This document discusses moment closure inference for stochastic kinetic models. It begins with an introduction to moment closure techniques using a simple birth-death process as a case study. It then discusses how to derive moment equations from the chemical master equation and how the deterministic model can be viewed as an approximation of the stochastic model by setting the variance to zero. The document also examines some limitations of moment closure approximations using examples of heat shock and p53-Mdm2 oscillation models. Finally, it presents a case study of using moment closure to model cotton aphid populations based on field data.
This document discusses harmonic and subharmonic functions. It begins by defining harmonic functions and providing examples to show properties like the maximum/minimum principle and mean-value property. It then introduces the Dirichlet problem for the disk and proves there is always a solution. Harnack's inequality relating values of positive harmonic functions is presented. Finally, it defines subharmonic functions and provides examples to show properties like the submean-value property.
The document summarizes key concepts of quantum mechanics from chapters 3 and 4 of McQuarrie, including:
1) The Schrodinger equation and its solutions in 1D and 3D.
2) Solving the time-independent Schrodinger equation involves finding the general solution, applying boundary conditions, and normalizing the wavefunction.
3) Wavefunctions represent probability distributions, and the probability of finding a particle in a region is calculated by integrating the wavefunction.
On Application of the Fixed-Point Theorem to the Solution of Ordinary Differe...BRNSS Publication Hub
We know that a large number of problems in differential equations can be reduced to finding the solution x to an equation of the form Tx=y. The operator T maps a subset of a Banach space X into another Banach space Y and y is a known element of Y. If y=0 and Tx=Ux−x, for another operator U, the equation Tx=y is equivalent to the equation Ux=x. Naturally, to solve Ux=x, we must assume that the range R (U) and the domain D (U) have points in common. Points x for which Ux=x are called fixed points of the operator U. In this work, we state the main fixed-point theorems that are most widely used in the field of differential equations. These are the Banach contraction principle, the Schauder–Tychonoff theorem, and the Leray–Schauder theorem. We will only prove the first theorem and then proceed.
Electromagnetic Scattering from Objects with Thin Coatings.2016.05.04.02Luke Underwood
This document discusses a numerical method for modeling electromagnetic scattering from objects with thin coatings. It presents a method that properly weights the integration of near-singular behaviors introduced by thin coatings. The weighting is applied to the Fourier coefficients of Dirac delta functions. This allows the modeling of how varying a thin coating's properties alters an object's far-field scattering pattern.
The document summarizes numerical integration methods for solving equations of motion directly in the time domain, including explicit and implicit methods. It describes Newmark's β method, the central difference method, and Wilson-θ method. Key steps involve discretizing the equations of motion and relating response parameters at different time steps using finite difference approximations. Stability, accuracy, and error considerations are also discussed.
I am Grey N. I am a Physical Chemistry Assignment Expert at eduassignmenthelp.com. I hold a Ph.D. in Physical Chemistry, from Calgary, Canada. I have been helping students with their homework for the past 6 years. I solve assignments related to Physical Chemistry.
Visit eduassignmenthelp.com or email info@eduassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Physical Chemistry Assignments.
This document discusses the application of partial differential equations. It begins by classifying partial differential equations according to their mathematical form as either boundary value problems or steady-state equations. Some common partial differential equations are then presented, including the wave equation, heat equations, and Laplace's equation. Solution methods like separation of variables are introduced. Specific examples of the 1D wave equation and 1D heat equation are then covered. Finally, the document discusses the Laplace equation in 2D and 3D.
This document discusses harmonic and subharmonic functions. It begins by defining harmonic functions and providing examples to show properties like the maximum/minimum principle and mean-value property. It then introduces the Dirichlet problem for the disk and proves there is always a solution. Harnack's inequality relating values of positive harmonic functions is presented. Finally, it defines subharmonic functions and provides examples to show properties like the submean-value property.
The document summarizes key concepts of quantum mechanics from chapters 3 and 4 of McQuarrie, including:
1) The Schrodinger equation and its solutions in 1D and 3D.
2) Solving the time-independent Schrodinger equation involves finding the general solution, applying boundary conditions, and normalizing the wavefunction.
3) Wavefunctions represent probability distributions, and the probability of finding a particle in a region is calculated by integrating the wavefunction.
On Application of the Fixed-Point Theorem to the Solution of Ordinary Differe...BRNSS Publication Hub
We know that a large number of problems in differential equations can be reduced to finding the solution x to an equation of the form Tx=y. The operator T maps a subset of a Banach space X into another Banach space Y and y is a known element of Y. If y=0 and Tx=Ux−x, for another operator U, the equation Tx=y is equivalent to the equation Ux=x. Naturally, to solve Ux=x, we must assume that the range R (U) and the domain D (U) have points in common. Points x for which Ux=x are called fixed points of the operator U. In this work, we state the main fixed-point theorems that are most widely used in the field of differential equations. These are the Banach contraction principle, the Schauder–Tychonoff theorem, and the Leray–Schauder theorem. We will only prove the first theorem and then proceed.
Electromagnetic Scattering from Objects with Thin Coatings.2016.05.04.02Luke Underwood
This document discusses a numerical method for modeling electromagnetic scattering from objects with thin coatings. It presents a method that properly weights the integration of near-singular behaviors introduced by thin coatings. The weighting is applied to the Fourier coefficients of Dirac delta functions. This allows the modeling of how varying a thin coating's properties alters an object's far-field scattering pattern.
The document summarizes numerical integration methods for solving equations of motion directly in the time domain, including explicit and implicit methods. It describes Newmark's β method, the central difference method, and Wilson-θ method. Key steps involve discretizing the equations of motion and relating response parameters at different time steps using finite difference approximations. Stability, accuracy, and error considerations are also discussed.
I am Grey N. I am a Physical Chemistry Assignment Expert at eduassignmenthelp.com. I hold a Ph.D. in Physical Chemistry, from Calgary, Canada. I have been helping students with their homework for the past 6 years. I solve assignments related to Physical Chemistry.
Visit eduassignmenthelp.com or email info@eduassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Physical Chemistry Assignments.
This document discusses the application of partial differential equations. It begins by classifying partial differential equations according to their mathematical form as either boundary value problems or steady-state equations. Some common partial differential equations are then presented, including the wave equation, heat equations, and Laplace's equation. Solution methods like separation of variables are introduced. Specific examples of the 1D wave equation and 1D heat equation are then covered. Finally, the document discusses the Laplace equation in 2D and 3D.
