This document discusses Markov chains and provides an example of modeling the distribution of red squirrels and gray squirrels in Great Britain over time.
1) A Markov chain models transitions between states based on conditional probabilities, with the transition matrix representing the probabilities of moving between states. This is used to model the distribution of red and gray squirrels across different regions.
2) Squares of land are classified into four states based on which squirrel species are present. Transition counts between states over time are used to construct a transition matrix representing conditional probabilities.
3) The steady-state matrix and distribution indicate that in the long run, around 17% of regions will contain only red squirrels, 6% only gray
Cellular automata are algorithms that model the discrete spatial and temporal evolution of complex systems using local transformation rules. They consist of a lattice of cells that can have different states, with the state of each cell updated synchronously based on the states of neighboring cells. There are various types of cellular automata depending on factors like the neighborhood considered and whether the rules are deterministic or probabilistic. Cellular automata have been widely used to model microstructure evolution in materials science applications like recrystallization simulation.
1. The document discusses spacey random walks, which are a type of stochastic process that can be used to model higher-order Markov chains.
2. A spacey random walk is defined based on the transition probabilities of a higher-order Markov chain, but "forgets" its history and pretends to come from a random previous state.
3. The stationary distributions of spacey random walks are given by tensor eigenvectors of the transition tensor for the higher-order Markov chain. This provides a connection between higher-order Markov chains and tensor eigenvectors.
The document discusses Markov chain Monte Carlo (MCMC) methods, which use Markov chains to generate dependent samples from probability distributions that are difficult to directly sample from. It introduces Gibbs sampling and the Metropolis-Hastings algorithm as two common MCMC techniques. Gibbs sampling works by iteratively sampling each parameter from its conditional distribution given current values of other parameters. Metropolis-Hastings also iteratively proposes new parameter values but only accepts them probabilistically, based on the target distribution. Both techniques generate Markov chains that can be used to approximate integrals and obtain quantities of interest from complex distributions.
The document summarizes research on spacey random walks, which are a type of stochastic process that can model higher-order Markov chains. Key points:
1. Spacey random walks generalize higher-order Markov chains by forgetting history but pretending to remember a random previous state, with the stationary distribution given by a tensor eigenvector of the transition tensor.
2. This connects higher-order Markov chains to tensor eigenvectors and provides a stochastic interpretation of tensor eigenvectors as stationary distributions.
3. The dynamics of spacey random walks can be modeled as an ordinary differential equation, allowing tensor eigenvectors to be computed by numerically integrating the dynamical system.
This 10 hours class is intended to give students the basis to empirically solve statistical problems. Talk 1 serves as an introduction to the statistical software R, and presents how to calculate basic measures such as mean, variance, correlation and gini index. Talk 2 shows how the central limit theorem and the law of the large numbers work empirically. Talk 3 presents the point estimate, the confidence interval and the hypothesis test for the most important parameters. Talk 4 introduces to the linear regression model and Talk 5 to the bootstrap world. Talk 5 also presents an easy example of a markov chains.
All the talks are supported by script codes, in R language.
The document discusses Markov chains and their relationship to random walks on graphs and electrical networks. Some key points:
- A Markov chain is a process that transitions between a finite set of states based on transition probabilities that depend only on the current state.
- For a strongly connected Markov chain, there exists a unique stationary distribution that the long-term probabilities of the chain converge to, regardless of the starting state.
- Random walks on undirected graphs can be modeled as Markov chains, where the transition probabilities are proportional to edge conductances in an analogous electrical network. The stationary distribution of such a random walk is proportional to vertex degrees or conductances.
This document discusses Markov chains, which are a type of stochastic process used to model randomly changing systems. It defines Markov chains and their key properties, like the Markov property and transition probabilities. It provides examples like modeling customer purchases over time and inventory management. It also covers concepts like steady state probabilities, transition matrices, and mean first passage times.
Markov analysis examines dependent random events where the likelihood of future events depends on past events. It models this using a transition matrix showing the probabilities of moving between states. The document discusses Markov analysis of accounts receivable to predict future payment categories. It defines states like paid, overdue 1-3 months, etc. and a transition matrix showing the probabilities of moving between states. Markov analysis can then predict future distributions of accounts among the states by multiplying the current distribution by the transition matrix repeatedly.
Cellular automata are algorithms that model the discrete spatial and temporal evolution of complex systems using local transformation rules. They consist of a lattice of cells that can have different states, with the state of each cell updated synchronously based on the states of neighboring cells. There are various types of cellular automata depending on factors like the neighborhood considered and whether the rules are deterministic or probabilistic. Cellular automata have been widely used to model microstructure evolution in materials science applications like recrystallization simulation.
1. The document discusses spacey random walks, which are a type of stochastic process that can be used to model higher-order Markov chains.
2. A spacey random walk is defined based on the transition probabilities of a higher-order Markov chain, but "forgets" its history and pretends to come from a random previous state.
3. The stationary distributions of spacey random walks are given by tensor eigenvectors of the transition tensor for the higher-order Markov chain. This provides a connection between higher-order Markov chains and tensor eigenvectors.
The document discusses Markov chain Monte Carlo (MCMC) methods, which use Markov chains to generate dependent samples from probability distributions that are difficult to directly sample from. It introduces Gibbs sampling and the Metropolis-Hastings algorithm as two common MCMC techniques. Gibbs sampling works by iteratively sampling each parameter from its conditional distribution given current values of other parameters. Metropolis-Hastings also iteratively proposes new parameter values but only accepts them probabilistically, based on the target distribution. Both techniques generate Markov chains that can be used to approximate integrals and obtain quantities of interest from complex distributions.
