Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 4702

Search results for: stochastic processes

4702 Gamification Using Stochastic Processes: Engage Children to Have Healthy Habits

Authors: Andre M. Carvalho, Pedro Sebastiao

Abstract:

This article is based on a dissertation that intends to analyze and make a model, intelligently, algorithms based on stochastic processes of a gamification application applied to marketing. Gamification is used in our daily lives to engage us to perform certain actions in order to achieve goals and gain rewards. This strategy is an increasingly adopted way to encourage and retain customers through game elements. The application of gamification aims to encourage children between 6 and 10 years of age to have healthy habits and the purpose of serving as a model for use in marketing. This application was developed in unity; we implemented intelligent algorithms based on stochastic processes, web services to respond to all requests of the application, a back-office website to manage the application and the database. The behavioral analysis of the use of game elements and stochastic processes in children’s motivation was done. The application of algorithms based on stochastic processes in-game elements is very important to promote cooperation and to ensure fair and friendly competition between users which consequently stimulates the user’s interest and their involvement in the application and organization.

Keywords: engage, games, gamification, randomness, stochastic processes

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4701 Numerical Simulation of Wishart Diffusion Processes

Authors: Raphael Naryongo, Philip Ngare, Anthony Waititu

Abstract:

This paper deals with numerical simulation of Wishart processes for a single asset risky pricing model whose volatility is described by Wishart affine diffusion processes. The multi-factor specification of volatility will make the model more flexible enough to fit the stock market data for short or long maturities for better returns. The Wishart process is a stochastic process which is a positive semi-definite matrix-valued generalization of the square root process. The aim of the study is to model the log asset stock returns under the double Wishart stochastic volatility model. The solution of the log-asset return dynamics for Bi-Wishart processes will be obtained through Euler-Maruyama discretization schemes. The numerical results on the asset returns are compared to the existing models returns such as Heston stochastic volatility model and double Heston stochastic volatility model

Keywords: euler schemes, log-asset return, infinitesimal generator, wishart diffusion affine processes

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4700 Weak Solutions Of Stochastic Fractional Differential Equations

Authors: Lev Idels, Arcady Ponosov

Abstract:

Stochastic fractional differential equations have recently attracted considerable attention, as they have been used to model real-world processes, which are subject to natural memory effects and measurement uncertainties. Compared to conventional hereditary differential equations, one of the advantages of fractional differential equations is related to more realistic geometric properties of their trajectories that do not intersect in the phase space. In this report, a Peano-like existence theorem for nonlinear stochastic fractional differential equations is proven under very general hypotheses. Several specific classes of equations are checked to satisfy these hypotheses, including delay equations driven by the fractional Brownian motion, stochastic fractional neutral equations and many others.

Keywords: delay equations, operator methods, stochastic noise, weak solutions

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4699 Dynamical Relation of Poisson Spike Trains in Hodkin-Huxley Neural Ion Current Model and Formation of Non-Canonical Bases, Islands, and Analog Bases in DNA, mRNA, and RNA at or near the Transcription

Authors: Michael Fundator

Abstract:

Groundbreaking application of biomathematical and biochemical research in neural networks processes to formation of non-canonical bases, islands, and analog bases in DNA and mRNA at or near the transcription that contradicts the long anticipated statistical assumptions for the distribution of bases and analog bases compounds is implemented through statistical and stochastic methods apparatus with addition of quantum principles, where the usual transience of Poisson spike train becomes very instrumental tool for finding even almost periodical type of solutions to Fokker-Plank stochastic differential equation. Present article develops new multidimensional methods of finding solutions to stochastic differential equations based on more rigorous approach to mathematical apparatus through Kolmogorov-Chentsov continuity theorem that allows the stochastic processes with jumps under certain conditions to have γ-Holder continuous modification that is used as basis for finding analogous parallels in dynamics of neutral networks and formation of analog bases and transcription in DNA.

Keywords: Fokker-Plank stochastic differential equation, Kolmogorov-Chentsov continuity theorem, neural networks, translation and transcription

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4698 A Study on Stochastic Integral Associated with Catastrophes

Authors: M. Reni Sagayaraj, S. Anand Gnana Selvam, R. Reynald Susainathan

Abstract:

We analyze stochastic integrals associated with a mutation process. To be specific, we describe the cell population process and derive the differential equations for the joint generating functions for the number of mutants and their integrals in generating functions and their applications. We obtain first-order moments of the processes of the two-way mutation process in first-order moment structure of X (t) and Y (t) and the second-order moments of a one-way mutation process. In this paper, we obtain the limiting behaviour of the integrals in limiting distributions of X (t) and Y (t).