1. The document discusses differential equations, which relate functions and their derivatives. First order equations relate a function to its first derivative, while second order equations relate it to its second derivative.
2. Differential equations are used in physics, such as Newton's second law relating force, mass and acceleration. They can have many solutions or no solutions. Simultaneous differential equations involve multiple dependent variables.
3. Examples of simultaneous differential equations include models of survival with AIDS, earthquake effects on buildings with multiple floors, and harvesting renewable resources like fish populations over time.
3 examples of PDE, for Laplace, Diffusion of Heat and Wave function. A brief definition of Fouriers Series. Slides created and compiled using LaTeX, beamer package.
I. A power series is a polynomial with infinitely many terms of the form Σn=0∞anxn.
II. The radius of convergence R determines the values of x where a power series converges absolutely (for |x|<R), diverges (for |x|>R), or may converge or diverge (for |x|=R).
III. Tests like the ratio test and root test can be used to calculate the radius of convergence R.
International journal of engineering issues vol 2015 - no 2 - paper5sophiabelthome
This document summarizes key duality theorems for optimization problems involving n-set functions:
1) The weak duality theorem shows that the objective value of any feasible dual problem solution provides a lower bound on the objective of any primal problem solution.
2) The strong duality theorem establishes that under suitable convexity conditions, the optimal objective values of the primal and dual problems are equal.
3) Mangasarian's strict converse duality theorem proves that if a solution satisfies the Kuhn-Tucker conditions and strong duality holds, then the objective function is strictly convex at that solution.
1. A bi-variate random variable has a joint probability distribution function (PDF) that defines the probability of two random variables occurring together. The marginal PDF defines the probability of each variable individually, while the conditional PDF defines the probability of one variable given the other.
2. A multi-variate random variable contains multiple random variables defined by a mean vector and covariance matrix. A linear transformation of a Gaussian multi-variate random variable remains Gaussian with a transformed mean vector and covariance matrix.
3. If two random vectors are jointly Gaussian and uncorrelated, they are also independent, as their joint PDF can be written as the product of their individual PDFs.
How to Solve a Partial Differential Equation on a surfacetr1987
Familiar techniques of separation of variables and Fourier series can be used to solve a variety of pde based on domains in the plane, however these techniques do not extend naturally to surface problems. Instead we look to take a computational approach. The talk will cover the basics of finite difference and finite element approximations of the one dimensional heat equation and show how to extend these ideas on to surfaces. If time allows, we will show numerical results of an optimal partition problem based on a sphere. No background knowledge of pde or computation is required.
This document discusses fuzzy relations, reasoning, and linguistic variables. It defines fuzzy relations as membership functions between elements of Cartesian product spaces. It describes the extension principle for mapping fuzzy sets through functions. Max-min and max-product composition are defined for combining fuzzy relations. Linguistic variables allow information to be expressed using fuzzy linguistic terms rather than numerical values. Operations on linguistic variables like concentration and dilation are discussed. Fuzzy if-then rules are defined using implication functions to model "if A then B" statements where A and B are linguistic values. Fuzzy reasoning uses these rules and facts to derive conclusions.
This document discusses ordinary differential equations (ODEs). It defines ODEs and differentiates them from partial differential equations. ODEs can be classified by type, order, and linearity. Initial value problems involve solving an ODE with initial conditions specified at a point, while boundary value problems involve conditions at boundary points. The document provides examples of solving first- and second-order initial value problems. It also discusses the existence and uniqueness of solutions to initial value problems under certain continuity conditions on the functions defining the ODE.
The document discusses partial differential equations and their solutions. It can be summarized as:
1) A partial differential equation involves a function of two or more variables and some of its partial derivatives, with one dependent variable and one or more independent variables. Standard notation is presented for partial derivatives.
2) Partial differential equations can be formed by eliminating arbitrary constants or arbitrary functions from an equation relating the dependent and independent variables. Examples of each method are provided.
3) Solutions to partial differential equations can be complete, containing the maximum number of arbitrary constants allowed, particular where the constants are given specific values, or singular where no constants are present. Methods for determining the general solution are described.
All of material inside is un-licence, kindly use it for educational only but please do not to commercialize it.
Based on 'ilman nafi'an, hopefully this file beneficially for you.
Thank you.
This document discusses moment-generating functions and their properties and applications in probability theory. It defines the moment-generating function for both discrete and continuous random variables. Some key properties of moment-generating functions are that each probability distribution has a unique moment-generating function, and they can be used to find the distribution of sums of random variables. The document also describes several common probability distributions and derives their corresponding moment-generating functions.
This document introduces the concept of average sensitivity of algorithms and summarizes results for several graph algorithms. It defines average sensitivity as the average change in an algorithm's output when a single input element is changed. The document presents algorithms for minimum spanning tree, minimum cuts, and matching problems that have low average sensitivity. It argues that average sensitivity is an important dimension for understanding the stability of algorithms and their practical use with noisy real-world data.
The document discusses partial differentiation and its applications. It covers functions of two variables, first and second partial derivatives, and applications including the Cobb-Douglas production function and finding marginal productivity from a production function. Examples are provided to demonstrate calculating partial derivatives of various functions and applying partial derivatives in contexts like production analysis.
This presentation gives the basic idea about the methods of solving ODEs
The methods like variation of parameters, undetermined coefficient method, 1/f(D) method, Particular integral and complimentary functions of an ODE
This document discusses using ODEs (ordinary differential equations) in MATLAB. It begins by introducing initial value problems for ODEs and numerical solutions. It then describes Euler's method for solving first-order ODEs and provides an example of using it to model bacterial growth. Built-in ODE solvers like ode23 and ode45 are introduced. The document also covers solving second-order ODEs by converting them to a system of first-order equations, solving systems of ODEs, stiffness, and passing additional parameters to ODE functions.
The document summarizes key points about equality constrained minimization problems and Newton's method for solving them. It discusses:
1) Equality constrained minimization problems and their equivalent forms via eliminating constraints or using the dual problem.
2) Newton's method extended to include equality constraints, where the Newton step is defined to satisfy the linearized optimality conditions and ensures feasible descent.
3) An infeasible start Newton method that computes steps to reduce the primal-dual residual norm, ensuring iterates become feasible within a finite number of steps.
The document provides examples of sentences written in both active and passive voice. In active voice, the subject performs the action stated by the verb. In passive voice, the subject receives the action. For each example provided, there is a rewriting of the sentence from active to passive voice or vice versa. Various tenses including present, future, and past are used in the examples.
