. </para>

Insufficiency of chemical network model integration using a high-order Taylor series method

The rate of change for the concentrations of chemical substances in a set of reactions is modeled by a nonlinear dynamical system, which warrants the use of numerical integration methods for differential equations. Previous work advocates the use of a specialized high-order Taylor series method because of an observed reduction in computation time. Contrastingly, we show combinatorial and computational difficulties of the standard Taylor series method, which may dramatically increase computational time or reduce the quality of output. We provide two implementations, a naive algorithm and an algorithm employing dynamic programming; we are able to overcome only some numerical obstacles and therefore conclude that the Taylor series approach is insufficient for large sets of reactions having many chemical substances.

Structural Identifiability of the Minimal Model and a Euglycemic Hyperinsulinemic Clamp Model for glucose-insulin dynamics

Many mathematical models have been developed to describe glucose and insulin kinetics as a means of analysing the effective control of diabetes. Of these probably the most widely accepted is the so-called “Minimal Model”. This paper concentrates on structural identifiability analyses of two well-established mathematical models that have been developed to characterise glucose-insulin kinetics under different experimental scenarios. Such analysis is a pre-requisite to experiment design and parameter estimation and is applied for the first time to these models with the structures considered. The analysis is initially applied to a basic form of the Minimal Model using the Taylor Series approach, demonstrating global identifiability contrary to previously published results. A now well-accepted extended form of the Minimal model is then considered where the model is provided in an augmented form in order to apply the Similarity Transformation approach. This augmentation was necessary to cater for the particular structure for the Minimal Model that includes a time term. Once more the system, proved to be globally identifiable when both glucose and insulin are observed. Due to the inappropriate nature of the Minimal Model with regard to glucose clamping an alternative model describing the glucose-insulin dynamics during Euglycemic Hyperinsulinemic Clamp was considered. The structural identifiability analysis of the Euglycemic Hyperinsulinemic Clamp model is also performed using the Taylor Series Approach and results show that, with glucose infusion as input alone, the model is structurally globally identifiable.

Inverse Estimation of the Thermal Conductivity in a One-Dimensional Domain by Taylor Series Approach

The Taylor series approximation is developed for the inverse estimation of thermal conductivity in a one-dimensional domain. The differential governing equation of heat conduction is converted to a discrete system of linear equations in matrix form using the temperature measurement and heat generation at the grid points as well as surface heat flux. The unknown thermal conductivity is estimated by solving the linear algebraic equations directly without iterations. The features of the present method are that no prior information about the functional form of the thermal conductivity is required, nor are any initial guesses or iterations in the calculation process needed. The accuracy and robustness of the present method are verified by comparing the results with the analytical solutions for constant, spatial- and temperature-dependent thermal conductivities. The results show that the inverse solutions are in good agreement with the exact solutions.

First‐order error propagation formulae for the Gassmann relations

ABSTRACTThe Gassmann relations of poroelasticity provide a connection between the dry and the saturated elastic moduli of porous rock and are useful in a variety of petroleum geoscience applications. Because some uncertainty is usually associated with the input parameters, the propagation of error in the inputs into the final moduli estimates is immediately of interest. Two common approaches to error propagation include: a first‐order Taylor series expansion and Monte‐Carlo methods. The Taylor series approach requires derivatives, which are obtained either analytically or numerically and is usually limited to a first‐order analysis. The formulae for analytical derivatives were often prohibitively complicated before modern symbolic computation packages became prevalent but they are now more accessible. We apply this method and present formulae for uncertainty in the predicted bulk and shear moduli for two forms of the Gassmann relations. Numerical results obtained with these uncertainty formulae are compared with Monte‐Carlo calculations as a form of validation and to illustrate the relative characteristics of the two approaches. Particular emphasis is given to the problem of correlated variables, which are often ignored in naïve approaches to error analysis. Going out to the error level that the two methods were compared, the means agree and the variance of the Monte Carlo method for bulk modulus grows with input error.

