A fractional calculus model for HIV dynamics: real data, parameter estimation and computational strategies

Abstract The ability of the so-called Caputo-Fabrizio (CF) and Atangana-Baleanu (AB) operators to create suitable models for real data is tested with real world data. Two alternative models based on the CF and AB operators are assessed and compared with known models for data sets obtained from electrochemical capacitors and the human body electrical impedance. The results show that the CF and AB descriptions perform poorly when compared with the classical fractional derivatives.

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Continuous Autoregressive Moving Average Models

Methodologies and Applications of Computational Statistics for Machine Intelligence - Advances in Systems Analysis, Software Engineering, and High Performance Computing ◽

10.4018/978-1-7998-7701-1.ch007 ◽

2021 ◽

pp. 118-141

Author(s):

Yakup Ari

Keyword(s):

Time Series ◽

Parameter Estimation ◽

Differential Equations ◽

Stochastic Differential Equations ◽

Discrete Time ◽

Moving Average ◽

Likelihood Estimation ◽

Real Data ◽

Autoregressive Moving Average ◽

The Difference

The financial time series have a high frequency and the difference between their observations is not regular. Therefore, continuous models can be used instead of discrete-time series models. The purpose of this chapter is to define Lévy-driven continuous autoregressive moving average (CARMA) models and their applications. The CARMA model is an explicit solution to stochastic differential equations, and also, it is analogue to the discrete ARMA models. In order to form a basis for CARMA processes, the structures of discrete-time processes models are examined. Then stochastic differential equations, Lévy processes, compound Poisson processes, and variance gamma processes are defined. Finally, the parameter estimation of CARMA(2,1) is discussed as an example. The most common method for the parameter estimation of the CARMA process is the pseudo maximum likelihood estimation (PMLE) method by mapping the ARMA coefficients to the corresponding estimates of the CARMA coefficients. Furthermore, a simulation study and a real data application are given as examples.

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Regularized Parameter Estimation in High-Dimensional Gaussian Mixture Models

Neural Computation ◽

10.1162/neco_a_00128 ◽

2011 ◽

Vol 23 (6) ◽

pp. 1605-1622 ◽

Cited By ~ 12

Author(s):

Lingyan Ruan ◽

Ming Yuan ◽

Hui Zou

Keyword(s):

Parameter Estimation ◽

Mixture Models ◽

Gaussian Mixture Models ◽

Expectation Maximization Algorithm ◽

Real Data ◽

Gaussian Mixture ◽

High Dimensional ◽

Model Based Clustering ◽

Text Type ◽

Effective Dimensionality

Finite gaussian mixture models are widely used in statistics thanks to their great flexibility. However, parameter estimation for gaussian mixture models with high dimensionality can be challenging because of the large number of parameters that need to be estimated. In this letter, we propose a penalized likelihood estimator to address this difficulty. The [Formula: see text]-type penalty we impose on the inverse covariance matrices encourages sparsity on its entries and therefore helps to reduce the effective dimensionality of the problem. We show that the proposed estimate can be efficiently computed using an expectation-maximization algorithm. To illustrate the practical merits of the proposed method, we consider its applications in model-based clustering and mixture discriminant analysis. Numerical experiments with both simulated and real data show that the new method is a valuable tool for high-dimensional data analysis.

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Fractional Calculus as a Simple Tool for Modeling and Analysis of Long Memory Process in Industry

Mathematics ◽

10.3390/math7060511 ◽

2019 ◽

Vol 7 (6) ◽

pp. 511 ◽

Cited By ~ 6

Author(s):

Ivo Petráš ◽

Ján Terpák

Keyword(s):

Time Series ◽

Fractional Calculus ◽

Fractional Order ◽

Long Memory ◽

Real Data ◽

Data Set ◽

Modeling And Analysis ◽

Long Memory Process ◽

Real Process ◽

Definition Of

This paper deals with the application of the fractional calculus as a tool for mathematical modeling and analysis of real processes, so called fractional-order processes. It is well-known that most real industrial processes are fractional-order ones. The main purpose of the article is to demonstrate a simple and effective method for the treatment of the output of fractional processes in the form of time series. The proposed method is based on fractional-order differentiation/integration using the Grünwald–Letnikov definition of the fractional-order operators. With this simple approach, we observe important properties in the time series and make decisions in real process control. Finally, an illustrative example for a real data set from a steelmaking process is presented.

