Autoregressive (AR) model and conditional autoregressive (CAR) model are specific regressive models in which independent variables and dependent variable imply the same object. They are powerful statistical tools to predict values based on correlation of time domain and space domain, which are useful in epidemiology analysis. In this research, I combine them by the simple way in which AR and CAR is estimated in two separate steps so as to cover time domain and space domain in spatial-temporal data analysis. Moreover, I integrate logistic model into CAR model, which aims to improve competence of autoregressive models.
Comparative study of results obtained by analysis of structures using ANSYS, ...IOSR Journals
The analysis of complex structures like frames, trusses and beams is carried out using the Finite
Element Method (FEM) in software products like ANSYS and STAAD. The aim of this paper is to compare the
deformation results of simple and complex structures obtained using these products. The same structures are
also analyzed by a MATLAB program to provide a common reference for comparison. STAAD is used by civil
engineers to analyze structures like beams and columns while ANSYS is generally used by mechanical engineers
for structural analysis of machines, automobile roll cage, etc. Since both products employ the same fundamental
principle of FEM, there should be no difference in their results. Results however, prove contradictory to this for
complex structures. Since FEM is an approximate method, accuracy of the solutions cannot be a basis for their
comparison and hence, none of the varying results can be termed as better or worse. Their comparison may,
however, point to conservative results, significant digits and magnitude of difference so as to enable the analyst
to select the software best suited for the particular application of his or her structure.
Extreme bound analysis based on correlation coefficient for optimal regressio...Loc Nguyen
Regression analysis is an important tool in statistical analysis, in which there is a demand of discovering essential independent variables among many other ones, especially in case that there is a huge number of random variables. Extreme bound analysis is a powerful approach to extract such important variables called robust regressors. In this research, a so-called Regressive Expectation Maximization with RObust regressors (REMRO) algorithm is proposed as an alternative method beside other probabilistic methods for analyzing robust variables. By the different ideology from other probabilistic methods, REMRO searches for robust regressors forming optimal regression model and sorts them according to descending ordering given their fitness values determined by two proposed concepts of local correlation and global correlation. Local correlation represents sufficient explanatories to possible regressive models and global correlation reflects independence level and stand-alone capacity of regressors. Moreover, REMRO can resist incomplete data because it applies Regressive Expectation Maximization (REM) algorithm into filling missing values by estimated values based on ideology of expectation maximization (EM) algorithm. From experimental results, REMRO is more accurate for modeling numeric regressors than traditional probabilistic methods like Sala-I-Martin method but REMRO cannot be applied in case of nonnumeric regression model yet in this research.
I am Ben R. I am a Statistics Assignment Expert at statisticshomeworkhelper.com. I hold a Ph.D. in Statistics, from University of Denver, USA. I have been helping students with their homework for the past 5 years. I solve assignments related to Statistics.
Visit statisticshomeworkhelper.com or email info@statisticshomeworkhelper.com.
You can also call on +1 678 648 4277 for any assistance with Statistics Assignments.
Development, Optimization, and Analysis of Cellular Automaton Algorithms to S...IRJET Journal
This document summarizes research on using cellular automaton algorithms to solve stochastic partial differential equations (SPDEs). It proposes a finite-difference method to approximate an SPDE modeling a random walk with angular diffusion. A Monte Carlo algorithm is also developed for comparison. Analysis finds a moderate correlation between the two methods, suggesting the finite-difference approach is reasonably accurate. It also identifies an inverse-square relationship between variables, linking to a foundational stochastic analysis concept. The research concludes the finite-difference method shows promise for approximating SPDEs while considering boundary conditions.
The document discusses pseudospectra as an alternative to eigenvalues for analyzing non-normal matrices and operators. It defines three equivalent definitions of pseudospectra: (1) the set of points where the resolvent is larger than ε-1, (2) the set of points that are eigenvalues of a perturbed matrix with perturbation smaller than ε, and (3) the set of points where the resolvent applied to a unit vector is larger than ε. It also shows that pseudospectra are nested sets and their intersection is the spectrum. The definitions extend to operators on Hilbert spaces using singular values.
USING LEARNING AUTOMATA AND GENETIC ALGORITHMS TO IMPROVE THE QUALITY OF SERV...IJCSEA Journal
A hybrid learning automata–genetic algorithm (HLGA) is proposed to solve QoS routing optimization problem of next generation networks. The algorithm complements the advantages of the learning Automato Algorithm(LA) and Genetic Algorithm(GA). It firstly uses the good global search capability of LA to generate initial population needed by GA, then it uses GA to improve the Quality of Service(QoS) and acquiring the optimization tree through new algorithms for crossover and mutation operators which are an NP–Complete problem. In the proposed algorithm, the connectivity matrix of edges is used for genotype representation. Some novel heuristics are also proposed for mutation, crossover, and creation of random individuals. We evaluate the performance and efficiency of the proposed HLGA-based algorithm in comparison with other existing heuristic and GA-based algorithms by the result of simulation. Simulation results demonstrate that this paper proposed algorithm not only has the fast calculating speed and high accuracy but also can improve the efficiency in Next Generation Networks QoS routing. The proposed algorithm has overcome all of the previous algorithms in the literature..
Computer Science
Active and Programmable Networks
Active safety systems
Ad Hoc & Sensor Network
Ad hoc networks for pervasive communications
Adaptive, autonomic and context-aware computing
Advance Computing technology and their application
Advanced Computing Architectures and New Programming Models
Advanced control and measurement
Aeronautical Engineering,
Agent-based middleware
Alert applications
Automotive, marine and aero-space control and all other control applications
Autonomic and self-managing middleware
Autonomous vehicle
Biochemistry
Bioinformatics
BioTechnology(Chemistry, Mathematics, Statistics, Geology)
Broadband and intelligent networks
Broadband wireless technologies
CAD/CAM/CAT/CIM
Call admission and flow/congestion control
Capacity planning and dimensioning
Changing Access to Patient Information
Channel capacity modelling and analysis
Civil Engineering,
Cloud Computing and Applications
Collaborative applications
Communication application
Communication architectures for pervasive computing
Communication systems
Computational intelligence
Computer and microprocessor-based control
Computer Architecture and Embedded Systems
Computer Business
Computer Sciences and Applications
Computer Vision
Computer-based information systems in health care
Computing Ethics
Computing Practices & Applications
Congestion and/or Flow Control
Content Distribution
Context-awareness and middleware
Creativity in Internet management and retailing
Cross-layer design and Physical layer based issue
Cryptography
Data Base Management
Data fusion
Data Mining
Data retrieval
Data Storage Management
Decision analysis methods
Decision making
Digital Economy and Digital Divide
Digital signal processing theory
Distributed Sensor Networks
Drives automation
Drug Design,
Drug Development
DSP implementation
E-Business
E-Commerce
E-Government
Electronic transceiver device for Retail Marketing Industries
Electronics Engineering,
Embeded Computer System
Emerging advances in business and its applications
Emerging signal processing areas
Enabling technologies for pervasive systems
Energy-efficient and green pervasive computing
Environmental Engineering,
Estimation and identification techniques
Evaluation techniques for middleware solutions
Event-based, publish/subscribe, and message-oriented middleware
Evolutionary computing and intelligent systems
Expert approaches
Facilities planning and management
Flexible manufacturing systems
Formal methods and tools for designing
Fuzzy algorithms
Fuzzy logics
GPS and location-based app
Spacey random walks from Householder Symposium XX 2017Austin Benson
1. Spacey random walks are a stochastic process that provides a probabilistic interpretation of tensor eigenvectors. The spacey random walk forgets its previous state but guesses it randomly, resulting in a limiting distribution that is a tensor eigenvector.
2. Higher-order Markov chains can be modeled as spacey random walks, which converge to tensor eigenvectors. This provides an algorithm for computing eigenvectors via numerical integration rather than algebraic methods.
3. Spacey random walks generalize Pólya urn processes and have applications in transportation modeling, clustering multi-relational data, and ranking. Learning the transition tensor from taxi trajectory data supports the spacey random walk hypothesis.