Keywords: stochastic integrals, single–server queue model, catastrophes, busy period

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4697 Stochastic Age-Structured Population Models

Authors: Arcady Ponosov

Abstract:

Many well-known age-structured population models are derived from the celebrated McKendrick-von Foerster equation (MFE), also called the biological conservation law. A similar technique is suggested for the stochastically perturbed MFE. This technique is shown to produce stochastic versions of the deterministic population models, which appear to be very different from those one can construct by simply appending additive stochasticity to deterministic equations. In particular, it is shown that stochastic Nicholson’s blowflies model should contain both additive and multiplicative stochastic noises. The suggested transformation technique is similar to that used in the deterministic case. The difference is hidden in the formulas for the exact solutions of the simplified boundary value problem for the stochastically perturbed MFE. The analysis is also based on the theory of stochastic delay differential equations.

Keywords: boundary value problems, population models, stochastic delay differential equations, stochastic partial differential equation

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4696 On Stochastic Models for Fine-Scale Rainfall Based on Doubly Stochastic Poisson Processes

Authors: Nadarajah I. Ramesh

Abstract:

Much of the research on stochastic point process models for rainfall has focused on Poisson cluster models constructed from either the Neyman-Scott or Bartlett-Lewis processes. The doubly stochastic Poisson process provides a rich class of point process models, especially for fine-scale rainfall modelling. This paper provides an account of recent development on this topic and presents the results based on some of the fine-scale rainfall models constructed from this class of stochastic point processes. Amongst the literature on stochastic models for rainfall, greater emphasis has been placed on modelling rainfall data recorded at hourly or daily aggregation levels. Stochastic models for sub-hourly rainfall are equally important, as there is a need to reproduce rainfall time series at fine temporal resolutions in some hydrological applications. For example, the study of climate change impacts on hydrology and water management initiatives requires the availability of data at fine temporal resolutions. One approach to generating such rainfall data relies on the combination of an hourly stochastic rainfall simulator, together with a disaggregator making use of downscaling techniques. Recent work on this topic adopted a different approach by developing specialist stochastic point process models for fine-scale rainfall aimed at generating synthetic precipitation time series directly from the proposed stochastic model. One strand of this approach focused on developing a class of doubly stochastic Poisson process (DSPP) models for fine-scale rainfall to analyse data collected in the form of rainfall bucket tip time series. In this context, the arrival pattern of rain gauge bucket tip times N(t) is viewed as a DSPP whose rate of occurrence varies according to an unobserved finite state irreducible Markov process X(t). Since the likelihood function of this process can be obtained, by conditioning on the underlying Markov process X(t), the models were fitted with maximum likelihood methods. The proposed models were applied directly to the raw data collected by tipping-bucket rain gauges, thus avoiding the need to convert tip-times to rainfall depths prior to fitting the models. One advantage of this approach was that the use of maximum likelihood methods enables a more straightforward estimation of parameter uncertainty and comparison of sub-models of interest. Another strand of this approach employed the DSPP model for the arrivals of rain cells and attached a pulse or a cluster of pulses to each rain cell. Different mechanisms for the pattern of the pulse process were used to construct variants of this model. We present the results of these models when they were fitted to hourly and sub-hourly rainfall data. The results of our analysis suggest that the proposed class of stochastic models is capable of reproducing the fine-scale structure of the rainfall process, and hence provides a useful tool in hydrological modelling.

Keywords: fine-scale rainfall, maximum likelihood, point process, stochastic model

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4695 Non-Stationary Stochastic Optimization of an Oscillating Water Column

Authors: María L. Jalón, Feargal Brennan

Abstract:

A non-stationary stochastic optimization methodology is applied to an OWC (oscillating water column) to find the design that maximizes the wave energy extraction. Different temporal cycles are considered to represent the long-term variability of the wave climate at the site in the optimization problem. The results of the non-stationary stochastic optimization problem are compared against those obtained by a stationary stochastic optimization problem. The comparative analysis reveals that the proposed non-stationary optimization provides designs with a better fit to reality. However, the stationarity assumption can be adequate when looking at averaged system response.