Taylor series approach to fuzzy multiobjective linear fractional programming

This paper presents the use of a Taylor series for fuzzy multiobjective linear fractional programming problems (FMOLFP). The Taylor series is a series expansion that a representation of a function. In the proposed approach, membership functions associated with each objective of fuzzy multiobjective linear fractional programming problem transformed by using a Taylor series are unified. Thus, the problem is reduced to a single objective. Practical applications and numerical examples are used in order to show the efficiency and superiority of the proposed approach.

Developing new correlations and uncertainty analysis for the prediction of ground-level and maximum ground-level concentrations

The determination of Ground Level Concentration (GLC) and Maximum Ground Level Concentration (MGLC) is useful for the prediction of violations for air quality regulations. In this study, the predictions of GLC and MGLC when using different existing empirical formulas are analysed and the effect of dispersion parameters is investigated. Sensitivity analysis studies using both a first order and a second order Taylor series approach are carried out. The dispersion coefficients are assumed to be of a power law form and are used in conjunction with the Gaussian plume model in order to calculate the sensitivity coefficients.

Nonlinear Mapping of Gaussian Statistics: Theory and Applications to Spacecraft Trajectory Design

This paper discusses the nonlinear propagation of spacecraft trajectory uncertainties via solutions of the Fokker– Planck equation. We first discuss the solutions of the Fokker–Planck equation for a deterministic system with a Gaussian boundary condition. Next, we derive an analytic expression of a nonlinear trajectory solution using a higher-order Taylor series approach, discuss the region of convergence for the solutions, and apply the result to spacecraft applications. Such applications consist of nonlinear propagation of the mean and covariance matrix, design of statistically correct trajectories, and nonlinear statistical targeting. The two-body and Hill three-body problems are chosen as examples and realistic initial uncertainty models are considered. The results show that the nonlinear map of the trajectory uncertainties can be approximated in an analytic form, and there exists an optimal place to perform a correction maneuver, which is not found using the linear method.

Quantifying uncertainties associated with the measurement of primary production

Depth-integrated primary production was estimated in the Celtic Sea (European Shelf) by 2 independent methods. Daily production rates were derived from photosynthetic parameters determined by photosynthesis/irradiance (P/E) experiments on samples incubated with 14 C-bicarbonate ( 14 C method) and from in situ profiles with the fast repetitive rate fluorometer (FRRF) method. Production was estimated to be between 332 and 910 mg C m -2 d -1 by the 14 C method and from 351 to 1278 mg C m -2 d -1 by the FRRF method. The uncertainty, or error, of these estimates was determined by calculating 95% confidence intervals using Taylor series and Monte Carlo modelling approaches. Both approaches revealed significant uncertainty in estimates. The Taylor series analysis of the 14 C method indicated that 95% confidence interval widths were between 4 and 20% of the estimated production. The smallest error in this study, calculated using a Taylor series analysis, gave a 95% confidence interval of (578, 622) mg C m -2 d -1 for an estimated production rate of 600 mg C m -2 d -1 . The largest error was calculated using a Monte Carlo analysis and found that a production rate of 377 mg C m -2 d -1 had 95% confidence limits of (300, 453) mg C m -2 d -1 . Larger uncertainties were associated with the FRRF method, but reflect only the increased complexity of equations used to derive photosynthetic parameters and not a known greater degree of error in the approach. The precision of primary production measurements is a neglected but important factor that should be considered in primary production experiments and in testing satellite remote sensing algorithms.

MODELLING OF HAEMODIALYSIS IN LIMITING SERUM FREE LIGHT CHAINS IN PATIENTS WITH RENAL FAILURE

A mathematical model for the in vivo kinetics of free light chains is developed. The model consists of two compartments, one containing the plasma and the other the interstitial fluid. It is used to examine the effects of dialysis on myeloma, a form of cancer that results in high free light chain concentrations. The structural identifiability of the model is analysed using the Taylor series approach, and it is confirmed that the model is structurally globally identifiable provided data are collected both during dialysis and after it. The model is then used to fit clinical data from a myeloma patient suffering chronic renal failure. The data show that dialysis causes a reduction in free light chain concentrations, and this is reflected in the model responses.

