This document provides an introduction to hidden Markov models (HMMs). It begins by explaining Markov models and how they can be used to model weather prediction based on the previous day's weather. It then introduces HMMs, where the true state is hidden and can only be observed through probabilistic emissions or observations. An example HMM is provided for weather prediction where the true weather is hidden but can be observed through whether someone brings an umbrella. The key concepts of HMMs, including states, transition probabilities, emission probabilities, and the likelihood of state sequences given observations, are defined.
Quantum mechanics and the square root of the Brownian motionMarco Frasca
The document discusses taking the square root of Brownian motion and how it relates to quantum mechanics. It shows that defining the square root through stochastic integration reproduces the heat kernel and Schrodinger's equation. This indicates the process is doing quantum mechanics. The approach is generalized to include potentials, deriving the harmonic oscillator case. Finally, using Dirac's algebra trick and introducing additional Brownian motions, the formalism reproduces the Dirac equation and introduces spin naturally through stochastic behavior.
International Conference on Monte Carlo techniques
Closing conference of thematic cycle
Paris July 5-8th 2016
Campus les cordeliers
Jere Koskela's slides
Proceedings A Method For Finding Complete Observables In Classical Mechanicsvcuesta
1. The document presents a new method for finding complete observables in classical mechanics, which are gauge invariant quantities.
2. The method starts with partial observables and clocks, which are non-gauge invariant phase space functions. Using constants of motion, the partial observables can be written in terms of the clocks to obtain complete observables.
3. As an example, the method is applied to a particle in a gravitational field, where the Hamiltonian is used as a constant of motion to write the position variable as a function of the momentum and time.
The document discusses using k-nearest neighbors and KD-trees to create a computationally cheap approximation (πa) of an expensive-to-evaluate target distribution π. This approximation allows the use of delayed acceptance in a Metropolis-Hastings or pseudo-marginal Metropolis-Hastings algorithm to potentially reduce computation cost per iteration. Specifically, it describes:
1) Using a weighted average of the k nearest neighbor π values to define the approximation πa.
2) How delayed acceptance preserves the stationary distribution while mixing more slowly than standard MH.
3) Storing the evaluated π values in a KD-tree to enable fast lookup of the k nearest neighbors.
The document provides an outline for a course on quantum mechanics. It discusses key topics like the time-dependent Schrodinger equation, eigenvalues and eigenfunctions, boundary conditions for wave functions, and applications like the particle in a box model. Specific solutions to the Schrodinger equation are explored for stationary states with definite energy, including the wave function for a free particle and the quantization of energy for a particle confined to a one-dimensional box.
1) A quantum particle is described by a wave function ψ(x) which is a function of position. This provides a complete description of the particle's state.
2) The wave function does not indicate a precise position for the particle. Instead, the particle is considered delocalized, meaning it does not have a well-defined position more precise than the spread of the wave function.
3) Certain properties of the wave function, like whether it is nonzero in a particular region of space, provide some information about where the particle might be found if its position were measured. But the particle does not have a well-defined position until a measurement is made.
TIME-ABSTRACTING BISIMULATION FOR MARKOVIAN TIMED AUTOMATAijseajournal
Markovian timed automata (MTA) has been proposed as an expressive formalism for specification of realtime
properties in Markovian stochastic processes. In this paper, we define bisimulation relation for
deterministic MTA. This definition provides a basis for developing effective algorithms for deciding
bisimulation relation for such automata.
On Application of Unbounded Hilbert Linear Operators in Quantum MechanicsBRNSS Publication Hub
This research work presents an important Banach space in functional analysis which is known and called
Hilbert space. We verified the crucial operations in this space and their applications in physics, particularly
in quantum mechanics. The operations are restricted to the unbounded linear operators densely defined
in Hilbert space which is the case of prime interest in physics, precisely in quantum machines. Precisely,
we discuss the role of unbounded linear operators in quantum mechanics, particularly, in the study of
Heisenberg uncertainty principle, time-independent Schrödinger equation, Harmonic oscillation, and
finally, the application of Hamilton operator. To make these analyses fruitful, the knowledge of Hilbert
spaces was first investigated followed by the spectral theory of unbounded operators, which are claimed
to be densely defined in Hilbert space. Consequently, the theory of probability is also employed to study
some systems since the operators used in studying these systems are only dense in H (i.e., they must (or
probably) be in the domain of H defined by L2 ( ) −∞,+∞ ).
Quantum mechanics and the square root of the Brownian motionMarco Frasca
The document discusses taking the square root of Brownian motion and how it relates to quantum mechanics. It shows that defining the square root through stochastic integration reproduces the heat kernel and Schrodinger's equation. This indicates the process is doing quantum mechanics. The approach is generalized to include potentials, deriving the harmonic oscillator case. Finally, using Dirac's algebra trick and introducing additional Brownian motions, the formalism reproduces the Dirac equation and introduces spin naturally through stochastic behavior.
International Conference on Monte Carlo techniques
Closing conference of thematic cycle
Paris July 5-8th 2016
Campus les cordeliers
Jere Koskela's slides
Proceedings A Method For Finding Complete Observables In Classical Mechanicsvcuesta
1. The document presents a new method for finding complete observables in classical mechanics, which are gauge invariant quantities.
2. The method starts with partial observables and clocks, which are non-gauge invariant phase space functions. Using constants of motion, the partial observables can be written in terms of the clocks to obtain complete observables.
3. As an example, the method is applied to a particle in a gravitational field, where the Hamiltonian is used as a constant of motion to write the position variable as a function of the momentum and time.
The document discusses using k-nearest neighbors and KD-trees to create a computationally cheap approximation (πa) of an expensive-to-evaluate target distribution π. This approximation allows the use of delayed acceptance in a Metropolis-Hastings or pseudo-marginal Metropolis-Hastings algorithm to potentially reduce computation cost per iteration. Specifically, it describes:
1) Using a weighted average of the k nearest neighbor π values to define the approximation πa.
2) How delayed acceptance preserves the stationary distribution while mixing more slowly than standard MH.
3) Storing the evaluated π values in a KD-tree to enable fast lookup of the k nearest neighbors.
The document provides an outline for a course on quantum mechanics. It discusses key topics like the time-dependent Schrodinger equation, eigenvalues and eigenfunctions, boundary conditions for wave functions, and applications like the particle in a box model. Specific solutions to the Schrodinger equation are explored for stationary states with definite energy, including the wave function for a free particle and the quantization of energy for a particle confined to a one-dimensional box.
1) A quantum particle is described by a wave function ψ(x) which is a function of position. This provides a complete description of the particle's state.
2) The wave function does not indicate a precise position for the particle. Instead, the particle is considered delocalized, meaning it does not have a well-defined position more precise than the spread of the wave function.
3) Certain properties of the wave function, like whether it is nonzero in a particular region of space, provide some information about where the particle might be found if its position were measured. But the particle does not have a well-defined position until a measurement is made.
TIME-ABSTRACTING BISIMULATION FOR MARKOVIAN TIMED AUTOMATAijseajournal
Markovian timed automata (MTA) has been proposed as an expressive formalism for specification of realtime
properties in Markovian stochastic processes. In this paper, we define bisimulation relation for
deterministic MTA. This definition provides a basis for developing effective algorithms for deciding
bisimulation relation for such automata.
On Application of Unbounded Hilbert Linear Operators in Quantum MechanicsBRNSS Publication Hub
This research work presents an important Banach space in functional analysis which is known and called
Hilbert space. We verified the crucial operations in this space and their applications in physics, particularly
in quantum mechanics. The operations are restricted to the unbounded linear operators densely defined
in Hilbert space which is the case of prime interest in physics, precisely in quantum machines. Precisely,
we discuss the role of unbounded linear operators in quantum mechanics, particularly, in the study of
Heisenberg uncertainty principle, time-independent Schrödinger equation, Harmonic oscillation, and
finally, the application of Hamilton operator. To make these analyses fruitful, the knowledge of Hilbert
spaces was first investigated followed by the spectral theory of unbounded operators, which are claimed
to be densely defined in Hilbert space. Consequently, the theory of probability is also employed to study
some systems since the operators used in studying these systems are only dense in H (i.e., they must (or
probably) be in the domain of H defined by L2 ( ) −∞,+∞ ).