Keywords: non-stationary stochastic optimization, oscillating water, temporal variability, wave energy

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4694 Lyapunov and Input-to-State Stability of Stochastic Differential Equations

Authors: Arcady Ponosov, Ramazan Kadiev

Abstract:

Input-to-State Stability (ISS) is widely used in deterministic control theory but less known in the stochastic case. Roughly speaking, the theory explains when small perturbations of the right-hand sides of the system on the entire semiaxis cause only small changes in the solutions of the system, again on the entire semiaxis. This property is crucial in many applications. In the report, we explain how to define and study ISS for systems of linear stochastic differential equations with or without delays. The central result connects ISS with the property of Lyapunov stability. This relationship is well-known in the deterministic setting, but its stochastic version is new. As an application, a method of studying asymptotic Lyapunov stability for stochastic delay equations is described and justified. Several examples are provided that confirm the efficiency and simplicity of the framework.

Keywords: asymptotic stability, delay equations, operator methods, stochastic perturbations

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4693 A Review on Enhancing Heat Transfer Processes by Open-Cell Metal Foams and Industrial Applications

Authors: S. Cheragh Dar, M. Saljooghi, A. Babrgir

Abstract:

In the last couple of decades researchers' attitudes were focused on developing and enhancing heat transfer processes by using new components or cellular solids that divide into stochastic structures and periodic structures. Open-cell metal foams are part of stochastic structures families that they can be considered as an avant-garde technology and they have unique properties, this porous media can have tremendous achievements in thermal processes. This paper argues and surveys postulating possible in industrial thermal issues which include: compact electronic cooling, heat exchanger, aerospace, fines, turbo machinery, automobiles, crygen tanks, biomechanics, high temperature filters and etc. Recently, by surveying exponential rate of publications in thermal open-cell metal foams, all can be demonstrated in a holistic view which can lead researchers to a new level of understanding in different industrial thermal sections.

Keywords: heat transfer, industrial thermal, cellular solids, open cell metal foam

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4692 On Stability of Stochastic Differential Equations with Non Trivial Solutions

Authors: Fakhreddin Abedi, Wah June Leong

Abstract:

Exponential stability of stochastic differential equations with non-trivial solutions is provided in terms of Lyapunov functions. The main result of this paper establishes that, under certain hypotheses for the dynamics f (.) and g(.), practical exponential stability in probability at the small neighborhood of the origin is equivalent to the existence of an appropriate Lyapunov function. Indeed, we establish exponential stability of stochastic differential equations when almost all the state trajectories are bounded and approach a sufficiently small neighborhood of the origin. We derive sufficient conditions for the exponential stability of stochastic differential equations. Finally, we give a numerical example illustrating our results.

Keywords: exponential stability in probability, stochastic differential equations, Lyapunov technique, Ito’s formula

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4691 A Reactive Flexible Job Shop Scheduling Model in a Stochastic Environment

Authors: Majid Khalili, Hamed Tayebi

Abstract:

This paper considers a stochastic flexible job-shop scheduling (SFJSS) problem in the presence of production disruptions, and reactive scheduling is implemented in order to find the optimal solution under uncertainty. In this problem, there are two main disruptions including machine failure which influences operation time, and modification or cancellation of the order delivery date during production. In order to decrease the negative effects of these difficulties, two derived strategies from reactive scheduling are used; the first one is relevant to being able to allocate multiple machine to each job, and the other one is related to being able to select the best alternative process from other job while some disruptions would be created in the processes of a job. For this purpose, a Mixed Integer Linear Programming model is proposed.

Keywords: flexible job-shop scheduling, reactive scheduling, stochastic environment, mixed integer linear programming

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4690 Stochastic Energy and Reserve Scheduling with Wind Generation and Generic Energy Storage Systems

Authors: Amirhossein Khazali, Mohsen Kalantar

Abstract:

Energy storage units can play an important role to provide an economic and secure operation of future energy systems. In this paper, a stochastic energy and reserve market clearing scheme is presented considering storage energy units. The approach is proposed to deal with stochastic and non-dispatchable renewable sources with a high level of penetration in the energy system. A two stage stochastic programming scheme is formulated where in the first stage the energy market is cleared according to the forecasted amount of wind generation and demands and in the second stage the real time market is solved according to the assumed scenarios.

