scholarly journals The damped pendulum random differential equation: A comprehensive stochastic analysis via the computation of the probability density function

2018 ◽  
Vol 512 ◽  
pp. 261-279 ◽  
Author(s):  
J. Calatayud ◽  
J.-C. Cortés ◽  
M. Jornet
Keyword(s):  
Differential Equation ◽  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Stochastic Analysis ◽  
Random Differential Equation

scholarly journals Numerical simulation of the two-locus Wright-Fisher stochastic differential equation with application to approximating transition probability densities

2020 ◽  
Author(s):  
Zhangyi He ◽  
Mark Beaumont ◽  
Feng Yu
Keyword(s):  
Differential Equation ◽  
Natural Selection ◽  
Probability Density Function ◽  
Stochastic Differential Equation ◽  
Probability Density ◽  
Density Function ◽  
Genetic Recombination ◽  
Transition Probability ◽  
Superior Performance ◽  
Transition Probability Density

AbstractOver the past decade there has been an increasing focus on the application of the Wright-Fisher diffusion to the inference of natural selection from genetic time series. A key ingredient for modelling the trajectory of gene frequencies through the Wright-Fisher diffusion is its transition probability density function. Recent advances in DNA sequencing techniques have made it possible to monitor genomes in great detail over time, which presents opportunities for investigating natural selection while accounting for genetic recombination and local linkage. However, most existing methods for computing the transition probability density function of the Wright-Fisher diffusion are only applicable to one-locus problems. To address two-locus problems, in this work we propose a novel numerical scheme for the Wright-Fisher stochastic differential equation of population dynamics under natural selection at two linked loci. Our key innovation is that we reformulate the stochastic differential equation in a closed form that is amenable to simulation, which enables us to avoid boundary issues and reduce computational costs. We also propose an adaptive importance sampling approach based on the proposal introduced by Fearnhead (2008) for computing the transition probability density of the Wright-Fisher diffusion between any two observed states. We show through extensive simulation studies that our approach can achieve comparable performance to the method of Fearnhead (2008) but can avoid manually tuning the parameter ρ to deliver superior performance for different observed states.

scholarly journals The Impact of Land–Atmosphere Interactions on the Temporal Variability of Soil Moisture at the Regional Scale

2005 ◽  
Vol 6 (1) ◽  
pp. 53-67 ◽  
Author(s):  
John P. Kochendorfer ◽  
Jorge A. Ramírez
Keyword(s):  
Differential Equation ◽  
Soil Moisture ◽  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Temporal Variability ◽  
Regional Scale ◽  
Precipitation Efficiency ◽  
Precipitation Recycling ◽  
The Impact

Abstract This study examines the impact of the nonlinear dynamics of soil-moisture feedbacks to precipitation on the temporal variability of soil moisture at the regional scale. It is a modeling study in which the large-scale soil-water balance is first formulated as an ordinary differential equation and then recast as a stochastic differential equation by incorporating colored noise representing the high-frequency temporal variability and correlation of precipitation. The underlying model couples the atmospheric and surface-water balances and accounts for both precipitation recycling and precipitation-efficiency feedbacks, which arise from the surface energy balance. Based on the governing Fokker–Planck equation, three different analytical solutions (corresponding to differing forms and combinations of feedbacks) are derived for the steady-state probability density function of soil moisture. Using NCEP–NCAR reanalysis data, estimates of potential evapotranspiration, and long-term observations of precipitation, streamflow, and soil moisture, the model is parameterized for a 5° × 5° region encompassing the state of Illinois. It is shown that precipitation-efficiency feedbacks can be significant contributors to the variability of soil moisture at the regional scale. Precipitation recycling, on the other hand, increases the variability by a negligible amount. For all feedback cases, the probability density function is unimodal and nearly symmetric. The analysis concludes with an examination of the dependence of the shape of the probability density functions on spatial scale. It is shown that the associated increases in either the correlation time scale or the variance of the noise will produce a bimodal distribution when precipitation-efficiency feedbacks are included. However, the magnitudes of the necessary increases are of an unrealistic magnitude.

scholarly journals Construction of Reducible Stochastic Differential Equation Systems for Tree Height–Diameter Connections

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1363 ◽  
Author(s):  
Martynas Narmontas ◽  
Petras Rupšys ◽  
Edmundas Petrauskas
Keyword(s):  
Differential Equation ◽  
Probability Density Function ◽  
Stochastic Differential Equation ◽  
Probability Density ◽  
Density Function ◽  
Tree Height ◽  
Bivariate Distribution ◽  
General Purpose ◽  
Mixed Effect ◽  
Bivariate Probability