Comparative study of results obtained by analysis of structures using ANSYS, ...IOSR Journals
The analysis of complex structures like frames, trusses and beams is carried out using the Finite
Element Method (FEM) in software products like ANSYS and STAAD. The aim of this paper is to compare the
deformation results of simple and complex structures obtained using these products. The same structures are
also analyzed by a MATLAB program to provide a common reference for comparison. STAAD is used by civil
engineers to analyze structures like beams and columns while ANSYS is generally used by mechanical engineers
for structural analysis of machines, automobile roll cage, etc. Since both products employ the same fundamental
principle of FEM, there should be no difference in their results. Results however, prove contradictory to this for
complex structures. Since FEM is an approximate method, accuracy of the solutions cannot be a basis for their
comparison and hence, none of the varying results can be termed as better or worse. Their comparison may,
however, point to conservative results, significant digits and magnitude of difference so as to enable the analyst
to select the software best suited for the particular application of his or her structure.
Extreme bound analysis based on correlation coefficient for optimal regressio...Loc Nguyen
Regression analysis is an important tool in statistical analysis, in which there is a demand of discovering essential independent variables among many other ones, especially in case that there is a huge number of random variables. Extreme bound analysis is a powerful approach to extract such important variables called robust regressors. In this research, a so-called Regressive Expectation Maximization with RObust regressors (REMRO) algorithm is proposed as an alternative method beside other probabilistic methods for analyzing robust variables. By the different ideology from other probabilistic methods, REMRO searches for robust regressors forming optimal regression model and sorts them according to descending ordering given their fitness values determined by two proposed concepts of local correlation and global correlation. Local correlation represents sufficient explanatories to possible regressive models and global correlation reflects independence level and stand-alone capacity of regressors. Moreover, REMRO can resist incomplete data because it applies Regressive Expectation Maximization (REM) algorithm into filling missing values by estimated values based on ideology of expectation maximization (EM) algorithm. From experimental results, REMRO is more accurate for modeling numeric regressors than traditional probabilistic methods like Sala-I-Martin method but REMRO cannot be applied in case of nonnumeric regression model yet in this research.
I am Ben R. I am a Statistics Assignment Expert at statisticshomeworkhelper.com. I hold a Ph.D. in Statistics, from University of Denver, USA. I have been helping students with their homework for the past 5 years. I solve assignments related to Statistics.
Visit statisticshomeworkhelper.com or email info@statisticshomeworkhelper.com.
You can also call on +1 678 648 4277 for any assistance with Statistics Assignments.
Development, Optimization, and Analysis of Cellular Automaton Algorithms to S...IRJET Journal
This document summarizes research on using cellular automaton algorithms to solve stochastic partial differential equations (SPDEs). It proposes a finite-difference method to approximate an SPDE modeling a random walk with angular diffusion. A Monte Carlo algorithm is also developed for comparison. Analysis finds a moderate correlation between the two methods, suggesting the finite-difference approach is reasonably accurate. It also identifies an inverse-square relationship between variables, linking to a foundational stochastic analysis concept. The research concludes the finite-difference method shows promise for approximating SPDEs while considering boundary conditions.
The document discusses pseudospectra as an alternative to eigenvalues for analyzing non-normal matrices and operators. It defines three equivalent definitions of pseudospectra: (1) the set of points where the resolvent is larger than ε-1, (2) the set of points that are eigenvalues of a perturbed matrix with perturbation smaller than ε, and (3) the set of points where the resolvent applied to a unit vector is larger than ε. It also shows that pseudospectra are nested sets and their intersection is the spectrum. The definitions extend to operators on Hilbert spaces using singular values.
USING LEARNING AUTOMATA AND GENETIC ALGORITHMS TO IMPROVE THE QUALITY OF SERV...IJCSEA Journal
A hybrid learning automata–genetic algorithm (HLGA) is proposed to solve QoS routing optimization problem of next generation networks. The algorithm complements the advantages of the learning Automato Algorithm(LA) and Genetic Algorithm(GA). It firstly uses the good global search capability of LA to generate initial population needed by GA, then it uses GA to improve the Quality of Service(QoS) and acquiring the optimization tree through new algorithms for crossover and mutation operators which are an NP–Complete problem. In the proposed algorithm, the connectivity matrix of edges is used for genotype representation. Some novel heuristics are also proposed for mutation, crossover, and creation of random individuals. We evaluate the performance and efficiency of the proposed HLGA-based algorithm in comparison with other existing heuristic and GA-based algorithms by the result of simulation. Simulation results demonstrate that this paper proposed algorithm not only has the fast calculating speed and high accuracy but also can improve the efficiency in Next Generation Networks QoS routing. The proposed algorithm has overcome all of the previous algorithms in the literature..
Computer Science
Active and Programmable Networks
Active safety systems
Ad Hoc & Sensor Network
Ad hoc networks for pervasive communications
Adaptive, autonomic and context-aware computing
Advance Computing technology and their application
Advanced Computing Architectures and New Programming Models
Advanced control and measurement
Aeronautical Engineering,
Agent-based middleware
Alert applications
Automotive, marine and aero-space control and all other control applications
Autonomic and self-managing middleware
Autonomous vehicle
Biochemistry
Bioinformatics
BioTechnology(Chemistry, Mathematics, Statistics, Geology)
Broadband and intelligent networks
Broadband wireless technologies
CAD/CAM/CAT/CIM
Call admission and flow/congestion control
Capacity planning and dimensioning
Changing Access to Patient Information
Channel capacity modelling and analysis
Civil Engineering,
Cloud Computing and Applications
Collaborative applications
Communication application
Communication architectures for pervasive computing
Communication systems
Computational intelligence
Computer and microprocessor-based control
Computer Architecture and Embedded Systems
Computer Business
Computer Sciences and Applications
Computer Vision
Computer-based information systems in health care
Computing Ethics
Computing Practices & Applications
Congestion and/or Flow Control
Content Distribution
Context-awareness and middleware
Creativity in Internet management and retailing
Cross-layer design and Physical layer based issue
Cryptography
Data Base Management
Data fusion
Data Mining
Data retrieval
Data Storage Management
Decision analysis methods
Decision making
Digital Economy and Digital Divide
Digital signal processing theory
Distributed Sensor Networks
Drives automation
Drug Design,
Drug Development
DSP implementation
E-Business
E-Commerce
E-Government
Electronic transceiver device for Retail Marketing Industries
Electronics Engineering,
Embeded Computer System
Emerging advances in business and its applications
Emerging signal processing areas
Enabling technologies for pervasive systems
Energy-efficient and green pervasive computing
Environmental Engineering,
Estimation and identification techniques
Evaluation techniques for middleware solutions
Event-based, publish/subscribe, and message-oriented middleware
Evolutionary computing and intelligent systems
Expert approaches
Facilities planning and management
Flexible manufacturing systems
Formal methods and tools for designing
Fuzzy algorithms
Fuzzy logics
GPS and location-based app
Spacey random walks from Householder Symposium XX 2017Austin Benson
1. Spacey random walks are a stochastic process that provides a probabilistic interpretation of tensor eigenvectors. The spacey random walk forgets its previous state but guesses it randomly, resulting in a limiting distribution that is a tensor eigenvector.
2. Higher-order Markov chains can be modeled as spacey random walks, which converge to tensor eigenvectors. This provides an algorithm for computing eigenvectors via numerical integration rather than algebraic methods.
3. Spacey random walks generalize Pólya urn processes and have applications in transportation modeling, clustering multi-relational data, and ranking. Learning the transition tensor from taxi trajectory data supports the spacey random walk hypothesis.
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.
---
Contributed talk at Quantum Techniques in Machine Learning 2021, Tokyo, November 8-12 2021.
By Quoc Hoan Tran, Takahiro Goto and Kohei Nakajima
This document proposes an improved particle swarm optimization (PSO) algorithm for data clustering that incorporates Gauss chaotic map. PSO is often prone to premature convergence, so the proposed method uses Gauss chaotic map to generate random sequences that substitute the random parameters in PSO, providing more exploration of the search space. The algorithm is tested on six real-world datasets and shown to outperform K-means, standard PSO, and other hybrid clustering algorithms. The key aspects of the proposed GaussPSO method and experimental results demonstrating its effectiveness are described.