How to develop a successful marketing and communication strategy:
1) Concept: Develop a Sustainable Marketing Strategy
2) Modes of Expression: Address all Senses Creatively
3) Channels: Find Effective Communication Routes.
"No magic formula, but a a very valuable tool"
1. The document discusses differential equations, which relate functions and their derivatives. First order equations relate a function to its first derivative, while second order equations relate it to its second derivative.
2. Differential equations are used in physics, such as Newton's second law relating force, mass and acceleration. They can have many solutions or no solutions. Simultaneous differential equations involve multiple dependent variables.
3. Examples of simultaneous differential equations include models of survival with AIDS, earthquake effects on buildings with multiple floors, and harvesting renewable resources like fish populations over time.
3 examples of PDE, for Laplace, Diffusion of Heat and Wave function. A brief definition of Fouriers Series. Slides created and compiled using LaTeX, beamer package.
I. A power series is a polynomial with infinitely many terms of the form Σn=0∞anxn.
II. The radius of convergence R determines the values of x where a power series converges absolutely (for |x|<R), diverges (for |x|>R), or may converge or diverge (for |x|=R).
III. Tests like the ratio test and root test can be used to calculate the radius of convergence R.
International journal of engineering issues vol 2015 - no 2 - paper5sophiabelthome
This document summarizes key duality theorems for optimization problems involving n-set functions:
1) The weak duality theorem shows that the objective value of any feasible dual problem solution provides a lower bound on the objective of any primal problem solution.
2) The strong duality theorem establishes that under suitable convexity conditions, the optimal objective values of the primal and dual problems are equal.
3) Mangasarian's strict converse duality theorem proves that if a solution satisfies the Kuhn-Tucker conditions and strong duality holds, then the objective function is strictly convex at that solution.
1. A bi-variate random variable has a joint probability distribution function (PDF) that defines the probability of two random variables occurring together. The marginal PDF defines the probability of each variable individually, while the conditional PDF defines the probability of one variable given the other.
2. A multi-variate random variable contains multiple random variables defined by a mean vector and covariance matrix. A linear transformation of a Gaussian multi-variate random variable remains Gaussian with a transformed mean vector and covariance matrix.
3. If two random vectors are jointly Gaussian and uncorrelated, they are also independent, as their joint PDF can be written as the product of their individual PDFs.
How to Solve a Partial Differential Equation on a surfacetr1987
Familiar techniques of separation of variables and Fourier series can be used to solve a variety of pde based on domains in the plane, however these techniques do not extend naturally to surface problems. Instead we look to take a computational approach. The talk will cover the basics of finite difference and finite element approximations of the one dimensional heat equation and show how to extend these ideas on to surfaces. If time allows, we will show numerical results of an optimal partition problem based on a sphere. No background knowledge of pde or computation is required.
This document discusses fuzzy relations, reasoning, and linguistic variables. It defines fuzzy relations as membership functions between elements of Cartesian product spaces. It describes the extension principle for mapping fuzzy sets through functions. Max-min and max-product composition are defined for combining fuzzy relations. Linguistic variables allow information to be expressed using fuzzy linguistic terms rather than numerical values. Operations on linguistic variables like concentration and dilation are discussed. Fuzzy if-then rules are defined using implication functions to model "if A then B" statements where A and B are linguistic values. Fuzzy reasoning uses these rules and facts to derive conclusions.
This document discusses ordinary differential equations (ODEs). It defines ODEs and differentiates them from partial differential equations. ODEs can be classified by type, order, and linearity. Initial value problems involve solving an ODE with initial conditions specified at a point, while boundary value problems involve conditions at boundary points. The document provides examples of solving first- and second-order initial value problems. It also discusses the existence and uniqueness of solutions to initial value problems under certain continuity conditions on the functions defining the ODE.
The document discusses partial differential equations and their solutions. It can be summarized as:
1) A partial differential equation involves a function of two or more variables and some of its partial derivatives, with one dependent variable and one or more independent variables. Standard notation is presented for partial derivatives.
2) Partial differential equations can be formed by eliminating arbitrary constants or arbitrary functions from an equation relating the dependent and independent variables. Examples of each method are provided.
3) Solutions to partial differential equations can be complete, containing the maximum number of arbitrary constants allowed, particular where the constants are given specific values, or singular where no constants are present. Methods for determining the general solution are described.
All of material inside is un-licence, kindly use it for educational only but please do not to commercialize it.
Based on 'ilman nafi'an, hopefully this file beneficially for you.
Thank you.
This document discusses moment-generating functions and their properties and applications in probability theory. It defines the moment-generating function for both discrete and continuous random variables. Some key properties of moment-generating functions are that each probability distribution has a unique moment-generating function, and they can be used to find the distribution of sums of random variables. The document also describes several common probability distributions and derives their corresponding moment-generating functions.
This document introduces the concept of average sensitivity of algorithms and summarizes results for several graph algorithms. It defines average sensitivity as the average change in an algorithm's output when a single input element is changed. The document presents algorithms for minimum spanning tree, minimum cuts, and matching problems that have low average sensitivity. It argues that average sensitivity is an important dimension for understanding the stability of algorithms and their practical use with noisy real-world data.
The document discusses partial differentiation and its applications. It covers functions of two variables, first and second partial derivatives, and applications including the Cobb-Douglas production function and finding marginal productivity from a production function. Examples are provided to demonstrate calculating partial derivatives of various functions and applying partial derivatives in contexts like production analysis.
This presentation gives the basic idea about the methods of solving ODEs
The methods like variation of parameters, undetermined coefficient method, 1/f(D) method, Particular integral and complimentary functions of an ODE
This document discusses using ODEs (ordinary differential equations) in MATLAB. It begins by introducing initial value problems for ODEs and numerical solutions. It then describes Euler's method for solving first-order ODEs and provides an example of using it to model bacterial growth. Built-in ODE solvers like ode23 and ode45 are introduced. The document also covers solving second-order ODEs by converting them to a system of first-order equations, solving systems of ODEs, stiffness, and passing additional parameters to ODE functions.
The document summarizes key points about equality constrained minimization problems and Newton's method for solving them. It discusses:
1) Equality constrained minimization problems and their equivalent forms via eliminating constraints or using the dual problem.
2) Newton's method extended to include equality constraints, where the Newton step is defined to satisfy the linearized optimality conditions and ensures feasible descent.
