Adaptive estimation of hidden semi-Markov chains with parameterised transition probabilities and exponential decaying states

This paper develops maximum likelihood (ML) estimation schemes for finite-state semi-Markov chains in white Gaussian noise. We assume that the semi-Markov chain is characterised by transition probabilities of known parametric from with unknown parameters. We reformulate this hidden semi-Markov model (HSM) problem in the scalar case as a two-vector homogeneous hidden Markov model (HMM) problem in which the state consist of the signal augmented by the time to last transition. With this reformulation we apply the expectation Maximumisation (EM ) algorithm to obtain ML estimates of the transition probabilities parameters, Markov state levels and noise variance. To demonstrate our proposed schemes, motivated by neuro-biological applications, we use a damped sinusoidal parameterised function for the transition probabilities.

Maximum principle and numerical method for the multi-term time–space Riesz–Caputo fractional differential equations

The maximum principle for the space and time–space fractional partial differential equations is still an open problem. In this paper, we consider a multi-term time–space Riesz–Caputo fractional differential equations over an open bounded domain. A maximum principle for the equation is proved. The uniqueness and continuous dependence of the solution are derived. Using a fractional predictor–corrector method combining the L1 and L2 discrete schemes, we present a numerical method for the specified equation. Two examples are given to illustrate the obtained results.

A Novel High Order Space-Time Spectral Method for the Time Fractional Fokker--Planck Equation

The fractional Fokker-Planck equation is an important physical model for simulating anomalous diffusions with external forces. Because of the non-local property of the fractional derivative an interesting problem is to explore high accuracy numerical methods for fractional differential equations. In this paper, a space-time spectral method is presented for the numerical solution of the time fractional Fokker-Planck initial-boundary value problem. The proposed method employs the Jacobi polynomials for the temporal discretization and Fourier-like basis functions for the spatial discretization. Due to the diagonalizable trait of the Fourier-like basis functions, this leads to a reduced representation of the inner product in the Galerkin analysis. We prove that the time fractional Fokker-Planck equation attains the same approximation order as the time fractional diffusion equation developed in [23] by using the present method. That indicates an exponential decay may be achieved if the exact solution is sufficiently smooth. Finally, some numerical results are given to demonstrate the high order accuracy and efficiency of the new numerical scheme. The results show that the errors of the numerical solutions obtained by the space-time spectral method decay exponentially.

Assessment of body composition of Sri Lankan Australian children using ethnic specific equations

Since ethnic differences exist in body composition, assessment methods need to be validated prior to use in different populations. This study attempts to validate the use of Sri Lankan based body composition assessment tools on a group of 5 - 15 year old Australian children of Sri Lankan origin. The study was conducted at the Body Composition Laboratory of the Children’s Nutrition Research Centre at the Royal Children’s Hospital, Brisbane, Australia. Height (Ht), weight (Wt), segmental length (Lsegment name) and skinfold thickness (SFT) were measured. The whole body and segmental bio impedance analysis (BIA) were also measured. The body composition determined by the deuterium dilution technique (criterion method) was compared with the assessments done using prediction equations developed on Sri Lankan children. 27 boys and 15 girls were studied. All predictions of body composition parameters, except percentage fat mass (FM) assessed by the SFT-FM equation in girls gave statistically significant correlations with the criterion method. They had a low mean bias and most were not influenced by the measured parameter. Although living in a different socioeconomic state, the equations developed on children of the same ethnic background gives a better predictive value of body composition. This highlights the ethnic influence on body composition.

Use of Skin-Fold Thickness in Sri Lankan Children: Comparison of Several Prediction Equations

Objective There are many prediction equations available in the literature for the assessment of body composition from skinfold thickness (SFT). This study aims to cross validate some of those prediction equations to determine the suitability of their use on Sri Lankan children. Methods Height, weight and SFT of 5 different sites were measured. Total body water was assessed using the isotope dilution method (D2O). Percentage Fat mass (%FM) was estimated from SFT using prediction equations described by five authors in the literature. Results Five to 15 year old healthy, 282 Sri Lankan children were studied. The equation of Brook gave Ihe lowest bias but limits of agreement were high. Equations described by Deurenberg et al gave slightly higher bias but limits of agreement were narrowest and bias was not influence by extremes of body fat. Although prediction equations did not estimate %FM adequately, the association between %FM and SFT measures, were quite satisfactory. Conclusion We conclude that SFT can be used effectively in the assessment of body composition in children. However, for the assessment of body composition using SFT, either prediction equations should be derived to suit the local populations or existing equations should be cross-validated to determine the suitability before its application.

