The document discusses the history and development of hidden Markov models (HMMs). It describes key concepts such as HMMs consisting of hidden states that produce observable outputs, and how they can be used to model sequential data. The document also provides examples of applying HMMs to problems such as gene finding, multiple sequence alignment, and protein secondary structure prediction. It summarizes algorithms like forward-backward, Viterbi, and Baum-Welch that are used to train and make predictions from HMMs. Finally, it mentions some popular HMM software tools like HMMER and SAM.
An AsmL model for an Intelligent Vehicle Control Systeminfopapers
F. Stoica, An AsmL model for an Intelligent Vehicle Control System, Proceedings of the 11th WSEAS Int. Conf. on COMPUTERS: Computer Science and Technology, vol. 4, Crete Island, Greece, ISBN: 978-960-8457-92-8, pp. 323-328, July 2007
This document discusses state-space representation of linear time-invariant (LTI) systems. It defines system state, state equations, and output equations. The key points are:
1) State equations describe the dynamics of a system using first-order differential equations relating state variables. Output equations relate outputs to state variables and inputs.
2) For LTI systems, the state equations can be written in matrix form as dx/dt = Ax + Bu, and output equations as y = Cx + Du.
3) Block diagrams can be constructed from the state-space model, with integrators for each state variable and blocks representing the A, B, C, and D matrices.
Modern Control - Lec07 - State Space Modeling of LTI SystemsAmr E. Mohamed
The document provides an overview of state-space representation of linear time-invariant (LTI) systems. It defines key concepts such as state variables, state vector, state equations, and output equations. Examples are given to show how to derive the state-space models from differential equations describing dynamical systems. Specifically, it shows how to 1) select state variables, 2) write first-order differential equations as state equations, and 3) obtain output equations to fully represent LTI systems in state-space form.
The document discusses maximum likelihood estimation. It begins by explaining that maximum likelihood chooses parameter values that make the observed data most probable given a statistical model. This provides a justification for estimation techniques like least squares regression. The document provides an example of estimating a population proportion from a sample. It then generalizes maximum likelihood to cover a wide range of models and estimation problems. It discusses properties like consistency, efficiency, and how to conduct hypothesis tests based on maximum likelihood. Numerical optimization techniques are often required to find maximum likelihood estimates for complex models.
Fixed Point Theorm In Probabilistic Analysisiosrjce
Probabilistic operator theory is the branch of probabilistic analysis which is concerned with the study of
operator-valued random variables and their properties. The development of a theory of random operators is of
interest in its own right as a probabilistic generalization of (deterministic) operator theory and just as operator
theory is of fundamental importance in the study of operator equations, the development of probabilistic operator
theory is required for the study of various classes of random equations
The document discusses using the Nelder-Mead search algorithm to optimize parameters in the Fuzzy BEXA machine learning algorithm. Specifically, it aims to optimize parameters related to converting data files, defining membership functions, and setting threshold cutoffs, to maximize classification accuracy. The author developed a Java program to optimize two threshold parameters (αa and αc) using Nelder-Mead to search the parameter space and call Fuzzy BEXA to evaluate classification accuracy as the objective function. While Nelder-Mead works well for this optimization, initial parameter guesses can impact finding the true global optimum.
An AsmL model for an Intelligent Vehicle Control Systeminfopapers
F. Stoica, An AsmL model for an Intelligent Vehicle Control System, Proceedings of the 11th WSEAS Int. Conf. on COMPUTERS: Computer Science and Technology, vol. 4, Crete Island, Greece, ISBN: 978-960-8457-92-8, pp. 323-328, July 2007
This document discusses state-space representation of linear time-invariant (LTI) systems. It defines system state, state equations, and output equations. The key points are:
1) State equations describe the dynamics of a system using first-order differential equations relating state variables. Output equations relate outputs to state variables and inputs.
2) For LTI systems, the state equations can be written in matrix form as dx/dt = Ax + Bu, and output equations as y = Cx + Du.
3) Block diagrams can be constructed from the state-space model, with integrators for each state variable and blocks representing the A, B, C, and D matrices.
Modern Control - Lec07 - State Space Modeling of LTI SystemsAmr E. Mohamed
The document provides an overview of state-space representation of linear time-invariant (LTI) systems. It defines key concepts such as state variables, state vector, state equations, and output equations. Examples are given to show how to derive the state-space models from differential equations describing dynamical systems. Specifically, it shows how to 1) select state variables, 2) write first-order differential equations as state equations, and 3) obtain output equations to fully represent LTI systems in state-space form.
The document discusses maximum likelihood estimation. It begins by explaining that maximum likelihood chooses parameter values that make the observed data most probable given a statistical model. This provides a justification for estimation techniques like least squares regression. The document provides an example of estimating a population proportion from a sample. It then generalizes maximum likelihood to cover a wide range of models and estimation problems. It discusses properties like consistency, efficiency, and how to conduct hypothesis tests based on maximum likelihood. Numerical optimization techniques are often required to find maximum likelihood estimates for complex models.
Fixed Point Theorm In Probabilistic Analysisiosrjce
Probabilistic operator theory is the branch of probabilistic analysis which is concerned with the study of
operator-valued random variables and their properties. The development of a theory of random operators is of
interest in its own right as a probabilistic generalization of (deterministic) operator theory and just as operator
theory is of fundamental importance in the study of operator equations, the development of probabilistic operator
theory is required for the study of various classes of random equations
The document discusses using the Nelder-Mead search algorithm to optimize parameters in the Fuzzy BEXA machine learning algorithm. Specifically, it aims to optimize parameters related to converting data files, defining membership functions, and setting threshold cutoffs, to maximize classification accuracy. The author developed a Java program to optimize two threshold parameters (αa and αc) using Nelder-Mead to search the parameter space and call Fuzzy BEXA to evaluate classification accuracy as the objective function. While Nelder-Mead works well for this optimization, initial parameter guesses can impact finding the true global optimum.
