Showing papers on "Mathematical model published in 2016"

Effective drifts in dynamical systems with multiplicative noise: a review of recent progress.

Abstract: Noisy dynamical models are employed to describe a wide range of phenomena. Since exact modeling of these phenomena requires access to their microscopic dynamics, whose time scales are typically much shorter than the observable time scales, there is often need to resort to effective mathematical models such as stochastic differential equations (SDEs). In particular, here we consider effective SDEs describing the behavior of systems in the limits when natural time scales become very small. In the presence of multiplicative noise (i.e. noise whose intensity depends upon the system's state), an additional drift term, called noise-induced drift or effective drift, appears. The nature of this noise-induced drift has been recently the subject of a growing number of theoretical and experimental studies. Here, we provide an extensive review of the state of the art in this field. After an introduction, we discuss a minimal model of how multiplicative noise affects the evolution of a system. Next, we consider several case studies with a focus on recent experiments: the Brownian motion of a microscopic particle in thermal equilibrium with a heat bath in the presence of a diffusion gradient; the limiting behavior of a system driven by a colored noise modulated by a multiplicative feedback; and the behavior of an autonomous agent subject to sensorial delay in a noisy environment. This allows us to present the experimental results, as well as mathematical methods and numerical techniques, that can be employed to study a wide range of systems. At the end we give an application-oriented overview of future projects involving noise-induced drifts, including both theory and experiment.

Identification of time‐varying OE models in presence of non‐Gaussian noise: Application to pneumatic servo drives

Abstract: Summary Intensive research in the field of mathematical modeling of pneumatic servo drives has shown that their mathematical models are nonlinear in which many important details cannot be included in the model. Owing the influence of the combination of heat coefficient, unknown discharge coefficient, and change of temperature, it was supposed that parameters of the pneumatic cylinder are random (stochastic parameters). On the other side, it has been well known that the nonlinear model can be approximated by a linear model with time-varying parameters. Due to the aforementioned reasons, it can be assumed that the pneumatic cylinder model is a linear stochastic model with variable parameters. In practical conditions, in measurements, there are rare, inconsistent observations with the largest part of population of observations (outliers). Therefore, synthesis of robust algorithms is of primary interest. In this paper, the robust recursive algorithm for output error models with time-varying parameters is proposed. The convergence property of the proposed robust algorithm is analyzed using the methodology of an associated ordinary differential equation system. Because ad hoc selection of model orders leads to overparameterization or parsimony problem, the robust Akaike's criterion is proposed to overcome these problems. By determining the least favorable probability density for a given class of probability distribution represents a base for design of the robust version of Akaike's criterion. The behavior of the proposed robust identification algorithm is considered through intensive simulations that demonstrate the superiority of the robust algorithm in relation to the linear algorithms (derived under an assumption that the stochastic disturbance has a Gaussian distribution). The good practical values of the proposed robust algorithm to identification of the pneumatic cylinder are illustrated by experimental results. Copyright © 2016 John Wiley & Sons, Ltd.

Efficient Wireless Powering of Biomedical Sensor Systems for Multichannel Brain Implants

Abstract: This paper describes the complete mathematical optimization process of an inductive powering system suitable for the application within implanted biomedical systems. The optimization objectives are thereby size, energy efficiency, and tissue absorption. Within the first step, the influence of the operational frequency on the given quantities is computed by means of finite element simulations, yielding a compromise of power transfer efficiency of the wireless link and acceptable tissue heating in terms of the specific absorption rate. All simulations account for the layered structure of the human head, modeling the dielectric properties with Cole-Cole dispersion effects. In the second step, the relevant coupling and loss effects of the transmission coils are modeled as a function of the geometrical design parameters, enabling a noniterative and comprehensible mathematical derivation of the optimum coil geometry given an external size constraint. Further investigations of the optimum link design also consider high-permeability structures being applied to the primary coil, enhancing the efficiency by means of an increased mutual inductance. Thereby, a final link efficiency of 80% at a coil separation distance of 5 mm and 20% at 20 mm using a 10-mm planar receiving coil can be achieved, contributing to a higher integration density of multichannel brain implanted sensors. Moreover, the given procedure does not only give insight into the optimization of the coil design, but also provides a minimized set of mathematical expressions for designing a highly efficient primary side coil driver and for selecting the components of the secondary side impedance matching. All mathematical models and descriptions have been verified by simulation and concluding measurements.