3) An infeasible start Newton method that computes steps to reduce the primal-dual residual norm, ensuring iterates become feasible within a finite number of steps.
The document provides examples of sentences written in both active and passive voice. In active voice, the subject performs the action stated by the verb. In passive voice, the subject receives the action. For each example provided, there is a rewriting of the sentence from active to passive voice or vice versa. Various tenses including present, future, and past are used in the examples.
How to develop a successful marketing and communication strategy:
1) Concept: Develop a Sustainable Marketing Strategy
2) Modes of Expression: Address all Senses Creatively
3) Channels: Find Effective Communication Routes.
"No magic formula, but a a very valuable tool"
This document discusses how Scilab handles numerical computations differently than mathematical formulas. It analyzes Scilab's implementation of solving the quadratic equation, calculating numerical derivatives, and performing complex division. For each, it shows that Scilab uses robust algorithms that account for issues like rounding errors, underflow, and overflow, unlike a naive implementation directly from the mathematical formulas.
S. Venkata Rao has over 15 years of experience in trade finance and banking. He currently works as an Assistant Manager of Trade Finance at First Gulf Bank in Abu Dhabi, UAE. Previously he held roles at First Gulf Bank, Calyon Bank, State Bank of Mauritius Limited, and IndusInd Bank Limited in India. He has numerous qualifications in trade finance, operations, and banking. His expertise includes trade finance products, foreign currency payments, clearing operations, and ensuring compliance.
Bayesian Experimental Design for Stochastic Kinetic ModelsColin Gillespie
In recent years, the use of the Bayesian paradigm for estimating the optimal experimental design has increased. However, standard techniques are
computationally intensive for even relatively small stochastic kinetic models. One solution to this problem is to couple cloud computing with a model emulator.
By running simulations simultaneously in the cloud, the large design space can be explored. A Gaussian process is then fitted to this output, enabling the
optimal design parameters to be estimated.
This document provides an introduction to moment closure techniques for approximating stochastic models using systems of ordinary differential equations (ODEs). It discusses how moment closure works by deriving the moment equations for simple birth-death and dimerization models. While moment closure can provide fast approximations, it may fail for models with strong correlations between species, like the p53-Mdm2 oscillations model. Software exists to automatically generate moment equations from SBML models.
This document discusses the application of fixed-point theorems to solve ordinary differential equations. It begins by introducing the Banach contraction principle and proving it. It then states two other important fixed-point theorems - the Schauder-Tychonoff theorem and the Leray-Schauder theorem. The rest of the document focuses on proving the Schauder-Tychonoff theorem, which characterizes compact subsets of function spaces and shows that if an operator maps into a relatively compact subset, it has a fixed point. This allows the fixed-point theorems to be applied to finding solutions to differential equations.
This document discusses the application of fixed-point theorems to solve ordinary differential equations. It begins by introducing the Banach contraction principle and proving it. It then states two other important fixed-point theorems - the Schauder-Tychonoff theorem and the Leray-Schauder theorem. The rest of the document focuses on proving the Schauder-Tychonoff theorem, which characterizes compact subsets of function spaces and shows that if an operator maps into a relatively compact subset, it has a fixed point. This allows the fixed-point theorems to be applied to finding solutions to differential equations.
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
This document presents a common fixed point theorem for six self-maps (A, B, S, T, L, M) on a Menger space using the concept of weak compatibility. It proves that if the maps satisfy certain conditions, including being weakly compatible and their images being complete subspaces, then the maps have a unique common fixed point. The proof constructs sequences to show the maps have a coincidence point, then uses weak compatibility and lemmas to show this point is the unique common fixed point.
EXPERT SYSTEMS AND SOLUTIONS
Project Center For Research in Power Electronics and Power Systems
IEEE 2010 , IEEE 2011 BASED PROJECTS FOR FINAL YEAR STUDENTS OF B.E
Email: expertsyssol@gmail.com,
Cell: +919952749533, +918608603634
www.researchprojects.info
OMR, CHENNAI
IEEE based Projects For
Final year students of B.E in
EEE, ECE, EIE,CSE
M.E (Power Systems)
M.E (Applied Electronics)
M.E (Power Electronics)
Ph.D Electrical and Electronics.
Training
Students can assemble their hardware in our Research labs. Experts will be guiding the projects.
EXPERT GUIDANCE IN POWER SYSTEMS POWER ELECTRONICS
We provide guidance and codes for the for the following power systems areas.
1. Deregulated Systems,
2. Wind power Generation and Grid connection
3. Unit commitment
4. Economic Dispatch using AI methods
5. Voltage stability
6. FLC Control
7. Transformer Fault Identifications
8. SCADA - Power system Automation
we provide guidance and codes for the for the following power Electronics areas.
1. Three phase inverter and converters
2. Buck Boost Converter
3. Matrix Converter
4. Inverter and converter topologies
5. Fuzzy based control of Electric Drives.
6. Optimal design of Electrical Machines
7. BLDC and SR motor Drives
Low rank tensor approximation of probability density and characteristic funct...Alexander Litvinenko
This document summarizes a presentation on computing divergences and distances between high-dimensional probability density functions (pdfs) represented using tensor formats. It discusses:
1) Motivating the problem using examples from stochastic PDEs and functional representations of uncertainties.
2) Computing Kullback-Leibler divergence and other divergences when pdfs are not directly available.
3) Representing probability characteristic functions and approximating pdfs using tensor decompositions like CP and TT formats.
4) Numerical examples computing Kullback-Leibler divergence and Hellinger distance between Gaussian and alpha-stable distributions using these tensor approximations.
1) The document discusses solving partial differential equations using variable separation. It presents the one-dimensional wave equation and solves it using variable separation.
2) It derives three cases for the solution based on whether k is positive, negative, or zero. It then presents the general solution as a summation involving sines and cosines.
3) It applies the general solution to two example problems of a vibrating string, finding the displacement as a function of position and time by satisfying the boundary conditions.
The Multivariate Gaussian Probability DistributionPedro222284
The document discusses the multivariate Gaussian probability distribution. It defines the distribution and provides its probability density function. It then discusses various properties including: functions of Gaussian variables such as linear transformations and addition; the characteristic function and how to calculate moments; marginalization and conditional distributions. It also provides some tips and tricks for working with Gaussian distributions including how to calculate products.
I am Frank P. I am a Physical Chemistry Assignment Expert at eduassignmenthelp.com. I hold a Ph.D. in Physical Chemistry, from Malacca, Malaysia. I have been helping students with their homework for the past 6 years. I solve assignments related to Physical Chemistry.