The document summarizes research on spacey random walks, which are a type of stochastic process that can model higher-order Markov chains. Key points:
1. Spacey random walks generalize higher-order Markov chains by forgetting history but pretending to remember a random previous state, with the stationary distribution given by a tensor eigenvector of the transition tensor.
2. This connects higher-order Markov chains to tensor eigenvectors and provides a stochastic interpretation of tensor eigenvectors as stationary distributions.
3. The dynamics of spacey random walks can be modeled as an ordinary differential equation, allowing tensor eigenvectors to be computed by numerically integrating the dynamical system.
This 10 hours class is intended to give students the basis to empirically solve statistical problems. Talk 1 serves as an introduction to the statistical software R, and presents how to calculate basic measures such as mean, variance, correlation and gini index. Talk 2 shows how the central limit theorem and the law of the large numbers work empirically. Talk 3 presents the point estimate, the confidence interval and the hypothesis test for the most important parameters. Talk 4 introduces to the linear regression model and Talk 5 to the bootstrap world. Talk 5 also presents an easy example of a markov chains.
All the talks are supported by script codes, in R language.
The document discusses Markov chains and their relationship to random walks on graphs and electrical networks. Some key points:
- A Markov chain is a process that transitions between a finite set of states based on transition probabilities that depend only on the current state.
- For a strongly connected Markov chain, there exists a unique stationary distribution that the long-term probabilities of the chain converge to, regardless of the starting state.
- Random walks on undirected graphs can be modeled as Markov chains, where the transition probabilities are proportional to edge conductances in an analogous electrical network. The stationary distribution of such a random walk is proportional to vertex degrees or conductances.
This document discusses Markov chains, which are a type of stochastic process used to model randomly changing systems. It defines Markov chains and their key properties, like the Markov property and transition probabilities. It provides examples like modeling customer purchases over time and inventory management. It also covers concepts like steady state probabilities, transition matrices, and mean first passage times.
Markov analysis examines dependent random events where the likelihood of future events depends on past events. It models this using a transition matrix showing the probabilities of moving between states. The document discusses Markov analysis of accounts receivable to predict future payment categories. It defines states like paid, overdue 1-3 months, etc. and a transition matrix showing the probabilities of moving between states. Markov analysis can then predict future distributions of accounts among the states by multiplying the current distribution by the transition matrix repeatedly.
This document provides an introduction to hidden Markov models (HMMs). It defines HMMs as an extension of Markov models that allows for observations that are probabilistic functions of hidden states. The core problems of HMMs are finding the probability of an observed sequence and determining the most probable hidden state sequence that produced an observation. HMMs have applications in areas like speech recognition by finding the most likely string of words given acoustic input using the Viterbi and forward algorithms.
This document proposes a method for estimating k sample survival functions under stochastic ordering constraints. It begins by reviewing existing work on estimating survival functions for two samples and extends this to k samples. The proposed method uses benchmark functions to estimate survival curves in a way that maintains stochastic ordering. It was tested on both uncensored and censored data and was shown to have low mean squared error and bias. The method was also applied to a real-world dataset with results comparable to previous work.
Supervised Hidden Markov Chains.
Here, we used the paper by Rabiner as a base for the presentation. Thus, we have the following three problems:
1.- How efficiently compute the probability given a model.
2.- Given an observation to which class it belongs
3.- How to find the parameters given data for training.
The first two follow Rabiner's explanation, but in the third one I used the Lagrange Multiplier Optimization because Rabiner lacks a clear explanation about how solving the issue.
This document discusses using transition diagrams to teach concepts in Markov chains. It defines key terms like transition matrix, transition diagram, n-step transition probabilities, first passage times, persistent/transient states, mean recurrence times, and periodic/aperiodic states. Examples are provided to show how these concepts can be understood and calculated from a transition diagram. The document argues transition diagrams provide a good technique for students to visualize and solve problems involving Markov chains.
This document provides a summary of Markov chains. It begins by defining stochastic processes and Markov chains. A Markov chain is a stochastic process where the probability of the next state depends only on the current state, not on the sequence of events that preceded it. The document discusses n-step transition probabilities, classification of states, and steady-state probabilities. It provides examples of Markov chains for cola purchases and camera store inventory to illustrate the concepts.
This document discusses Markov chains and Hidden Markov Models. It defines key properties of Markov chains including the Markov property and transition matrices. It provides examples of Markov chains for weather prediction and DNA sequences. Hidden Markov Models are introduced as having hidden states that can only be observed through output tokens. The difference between Markov chains and HMMs is explained. The document shows an example HMM for correlating tree ring size to temperature. It finds the optimal state sequences for this HMM using dynamic programming and the HMM equations. R code examples are provided for Markov chain transition matrices for DNA sequences.
This document provides an introduction to hidden Markov models (HMMs). It explains what HMMs are, where they are used, and why they are useful. Key aspects of HMMs covered include the Markov chain process, notation used in HMMs, an example of applying an HMM to temperature data, and the three main problems HMMs are used to solve: scoring observation sequences, finding optimal state sequences, and training a model. The document also outlines the forward, backward, and other algorithms used to efficiently solve these three problems.
This document summarizes key concepts related to Markov chains and linear algebra. It provides an example of using a transition matrix to model the probabilities of television viewers switching between two stations over time. The transition matrix allows calculating the probability vectors for future weeks through matrix multiplication. A steady-state vector can also be determined by solving the equation A*p=p, representing the long-term probabilities once the system reaches equilibrium.
This document discusses discrete-time Markov chains (DTMCs) and provides examples of DTMC modeling. It begins by defining DTMCs and contrasting them with continuous-time Markov chains (CTMCs). DTMCs can only transition between states at discrete time steps, while CTMCs can transition at any time. The document then defines DTMCs mathematically and discusses properties like homogeneity, the Markov property, and transition probability matrices. Examples of DTMC modeling are provided for problems like a machine repair facility and someone carrying umbrellas. The document concludes by discussing concepts like n-step transition probabilities, stationary distributions, limiting probabilities, and steady states.