Monte Carlo methods use random sampling to solve quantitative problems. They were first used by Stanislaw Ulam and Nicholas Metropolis to solve non-random problems by transforming them into random forms. Monte Carlo simulations play a major role in experimental physics by designing experiments, evaluating potential outputs and risks, and validating results. Random numbers are generated using pseudorandom number generators or by transforming uniform random variables using probability distribution functions. The accuracy of Monte Carlo simulations improves as the number of samples increases, with the standard error declining proportionally with the square root of the number of samples.
Inference & Learning in Linear Chain Conditional Random Fields (CRFs)Anmol Dwivedi
This mini-project will consider performing inference and learning in Linear Chain CRFs. In particular, it will consider an application to hand-written word recognition. Handwritten word recognition is a task many have explored with different methods of machine learning. Some written characters can be evaluated individually or as a whole word to account for the context in characters. In this mini-project, we use linear chain CRF models to account for context between the characters of a word to improve word recognition accuracy.
Tutorial on Markov Random Fields (MRFs) for Computer Vision ApplicationsAnmol Dwivedi
The goal of this mini-project is to implement a pairwise binary label-observation Markov Random Field
model for bi-level image segmentation. Specifically, two inference algorithms, i.e., the Iterative
Conditional Mode (ICM) and Gibbs sampling methods will be implemented to perform image segmentation.
Research Inventy : International Journal of Engineering and Scienceresearchinventy
This document summarizes research on solving simultaneous equations. It discusses how solving simultaneous equations is an NP-complete problem that can be solved deterministically using Gaussian elimination. The paper presents the Gaussian elimination algorithm, provides pseudocode for its implementation, and analyzes its time complexity of O(n3). It also describes other methods for solving simultaneous equations like substitution and discusses how Gaussian elimination transforms the NP-complete problem into a polynomial time problem.
A short list of the most useful R commands
reference: http://www.personality-project.org/r/r.commands.html
R programı ile ilgilenen veya yeni öğrenmeye başlayan herkes için hazırlanmıştır.
This document summarizes a study on evaluating the rate of convergence of the Newton-Raphson method. A computer program was coded in Java to calculate cube roots from 1 to 25 using Newton-Raphson. The lowest rate of convergence was for the cube root of 16, and the highest was for 3. The average rate of convergence was found to be 0.217920. Formulas for estimating the rate of convergence from successive approximations are also presented.
Stochastic modelling and its applicationsKartavya Jain
Stochastic processes and modelling have various applications in telecommunications. Token rings, continuous-time Markov chains, and fluid-flow models are used to model traffic flow and network performance. Aggregate dynamic stochastic models can model air traffic control by representing aircraft arrivals as Poisson processes. Disturbances like weather can be incorporated by altering flow rates. Wireless network models use search algorithms and location stochastic processes to track mobile users.
Fractional Newton-Raphson Method and Some Variants for the Solution of Nonlin...mathsjournal
The following document presents some novel numerical methods valid for one and several variables, which
using the fractional derivative, allow us to find solutions for some nonlinear systems in the complex space using
real initial conditions. The origin of these methods is the fractional Newton-Raphson method, but unlike the
latter, the orders proposed here for the fractional derivatives are functions. In the first method, a function is
used to guarantee an order of convergence (at least) quadratic, and in the other, a function is used to avoid the
discontinuity that is generated when the fractional derivative of the constants is used, and with this, it is possible
that the method has at most an order of convergence (at least) linear.
The document discusses various numerical methods for finding roots of functions, including:
- Bracketing methods like bisection and false position that search between initial lower and upper bounds.
- Open methods like Newton-Raphson and secant that do not require bracketing but may not converge.
- Techniques for polynomials like Müller's and Bairstow's methods.
Examples demonstrate applying bisection, false position, and Newton-Raphson to find the mass in a falling object problem. The convergence properties and relative performance of the different methods are analyzed.
This document discusses numerical methods in MATLAB, including root finding, interpolation, integration, and solving ordinary differential equations. It provides examples of using MATLAB functions like fzero, roots, interp1, quad, and ode45. The key functions and methods covered are:
- fzero finds roots of univariate functions numerically. roots finds roots of polynomials.
- interp1 performs one-dimensional interpolation using methods like nearest, linear, spline, and cubic interpolation.
- quad and quad8 numerically evaluate integrals of varying accuracy and order.
- ode23, ode45, ode113, ode15s, and ode23s solve non-stiff and stiff ordinary differential equations.
- Estimation theory involves using observed data to determine unknown parameters of a system. This includes problems like estimating locations/velocities from radar signals or inferring transmitted signals from received noisy data.
- Estimation includes parametric estimation, which assumes a model and estimates parameters like mean/variance, and non-parametric estimation, which directly estimates probability densities without assuming a model.
- An estimator is a rule for guessing the value of an unknown parameter based on observed data. Good estimators are unbiased, have low variance, are consistent as more data is observed, and have minimum mean squared error. The minimum variance unbiased estimator is preferred.
The Cramer-Rao Inequality provides us with a lower bound on the variance of an unbiased estimator for a parameter.
The Cramer-Rao Inequality Let X = (X1,X2,. . ., Xn) be a random sample from a distribution with d.f. f(x|θ), where θ is a scalar parameter. Under certain regularity conditions on f(x|θ), for any unbiased estimator φˆ (X) of φ (θ)
Lambda calculus (also written as λ-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computation that can be used to simulate any Turing machine and was first introduced by mathematician Alonzo Church in the 1930s as part of his research of the foundations of mathematics.
Lambda calculus consists of constructing lambda terms and performing reduction operations on them
Senior Seminar: Systems of Differential EquationsJDagenais
This document discusses solving systems of differential equations. It begins by introducing systems of differential equations and their importance in modeling natural processes. It then outlines the key concepts needed to solve systems, including matrices, eigenvalues, and diagonalization. The document focuses on solving homogeneous systems where the eigenvalues are distinct and real. It presents the process of writing systems in matrix form and looking for solutions of the form x=e^rt to find the eigenvalues from the characteristic equation.