This document summarizes several iterative methods for solving systems of linear equations. It discusses stationary methods like Jacobi, Gauss-Seidel, and SOR, as well as non-stationary methods like conjugate gradient and preconditioned conjugate gradient. Matlab programs are provided for each method. The results show that non-stationary methods converge faster than stationary methods. Specifically, the preconditioned conjugate gradient method approximates the solution to five decimal places within 5 iterations for the example problem. The document also discusses properties of the conjugate gradient method and how preconditioning can improve convergence. These iterative methods have applications in solving partial differential equations.
ER Publication,
IJETR, IJMCTR,
Journals,
International Journals,
High Impact Journals,
Monthly Journal,
Good quality Journals,
Research,
Research Papers,
Research Article,
Free Journals, Open access Journals,
erpublication.org,
Engineering Journal,
Science Journals,
Erwin. e. obermayer k._schulten. k. _1992: self-organising maps_stationary st...ArchiLab 7
This document summarizes research investigating how the shape of neighborhood functions affects convergence rates and the presence of metastable states in Kohonen's self-organizing feature map algorithm. The key findings are:
1) For neighborhood functions that are convex over a large interval, there exist no metastable states, while other functions allow metastable states regardless of parameters.
2) For Gaussian functions, there is a threshold width above which metastable states cannot exist.
3) Convergence is fastest using functions that are convex over a large range but differ greatly between neighbors, such as Gaussian functions with width near the number of neurons. Metastable states and neighborhood function shape strongly influence convergence time.
International Journal of Engineering Research and DevelopmentIJERD Editor
Electrical, Electronics and Computer Engineering,
Information Engineering and Technology,
Mechanical, Industrial and Manufacturing Engineering,
Automation and Mechatronics Engineering,
Material and Chemical Engineering,
Civil and Architecture Engineering,
Biotechnology and Bio Engineering,
Environmental Engineering,
Petroleum and Mining Engineering,
Marine and Agriculture engineering,
Aerospace Engineering.
This document discusses strategies for parallelizing spectral methods. Spectral methods are global in nature due to their use of global basis functions, making them challenging to parallelize on fine-grained architectures. However, the document finds that spectral methods can be effectively parallelized. The main computational steps in spectral methods are the calculation of differential operators on functions and solving linear systems, both of which can exploit parallelism. Domain decomposition techniques may also help parallelize computations over non-Cartesian domains.
Robust Fuzzy Data Clustering In An Ordinal Scale Based On A Similarity MeasureIJRES Journal
This paper is devoted to processing data given in an ordinal scale. A new objective function of a
special type is introduced. A group of robust fuzzy clustering algorithms based on the similarity measure is
introduced.
This summarizes a document about a filter-and-refine approach for reducing computational cost when performing correlation analysis on pairs of spatial time series datasets. It groups similar time series within each dataset into "cones" based on spatial autocorrelation. Cone-level correlation computation can then filter out many element pairs whose correlation is clearly below a threshold. The remaining pairs require individual correlation computation in the refinement phase. Experiments on Earth science datasets showed significant computational savings, especially with high correlation thresholds.
This document summarizes work exploring the use of CUDA GPUs and Cell processors to accelerate a gravitational wave source-modelling application called the EMRI Teukolsky code. The code models gravitational waves generated by a small compact object orbiting a supermassive black hole. The authors implemented the code on a Cell processor and Nvidia GPU using CUDA. They were able to achieve over an order of magnitude speedup compared to a CPU implementation by leveraging the parallelism of these hardware accelerators.
1) The document discusses dynamics modeling for robotic manipulators using the Denavit-Hartenberg representation and Lagrangian mechanics. It describes using the Euler-Lagrange method to derive equations of motion for robotic links by computing kinetic and potential energy terms.
2) As an example, dynamics equations are derived for a simple 1 degree-of-freedom robotic arm. Kinetic and potential energy expressions are written and the Lagrangian is computed to obtain the equation of motion.
3) State-space modeling basics are reviewed using the example of a damped spring-mass system, showing how to write the system dynamics as state-space matrices to evaluate responses like step response.
This document summarizes Chapter 2 of a textbook on functional analysis in mechanics. It introduces Sobolev spaces, which are function spaces used to model mechanical problems. Sobolev spaces allow for generalized notions of derivatives of functions. The chapter discusses imbedding theorems for Sobolev spaces, which describe how functions in one Sobolev space can be mapped continuously or compactly to other function spaces. It provides examples of imbedding properties for specific Sobolev spaces over different domains.
This document summarizes key concepts from Sobolev spaces and their applications in mechanics problems. It introduces Sobolev spaces Wm,p(Ω) whose norms involve integrals of function and derivative values. These spaces allow generalized notions of derivatives. Sobolev's imbedding theorem establishes continuity properties of mappings between Sobolev and other function spaces. These properties are important for analyzing mechanical models that involve elements in Sobolev spaces.
This document presents results from a lattice QCD calculation of the proton isovector scalar charge (gs) at two light quark masses. The calculation uses domain-wall fermions and Iwasaki gauge actions on a 323x64 lattice with a spacing of 0.144 fm. Ratios of three-point to two-point correlation functions are formed and fit to a plateau to extract gs. Values of gs are obtained for quark masses of 0.0042 and 0.001, and all-mode averaging is used for the lighter mass. Chiral perturbation theory will be used to extrapolate gs to the physical quark mass. Preliminary results for gs at the unphysical quark masses are reported in lattice units.
A COMPARISON OF PARTICLE SWARM OPTIMIZATION AND DIFFERENTIAL EVOLUTIONijsc
Two modern optimization methods including Particle Swarm Optimization and Differential Evolution are
compared on twelve constrained nonlinear test functions. Generally, the results show that Differential
Evolution is better than Particle Swarm Optimization in terms of high-quality solutions, running time and
robustness.
This paper proposes a new method called Local Collaborative Ranking (LCR) for recommender systems. LCR assumes the user-item rating matrix is locally low-rank, meaning the matrix is low-rank within neighborhoods defined by a distance metric on user-item pairs. LCR combines a recent local low-rank approximation approach with empirical risk minimization for ranking losses. Experiments show LCR outperforms other state-of-the-art recommendation methods. LCR is also easily parallelizable, making it suitable for large-scale industrial applications.
The document presents a new approach for dynamic analysis of parallel manipulators based on the principle of virtual work. It illustrates the approach using a simple 4-bar linkage example, calculating the inertial forces and moments, virtual displacements, and input torque. It then generalizes the approach for dynamic analysis of a 6 degree-of-freedom parallel manipulator like a Gough-Stewart platform. The approach leads to faster computation than traditional Newton-Euler methods by not requiring calculation of constraint forces between links.
The document describes adaptive filters and the least mean squares (LMS) algorithm. Adaptive filters are filters whose coefficients are adjusted over time based on an optimization algorithm to minimize a cost function. The LMS algorithm is commonly used to update the filter coefficients to minimize the mean squared error between the filter output and a desired response. It does this by iteratively adjusting each coefficient proportional to the input signal and the error at each time step in an efficient way that does not require knowledge of complete statistics. The LMS algorithm and its application are summarized.
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.
Adversarial Variational Autoencoders to extend and improve generative model -...Loc Nguyen
Generative artificial intelligence (GenAI) has been developing with many incredible achievements like ChatGPT and Bard. Deep generative model (DGM) is a branch of GenAI, which is preeminent in generating raster data such as image and sound due to strong points of deep neural network (DNN) in inference and recognition. The built-in inference mechanism of DNN, which simulates and aims to synaptic plasticity of human neuron network, fosters generation ability of DGM which produces surprised results with support of statistical flexibility. Two popular approaches in DGM are Variational Autoencoders (VAE) and Generative Adversarial Network (GAN). Both VAE and GAN have their own strong points although they share and imply underline theory of statistics as well as incredible complex via hidden layers of DNN when DNN becomes effective encoding/decoding functions without concrete specifications. In this research, I try to unify VAE and GAN into a consistent and consolidated model called Adversarial Variational Autoencoders (AVA) in which VAE and GAN complement each other, for instance, VAE is a good data generator by encoding data via excellent ideology of Kullback-Leibler divergence and GAN is a significantly important method to assess reliability of data which is realistic or fake. In other words, AVA aims to improve accuracy of generative models, besides AVA extends function of simple generative models. In methodology this research focuses on combination of applied mathematical concepts and skillful techniques of computer programming in order to implement and solve complicated problems as simply as possible.