This document discusses Markov chain Monte Carlo (MCMC) methods. It begins with an outline of the Metropolis-Hastings algorithm, which is a generic MCMC method for obtaining a sequence of random samples from a probability distribution when direct sampling is difficult. The document then provides details on the Metropolis-Hastings algorithm, including its convergence properties. It also discusses the independent Metropolis-Hastings algorithm as a special case and provides an example to illustrate it.
This document provides an introduction and overview of quantum Monte Carlo methods. It begins by reviewing the Metropolis algorithm and how it can be used to evaluate integrals and quantum mechanical operators. It then outlines the key topics which will be covered, including the path integral formulation of quantum mechanics, diffusion Monte Carlo, and calculating the one-body density matrix and excitation energies. The document proceeds to explain how the path integral formulation leads to the Schrodinger equation in the limit of small time steps, and how imaginary time evolution can be used to project out the ground state wavefunction. It concludes by providing examples of applying these methods to calculate properties of hydrogen, molecular hydrogen, and the one-body density matrix of silicon.
This paper studies an approximate dynamic programming (ADP) strategy of a group of nonlinear switched systems, where the external disturbances are considered. The neural network (NN) technique is regarded to estimate the unknown part of actor as well as critic to deal with the corresponding nominal system. The training technique is simul-taneously carried out based on the solution of minimizing the square error Hamilton function. The closed system’s tracking error is analyzed to converge to an attraction region of origin point with the uniformly ultimately bounded (UUB) description. The simulation results are implemented to determine the effectiveness of the ADP based controller.
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.
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 developing near-optimal state feedback controllers for nonlinear discrete-time systems using iterative approximate dynamic programming (ADP) algorithms. Specifically:
1) An infinite-horizon optimal state feedback controller is developed for discrete-time systems based on the dual heuristic programming (DHP) algorithm.
2) A new optimal control scheme is developed using the generalized DHP (GDHP) algorithm and a discounted cost functional.
3) An infinite-horizon optimal stabilizing state feedback controller is designed based on the globalized dual heuristic programming (GHJB) algorithm.
4) Finite-horizon optimal controllers with an ε-error bound are proposed, where the number of optimal control steps can be determined
I am Keziah D. I am a Mechanical Engineering Assignment Expert at matlabassignmentexperts.com. I hold a Ph.D. Matlab, University of North Carolina, USA. I have been helping students with their homework for the past 8 years. I solve assignments related to Mechanical Engineering.
Visit matlabassignmentexperts.com or email info@matlabassignmentexperts.com.
You can also call on +1 678 648 4277 for any assistance with Mechanical Engineering Assignments.
1) The document discusses the exact solution to the Klein-Gordon shutter problem, finding that the wave function does not resemble the optical expression for diffraction but the charge density does show transient oscillations resembling a diffraction pattern.
2) It presents the exact solution for the Klein-Gordon shutter problem using discontinuous initial conditions, finding the wave function solution differs from Moshinsky's approximation.
3) When the exact relativistic charge density is plotted over time, it shows transient oscillations that resemble a diffraction pattern, despite some relativistic differences, demonstrating that diffraction in time exists in relativistic scenarios.
Markov chain Monte Carlo methods and some attempts at parallelizing themPierre Jacob
Markov chain Monte Carlo (MCMC) methods are commonly used to approximate properties of target probability distributions. However, MCMC estimators are generally biased for any fixed number of samples. The document discusses various techniques for constructing unbiased estimators from MCMC output, including regeneration, sequential Monte Carlo samplers, and coupled Markov chains. Specifically, running two Markov chains in parallel and taking the difference in their values at meeting times can yield an unbiased estimator, though certain conditions must hold.
This document summarizes several numerical methods for solving the advection and wave equations, including:
1) FTCS (Forward Time Centered Space), which is unconditionally unstable. Lax and Lax-Wendroff add diffusion terms to stabilize FTCS.
2) CTCS (Centered Time Centered Space), which is conditionally stable for Courant numbers ≤ 1.
3) Upwinding and Beam-Warming methods, which use points trailing the wave to ensure stability for large Courant numbers.
4) The Box method, which is stable for any Courant number by using points at multiple time levels.
Boundary conditions for the wave equation
The document provides an introduction to Brownian motion by starting with a one-dimensional discrete case modeled as a drunk walking randomly. It shows that Brownian motion has the properties of being memory-less, homogeneous in time and space. By taking the limit of discrete steps, the model arrives at continuous Brownian motion described by a partial differential equation. The document then briefly outlines the history of Brownian motion from its discovery to developments in modeling it as a stochastic process.
EM 알고리즘을 jensen's inequality부터 천천히 잘 설명되어있다
이것을 보면, LDA의 Variational method로 학습하는 방식이 어느정도 이해가 갈 것이다.
옛날 Andrew Ng 선생님의 강의노트에서 발췌한 건데 5년전에 본 것을
아직도 찾아가면서 참고하면서 해야 된다는 게 그 강의가 얼마나 명강의였는지 새삼 느끼게 된다.
This document provides an introduction to quantum Monte Carlo methods. It discusses using Monte Carlo integration to evaluate multi-dimensional integrals that arise in quantum mechanical problems. Variational Monte Carlo is introduced as using a trial wavefunction to sample configuration space and estimate observables, like the energy. The Metropolis algorithm is described as a way to generate Markov chains that sample a given probability distribution. This allows using Monte Carlo methods to solve the electronic structure problem by approximating many-body wavefunctions and integrals over configuration space.
Paolo Creminelli "Dark Energy after GW170817"SEENET-MTP
GW170817 and GRB170817A were detected simultaneously, originating from the same source. This provides a tight limit on the speed of gravitational waves and photons, showing they travel at the same speed to a high degree of precision. The effective field theory of dark energy provides a framework to parametrize possible deviations from general relativity at cosmological scales. It allows gravitational waves and photons to potentially travel at different speeds depending on the coefficients in the effective field theory action. The detection of GW170817 and GRB170817A from the same source places strong constraints on these coefficients and possible dark energy models.
Application of Schrodinger Equation to particle in one Dimensional Box: Energ...limbraj Ravangave
It is very interesting application of Schrodinger equation.. we find the solution to Schrodinger equation for particle moving under some type of interaction. the motion of particle in one dimensional box is to and fro motion in uniform potential. the problem is explore in easy way and better for understanding to the students.
visit our blog
https://elearnerphysics.blogspot.com/
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.
This document discusses linear response theory and time-dependent density functional theory (TDDFT) for calculating absorption spectroscopy. It begins by motivating the use of absorption spectroscopy to study many-body effects. It then outlines how to calculate the response of a system to a perturbation within linear response theory and the Kubo formula. The document discusses using TDDFT to include electron correlation effects beyond the independent particle and time-dependent Hartree approximations. It emphasizes that TDDFT provides an exact framework for calculating neutral excitations if the correct exchange-correlation functional is used.
A Fast Algorithm for Solving Scalar Wave Scattering Problem by Billions of Pa...A G
This document proposes a fast algorithm for solving wave scattering problems involving billions of particles using the convolution theorem and fast Fourier transforms (FFTs). The algorithm represents the Green's function as a vector and stores particle positions on a uniform grid, allowing the scattering calculation to be computed as a 3D convolution. This convolution can be rapidly evaluated using FFTs, significantly improving the efficiency over direct matrix-vector multiplication. The algorithm distributes data across multiple machines in a cluster to parallelize the computations.
