Opinion dynamics with continuous age structure

Two algorithms that combine Brownian dynami cs (BD) simulations with mean-field partial differential equations (PDEs) are presented. This PDE-assisted Brownian dynamics (PBD) methodology provides exact particle tracking data in parts of the domain, whilst making use of a mean-field reaction-diffusion PDE description elsewhere. The first PBD algorithm couples BD simulations with PDEs by randomly creating new particles close to the interface, which partitions the domain, and by reincorporating particles into the continuum PDE-description when they cross the interface. The second PBD algorithm introduces an overlap region, where both descriptions exist in parallel. It is shown that the overlap region is required to accurately compute variances using PBD simulations. Advantages of both PBD approaches are discussed and illustrative numerical examples are presented. © 2013 Society for Industrial and Applied Mathematics.

Stochastic differential equation (SDE) models are the foundation for pricing and hedging financial derivatives. The drift and volatility functions in SDE models are typically chosen to be algebraic functions with a small number ( < 5 ) of parameters which can be calibrated to market data. A more flexible approach is to use neural networks to model the drift and volatility functions, which provides more degrees-of-freedom to match observed market data. Training of models requires optimizing over an SDE, which is computationally challenging. For European options, we develop a fast stochastic gradient descent (SGD) algorithm for training the neural network-SDE model. Our SGD algorithm uses two independent SDE paths to obtain an unbiased estimate of the direction of steepest descent. For American options, we optimize over the corresponding Kolmogorov partial differential equation (PDE). The neural network appears as coefficient functions in the PDE. Models are trained on large datasets (many contracts), requiring either large simulations (many Monte Carlo samples for the stock price paths) or large numbers of PDEs (a PDE must be solved for each contract). Numerical results are presented for real market data including S&P 500 index options, S&P 100 index options, and single-stock American options. The neural-network-based SDE models are compared against the Black-Scholes model, the Dupire's local volatility model, and the Heston model. Models are evaluated in terms of how accurate they are at pricing out-of-sample financial derivatives, which is a core task in derivative pricing at financial institutions. Specifically, we calibrate a neural network-SDE model to market data for a financial derivative on an asset with price S t with a payoff function g ( s ) , and we then evaluate its generalization accuracy for a financial derivative on the same asset S t but with a different payoff function f ( s ) . In addition to comparing out-of-sample pricing accuracy, we evaluate the hedging performance of the neural network-SDE model.

In this study, we propose a new method that is useful for estimating unknown parameter values of stochastic differential equation (SDE) models, based on probability density function (PDF) data measured from random dynamical systems. As our method does not require explicit description of PDF, it can be applied to the SDE models even when their PDFs are hardly derived in explicit forms due to multiplicative-noise terms, nonlinear terms, and so on. Therefore, our method is expected to provide a versatile tool to dynamically parameterize measured PDF data. In our proposed method, it is assumed that a measured PDF is obtained from a random dynamical system whose structure is described by a known SDE model with unknown parameter values. With the help of It\^o calculus, the Fokker-Planck equation (FPE) is derived from the SDE model. The measured PDF and a candidate of parameter values are substituted into the FPE to calculate a FPE residual. Our method is applied to two random vibration systems. Their FPE residuals tend to zero as the parameter values tend to exact values, showing that our proposed FPE residual can be utilized for unknown parameter estimation of SDE models.

Background: Growth trajectories are highly variable between children, making epidemiological analyses challenging both to the identification of malnutrition interventions at the population level and also risk assessment at individual level. We introduce stochastic differential equation (SDE) models into child growth research. SDEs describe flexible dynamic processes comprising: drift - gradual smooth changes – such as physiology or gut microbiome, and diffusion - sudden perturbations, such as illness or infection. Methods: We present a case study applying SDE models to child growth trajectory data from the Haydom, Tanzania and Venda, South Africa sites within the MAL-ED cohort. These data comprise n=460 children aged 0-24 months. A comparison with classical curve fitting (linear mixed models) is also presented. Results: The SDE models offered a wide range of new flexible shapes and parameterizations compared to classical additive models, with performance as good or better than standard approaches. The predictions from the SDE models suggest distinct longitudinal clusters that form distinct ‘streams’ hidden by the large between-child variability. Conclusions: Using SDE models to predict future growth trajectories revealed new insights in the observed data, where trajectories appear to cluster together in bands, which may have a future risk assessment application. SDEs offer an attractive approach for child growth modelling and potentially offer new insights.