Markov chain analysis uses Markov models to analyze randomly changing systems where future states only depend on the present state, not past states. A Markov chain has a fixed set of states, transition probabilities between states, and will converge to a unique long-run distribution. Markov chains assume states are fully observable and the system is autonomous. Common examples include weather patterns and gambling. Markov chains can be modeled and simulated using R packages like msm and markovchain.
This document provides an introduction to bootstrap methods and Markov chains. It discusses how bootstrap can be used to estimate properties of a statistic like mean or variance when the sample is small and assumptions of the central limit theorem may not apply. The basic bootstrap approach resamples the original sample with replacement to create new bootstrap samples and estimates the statistic for each. Markov chains are defined as stochastic processes where the next state only depends on the current state. An example of a 2-state Markov chain is provided along with notation for transition probabilities and computing unconditional probabilities. The document also discusses stationary distributions for Markov chains.
The document discusses Markov chains and their application to modeling transitions between states over time. It defines Markov chains as processes where the probability of the next state depends only on the current state. Transition matrices are used to represent the probabilities of moving between states. The powers of a transition matrix converge to a steady state as time increases, with all columns being identical, representing the long-term probabilities of being in each state. Finding the steady state vector involves solving the equation Tu=u. An example of modeling class attendance as a Markov chain is presented.
Hidden Markov models (HMMs) are probabilistic graphical models that allow prediction of a sequence of hidden states from observed variables. HMMs make the Markov assumption that the next state depends only on the current state, not past states. They require specification of transition probabilities between hidden states, emission probabilities of observations given states, and initial state probabilities to compute the joint probability of state sequences given observations. The most probable hidden state sequence, determined from these probabilities, is taken as the best inference.
1. Markov processes can be used to model systems that transition between states based on probabilities. The document discusses several applications of Markov processes including calculating steady state probabilities, absorption probabilities, and expected times to absorption.
2. As an example, the document examines a phone company problem where calls arrive as a Poisson process and call durations are exponentially distributed. It shows how to set up and solve the balance equations to find steady state probabilities.
3. Other examples covered include finding the probability of an absent-minded professor getting wet during rain and calculating absorption probabilities and expected times for an example Markov chain. The document also discusses mean first passage and recurrence times.
Effect of a Linear Potential on the Temporal Diffraction of Particle in a BoxJohn Ray Martinez
The paper was accepted for oral presentation and published in SPP (Samahang Pisika ng Pilipinas) publication. This was presented during 27th SPP Congress.
The document discusses von Neumann entropy in quantum computation. It provides definitions of key terms like von Neumann entropy, density matrix, and computational complexity theory. Von Neumann entropy extends concepts of classical entropy to quantum mechanics and characterizes the classical and quantum information capacities of an ensemble. It quantifies the degree of mixing of a quantum state and how much a state departs from a pure state. The von Neumann entropy of a system is computed using the density matrix and eigendecomposition of the system's quantum state.
Parametric time domain system identification of a mass spring-damperMidoOoz
This document describes a laboratory experiment for an undergraduate system dynamics course to identify physical parameters of a mass-spring-damper system using parametric system identification. Students will collect step response data from the system under different mass configurations and use the data to determine damped natural frequency and damping ratio. Equations relating these parameters to the physical stiffness, mass, and damping values will then allow the students to estimate the physical parameters without disassembling the system. The goal is for students to understand that lumped parameter models are an approximation and will not perfectly match experimental data due to small nonlinearities in real systems.
Markov chains can be used for economic modeling. A Markov chain is characterized by: (1) possible states of a system, (2) a transition matrix showing the probability of moving between states, and (3) initial state probabilities. The transition matrix specifies the one-step probabilities between each pair of states. Markov chains can converge to a stationary distribution over time. Recurrent states will be revisited, while transient states will not. Markov chains can model topics like industry investment, consumption/saving over a lifecycle, and regime-switching in economic time series.
Invariant Manifolds, Passage through Resonance, Stability and a Computer Assi...Diego Tognola
1) The document is a dissertation submitted to ETH Zurich that studies invariant manifolds, passage through resonance, stability, and applies these concepts to a synchronous motor model.
2) It first develops theory for a general Hamiltonian system coupled to a linear system by weak periodic perturbations, showing the persistence of invariant manifolds. It then uses averaging techniques to analyze global dynamics, assuming a finite number of resonances.
3) It represents the reduced system in a way suitable for stability analysis, covering both non-degenerate and degenerate cases.
4) The second part applies these methods to explicitly model a miniature synchronous motor, analytically deriving approximations and numerically simulating and confirming the dynamics, showing approach
This document provides a summary of key concepts in nonlinear systems and control theory that are necessary background for subsequent chapters. It introduces notation used throughout the book and defines stability concepts such as Lyapunov stability, asymptotic stability, and exponential stability. It also summarizes Lyapunov's direct method, which allows determining stability properties of an equilibrium point from the properties of the system function f(x) and its relationship to a positive definite function V(x).
This document discusses discrete state space models. It begins with an introduction to state variable models and their generic structure. It then discusses various canonical forms for state space models including controllable, observable, and Jordan canonical forms. It also covers computing the characteristic equation, eigenvalues, state transition matrices using different techniques like inverse Laplace transform, similarity transformations, and Cayley-Hamilton theorem. Examples are provided to illustrate finding state space models from transfer functions and computing the state transition matrix.