Visit eduassignmenthelp.com or email info@eduassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Physical Chemistry Assignments.
The first report of Machine Learning Seminar organized by Computational Linguistics Laboratory at Kazan Federal University. See http://cll.niimm.ksu.ru/cms/lang/en_US/main/seminars/mlseminar
This document contains solutions to 4 problems involving ordinary differential equations:
1) Solving an initial value problem leads to the solution y(x) = x^4.
2) Solving a homogeneous first order ODE leaves the solution in implicit form as 3x^2 - ln|y/x| = ln|x| + C.
3) Solving an exact ODE leaves the solution in implicit form as 3x + cos(2x + 3y) + 2y = C.
4) Solving a cooling model ODE determines that the time for a cake's temperature to reach 100°F is approximately 66.67 minutes.
1. The document discusses local volatility and the "smile effect" where implied volatility depends on strike price and maturity. It presents the derivation of the local volatility partial differential equation from no-arbitrage arguments.
2. It notes some limitations of the local volatility model, including that stock returns are not additive and the model does not have a clear pathwise connection to the underlying stock process.
3. A new remark is added discussing how the local volatility PDE can be reformulated as a classical Cauchy problem by changing variables, and clarifying the relationship between the local volatility diffusion and the underlying stock process.
What happens when the Kolmogorov-Zakharov spectrum is nonlocal?Colm Connaughton
This document summarizes research on the behavior of the Kolmogorov-Zakharov (KZ) spectrum when it is nonlocal. It examines a model of cluster-cluster aggregation described by the Smoluchowski equation, which can be viewed as a model of 3-wave turbulence without backscatter. The research finds that when the exponents in the interaction term satisfy certain conditions, the KZ spectrum is nonlocal. In this case, the stationary state has a novel functional form and can become unstable, leading to oscillatory behavior in the cascade dynamics at long times. Open questions remain about whether physical systems exhibit this behavior and how the results are affected by including backscatter terms.
1) The document contains an exam with 4 problems related to ordinary differential equations.
2) The first problem involves solving a homogeneous Cauchy-Euler equation and results in a general solution involving exponential terms.
3) The second problem involves solving an initial value problem using the method of undetermined coefficients, resulting in a particular solution added to the general solution.
4) The third problem uses the method of variation of parameters to solve a non-homogeneous differential equation involving trigonometric functions.
5) The fourth problem involves applying an equation of motion to a physical system of a mass on a spring to determine the velocity at a specific time.
This document provides an overview of second order ordinary differential equations (ODEs). It discusses the method of variation of parameters for finding particular solutions to inhomogeneous second order linear ODEs. It provides examples of applying this method, including a spring-mass system and an electric circuit. It also discusses applications of second order ODEs to modeling spring-mass systems and electric circuits.
Computing f-Divergences and Distances of\\ High-Dimensional Probability Densi...Alexander Litvinenko
Talk presented on SIAM IS 2022 conference.
Very often, in the course of uncertainty quantification tasks or
data analysis, one has to deal with high-dimensional random variables (RVs)
(with values in $\Rd$). Just like any other RV,
a high-dimensional RV can be described by its probability density (\pdf) and/or
by the corresponding probability characteristic functions (\pcf),
or a more general representation as
a function of other, known, random variables.
Here the interest is mainly to compute characterisations like the entropy, the Kullback-Leibler, or more general
$f$-divergences. These are all computed from the \pdf, which is often not available directly,
and it is a computational challenge to even represent it in a numerically
feasible fashion in case the dimension $d$ is even moderately large. It
is an even stronger numerical challenge to then actually compute said characterisations
in the high-dimensional case.
In this regard, in order to achieve a computationally feasible task, we propose
to approximate density by a low-rank tensor.
PaperNo14-Habibi-IJMA-n-Tuples and ChaoticityMezban Habibi
This document presents theorems and definitions related to n-tuples of operators on a Frechet space and conditions for chaoticity. It begins with definitions of key concepts such as the orbit of a vector under an n-tuple of operators and what it means for an n-tuple to be hypercyclic or for a vector to be periodic. The main results section presents two theorems, the first characterizing when an n-tuple satisfies the hypercyclicity criterion and the second proving conditions under which an n-tuple of weighted backward shifts is chaotic. The second theorem shows the equivalence of an n-tuple being chaotic, hypercyclic with a non-trivial periodic point, having a non-trivial periodic point, and a
This document discusses Hilbert-Schmidt n-tuples of operators on a Banach space. It presents two main results: 1) the Hypercyclicity Criterion, which provides conditions for an n-tuple of operators to be hypercyclic, and 2) conditions under which an n-tuple of unilateral weighted backward shifts is chaotic or has a non-trivial periodic point. It also references several other works studying properties of n-tuples and hypercyclic operators.
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The tau-leap method for simulating stochastic kinetic modelsColin Gillespie
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Moment closure inference for stochastic kinetic models
1. Moment closure inference for
stochastic kinetic models
Colin Gillespie
School of Mathematics & Statistics
2. Talk outline
An introduction to moment closure
Case study: Aphids
Conclusion
2/43
3. Birth-death process
Birth-death model
X −→ 2X and 2X −→ X
which has the propensity functions λX and µX .
Deterministic representation
The deterministic model is
dX (t )
= ( λ − µ )X (t ) ,
dt
which can be solved to give X (t ) = X (0) exp[(λ − µ)t ].
3/43
4. Birth-death process
Birth-death model
X −→ 2X and 2X −→ X
which has the propensity functions λX and µX .
Deterministic representation
The deterministic model is
dX (t )
= ( λ − µ )X (t ) ,
dt
which can be solved to give X (t ) = X (0) exp[(λ − µ)t ].