Conditional mixture model and its application for regression modelLoc Nguyen
Expectation maximization (EM) algorithm is a powerful mathematical tool for estimating statistical parameter when data sample contains hidden part and observed part. EM is applied to learn finite mixture model in which the whole distribution of observed variable is average sum of partial distributions. Coverage ratio of every partial distribution is specified by the probability of hidden variable. An application of mixture model is soft clustering in which cluster is modeled by hidden variable whereas each data point can be assigned to more than one cluster and degree of such assignment is represented by the probability of hidden variable. However, such probability in traditional mixture model is simplified as a parameter, which can cause loss of valuable information. Therefore, in this research I propose a so-called conditional mixture model (CMM) in which the probability of hidden variable is modeled as a full probabilistic density function (PDF) that owns individual parameter. CMM aims to extend mixture model. I also propose an application of CMM which is called adaptive regression model (ARM). Traditional regression model is effective when data sample is scattered equally. If data points are grouped into clusters, regression model tries to learn a unified regression function which goes through all data points. Obviously, such unified function is not effective to evaluate response variable based on grouped data points. The concept “adaptive” of ARM means that ARM solves the ineffectiveness problem by selecting the best cluster of data points firstly and then evaluating response variable within such best cluster. In orther words, ARM reduces estimation space of regression model so as to gain high accuracy in calculation.
Keywords: expectation maximization (EM) algorithm, finite mixture model, conditional mixture model, regression model, adaptive regression model (ARM).
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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.
---
Contributed talk at Quantum Techniques in Machine Learning 2021, Tokyo, November 8-12 2021.
By Quoc Hoan Tran, Takahiro Goto and Kohei Nakajima
This document proposes an improved particle swarm optimization (PSO) algorithm for data clustering that incorporates Gauss chaotic map. PSO is often prone to premature convergence, so the proposed method uses Gauss chaotic map to generate random sequences that substitute the random parameters in PSO, providing more exploration of the search space. The algorithm is tested on six real-world datasets and shown to outperform K-means, standard PSO, and other hybrid clustering algorithms. The key aspects of the proposed GaussPSO method and experimental results demonstrating its effectiveness are described.
This document summarizes several iterative methods for solving systems of linear equations. It discusses stationary methods like Jacobi, Gauss-Seidel, and SOR, as well as non-stationary methods like conjugate gradient and preconditioned conjugate gradient. Matlab programs are provided for each method. The results show that non-stationary methods converge faster than stationary methods. Specifically, the preconditioned conjugate gradient method approximates the solution to five decimal places within 5 iterations for the example problem. The document also discusses properties of the conjugate gradient method and how preconditioning can improve convergence. These iterative methods have applications in solving partial differential equations.
ER Publication,
IJETR, IJMCTR,
Journals,
International Journals,
High Impact Journals,
Monthly Journal,
Good quality Journals,
Research,
Research Papers,
Research Article,
Free Journals, Open access Journals,
erpublication.org,
Engineering Journal,
Science Journals,
Erwin. e. obermayer k._schulten. k. _1992: self-organising maps_stationary st...ArchiLab 7
This document summarizes research investigating how the shape of neighborhood functions affects convergence rates and the presence of metastable states in Kohonen's self-organizing feature map algorithm. The key findings are:
1) For neighborhood functions that are convex over a large interval, there exist no metastable states, while other functions allow metastable states regardless of parameters.
2) For Gaussian functions, there is a threshold width above which metastable states cannot exist.
3) Convergence is fastest using functions that are convex over a large range but differ greatly between neighbors, such as Gaussian functions with width near the number of neurons. Metastable states and neighborhood function shape strongly influence convergence time.
International Journal of Engineering Research and DevelopmentIJERD Editor
Electrical, Electronics and Computer Engineering,
Information Engineering and Technology,
Mechanical, Industrial and Manufacturing Engineering,
Automation and Mechatronics Engineering,
Material and Chemical Engineering,
Civil and Architecture Engineering,
Biotechnology and Bio Engineering,
Environmental Engineering,
Petroleum and Mining Engineering,
Marine and Agriculture engineering,
Aerospace Engineering.
This document discusses strategies for parallelizing spectral methods. Spectral methods are global in nature due to their use of global basis functions, making them challenging to parallelize on fine-grained architectures. However, the document finds that spectral methods can be effectively parallelized. The main computational steps in spectral methods are the calculation of differential operators on functions and solving linear systems, both of which can exploit parallelism. Domain decomposition techniques may also help parallelize computations over non-Cartesian domains.
Robust Fuzzy Data Clustering In An Ordinal Scale Based On A Similarity MeasureIJRES Journal
This paper is devoted to processing data given in an ordinal scale. A new objective function of a
special type is introduced. A group of robust fuzzy clustering algorithms based on the similarity measure is
introduced.
This summarizes a document about a filter-and-refine approach for reducing computational cost when performing correlation analysis on pairs of spatial time series datasets. It groups similar time series within each dataset into "cones" based on spatial autocorrelation. Cone-level correlation computation can then filter out many element pairs whose correlation is clearly below a threshold. The remaining pairs require individual correlation computation in the refinement phase. Experiments on Earth science datasets showed significant computational savings, especially with high correlation thresholds.
This document summarizes work exploring the use of CUDA GPUs and Cell processors to accelerate a gravitational wave source-modelling application called the EMRI Teukolsky code. The code models gravitational waves generated by a small compact object orbiting a supermassive black hole. The authors implemented the code on a Cell processor and Nvidia GPU using CUDA. They were able to achieve over an order of magnitude speedup compared to a CPU implementation by leveraging the parallelism of these hardware accelerators.
1) The document discusses dynamics modeling for robotic manipulators using the Denavit-Hartenberg representation and Lagrangian mechanics. It describes using the Euler-Lagrange method to derive equations of motion for robotic links by computing kinetic and potential energy terms.
2) As an example, dynamics equations are derived for a simple 1 degree-of-freedom robotic arm. Kinetic and potential energy expressions are written and the Lagrangian is computed to obtain the equation of motion.
3) State-space modeling basics are reviewed using the example of a damped spring-mass system, showing how to write the system dynamics as state-space matrices to evaluate responses like step response.
This document summarizes Chapter 2 of a textbook on functional analysis in mechanics. It introduces Sobolev spaces, which are function spaces used to model mechanical problems. Sobolev spaces allow for generalized notions of derivatives of functions. The chapter discusses imbedding theorems for Sobolev spaces, which describe how functions in one Sobolev space can be mapped continuously or compactly to other function spaces. It provides examples of imbedding properties for specific Sobolev spaces over different domains.
This document summarizes key concepts from Sobolev spaces and their applications in mechanics problems. It introduces Sobolev spaces Wm,p(Ω) whose norms involve integrals of function and derivative values. These spaces allow generalized notions of derivatives. Sobolev's imbedding theorem establishes continuity properties of mappings between Sobolev and other function spaces. These properties are important for analyzing mechanical models that involve elements in Sobolev spaces.
This document presents results from a lattice QCD calculation of the proton isovector scalar charge (gs) at two light quark masses. The calculation uses domain-wall fermions and Iwasaki gauge actions on a 323x64 lattice with a spacing of 0.144 fm. Ratios of three-point to two-point correlation functions are formed and fit to a plateau to extract gs. Values of gs are obtained for quark masses of 0.0042 and 0.001, and all-mode averaging is used for the lighter mass. Chiral perturbation theory will be used to extrapolate gs to the physical quark mass. Preliminary results for gs at the unphysical quark masses are reported in lattice units.
A COMPARISON OF PARTICLE SWARM OPTIMIZATION AND DIFFERENTIAL EVOLUTIONijsc
Two modern optimization methods including Particle Swarm Optimization and Differential Evolution are
compared on twelve constrained nonlinear test functions. Generally, the results show that Differential
Evolution is better than Particle Swarm Optimization in terms of high-quality solutions, running time and
robustness.