Synchronizing Chaotic Systems - Karl DutsonKarl Dutson
The document discusses synchronizing chaotic systems like the logistic map and Lorenz system. It aims to investigate coupling more than two copies of a dynamical system and determine if synchronization can be described by higher-order Lyapunov exponents. The research will first examine the logistic map, find bifurcation points and fixed points analytically. It will then consider two coupled logistic maps and the parameter values that synchronize them in relation to the Lyapunov exponent. Finally, it will look at higher-dimensional coupled systems and relations between their synchronization and higher-order Lyapunov exponents.
This document provides an introduction to hidden Markov models (HMMs). It defines HMMs as an extension of Markov models that allows for observations that are probabilistic functions of hidden states. The core problems of HMMs are finding the probability of an observed sequence and determining the most probable hidden state sequence that produced an observation. HMMs have applications in areas like speech recognition by finding the most likely string of words given acoustic input using the Viterbi and forward algorithms.
This document proposes a method for estimating k sample survival functions under stochastic ordering constraints. It begins by reviewing existing work on estimating survival functions for two samples and extends this to k samples. The proposed method uses benchmark functions to estimate survival curves in a way that maintains stochastic ordering. It was tested on both uncensored and censored data and was shown to have low mean squared error and bias. The method was also applied to a real-world dataset with results comparable to previous work.
Supervised Hidden Markov Chains.
Here, we used the paper by Rabiner as a base for the presentation. Thus, we have the following three problems:
1.- How efficiently compute the probability given a model.
2.- Given an observation to which class it belongs
3.- How to find the parameters given data for training.
The first two follow Rabiner's explanation, but in the third one I used the Lagrange Multiplier Optimization because Rabiner lacks a clear explanation about how solving the issue.
This document discusses using transition diagrams to teach concepts in Markov chains. It defines key terms like transition matrix, transition diagram, n-step transition probabilities, first passage times, persistent/transient states, mean recurrence times, and periodic/aperiodic states. Examples are provided to show how these concepts can be understood and calculated from a transition diagram. The document argues transition diagrams provide a good technique for students to visualize and solve problems involving Markov chains.
This document provides a summary of Markov chains. It begins by defining stochastic processes and Markov chains. A Markov chain is a stochastic process where the probability of the next state depends only on the current state, not on the sequence of events that preceded it. The document discusses n-step transition probabilities, classification of states, and steady-state probabilities. It provides examples of Markov chains for cola purchases and camera store inventory to illustrate the concepts.
This document discusses Markov chains and Hidden Markov Models. It defines key properties of Markov chains including the Markov property and transition matrices. It provides examples of Markov chains for weather prediction and DNA sequences. Hidden Markov Models are introduced as having hidden states that can only be observed through output tokens. The difference between Markov chains and HMMs is explained. The document shows an example HMM for correlating tree ring size to temperature. It finds the optimal state sequences for this HMM using dynamic programming and the HMM equations. R code examples are provided for Markov chain transition matrices for DNA sequences.
This document provides an introduction to hidden Markov models (HMMs). It explains what HMMs are, where they are used, and why they are useful. Key aspects of HMMs covered include the Markov chain process, notation used in HMMs, an example of applying an HMM to temperature data, and the three main problems HMMs are used to solve: scoring observation sequences, finding optimal state sequences, and training a model. The document also outlines the forward, backward, and other algorithms used to efficiently solve these three problems.
This document summarizes key concepts related to Markov chains and linear algebra. It provides an example of using a transition matrix to model the probabilities of television viewers switching between two stations over time. The transition matrix allows calculating the probability vectors for future weeks through matrix multiplication. A steady-state vector can also be determined by solving the equation A*p=p, representing the long-term probabilities once the system reaches equilibrium.
This document discusses discrete-time Markov chains (DTMCs) and provides examples of DTMC modeling. It begins by defining DTMCs and contrasting them with continuous-time Markov chains (CTMCs). DTMCs can only transition between states at discrete time steps, while CTMCs can transition at any time. The document then defines DTMCs mathematically and discusses properties like homogeneity, the Markov property, and transition probability matrices. Examples of DTMC modeling are provided for problems like a machine repair facility and someone carrying umbrellas. The document concludes by discussing concepts like n-step transition probabilities, stationary distributions, limiting probabilities, and steady states.
Markov chain analysis uses Markov models to analyze randomly changing systems where future states only depend on the present state, not past states. A Markov chain has a fixed set of states, transition probabilities between states, and will converge to a unique long-run distribution. Markov chains assume states are fully observable and the system is autonomous. Common examples include weather patterns and gambling. Markov chains can be modeled and simulated using R packages like msm and markovchain.
This document provides an introduction to bootstrap methods and Markov chains. It discusses how bootstrap can be used to estimate properties of a statistic like mean or variance when the sample is small and assumptions of the central limit theorem may not apply. The basic bootstrap approach resamples the original sample with replacement to create new bootstrap samples and estimates the statistic for each. Markov chains are defined as stochastic processes where the next state only depends on the current state. An example of a 2-state Markov chain is provided along with notation for transition probabilities and computing unconditional probabilities. The document also discusses stationary distributions for Markov chains.
The document discusses Markov chains and their application to modeling transitions between states over time. It defines Markov chains as processes where the probability of the next state depends only on the current state. Transition matrices are used to represent the probabilities of moving between states. The powers of a transition matrix converge to a steady state as time increases, with all columns being identical, representing the long-term probabilities of being in each state. Finding the steady state vector involves solving the equation Tu=u. An example of modeling class attendance as a Markov chain is presented.
Hidden Markov models (HMMs) are probabilistic graphical models that allow prediction of a sequence of hidden states from observed variables. HMMs make the Markov assumption that the next state depends only on the current state, not past states. They require specification of transition probabilities between hidden states, emission probabilities of observations given states, and initial state probabilities to compute the joint probability of state sequences given observations. The most probable hidden state sequence, determined from these probabilities, is taken as the best inference.