3/43
5. Stochastic representation
In the stochastic framework, each
reaction has a probability of occurring
50
The analogous version of the
40
birth-death process is the difference
Population
equation 30
20
dpn
= λ(n − 1)pn−1 + µ(n + 1)pn+1 10
dt
− (λ + µ)npn 0
0 1 2 3 4
Time
Usually called the forward Kolmogorov
equation or chemical master equation
4/43
6. Moment equations
Multiply the CME by enθ and sum over n, to obtain
∂M ∂M
= [λ(eθ − 1) + µ(e−θ − 1)]
∂t ∂θ
where
∞
M (θ; t ) = ∑ e n θ pn ( t )
n =0
If we differentiate this p.d.e. w.r.t θ and set θ = 0, we get
dE[N (t )]
= (λ − µ)E[N (t )]
dt
where E[N (t )] is the mean
5/43
7. The mean equation
dE[N (t )]
= (λ − µ)E[N (t )]
dt
This ODE is solvable - the associated forward Kolmogorov equation is
also solvable
The equation for the mean and deterministic ODE are identical
When the rate laws are linear, the stochastic mean and deterministic
solution always correspond
6/43
8. The variance equation
If we differentiate the p.d.e. w.r.t θ twice and set θ = 0, we get:
dE[N (t )2 ]
= (λ − µ)E[N (t )] + 2(λ − µ)E[N (t )2 ]
dt
and hence the variance Var[N (t )] = E[N (t )2 ] − E[N (t )]2 .
Differentiating three times gives an expression for the skewness, etc
7/43
10. Dimerisation moment equations
We formulate the dimer model in terms of moment equations
dE[X1 ] 2
= 0.5k1 (E[X1 ] − E[X1 ]) − k2 E[X1 ]
dt
2
dE[X1 ] 2 2
= k1 (E[X1 X2 ] − E[X1 X2 ]) + 0.5k1 (E[X1 ] − E[X1 ])
dt
2
+ k2 (E[X1 ] − 2E[X1 ])
where E[X1 ] is the mean of X1 and E[X1 ] − E[X1 ]2 is the variance
2
The i th moment equation depends on the (i + 1)th equation
9/43
11. Deterministic approximates stochastic
Rewriting
dE[X1 ] 2
= 0.5k1 (E[X1 ] − E[X1 ]) − k2 E[X1 ]
dt
in terms of its variance, i.e. E[X1 ] = Var[X1 ] + E[X1 ]2 , we get
2
dE[X1 ]
= 0.5k1 E [X1 ](E[X1 ] − 1) + 0.5k1 Var[X1 ] − k2 E[X1 ] (1)
dt
Setting Var[X1 ] = 0 in (1), recovers the deterministic equation
So we can consider the deterministic models as an approximation to
the stochastic
When we have polynomial rate laws, setting the variance to zero
results in the deterministic equation
10/43
12. Deterministic approximates stochastic
Rewriting
dE[X1 ] 2
= 0.5k1 (E[X1 ] − E[X1 ]) − k2 E[X1 ]
dt
in terms of its variance, i.e. E[X1 ] = Var[X1 ] + E[X1 ]2 , we get
2
dE[X1 ]
= 0.5k1 E [X1 ](E[X1 ] − 1) + 0.5k1 Var[X1 ] − k2 E[X1 ] (1)
dt
Setting Var[X1 ] = 0 in (1), recovers the deterministic equation
So we can consider the deterministic models as an approximation to
the stochastic
When we have polynomial rate laws, setting the variance to zero
results in the deterministic equation
10/43
13. Simple dimerisation model
To close the equations, we assume an underlying distribution
The easiest option is to assume an underlying Normal distribution, i.e.
E[X1 ] = 3E[X1 ]E[X1 ] − 2E[X1 ]3
3 2
But we could also use, the Poisson
3
E[X1 ] = E[X1 ] + 3E[X1 ]2 + E[X1 ]3
or the Log normal
2 3
3 E [ X1 ]
E [ X1 ] =
E [ X1 ]
11/43
14. Simple dimerisation model
To close the equations, we assume an underlying distribution
The easiest option is to assume an underlying Normal distribution, i.e.
E[X1 ] = 3E[X1 ]E[X1 ] − 2E[X1 ]3
3 2
But we could also use, the Poisson
3
E[X1 ] = E[X1 ] + 3E[X1 ]2 + E[X1 ]3
or the Log normal
2 3
3 E [ X1 ]
E [ X1 ] =
E [ X1 ]
11/43
15. Heat shock model
Proctor et al, 2005. Stochastic kinetic model of the heat shock system
twenty-three reactions
seventeen chemical species
A single stochastic simulation up to t = 2000 takes about 35 minutes.
If we convert the model to moment equations, we get 139 equations
ADP Native Protein
1200 6000000
5950000
1000
5900000
800
Population
5850000
600
5800000
400
5750000
200
5700000
0
0 500 1000 1500 2000 0 500 1000 1500 2000
Time
Gillespie, CS, 2009
12/43
16. Density plots: heat shock model
Time t=200 Time t=2000
0.006
Density
0.004
0.002
0.000
600 800 1000 1200 1400 600 800 1000 1200 1400
ADP population
13/43
17. P53-Mdm2 oscillation model
Proctor and Grey, 2008 300
16 chemical species
250
Around a dozen reactions
200
Population
The model contains an events
At t = 1, set X = 0 150
If we convert the model to moment 100
equations, we get 139 equations. 50
However, in this case the moment 0
closure approximation doesn’t do to 0 5 10 15 20 25 30
Time
well!
14/43
18. P53-Mdm2 oscillation model
Proctor and Grey, 2008
300
16 chemical species
Around a dozen reactions 250
The model contains an events 200
Population
At t = 1, set X = 0 150
If we convert the model to moment 100
equations, we get 139 equations.
50
However, in this case the moment
0
closure approximation doesn’t do to
0 5 10 15 20 25 30
well! Time
14/43
19. P53-Mdm2 oscillation model
Proctor and Grey, 2008
300
16 chemical species
Around a dozen reactions 250
The model contains an events 200
Population
At t = 1, set X = 0 150
If we convert the model to moment 100
equations, we get 139 equations.