- Dr. Harish Garg has a Ph.D. in applied mathematics from IIT Roorkee and works as an assistant professor teaching stochastic matrices.
- A stochastic matrix models a randomly changing system over time and can be represented by a transition matrix where each row sums to 1.
- Markov chains are a type of stochastic process where the next state only depends on the present state and not on past states. The transitions between states are represented using a transition probability matrix.
Recursion is a technique where a method calls itself directly or indirectly. It is useful for solving problems that involve repeating patterns or combinatorial algorithms. The document provides examples of calculating factorials, generating all binary vectors, and finding all paths in a labyrinth recursively. It discusses how to avoid harmful recursion that uses excessive memory and discusses when recursion is preferable to iteration, such as for problems that require exploring multiple continuations at each step. Exercises are provided to help practice implementing various recursive algorithms.
Principles of functional progrmming in scalaehsoon
a short outline on necessity of functional programming and principles of functional programming in Scala.
In the article some keyword are used but not explained (to keep the article short and simple), the interested reader can look them up in internet.
This document provides a brief list of basic MATLAB commands for loading and quitting MATLAB, listing variables, clearing variables, saving and loading workspaces, plotting data, performing equation fitting and data analysis, and some notes on matrix algebra operations in MATLAB. Some key commands include matlab to load MATLAB, quit or exit to quit MATLAB, plot for plotting data, and inv or \ for solving simultaneous equations. MATLAB uses matrices to store numeric data and performs operations element-wise on matrices.
Lecture 2&3 Computer vision image formation ,filters&edge detectioncairo university
This document provides information about a computer vision course including the professor, TAs, and topics to be covered such as image formation, filters, and edge detection. It also includes diagrams explaining image coordinate systems, the pinhole camera model, and how world coordinates are projected and mapped to image coordinates. Key concepts covered are the point spread function, image and object domains, and how a pinhole camera forms images by blocking light rays except through the aperture.
This document summarizes an academic paper that proposes modifying well-known local linear models for system identification by replacing their original recursive learning rules with outlier-robust variants based on M-estimation. It describes three existing local linear models - local linear map (LLM), radial basis function network (RBFN), and local model network (LMN) - and then introduces the concept of M-estimation as a way to make the learning rules of these models more robust to outliers. The performance of the proposed outlier-robust variants is evaluated on three benchmark datasets and is found to provide considerable improvement in the presence of outliers compared to the original models.
On the Principle of Optimality for Linear Stochastic Dynamic System ijfcstjournal
In this work, processes represented by linear stochastic dynamic system are investigated and by
considering optimal control problem, principle of optimality is proven. Also, for existence of optimal
control and corresponding optimal trajectory, proofs of theorems of necessity and sufficiency condition are
attained.
Hidden Markov Models with applications to speech recognitionbutest
This document provides an introduction to hidden Markov models (HMMs). It discusses how HMMs can be used to model sequential data where the underlying states are not directly observable. The key aspects of HMMs are: (1) the model has a set of hidden states that evolve over time according to transition probabilities, (2) observations are emitted based on the current hidden state, (3) the four basic problems of HMMs are evaluation, decoding, training, and model selection. Examples discussed include modeling coin tosses, balls in urns, and speech recognition. Learning algorithms for HMMs like Baum-Welch and Viterbi are also summarized.
Monte Carlo methods use random sampling to solve quantitative problems. They were first used by Stanislaw Ulam and Nicholas Metropolis to solve non-random problems by transforming them into random forms. Monte Carlo simulations play a major role in experimental physics by designing experiments, evaluating potential outputs and risks, and validating results. Random numbers are generated using pseudorandom number generators or by transforming uniform random variables using probability distribution functions. The accuracy of Monte Carlo simulations improves as the number of samples increases, with the standard error declining proportionally with the square root of the number of samples.
Inference & Learning in Linear Chain Conditional Random Fields (CRFs)Anmol Dwivedi
This mini-project will consider performing inference and learning in Linear Chain CRFs. In particular, it will consider an application to hand-written word recognition. Handwritten word recognition is a task many have explored with different methods of machine learning. Some written characters can be evaluated individually or as a whole word to account for the context in characters. In this mini-project, we use linear chain CRF models to account for context between the characters of a word to improve word recognition accuracy.
Tutorial on Markov Random Fields (MRFs) for Computer Vision ApplicationsAnmol Dwivedi
The goal of this mini-project is to implement a pairwise binary label-observation Markov Random Field
model for bi-level image segmentation. Specifically, two inference algorithms, i.e., the Iterative
Conditional Mode (ICM) and Gibbs sampling methods will be implemented to perform image segmentation.
Research Inventy : International Journal of Engineering and Scienceresearchinventy
This document summarizes research on solving simultaneous equations. It discusses how solving simultaneous equations is an NP-complete problem that can be solved deterministically using Gaussian elimination. The paper presents the Gaussian elimination algorithm, provides pseudocode for its implementation, and analyzes its time complexity of O(n3). It also describes other methods for solving simultaneous equations like substitution and discusses how Gaussian elimination transforms the NP-complete problem into a polynomial time problem.
A short list of the most useful R commands
reference: http://www.personality-project.org/r/r.commands.html
R programı ile ilgilenen veya yeni öğrenmeye başlayan herkes için hazırlanmıştır.
This document summarizes a study on evaluating the rate of convergence of the Newton-Raphson method. A computer program was coded in Java to calculate cube roots from 1 to 25 using Newton-Raphson. The lowest rate of convergence was for the cube root of 16, and the highest was for 3. The average rate of convergence was found to be 0.217920. Formulas for estimating the rate of convergence from successive approximations are also presented.