This paper proposes a new method called Local Collaborative Ranking (LCR) for recommender systems. LCR assumes the user-item rating matrix is locally low-rank, meaning the matrix is low-rank within neighborhoods defined by a distance metric on user-item pairs. LCR combines a recent local low-rank approximation approach with empirical risk minimization for ranking losses. Experiments show LCR outperforms other state-of-the-art recommendation methods. LCR is also easily parallelizable, making it suitable for large-scale industrial applications.
The document presents a new approach for dynamic analysis of parallel manipulators based on the principle of virtual work. It illustrates the approach using a simple 4-bar linkage example, calculating the inertial forces and moments, virtual displacements, and input torque. It then generalizes the approach for dynamic analysis of a 6 degree-of-freedom parallel manipulator like a Gough-Stewart platform. The approach leads to faster computation than traditional Newton-Euler methods by not requiring calculation of constraint forces between links.
The document describes adaptive filters and the least mean squares (LMS) algorithm. Adaptive filters are filters whose coefficients are adjusted over time based on an optimization algorithm to minimize a cost function. The LMS algorithm is commonly used to update the filter coefficients to minimize the mean squared error between the filter output and a desired response. It does this by iteratively adjusting each coefficient proportional to the input signal and the error at each time step in an efficient way that does not require knowledge of complete statistics. The LMS algorithm and its application are summarized.
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.
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Adversarial Variational Autoencoders to extend and improve generative model -...Loc Nguyen
Generative artificial intelligence (GenAI) has been developing with many incredible achievements like ChatGPT and Bard. Deep generative model (DGM) is a branch of GenAI, which is preeminent in generating raster data such as image and sound due to strong points of deep neural network (DNN) in inference and recognition. The built-in inference mechanism of DNN, which simulates and aims to synaptic plasticity of human neuron network, fosters generation ability of DGM which produces surprised results with support of statistical flexibility. Two popular approaches in DGM are Variational Autoencoders (VAE) and Generative Adversarial Network (GAN). Both VAE and GAN have their own strong points although they share and imply underline theory of statistics as well as incredible complex via hidden layers of DNN when DNN becomes effective encoding/decoding functions without concrete specifications. In this research, I try to unify VAE and GAN into a consistent and consolidated model called Adversarial Variational Autoencoders (AVA) in which VAE and GAN complement each other, for instance, VAE is a good data generator by encoding data via excellent ideology of Kullback-Leibler divergence and GAN is a significantly important method to assess reliability of data which is realistic or fake. In other words, AVA aims to improve accuracy of generative models, besides AVA extends function of simple generative models. In methodology this research focuses on combination of applied mathematical concepts and skillful techniques of computer programming in order to implement and solve complicated problems as simply as possible.
Conditional mixture model and its application for regression modelLoc Nguyen
Expectation maximization (EM) algorithm is a powerful mathematical tool for estimating statistical parameter when data sample contains hidden part and observed part. EM is applied to learn finite mixture model in which the whole distribution of observed variable is average sum of partial distributions. Coverage ratio of every partial distribution is specified by the probability of hidden variable. An application of mixture model is soft clustering in which cluster is modeled by hidden variable whereas each data point can be assigned to more than one cluster and degree of such assignment is represented by the probability of hidden variable. However, such probability in traditional mixture model is simplified as a parameter, which can cause loss of valuable information. Therefore, in this research I propose a so-called conditional mixture model (CMM) in which the probability of hidden variable is modeled as a full probabilistic density function (PDF) that owns individual parameter. CMM aims to extend mixture model. I also propose an application of CMM which is called adaptive regression model (ARM). Traditional regression model is effective when data sample is scattered equally. If data points are grouped into clusters, regression model tries to learn a unified regression function which goes through all data points. Obviously, such unified function is not effective to evaluate response variable based on grouped data points. The concept “adaptive” of ARM means that ARM solves the ineffectiveness problem by selecting the best cluster of data points firstly and then evaluating response variable within such best cluster. In orther words, ARM reduces estimation space of regression model so as to gain high accuracy in calculation.
Keywords: expectation maximization (EM) algorithm, finite mixture model, conditional mixture model, regression model, adaptive regression model (ARM).
Nghịch dân chủ luận (tổng quan về dân chủ và thể chế chính trị liên quan đến ...Loc Nguyen
Vũ trụ có vật chất và phản vật chất, xã hội có xung đột và hữu hão để phát triển và suy tàn rồi suy tàn và phát triển. Tôi dựa vào đó để biện minh cho một bài viết có tính chất phản động nghịch chuyển thời cuộc nhưng bạn đọc sẽ tự tìm ra ý nghĩa bất ly của các hình thái xã hội. Ngoài ra bài viết này không đi sâu vào nghiên cứu pháp luật, chỉ đưa ra một cách nhìn tổng quan về dân chủ và thể chế chính trị liên quan đến triết học và tôn giáo, mà theo đó đóng góp của bài viết là khái niệm “nương tạm” của tư pháp không thật sự từ bầu cử và cũng không thật sự từ bổ nhiệm.
A Novel Collaborative Filtering Algorithm by Bit Mining Frequent ItemsetsLoc Nguyen
Collaborative filtering (CF) is a popular technique in recommendation study. Concretely, items which are recommended to user are determined by surveying her/his communities. There are two main CF approaches, which are memory-based and model-based. I propose a new CF model-based algorithm by mining frequent itemsets from rating database. Hence items which belong to frequent itemsets are recommended to user. My CF algorithm gives immediate response because the mining task is performed at offline process-mode. I also propose another so-called Roller algorithm for improving the process of mining frequent itemsets. Roller algorithm is implemented by heuristic assumption “The larger the support of an item is, the higher it’s likely that this item will occur in some frequent itemset”. It models upon doing white-wash task, which rolls a roller on a wall in such a way that is capable of picking frequent itemsets. Moreover I provide enhanced techniques such as bit representation, bit matching and bit mining in order to speed up recommendation process. These techniques take advantages of bitwise operations (AND, NOT) so as to reduce storage space and make algorithms run faster.
Simple image deconvolution based on reverse image convolution and backpropaga...Loc Nguyen
Deconvolution task is not important in convolutional neural network (CNN) because it is not imperative to recover convoluted image when convolutional layer is important to extract features. However, the deconvolution task is useful in some cases of inspecting and reflecting a convolutional filter as well as trying to improve a generated image when information loss is not serious with regard to trade-off of information loss and specific features such as edge detection and sharpening. This research proposes a duplicated and reverse process of recovering a filtered image. Firstly, source layer and target layer are reversed in accordance with traditional image convolution so as to train the convolutional filter. Secondly, the trained filter is reversed again to derive a deconvolutional operator for recovering the filtered image. The reverse process is associated with backpropagation algorithm which is most popular in learning neural network. Experimental results show that the proposed technique in this research is better to learn the filters that focus on discovering pixel differences. Therefore, the main contribution of this research is to inspect convolutional filters from data.
Technological Accessibility: Learning Platform Among Senior High School StudentsLoc Nguyen
The document discusses technological accessibility among senior high school students. It covers several topics:
- Returning to nature through technology and ensuring technology accessibility is a life skill strengthened by advantages while avoiding problems for youth.
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- Researching career is optional but following passion and balancing positive and negative emotions can contribute to success.
The document discusses how engineering and technology can be used for social impact. It makes three key points:
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2. While technology increases wealth, it also widens inequality gaps. However, technology can indirectly address inequality through education by providing universal access to learning and bringing universities to people everywhere.
3. Diversity among countries should be embraced, and late innovations building on existing technologies can be particularly impactful, as was the strategy of a racing champion who overtook opponents in the final round through close observation and
Harnessing Technology for Research EducationLoc Nguyen
The document discusses harnessing technology for research and education. It notes some paradoxes of technology, such as how it can both increase inequality but also help solve it through education. It discusses the future of education being supported by distance learning tools, AI, and virtual/augmented reality. Open universities are seen as an intermediate step towards "home universities" and as a place where students can become lifelong learners and teachers. Understanding, emotion, and compassion are discussed as being more important than just remembering facts. The document ends by discussing connecting different subjects and technologies like a jigsaw puzzle.