50
However, in this case the moment
0
closure approximation doesn’t do to
0 5 10 15 20 25 30
well! Time
14/43
20. What went wrong?
The Moment closure (tends) to fail when there is a large difference
between the deterministic and stochastic formulations
In this particular case, strongly correlated species
Typically when the MC approximation fails, it gives a negative
variance
The MC approximation does work well for other parameter values for
the p53 model
15/43
22. Cotton aphids
Aphid infestation (G & Golightly, 2010)
A cotton aphid infestation of a cotton plant can result in:
leaves that curl and pucker
seedling plants become stunted and may die
a late season infestation can result in stained cotton
cotton aphids have developed resistance to many chemical
treatments and so can be difficult to treat
Basically it costs someone a lot of money
17/43
23. Cotton aphids
Aphid infestation (G & Golightly, 2010)
A cotton aphid infestation of a cotton plant can result in:
leaves that curl and pucker
seedling plants become stunted and may die
a late season infestation can result in stained cotton
cotton aphids have developed resistance to many chemical
treatments and so can be difficult to treat
Basically it costs someone a lot of money
17/43
24. Cotton aphids
The data consists of
five observations at each plot
the sampling times are t=0, 1.14, 2.29, 3.57 and 4.57 weeks (i.e.
every 7 to 8 days)
three blocks, each being in a distinct area
three irrigation treatments (low, medium and high)
three nitrogen levels (blanket, variable and none)
18/43
27. Some notation
Let
n (t ) to be the size of the aphid population at time t
c (t ) to be the cumulative aphid population at time t
1. We observe n (t ) at discrete time points
2. We don’t observe c (t )
3. c (t ) ≥ n (t )
20/43
28. The model
We assume, based on previous modelling (Matis et al., 2004)
An aphid birth rate of λn (t )
An aphid death rate of µn (t )c (t )
So extinction is certain, as eventually µnc > λn for large t
21/43
29. The model
Deterministic representation
Previous modelling efforts have focused on deterministic models:
dN (t )
= λN (t ) − µC (t )N (t )
dt
dC (t )
= λN (t )
dt
Some problems
Initial and final aphid populations are quite small
No allowance for ‘natural’ random variation
Solution: use a stochastic model
22/43
30. The model
Deterministic representation
Previous modelling efforts have focused on deterministic models:
dN (t )
= λN (t ) − µC (t )N (t )
dt
dC (t )
= λN (t )
dt
Some problems
Initial and final aphid populations are quite small
No allowance for ‘natural’ random variation
Solution: use a stochastic model
22/43
31. The model
Stochastic representation
Let pn,c (t ) denote the probability:
there are n aphids in the population at time t
a cumulative population size of c at time t
This gives the forward Kolmogorov equation
dpn,c (t )
= λ(n − 1)pn−1,c −1 (t ) + µc (n + 1)pn+1,c (t )
dt
− n ( λ + µ c ) p n ,c ( t )
Even though this equation is fairly simple, it still can’t be solved exactly.
23/43
32. Some simulations
800
600
Aphid pop.
400
200
0
0 2 4 6 8 10
Time (days)
Parameters: n (0) = c (0) = 1, λ = 1.7 and µ = 0.001 24/43
33. Some simulations
800
600
Aphid pop.
400
200
0
0 2 4 6 8 10
Time (days)
Parameters: n (0) = c (0) = 1, λ = 1.7 and µ = 0.001 24/43
34. Some simulations
800
600
Aphid pop.
400
200
0
0 2 4 6 8 10
Time (days)
Parameters: n (0) = c (0) = 1, λ = 1.7 and µ = 0.001 24/43
35. Stochastic parameter estimation
Let X(tu ) = (n (tu ), c (tu )) be the vector of observed aphid counts
and unobserved cumulative population size at time tu ;
To infer λ and µ, we need to estimate
Pr[X(tu )| X(tu −1 ), λ, µ]
i.e. the solution of the forward Kolmogorov equation
We will use moment closure to estimate this distribution
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36. Stochastic parameter estimation
Let X(tu ) = (n (tu ), c (tu )) be the vector of observed aphid counts
and unobserved cumulative population size at time tu ;
To infer λ and µ, we need to estimate
Pr[X(tu )| X(tu −1 ), λ, µ]
i.e. the solution of the forward Kolmogorov equation
We will use moment closure to estimate this distribution
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37. Moment equations for the means
dE[n (t )]
= λE[n(t )] − µ(E[n(t )]E[c (t )] + Cov[n(t ), c (t )])
dt
dE[c (t )]
= λE[n(t )]
dt
The equation for the E[n (t )] depends on the Cov[n (t ), c (t )]
Setting Cov[n (t ), c (t )]=0 gives the deterministic model
We obtain similar equations for higher-order moments
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38. Moment equations for the means
dE[n (t )]
= λE[n(t )] − µ(E[n(t )]E[c (t )] + Cov[n(t ), c (t )])
dt
dE[c (t )]
= λE[n(t )]
dt
The equation for the E[n (t )] depends on the Cov[n (t ), c (t )]
Setting Cov[n (t ), c (t )]=0 gives the deterministic model
We obtain similar equations for higher-order moments
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39. Parameter inference
Given
the parameters: {λ, µ}
the initial states: X(tu −1 ) = (n (tu −1 ), c (tu −1 ));
We have
X(tu ) | X(tu −1 ), λ, µ ∼ N (ψu −1 , Σu −1 )
where ψu −1 and Σu −1 are calculated using the moment closure
approximation
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40. Parameter inference
Summarising our beliefs about {λ, µ} and the unobserved
cumulative population c (t0 ) via priors p (λ, µ) and p (c (t0 ))
The joint posterior for parameters and unobserved states (for a single
data set) is
4
p (λ, µ, c | n) ∝ p (λ, µ) p (c(t0 )) ∏ p (x(tu ) | x(tu−1 ), λ, µ)
u =1
For the results shown, we used a simple random walk MH step to
explore the parameter and state spaces
For more complicated models, we can use a Durham & Gallant style
bridge (Milner, G & Wilkinson, 2012).
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41. Parameter inference
Summarising our beliefs about {λ, µ} and the unobserved
cumulative population c (t0 ) via priors p (λ, µ) and p (c (t0 ))
The joint posterior for parameters and unobserved states (for a single
data set) is
4
p (λ, µ, c | n) ∝ p (λ, µ) p (c(t0 )) ∏ p (x(tu ) | x(tu−1 ), λ, µ)
u =1
For the results shown, we used a simple random walk MH step to
explore the parameter and state spaces
For more complicated models, we can use a Durham & Gallant style
bridge (Milner, G & Wilkinson, 2012).