Stochastic modelling and its applicationsKartavya Jain
Stochastic processes and modelling have various applications in telecommunications. Token rings, continuous-time Markov chains, and fluid-flow models are used to model traffic flow and network performance. Aggregate dynamic stochastic models can model air traffic control by representing aircraft arrivals as Poisson processes. Disturbances like weather can be incorporated by altering flow rates. Wireless network models use search algorithms and location stochastic processes to track mobile users.
Fractional Newton-Raphson Method and Some Variants for the Solution of Nonlin...mathsjournal
The following document presents some novel numerical methods valid for one and several variables, which
using the fractional derivative, allow us to find solutions for some nonlinear systems in the complex space using
real initial conditions. The origin of these methods is the fractional Newton-Raphson method, but unlike the
latter, the orders proposed here for the fractional derivatives are functions. In the first method, a function is
used to guarantee an order of convergence (at least) quadratic, and in the other, a function is used to avoid the
discontinuity that is generated when the fractional derivative of the constants is used, and with this, it is possible
that the method has at most an order of convergence (at least) linear.
The document discusses various numerical methods for finding roots of functions, including:
- Bracketing methods like bisection and false position that search between initial lower and upper bounds.
- Open methods like Newton-Raphson and secant that do not require bracketing but may not converge.
- Techniques for polynomials like Müller's and Bairstow's methods.
Examples demonstrate applying bisection, false position, and Newton-Raphson to find the mass in a falling object problem. The convergence properties and relative performance of the different methods are analyzed.
This document discusses numerical methods in MATLAB, including root finding, interpolation, integration, and solving ordinary differential equations. It provides examples of using MATLAB functions like fzero, roots, interp1, quad, and ode45. The key functions and methods covered are:
- fzero finds roots of univariate functions numerically. roots finds roots of polynomials.
- interp1 performs one-dimensional interpolation using methods like nearest, linear, spline, and cubic interpolation.
- quad and quad8 numerically evaluate integrals of varying accuracy and order.
- ode23, ode45, ode113, ode15s, and ode23s solve non-stiff and stiff ordinary differential equations.
- Estimation theory involves using observed data to determine unknown parameters of a system. This includes problems like estimating locations/velocities from radar signals or inferring transmitted signals from received noisy data.
- Estimation includes parametric estimation, which assumes a model and estimates parameters like mean/variance, and non-parametric estimation, which directly estimates probability densities without assuming a model.
- An estimator is a rule for guessing the value of an unknown parameter based on observed data. Good estimators are unbiased, have low variance, are consistent as more data is observed, and have minimum mean squared error. The minimum variance unbiased estimator is preferred.
The Cramer-Rao Inequality provides us with a lower bound on the variance of an unbiased estimator for a parameter.
The Cramer-Rao Inequality Let X = (X1,X2,. . ., Xn) be a random sample from a distribution with d.f. f(x|θ), where θ is a scalar parameter. Under certain regularity conditions on f(x|θ), for any unbiased estimator φˆ (X) of φ (θ)
Lambda calculus (also written as λ-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computation that can be used to simulate any Turing machine and was first introduced by mathematician Alonzo Church in the 1930s as part of his research of the foundations of mathematics.
Lambda calculus consists of constructing lambda terms and performing reduction operations on them
Senior Seminar: Systems of Differential EquationsJDagenais
This document discusses solving systems of differential equations. It begins by introducing systems of differential equations and their importance in modeling natural processes. It then outlines the key concepts needed to solve systems, including matrices, eigenvalues, and diagonalization. The document focuses on solving homogeneous systems where the eigenvalues are distinct and real. It presents the process of writing systems in matrix form and looking for solutions of the form x=e^rt to find the eigenvalues from the characteristic equation.
- Dr. Harish Garg has a Ph.D. in applied mathematics from IIT Roorkee and works as an assistant professor teaching stochastic matrices.
- A stochastic matrix models a randomly changing system over time and can be represented by a transition matrix where each row sums to 1.
- Markov chains are a type of stochastic process where the next state only depends on the present state and not on past states. The transitions between states are represented using a transition probability matrix.
Recursion is a technique where a method calls itself directly or indirectly. It is useful for solving problems that involve repeating patterns or combinatorial algorithms. The document provides examples of calculating factorials, generating all binary vectors, and finding all paths in a labyrinth recursively. It discusses how to avoid harmful recursion that uses excessive memory and discusses when recursion is preferable to iteration, such as for problems that require exploring multiple continuations at each step. Exercises are provided to help practice implementing various recursive algorithms.
Principles of functional progrmming in scalaehsoon
a short outline on necessity of functional programming and principles of functional programming in Scala.
In the article some keyword are used but not explained (to keep the article short and simple), the interested reader can look them up in internet.
This document provides a brief list of basic MATLAB commands for loading and quitting MATLAB, listing variables, clearing variables, saving and loading workspaces, plotting data, performing equation fitting and data analysis, and some notes on matrix algebra operations in MATLAB. Some key commands include matlab to load MATLAB, quit or exit to quit MATLAB, plot for plotting data, and inv or \ for solving simultaneous equations. MATLAB uses matrices to store numeric data and performs operations element-wise on matrices.
Lecture 2&3 Computer vision image formation ,filters&edge detectioncairo university
This document provides information about a computer vision course including the professor, TAs, and topics to be covered such as image formation, filters, and edge detection. It also includes diagrams explaining image coordinate systems, the pinhole camera model, and how world coordinates are projected and mapped to image coordinates. Key concepts covered are the point spread function, image and object domains, and how a pinhole camera forms images by blocking light rays except through the aperture.