Carrera Unified Formulation for Free-Vibration Analysis of Aircraft Structures

Abstract: Advanced structural models, based on variable one-, two-, and three-dimensional kinematics, are proposed in this paper and applied to the analysis of the free vibration of reinforced aircraft shell structures. The used models go beyond classical structural theories, that is, Euler–Bernoulli (for one-dimensional beams) and Kirchhoff (for two-dimensional plates) type assumptions. The order of the expansion of the displacement fields over the cross section (one-dimensional case) and along the plate thickness (two-dimensional case) is, in fact, a free parameter of the problem. In this paper, Lagrange polynomials are used to build such expansions, and as a consequence, only displacements are used as the problem unknowns (no rotations or derivatives of displacements, which are typical of one-dimensional/two-dimensional classical theories, are introduced). The finite-element method is used to provide numerical solutions. The related arrays and the governing dynamical equations are written in terms of a few funda...

A cross-entropy method and probabilistic sensitivity analysis framework for calibrating microscopic traffic models

Abstract: Car following modeling framework seeks for a more realistic representation of car following behavior in complex driving situations to improve traffic safety and to better understand several puzzling traffic flow phenomena, such as stop-and-go oscillations. Calibration and validation techniques pave the way towards the descriptive power of car-following models and their applicability for analyzing traffic flow. However, calibrating these models is never a trivial task. This is caused by the fact that some parameters, such as reaction time, are generally not directly observable from traffic data. On the other hand, traffic data might be subject to various errors and noises. This contribution puts forward a Cross-Entropy Method (CEM) based approach to identify parameters of deterministic car-following models under noisy data by formulating it as a stochastic optimization problem. This approach allows for statistical analysis of the parameter estimations. Another challenge arising in the calibration of car following models concerns the selection of the most important parameters. This paper introduces a relative entropy based Probabilistic Sensitivity Analysis (PSA) algorithm to identify the important parameters so as to reduce the complexity, data requirement and computational effort of the calibration process. Since the CEM and the PSA are based on the Kullback–Leibler (K–L) distance, they can be simultaneously integrated into a unified framework to further reduce the computational burden. The proposed framework is applied to calibrate the intelligent driving model using vehicle trajectories data from the NGSIM project. Results confirm the great potential of this approach.

Introducing Modeling Transition Diagrams as a Tool to Connect Mathematical Modeling to Mathematical Thinking

Abstract: This study contributes a methodological tool to reconstruct the cognitive processes and mathematical activities carried out by mathematical modelers. Represented as Modeling Transition Diagrams (MTDs), individual modeling routes were constructed for four engineering undergraduate students. Findings stress the importance and limitations of using micro-analysis to examine modeling processes. The findings and the MTDs were used to critically question the implications of using modeling cycles as a theory of mathematical modeling processes.

Nonlinear Model Analysis and “Switching Model” of AC–DC Three-Degree-of-Freedom Hybrid Magnetic Bearing

Abstract: The most widely studied linear models of the suspension forces in ac–dc three-degree-of-freedom hybrid magnetic bearings may not be practical for application to the precision control of bearings with larger air gaps for larger suspension forces. In addition, the simplified linear and the real nonlinear models in linear zone of current and displacements have not been compared before. In this study, the two most commonly used nonlinear models are analyzed comprehensively to address the lack of nonlinear model analysis in previous research and to seek the most appropriate model for both nonlinear and linear zones (full zone). The results show that no single model completely approximates the test results over the entire zone. A composite model, such as the “switching model,” is more accurate. To illustrate the applicability of the results to both nonlinear model analysis and the proposed “switching model,” nonlinear stiffness and performance test experiments were conducted. It is concluded that the model analysis and the proposed “switching model” can provide a control system with the most suitable mathematical models of the suspension force.

Strong Inference in Mathematical Modeling: A Method for Robust Science in the Twenty-First Century

Abstract: While there are many opinions on what mathematical modeling in biology is, in essence, modeling is a mathematical tool, like a microscope, which allows consequences to logically follow from a set of assumptions. Only when this tool is applied appropriately, as microscope is used to look at small items, it may allow to understand importance of specific mechanisms/assumptions in biological processes. Mathematical modeling can be less useful or even misleading if used inappropriately, for example, when a microscope is used to study stars. According to some philosophers (Oreskes et al., 1994), the best use of mathematical models is not when a model is used to confirm a hypothesis but rather when a model shows inconsistency of the model (defined by a specific set of assumptions) and data. Following the principle of strong inference for experimental sciences proposed by Platt (1964), I suggest "strong inference in mathematical modeling" as an effective and robust way of using mathematical modeling to understand mechanisms driving dynamics of biological systems. The major steps of strong inference in mathematical modeling are (1) to develop multiple alternative models for the phenomenon in question; (2) to compare the models with available experimental data and to determine which of the models are not consistent with the data; (3) to determine reasons why rejected models failed to explain the data, and (4) to suggest experiments which would allow to discriminate between remaining alternative models. The use of strong inference is likely to provide better robustness of predictions of mathematical models and it should be strongly encouraged in mathematical modeling-based publications in the Twenty-First century.