Future of education with support of technologyLoc Nguyen
The document discusses the future of education with the support of technology. It notes that while technology deepens inequality by benefiting the rich, it can indirectly address inequality through education by allowing anyone to access knowledge and study from anywhere. Emerging forms of educational support include distance learning using tools like Zoom, artificial intelligence assistants, and virtual/augmented reality. Blended teaching combining different methods is emphasized. Understanding is seen as more important than memorization, and education should foster understanding, emotional development, and compassion through nature-based activities. The document argues technology can help education promote love by connecting various topics through both fusion and by assembling them like a jigsaw puzzle.
The document discusses generative artificial intelligence (AI) and its applications, using the metaphor of a dragon's flight. It introduces digital generative AI models that can generate images, sounds, and motions. As an example, it describes a model that could generate possible orbits or movements for a dragon and tiger in an image along with a bamboo background. The document then discusses how generative AI could be applied creatively in education to generate new learning materials and methods. It argues that while technology and societies are changing rapidly, climate change is a unifying challenge that nations must work together to overcome.
Adversarial Variational Autoencoders to extend and improve generative modelLoc Nguyen
Generative artificial intelligence (GenAI) has been developing with many incredible achievements like ChatGPT and Bard. Deep generative model (DGM) is a branch of GenAI, which is preeminent in generating raster data such as image and sound due to strong points of deep neural network (DNN) in inference and recognition. The built-in inference mechanism of DNN, which simulates and aims to synaptic plasticity of human neuron network, fosters generation ability of DGM which produces surprised results with support of statistical flexibility. Two popular approaches in DGM are Variational Autoencoders (VAE) and Generative Adversarial Network (GAN). Both VAE and GAN have their own strong points although they share and imply underline theory of statistics as well as incredible complex via hidden layers of DNN when DNN becomes effective encoding/decoding functions without concrete specifications. In this research, I try to unify VAE and GAN into a consistent and consolidated model called Adversarial Variational Autoencoders (AVA) in which VAE and GAN complement each other, for instance, VAE is good at generator by encoding data via excellent ideology of Kullback-Leibler divergence and GAN is a significantly important method to assess reliability of data which is realistic or fake. In other words, AVA aims to improve accuracy of generative models, besides AVA extends function of simple generative models. In methodology this research focuses on combination of applied mathematical concepts and skillful techniques of computer programming in order to implement and solve complicated problems as simply as possible.
Learning dyadic data and predicting unaccomplished co-occurrent values by mix...Loc Nguyen
Dyadic data which is also called co-occurrence data (COD) contains co-occurrences of objects. Searching for statistical models to represent dyadic data is necessary. Fortunately, finite mixture model is a solid statistical model to learn and make inference on dyadic data because mixture model is built smoothly and reliably by expectation maximization (EM) algorithm which is suitable to inherent spareness of dyadic data. This research summarizes mixture models for dyadic data. When each co-occurrence in dyadic data is associated with a value, there are many unaccomplished values because a lot of co-occurrences are inexistent. In this research, these unaccomplished values are estimated as mean (expectation) of random variable given partial probabilistic distributions inside dyadic mixture model.
Machine learning forks into three main branches such as supervised learning, unsupervised learning, and reinforcement learning where reinforcement learning is much potential to artificial intelligence (AI) applications because it solves real problems by progressive process in which possible solutions are improved and finetuned continuously. The progressive approach, which reflects ability of adaptation, is appropriate to the real world where most events occur and change continuously and unexpectedly. Moreover, data is getting too huge for supervised learning and unsupervised learning to draw valuable knowledge from such huge data at one time. Bayesian optimization (BO) models an optimization problem as a probabilistic form called surrogate model and then directly maximizes an acquisition function created from such surrogate model in order to maximize implicitly and indirectly the target function for finding out solution of the optimization problem. A popular surrogate model is Gaussian process regression model. The process of maximizing acquisition function is based on updating posterior probability of surrogate model repeatedly, which is improved after every iteration. Taking advantages of acquisition function or utility function is also common in decision theory but the semantic meaning behind BO is that BO solves problems by progressive and adaptive approach via updating surrogate model from a small piece of data at each time, according to ideology of reinforcement learning. Undoubtedly, BO is a reinforcement learning algorithm with many potential applications and thus it is surveyed in this research with attention to its mathematical ideas. Moreover, the solution of optimization problem is important to not only applied mathematics but also AI.
Support vector machine is a powerful machine learning method in data classification. Using it for applied researches is easy but comprehending it for further development requires a lot of efforts. This report is a tutorial on support vector machine with full of mathematical proofs and example, which help researchers to understand it by the fastest way from theory to practice. The report focuses on theory of optimization which is the base of support vector machine.
The document proposes a "jagged stock investment" strategy where stocks are repeatedly purchased at intervals where the price is increasing, similar to rises and falls in a saw blade. It shows mathematically that this strategy can achieve equal or higher returns than bank depositing if the average growth rate and interest rate are the same. The strategy models purchasing a share and its replications over time periods where each replication is bought using the profits from the previous rise. It provides formulas to calculate the overall value and return on investment from this jagged stock investment approach.
This document presents a study on minima distribution and its application to explaining the convergence and convergence speed of optimization algorithms. The author proposes weaker conditions for the convergence and monotonicity properties of minima distribution compared to previous work. Specifically, the convergence condition requires the function τ(x) to be positive and non-minimizing everywhere except at minimizers of the target function f(x). The monotonicity condition requires f(x) and the derivative of τ(x) to be inversely proportional. The author then defines convergence speed as the derivative of the integral of f(x) with respect to the optimization iterations and shows it depends on the slope of the derivative of τ(x).
This is chapter 4 “Variants of EM algorithm” in my book “Tutorial on EM algorithm”, which focuses on EM variants. The main purpose of expectation maximization (EM) algorithm, also GEM algorithm, is to maximize the log-likelihood L(Θ) = log(g(Y|Θ)) with observed data Y by maximizing the conditional expectation Q(Θ’|Θ). Such Q(Θ’|Θ) is defined fixedly in E-step. Therefore, most variants of EM algorithm focus on how to maximize Q(Θ’|Θ) in M-step more effectively so that EM is faster or more accurate.
This is the chapter 3 "Properties and convergence of EM algorithm" in my book “Tutorial on EM algorithm”, which focuses on mathematical explanation of the convergence of GEM algorithm given by DLR (Dempster, Laird, & Rubin, 1977, pp. 6-9).
Local optimization with convex function is solved perfectly by traditional mathematical methods such as Newton-Raphson and gradient descent but it is not easy to solve the global optimization with arbitrary function although there are some purely mathematical approaches such as approximation, cutting plane, branch and bound, and interval method which can be impractical because of their complexity and high computation cost. Recently, some evolutional algorithms which are inspired from biological activities are proposed to solve the global optimization by acceptable heuristic level. Among them is particle swarm optimization (PSO) algorithm which is proved as an effective and feasible solution for global optimization in real applications. Although the ideology of PSO is not complicated, it derives many variants, which can make new researchers confused. Therefore, this tutorial focuses on describing, systemizing, and classifying PSO by succinct and straightforward way. Moreover, a combination of PSO and another evolutional algorithm as artificial bee colony (ABC) algorithm for improving PSO itself or solving other advanced problems are mentioned too.
Maximum likelihood estimation (MLE) is a popular method for parameter estimation in both applied probability and statistics but MLE cannot solve the problem of incomplete data or hidden data because it is impossible to maximize likelihood function from hidden data. Expectation maximum (EM) algorithm is a powerful mathematical tool for solving this problem if there is a relationship between hidden data and observed data. Such hinting relationship is specified by a mapping from hidden data to observed data or by a joint probability between hidden data and observed data (showing MLE, EM, and practical EM, hidden info implies the hinting relationship).
The essential ideology of EM is to maximize the expectation of likelihood function over observed data based on the hinting relationship instead of maximizing directly the likelihood function of hidden data (showing the full EM with proof along with two steps).
An important application of EM is (finite) mixture model which in turn is developed towards two trends such as infinite mixture model and semiparametric mixture model. Especially, in semiparametric mixture model, component probabilistic density functions are not parameterized. Semiparametric mixture model is interesting and potential for other applications where probabilistic components are not easy to be specified (showing mixture models).
I raise a question that whether it is possible to backward discover semiparametric EM from semiparametric mixture model. I hope that this question will open a new trend or new extension for EM algorithm (showing the question).