A preconditioned numerical solver for stiff nonlinear reaction-diffusion equations with fractional Laplacians that avoids dense matrices

The numerical solution of fractional partial differential equations poses significant computational challenges in regard to efficiency as a result of the spatial nonlocality of the fractional differential operators. The dense coefficient matrices that arise from spatial discretisation of these operators mean that even one-dimensional problems can be difficult to solve using standard methods on grids comprising thousands of nodes or more. In this work we address this issue of efficiency for one-dimensional, nonlinear space-fractional reaction–diffusion equations with fractional Laplacian operators. We apply variable-order, variable-stepsize backward differentiation formulas in a Jacobian-free Newton–Krylov framework to advance the solution in time. A key advantage of this approach is the elimination of any requirement to form the dense matrix representation of the fractional Laplacian operator. We show how a banded approximation to this matrix, which can be formed and factorised efficiently, can be used as part of an effective preconditioner that accelerates convergence of the Krylov subspace iterative solver. Our approach also captures the full contribution from the nonlinear reaction term in the preconditioner, which is crucial for problems that exhibit stiff reactions. Numerical examples are presented to illustrate the overall effectiveness of the solver.

Fourier spectral methods for fractional-in-space reaction-diffusion equations

Fractional differential equations are becoming increasingly used as a powerful modelling approach for understanding the many aspects of nonlocality and spatial heterogeneity. However, the numerical approximation of these models is demanding and imposes a number of computational constraints. In this paper, we introduce Fourier spectral methods as an attractive and easy-to-code alternative for the integration of fractional-in-space reaction-diffusion equations described by the fractional Laplacian in bounded rectangular domains ofRn. The main advantages of the proposed schemes is that they yield a fully diagonal representation of the fractional operator, with increased accuracy and efficiency when compared to low-order counterparts, and a completely straightforward extension to two and three spatial dimensions. Our approach is illustrated by solving several problems of practical interest, including the fractional Allen–Cahn, FitzHugh–Nagumo and Gray–Scott models, together with an analysis of the properties of these systems in terms of the fractional power of the underlying Laplacian operator.

The subordinated processes controlled by a family of subordinators and corresponding Fokker-Planck type equations

In this work, we consider subordinated processes controlled by a family of subordinators which consist of a power function of a time variable and a negative power function of an α-stable random variable. The effect of parameters in the subordinators on the subordinated process is discussed. By suitable variable substitutions and the Laplace transform technique, the corresponding fractional Fokker–Planck-type equations are derived. We also compute their mean square displacements in a free force field. By choosing suitable ranges of parameters, the resulting subordinated processes may be subdiffusive, normal diffusive or superdiffusive

Ability of commonly used prediction equations to predict resting energy expenditure in children with inflammatory bowel disease

Background: Paediatric onset inflammatory bowel disease (IBD) may cause alterations in energy requirements and invalidate the use of standard prediction equations. Our aim was to evaluate four commonly used prediction equations for resting energy expenditure (REE) in children with IBD. Methods: Sixty-three children had repeated measurements of REE as part of a longitudinal research study yielding a total of 243 measurements. These were compared with predicted REE from Schofield, Oxford, FAO/WHO/UNU, and Harris-Benedict equations using the Bland-Altman method. Results: Mean (±SD) age of the patients was 14.2 (2.4) years. Mean measured REE was 1566 (336) kcal per day compared with 1491 (236), 1441 (255), 1481 (232), and 1435 (212) kcal per day calculated from Schofield, Oxford, FAO/WHO/UNU, and Harris-Benedict, respectively. While the Schofield equation demonstrated the least difference between measured and predicted REE, it, along with the other equations tested, did not perform uniformly across all subjects, indicating greater errors at either end of the spectrum of energy expenditure. Smaller differences were found for all prediction equations for Crohn's disease compared with ulcerative colitis. Conclusions: Of the commonly used equations, the equation of Schofield should be used in pediatric patients with IBD when measured values are not able to be obtained. (Inflamm Bowel Dis 2010;) Copyright © 2010 Crohn's & Colitis Foundation of America, Inc.