Keywords: energy and reserve market, energy storage device, stochastic programming, wind generation

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4689 Finding DEA Targets Using Multi-Objective Programming

Authors: Farzad Sharifi, Raziyeh Shamsi

Abstract:

In this paper, we obtain the projection of inefficient units in data envelopment analysis (DEA) in the case of stochastic inputs and outputs using the multi-objective programming (MOP) structure. In some problems, the inputs might be stochastic while the outputs are deterministic, and vice versa. In such cases, we propose molti-objective DEA-R model, because in some cases (e.g., when unnecessary and irrational weights by the BCC model reduces the efficiency score), an efficient DMU is introduced as inefficient by the BCC model, whereas the DMU is considered efficient by the DEA-R model. In some other case, only the ratio of stochastic data may be available (e.g; the ratio of stochastic inputs to stochastic outputs). Thus, we provide multi objective DEA model without explicit outputs and prove that in-put oriented MOP DEA-R model in the invariable return to scale case can be replacing by MOP- DEA model without explicit outputs in the variable return to scale and vice versa. Using the interactive methods for solving the proposed model, yields a projection corresponding to the viewpoint of the DM and the analyst, which is nearer to reality and more practical. Finally, an application is provided.

Keywords: DEA, MOLP, STOCHASTIC, DEA-R

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4688 A Multi-Objective Programming Model to Supplier Selection and Order Allocation Problem in Stochastic Environment

Authors: Rouhallah Bagheri, Morteza Mahmoudi, Hadi Moheb-Alizadeh

Abstract:

This paper aims at developing a multi-objective model for supplier selection and order allocation problem in stochastic environment, where purchasing cost, percentage of delivered items with delay and percentage of rejected items provided by each supplier are supposed to be stochastic parameters following any arbitrary probability distribution. In this regard, dependent chance programming is used which maximizes probability of the event that total purchasing cost, total delivered items with delay and total rejected items are less than or equal to pre-determined values given by decision maker. The abovementioned stochastic multi-objective programming problem is then transformed into a stochastic single objective programming problem using minimum deviation method. In the next step, the further problem is solved applying a genetic algorithm, which performs a simulation process in order to calculate the stochastic objective function as its fitness function. Finally, the impact of stochastic parameters on the given solution is examined via a sensitivity analysis exploiting coefficient of variation. The results show that whatever stochastic parameters have greater coefficients of variation, the value of the objective function in the stochastic single objective programming problem is deteriorated.

Keywords: supplier selection, order allocation, dependent chance programming, genetic algorithm

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4687 Supplier Selection and Order Allocation Using a Stochastic Multi-Objective Programming Model and Genetic Algorithm

Authors: Rouhallah Bagheri, Morteza Mahmoudi, Hadi Moheb-Alizadeh

Abstract:

In this paper, we develop a supplier selection and order allocation multi-objective model in stochastic environment in which purchasing cost, percentage of delivered items with delay and percentage of rejected items provided by each supplier are supposed to be stochastic parameters following any arbitrary probability distribution. To do so, we use dependent chance programming (DCP) that maximizes probability of the event that total purchasing cost, total delivered items with delay and total rejected items are less than or equal to pre-determined values given by decision maker. After transforming the above mentioned stochastic multi-objective programming problem into a stochastic single objective problem using minimum deviation method, we apply a genetic algorithm to get the later single objective problem solved. The employed genetic algorithm performs a simulation process in order to calculate the stochastic objective function as its fitness function. At the end, we explore the impact of stochastic parameters on the given solution via a sensitivity analysis exploiting coefficient of variation. The results show that as stochastic parameters have greater coefficients of variation, the value of objective function in the stochastic single objective programming problem is worsened.

Keywords: dependent chance programming, genetic algorithm, minimum deviation method, order allocation, supplier selection

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4686 Stochastic Variation of the Hubble's Parameter Using Ornstein-Uhlenbeck Process

Authors: Mary Chriselda A

Abstract:

This paper deals with the fact that the Hubble's parameter is not constant and tends to vary stochastically with time. This premise has been proven by converting it to a stochastic differential equation using the Ornstein-Uhlenbeck process. The formulated stochastic differential equation is further solved analytically using the Euler and the Kolmogorov Forward equations, thereby obtaining the probability density function using the Fourier transformation, thereby proving that the Hubble's parameter varies stochastically. This is further corroborated by simulating the observations using Python and R-software for validation of the premise postulated. We can further draw conclusion that the randomness in forces affecting the white noise can eventually affect the Hubble’s Parameter leading to scale invariance and thereby causing stochastic fluctuations in the density and the rate of expansion of the Universe.