This study proposes a general bivariate stochastic differential equation model of population growth which includes random forces governing the dynamics of the bivariate distribution of size variables. The dynamics of the bivariate probability density function of the size variables in a population are described by the mixed-effect parameters Vasicek, Gompertz, Bertalanffy, and the gamma-type bivariate stochastic differential equations (SDEs). The newly derived bivariate probability density function and its marginal univariate, as well as the conditional univariate function, can be applied for the modeling of population attributes such as the mean value, quantiles, and much more. The models presented here are the basis for further developments toward the tree diameter–height and height–diameter relationships for general purpose in forest management. The present study experimentally confirms the effectiveness of using bivariate SDEs to reconstruct diameter–height and height–diameter relationships by using measurements obtained from mountain pine tree (Pinus mugo Turra) species dataset in Lithuania.

Wiener–Hermite expansion of a process generated by an Itô stochastic differential equation

1983 ◽  
Vol 20 (04) ◽  
pp. 754-765 ◽  
Author(s):  
Etsuo Isobe ◽  
Shunsuke Sato
Keyword(s):  
Differential Equation ◽  
Probability Density Function ◽  
Stochastic Differential Equation ◽  
Probability Density ◽  
Density Function ◽  
Series Expansion ◽  
Transition Probability ◽  
Transition Probability Density ◽  
Hermite Expansion ◽  
Functional Series

In this paper we deal with the Wiener–Hermite expansion of a process generated by an Itô stochastic differential equation. The so-called Wiener kernels which appear in the functional series expansion are expressed in terms of the transition probability density function of the process.

Wiener–Hermite expansion of a process generated by an Itô stochastic differential equation

1983 ◽  
Vol 20 (4) ◽  
pp. 754-765 ◽  
Author(s):  
Etsuo Isobe ◽  
Shunsuke Sato
Keyword(s):  
Differential Equation ◽  
Probability Density Function ◽  
Stochastic Differential Equation ◽  
Probability Density ◽  
Density Function ◽  
Series Expansion ◽  
Transition Probability ◽  
Transition Probability Density ◽  
Hermite Expansion ◽  
Functional Series

In this paper we deal with the Wiener–Hermite expansion of a process generated by an Itô stochastic differential equation. The so-called Wiener kernels which appear in the functional series expansion are expressed in terms of the transition probability density function of the process.

The Generalized Density Evolution Equation for the Dynamic Analysis of Slender Masonry Structures

2019 ◽  
Vol 817 ◽  
pp. 350-355
Author(s):  
Massimiliano Lucchesi ◽  
Barbara Pintucchi ◽  
Nicola Zani
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Structural Dynamics ◽  
Linear Partial Differential Equation ◽  
Masonry Structures ◽  
Structure Characteristics ◽  
Quantities Of Interest

Within the framework of structural dynamics, the article deals with the problem ofdetermining at a given moment the probability density function of certain quantities of interest,based on the uncertainties about the initial data, the structure characteristics and the applied loads.The proposed method uses the so-called principle of preservation of probability, and leads towriting a linear partial differential equation for any quantity whose probability density function hasto be determined.

MULTIVARIATE PARTIAL DIFFERENTIAL EQUATION DESCRIBING THE EVOLUTION OF A GAUSSIAN PROCESS

2009 ◽  
Vol 09 (04) ◽  
pp. 493-518
Author(s):  
MURAD S. TAQQU ◽  
MARK VEILLETTE
Keyword(s):  
Differential Equation ◽  
Probability Density Function ◽  
Differential Equations ◽  
Gaussian Process ◽  
Probability Density ◽  
Density Function ◽  
Single Equation ◽  
Marginal Probability ◽  
Partial Differential ◽  
Finite Dimensional

If {X(t), t ≥ 0} is a Gaussian process, the diffusion equation characterizes its marginal probability density function. How about finite-dimensional distributions? For each n ≥ 1, we derive a system of partial differential equations which are satisfied by the probability density function of the vector (X(t1), …, X(tn)). We then show that these differential equations determine uniquely the finite-dimensional distributions of Gaussian processes. We also discuss situations where the system can be replaced by a single equation, which is either one member of the system, or an aggregate equation obtained by summing all the equations in the system.

On the Probability Density Function of an Air Conditioner Duty Cycle

1991 ◽  
Author(s):  
A. H. Noureddine ◽  
A. T. Alouani ◽  
S. A. Patil
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Duty Cycle ◽  
Air Conditioner
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