1. Markov processes can be used to model systems that transition between states based on probabilities. The document discusses several applications of Markov processes including calculating steady state probabilities, absorption probabilities, and expected times to absorption.
2. As an example, the document examines a phone company problem where calls arrive as a Poisson process and call durations are exponentially distributed. It shows how to set up and solve the balance equations to find steady state probabilities.
3. Other examples covered include finding the probability of an absent-minded professor getting wet during rain and calculating absorption probabilities and expected times for an example Markov chain. The document also discusses mean first passage and recurrence times.
Effect of a Linear Potential on the Temporal Diffraction of Particle in a BoxJohn Ray Martinez
The paper was accepted for oral presentation and published in SPP (Samahang Pisika ng Pilipinas) publication. This was presented during 27th SPP Congress.
The document discusses von Neumann entropy in quantum computation. It provides definitions of key terms like von Neumann entropy, density matrix, and computational complexity theory. Von Neumann entropy extends concepts of classical entropy to quantum mechanics and characterizes the classical and quantum information capacities of an ensemble. It quantifies the degree of mixing of a quantum state and how much a state departs from a pure state. The von Neumann entropy of a system is computed using the density matrix and eigendecomposition of the system's quantum state.
Parametric time domain system identification of a mass spring-damperMidoOoz
This document describes a laboratory experiment for an undergraduate system dynamics course to identify physical parameters of a mass-spring-damper system using parametric system identification. Students will collect step response data from the system under different mass configurations and use the data to determine damped natural frequency and damping ratio. Equations relating these parameters to the physical stiffness, mass, and damping values will then allow the students to estimate the physical parameters without disassembling the system. The goal is for students to understand that lumped parameter models are an approximation and will not perfectly match experimental data due to small nonlinearities in real systems.
Markov chains can be used for economic modeling. A Markov chain is characterized by: (1) possible states of a system, (2) a transition matrix showing the probability of moving between states, and (3) initial state probabilities. The transition matrix specifies the one-step probabilities between each pair of states. Markov chains can converge to a stationary distribution over time. Recurrent states will be revisited, while transient states will not. Markov chains can model topics like industry investment, consumption/saving over a lifecycle, and regime-switching in economic time series.
Invariant Manifolds, Passage through Resonance, Stability and a Computer Assi...Diego Tognola
1) The document is a dissertation submitted to ETH Zurich that studies invariant manifolds, passage through resonance, stability, and applies these concepts to a synchronous motor model.
2) It first develops theory for a general Hamiltonian system coupled to a linear system by weak periodic perturbations, showing the persistence of invariant manifolds. It then uses averaging techniques to analyze global dynamics, assuming a finite number of resonances.
3) It represents the reduced system in a way suitable for stability analysis, covering both non-degenerate and degenerate cases.
4) The second part applies these methods to explicitly model a miniature synchronous motor, analytically deriving approximations and numerically simulating and confirming the dynamics, showing approach
This document provides a summary of key concepts in nonlinear systems and control theory that are necessary background for subsequent chapters. It introduces notation used throughout the book and defines stability concepts such as Lyapunov stability, asymptotic stability, and exponential stability. It also summarizes Lyapunov's direct method, which allows determining stability properties of an equilibrium point from the properties of the system function f(x) and its relationship to a positive definite function V(x).
This document discusses discrete state space models. It begins with an introduction to state variable models and their generic structure. It then discusses various canonical forms for state space models including controllable, observable, and Jordan canonical forms. It also covers computing the characteristic equation, eigenvalues, state transition matrices using different techniques like inverse Laplace transform, similarity transformations, and Cayley-Hamilton theorem. Examples are provided to illustrate finding state space models from transfer functions and computing the state transition matrix.
Synchronizing Chaotic Systems - Karl DutsonKarl Dutson
The document discusses synchronizing chaotic systems like the logistic map and Lorenz system. It aims to investigate coupling more than two copies of a dynamical system and determine if synchronization can be described by higher-order Lyapunov exponents. The research will first examine the logistic map, find bifurcation points and fixed points analytically. It will then consider two coupled logistic maps and the parameter values that synchronize them in relation to the Lyapunov exponent. Finally, it will look at higher-dimensional coupled systems and relations between their synchronization and higher-order Lyapunov exponents.
This document discusses state space analysis and related concepts. It defines state as a group of variables that summarize a system's history to predict future outputs. The minimum number of state variables required is equal to the number of storage elements in the system. These state variables form a state vector. The document also covers state space representation, diagonalization, solving state equations, the state transition matrix, and concepts of controllability and observability.
This document provides an overview of Lagrangian mechanics and constraints in classical mechanics. It defines different types of constraints including holonomic, non-holonomic, rheonomic, and scleronomic constraints. Generalized coordinates are introduced as a set of independent parameters that can describe the motion of a mechanical system with constraints. The configuration space is defined as a 3N-dimensional space where a point represents the configuration of a system of N particles. Constraints reduce the number of degrees of freedom from 3N coordinates to n generalized coordinates.
A Markov model assumes that the current state captures all relevant information for predicting the future. It can be used for language modeling by assigning probabilities to word sequences. Google's PageRank algorithm ranks web pages based on the principle that more authoritative pages, as determined by other pages linking to them, should rank higher. It models the probability of being on a page as a stationary distribution of a Markov chain defined by the link structure of the web.
The document discusses calculating and plotting the allocation of entropy for bipartite and tripartite quantum systems. It provides tables of entropy calculations for a bipartite system of |0> and |1> states, and checks that the results satisfy the subadditivity inequality. It also outlines the methodology to perform similar calculations and plots for other systems to visualize the convex cones of entropy allocation.
This document discusses additional topics related to discrete-time Markov chains, including:
1) Classifying states as recurrent, transient, periodic, or aperiodic;
2) Economic analysis of Markov chains by considering costs of states and transitions;
3) Calculating first passage times and steady-state probabilities.