I am Leonard K. I am a Stochastic Processes Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from, University of Arkansas, USA.
I have been helping students with their homework for the past 6 years. I solve assignments related to Stochastic Processes.
Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Stochastic Processes Assignments.
1) The theorem of least work states that for statically indeterminate structures, the partial derivative of the total strain energy with respect to redundant/statically indeterminate actions must be equal to zero.
2) This is because redundant forces act to prevent any displacement at their point of application. The forces developed in a redundant structure minimize the total internal strain energy.
3) The theorem is proved by analyzing a statically indeterminate beam as the superposition of a determinate beam with applied loads and a determinate beam with the redundant reaction. Equating the deflections caused by each case results in the condition that the strain energy is minimized.
This document discusses hidden Markov models (HMMs). It begins by covering Markov chains and Markov models. It then defines HMMs, noting that they have hidden states that can produce observable outputs. The key components of an HMM - the transition probabilities, observation probabilities, and initial probabilities - are explained. Examples of HMMs for weather and marble jars are provided to illustrate calculating probabilities. The main issues in using HMMs are identified as evaluation, decoding, and learning. Evaluation calculates the probability of an observation sequence given a model. Decoding finds the most likely state sequence that produced an observation sequence. Learning determines model parameters to best fit training data.
This document provides an introduction to Markov models and hidden Markov models. It explains that Markov models make the assumption that the probability of future states depends only on the present state, not on the sequence of events that preceded it. This allows weather prediction to be modeled based on the probability of today's weather given only yesterday's weather. Hidden Markov models add the complexity that the true states are hidden and can only be inferred from observable events, like whether an umbrella was carried based on the actual sunny, rainy, or foggy weather. The document gives examples of calculating state probabilities using these types of models.
This document discusses Markov chain Monte Carlo (MCMC) methods. It begins with an outline of the Metropolis-Hastings algorithm, which is a generic MCMC method for obtaining a sequence of random samples from a probability distribution when direct sampling is difficult. The document then provides details on the Metropolis-Hastings algorithm, including its convergence properties. It also discusses the independent Metropolis-Hastings algorithm as a special case and provides an example to illustrate it.
This document provides an introduction and overview of quantum Monte Carlo methods. It begins by reviewing the Metropolis algorithm and how it can be used to evaluate integrals and quantum mechanical operators. It then outlines the key topics which will be covered, including the path integral formulation of quantum mechanics, diffusion Monte Carlo, and calculating the one-body density matrix and excitation energies. The document proceeds to explain how the path integral formulation leads to the Schrodinger equation in the limit of small time steps, and how imaginary time evolution can be used to project out the ground state wavefunction. It concludes by providing examples of applying these methods to calculate properties of hydrogen, molecular hydrogen, and the one-body density matrix of silicon.
This paper studies an approximate dynamic programming (ADP) strategy of a group of nonlinear switched systems, where the external disturbances are considered. The neural network (NN) technique is regarded to estimate the unknown part of actor as well as critic to deal with the corresponding nominal system. The training technique is simul-taneously carried out based on the solution of minimizing the square error Hamilton function. The closed system’s tracking error is analyzed to converge to an attraction region of origin point with the uniformly ultimately bounded (UUB) description. The simulation results are implemented to determine the effectiveness of the ADP based controller.
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.
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 developing near-optimal state feedback controllers for nonlinear discrete-time systems using iterative approximate dynamic programming (ADP) algorithms. Specifically:
1) An infinite-horizon optimal state feedback controller is developed for discrete-time systems based on the dual heuristic programming (DHP) algorithm.
2) A new optimal control scheme is developed using the generalized DHP (GDHP) algorithm and a discounted cost functional.
3) An infinite-horizon optimal stabilizing state feedback controller is designed based on the globalized dual heuristic programming (GHJB) algorithm.
4) Finite-horizon optimal controllers with an ε-error bound are proposed, where the number of optimal control steps can be determined
I am Keziah D. I am a Mechanical Engineering Assignment Expert at matlabassignmentexperts.com. I hold a Ph.D. Matlab, University of North Carolina, USA. I have been helping students with their homework for the past 8 years. I solve assignments related to Mechanical Engineering.
Visit matlabassignmentexperts.com or email info@matlabassignmentexperts.com.
You can also call on +1 678 648 4277 for any assistance with Mechanical Engineering Assignments.
1) The document discusses the exact solution to the Klein-Gordon shutter problem, finding that the wave function does not resemble the optical expression for diffraction but the charge density does show transient oscillations resembling a diffraction pattern.
2) It presents the exact solution for the Klein-Gordon shutter problem using discontinuous initial conditions, finding the wave function solution differs from Moshinsky's approximation.
3) When the exact relativistic charge density is plotted over time, it shows transient oscillations that resemble a diffraction pattern, despite some relativistic differences, demonstrating that diffraction in time exists in relativistic scenarios.
Markov chain Monte Carlo methods and some attempts at parallelizing themPierre Jacob
Markov chain Monte Carlo (MCMC) methods are commonly used to approximate properties of target probability distributions. However, MCMC estimators are generally biased for any fixed number of samples. The document discusses various techniques for constructing unbiased estimators from MCMC output, including regeneration, sequential Monte Carlo samplers, and coupled Markov chains. Specifically, running two Markov chains in parallel and taking the difference in their values at meeting times can yield an unbiased estimator, though certain conditions must hold.
This document summarizes several numerical methods for solving the advection and wave equations, including:
1) FTCS (Forward Time Centered Space), which is unconditionally unstable. Lax and Lax-Wendroff add diffusion terms to stabilize FTCS.
2) CTCS (Centered Time Centered Space), which is conditionally stable for Courant numbers ≤ 1.
3) Upwinding and Beam-Warming methods, which use points trailing the wave to ensure stability for large Courant numbers.
4) The Box method, which is stable for any Courant number by using points at multiple time levels.
Boundary conditions for the wave equation
The document provides an introduction to Brownian motion by starting with a one-dimensional discrete case modeled as a drunk walking randomly. It shows that Brownian motion has the properties of being memory-less, homogeneous in time and space. By taking the limit of discrete steps, the model arrives at continuous Brownian motion described by a partial differential equation. The document then briefly outlines the history of Brownian motion from its discovery to developments in modeling it as a stochastic process.
EM 알고리즘을 jensen's inequality부터 천천히 잘 설명되어있다
이것을 보면, LDA의 Variational method로 학습하는 방식이 어느정도 이해가 갈 것이다.
옛날 Andrew Ng 선생님의 강의노트에서 발췌한 건데 5년전에 본 것을
아직도 찾아가면서 참고하면서 해야 된다는 게 그 강의가 얼마나 명강의였는지 새삼 느끼게 된다.
This document provides an introduction to quantum Monte Carlo methods. It discusses using Monte Carlo integration to evaluate multi-dimensional integrals that arise in quantum mechanical problems. Variational Monte Carlo is introduced as using a trial wavefunction to sample configuration space and estimate observables, like the energy. The Metropolis algorithm is described as a way to generate Markov chains that sample a given probability distribution. This allows using Monte Carlo methods to solve the electronic structure problem by approximating many-body wavefunctions and integrals over configuration space.
Paolo Creminelli "Dark Energy after GW170817"SEENET-MTP
GW170817 and GRB170817A were detected simultaneously, originating from the same source. This provides a tight limit on the speed of gravitational waves and photons, showing they travel at the same speed to a high degree of precision. The effective field theory of dark energy provides a framework to parametrize possible deviations from general relativity at cosmological scales. It allows gravitational waves and photons to potentially travel at different speeds depending on the coefficients in the effective field theory action. The detection of GW170817 and GRB170817A from the same source places strong constraints on these coefficients and possible dark energy models.