Keywords: Chapman Kolmogorov forward differential equations, fourier transformation, hubble's parameter, ornstein-uhlenbeck process , stochastic differential equations

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4685 Stochastic Prioritization of Dependent Actuarial Risks: Preferences among Prospects

Authors: Ezgi Nevruz, Kasirga Yildirak, Ashis SenGupta

Abstract:

Comparing or ranking risks is the main motivating factor behind the human trait of making choices. Cumulative prospect theory (CPT) is a preference theory approach that evaluates perception and bias in decision making under risk and uncertainty. We aim to investigate the aggregate claims of different risk classes in terms of their comparability and amenability to ordering when the impact of risk perception is considered. For this aim, we prioritize the aggregate claims taken as actuarial risks by using various stochastic ordering relations. In order to prioritize actuarial risks, we use stochastic relations such as stochastic dominance and stop-loss dominance that are proposed in the frame of partial order theory. We take into account the dependency of the individual claims exposed to similar environmental risks. At first, we modify the zero-utility premium principle in order to obtain a solution for the stop-loss premium under CPT. Then, we propose a stochastic stop-loss dominance of the aggregate claims and find a relation between the stop-loss dominance and the first-order stochastic dominance under the dependence assumption by using properties of the familiar as well as some emerging multivariate claim distributions.

Keywords: cumulative prospect theory, partial order theory, risk perception, stochastic dominance, stop-loss dominance

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4684 Formulating the Stochastic Finite Elements for Free Vibration Analysis of Plates with Variable Elastic Modulus

Authors: Mojtaba Aghamiri Esfahani, Mohammad Karkon, Seyed Majid Hosseini Nezhad, Reza Hosseini-Ara

Abstract:

In this study, the effect of uncertainty in elastic modulus of a plate on free vibration response is investigated. For this purpose, the elastic modulus of the plate is modeled as stochastic variable with normal distribution. Moreover, the distance autocorrelation function is used for stochastic field. Then, by applying the finite element method and Monte Carlo simulation, stochastic finite element relations are extracted. Finally, with a numerical test, the effect of uncertainty in the elastic modulus on free vibration response of a plate is studied. The results show that the effect of uncertainty in elastic modulus of the plate cannot play an important role on the free vibration response.

Keywords: stochastic finite elements, plate bending, free vibration, Monte Carlo, Neumann expansion method.

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4683 Portfolio Optimization under a Hybrid Stochastic Volatility and Constant Elasticity of Variance Model

Authors: Jai Heui Kim, Sotheara Veng

Abstract:

This paper studies the portfolio optimization problem for a pension fund under a hybrid model of stochastic volatility and constant elasticity of variance (CEV) using asymptotic analysis method. When the volatility component is fast mean-reverting, it is able to derive asymptotic approximations for the value function and the optimal strategy for general utility functions. Explicit solutions are given for the exponential and hyperbolic absolute risk aversion (HARA) utility functions. The study also shows that using the leading order optimal strategy results in the value function, not only up to the leading order, but also up to first order correction term. A practical strategy that does not depend on the unobservable volatility level is suggested. The result is an extension of the Merton's solution when stochastic volatility and elasticity of variance are considered simultaneously.

Keywords: asymptotic analysis, constant elasticity of variance, portfolio optimization, stochastic optimal control, stochastic volatility

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4682 Use the Null Space to Create Starting Point for Stochastic Programming

Authors: Ghussoun Al-Jeiroudi

Abstract:

Stochastic programming is one of the powerful technique which is used to solve real-life problems. Hence, the data of real-life problems is subject to significant uncertainty. Uncertainty is well studied and modeled by stochastic programming. Each day, problems become bigger and bigger and the need for a tool, which does deal with large scale problems, increase. Interior point method is a perfect tool to solve such problems. Interior point method is widely employed to solve the programs, which arise from stochastic programming. It is an iterative technique, so it is required a starting point. Well design starting point plays an important role in improving the convergence speed. In this paper, we propose a starting point for interior point method for multistage stochastic programming. Usually, the optimal solution of stage k+1 is used as starting point for the stage k. This point has the advantage of being close to the solution of the current program. However, it has a disadvantage; it is not in the feasible region of the current program. So, we suggest to take this point and modifying it. That is by adding to it a vector in the null space of the matrix of the unchanged constraints because the solution will change only in the null space of this matrix.