This document summarizes an academic paper that proposes modifying well-known local linear models for system identification by replacing their original recursive learning rules with outlier-robust variants based on M-estimation. It describes three existing local linear models - local linear map (LLM), radial basis function network (RBFN), and local model network (LMN) - and then introduces the concept of M-estimation as a way to make the learning rules of these models more robust to outliers. The performance of the proposed outlier-robust variants is evaluated on three benchmark datasets and is found to provide considerable improvement in the presence of outliers compared to the original models.
On the Principle of Optimality for Linear Stochastic Dynamic System ijfcstjournal
In this work, processes represented by linear stochastic dynamic system are investigated and by
considering optimal control problem, principle of optimality is proven. Also, for existence of optimal
control and corresponding optimal trajectory, proofs of theorems of necessity and sufficiency condition are
attained.
Hidden Markov Models with applications to speech recognitionbutest
This document provides an introduction to hidden Markov models (HMMs). It discusses how HMMs can be used to model sequential data where the underlying states are not directly observable. The key aspects of HMMs are: (1) the model has a set of hidden states that evolve over time according to transition probabilities, (2) observations are emitted based on the current hidden state, (3) the four basic problems of HMMs are evaluation, decoding, training, and model selection. Examples discussed include modeling coin tosses, balls in urns, and speech recognition. Learning algorithms for HMMs like Baum-Welch and Viterbi are also summarized.
Hidden Markov Models with applications to speech recognitionbutest
This document provides an introduction to hidden Markov models (HMMs). It discusses how HMMs can be used to model sequential data where the underlying states are not directly observable. The key aspects of HMMs are: (1) the model has a set of hidden states that evolve over time according to transition probabilities, (2) observations are emitted depending on the current hidden state, (3) the four basic problems of HMMs are evaluation, decoding, training, and model selection. Examples discussed include modeling coin tosses, balls in urns, and speech recognition. Learning algorithms for HMMs like Baum-Welch and Viterbi training are also summarized.
In this presentation we describe the formulation of the HMM model as consisting of states that are hidden that generate the observables. We introduce the 3 basic problems: Finding the probability of a sequence of observation given the model, the decoding problem of finding the hidden states given the observations and the model and the training problem of determining the model parameters that generate the given observations. We discuss the Forward, Backward, Viterbi and Forward-Backward algorithms.
This document summarizes key concepts from CS 221 lecture 5 on hidden Markov models and temporal filtering. The lecture covered Markov chains, hidden Markov models, and particle filtering for approximate inference in hidden Markov models. Hidden Markov models extend Markov chains to allow for hidden states that are observed indirectly through emissions. Particle filtering uses samples or "particles" to represent the distribution over hidden states and approximate inference.
The document discusses pairwise sequence alignment and dynamic programming algorithms for computing optimal alignments. It covers:
- Assumptions of sequence evolution including substitutions, insertions, deletions, duplications, and domain reuse.
- Using sequence comparison to discover functional and evolutionary relationships by identifying similar sequences and orthologs with similar functions.
- The dot plot method for discovering sequence similarity by plotting sequences against each other in a matrix and identifying diagonals of matches.
- Dynamic programming algorithms that compute the optimal alignment score in quadratic time and linear space by breaking the problem into overlapping subproblems.
- Extensions of the basic algorithm to handle affine gap penalties by introducing three matrices to track alignments ending in matches, gaps
This document provides information about a Digital Signal Processing course taught at Beirut Arab University in Spring 2015. It includes the course title and instructor, a list of topics to be covered such as discrete signals, impulse response, linear systems, and MATLAB scripts for sampling an analog signal and generating echoes. Examples of MATLAB code are provided for adding, shifting, and processing discrete sequences.
An Improved Iterative Method for Solving General System of Equations via Gene...Zac Darcy
Various algorithms are known for solving linear system of equations. Iteration methods for solving the
large sparse linear systems are recommended. But in the case of general n× m matrices the classic
iterative algorithms are not applicable except for a few cases. The algorithm presented here is based on the
minimization of residual of solution and has some genetic characteristics which require using Genetic
Algorithms. Therefore, this algorithm is best applicable for construction of parallel algorithms. In this
paper, we describe a sequential version of proposed algorithm and present its theoretical analysis.
Moreover we show some numerical results of the sequential algorithm and supply an improved algorithm
and compare the two algorithms.
An Improved Iterative Method for Solving General System of Equations via Gene...Zac Darcy
Various algorithms are known for solving linear system of equations. Iteration methods for solving the
large sparse linear systems are recommended. But in the case of general n× m matrices the classic
iterative algorithms are not applicable except for a few cases. The algorithm presented here is based on the
minimization of residual of solution and has some genetic characteristics which require using Genetic
Algorithms. Therefore, this algorithm is best applicable for construction of parallel algorithms. In this
paper, we describe a sequential version of proposed algorithm and present its theoretical analysis.
Moreover we show some numerical results of the sequential algorithm and supply an improved algorithm
and compare the two algorithms.
Statement of stochastic programming problemsSSA KPI
AACIMP 2010 Summer School lecture by Leonidas Sakalauskas. "Applied Mathematics" stream. "Stochastic Programming and Applications" course. Part 1.
More info at http://summerschool.ssa.org.ua
Conditional random fields (CRFs) are probabilistic models for segmenting and labeling sequence data. CRFs address limitations of previous models like hidden Markov models (HMMs) and maximum entropy Markov models (MEMMs). CRFs allow incorporation of arbitrary, overlapping features of the observation sequence and label dependencies. Parameters are estimated to maximize the conditional log-likelihood using iterative scaling or tracking partial feature expectations. Experiments show CRFs outperform HMMs and MEMMs on synthetic and real-world tasks by addressing label bias problems and modeling dependencies beyond the previous label.
Forecasting With An Adaptive Control Algorithmshwetakarsh
This working paper describes using a recursive least squares estimation method with exponential forgetting to estimate coefficients of a state space model of the US macroeconomy. The model uses 12 state variables including consumption, investment, industrial production, and interest rates. Out-of-sample forecasts show lower root mean square errors than OLS forecasts, indicating the adaptive method provides better tracking. Sensitivity analysis found a forgetting factor of 0.96411 produced the most accurate in-sample and out-of-sample forecasts compared to other values tested.