Mathematical models are routinely calibrated to experimental data, with goals ranging from building predictive models to quantifying parameters that cannot be measured. Whether or not reliable parameter estimates are obtainable from the available data can easily be overlooked. Such issues of parameter identifiability have important ramifications for both the predictive power of a model, and the mechanistic insight that can be obtained. Identifiability analysis is well-established for deterministic, ordinary differential equation (ODE) models, but there are no commonly adopted methods for analysing identifiability in stochastic models. We provide an accessible introduction to identifiability analysis and demonstrate how existing ideas for analysis of ODE models can be applied to stochastic differential equation (SDE) models through four practical case studies. To assess structural identifiability, we study ODEs that describe the statistical moments of the stochastic process using open-source software tools. Using practically motivated synthetic data and Markov chain Monte Carlo methods, we assess parameter identifiability in the context of available data. Our analysis shows that SDE models can often extract more information about parameters than deterministic descriptions. All code used to perform the analysis is available on Github.

Event Abstract Back to Event Stochastic modeling of neuronal signaling Jukka Intosalmi1*, Tiina Manninen1, Keijo Ruohonen1 and Marja-Leena Linne1 1 Tampere University of Technology, Finland The time evolution of biochemical systems in neurons is traditionally modeled using deterministic ordinary differential equations (ODEs). Chemical reactions, however, are random in nature, and the deterministic approach is valid only for a restricted class of systems. Stochastic models take random fluctuations into account and are thus more realistic. Biochemical reactions can be modeled stochastically using numerous different methods. An ideal model would have the following three important properties. First, the model should be as realistic as possible, second, the mathematical method should be easily implementable as a computer algorithm, and third, the algorithm should be computationally effective. Naturally, all these three conditions cannot be fulfilled at the same time. Some realistic modeling approaches can be derived directly from chemical kinetics without making any approximations. Such approaches are called exact. A good example of an exact modeling approach is the stochastic simulation algorithm (SSA) developed by Gillespie. The SSA is applicable when the molecular populations in the system are small, but it becomes computationally inefficient when the numbers of molecules increase. In order to construct stochastic models that can be effectively simulated, new mathematical approaches have to be explored. As an approximate method also stochastic differential equations (SDEs) have been considered a promising way to model biochemical reactions stochastically. The SDE approach is attractive especially if we consider a system for which the SSA is computationally inefficient and the traditional deterministic ODE approach cannot be used as a good approximation. SDE models treat the chemical populations as real numbers and the construction of the model is based on the law of mass action, similarly as the construction of the traditional deterministic modeling approach. The SDE model, however, takes the random fluctuations into account by describing the time evolution of the system with the stochastic Itô process instead of the deterministic set of ODEs. The Itô process is basically a set of coupled stochastic differential equations. Although SDE modeling usually leads to equations that cannot be solved analytically, the solutions can be approximated numerically using different numerical integration methods. Thus, SDEs provide a mathematically rigorous way to construct models that can be simulated effectively. In order to investigate how well the SDE approach models biochemical systems, the results have to be compared with experimental data or with the simulation results from some exact simulation procedure. In this study, we use the methodology of spectral analysis to compare the simulation results of the SDE model and the SSA. As case studies, we consider e.g. calcium binding and protein kinase C (PKC) signal transduction pathway. We investigate the nature of noise in different modeling approaches and try to interpret the meaning of the noise in real biological systems. The main goal of our study is to find the most essential components and parameters in the SDE model and to adjust them so that the model is capable of giving similar results as the SSA. The simulations are carried out using the distributed computing resources (GRID) provided by Techila Technologies Ltd. Conference: Neuroinformatics 2008, Stockholm, Sweden, 7 Sep - 9 Sep, 2008. Presentation Type: Poster Presentation Topic: Computational Neuroscience Citation: Intosalmi J, Manninen T, Ruohonen K and Linne M (2008). Stochastic modeling of neuronal signaling. Front. Neuroinform. Conference Abstract: Neuroinformatics 2008. doi: 10.3389/conf.neuro.11.2008.01.023 Copyright: The abstracts in this collection have not been subject to any Frontiers peer review or checks, and are not endorsed by Frontiers. They are made available through the Frontiers publishing platform as a service to conference organizers and presenters. The copyright in the individual abstracts is owned by the author of each abstract or his/her employer unless otherwise stated. Each abstract, as well as the collection of abstracts, are published under a Creative Commons CC-BY 4.0 (attribution) licence (https://creativecommons.org/licenses/by/4.0/) and may thus be reproduced, translated, adapted and be the subject of derivative works provided the authors and Frontiers are attributed. For Frontiers’ terms and conditions please see https://www.frontiersin.org/legal/terms-and-conditions. Received: 28 Jul 2008; Published Online: 28 Jul 2008. * Correspondence: Jukka Intosalmi, Tampere University of Technology, Tampere, Finland, jukka.intosalmi@tut.fi Login Required This action requires you to be registered with Frontiers and logged in. To register or login click here. Abstract Info Abstract The Authors in Frontiers Jukka Intosalmi Tiina Manninen Keijo Ruohonen Marja-Leena Linne Google Jukka Intosalmi Tiina Manninen Keijo Ruohonen Marja-Leena Linne Google Scholar Jukka Intosalmi Tiina Manninen Keijo Ruohonen Marja-Leena Linne PubMed Jukka Intosalmi Tiina Manninen Keijo Ruohonen Marja-Leena Linne Related Article in Frontiers Google Scholar PubMed Abstract Close Back to top Javascript is disabled. Please enable Javascript in your browser settings in order to see all the content on this page.