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FracFit: A robust parameter estimation tool for fractional calculus models

Water Resources Research ◽

10.1002/2016wr019748 ◽

2017 ◽

Vol 53 (3) ◽

pp. 2559-2567 ◽

Cited By ~ 28

Author(s):

James F. Kelly ◽

Diogo Bolster ◽

Mark M. Meerschaert ◽

Jennifer D. Drummond ◽

Aaron I. Packman

Keyword(s):

Parameter Estimation ◽

Fractional Calculus ◽

Robust Parameter ◽

Robust Parameter Estimation

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Detection of arterial wall abnormalities via Bayesian model selection

10.1101/422485 ◽

2018 ◽

Author(s):

Karen Larson ◽

Clark Bowman ◽

Costas Papadimitriou ◽

Petros Koumoutsakos ◽

Anastasios Matzavinos

Keyword(s):

Parameter Estimation ◽

Model Selection ◽

Parallel Implementation ◽

Real Data ◽

Physical Models ◽

Patient Specific ◽

Small Scale ◽

Tree Model ◽

Arterial Tree ◽

Flow Models

AbstractPatient-specific modeling of hemodynamics in arterial networks has so far relied on parameter estimation for inexpensive or small-scale models. We describe here a Bayesian uncertainty quantification framework which makes two major advances: an efficient parallel implementation, allowing parameter estimation for more complex forward models, and a system for practical model selection, allowing evidence-based comparison between distinct physical models. We demonstrate the proposed methodology by generating simulated noisy flow velocity data from a branching arterial tree model in which a structural defect is introduced at an unknown location; our approach is shown to accurately locate the abnormality and estimate its physical properties even in the presence of significant observational and systemic error. As the method readily admits real data, it shows great potential in patient-specific parameter fitting for hemodynamical flow models.

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Fractional dynamic system simulating the growth of microbe

Advances in Difference Equations ◽

10.1186/s13662-021-03498-3 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Samir B. Hadid ◽

Rabha W. Ibrahim

Keyword(s):

Differential Equations ◽

Fractional Calculus ◽

Dynamic System ◽

Numerical Solution ◽

Optimal Solution ◽

Real Data ◽

Fractional Dynamic System ◽

Tutte Polynomials ◽

Fractional System

AbstractThere are different approaches that indicate the dynamic of the growth of microbe. In this research, we simulate the growth by utilizing the concept of fractional calculus. We investigate a fractional system of integro-differential equations, which covers the subtleties of the diffusion between infected and asymptomatic cases. The suggested system is applicable to distinguish the presentation of growth level of the infection and to approve if its mechanism is positively active. An optimal solution under simulation mapping assets is considered. The estimated numerical solution is indicated by employing the fractional Tutte polynomials. Our methodology is based on the Atangana–Baleanu calculus (ABC). We assess the recommended system by utilizing real data.

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Detection of arterial wall abnormalities via Bayesian model selection

Royal Society Open Science ◽

10.1098/rsos.182229 ◽

2019 ◽

Vol 6 (10) ◽

pp. 182229

Author(s):

Karen Larson ◽

Clark Bowman ◽

Costas Papadimitriou ◽

Petros Koumoutsakos ◽

Anastasios Matzavinos

Keyword(s):

Parameter Estimation ◽

Model Selection ◽

Parallel Implementation ◽

Real Data ◽

Physical Models ◽

Patient Specific ◽

Small Scale ◽

Tree Model ◽

Arterial Tree ◽

Flow Models

Patient-specific modelling of haemodynamics in arterial networks has so far relied on parameter estimation for inexpensive or small-scale models. We describe here a Bayesian uncertainty quantification framework which makes two major advances: an efficient parallel implementation, allowing parameter estimation for more complex forward models, and a system for practical model selection, allowing evidence-based comparison between distinct physical models. We demonstrate the proposed methodology by generating simulated noisy flow velocity data from a branching arterial tree model in which a structural defect is introduced at an unknown location; our approach is shown to accurately locate the abnormality and estimate its physical properties even in the presence of significant observational and systemic error. As the method readily admits real data, it shows great potential in patient-specific parameter fitting for haemodynamical flow models.