Identification of Wave Energy Device Models From Numerical Wave Tank Data—Part 2: Data-Based Model Determination

Abstract: In this paper and its companion, the identification of mathematical models describing the behaviour of wave energy devices (WECs) in the ocean is investigated through the use of numerical wave tank experiments. When the wave amplitude and the WEC displacement are not negligible with respect to the WEC dimensions, nonlinear hydrodynamic effects may appear, and the accuracy of linear hydrodynamic models is reduced, leading to the necessity of introducing some nonlinearities in the model structure. This paper proposes, for WEC modelling, the use of discrete-time nonlinear autoregressive with exogenous input (NARX) models, as an alternative to continuous-time models. Techniques of model identification are also explained and applied to a case study.

Multiscale modeling of brain dynamics: from single neurons and networks to mathematical tools

Abstract: The extreme complexity of the brain naturally requires mathematical modeling approaches on a large variety of scales; the spectrum ranges from single neuron dynamics over the behavior of groups of neurons to neuronal network activity. Thus, the connection between the microscopic scale (single neuron activity) to macroscopic behavior (emergent behavior of the collective dynamics) and vice versa is a key to understand the brain in its complexity. In this work, we attempt a review of a wide range of approaches, ranging from the modeling of single neuron dynamics to machine learning. The models include biophysical as well as data-driven phenomenological models. The discussed models include Hodgkin-Huxley, FitzHugh-Nagumo, coupled oscillators (Kuramoto oscillators, Rossler oscillators, and the Hindmarsh-Rose neuron), Integrate and Fire, networks of neurons, and neural field equations. In addition to the mathematical models, important mathematical methods in multiscale modeling and reconstruction of the causal connectivity are sketched. The methods include linear and nonlinear tools from statistics, data analysis, and time series analysis up to differential equations, dynamical systems, and bifurcation theory, including Granger causal connectivity analysis, phase synchronization connectivity analysis, principal component analysis (PCA), independent component analysis (ICA), and manifold learning algorithms such as ISOMAP, and diffusion maps and equation-free techniques. WIREs Syst Biol Med 2016, 8:438-458. doi: 10.1002/wsbm.1348 For further resources related to this article, please visit the WIREs website.

Frequency-Domain In-Vehicle UWB Channel Modeling

Abstract: This paper aims to present a simple but robust model characterizing the frequency-dependent transfer function of an in-vehicle ultrawideband (UWB) channel. A large number of transfer functions spanning the UWB (3–11 GHz) are recorded inside the passenger compartment of a four-seated sedan. It is found that the complex transfer function can be decomposed into two terms, the first term being a real-valued long-term trend that characterizes frequency dependence with a power law and the second term forming a complex correlative discrete series that may be represented via an autoregressive (AR) model. An exhaustive simulation framework is laid out based on empirical equations characterizing trend parameters and AR process coefficients. The simulation of the transfer function is straightforward as it involves only a handful of variables; however, it is in good agreement with the actual measured data. The proposed model is further validated by comparing different channel parameters, such as coherence bandwidth, power delay profile, and root-mean-square delay spread, obtained from raw and synthetic data sets. It is also shown how the model can be compared with existing time-domain Saleh–Valenzuela-influenced models and related IEEE standards.

Traveling Pulses for a Two-Species Chemotaxis Model.

Abstract: Mathematical models have been widely used to describe the collective movement of bacteria by chemotaxis. In particular, bacterial concentration waves traveling in a narrow channel have been experimentally observed and can be precisely described thanks to a mathematical model at the macroscopic scale. Such model was derived in [1] using a kinetic model based on an accurate description of the mesoscopic run-and-tumble process. We extend this approach to study the behavior of the interaction between two populations of E. Coli. Separately, each population travels with its own speed in the channel. When put together, a synchronization of the speed of the traveling pulses can be observed. We show that this synchronization depends on the fraction of the fast population. Our approach is based on mathematical analysis of a macroscopic model of partial differential equations. Numerical simulations in comparison with experimental observations show qualitative agreement.