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.
Universal Approximation Property via Quantum Feature Maps
----
The quantum Hilbert space can be used as a quantum-enhanced feature space in machine learning (ML) via the quantum feature map to encode classical data into quantum states. We prove the ability to approximate any continuous function with optimal approximation rate via quantum ML models in typical quantum feature maps.
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Contributed talk at Quantum Techniques in Machine Learning 2021, Tokyo, November 8-12 2021.
By Quoc Hoan Tran, Takahiro Goto and Kohei Nakajima
PROGRAMMA ATTIVITA’ DIDATTICA A.A. 2016/17
DOTTORATO DI RICERCA IN INGEGNERIA STRUTTURALE E GEOTECNICA
____________________________________________________________
STOCHASTIC DYNAMICS AND MONTE CARLO SIMULATION IN EARTHQUAKE ENGINEERING APPLICATIONS
Lecture Series by
Agathoklis Giaralis, Ph.D., M.ASCE., P.E. City, University of London
Visiting Professor Sapienza University of Rome
- The document details a state space solver approach for analog mixed-signal simulations using SystemC. It models analog circuits as sets of linear differential equations and solves them using the Runge-Kutta method of numerical integration.
- Two examples are provided: a digital voltage regulator simulation and a digital phase locked loop simulation. Both analog circuits are modeled in state space and simulated alongside a digital design to verify mixed-signal behavior.
- The state space approach allows modeling analog circuits without transistor-level details, improving simulation speed over traditional mixed-mode simulations while still capturing system-level behavior.
Chaotic system and its Application in CryptographyMuhammad Hamid
A seminar on Chaotic System and Its application in cryptography specially in image encryption. Slide covers
Introduction
Bifurcation Diagram
Lyapnove Exponent
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
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Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
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There are 4 phases of wound healing: hemostasis, inflammation, proliferation, and remodeling. This document also describes the mechanism of wound healing. Factors that affect healing include infection, uncontrolled diabetes, poor nutrition, age, anemia, the presence of foreign bodies, etc.
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Philippine Edukasyong Pantahanan at Pangkabuhayan (EPP) CurriculumMJDuyan
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𝐄𝐱𝐩𝐥𝐚𝐢𝐧 𝐭𝐡𝐞 𝐍𝐚𝐭𝐮𝐫𝐞 𝐚𝐧𝐝 𝐒𝐜𝐨𝐩𝐞 𝐨𝐟 𝐚𝐧 𝐄𝐧𝐭𝐫𝐞𝐩𝐫𝐞𝐧𝐞𝐮𝐫:
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Temple of Asclepius in Thrace. Excavation resultsKrassimira Luka
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Hmm and neural networks
1. BY
I P G B I O I N FO R M AT I C S
R . JA N A N I
1 9 P B I 0 0 3
Tertiary structure prediction
2. HMM HISTORY
HMM developed and published in 1960s and 1970s
Not widespread until late 80s
Theory published in mathematical journals.
Insufficient tutorial material for readers to understand and
apply concepts.
Andrey Andreyevich Markov was a Russian mathematician,
known for work on stochastic process.
His primary subject of research later became known as
Markov chains and Markov processes.
3. HIDDEN MARKOV MODEL
It’s a statistical analysis of sequences, especially for signal
models in which the system is being modeled is assumed to
be a Markov process with hidden states.
It states that the evolution of observable events depend on
internal factors, which are not directly observable.
It offer a mathematical description of current state of system
whose internal state is not known, only its output.
It is one among the various signal processing models and
algorithms have been used in biological sequence analysis.
It considers the real world problems structure dealing
classifying raw observations
They are sequential and cannot see the event producing the
output.
4. Observed event ‘symbol’ and invisible factor underlying
‘state’.
Consists of two stochastic process
1. Invisible process of hidden states.
2. Visible process of observable symbols.
The hidden states of markov chain and the probability
distribution of observed symbol depends on underlying
states.
It is also called doubly-embedded stochastic process.
It is well known for effectiveness in modeling the
correlations between adjacent symbols, domains or events
used in various fields.
5. Consist of finite number of set of states, an alphabet of output
symbols, a set of transition probabilities, a set of emission
probabilities.
Emission probabilities specify distribution of output symbols
that may be emitted from each state.
Two stochastic process the process of moving between states
and the process of emitting an output sequence.
Sequence of state transition is a hidden process and is
observed through the sequence of emitted symbols.
6. Two states: ’rain’ and ‘dry’
Transition probabilities: P(‘rain’/’rain’)=0.3
P(‘dry’ ’rain’)=0.7,P(‘ra’)=0.6
In dry)=0.2, P(‘dry ‘’dry’)=0.8
Initial probabilities : say P(‘rain’)=0.4, p(‘dry’)
Suppose calculate the probability of
sequence of state in our example{dry dry
rain rain}
P({‘dry ‘’dry ‘’rain ‘’rain’})=P(rain rain)P(rain
dry)p(dry dry)P(dry rain)
• =
=0.3*0.2*0.8*0.6
7. Forward-Backward procedure
Forward Algorithm -Intuition
Our goal is to determine the probability of a
sequence of observations (X1, X2, …, Xn)
given 𝜆
In the forward algorithm approach, divide
the sequence X in to sub-sequences,
compute the probabilities, store them in the
table for later use.
The probability of a larger sequence is
obtained by combining the probabilities of
these smaller sequences.
Specifically, compute the joint probability of
a sub-sequence starting from time t = 1
where the sub-sequence ends on a state y.
compute: P(X1:t, Yt| 𝜆)
then compute P(X1:n| 𝜆) by marginalizing Y
8. Forward Algorithm
Goal: Compute P(Yk, X1:k) assuming the model parameters to be
known
Approach: known emission and transition probabilities, factorize the
joint distribution P(Yk, X1:k) in terms of the known parameters and
solve. In order to implement efficiently use dynamic programming
where a large problem is solved by solving the overlapping sub-
problems and combining the solution.To do this set up the recursion.