Polynomial approach for the most general linear Fredholm integrodifferential‐difference equations using Taylor matrix method

A Taylor matrix method is developed to find an approximate solution of the most general linear Fredholm integrodifferential‐difference equations with variable coefficients under the mixed conditions in terms of Taylor polynomials. This method transforms the given general linear Fredholm integrodifferential‐difference equations and the mixed conditions to matrix equations with unknown Taylor coefficients. By means of the obtained matrix equations, the Taylor coefficients can be easily computed. Hence, the finite Taylor series approach is obtained. Also, examples are presented and the results are discussed.

Approximate solution of complex differential equations for a rectangular domain with Taylor collocation method

In this study, Taylor collocation method is investigated for the approximate computation of high-order linear complex differential equations. Using the collocation points on any rectangular domain in the complex plane, the method transforms the given complex differential equation and the mixed conditions to matrix equation with unknown Taylor coefficients. By means of the obtained matrix equations, the Taylor coefficients can be easily computed. Hence, the finite Taylor series approach is obtained. Also, examples are presented and the results are discussed.

Hybrid grey box modelling of a pickling process

This paper deals with grey box modelling of an industrial process, in which known parts are modelled using a priori information and significant unknown parts are described as general continuous nonlinear functions. The modelling procedure follows a structured approach, which includes basic modelling, data acquisition, model calibration, hybrid expanded modelling, stochastic modelling and model appraisal. The general functions are approximated by means of the Taylor series including higher order terms, where the partial derivatives are estimated from measured data by minimising the maximum likelihood function. The Taylor series approach is used to keep the number of estimated parameters low in comparison with other nonlinear black box identification methods. The model is suitable for formulating algorithms to control the process, for example, a model predictive controller. The model can also be used to simulate various different production situations in order to improve the capacity of the total production line. Further, the relevant parts of the Taylor series can be used to explain in what way unknown process parts influence the behaviour of the process and give ideas for further investigation concerning the studied process.

Factors contributing to increased emergency department use among patients with chronic conditions

Study objectives: The purpose of the study is to identify risk factors for emergency department (ED) utilization among patients with chronic conditions. Methods: This is a descriptive, population-based study using data from the Medical Expenditure Panel Survey 2000. Individuals aged 18 years or older and with diagnoses of chronic obstructive pulmonary disease, congestive heart failure, diabetes mellitus, or hypertension were included as having chronic conditions. Those patients were classified as ED users and non-ED users. Variables representing characteristics such as demographics, socioeconomic status, health care utilization, and number of chronic conditions were used in the analysis. STATA software was used for Taylor-series approach to estimate SE for weighted survey estimates. Multivariable analysis (stratification) and regression model were used to compare the 2 groups and to assess the factors associated with visits to the ED. Confidence intervals were calculated in each analysis. Results: Of 25,095 cases reviewed, 3,997 (16%) were identified as having at least 1 of the 4 chronic conditions, and 18% of those were ED users. There were no statistically significant differences between ED users and non-ED users in variables such as age and insurance status. When the regression model was analyzed, statistically significant factors associated with ED users were being female, nonwhite, or unmarried; lacking prescription medical insurance; having a higher incidence of outpatient physician visits; and having a higher incidence of prescriptions drug refills (P<.01). Conclusion: Individuals who have chronic conditions and are poor, lack prescription insurance, are female, are nonwhite, have less than a college education, or have a poor perception of their mental and physical health status are more likely to use the ED for care. Study objectives: The purpose of the study is to identify risk factors for emergency department (ED) utilization among patients with chronic conditions. Methods: This is a descriptive, population-based study using data from the Medical Expenditure Panel Survey 2000. Individuals aged 18 years or older and with diagnoses of chronic obstructive pulmonary disease, congestive heart failure, diabetes mellitus, or hypertension were included as having chronic conditions. Those patients were classified as ED users and non-ED users. Variables representing characteristics such as demographics, socioeconomic status, health care utilization, and number of chronic conditions were used in the analysis. STATA software was used for Taylor-series approach to estimate SE for weighted survey estimates. Multivariable analysis (stratification) and regression model were used to compare the 2 groups and to assess the factors associated with visits to the ED. Confidence intervals were calculated in each analysis. Results: Of 25,095 cases reviewed, 3,997 (16%) were identified as having at least 1 of the 4 chronic conditions, and 18% of those were ED users. There were no statistically significant differences between ED users and non-ED users in variables such as age and insurance status. When the regression model was analyzed, statistically significant factors associated with ED users were being female, nonwhite, or unmarried; lacking prescription medical insurance; having a higher incidence of outpatient physician visits; and having a higher incidence of prescriptions drug refills (P<.01). Conclusion: Individuals who have chronic conditions and are poor, lack prescription insurance, are female, are nonwhite, have less than a college education, or have a poor perception of their mental and physical health status are more likely to use the ED for care.