Keywords: interior point methods, stochastic programming, null space, starting points

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4681 Stochastic Programming and C-Somga: Animal Ration Formulation

Authors: Pratiksha Saxena, Dipti Singh, Neha Khanna

Abstract:

A self-organizing migrating genetic algorithm(C-SOMGA) is developed for animal diet formulation. This paper presents animal diet formulation using stochastic and genetic algorithm. Tri-objective models for cost minimization and shelf life maximization are developed. These objectives are achieved by combination of stochastic programming and C-SOMGA. Stochastic programming is used to introduce nutrient variability for animal diet. Self-organizing migrating genetic algorithm provides exact and quick solution and presents an innovative approach towards successful application of soft computing technique in the area of animal diet formulation.

Keywords: animal feed ration, feed formulation, linear programming, stochastic programming, self-migrating genetic algorithm, C-SOMGA technique, shelf life maximization, cost minimization, nutrient maximization

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4680 Identification of Wiener Model Using Iterative Schemes

Authors: Vikram Saini, Lillie Dewan

Abstract:

This paper presents the iterative schemes based on Least square, Hierarchical Least Square and Stochastic Approximation Gradient method for the Identification of Wiener model with parametric structure. A gradient method is presented for the parameter estimation of wiener model with noise conditions based on the stochastic approximation. Simulation results are presented for the Wiener model structure with different static non-linear elements in the presence of colored noise to show the comparative analysis of the iterative methods. The stochastic gradient method shows improvement in the estimation performance and provides fast convergence of the parameters estimates.

Keywords: hard non-linearity, least square, parameter estimation, stochastic approximation gradient, Wiener model

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4679 Prediction of Gully Erosion with Stochastic Modeling by using Geographic Information System and Remote Sensing Data in North of Iran

Authors: Reza Zakerinejad

Abstract:

Gully erosion is a serious problem that threading the sustainability of agricultural area and rangeland and water in a large part of Iran. This type of water erosion is the main source of sedimentation in many catchment areas in the north of Iran. Since in many national assessment approaches just qualitative models were applied the aim of this study is to predict the spatial distribution of gully erosion processes by means of detail terrain analysis and GIS -based logistic regression in the loess deposition in a case study in the Golestan Province. This study the DEM with 25 meter result ion from ASTER data has been used. The Landsat ETM data have been used to mapping of land use. The TreeNet model as a stochastic modeling was applied to prediction the susceptible area for gully erosion. In this model ROC we have set 20 % of data as learning and 20 % as learning data. Therefore, applying the GIS and satellite image analysis techniques has been used to derive the input information for these stochastic models. The result of this study showed a high accurate map of potential for gully erosion.

Keywords: TreeNet model, terrain analysis, Golestan Province, Iran

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4678 Closed-Form Sharma-Mittal Entropy Rate for Gaussian Processes

Authors: Septimia Sarbu

Abstract:

The entropy rate of a stochastic process is a fundamental concept in information theory. It provides a limit to the amount of information that can be transmitted reliably over a communication channel, as stated by Shannon's coding theorems. Recently, researchers have focused on developing new measures of information that generalize Shannon's classical theory. The aim is to design more efficient information encoding and transmission schemes. This paper continues the study of generalized entropy rates, by deriving a closed-form solution to the Sharma-Mittal entropy rate for Gaussian processes. Using the squeeze theorem, we solve the limit in the definition of the entropy rate, for different values of alpha and beta, which are the parameters of the Sharma-Mittal entropy. In the end, we compare it with Shannon and Rényi's entropy rates for Gaussian processes.

Keywords: generalized entropies, Sharma-Mittal entropy rate, Gaussian processes, eigenvalues of the covariance matrix, squeeze theorem

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4677 Method to Find a ε-Optimal Control of Stochastic Differential Equation Driven by a Brownian Motion

Authors: Francys Souza, Alberto Ohashi, Dorival Leao

Abstract:

We present a general solution for finding the ε-optimal controls for non-Markovian stochastic systems as stochastic differential equations driven by Brownian motion, which is a problem recognized as a difficult solution. The contribution appears in the development of mathematical tools to deal with modeling and control of non-Markovian systems, whose applicability in different areas is well known. The methodology used consists to discretize the problem through a random discretization. In this way, we transform an infinite dimensional problem in a finite dimensional, thereafter we use measurable selection arguments, to find a control on an explicit form for the discretized problem. Then, we prove the control found for the discretized problem is a ε-optimal control for the original problem. Our theory provides a concrete description of a rather general class, among the principals, we can highlight financial problems such as portfolio control, hedging, super-hedging, pairs-trading and others. Therefore, our main contribution is the development of a tool to explicitly the ε-optimal control for non-Markovian stochastic systems. The pathwise analysis was made through a random discretization jointly with measurable selection arguments, has provided us with a structure to transform an infinite dimensional problem into a finite dimensional. The theory is applied to stochastic control problems based on path-dependent stochastic differential equations, where both drift and diffusion components are controlled. We are able to explicitly show optimal control with our method.