We can write: X1:k= X1,X2…Xk-1, Xk
From sum rule we know: P(X = xi) = 𝑗𝑃(𝑋=𝑥𝑖,𝑌=𝑦𝑗)
𝑃𝑌𝑘,𝑋1:𝑘= 𝑦𝑘−1𝑚𝑃𝑌𝑘,𝑌𝑘−1,𝑋1:𝑘
𝑃𝑌𝑘,𝑋1:𝑘= 𝑦𝑘−1𝑚𝑃(𝑋1:𝑘−1,𝑌𝑘−1,𝑌𝑘,𝑋𝑘)
From product rule the above factorizes to:
𝑦𝑘−1𝑚𝑃𝑋1:𝑘−1𝑃𝑌𝑘−1𝑋1:𝑘−1𝑃(𝑌𝑘𝑌𝑘−1,𝑋1:𝑘−1)𝑃(𝑋𝑘𝑌𝑘,𝑌𝑘−1,𝑋1:𝑘−1)
= 𝑦𝑘−1𝑚𝑃𝑋1:𝑘−1𝑃𝑌𝑘−1𝑋1:𝑘−1𝑃(𝑌𝑘𝑌𝑘−1)𝑃(𝑋𝑘𝑌𝑘)
We can write: 𝛼𝑘𝑌𝑘= 𝑦𝑘−1𝑚𝑃(𝑌𝑘𝑌𝑘−1)𝑃(𝑋𝑘𝑌𝑘)𝛼𝑘−1(𝑌𝑘−1)
Initialization: 𝛼1𝑌1=𝑃𝑌1,𝑋1=𝑃(𝑌1) P(X1|Y1)
can now compute the different αvalues
9. Forward Algorithm: Implementation
defforward(self, obs):
self.fwd= [{}]
for y in self.states:
self.fwd[0][y] = self.pi[y] * self.B[y][obs[0]] # Initialize
base cases
for t in range(1, len(obs)):
self.fwd.append({})
for y in self.states:
self.fwd[t][y] = sum((self.fwd[t-1][y0] * self.A[y0][y] *
self.B[y][obs[t]]) for y0 in self.states)
prob= sum((self.fwd[len(obs) -1][s]) for s in self.states)
return prob
10. Backward Algorithm -Intuition
Our goal is to determine the probability of
a sequence of observations (Xk+1, Xk+2,
…, Xn|Yk,𝜆)
Given that the HMM has seen k
observations and ended up in a state Yk=
y, compute the probability of the
remaining part: Xk+1, Xk+2, …, Xn
Form the sub-sequences starting from the
last observation Xn and proceed
backward to the first.
Specifically, compute the conditional
probability of a sub-sequence starting
from k+1 and ending in n, where the state
at k is given.
can compute P(X1:n| 𝜆) by marginalizing
Y. The probability of an observation
sequence computed by backward
algorithm will be equal to that computed
with forward algorithm.
11. Backward Algorithm Implementation
defbackward(self, obs):
self.bwk= [{} for t in range(len(obs))]
T = len(obs)
for y in self.states:
self.bwk[T-1][y] = 1
for t in reversed(range(T-1)):
for y in self.states:
self.bwk[t][y] = sum((self.bwk[t+1][y1] * self.A[y][y1]
* self.B[y1][obs[t+1]]) for y1 in self.states)
prob= sum((self.pi[y]* self.B[y][obs[0]] *
self.bwk[0][y]) for y in self.states)
return prob
12. Viterbi Algorithm -Intuition
Our goal is to determine the most probable
state sequence for a given sequence of
observations (X1, X2, …, Xn) given 𝜆
This is a decoding process where we
discover the hidden state sequence looking
at the observations
Specifically, we need: argmaxYP(X1:t|Y1:t,
𝜆). This is equivalent to finding
argmaxYP(X1:t, Y1:t| 𝜆)
In the forward algorithm approach, we
computed the probabilities along each path
that led to the given state and summed the
probabilities to get the probability of
reaching that state regardless of the path
taken.
In Viterbi we are interested in only a
specific path that maximizes the
probability of reaching the required
state. Each state along this path (the one
that yields max probability) forms the
sequence of hidden states that are
interested in.
13. Viterbi Implementation
defviterbi(self, obs):
vit= [{}]
path = {}
for y in self.states:
vit[0][y] = self.pi[y] * self.B[y][obs[0]]
path[y] = [y]
for t in range(1, len(obs)):
vit.append({})
newpath= {}
for y in self.states:
(prob, state) = max((vit[t-1][y0] * self.A[y0][y] * self.B[y][obs[t]], y0)
for y0 in self.states)
vit[t][y] = prob
newpath[y] = path[state] + [y]
path = newpath
(prob, state) = max((vit[len(obs) -1][y], y) for y in self.states)
return (prob, path[state])
14. Application to HMMS to specific problem
Constructing genetic linkage maps.
Identifying non-coding DNA
Identifying protein-binding sites on DNA
Modelling helical caps
Protein secondary structure prediction
Protein domain classification
Problem of gene finding
Given DNA sequence the problem is to be determine the
location of genes
Input sequence of DNA X=(X1,… .Xn) £, where£=A, G, C, T
The output gives correctly labelled elements in X belonging to
coding, non-coding or inter-genic region.
Tools available Genie, GeneID, and HMMGene.
Matching of known set of DNA against a set of known genes.
15. HMM and multiple sequence alignment
HMM can be automatically create a multiple alignment from a group of unaligned sequences.
It is useful for prediction of history of evolution.
One of the major advantage of HMM can be estimated from sequence without aligning the sequence
first.
