Stochastic Modeling: New Applications
https://www.maplesoft.com/applications/category.aspx?cid=236
en-us2018 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemMon, 22 Jan 2018 22:00:50 GMTMon, 22 Jan 2018 22:00:50 GMTNew applications in the Stochastic Modeling categoryhttps://www.maplesoft.com/images/Application_center_hp.jpgStochastic Modeling: New Applications
https://www.maplesoft.com/applications/category.aspx?cid=236
Jump-diffusion stochastic processes with Maple
https://www.maplesoft.com/applications/view.aspx?SID=153516&ref=Feed
<p>The application presents and definition, creation and handling of jump-diffusion processes. In general, jump-diffusion is an extension to the theory of stochastic processes where the underlying parameters exhibit shocks and "jump" to their new values. Stochasticity with jumps is well recognised in several scientific branches including physics, chemistry, biology, but also economic and finance. The application looks at the example of the last-mentioned fields where the theory of jump-diffusions has been particularly actively researched and applied.</p><img src="https://www.maplesoft.com/view.aspx?si=153516/JD_Process.jpg" alt="Jump-diffusion stochastic processes with Maple" style="max-width: 25%;" align="left"/><p>The application presents and definition, creation and handling of jump-diffusion processes. In general, jump-diffusion is an extension to the theory of stochastic processes where the underlying parameters exhibit shocks and "jump" to their new values. Stochasticity with jumps is well recognised in several scientific branches including physics, chemistry, biology, but also economic and finance. The application looks at the example of the last-mentioned fields where the theory of jump-diffusions has been particularly actively researched and applied.</p>https://www.maplesoft.com/applications/view.aspx?SID=153516&ref=FeedSat, 08 Mar 2014 05:00:00 ZIgor HlivkaIgor HlivkaGreat Expectations
https://www.maplesoft.com/applications/view.aspx?SID=127116&ref=Feed
<p>An investor is offered what appears to be a great investment opportunity. Unfortunately it doesn't turn out to be so great in the long run. This interactive Maple document explores the situation using simulation and analysis, and suggests a new strategy that would produce better results.</p>
<p>This is an example suitable for presentation in an undergraduate course on probability. No knowledge of Maple is required.</p><img src="https://www.maplesoft.com/view.aspx?si=127116/expectation_thum.png" alt="Great Expectations" style="max-width: 25%;" align="left"/><p>An investor is offered what appears to be a great investment opportunity. Unfortunately it doesn't turn out to be so great in the long run. This interactive Maple document explores the situation using simulation and analysis, and suggests a new strategy that would produce better results.</p>
<p>This is an example suitable for presentation in an undergraduate course on probability. No knowledge of Maple is required.</p>https://www.maplesoft.com/applications/view.aspx?SID=127116&ref=FeedThu, 27 Oct 2011 04:00:00 ZCopula function in multivariate dependency analysis
https://www.maplesoft.com/applications/view.aspx?SID=100528&ref=Feed
<p>Copula is a constructor function for multivariate distribution from univariate marginals. It is a method to link univariate samples, not necessarily from identical distributions, into joint multivariate distributions. In this way, copulas are more generic and flexible functions to study dependency arising from multivariate distributions.</p>
<p>Conceptually, copulas are based on transformation of the underlying marginal into new derived variable with uniform distribution. Consequently, any multivariate distribution can be expressed in the form of copula function. If each marginal is continuous then copula is unique. Sklar in 1959 was the first to point this out.</p>
<p>Copulas represent a broad set of functions and they generally differ by (i) number of dependency factors and (ii) construction complexity. The choose of copula depends on the nature of the multivariate study and fitting objectives to an underlying data.</p><img src="https://www.maplesoft.com/view.aspx?si=100528/maple_icon.jpg" alt="Copula function in multivariate dependency analysis" style="max-width: 25%;" align="left"/><p>Copula is a constructor function for multivariate distribution from univariate marginals. It is a method to link univariate samples, not necessarily from identical distributions, into joint multivariate distributions. In this way, copulas are more generic and flexible functions to study dependency arising from multivariate distributions.</p>
<p>Conceptually, copulas are based on transformation of the underlying marginal into new derived variable with uniform distribution. Consequently, any multivariate distribution can be expressed in the form of copula function. If each marginal is continuous then copula is unique. Sklar in 1959 was the first to point this out.</p>