As an example, it analyzes an insurance company Markov chain model with four states representing customer accident history to find the long-run average annual premium.
This document provides an introduction to Cartesian tensor analysis. It defines scalars, vectors, and second-order tensors, and explains how they can be represented by their components in a rectangular Cartesian coordinate system. Vectors transform according to the transformation matrix between two coordinate systems. The document also introduces suffix notation for representing tensor components and the summation convention for repeated indices.
A SURVEY OF MARKOV CHAIN MODELS IN LINGUISTICS APPLICATIONScsandit
Markov chain theory isan important tool in applied probability that is quite useful in modeling real-world computing applications.For a long time, rresearchers have used Markov chains for data modeling in a wide range of applications that belong to different fields such as computational linguists, image processing, communications,bioinformatics, finance systems, etc. This paper explores the Markov chain theory and its extension hidden Markov models (HMM) in natural language processing (NLP) applications. This paper also presents some aspects related to Markov chains and HMM such as creating transition matrices, calculating data sequence probabilities, and extracting the hidden states.
This document provides information about a unit on state-space analysis for an electrical engineering course. It includes the topics that will be covered such as state variables, state-space representation of transfer functions, state transition matrices, and controllability and observability. It defines key concepts like state, state vector, and state space. It explains the importance and advantages of state-space analysis over other methods like using transfer functions. The outcomes of the unit are to learn how to model systems in state-space form and analyze properties like controllability and observability.
This document summarizes key concepts from a lecture on stability and bifurcation theory of dynamical systems. It defines fundamental concepts like orbits, stability of equilibria and periodic orbits, and introduces bifurcation theory. Bifurcations occur when stability is lost as parameters change, resulting in qualitative changes to dynamics. The document discusses linear stability analysis and how bifurcations can involve changes in the number or stability of equilibria or periodic orbits. It distinguishes between static, dynamic, and mixed bifurcations that can occur at non-hyperbolic equilibrium points of parameter-dependent systems.
The document summarizes three statistical ensembles:
1) The microcanonical ensemble describes systems with a fixed number of particles, volume, and energy range. The entropy is determined from the number of accessible microstates.
2) The canonical ensemble describes systems with a fixed number of particles, volume, and temperature. The probability of a given energy state is determined by the Boltzmann factor.
3) The grand canonical ensemble describes systems with a variable number of particles. In addition to commuting with the Hamiltonian, the density operator must commute with the number operator. It is characterized by chemical potential and fugacity.
A system of linear equations determines the intersection of hyperplanes in an n-dimensional space, with solutions being a flat of any dimension. The behavior of a linear system depends on the number of equations and unknowns, with fewer equations than unknowns usually having infinitely many solutions, equal numbers usually having a single unique solution, and more equations than unknowns usually having no solution. Standard methods for solving systems include Gaussian elimination, Cramer's rule, and iterative methods for large systems.
The document summarizes systems of linear equations. It discusses how a system determines a collection of planes or hyperplanes in space, with the intersection point being the solution. It describes how a system can have infinitely many solutions, a single unique solution, or no solution, depending on the number of equations and variables. It also covers key concepts like independence, consistency, and equivalence regarding linear systems.
This lecture discusses oscillations in linear systems near equilibrium. It introduces the formulation of the eigenvalue equation to determine the normal modes and frequencies of free vibration. As an example, it analyzes the free vibrations of a linear triatomic molecule, modeling it as three masses connected by springs. Solving the eigenvalue equation yields three normal mode frequencies, one of which is zero corresponding to the center of mass motion.
This document contains 51 multiple choice questions about Markov analysis. Markov analysis is a technique that deals with predicting future probabilities based on currently known probabilities. It involves defining a set of possible states, determining transition probabilities between states, and using these to predict future state probabilities. Key concepts covered include the matrix of transition probabilities, equilibrium conditions, absorbing states, and using Markov analysis to model situations like market share changes, machine reliability, and weather patterns.
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1. 3.8 Fundamentals of Markov Chains
A Markov chain is a special class of state model. As with earlier state models, it consists of a
collection of states, only now we are modeling probabilities of transitions between states. The
weight assigned to each arrow is now interpreted as either the probability that something in the state
at the arrow's tail moves to the state at the arrow's head, or the percentage of things at the arrow's
tail which move to the state at the arrow's head. At each time step, something in one state must
either remain where it is or move to another state. Thus the sum of the arrows into (or out of) a state
must be one. The state vector X(i) in a Markov model traditionally lists either the probability that a
system is in a particular state at a particular time, or the percentage of the system which is in each
state at a given time. Thus X(t) is a probability distribution vector and must sum to one. We have
occasionally mentioned such vectors in what we have done before, but when dealing with a Markov
model we deal with probability distribution vectors exclusively. Recapping, there are three
properties which identify a state model as being a Markov model: 1) The Markov assumption: the
probability of one's moving from state i to state j is independent of what happened before moving to
state j and of how one got to state i. 2) Conservation: the sum of the probabilities out of a state must
be one. 3) The vector X(t) is a probability distribution vector which describes the probability of the
system's being in each of the states at time n.
In some sense, we have been assuming the Markov assumption all along. By this we mean that we
have been assuming that the number being assigned to a state during a time step depends only on
the way things were distributed during the prior time step and not any further back than that. This
was the fourth convention we made when defining state diagrams. Essentially it says that we are
considering only first-order recurrence relations. Strictly speaking the Markov assumption refers to
only probabilities, but we used equivalents of it with birth rates that were greater than one. When
discussing the probabilities associated with a Markov chain, the term conditional probability is
often used. Conditional probability means just the probability of something's happening given that
something else has already happened. In our case the probability of moving from state i to state j
assumes we were in state i to begin with, so, technically, this is a conditional probability.