Application of Schrodinger Equation to particle in one Dimensional Box: Energ...limbraj Ravangave
It is very interesting application of Schrodinger equation.. we find the solution to Schrodinger equation for particle moving under some type of interaction. the motion of particle in one dimensional box is to and fro motion in uniform potential. the problem is explore in easy way and better for understanding to the students.
visit our blog
https://elearnerphysics.blogspot.com/
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.
This document discusses linear response theory and time-dependent density functional theory (TDDFT) for calculating absorption spectroscopy. It begins by motivating the use of absorption spectroscopy to study many-body effects. It then outlines how to calculate the response of a system to a perturbation within linear response theory and the Kubo formula. The document discusses using TDDFT to include electron correlation effects beyond the independent particle and time-dependent Hartree approximations. It emphasizes that TDDFT provides an exact framework for calculating neutral excitations if the correct exchange-correlation functional is used.
A Fast Algorithm for Solving Scalar Wave Scattering Problem by Billions of Pa...A G
This document proposes a fast algorithm for solving wave scattering problems involving billions of particles using the convolution theorem and fast Fourier transforms (FFTs). The algorithm represents the Green's function as a vector and stores particle positions on a uniform grid, allowing the scattering calculation to be computed as a 3D convolution. This convolution can be rapidly evaluated using FFTs, significantly improving the efficiency over direct matrix-vector multiplication. The algorithm distributes data across multiple machines in a cluster to parallelize the computations.
I am Leonard K. I am a Stochastic Processes Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from, University of Arkansas, USA.
I have been helping students with their homework for the past 6 years. I solve assignments related to Stochastic Processes.
Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Stochastic Processes Assignments.
1) The theorem of least work states that for statically indeterminate structures, the partial derivative of the total strain energy with respect to redundant/statically indeterminate actions must be equal to zero.
2) This is because redundant forces act to prevent any displacement at their point of application. The forces developed in a redundant structure minimize the total internal strain energy.
3) The theorem is proved by analyzing a statically indeterminate beam as the superposition of a determinate beam with applied loads and a determinate beam with the redundant reaction. Equating the deflections caused by each case results in the condition that the strain energy is minimized.
This document discusses hidden Markov models (HMMs). It begins by covering Markov chains and Markov models. It then defines HMMs, noting that they have hidden states that can produce observable outputs. The key components of an HMM - the transition probabilities, observation probabilities, and initial probabilities - are explained. Examples of HMMs for weather and marble jars are provided to illustrate calculating probabilities. The main issues in using HMMs are identified as evaluation, decoding, and learning. Evaluation calculates the probability of an observation sequence given a model. Decoding finds the most likely state sequence that produced an observation sequence. Learning determines model parameters to best fit training data.
This document provides an introduction to Markov models and hidden Markov models. It explains that Markov models make the assumption that the probability of future states depends only on the present state, not on the sequence of events that preceded it. This allows weather prediction to be modeled based on the probability of today's weather given only yesterday's weather. Hidden Markov models add the complexity that the true states are hidden and can only be inferred from observable events, like whether an umbrella was carried based on the actual sunny, rainy, or foggy weather. The document gives examples of calculating state probabilities using these types of models.
Hidden Markov Model - The Most Probable PathLê Hòa
This document provides an overview of hidden Markov models including:
- The components of hidden Markov models including states, transition probabilities, emission probabilities, and observation sequences.
- How the Viterbi algorithm can be used to find the most probable hidden state sequence that explains an observed sequence by calculating likelihoods recursively and backtracking through the model.
- An example application of the Viterbi algorithm to find the most probable hidden weather sequence given observed data from a weather HMM model.
This document provides an overview of Markov models and hidden Markov models (HMMs). It begins by introducing Markov chains, which are probabilistic state machines where the probability of transitioning to the next state depends only on the current state. Hidden Markov models extend Markov chains by adding hidden states that are not directly observable. The key aspects of HMMs are defined, including the hidden states, observed outputs, transition probabilities, and output probabilities. The document then discusses how to compute the likelihood of an observed sequence given an HMM, including using the forward algorithm to efficiently sum over all possible state sequences. Overall, the document provides a conceptual introduction to Markov models and HMMs, focusing on their structure, assumptions, and the forward algorithm
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2.- Given an observation to which class it belongs
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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.
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Characterization of Subsurface Heterogeneity: Integration of Soft and Hard In...Amro Elfeki
Park, E., Elfeki, A. M. M., Dekking, F.M. (2003). Characterization of subsurface heterogeneity: Integration of soft and hard information using multi-dimensional Coupled Markov chain approach. Underground Injection Science and Technology Symposium, Lawrence Berkeley National Lab., October 22-25, 2003. p.49. Eds. Tsang, Chin.-Fu and Apps, John A.
http://www.lbl.gov/Conferences/UIST/index.html#topics
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Introduction to Bootstrap and elements of Markov Chains
Hmm tutexamplesalgo
1. Hidden Markov Models
A Tutorial for the Course Computational Intelligence
http://www.igi.tugraz.at/lehre/CI
Barbara Resch (modified by Erhard and Car line Rank)
Signal Processing and Speech Communication Laboratory
Inffeldgasse 16c/II
phone 873–4436
Abstract
This tutorial gives a gentle introduction to Markov models and hidden Markov models (HMMs) and relates
them to their use in automatic speech recognition. Additionally, the Viterbi algorithm is considered,
relating the most likely state sequence of a HMM to a given sequence of observations.
Usage
To make full use of this tutorial you should
1. Download the file HMM.zip1 which contains this tutorial and the accompanying Matlab programs.
2. Unzip HMM.zip which will generate a subdirectory named HMM/matlab where you can find all the
Matlab programs.
3. Add the folder HMM/matlab and the subfolders to the Matlab search path with a command
like addpath(’C:WorkHMMmatlab’) if you are using a Windows machine or addpath(’/home/-
jack/HMM/matlab’) if you are using a Unix/Linux machine.
Sources
This tutorial is based on
• “Markov Models and Hidden Markov Models - A Brief Tutorial” International Computer Science
Institute Technical Report TR-98-041, by Eric Fosler-Lussier,
• EPFL lab notes “Introduction to Hidden Markov Models” by Herv´ Bourlard, Sacha Krstulovi´,
e c
and Mathew Magimai-Doss, and
• HMM-Toolbox (also included in BayesNet Toolbox) for Matlab by Kevin Murphy.
1 Markov Models
Let’s talk about the weather. Let’s say in Graz, there are three types of weather: sunny , rainy , and
foggy . Let’s assume for the moment that the weather lasts all day, i.e., it doesn’t change from rainy
to sunny in the middle of the day.
Weather prediction is about trying to guess what the weather will be like tomorrow based on the ob-
servations of the weather in the past (the history). Let’s set up a statistical model for weather prediction:
We collect statistics on what the weather qn is like today (on day n) depending on what the weather
1 http://www.igi.tugraz.at/lehre/CI/tutorials/HMM.zip
1
2. was like yesterday qn−1 , the day before qn−2 , and so forth. We want to find the following conditional
probabilities
P (qn |qn−1 , qn−2 , ..., q1 ), (1)
meaning, the probability of the unknown weather at day n, qn ∈ { , , }, depending on the (known)
weather qn−1 , qn−2 , . . . of the past days.
Using the probability in eq. 1, we can make probabilistic predictions of the type of weather for
tomorrow and the next days using the observations of the weather history. For example, if we knew that
the weather for the past three days was { , , } in chronological order, the probability that tomorrow
would be is given by:
P (q4 = |q3 = , q2 = , q1 = ). (2)
This probability could be inferred from the relative frequency (the statistics) of past observations of
weather sequences { , , , }.