ANAMOLOUS SECONDARY GROWTH IN DICOT ROOTS.pptxRASHMI M G
Abnormal or anomalous secondary growth in plants. It defines secondary growth as an increase in plant girth due to vascular cambium or cork cambium. Anomalous secondary growth does not follow the normal pattern of a single vascular cambium producing xylem internally and phloem externally.
The ability to recreate computational results with minimal effort and actionable metrics provides a solid foundation for scientific research and software development. When people can replicate an analysis at the touch of a button using open-source software, open data, and methods to assess and compare proposals, it significantly eases verification of results, engagement with a diverse range of contributors, and progress. However, we have yet to fully achieve this; there are still many sociotechnical frictions.
Inspired by David Donoho's vision, this talk aims to revisit the three crucial pillars of frictionless reproducibility (data sharing, code sharing, and competitive challenges) with the perspective of deep software variability.
Our observation is that multiple layers — hardware, operating systems, third-party libraries, software versions, input data, compile-time options, and parameters — are subject to variability that exacerbates frictions but is also essential for achieving robust, generalizable results and fostering innovation. I will first review the literature, providing evidence of how the complex variability interactions across these layers affect qualitative and quantitative software properties, thereby complicating the reproduction and replication of scientific studies in various fields.
I will then present some software engineering and AI techniques that can support the strategic exploration of variability spaces. These include the use of abstractions and models (e.g., feature models), sampling strategies (e.g., uniform, random), cost-effective measurements (e.g., incremental build of software configurations), and dimensionality reduction methods (e.g., transfer learning, feature selection, software debloating).
I will finally argue that deep variability is both the problem and solution of frictionless reproducibility, calling the software science community to develop new methods and tools to manage variability and foster reproducibility in software systems.
Exposé invité Journées Nationales du GDR GPL 2024
ESR spectroscopy in liquid food and beverages.pptxPRIYANKA PATEL
With increasing population, people need to rely on packaged food stuffs. Packaging of food materials requires the preservation of food. There are various methods for the treatment of food to preserve them and irradiation treatment of food is one of them. It is the most common and the most harmless method for the food preservation as it does not alter the necessary micronutrients of food materials. Although irradiated food doesn’t cause any harm to the human health but still the quality assessment of food is required to provide consumers with necessary information about the food. ESR spectroscopy is the most sophisticated way to investigate the quality of the food and the free radicals induced during the processing of the food. ESR spin trapping technique is useful for the detection of highly unstable radicals in the food. The antioxidant capability of liquid food and beverages in mainly performed by spin trapping technique.
EWOCS-I: The catalog of X-ray sources in Westerlund 1 from the Extended Weste...Sérgio Sacani
Context. With a mass exceeding several 104 M⊙ and a rich and dense population of massive stars, supermassive young star clusters
represent the most massive star-forming environment that is dominated by the feedback from massive stars and gravitational interactions
among stars.
Aims. In this paper we present the Extended Westerlund 1 and 2 Open Clusters Survey (EWOCS) project, which aims to investigate
the influence of the starburst environment on the formation of stars and planets, and on the evolution of both low and high mass stars.
The primary targets of this project are Westerlund 1 and 2, the closest supermassive star clusters to the Sun.
Methods. The project is based primarily on recent observations conducted with the Chandra and JWST observatories. Specifically,
the Chandra survey of Westerlund 1 consists of 36 new ACIS-I observations, nearly co-pointed, for a total exposure time of 1 Msec.
Additionally, we included 8 archival Chandra/ACIS-S observations. This paper presents the resulting catalog of X-ray sources within
and around Westerlund 1. Sources were detected by combining various existing methods, and photon extraction and source validation
were carried out using the ACIS-Extract software.
Results. The EWOCS X-ray catalog comprises 5963 validated sources out of the 9420 initially provided to ACIS-Extract, reaching a
photon flux threshold of approximately 2 × 10−8 photons cm−2
s
−1
. The X-ray sources exhibit a highly concentrated spatial distribution,
with 1075 sources located within the central 1 arcmin. We have successfully detected X-ray emissions from 126 out of the 166 known
massive stars of the cluster, and we have collected over 71 000 photons from the magnetar CXO J164710.20-455217.
Comparing Evolved Extractive Text Summary Scores of Bidirectional Encoder Rep...University of Maribor
Slides from:
11th International Conference on Electrical, Electronics and Computer Engineering (IcETRAN), Niš, 3-6 June 2024
Track: Artificial Intelligence
https://www.etran.rs/2024/en/home-english/
Nucleophilic Addition of carbonyl compounds.pptxSSR02
Nucleophilic addition is the most important reaction of carbonyls. Not just aldehydes and ketones, but also carboxylic acid derivatives in general.
Carbonyls undergo addition reactions with a large range of nucleophiles.
Comparing the relative basicity of the nucleophile and the product is extremely helpful in determining how reversible the addition reaction is. Reactions with Grignards and hydrides are irreversible. Reactions with weak bases like halides and carboxylates generally don’t happen.
Electronic effects (inductive effects, electron donation) have a large impact on reactivity.
Large groups adjacent to the carbonyl will slow the rate of reaction.
Neutral nucleophiles can also add to carbonyls, although their additions are generally slower and more reversible. Acid catalysis is sometimes employed to increase the rate of addition.
When I was asked to give a companion lecture in support of ‘The Philosophy of Science’ (https://shorturl.at/4pUXz) I decided not to walk through the detail of the many methodologies in order of use. Instead, I chose to employ a long standing, and ongoing, scientific development as an exemplar. And so, I chose the ever evolving story of Thermodynamics as a scientific investigation at its best.
Conducted over a period of >200 years, Thermodynamics R&D, and application, benefitted from the highest levels of professionalism, collaboration, and technical thoroughness. New layers of application, methodology, and practice were made possible by the progressive advance of technology. In turn, this has seen measurement and modelling accuracy continually improved at a micro and macro level.
Perhaps most importantly, Thermodynamics rapidly became a primary tool in the advance of applied science/engineering/technology, spanning micro-tech, to aerospace and cosmology. I can think of no better a story to illustrate the breadth of scientific methodologies and applications at their best.
Phenomics assisted breeding in crop improvementIshaGoswami9
As the population is increasing and will reach about 9 billion upto 2050. Also due to climate change, it is difficult to meet the food requirement of such a large population. Facing the challenges presented by resource shortages, climate
change, and increasing global population, crop yield and quality need to be improved in a sustainable way over the coming decades. Genetic improvement by breeding is the best way to increase crop productivity. With the rapid progression of functional
genomics, an increasing number of crop genomes have been sequenced and dozens of genes influencing key agronomic traits have been identified. However, current genome sequence information has not been adequately exploited for understanding
the complex characteristics of multiple gene, owing to a lack of crop phenotypic data. Efficient, automatic, and accurate technologies and platforms that can capture phenotypic data that can
be linked to genomics information for crop improvement at all growth stages have become as important as genotyping. Thus,
high-throughput phenotyping has become the major bottleneck restricting crop breeding. Plant phenomics has been defined as the high-throughput, accurate acquisition and analysis of multi-dimensional phenotypes
during crop growing stages at the organism level, including the cell, tissue, organ, individual plant, plot, and field levels. With the rapid development of novel sensors, imaging technology,
and analysis methods, numerous infrastructure platforms have been developed for phenotyping.
This presentation explores a brief idea about the structural and functional attributes of nucleotides, the structure and function of genetic materials along with the impact of UV rays and pH upon them.
Nucleic Acid-its structural and functional complexity.
A Proposal of Two-step Autoregressive Model
1. A Proposal of Two-step Autoregressive Model
Prof. Dr. Loc Nguyen PhD, PostDoc
Loc Nguyen’s Academic Network, Vietnam
Email: ng_phloc@yahoo.com
Homepage: www.locnguyen.net
Two-step Autoregressive Model - Loc Nguyen
14/12/2022 1
2. Abstract
Autoregressive (AR) model and conditional autoregressive (CAR)
model are specific regressive models in which independent
variables and dependent variable imply the same object. They are
powerful statistical tools to predict values based on correlation of
time domain and space domain, which are useful in epidemiology
analysis. In this research, I combine them by the simple way in
which AR and CAR is estimated in two separate steps so as to
cover time domain and space domain in spatial-temporal data
analysis. Moreover, I integrate logistic model into CAR model,
which aims to improve competence of autoregressive models.