Equivalence and identifiability analysis of uncontrolled nonlinear dynamical systems

The problem of parameter identifiability has been considered from different points of view in the case of nonlinear dynamical systems. For analytic systems the standard approach for uncontrolled systems is the Taylor series approach (Pohjanpalo, Math. Biosciences 41 (1978) 21), or the approaches based on differential algebra for polynomial and rational systems. The similarity transformation approach, based on the local state isomorphism theorem, gives a sufficient and necessary condition for global identifiability of nonlinear controlled systems. But it leads only to a necessary condition for identifiability in the case of some uncontrolled systems. Our contribution consists in using the equivalence of systems, based on the straightening out theorem, to analyse the identifiability of uncontrolled systems. From this theory, we state the necessary or sufficient identifiability conditions, some of them depending on the state variable dimension.

A differential quadrature solution of the two-dimensional inverse heat conduction problem

A numerical procedure using a combined differential quadrature and Taylor series approach is presented for the solution of the transient two-dimensional inverse heat conduction problem. The method is relatively easy to program, compared to other more complex solutions currently available, and gives good results for both exact and inexact (noisy) data. The method is suitable to provide estimates of unknown boundary conditions, in advanced inverse procedures.

Analysis of Performance of BPSK in an Additive Combination of Impulsive and Gaussian Noise

We analyze the bit error rate (BER) performance of a correlation receiver subject to impulsive plus Gaussian noise with coherent binary phase-shift keying (BPSK). The impulsive component of the noise is assumed to be due to random occurrences of impulses following a Poisson arrival process. Using the moments of the impulsive component, a series expression for the BER is derived by a Taylor's series approach. From this expression, approximate formulae for extreme values of signal-to-noise ratio (SNR) and impulse arrival rate (IAR) are obtained. Numerical results show that when the square root raised cosine pulse is used, (1) the BER performance improves with increase of the rolloff factor, but the improvement becomes more significant with decrease of the Gaussian-to-total noise ratio or increase of the IAR, (2) as the rolloff factor increases, a decrease of the IAR degrades the BER performance for low SNRs, but improves it for high SNRs.

Asymptotic properties of the Eulerian truncation approximation: Analysis of the perfectly stratified transport problem

It is well known that stochastic groundwater problems are difficult to solve without making approximations of one kind or another. A popular approximation used in both subsurface flow and transport applications is the so‐called “Eulerian truncation.” This approximation, which relies on a perturbation expansion of the governing equation, neglects certain terms involving products of small fluctuations. Several recent publications [e.g., Dagan and Neuman, 1991; Neuman and Orr, 1993; Neuman, 1993] question the validity of Eulerian truncation, arguing that it is inconsistent because neglected terms are of the same order as those retained. In this paper we analyze the asymptotic properties of the Eulerian truncation solution, using a Taylor series approach which clearly reveals the properties of different order approximations. We show that the Eulerian truncation does, in fact, generate an asymptotic ensemble mean expansion for the perfectly stratified example considered by Dagan and Neuman [1991]. Moreover, we show that the Eulerian truncation solution to this problem is identical to the well‐known cumulant expansion approximation, which is also based on an asymptotic expansion.