The sequence used to estimate or train the model are called training sequences.
Estimation done with the iterative forward-backward algorithm, also known as Baum-Welch
algorithm.
It maximizes the likelihood of training sequence.
Protein secondary structure prediction using HMMs
HMM is used to analyse the amino-acid sequence of proteins, studying secondary structures(helix,
sheet, and turn) and predicting the secondary structure of sequence.
The sequence contains the secondary structure whose HMM shows the highest probability
Profile-profile HMMs
HMMs built by analysing the distribution of amino acids in the training set of related proteins.
It’s a statistical model of protein family.
A state shown by diamond shaped box model insertions of random letters between two alignment
position.
A state shown by circle model deletions corresponding to gap in an alignment.
States of neighbouring portions are connected by lines.
For each line there is transition probability.
The repository of protein-profile HMMs found in PRAY database (http://www.pfam.wustl.edu) . It’s a
protein family database.
16. HMM software
HMMER
Is a package of nine programs use HMMs for sequence database search
Freely distributed.
Implementation of profile HMM method for sensitive database searches
using MSA queries.
Takes MSA inputs and build statistical model
17. SAM(sequence alignment and modelling)
Collection of flexible software tools for creating, refining,
using linear HMM for biological sequence analysis.
Model states can be viewed as representing the sequence
of columns in a MSA with arbitrary position dependent
insertions and deletions in each sequence.
Trained on a family of protein or nucleic acid sequence
using expectation-maximization algorithm and variety of
algorithmic heuristics.
18. Advantages:
Handle sequence of variable length.
Used in biological data analysis, machine learning techniques,
which requires fixed length input, such as neural network or
support vector machine
Allows position dependent gap penalties.
HMMs treat insertions and deletions in a statistical manner
that is dependent on position.
Limitations of HMMs
Linear model, unable to capture higher order correlation
among amino acids.
There is a standard machine learning problem with HMMs.
20. Structure prediction by neural network
model
Neural networks:
Also called artificial neural network are parallel distributed information
structure.
Feed forward network or multi layer perception(MLP)
Accurate
Building of the initial random net
Involves
Random selection of the type of node
Random selection of parameters of node
Random selection of number of inputs
Connecting the input and output until the net is larger
Running the training set over the net
Selecting the proper output
Removal of all nodes which do not contribute to output
23. A method computing, based on the interaction of multiple
connected processing elements
Solve many problems
The ability to learn from experience in order to improve their
performance
Ability to deal with incomplete information
Biological approach to AI
Developed in 1943
Comprised one or more layers of neuron
Several types they are feed-forward and feedback networks.
25. Classification based on learning
Supervised
Each training pattern: input+desired output
Adopt weights
After many epochs convergences to a local minimum
Unsupervised learning
No help from the outside
No training data, no information available on the desired output
Learning by doing
used to pick out structure in the input
Clustering
Reduction of dimensionality
E.g., Kohonen’s learning laws
Reinforcement learning
Teacher : training data
The teacher scores the performance of the training examples
Use performance score to shuffle weights “randomly”
Relatively slow learning due to “randomness”
26. Training the network
Alter the parameters.
Add/delete connections
Add/delete nodes with their connection
Post processing steps:
Removal of unused edges or nodes of training set
To obtain better result the nets are combined
Each training pair is of the form
Pattern **LSADQISTVASFDK
Target H
Short protein chain centred on the residue to be predicted
The estranged symbol*is used for windows that overlap the N-orC-
terminal of chain
27. 3 target classes
Helix, strands, and coil defined by collapsing the eight structural
classes given in DESPITE(Definition of Secondary Structure of
Protein)
DSSP classes prediction class
H, G helix
E strand
B, I, S, T, e, g, h coil
Both residues and target classes are encoded in unary format
Alanine 100000000000000
Helix 100
Every amino acid secondary structure type is given equal weight
age
28. Evaluating prediction efficiency
Jack-knife test
Standard way to gain unbiased measure neural network
performance is to perform Jack-knife testing or n-flod cross
validation
Tuned using training database
P times for a training set containing P protein chains, each
time the network is trained
Computational cost high
With P=1, n-flod validation identical to Jack-knife testing
29. Percentage of correctly classified residues
Popular statistical measure of performance, known as Q3.
Typical score Q3=62%
Particular measure fails to penalize the network for over prediction (non helix residues to
be helix) and understand prediction s(helix residues predicted to be non helix)
Correlation coefficient for each target class
Rigorous measure which involves calculating the correlation coefficient for each target
class
Cn= pn-ou/
√(p+sigma) (p+u) (n+sigma) (n+u)
p=pattern correctly assigned to helix
n=pattern correctly assigned to non-helix
Sigma=pattern incorrectly assigned to helix
U=pattern incorrectly assigned to non helix
The correlation coefficient for helix(Ch), strand(Ce), and coil (Cc) range from +1 totally
correlated to -1 totally anti-correlated
A measure which does take the location of predicted segment into account is the
percentage of overlapping segments.
30. Reliability index (RI)
Measure proposed by Rost and Sander, 1993.
RI of given residues is calculated using the highest and second
highest output values
Integer[(net highest output-net next highest output) ×10]
Drawbacks
Prediction are based on limited local context.
Non local factors not taken into account
Predictions based on limited amount of biological information
Principles underlying protein structure not considered
The predictions are uncorrelated
Predictions based on performance of single network, with inherent
bias/noise.
31. Applications
Pattern recognition
Investment analysis
Control systems and monitoring
Mobile computing
Marketing and financial applications
Forecasting-sales, market research, Meteorology
32. Advantages
Perform task that linear program cannot do
When an element of neural network fails, it can continue
without problem
It learns and does not need to be reprogrammed
It can be implemented in any applications
It can be implemented without any problem
Disadvantages
Needs training to operate
Architecture is different from microprocessors
Requires high process time for large network