<p>Copulas represent a broad set of functions and they generally differ by (i) number of dependency factors and (ii) construction complexity. The choose of copula depends on the nature of the multivariate study and fitting objectives to an underlying data.</p>https://www.maplesoft.com/applications/view.aspx?SID=100528&ref=FeedWed, 29 Dec 2010 05:00:00 ZI. HlivkaI. HlivkaGeneration of correlated random numbers
https://www.maplesoft.com/applications/view.aspx?SID=99806&ref=Feed
<p>This application is an extension of an earlier document on multivariate distributions and demonstrates how Maple can be used to generate random samples from such distribution. In a narrow sense, it presents the tool for generation of correlated samples. The sampling need for multi-factor random variables (RV) with a given correlation structure arises in many applications in economics, finance, but also in natural sciences such as genetics, physics etc. and here we show that such task can be accomplished with ease using Maple’s <em>Statistic</em>s and <em>Linear Algebra</em> packages.</p><img src="https://www.maplesoft.com/view.aspx?si=99806/maple_icon.jpg" alt="Generation of correlated random numbers" style="max-width: 25%;" align="left"/><p>This application is an extension of an earlier document on multivariate distributions and demonstrates how Maple can be used to generate random samples from such distribution. In a narrow sense, it presents the tool for generation of correlated samples. The sampling need for multi-factor random variables (RV) with a given correlation structure arises in many applications in economics, finance, but also in natural sciences such as genetics, physics etc. and here we show that such task can be accomplished with ease using Maple’s <em>Statistic</em>s and <em>Linear Algebra</em> packages.</p>https://www.maplesoft.com/applications/view.aspx?SID=99806&ref=FeedFri, 03 Dec 2010 05:00:00 ZI. HlivkaI. HlivkaDetermine Integrals by Monte-Carlo method
https://www.maplesoft.com/applications/view.aspx?SID=96010&ref=Feed
<p>We build two procedures to determine approximately single variable and two variable integrals by Monte-Carlo method.</p><img src="https://www.maplesoft.com/applications/images/app_image_blank_lg.jpg" alt="Determine Integrals by Monte-Carlo method" style="max-width: 25%;" align="left"/><p>We build two procedures to determine approximately single variable and two variable integrals by Monte-Carlo method.</p>https://www.maplesoft.com/applications/view.aspx?SID=96010&ref=FeedSat, 14 Aug 2010 04:00:00 ZDuong Ngoc HaoDuong Ngoc HaoMaple for Commodity Finance
https://www.maplesoft.com/applications/view.aspx?SID=35126&ref=Feed
<p>In this application we show how Maple can handle financial options models that have become popular in commodity finance. The financial trading of commodities has dramatically increased over past years as finance community started to look for non-standard instruments uncorrelated with the traditional financial products such as stocks, bonds, rates and currencies. Commodities, unlike their financial counterparts, require different approach to the process modeling: (i) commodities exhibit seasonality effects, (ii) commodity futures are exposed to many deformation modes, (iii) futures volatility is driven by the "Samuelson" effect that causes its drop as the expiry time shortens.</p><img src="https://www.maplesoft.com/view.aspx?si=35126/thumb.jpg" alt="Maple for Commodity Finance" style="max-width: 25%;" align="left"/><p>In this application we show how Maple can handle financial options models that have become popular in commodity finance. The financial trading of commodities has dramatically increased over past years as finance community started to look for non-standard instruments uncorrelated with the traditional financial products such as stocks, bonds, rates and currencies. Commodities, unlike their financial counterparts, require different approach to the process modeling: (i) commodities exhibit seasonality effects, (ii) commodity futures are exposed to many deformation modes, (iii) futures volatility is driven by the "Samuelson" effect that causes its drop as the expiry time shortens.</p>https://www.maplesoft.com/applications/view.aspx?SID=35126&ref=FeedMon, 01 Feb 2010 05:00:00 ZI. HlivkaI. HlivkaStopping Patterns in random Sequences
https://www.maplesoft.com/applications/view.aspx?SID=5643&ref=Feed
Theory for one and two stopping patterns.
Programs for theoretical values and simulation.
Winning ways for a game with two stopping patterns, including graphs and strategy.
Asymptotic values of probabilities for one stopping pattern.
Relations between median and mean waiting times.<img src="https://www.maplesoft.com/view.aspx?si=5643/stoppingpatternsforrandomsequences_344.jpg" alt="Stopping Patterns in random Sequences" style="max-width: 25%;" align="left"/>Theory for one and two stopping patterns.
Programs for theoretical values and simulation.
Winning ways for a game with two stopping patterns, including graphs and strategy.
Asymptotic values of probabilities for one stopping pattern.