Here’s one problem: the larger n is, the more observations we must collect. Suppose that n = 6,
then we have to collect statistics for 3(6−1) = 243 past histories. Therefore, we will make a simplifying
assumption, called the Markov assumption:
For a sequence {q1 , q2 , ..., qn }:
P (qn |qn−1 , qn−2 , ..., q1 ) = P (qn |qn−1 ). (3)
This is called a first-order Markov assumption: we say that the probability of a certain observation at
time n only depends on the observation qn−1 at time n − 1. (A second-order Markov assumption would
have the probability of an observation at time n depend on qn−1 and qn−2 . In general, when people
talk about a Markov assumption, they usually mean the first-order Markov assumption.) A system for
which eq. 3 is true is a (first-order) Markov model, and an output sequence {qi } of such a system is a
(first-order) Markov chain.
We can also express the probability of a certain sequence {q1 , q2 , . . . , qn } (the joint probability of
certain past and current observations) using the Markov assumption:2
n
P (q1 , ..., qn ) = P (qi |qi−1 ). (4)
i=1
The Markov assumption has a profound affect on the number of histories that we have to find statistics
for – we now only need 3 · 3 = 9 numbers (P (qn |qn−1 ) for every possible combination of qn , qn−1 ∈
{ , , }) to characterize the probabilities of all possible sequences. The Markov assumption may or
may not be a valid assumption depending on the situation (in the case of weather, it’s probably not
valid), but it is often used to simplify modeling.
So let’s arbitrarily pick some numbers for P (qtomorrow |qtoday ), as given in table 1 (note, that – whatever
the weather is today – there certainly is some kind of weather tomorrow, so the probabilities in every
row of table 1 sum up to one).
Tomorrow’s weather
Today’s weather
0.8 0.05 0.15
0.2 0.6 0.2
0.2 0.3 0.5
Table 1: Probabilities p(qn+1 |qn ) of tomorrow’s weather based on today’s weather
For first-order Markov models, we can use these probabilities to draw a probabilistic finite state
automaton. For the weather domain, we would have three states, S = { , , }, and every day there
would be a possibility p(qn |qn−1 ) of a transition to a (possibly different) state according to the probabilities
in table 1. Such an automaton would look like shown in figure 1.
2 One question that comes to mind is “What is q ?” In general, one can think of q as the start word, so P (q |q )
0 0 1 0
is the probability that q1 can start a sentence. We can also just multiply a prior probability of q1 with the product of
n
i=2 P (qi |qi−1 ) , it’s just a matter of definitions.
2
3. 0.8 0.5
0.15
Sunny Foggy
0.2
0.05 0.3
0.2 0.2
Rainy
0.6
Figure 1: Markov model for the Graz weather with state transition probabilities according to table 1
1.0.1 Examples
1. Given that today the weather is , what’s the probability that tomorrow is and the day after is
?
Using the Markov assumption and the probabilities in table 1, this translates into:
P (q2 = , q3 = |q1 = ) = P (q3 = |q2 = , q1 = ) · P (q2 = |q1 = )
= P (q3 = |q2 = ) · P (q2 = |q1 = ) (Markov assumption)
= 0.05 · 0.8
= 0.04
You can also think about this as moving through the automaton (figure 1), multiplying the proba-
bilities along the path you go.
2. Assume, the weather yesterday was q1 = , and today it is q2 = , what is the probability that
tomorrow it will be q3 = ?
P (q3 = |q2 = , q1 = ) = P (q3 = |q2 = ) (Markov assumption)
= 0.2.
3. Given that the weather today is q1 = , what is the probability that it will be two days from
now: q3 = . (Hint: There are several ways to get from today to two days from now. You
have to sum over these paths.)
2 Hidden Markov Models (HMMs)
So far we heard of the Markov assumption and Markov models. So, what is a Hidden Markov Model ?
Well, suppose you were locked in a room for several days, and you were asked about the weather outside.
The only piece of evidence you have is whether the person who comes into the room bringing your daily
meal is carrying an umbrella ( ) or not ( ).
Let’s suppose the probabilities shown in table 2: The probability that your caretaker carries an
umbrella is 0.1 if the weather is sunny, 0.8 if it is actually raining, and 0.3 if it is foggy.
The equation for the weather Markov process before you were locked in the room was (eq. 4):
n
P (q1 , ..., qn ) = P (qi |qi−1 ).
i=1
3
4. Weather Probability of umbrella
Sunny 0.1
Rainy 0.8
Foggy 0.3
Table 2: Probability P (xi |qi ) of carrying an umbrella (xi = true) based on the weather qi on some day i
However, the actual weather is hidden from you. Finding the probability of a certain weather qi ∈
{ , , } can only be based on the observation xi , with xi = , if your caretaker brought an umbrella
on day i, and xi = if the caretaker did not bring an umbrella. This conditional probability P (qi |xi )
can be rewritten according to Bayes’ rule:
P (xi |qi )P (qi )
P (qi |xi ) = ,
P (xi )
or, for n days, and weather sequence Q = {q1 , . . . , qn }, as well as ‘umbrella sequence’ X = {x1 , , . . . , xn }
P (x1 , . . . , xn |q1 , . . . , qn )P (q1 , . . . , qn )
P (q1 , . . . , qn |x1 , . . . , xn ) = ,
P (x1 , . . . , xn )
using the probability P (q1 , . . . , qn ) of a Markov weather sequence from above, and the probability
P (x1 , . . . , xn ) of seeing a particular sequence of umbrella events (e.g., { , , }). The probability
n
P (x1 , . . . , xn |q1 , . . . , qn ) can be estimated as i=1 P (xi |qi ), if you assume that, for all i, the qi , xi are
independent of all xj and qj , for all j = i.
We want to draw conclusions from our observations (if the persons carries an umbrella or not) about
the weather outside. We can therefore omit the probability of seeing an umbrella P (x1 , . . . , xn ) as it
is independent of the weather, that we like to predict. We get a measure for the probability, which is
proportional to the probability, and which we will refer as the likelihood L.
P (q1 , . . . , qn |x1 , . . . , xn ) ∝
(5)
L(q1 , . . . , qn |x1 , . . . , xn ) = P (x1 , . . . , xn |q1 , . . . , qn ) · P (q1 , . . . , qn )
With our (first order) Markov assumption it turns to:
P (q1 , . . . , qn |x1 , . . . , xn ) ∝
n n
(6)
L(q1 , . . . , qn |x1 , . . . , xn ) = P (xi |qi ) · P (qi |qi−1 )
i=1 i=1
2.0.1 Examples
1. Suppose the day you were locked in it was sunny. The next day, the caretaker carried an umbrella
into the room. You would like to know, what the weather was like on this second day.
First we calculate the likelihood for the second day to be sunny:
L(q2 = |q1 = , x2 = ) = P (x2 = |q2 = ) · P (q2 = |q1 = )
= 0.1 · 0.8 = 0.08,
then for the second day to be rainy:
L(q2 = |q1 = , x2 = ) = P (x2 = |q2 = ) · P (q2 = |q1 = )
= 0.8 · 0.05 = 0.04,
and finally for the second day to be foggy:
L(q2 = |q1 = , x2 = ) = P (x2 = |q2 = ) · P (q2 = |q1 = )
= 0.3 · 0.15 = 0.045.
Thus, although the caretaker did carry an umbrella, it is most likely that on the second day the
weather was sunny.
4
5. 2. Suppose you do not know how the weather was when your were locked in. The following three days
the caretaker always comes without an umbrella. Calculate the likelihood for the weather on these
three days to have been {q1 = , q2 = , q3 = }. As you do not know how the weather is on
the first day, you assume the 3 weather situations are equi-probable on this day (cf. footnote on
page 2), and the prior probability for sun on day one is therefore P (q1 = |q0 ) = P (q1 = ) = 1/3.
L(q1 = , q2 = , q3 = |x1 = , x2 = , x3 = ) =
P (x1 = |q1 = ) · P (x2 = |q2 = ) · P (x3 = |q3 = )·
P (q1 = ) · P (q2 = |q1 = ) · P (q3 = |q2 = ) =
0.9 · 0.7 · 0.9 · 1/3 · 0.15 · 0.2 = 0.0057
(7)
2.1 HMM Terminology
A HMM Model is specified by:
- The set of states S = {s1 , s2 , . . . , sNs }, (corresponding to the three possible weather conditions
above),
and a set of parameters Θ = {π, A, B}:
- The prior probabilities πi = P (q1 = si ) are the probabilities of si being the first state of a state
sequence. Collected in a vector π. (The prior probabilities were assumed equi-probable in the last
example, πi = 1/Ns .)