Keywords: dynamic programming equation, optimal control, stochastic control, stochastic differential equation

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4676 Stability of Stochastic Model Predictive Control for Schrödinger Equation with Finite Approximation

Authors: Tomoaki Hashimoto

Abstract:

Recent technological advance has prompted significant interest in developing the control theory of quantum systems. Following the increasing interest in the control of quantum dynamics, this paper examines the control problem of Schrödinger equation because quantum dynamics is basically governed by Schrödinger equation. From the practical point of view, stochastic disturbances cannot be avoided in the implementation of control method for quantum systems. Thus, we consider here the robust stabilization problem of Schrödinger equation against stochastic disturbances. In this paper, we adopt model predictive control method in which control performance over a finite future is optimized with a performance index that has a moving initial and terminal time. The objective of this study is to derive the stability criterion for model predictive control of Schrödinger equation under stochastic disturbances.

Keywords: optimal control, stochastic systems, quantum systems, stabilization

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4675 Optimal Production and Maintenance Policy for a Partially Observable Production System with Stochastic Demand

Authors: Leila Jafari, Viliam Makis

Abstract:

In this paper, the joint optimization of the economic manufacturing quantity (EMQ), safety stock level, and condition-based maintenance (CBM) is presented for a partially observable, deteriorating system subject to random failure. The demand is stochastic and it is described by a Poisson process. The stochastic model is developed and the optimization problem is formulated in the semi-Markov decision process framework. A modification of the policy iteration algorithm is developed to find the optimal policy. A numerical example is presented to compare the optimal policy with the policy considering zero safety stock.

Keywords: condition-based maintenance, economic manufacturing quantity, safety stock, stochastic demand

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4674 Analysis of Two Methods to Estimation Stochastic Demand in the Vehicle Routing Problem

Authors: Fatemeh Torfi

Abstract:

Estimation of stochastic demand in physical distribution in general and efficient transport routs management in particular is emerging as a crucial factor in urban planning domain. It is particularly important in some municipalities such as Tehran where a sound demand management calls for a realistic analysis of the routing system. The methodology involved critically investigating a fuzzy least-squares linear regression approach (FLLRs) to estimate the stochastic demands in the vehicle routing problem (VRP) bearing in mind the customer's preferences order. A FLLR method is proposed in solving the VRP with stochastic demands. Approximate-distance fuzzy least-squares (ADFL) estimator ADFL estimator is applied to original data taken from a case study. The SSR values of the ADFL estimator and real demand are obtained and then compared to SSR values of the nominal demand and real demand. Empirical results showed that the proposed methods can be viable in solving problems under circumstances of having vague and imprecise performance ratings. The results further proved that application of the ADFL was realistic and efficient estimator to face the stochastic demand challenges in vehicle routing system management and solve relevant problems.

Keywords: fuzzy least-squares, stochastic, location, routing problems

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4673 CE Method for Development of Japan's Stochastic Earthquake Catalogue

Authors: Babak Kamrani, Nozar Kishi

Abstract:

Stochastic catalog represents the events module of the earthquake loss estimation models. It includes series of events with different magnitudes and corresponding frequencies/probabilities. For the development of the stochastic catalog, random or uniform sampling methods are used to sample the events from the seismicity model. For covering all the Magnitude Frequency Distribution (MFD), a huge number of events should be generated for the above-mentioned methods. Characteristic Event (CE) method chooses the events based on the interest of the insurance industry. We divide the MFD of each source into bins. We have chosen the bins based on the probability of the interest by the insurance industry. First, we have collected the information for the available seismic sources. Sources are divided into Fault sources, subduction, and events without specific fault source. We have developed the MFD for each of the individual and areal source based on the seismicity of the sources. Afterward, we have calculated the CE magnitudes based on the desired probability. To develop the stochastic catalog, we have introduced uncertainty to the location of the events too.

Keywords: stochastic catalogue, earthquake loss, uncertainty, characteristic event

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