In this paper, we propose a new model for continuous time opinion dynamics on an evolving network. As opposed to existing models, in which the network typically evolves by discretely adding or removing edges, we instead propose a model for opinion formation which is coupled to a network evolving through a system of ordinary differential equations for the edge weights. We interpret each edge weight as the strength of the relationship between a pair of individuals, with edges increasing in weight if pairs continually listen to each other’s opinions and decreasing if not. We investigate the impact of various edge dynamics at different timescales on the opinion dynamics itself. This is done partly through analytic results and partly through extensive numerical simulations of two case studies: one using bounded confidence interaction dynamics in the opinion formation process (as in the classical Hegselmann–Krause model) and one using an exponentially decaying interaction function. We find that the dynamic edge weights can have a significant impact on the opinion formation process, since they may result in consensus formation but can also reinforce polarisation. Overall, the proposed modelling approach allows us to quantify and investigate how the network and opinion dynamics influence each other.

Stochastic Differential Equation (SDE) models are often used to model the dynamics of complex biological systems. The stochastic nature of these models means that some behaviors are more likely than others. It is often the case that a model's primary purpose is to study rare but interesting or important behaviors, such as the formation of a tumor, or the failure of a cyber-physical system. Unfortunately, due to the limited availability of analytic methods for SDEs, stochastic simulations are the most common means for estimating (or bounding) the probability of rare behaviors. Naturally, the cost of stochastic simulations increases with the rarity of the behavior under consideration. To address this problem, we introduce a new algorithm, RESERCHE, that is specifically designed to quantify the likelihood of rare but interesting behaviors in SDE models. Our approach relies on the use of temporal logics for specifying rare behaviors of possible interest, and on the ability of bit-vector decision procedures to reason exhaustively about fixed precision arithmetic. We also compute the probability of an observed behavior under the assumption of Gaussian noise.