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42. Simulation study
Three treatments & two blocks
Baseline birth and death rates: {λ = 1.75, µ = 0.00095}
Treatment 2 increases µ by 0.0004
Treatment 3 increases λ by 0.35
The block effect reduces µ by 0.0003
Treatment 1 Treatment 2 Treatment 3
Block 1 {1.75, 0.00095} {1.75, 0.00135} {2.1, 0.00095}
Block 2 {1.75, 0.00065} {1.75, 0.00105} {2.1, 0.00065}
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43. Simulation study
Three treatments & two blocks
Baseline birth and death rates: {λ = 1.75, µ = 0.00095}
Treatment 2 increases µ by 0.0004
Treatment 3 increases λ by 0.35
The block effect reduces µ by 0.0003
Treatment 1 Treatment 2 Treatment 3
Block 1 {1.75, 0.00095} {1.75, 0.00135} {2.1, 0.00095}
Block 2 {1.75, 0.00065} {1.75, 0.00105} {2.1, 0.00065}
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44. Simulation study
Three treatments & two blocks
Baseline birth and death rates: {λ = 1.75, µ = 0.00095}
Treatment 2 increases µ by 0.0004
Treatment 3 increases λ by 0.35
The block effect reduces µ by 0.0003
Treatment 1 Treatment 2 Treatment 3
Block 1 {1.75, 0.00095} {1.75, 0.00135} {2.1, 0.00095}
Block 2 {1.75, 0.00065} {1.75, 0.00105} {2.1, 0.00065}
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45. Simulation study
Three treatments & two blocks
Baseline birth and death rates: {λ = 1.75, µ = 0.00095}
Treatment 2 increases µ by 0.0004
Treatment 3 increases λ by 0.35
The block effect reduces µ by 0.0003
Treatment 1 Treatment 2 Treatment 3
Block 1 {1.75, 0.00095} {1.75, 0.00135} {2.1, 0.00095}
Block 2 {1.75, 0.00065} {1.75, 0.00105} {2.1, 0.00065}
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47. Parameter structure
Let i , k represent the block and treatments level, i ∈ {1, 2} and
k ∈ {1, 2, 3}
For each data set, we assume birth rates of the form:
λik = λ + αi + β k
where α1 = β 1 = 0
So for block 1, treatment 1 we have:
λ11 = λ
and for block 2, treatment 1 we have:
λ21 = λ + α2
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48. MCMC scheme
Using the MCMC scheme described previously, we generated 2M
iterates and thinned by 1K
This took a few hours and convergence was fairly quick
We used independent proper uniform priors for the parameters
For the initial unobserved cumulative population, we had
c (t0 ) = n (t0 ) +
where has a Gamma distribution with shape 1 and scale 10.
This set up mirrors the scheme that we used for the real data set
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49. Marginal posterior distributions for
λ and µ
20000
6
15000
Density
Density
4
10000
2
5000
0
X 0
X
1.6 1.7 1.8 1.9 2.0 0.00090 0.00095 0.00100
Birth Rate Death Rate
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50. Marginal posterior distributions for birth
rates
−0.2 0.0 0.2 0.4
Block 2 Treatment 2 Treatment 3
6
Density
4
2
0 X X X
−0.2 0.0 0.2 0.4 −0.2 0.0 0.2 0.4
Birth Rate
We obtained similar densities for the death rates.
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51. Application to the cotton aphid data set
Recall that the data consists of
five observations on twenty randomly chosen leaves in each plot;
three blocks, each being in a distinct area;
three irrigation treatments (low, medium and high);
three nitrogen levels (blanket, variable and none);
the sampling times are t=0, 1.14, 2.29, 3.57 and 4.57 weeks (i.e.
every 7 to 8 days).
Following in the same vein as the simulated data, we are estimating 38
parameters (including interaction terms) and the latent cumulative aphid
population.
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52. Cotton aphid data
Marginal posterior distributions
6
15000
Density
Density
4
10000
2 5000
0 0
1.6 1.7 1.8 1.9 2.0 0.00090 0.00095 0.00100
Birth Rate Death Rate
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53. Does the model fit the data?
We simulate predictive distributions from the MCMC output, i.e. we
randomly sample parameter values (λ, µ) and the unobserved state
c and simulate forward
We simulate forward using the Gillespie simulator
not the moment closure approximation
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54. Does the model fit the data?
Predictive distributions for 6 of the 27 Aphid data sets
D 123 D 121 D131
2500
2000
1500
X
q
q q
q 1000
X
q
q
X q
q
q q
Aphid Population
q q
q
q q
q
q 500
X
q q
q
X
q q
q
q q X
q
q
q
q X X
q
q
q
X
q X q
q
q
X X 0
q
D 112 D 122 D 113
q
q
X
2500
q
q
2000
1500 q
q
X
q
q q
q
1000
q
q q
q
X q
q X
q
q
q
q
q
q
q
500 X q q
X q
q
X q
q
q q
q
X
q
q
q
X X q
X X
q
0
q
1.14 2.29 3.57 4.57 1.14 2.29 3.57 4.57 1.14 2.29 3.57 4.57
Time
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55. Summarising the results
Consider the additional number of aphids per treatment combination
Set c (0) = n (0) = 1 and tmax = 6
We now calculate the number of aphids we would see for each
parameter combination in addition to the baseline
For example, the effect due to medium water:
∗
λ211 = λ + αWater (M) and µ211 = µ + αWater (M)
So
i i
Additional aphids = cWater (M) − cbaseline
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56. Aphids over baseline
Main Effects
0 2000 6000 10000
Nitrogen (V) Water (H) Water (M)
0.0025
0.0020
0.0015
0.0010
0.0005
0.0000
Density
Block 3 Block 2 Nitrogen (Z)
0.0025
0.0020
0.0015
0.0010
0.0005
0.0000
0 2000 6000 10000 0 2000 6000 10000
Aphids
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58. Conclusions
The 95% credible intervals for the baseline birth and death rates are
(1.64, 1.86) and (0.00090, 0.00099).
Main effects have little effect by themselves
However block 2 appears to have a very strong interaction with
nitrogen
Moment closure parameter inference is a very useful technique for
estimating parameters in stochastic population models
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59. Future work
Aphid model
Other data sets suggest that there is aphid immigration in the early
stages
Model selection for stochastic models
Incorporate measurement error
Moment closure
Better closure techniques
Assessing the fit
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60. Acknowledgements
Andrew Golightly Richard Boys
Peter Milner
Darren Wilkinson Jim Matis (Texas A & M)
References
Gillespie, CS Moment closure approximations for mass-action models. IET Systems Biology 2009.
Gillespie, CS, Golightly, A Bayesian inference for generalized stochastic population growth models with application to aphids.
Journal of the Royal Statistical Society, Series C 2010.
Milner, P, Gillespie, CS, Wilkinson, DJ Moment closure approximations for stochastic kinetic models with rational rate laws.
Mathematical Biosciences 2011.
Milner, P, Gillespie, CS and Wilkinson, DJ Moment closure based parameter inference of stochastic kinetic models.
Statistics and Computing 2012.
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