Relations between median and mean waiting times.https://www.maplesoft.com/applications/view.aspx?SID=5643&ref=FeedWed, 06 Feb 2008 00:00:00 ZRoland EngdahlRoland EngdahlRanLip - black-box non-uniform random variate generator
https://www.maplesoft.com/applications/view.aspx?SID=4849&ref=Feed
RanLip is a toolbox for generation of nonuniform random variates from arbitrary Lipschitz-continuous distributions in Maple environment. It uses acceptance/ rejection approach, which is based on approximation of the probability density function from above with a "hat" function. RanLip provides very fast preprocessing and generation times, and yields small rejection constant. It exhibits good performance for up to five variables, and provides the user with a black box nonuniform random variate generator for a large class of distributions, in particular, multimodal distributions.<img src="https://www.maplesoft.com/view.aspx?si=4849/ranlib.jpg" alt="RanLip - black-box non-uniform random variate generator" style="max-width: 25%;" align="left"/>RanLip is a toolbox for generation of nonuniform random variates from arbitrary Lipschitz-continuous distributions in Maple environment. It uses acceptance/ rejection approach, which is based on approximation of the probability density function from above with a "hat" function. RanLip provides very fast preprocessing and generation times, and yields small rejection constant. It exhibits good performance for up to five variables, and provides the user with a black box nonuniform random variate generator for a large class of distributions, in particular, multimodal distributions.https://www.maplesoft.com/applications/view.aspx?SID=4849&ref=FeedWed, 13 Dec 2006 00:00:00 ZDr. Gleb BeliakovDr. Gleb BeliakovProbability Games - Parrondo's Paradox
https://www.maplesoft.com/applications/view.aspx?SID=1761&ref=Feed
This Worksheet contains a detailed explanation and analysis of Parrondo's paradox, which states that two losing games can couple into a winning one. It also includes an interactive simulation of the two original games used by J.M.R. Parrondo himself. It is written in Spanish.
Esta hoja contiene un estudio y un analisis detallados sobre la paradoja de Parrondo, que establece que dos juegos desfavorables pueden combinarse y producir resultados favorables para el jugador. Asi mismo, incluye una simulación interactiva de los dos juegos originales usados por J.M.R. Parrondo.<img src="https://www.maplesoft.com/view.aspx?si=1761/Probability Games - Parrondo's Paradox (Spanish)_8.gif" alt="Probability Games - Parrondo's Paradox" style="max-width: 25%;" align="left"/>This Worksheet contains a detailed explanation and analysis of Parrondo's paradox, which states that two losing games can couple into a winning one. It also includes an interactive simulation of the two original games used by J.M.R. Parrondo himself. It is written in Spanish.
Esta hoja contiene un estudio y un analisis detallados sobre la paradoja de Parrondo, que establece que dos juegos desfavorables pueden combinarse y producir resultados favorables para el jugador. Asi mismo, incluye una simulación interactiva de los dos juegos originales usados por J.M.R. Parrondo.https://www.maplesoft.com/applications/view.aspx?SID=1761&ref=FeedWed, 05 Jul 2006 00:00:00 ZMario MartínezMario MartínezRumors: Visualization and Analysis of a Stochastic Process
https://www.maplesoft.com/applications/view.aspx?SID=1702&ref=Feed
Simulation, visualization and analysis of a stochastic process and quantities associated with the process. Analysis of Markov processes using "first-step analysis" and limiting arguments.<img src="https://www.maplesoft.com/view.aspx?si=1702/popnvisual.jpg" alt="Rumors: Visualization and Analysis of a Stochastic Process" style="max-width: 25%;" align="left"/>Simulation, visualization and analysis of a stochastic process and quantities associated with the process. Analysis of Markov processes using "first-step analysis" and limiting arguments.https://www.maplesoft.com/applications/view.aspx?SID=1702&ref=FeedMon, 09 Jan 2006 00:00:00 ZProf. Steven DunbarProf. Steven DunbarIntroduction to Fuzzy Sets on a Real Domain
https://www.maplesoft.com/applications/view.aspx?SID=1410&ref=Feed
The RealDomain subpackage of FuzzySets allows the user to construct and work with fuzzy subsets of the real line.
The membership function of a fuzzy subset of the real line is represented by a piecewise function with a range of [0, 1].
The RealDomain subpackage includes a number of constructors of fuzzy subsets of R which simplify the construction of fuzzy sets and set operators and routines for working with fuzzy subsets of R .<img src="https://www.maplesoft.com/view.aspx?si=1410/FuzzySets_logo.gif" alt="Introduction to Fuzzy Sets on a Real Domain" style="max-width: 25%;" align="left"/>The RealDomain subpackage of FuzzySets allows the user to construct and work with fuzzy subsets of the real line.