The transition matrix for a Markov chain is then a matrix of probabilities (conditional probabilities
if we are perfectly correct) of moving from one state to another. Thus
We also require that the each column sums to one in order to satisfy the conservation property. The
system moves from states given by column indices to states given by row indices. For example, p21
is the probability of the system's moving from state 2 to state 1.
We can represent a Markov chain using a state diagram (Figure 3.12).
The transition probabilities pl:j are shown as the flows between states.
Stages, States, and Classes
1
2. FIGURE 3.12 General State diagram of a Markov Chain.
Consider the following transition matrix for a Markov chain:
There are three states for this chain, which we label i = 1,2,3. The state diagram for this chain is
shown in Figure 3.13.
Unlike models discussed earlier, the vector X(t) does not give the number of individuals in each
state at time t; rather it gives the probability that the system is in each state at time r. It is
conventional with Markov chains to denote X(t) as Xt. An initial distribution Xo, is a distribution for
the chance that the system is initially in each of the states. For instance, suppose
The interpretation X0 is that there is a 50% chance the system is initially in state 1, 30% chance it is
in state 2, and a 20% chance it is in state 3.
In this context, matrix multiplication gives the probability distribution one time step later. That is,
2
3. where X0 is an initial distribution. Using the transition matrix and initial distribution from above, we
have
so that after one time step, there is a 36% chance of the system's being in state 1, and 35% and 29%
chances of being in states 2 and 3 respectively. Using this notation, the distribution after t = n time
steps is given by
(62)
3
4. An important idea, which we make use of in the next two sections, is whether the sequence of
column vectors Xn, n > 1 converges to a steady-state (unchanging from time step to time step)
column vector . Determining allows us to answer long-term behavior questions we may
pose.
We observe here that if Xn —> as n —> oo, we must have the matrix Tn approaching some fixed
matrix L, that is
(63)
The matrix L, if it exists, is referred to as the "steady-state" matrix. The convergence of the matrix
Tn to the steady-state matrix L is independent of the initial distribution XQ, as-equation (63) shows.
The steady-state distribution and the steady-state matrix L can be shown to exist, provided that
the transition matrix T satisfies the property that some power has all positive entries. Matrices
satisfying this condition are called regular. If T is regular, we find the steady-state distribution
by solving the set of equations.
(64)
for , along with the condition that the sum of the entries in must be one. The matrix equation
(64) clearly conveys the idea that the steady-state distribution is a fixed point of the system of
equations (62). Equivalently, is an eigenvector with eigenvalue one. An intuitively appealing
method for determining the steady-state distribution , is to compute (or approximate) the steady-
state matrix L. Traditionally, this is done analytically using a method called matrix diagonalization.
Since
we approximate L by computing Tn for large values of n. This is easily done using a software
package, or, if the number of states is small, a calculator with matrix capabilities.
For our example, the steady-state matrix is approximately
(65)
The form of the matrix in equation (65) might at first glance appear surprising. If the steady-state
matrix L exists, it has the form given in (65) where each of the columns are identical. This fact
4
5. follows from the equation = LXo and recalling that the sum of the entries in the column vector
XQ isone. This equation also demonstrates that each column of L( ≈ Tn, for n large) is . Thus, for
our example, we have the steady-state distribution
There is another class of Markov chains which have important modeling properties. Consider the
following example of a transition matrix.
(66)
The state diagram for this transition matrix is shown in Figure 3.14.
This system has some important features. States 4 and 5 are called "absorbing" states. Once the
system enters an absorbing state, the system remains in that state from that time on. The absorbing
states are easily identified from the state diagram in that they have loops with weight one. States 4
and 5 both have loops with weight one.
Absorbing Markov chains are different in structure than those we have previously considered. An
absorbing state precludes the transition matrix from being regular. The assumption that the
transition matrix is regular is enough to ensure the existence of a steady-state matrix, but is not a
characterization. Steady-state matrices exist for absorbing.
FIGURE 3.14 State diagram for the Absorbing Markov Chain.
5
6. Markov chains, and the additional structure of the absorbing chains provides useful information.
The project section considers an example of a nonabsorbing, non-regular transition matrix for which
a steady-state matrix can be calculated.
If we compute the steady-state matrix for the above absorbing chain, we obtain
This matrix exhibits several properties that we need later on. Examining the structure of the
transition matrix T in equation (66), we see that it can be decomposed into blocks
of the form
The matrix I2x2 is just the 2 x 2 identity matrix, and if it was not obvious, we formed the blocks
around the identity matrix block. This decomposition is always possible for absorbing Markov
chains, though we may need to re-label the states so that the absorbing states are listed last (so the
identity matrix is in the proper position). In general, if a Markov chain has a absorbing states and b
nonabsorbing states, we can arrange the transition matrix to have the form
6
7. This block decomposition gives useful information about the absorbing Markov chain. The steady-
state matrix L has the form
The entries in the matrix represent the probability of being absorbed in ith
absorbing state if the system was initially in the jth nonabsorbing state. In the example,
These entries are viewed as "absorption" probabilities. For example, there is a 71.43% chance that
the system will be absorbed in state 4, given that it initially started in state 2. To understand which
state is which, refer back to the columns and rows of (67). The other entries have a similar
interpretation.
Further information is obtained from the fundamental matrix .The entries
of this matrix are the average number of times the process is in state j, given that it began in state i.
A proof of this result is in Olinick [48]. For our example, the fundamental matrix is
Recalling the block form of the transition matrix (68), the position of the submatrix A indicates that
i and j have values 1, 2, or 3, so that f1,1 = 1.25 is the average number of time steps that the system
is in state 1, given that it was initially in state 1. The other entries have analogous interpretations.