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Probabilistic moveout analysis by time warping

Geophysics ◽

10.1190/geo2018-0797.1 ◽

2020 ◽

Vol 85 (1) ◽

pp. U1-U20

Author(s):

Yanadet Sripanich ◽

Sergey Fomel ◽

Jeannot Trampert ◽

William Burnett ◽

Thomas Hess

Keyword(s):

Parameter Estimation ◽

Posterior Probability ◽

Model Building ◽

Probability Distributions ◽

Real Data ◽

Inversion Method ◽

Time Warping ◽

Anisotropic Models ◽

Trade Offs ◽

Set Up

Parameter estimation from reflection moveout analysis represents one of the most fundamental problems in subsurface model building. We have developed an efficient moveout inversion method based on the process of automatic flattening of common-midpoint (CMP) gathers using local slopes. We find that as a by-product of this flattening process, we can also estimate reflection traveltimes corresponding to the flattened CMP gathers. This traveltime information allows us to construct a highly overdetermined system and subsequently invert for moveout parameters including normal-moveout velocities and quartic coefficients related to anisotropy. We use the 3D generalized moveout approximation (GMA), which can accurately capture the effects of complex anisotropy on reflection traveltimes as the basis for our moveout inversion. Due to the cheap forward traveltime computations by GMA, we use a Monte Carlo inversion scheme for improved handling of the nonlinearity between the reflection traveltimes and moveout parameters. This choice also allows us to set up a probabilistic inversion workflow within a Bayesian framework, in which we can obtain the posterior probability distributions that contain valuable statistical information on estimated parameters such as uncertainty and correlations. We use synthetic and real data examples including the data from the SEAM Phase II unconventional reservoir model to demonstrate the performance of our method and discuss insights into the problem of moveout inversion gained from analyzing the posterior probability distributions. Our results suggest that the solutions to the problem of traveltime-only moveout inversion from 2D CMP gathers are relatively well constrained by the data. However, parameter estimation from 3D CMP gathers associated with more moveout parameters and complex anisotropic models are generally nonunique, and there are trade-offs among inverted parameters, especially the quartic coefficients.

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Parameter Estimation of a Reliability Model of Demand-Caused and Standby-Related Failures of Safety Components Exposed to Degradation by Demand Stress and Ageing That Undergo Imperfect Maintenance

Mathematical Problems in Engineering ◽

10.1155/2017/7042453 ◽

2017 ◽

Vol 2017 ◽

pp. 1-11 ◽

Cited By ~ 2

Author(s):

S. Martorell ◽

P. Martorell ◽

A. I. Sánchez ◽

R. Mullor ◽

I. Martón

Keyword(s):

Parameter Estimation ◽

Nuclear Power ◽

Likelihood Estimation ◽

Real Data ◽

Parameters Estimation ◽

Model Parameters ◽

Reliability Model ◽

Imperfect Maintenance ◽

Maintenance Policies

One can find many reliability, availability, and maintainability (RAM) models proposed in the literature. However, such models become more complex day after day, as there is an attempt to capture equipment performance in a more realistic way, such as, explicitly addressing the effect of component ageing and degradation, surveillance activities, and corrective and preventive maintenance policies. Then, there is a need to fit the best model to real data by estimating the model parameters using an appropriate tool. This problem is not easy to solve in some cases since the number of parameters is large and the available data is scarce. This paper considers two main failure models commonly adopted to represent the probability of failure on demand (PFD) of safety equipment: (1) by demand-caused and (2) standby-related failures. It proposes a maximum likelihood estimation (MLE) approach for parameter estimation of a reliability model of demand-caused and standby-related failures of safety components exposed to degradation by demand stress and ageing that undergo imperfect maintenance. The case study considers real failure, test, and maintenance data for a typical motor-operated valve in a nuclear power plant. The results of the parameters estimation and the adoption of the best model are discussed.

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