The membership function of a fuzzy subset of the real line is represented by a piecewise function with a range of [0, 1].
The RealDomain subpackage includes a number of constructors of fuzzy subsets of R which simplify the construction of fuzzy sets and set operators and routines for working with fuzzy subsets of R .https://www.maplesoft.com/applications/view.aspx?SID=1410&ref=FeedMon, 01 Nov 2004 00:00:00 ZDouglas HarderDouglas HarderMonte Carlo Simulation of the Brownian Bridge
https://www.maplesoft.com/applications/view.aspx?SID=4325&ref=Feed
This is a program that performs a monte carlo approximation of a Brownian path. This is actually termed a Brownian bridge. It functions along the conventionally accepted algorithm (available in much literature I would think)- take the interval (0,1) and succesively bisect. (0,1) -> (0,1/2,1) -> (0,1/4,1/2,3/4,1).... for each interval take the average of the two endpoints then add on a random element that is selected from a normal distribution and zero mean. Then do it again and again. You can change the statistical character. Realize that this has to be done with a normal distribution because: a) it makes physical sense that a particle would tend not to jump very far away in a small number of steps, b) we are actually approximating a functional integral defined for the Wiener measure. If one allows the output to be displayed for many of the lines while using a small number for N, the process will be much more clear than it is possible to describe verbally.
<img src="https://www.maplesoft.com/view.aspx?si=4325//applications/images/app_image_blank_lg.jpg" alt="Monte Carlo Simulation of the Brownian Bridge" style="max-width: 25%;" align="left"/>This is a program that performs a monte carlo approximation of a Brownian path. This is actually termed a Brownian bridge. It functions along the conventionally accepted algorithm (available in much literature I would think)- take the interval (0,1) and succesively bisect. (0,1) -> (0,1/2,1) -> (0,1/4,1/2,3/4,1).... for each interval take the average of the two endpoints then add on a random element that is selected from a normal distribution and zero mean. Then do it again and again. You can change the statistical character. Realize that this has to be done with a normal distribution because: a) it makes physical sense that a particle would tend not to jump very far away in a small number of steps, b) we are actually approximating a functional integral defined for the Wiener measure. If one allows the output to be displayed for many of the lines while using a small number for N, the process will be much more clear than it is possible to describe verbally.
https://www.maplesoft.com/applications/view.aspx?SID=4325&ref=FeedMon, 21 Oct 2002 14:37:06 ZJoseph BassettJoseph BassettStochastic Model of Filling a Box with Spheres
https://www.maplesoft.com/applications/view.aspx?SID=4282&ref=Feed
The task is to determine how many balls will fit into a given size box during the random filling of the box. This Maple document first calculates estimates to establish a lower and upper bound on the solution. Then, a stochastic model is programmed to randomly place balls into the box.
<img src="https://www.maplesoft.com/view.aspx?si=4282//applications/images/app_image_blank_lg.jpg" alt="Stochastic Model of Filling a Box with Spheres" style="max-width: 25%;" align="left"/>The task is to determine how many balls will fit into a given size box during the random filling of the box. This Maple document first calculates estimates to establish a lower and upper bound on the solution. Then, a stochastic model is programmed to randomly place balls into the box.
https://www.maplesoft.com/applications/view.aspx?SID=4282&ref=FeedMon, 24 Jun 2002 14:37:54 ZLee PartinLee PartinMarkov processes
https://www.maplesoft.com/applications/view.aspx?SID=4085&ref=Feed
We will construct transition matrices and Markov chains, automate the transition process, solve for equilibrium vectors, and see what happens visually as an initial vector transitions to new states, and ultimately converges to an equilibrium point.
<img src="https://www.maplesoft.com/view.aspx?si=4085//applications/images/app_image_blank_lg.jpg" alt="Markov processes " style="max-width: 25%;" align="left"/>We will construct transition matrices and Markov chains, automate the transition process, solve for equilibrium vectors, and see what happens visually as an initial vector transitions to new states, and ultimately converges to an equilibrium point.