On the equivalence between continuum and car-following models of traffic flow

Abstract: Recently different formulations of the first-order Lighthill-Whitham-Richards (LWR) model have been identified in different coordinates and state variables. However, relationships between higher-order continuum and car-following traffic flow models are still not well understood. In this study, we first categorize traffic flow models according to their coordinates, state variables, and orders in the three-dimensional representation of traffic flow and propose a unified approach to convert higher-order car-following models into continuum models and vice versa. The conversion method consists of two steps: equivalent transformations between the secondary Eulerian (E-S) formulations and the primary Lagrangian (L-P) formulations, and approximations of L-P derivatives with anisotropic (upwind) finite differences. We use the method to derive continuum models from general second- and third-order car-following models and derive car-following models from second-order continuum models. Furthermore, we demonstrate that corresponding higher-order continuum and car-following models have the same fundamental diagrams, and that the string stability conditions for vehicle-continuous car-following models are the same as the linear stability conditions for the corresponding continuum models. A numerical example verifies the analytical results. In a sense, we establish a weak equivalence between continuum and car-following models, subject to errors introduced by the finite difference approximation. Such an equivalence relation can help us to pick out anisotropic solutions of higher-order models with non-concave fundamental diagrams.

A new way of determining distance decay parameters in spatial interaction models with application to job accessibility analysis in Sweden

Abstract: In this paper we explore and compare various techniques for the calculation of distance decay parameters which are estimated using statistical methods with half-life decay parameters which are derived mathematically. Half-life models appear to be a valid alternative to traditional spatial interaction models, especially in the presence of spatially highly disaggregate data. Our results indicate that Half-life models are more accurate for the construction of decay parameters than are unconstrained spatial interaction models in ‘medium’ sized datasets but not as accurate as doubly-constrained models. However, using highly detailed and disaggregate datasets Half-life models may be viable alternatives to doubly-constrained spatial interaction models as the latter will be difficult to estimate when the number of origins and destinations increase. In addition, Half-life models rise in accuracy with increasing degrees of disaggregation due to reductions of systematic errors between observed individual level commuting distance and modelled distances between origins and destinations. In sum, our findings are as follows. First, since unconstrained and doubly-constrained spatial interaction models become increasingly difficult to estimate and/or less accurate to use compared to Half-life models as the spatial disaggregation increases choice of decay parameter estimation model should be considered in relation to level of disaggregation. Secondly, Half-life models are not affected by the systematic errors observed in the statistically derived models. Finally, using Half-life models for the estimation of decay parameters is simple which may make it easy to employ among practitioners lacking skills or computer means for the estimation of more complex statistically derived models.

Changing Cycle Lengths in State-Transition Models: Challenges and Solutions.

Abstract: The choice of a cycle length in state-transition models should be determined by the frequency of clinical events and interventions. Sometimes there is need to decrease the cycle length of an existing state-transition model to reduce error in outcomes resulting from discretization of the underlying continuous-time phenomena or to increase the cycle length to gain computational efficiency. Cycle length conversion is also frequently required if a new state-transition model is built using observational data that have a different measurement interval than the model's cycle length. We show that a commonly used method of converting transition probabilities to different cycle lengths is incorrect and can provide imprecise estimates of model outcomes. We present an accurate approach that is based on finding the root of a transition probability matrix using eigendecomposition. We present underlying mathematical challenges of converting cycle length in state-transition models and provide numerical approximation methods when the eigendecomposition method fails. Several examples and analytical proofs show that our approach is more general and leads to more accurate estimates of model outcomes than the commonly used approach. MATLAB codes and a user-friendly online toolkit are made available for the implementation of the proposed methods.

Travel Time Dynamics for Intelligent Transportation Systems: Theory and Applications

Abstract: This paper demonstrates the limitation of the flow-based travel time functions. This paper presents a density-based travel time function and further develops a fundamental model of travel time dynamics that is built from a given fundamental traffic relationship and vehicle characteristics. The travel time dynamics produce an asymmetric one-sided coupled system of hyperbolic partial differential equations, where the first equation represents the macroscopic traffic dynamics. The existence of the solution for the mathematical model is then presented. The main contribution of this paper is the mathematical development and analysis of the real-time model of travel time. Moreover, this paper also shows various intelligent transportation system applications where travel time is an important factor and where this new model would be extremely useful and important.