- The transition probabilities are the probabilities to go from state i to state j: ai,j = P (qn+1 =
sj |qn = si ). They are collected in the matrix A.
- The emission probabilities characterize the likelihood of a certain observation x, if the model is in
state si . Depending on the kind of observation x we have:
- for discrete observations, xn ∈ {v1 , . . . , vK }: bi,k = P (xn = vk |qn = si ), the probabilities to
observe vk if the current state is qn = si . The numbers bi,k can be collected in a matrix B.
(This would be the case for the weather model, with K = 2 possible observations v1 = and
v2 = .)
- for continuous valued observations, e.g., xn ∈ RD : A set of functions bi (xn ) = p(xn |qn = si )
describing the probability densities (probability density functions, pdfs) over the observation
space for the system being in state si . Collected in the vector B(x) of functions. Emission
pdfs are often parametrized, e.g, by mixtures of Gaussians.
The operation of a HMM is characterized by
- The (hidden) state sequence Q = {q1 , q2 , . . . , qN }, qn ∈ S, (the sequence of the weather conditions
from day 1 to N ).
- The observation sequence X = {x1 , x2 , . . . , xN }.
A HMM allowing for transitions from any emitting state to any other emitting state is called an
ergodic HMM. The other extreme, a HMM where the transitions only go from one state to itself or to a
unique follower is called a left-right HMM.
Useful formula:
- Probability of a state sequence: the probability of a state sequence Q = {q1 , q2 , . . . , qN } coming
from a HMM with parameters Θ corresponds to the product of the transition probabilities from
one state to the following:
N −1
P (Q|Θ) = πq1 · aqn ,qn+1 = πq1 · aq1 ,q2 · aq2 ,q3 · . . . · aqN −1 ,qN (8)
n=1
5
6. - Likelihood of an observation sequence given a state sequence, or likelihood of an observation sequence
along a single path: given an observation sequence X = {x1 , x2 , . . . , xN } and a state sequence Q =
{q1 , q2 , . . . , qN } (of the same length) determined from a HMM with parameters Θ, the likelihood
of X along the path Q is equal to:
N
P (X|Q, Θ) = P (xn |qn , Θ) = bq1 ,x1 · bq2 ,x2 · . . . · bqN ,xN (9)
n=1
i.e., it is the product of the emission probabilities computed along the considered path.
- Joint likelihood of an observation sequence X and a path Q: it is the probability that X and Q occur
simultaneously, p(X, Q|Θ), and decomposes into a product of the two quantities defined previously:
P (X, Q|Θ) = P (X|Q, Θ) · P (Q|Θ) (Bayes) (10)
- Likelihood of a sequence with respect to a HMM: the likelihood of an observation sequence X =
{x1 , x2 , . . . , xN } with respect to a Hidden Markov Model with parameters Θ expands as follows:
P (X|Θ) = P (X, Q|Θ)
all Q
i.e., it is the sum of the joint likelihoods of the sequence over all possible state sequences Q allowed
by the model.
2.2 Trellis Diagram
A trellis diagram can be used to visualize likelihood calculations of HMMs. Figure 2 shows such a diagram
for a HMM with 3 states.
b1,k b1,k b1,k b1,k
a1,1
State 1 .......... ..........
a1,2
b2,k b2,k b2,k b2,k
State 2 a1,3 .......... ..........
b3,k b3,k b3,k b3,k
.......... ..........
State 3
Sequence: x1 x2 xi xN
n=1 n=2 n=i n=N
time
Figure 2: Trellis diagram
Each column in the trellis shows the possible states of the weather at a certain time n. Each state
in one column is connected to each state in the adjacent columns by the transition likelihood given by
the the elements ai,j of the transition matrix A (shown for state 1 at time 1 in figure 2). At the bottom
is the observation sequence X = {x1 , . . . , xN }. bi,k is the likelihood of the observation xn = vk in state
qn = si at time n.
Figure 3 shows the trellis diagram for Example 2 in sect. 2.0.1. The likelihood of the state sequence
given the observation sequence can be found by simply following the path in the trellis diagram, multi-
plying the observation and transition likelihoods.
L=π ·b , ·a , ·b , ·a , ·b , (11)
= 1/3 · 0.9 · 0.15 · 0.7 · 0.2 · 0.9 (12)
6
7. b , = 0.9 b , = 0.9
a = 0.2
a , = 0.15 ,
STATES
b , = 0.7
Sequence: x1 = x2 = x3 =
n=1 n=2 n=3
time
Figure 3: Trellis diagram for Example 2 in sect. 2.0.1
2.3 Generating samples from Hidden Markov Models
2.3.1 Question
Find the parameter set Θ = {π, A, B} that describes the weather HMM. Let’s suppose that the prior
probabilities are the same for each state.
2.3.2 Experiment
Let’s generate a sample sequence X coming from our weather HMM. First you have to specify the
parameter set Θ = {π, A, B} that describes the model.
>> %Choose the prior probabilities as you like: E.g., first day sunny
>> Pi w=[1 0 0]
The transition matrix is given in table 1:
>> A w = [0.8 0.05 0.15; 0.2 0.6 0.2; 0.2 0.3 0.5]
As the observation probabilities in this case are discrete probabilities, they can be saved in a matrix:
>> B w = [0.1 0.9; 0.8 0.2; 0.3 0.7]
In this observation matrix B, using the numbers from table 2, the rows correspond to the states and the
columns to our 2 discrete observations, i.e., b1,1 = p(umbrella = ) and b1,2 = p(umbrella = ) is the
probability of seeing an umbrella, resp. not seeing an umbrella, if the weather is in state s1 : qn = .
Use the function sample dhmm to do several draws with the weather model. View the resulting samples
and state sequences with the help of the function plotseq w. In the matrix containing the data, 1 stands
for ‘umbrella= ’ and 2 for ‘umbrella= ’.
>> % drawing 1 sample of length 4 (4 days)
>> [data,hidden] = sample dhmm(Pi w,A w,B w,1,4)
>> plotseq w(data,hidden)
2.3.3 Task
Write a Matlab function prob path(), that computes the joint likelihood for a given observation se-
quence and an assumed path:
>> prob = prob path(data,path,Pi w,A w,B w)
Input arguments are the observed sequence data, the assumed path path, and the weather HMM param-
eters (prior probabilities Pi w, transition matrix A w, and emission matrix B w). Use the function obslik
= mk dhmm obs lik(data,obsmat) to produce an observation likelihood matrix where the entries corre-
spond to bi,j in a trellis diagram (see figures 2 and 3). Type help mk dhmm obs lik for more detailed
information.
• Produce same random sequences, plot them and calculate their likelihood.
7
8. You can test your function to make the likelihood calculation for the 2. Example from section 2.0.1
described above.(Input parameters: data=[2 2 2];path=[1 3 1];prior=[1/3 1/3 1/3].)
>> prob = prob path([2 2 2],[1 3 1], [1/3 1/3 1/3], A w, B w)
2.4 HMMs for speech recognition
Let’s forget about the weather and umbrellas for a moment and talk about speech. In automatic speech
ˆ
recognition, the task is to find the most likely sequence of words W given some acoustic input, or:
ˆ
W = arg max P (W |X). (13)
W ∈W
Here, X = {x1 , x2 , . . . , xN } is the sequence of “acoustic vectors” – or “feature vectors” – that are
ˆ
“extracted” from the speech signal, and we want to find W as the sequence of words W (out of all possible
word sequences W), that maximizes P (W |X). To compare this to the weather example, the acoustic
feature vectors are our observations, corresponding to the umbrella observations, and the word sequence
corresponds to the weather on successive days (a day corresponds to about 10 ms), i. e., to the hidden
state sequence of a HMM for speech production.