https://www.maplesoft.com/applications/view.aspx?SID=4085&ref=FeedFri, 17 Aug 2001 12:15:51 ZGregory MooreGregory MooreExplicit solutions of stochastic differential equations (SDEs)
https://www.maplesoft.com/applications/view.aspx?SID=3909&ref=Feed
This file contains instructions and examples which demonstrate the package "stochastic". The package contains commands which can be used to find explicit solutions of Stochastic Differential Equations (SDEs), construct numerical schemes up to strong order 2 and weak order 3.0, check for commutative noise of the first and second kind, and convert SDEs into their coloured noise form.<img src="https://www.maplesoft.com/view.aspx?si=3909//applications/images/app_image_blank_lg.jpg" alt="Explicit solutions of stochastic differential equations (SDEs)" style="max-width: 25%;" align="left"/>This file contains instructions and examples which demonstrate the package "stochastic". The package contains commands which can be used to find explicit solutions of Stochastic Differential Equations (SDEs), construct numerical schemes up to strong order 2 and weak order 3.0, check for commutative noise of the first and second kind, and convert SDEs into their coloured noise form.https://www.maplesoft.com/applications/view.aspx?SID=3909&ref=FeedWed, 20 Jun 2001 00:00:00 ZSasha CyganowskiSasha CyganowskiReliability of a power system: MTBF & availability
https://www.maplesoft.com/applications/view.aspx?SID=3772&ref=Feed
This worksheet demonstrates the use of Maple for computing system reliability and availability for a fault tolerant power system. It illustrates how symbolic expressions for these quantities can be readily calculated using the networks package and a few additional procedures.<img src="https://www.maplesoft.com/view.aspx?si=3772//applications/images/app_image_blank_lg.jpg" alt="Reliability of a power system: MTBF & availability" style="max-width: 25%;" align="left"/>This worksheet demonstrates the use of Maple for computing system reliability and availability for a fault tolerant power system. It illustrates how symbolic expressions for these quantities can be readily calculated using the networks package and a few additional procedures.https://www.maplesoft.com/applications/view.aspx?SID=3772&ref=FeedTue, 19 Jun 2001 00:00:00 ZJoseph RielJoseph RielPod-specific demography of killer-whales
https://www.maplesoft.com/applications/view.aspx?SID=3625&ref=Feed
This paper is an analysis of a stage-based model of the dynamics of killer whales, a modification of the Leslie model. In this case, the assumption is that the stage of an organism (yearling, juvenile, mature female, post-reproductive female) is a better indicator of reproductive performance and survival than age is. This assumption works well for many plants, and any organism in which size is correlated with survival and fecundity<img src="https://www.maplesoft.com/view.aspx?si=3625//applications/images/app_image_blank_lg.jpg" alt="Pod-specific demography of killer-whales" style="max-width: 25%;" align="left"/>This paper is an analysis of a stage-based model of the dynamics of killer whales, a modification of the Leslie model. In this case, the assumption is that the stage of an organism (yearling, juvenile, mature female, post-reproductive female) is a better indicator of reproductive performance and survival than age is. This assumption works well for many plants, and any organism in which size is correlated with survival and fecundityhttps://www.maplesoft.com/applications/view.aspx?SID=3625&ref=FeedMon, 18 Jun 2001 00:00:00 ZMatt MillerMatt MillerDemography of the vegetable ivory palm Pytelephas seemanii in Colombia, and the impact of seed harvesting
https://www.maplesoft.com/applications/view.aspx?SID=3626&ref=Feed
This paper is an analysis of a stage-based model of the dynamics of a palm tree species, using, a modification of the Leslie model. In this case, the assumption is that the stage of an organism (size class) is a better indicator of reproductive performance and survival than age is. This assumption works well for many plants, and any organism in which size is correlated with survival and fecundity. <img src="https://www.maplesoft.com/view.aspx?si=3626//applications/images/app_image_blank_lg.jpg" alt="Demography of the vegetable ivory palm Pytelephas seemanii in Colombia, and the impact of seed harvesting " style="max-width: 25%;" align="left"/>This paper is an analysis of a stage-based model of the dynamics of a palm tree species, using, a modification of the Leslie model. In this case, the assumption is that the stage of an organism (size class) is a better indicator of reproductive performance and survival than age is. This assumption works well for many plants, and any organism in which size is correlated with survival and fecundity. https://www.maplesoft.com/applications/view.aspx?SID=3626&ref=FeedMon, 18 Jun 2001 00:00:00 ZMatt MillerMatt MillerAll terminal reliability polynomial for a graph G with probabilty of edge failure p
https://www.maplesoft.com/applications/view.aspx?SID=3519&ref=Feed
Routine for the all terminal reliability polynomial for a graph G with probability of edge failure<img src="https://www.maplesoft.com/view.aspx?si=3519//applications/images/app_image_blank_lg.jpg" alt="All terminal reliability polynomial for a graph G with probabilty of edge failure p" style="max-width: 25%;" align="left"/>Routine for the all terminal reliability polynomial for a graph G with probability of edge failurehttps://www.maplesoft.com/applications/view.aspx?SID=3519&ref=FeedMon, 18 Jun 2001 00:00:00 ZMichael MonaganMichael Monagan