The problem of computing the maximum likelihood estimate of the parameters of a specific class of stochastic differential equation (SDE) models with linear drift whose sample paths are observed at discrete time points is considered. This estimate is obtained as in Cleur and Manfredi (1999) by discretizing the explicit expressions for the estimates which maximize the likelihood function in continuous time, by discretizing the likelihood function through a quadrature approximation before maximizing it, and by maximizing the likelihood function of the Euler scheme approximation to the underlying continuous process. Simulation results indicate that, for the constellation of parameter values considered, all three approaches lead to very similar results.

Parameter and uncertainty estimation in stochastic differential equation models with multi-rate data and nonstationary disturbances

Opinion dynamics in dynamical hypergraphs, mirroring the dynamical nature of high-order social interactions, are attracting increasing attention. Opinion dynamics on high-order interactions lead to non-trivial dynamical patterns compared with that on pairwise networks. How should we systematically understand the intrinsic differences between opinion dynamics in hypergraphs and that in networks? We establish a voter model in a dynamical hypergraph. We find that both opinion formation and transient topology are captured by a single multi-player game, provided that the hypergraphs evolve sufficiently fast. The Nash equilibrium analysis facilitates us to reveal the intrinsic differences between high-order interactions and pairwise interactions in the empirical system. Furthermore, we perform simulations to show how robust our theoretical results are beyond fast rewiring. Our work provides a game route for opinion dynamics and sheds a hidden connection between opinion formation on dynamical high-order interactions and multi-player games.

This article estimates a simple univariate model of expectation or opinion formation in continuous time adapting a ‘canonical’ stochastic model of collective opinion dynamics (Weidlich and Haag, 1983; Lux, 1995, 2009a). This framework is applied to a selected data set on survey-based expectations from the rich EU business and consumer survey database for 12 European countries. The model parameters are estimated through Maximum Likelihood (ML) and numerical solution of the transient probability density functions for the resulting stochastic process. The model's success is assessed with respect to its out-of-sample forecasting performance relative to univariate Time Series (TS) models of the Autoregressive Moving Average model, ARMA(p, q) and Autoregressive Fractionally Integrated Moving Average, ARFIMA(p, d, q) varieties. These tests speak for a slight superiority of the canonical opinion dynamics model over the alternatives in the majority of cases.

Reliable forecasting of wind power generation is crucial to optimal control of costs in generation of electricity with respect to the electricity demand. Here, we propose and analyze stochastic wind power forecast models described by parametrized stochastic differential equations, which introduce appropriate fluctuations in numerical forecast outputs. We use an approximate maximum likelihood method to infer the model parameters taking into account the time correlated sets of data. Furthermore, we study the validity and sensitivity of the parameters for each model. We applied our models to Uruguayan wind power production as determined by historical data and corresponding numerical forecasts for the period of March 1 to May 31, 2016.

Abstractsocial networks Online provide a globally available, large-scale infrastructure for people to exchange information and ideas. A topic of great interest in internet research is how to model this information exchange and, in particular, how to model and analyze the effects of interpersonal influence on processes such as information diffusion, influence propagation, and opinion formation. Recent empirical studies indicate that, in order to accurately model communication in online social networks, it is important to consider not just relationships between individuals, but also the frequency with which these individuals interact. We study a model of opinion formation in social networks proposed by De Groot and Lehrer and show how this model can be extended to include interaction frequency. We prove that, for the purposes of analysis and design, the opinion formation process with probabilistic interactions can be accurately approximated by a deterministic system where edge weights are adjusted for the probability of interaction. We also present simulations that illustrate the effects of different interaction frequencies on the opinion dynamics using real-world social network graphs.KeywordsOpinion DynamicsSocial InfluenceComplex NetworkAgent-based Modeling

Bounded confidence opinion dynamics with opinion leaders and environmental noises