Words are made of ordered sequences of phonemes: /h/ is followed by /e/ and then by /l/ and /O/
in the word “hello”. This structure can be adequately modeled by a left-right HMM, where each state
corresponds to a phone. Each phoneme can be considered to produce typical feature values according to
a particular probability density (possibly Gaussian) (Note, that the observed feature values xi now are
d-dimensional vectors and continuous valued, xi ∈ Rd , and no more discrete values, as for the weather
model where xi could only be true or false: xi ∈ { , }!).
In “real world” speech recognition, the phonemes themselves are often modeled as left-right HMMs
(e.g., to model separately the transition part at the begin of the phoneme, then the stationary part, and
finally the transition at the end). Words are then represented by large HMMs made of concatenations of
smaller phonetic HMMs.
Values used throughout the following experiments:
In the following experiments we will work with HMMs where the ‘observations’ are drawn from a Gaussian
probability distribution. Instead of an observation matrix (as in the weather example) the observations
of each state are described by the mean value and the variance of a Gaussian density.
The following 2-dimensional Gaussian pdfs will be used to model simulated observations of the vowels
/a/, /i/, and /y/3 . The observed features are the first two formants (maxima of the spectral envelope),
which are characteristic for the vowel identity, e.g., for the vowel /a/ the formants typically occur around
frequencies 730 Hz and 1090 Hz.
730 1625 5300
State s1 (for /a/): N/a/ : µ/a/ = , Σ/a/ =
1090 5300 53300
270 2525 1200
State s2 (for /i/): N/i/ : µ/i/ = , Σ/i/ =
2290 1200 36125
440 8000 8400
State s3 (for /y/): N/y/ : µ/y/ = , Σ/y/ =
1020 8400 18500
(Those densities have been used in the previous lab session.) The Gaussian pdfs now take the role of the
emission matrix B in the hidden Markov models for recognition of the three vowels /a/, /i/, and /y/.
The parameters of the densities and of the Markov models are stored in the file data.mat (Use: load
data). A Markov model named, e.g., hmm1 is stored as an object with fields hmm1.means, hmm1.vars,
hmm1.trans,hmm1.pi. The means fields contains a matrix of mean vectors, where each column of the
matrix corresponds to one state si of the HMM (e.g., to access the means of the second state of hmm1 use:
hmm1.means(:,2); the vars field contains a 3 dimensional array of variance matrices, where the third
3 Phonetic symbol for the usual pronunciation of ‘¨’
u
8
9. b1,k b1,k b1,k b1,k b1,k
State 1
.... ....
b2,k b2,k b2,k b2,k b2,k
State 2 .... ....
State 3
b3,k b3,k b3,k
.... b3,k
.... b3,k
Sequence: x1 x2 x3 xi xN
n=1 n=2 n=3 n=i n=N
time
Figure 4:
dimension corresponds to the state (e.g to access Σ/a/ for state 1 use hmm1.vars(:,:,1); the trans field
contains the transition matrix, and the pi field the prior probabilities.
2.4.1 Task
Load the HMMs and make a sketch of each of the models with the transition probabilities.
2.4.2 Experiment
Generate samples using the HMMs and plot them with plotseq and plotseq2.
Use the functions plotseq and plotseq2 to plot the obtained 2-dimensional data. In the resulting views,
the obtained sequences are represented by a yellow line where each point is overlaid with a colored dot.
The different colors indicate the state from which any particular point has been drawn.
>> %Example:generate sample from HMM1 of length N
>> [X,ST]=sample ghmm(hmm1,N)
>> plotseq(X,ST) % View of both dimensions as separate sequences
>> plotseq2(X,ST,hmm1) %2D view with location of gaussian states
Draw several samples with the same parameters and compare. Compare the Matlab figures with
your sketch of the model.
What is the effect of the different transition matrices of the HMMs on the sequences obtained during the
current experiment? Hence, what is the role of the transition probabilities in the HMM?
3 Pattern Recognition with HMM
In equation 10 we expressed the joint likelihood p(X, Q|Θ) of an observation sequence X and a path Q
given a model with parameters Θ.
The likelihood of a sequence with respect to a HMM (the likelihood of an observation sequence X =
{x1 , x2 , · · · , xN } for a given hidden Markov model with parameters Θ ) expands as follows:
p(X|Θ) = p(X, Q|Θ), (14)
every possible Q
i.e., it is the sum of the joint likelihoods of the sequence over all possible state sequences allowed by the
model (see Trellis diagram in figure 3).
Calculating the likelihood in this manner is computationally expensive, particularly for large models
or long sequences. It can be done with a recursive algorithm (forward-backward algorithm), which reduces
the complexity of the problem. For more information about this algorithm see [1]. It is very common
using log-likelihoods and log-probabilities, computing log p(X|Θ) instead of p(X|Θ).
9
10. 3.0.1 Experiment
Classify the sequences X1 , X2 , X3 , X4 , given in the file Xdata.mat, in a maximum likelihood sense with
respect to the four Markov models. Use the function loglik ghmm to compute the likelihood of a sequence
with respect to a HMM. Store the results in a matrix (they will be used in the next section).
>> load Xdata
>> % Example:
>> logProb(1,1) = loglik ghmm(X1,hmm1)
>> logProb(1,2) = loglik ghmm(X1,hmm2)
etc.
>> logProb(3,2) = loglik ghmm(X3,hmm2)
etc.
Instead of typing these commands for every combination of the four sequences and models, filling the
logProb matrix can be done automatically with the help of loops, using a command string composed of
fixed strings and strings containing the number of sequence/model:
>> for i=1:4,
for j=1:4,
stri = num2str(i);
strj = num2str(j);
cmd = [’logProb(’,stri,’,’,strj,’) = loglik_ghmm(X’,stri,’,hmm’,strj,’);’]
eval(cmd);
end;
end;
You can find the maximum value of each row i of the matrix, giving the index of the most likely model
for sequence Xi , with the Matlab function max:
for i=1:4;
[v,index]=max(logProb(i,:));
disp ([’X’,num2str(i),’ -> HMM’,num2str(index)]);
end
Most likely
i Sequence log p(Xi |Θ1 ) log p(Xi |Θ2 ) log p(Xi |Θ3 ) log p(Xi |Θ4 )
model
1 X1
2 X2
3 X3
4 X4
4 Optimal state sequence
In speech recognition and several other pattern recognition applications, it is useful to associate an
“optimal” sequence of states to a sequence of observations, given the parameters of a model. For instance,
in the case of speech recognition, knowing which frames of features “belong” to which state allows to
locate the word boundaries across time. This is called alignment of acoustic feature sequences.
A “reasonable” optimality criterion consists of choosing the state sequence (or path) that has the
maximum likelihood with respect to a given model. This sequence can be determined recursively via the
Viterbi algorithm.
This algorithm makes use of two variables:
• δn (i) is the highest likelihood of a single path among all the paths ending in state si at time n:
δn (i) = max p(q1 , q2 , . . . , qn−1 , qn = si , x1 , x2 , . . . xn |Θ) (15)
q1 ,q2 ,...,qn−1
10
11. • a variable ψn (i) which allows to keep track of the “best path” ending in state si at time n:
ψn (i) = arg max p(q1 , q2 , . . . , qn−1 , qn = si , x1 , x2 , . . . xn |Θ) (16)
q1 ,q2 ,...,qn−1
The idea of the Viterbi algorithm is to find the most probable path for each intermediate and finally
for the terminating state in the trellis. At each time n only the most likely path leading to each state si
‘survives’.
The Viterbi Algorithm
for a HMM with Ns states.