The sum of the entries of the jth column of the fundamental matrix F is the average number of time
steps for a process initially in state j to be absorbed. For example, if the system is initially in state 1,
it takes an average of 1.25 + 0.7143 + 0.7143 = 2.6786 time steps before the system enters an
absorbing state. The next two sections present models based upon Markov chains and use the above
analysis. The project section also contains some interesting Markov models, as well as some further
points of the theory of Markov chains.
3.9 Markovian Squirrels
The American gray squirrel (Sciurus carolinensis Gmelin) was introduced in Great Britain by a
series of releases from various sites starting in the late nineteenth century. In 1876, the first gray
7
8. squirrels were imported from North America, and have subsequently spread throughout England
and Wales, as well as parts of Scotland and Ireland.
Simultaneously, the native red squirrel (Sciurus vulgaris L.), considered the endemic subspecies,
has disappeared from most of the areas colonized by gray squirrels. Originally, the red squirrel was
distributed throughout Europe and eastward to northern China, Korea, and parts of the Japanese
archipelago. During the last century, the red squirrel has consistently declined, becoming extinct in
many areas of England and Wales, so that it is now confined almost solely to Northern England and
Scotland. A few isolated red squirrel populations exist on offshore islands in southern England and
mountainous Wales.
The introduction of the American gray squirrel continued until the early 1920s, by which time the
gray squirrels had rapidly spread throughout England. By 1930 it was apparent that the gray squirrel
was a pest in deciduous forests, and control measures were attempted. Once the pest status of the
gray squirrel was recognized, national distribution surveys were undertaken. The resulting
distribution maps clearly showed the tendency for the red squirrel to be lost from areas that had
been colonized by the gray squir during the preceding 15 to 20 years.
Since 1973, an annual questionnaire has been circulated to foresters by the British Forestry
Commission. The questionnaire concerns the presence or absence of the m squirrel species. It also
includes questions on the changes of squirrel abundance, details of tree damage, squirrel control
measures, and the number of squirrels killed. Using c data collected by the Forestry Commission,
we wish to construct a model to predict the trends in the distribution of both species of squirrels in
Great Britain.
Several researchers have studied the British squirrel populations, notably Reynolds [53]. and Usher
et al. [68]. The annual Forestry Commission data has been summarized in tl form of distribution
maps reflecting change over a two-year period.
Usher et al., [68] used an overlay technique to extract data from the distribution map Each 10-km
square on the overlay map that contained Forestry Commission land classified into one of four
states:
R: only red squirrels recorded in that year.
G: only gray squirrels recorded in that year.
B: both species of squirrels recorded in that year.
O: neither species of squirrels recorded in that year.
In order to satisfy the Markov assumption, squares that were present only in tv.: consecutive years
were counted. Counting the pairs of years, squares are allocated to any one of 16 classes, e.g., R ->
R, R -> G, G -> G, B -> O, etc.
A summary of these transition counts for each pair of years from 1973-74 to 1987-88 is given in
table 3.3 and is reprinted by permission of Blackwell Science Inc.
A frequency interpretation is required to employ the Markov chain analysis. If the entries in each
column are totaled, the corresponding matrix entry is found by division. For example, column R has
a total 2.529 + 61 + 282 + 3 = 2,875, so that the entry in the R, R position is 2,529/2,875 0.8797.
Care must be taken when calculating these frequencies. Inappropriate rounding will violate the
requirement that the columns sum to
8
9. TABLE 3.3 Red and Gray Squirrel Distribution Map Data for Great Britain.
3 Stages, States, and Classes
FIGURE 3.15 State diagram for the Markov Squirrels.
one. The transition matrix (rows and columns are in R, G, B, O order) is
The state diagram of this transition matrix T is given in Figure 3.15. We interpret these transition
frequencies as conditional probabilities. For example, there is an 87.97% chance that squares that
9
10. are currently in state R (red squirrels only) will remain in state R; similarly, there is a 2.73% chance
of squares that are currently occupied by both squirrel species, state G, will become occupied by
neither species, state B, after the next time step. Since the data taken from the annual Forestry
Commission survey is summarized as pairs of years, each time step represents a two-year period.
The matrix form of the transition probabilities is convenient for calculations. Using matrix
multiplication, we compute the two-time-step transition matrix as T2 = T x T, which is given by
The entries of this transition matrix are again interpreted as conditional probabilities. For instance,
there is a 17.33% chance that squares currently occupied by only red squirrels, state R, will be
occupied by both species, state B, in two time steps (four years).
Using the transition matrix T, it is possible to gain insight into the long-term behavior of the two
species of squirrels. We compute the steady-state matrix L for the two squirrel populations. The
question of interest in the study of the squirrel populations is what happens to the distribution of the
squirrel populations over a long period of time.
For our squirrel model, the steady-state matrix is approximately
Thus the steady-state distribution is
This result is interpreted as the long-term behavior of the squirrel populations in Great Britain as
follows: 17.05% of the squares will be in state R, containing only red squirrels. There will be 5.6%
of the squares in state G containing only gray squirrels. There will be populations of both squirrels
10
11. in 34.21% of the squares (state B), with the majority of the squares, 43.14%, being occupied by
neither species of squirrels (state O).
If the assumptions made in this model are correct, the red squirrel is not currently in danger. In fact,
it will have sole possession of more regions than the gray squirrel will have. In the long term, the
gray squirrels do not drive the reds to extinction. Actually this analysis says nothing about
population sizes, only about the number of regions controlled by each type of squirrel. While it
seems plausible that if the red squirrel territory (number of regions) is declining, then the population
is declining; the opposite may be true. A problem in the projects section asks you to perform this
analysis for the two squirrel species in Scotland, where the red squirrel is still widely distributed.
11