1. Initialization
δ1 (i) = πi · bi,x1 , i = 1, . . . , Ns
(17)
ψ1 (i) = 0
where πi is the prior probability of being in state si at time n = 1.
2. Recursion
2≤n≤N
δn (j) = max (δn−1 (i) · aij ) · bj,xn ,
1≤i≤Ns 1 ≤ j ≤ Ns
(18)
2≤n≤N
ψn (j) = arg max (δn−1 (i) · aij ) ,
1≤i≤Ns 1 ≤ j ≤ Ns
“Optimal policy is composed of optimal sub-policies”: find the path that leads to a maximum likeli-
hood considering the best likelihood at the previous step and the transitions from it; then multiply
by the current likelihood given the current state. Hence, the best path is found by induction.
3. Termination
p∗ (X|Θ) = max δN (i)
1≤i≤Ns
∗ (19)
qN = arg max δN (i)
1≤i≤Ns
Find the best likelihood when the end of the observation sequence t = T is reached.
4. Backtracking
Q∗ = {q1 , . . . , qN } so that qn = ψn+1 (qn+1 ),
∗ ∗ ∗ ∗
n = N − 1, N − 2, . . . , 1 (20)
Read (decode) the best sequence of states from the ψn vectors.
4.0.1 Example
Let’s get back to our weather HMM. You don’t know how the weather was when your were locked in. On
the first 3 days your umbrella observations are: {no umbrella, umbrella, umbrella} ({ , , }). Find
the most probable weather-sequence using the Viterbi algorithm (assume the 3 weather situations to be
equi-probable on day 1).
1. Initialization
n = 1:
δ1 ( ) = π · b , = 1/3 · 0.9 = 0.3
ψ1 ( ) = 0
δ1 ( ) = π · b , = 1/3 · 0.2 = 0.0667
ψ1 ( ) = 0
δ1 ( ) = π ·b , = 1/3 · 0.7 = 0.233
ψ1 ( ) = 0
11
12. δ2 ( ) = max(δ1 ( ) · a , , δ1 ( ) · a , , δ1 ( ) · a , )·b ,
δ1 = 0.3 ψ2 ( ) =
STATES
Sequence: x1 = x2 = x3 =
n=1 n=2 n=3
time
Figure 5: Viterbi algorithm to find the most likely weather sequence. Finding the most likely path to
state ‘sunny’ at n = 2.
2. Recursion
n=2:
We calculate the likelihood of getting to state ‘ ’ from all possible 3 predecessor states, and choose
the most likely one to go on with:
δ2 ( ) = max(δ1 ( ) · a , , δ1 ( ) · a , , δ1 ( ) · a , ) ·b ,
= max(0.3 · 0.8, 0.0667 · 0.2, 0.233 · 0.2) · 0.1 = 0.024
ψ2 ( ) =
The likelihood is stored in δ, the most likely predecessor in ψ. See figure 5.
The same procedure is executed with states and :
δ2 ( ) = max(δ1 ( ) · a , , δ1 ( ) · a , , δ1 ( ) · a , ) ·b ,
= max(0.3 · 0.05, 0.0667 · 0.6, 0.233 · 0.3) · 0.8 = 0.056
ψ2 ( ) =
δ2 ( ) = max(δ1 ( ) · a , , δ1 ( ) · a , , δ1 ( ) · a , ) · b ,
= max(0.3 · 0.15, 0.0667 · 0.2, 0.233 · 0.5) · 0.3 = 0.0350
ψ2 ( ) =
n=3:
δ3 ( ) = max(δ2 ( ) · a , , δ2 ( ) · a , , δ2 ( ) · a , ) ·b ,
= max(0.024 · 0.8, 0.056 · 0.2, 0.035 · 0.2) · 0.1 = 0.0019
ψ3 ( ) =
δ3 ( ) = max(δ2 ( ) · a , , δ2 ( ) · a , , δ2 ( ) · a , ) · b ,
= max(0.024 · 0.05, 0.056 · 0.6, 0.035 · 0.3) · 0.8 = 0.0269
ψ3 ( ) =
δ3 ( ) = max(δ2 ( ) · a , , δ2 ( ) · a , , δ2 ( ) · a , ) · b ,
= max(0.0024 · 0.15, 0.056 · 0.2, 0.035 · 0.5) · 0.3 = 0.0052
ψ3 ( ) =
Finally, we get one most likely path ending in each state of the model. See figure 6.
12
13. δ3 ( ) = 0.0019
STATES
δ3 ( ) = 0.0269
δ3 ( ) = 0.0052
Sequence: x1 = x2 = x3 =
n=1 n=2 n=3
time
Figure 6: Viterbi algorithm to find most likely weather sequence at n = 3.
δ3 ( ) = 0.0019
STATES
δ3 ( ) = 0.0269
δ3 ( ) = 0.0052
Sequence: x1 = x2 = x3 =
n=1 n=2 n=3
time
Figure 7: Viterbi algorithm to find most likely weather sequence. Backtracking.
3. Termination
The globally most likely path is determined, starting by looking for the last state of the most likely
sequence.
P ∗ (X|Θ) = max(δ3 (i)) = δ3 ( ) = 0.0269
∗
q3 = arg max(δ3 (i)) =
4. Backtracking
The best sequence of states can be read from the ψ vectors. See figure 7.
n = N − 1 = 2:
∗ ∗
q2 = ψ3 (q3 ) = ψ3 ( ) =
n = N − 1 = 1:
∗ ∗
q1 = ψ2 (q2 ) = ψ2 ( ) =
Thus the most likely weather sequence is: Q∗ = {q1 , q2 , q3 } = { , , }.
∗ ∗ ∗
4.0.2 Task
1. Write a Matlab function [loglik,path] = vit ghmm(data,HMM) to implement the Viterbi al-
gorithm for HMMs with Gaussian emissions. Use the function mk ghmm obs lik to calculate the
observation likelihood matrix for a given sequence. Store the δ and ψ vectors in a matrix in the
format of the observation likelihood matrix. You can either write the function with the algorithm
exactly as given before, or switch to make the calculations in the log likelihood domain, where the
13
14. multiplications of the parameters δ and ψ transform to additions. What are the advantages or dis-
advantages of the second method? (Think about how to implement it on a real system with limited
computational accuracy, and about a HMM with a large number of states where the probabilities
ai,j and bi,k might be small.)
2. Use the function vit ghmm to determine the most likely path for the sequences X1 , X2 , X3 and X4 .
Compare with the state sequence ST 1, . . . , ST 4 originally used to generate X1 , . . . , X4 . (Use the
function compseq, which provides a view of the first dimension of the observations as a time series,
and allows to compare the original alignment to the Viterbi solution).
>> [loglik,path] = vit ghmm(X1,hmm1); compseq(X1,ST1,path);
>> [loglik,path] = vit ghmm(X2,hmm1); compseq(X2,ST2,path);
Repeat for the remaining sequences.
3. Use the function vit ghmm to compute the probabilities of the sequences X1 , . . . , X4 along the best
paths with respect to each model Θ1 , . . . , Θ4 . Note down your results below. Compare with the
log-likelihoods obtained in section 3.0.1 using the forward procedure.
>> diffL=logProb-logProbViterbi
Likelihoods along the best path:
Most likely
i Sequence log p∗ (Xi |Θ1 ) log p∗ (Xi |Θ2 ) log p∗ (Xi |Θ3 ) log p∗ (Xi |Θ4 )
model
1 X1
2 X2
3 X3
4 X4
Difference between log-likelihoods and likelihoods along the best path:
Sequence HMM1 HMM2 HMM3 HMM4
X1
X2
X3
X4
Question:
Is the likelihood along the best path a good approximation of the real likelihood of a sequence given a
model?
References
[1] Rabiner, L. R. (1989). A tutorial on hidden Markov models and selected applications in speech
recognition. Proceedings of the IEEE, 77 (2), 257–286.
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