2
Two-step Autoregressive Model - Loc Nguyen
14/12/2022
3. Table of contents
1. Two-step autoregressive model
2. Integrated with logistic function
3. Conclusions
3
Two-step Autoregressive Model - Loc Nguyen
14/12/2022
4. 1. Two-step autoregressive model
Autoregressive (AR) model evaluates new value of a variable based on its old values; in other
words, independent variables called regressors and dependent variable called responsor refer to the
same concerned object (CO). AR model is often used to predict new values in time series, which is
called temporal autoregressive model. Conditional autoregressive (CAR) model also follows
ideology of AR model where regressors and responsor imply the same CO but CAR model added
more values of neighbors of CO. Therefore, CAR model is often used to predict new values in
space domain, which is called spatial autoregressive model. There are some researches like
(Mariella & Tarantino, 2010) to integrate AR model and CAR model but here I combine them
consequently by the simple way in which AR and CAR is estimated in two separate processes. In
other words, the research uses mainly autoregressive model through two steps:
1. Step 1: CAR model known as spatial autoregressive model is estimated to predict responsor
based on its neighbors and its environmental features like humidity, precipitation, etc. As a
convention, responsor is called case like disease case in epidemiology.
2. Step 2: AR model known as temporal autoregressive model is estimated to forecast next case
based on current cases.
14/12/2022 Two-step Autoregressive Model - Loc Nguyen 4
5. 1. Two-step autoregressive model
Let S = {1, 2,…, n} be spatial domain and so each site s can correspond to an area. Let nb(s) be set of neighbor
sites of site s. Let Xs = (xs1, xs2,…, xsk) be the vector of environmental features at site s and so xsj is the jth
environmental feature at site s. Let Ys be the outcome number of cases at site s. If time is concerned, Xs and Ys
are replaced by Xst and Yst, respectively. Suppose the time interval is T = 2, this research firstly uses spatial
autoregressive model to estimate the number of cases Yst = Ys at current timepoint t. Later on, the research uses
temporal autoregressive model to forecast the future number of cases Ys(t+1) at timepoint t+1 based on current
Yst. Therefore, the computational model of the research is called the two-step autoregressive model.
In the first step, spatial autoregressive model is estimated. Given current timepoint, we have Ys = Yst and Xs
= Xst. When building up spatial autoregressive model, we ignore timepoints in dataset. Because the number of
cases is moved around an average number, Poisson distribution is more suitable to build spatial autoregressive
model. However, for simplicity, I use normal distribution for spatial autoregressive model. Let W be the
symmetric weighted adjacency matrix (Mariella & Tarantino, 2010, p. 227):
𝑊 = 𝑤𝑠𝑠∗ where 𝑤𝑠𝑠∗ =
0 if 𝑠∗
= 𝑠
𝜑 𝑠, 𝑠∗
if 𝑠∗
∈ nb 𝑠
0 otherwise
(1.1)
Where φ(s, s*) measures the approximation of site s and site s* with note that nb(s) denotes the set of neighbors
of site s. Moreover, the set nb(s) does not contain s.
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6. 1. Two-step autoregressive model
Because W is symmetric, we have (Mariella & Tarantino, 2010, p. 227):
∀𝑠, 𝑠∗
∈ 𝑆, 𝑤𝑠𝑠∗ = 𝑤𝑠∗𝑠
Note, W is constant. Let WD be the diagonal matrix of normalization (Mariella & Tarantino, 2010, p.
227):
𝑊𝐷 = diag 𝑤1+, 𝑤2+, … , 𝑤𝑛+
𝑤𝑠+ =
𝑠∗∈nb 𝑠
𝑤𝑠𝑠∗
(1.2)
Let A be the matrix of interaction (Mariella & Tarantino, 2010, p. 227):
𝐴 = 𝑎𝑠𝑠∗ = 𝑊𝐷
−1
𝑊
𝑎𝑠𝑠∗ =
𝑤𝑠𝑠∗
𝑤𝑠+
(1.3)
Note, A is constant. Let ΣD be the diagonal matrix of variances (Mariella & Tarantino, 2010, p. 227):
Σ𝐷 = diag 𝜎1
2
, 𝜎2
2
, … , 𝜎𝑛
2
= 𝜎2
𝑊𝐷
−1
𝜎𝑠
2
=
𝜎2
𝑤𝑠+
(1.4)
Note, σ2 is called global variance.
14/12/2022 Two-step Autoregressive Model - Loc Nguyen 6
7. 1. Two-step autoregressive model
The probability density function (PDF) of the number of cases Ys at Xs given its neighbors Xs* is defined according to normal
distribution as follows (Mariella & Tarantino, 2010, p. 227):
𝑓 𝑌𝑠 𝑋𝑠, 𝑋𝑠∗, 𝛼𝑠, 𝜎2
=
1
2𝜋𝜎𝑠
2
exp −
𝑌𝑠 − 𝛼𝑠
𝑇𝑋𝑠 + 𝜌 𝑠∗∈nb 𝑠 𝑎𝑠𝑠∗ 𝑌𝑠∗ − 𝛼𝑠∗
𝑇
𝑋𝑠∗
2
2𝜎𝑠
2 (1.5)
Where αs = (αs1, αs2,…, αsk) is site regressive coefficient vector and σs
2 is site variance. The constant ρ is called effective
neighborhood constant, |ρ| < 1. Each site variance (local variance) σs
2 was defined.
𝜎𝑠
2
=
𝜎2
𝑤𝑠+
Therefore, only parameters αs and σ2 are parameters in the PDF of Ys at Xs given its neighbors. The mean and variance of Ys
at Xs given its neighbors are:
mean 𝑌𝑠 𝑋𝑠, 𝑋𝑠∗, 𝛼𝑠, 𝜎2
= 𝛼𝑠
𝑇
𝑋𝑠 + 𝜌
𝑠∗∈nb 𝑠
𝑎𝑠𝑠∗ 𝑌𝑠∗ − 𝛼𝑠∗
𝑇
𝑋𝑠∗
var 𝑌𝑠 𝑋𝑠, 𝑋𝑠∗, 𝛼𝑠, 𝜎2 = 𝜎𝑠
2
(1.6)
From the mean of Ys at Xs given its neighbors, the spatial autoregressive model of the number of cases is defined as follows:
𝑌𝑠 = 𝛼𝑠
𝑇𝑋𝑠 + 𝜌
𝑠∗∈nb 𝑠
𝑎𝑠𝑠∗ 𝑌𝑠∗ − 𝛼𝑠∗
𝑇
𝑋𝑠∗ (1.7)
14/12/2022 Two-step Autoregressive Model - Loc Nguyen 7
8. 1. Two-step autoregressive model
The essence of the first step in this research is to estimate all regressive coefficients αs when the
effective neighborhood constant ρ is pre-defined. Let xs
(i) be the ith instance of the vector of
environmental feature Xs at site s. Let ys
(i) be the ith instance of Ys at site s. Let Xs be the set of Ns
instances xs
(i) and let Ys be the set of Ns instances y(i). Each site s owns one dataset (Xs, Ys).
𝑿𝑠 =
𝒙𝑠
1
𝑇
𝒙𝑠
2
𝑇
⋮
𝒙𝑠
𝑁𝑠
𝑇
=
1 𝑥𝑠1
1
𝑥𝑠2
1
⋯ 𝑥𝑠𝑛
1
1 𝑥𝑠1
2
𝑥𝑠2
2
⋯ 𝑥𝑠𝑛
2
⋮ ⋮ ⋮ ⋱ ⋮
1 𝑥𝑠1
𝑁𝑠
𝑥𝑠2
𝑁𝑠
⋯ 𝑥𝑠𝑛
𝑁𝑠
𝒙𝑠
𝑖
=
1
𝑥𝑠1
𝑖
𝑥𝑠2
𝑖
⋮
𝑥𝑠𝑛
𝑖
, 𝒙𝑠
𝑗
=
1
𝑥𝑠1
𝑗
𝑥𝑠2
𝑗
⋮
𝑥𝑠𝑛
𝑗
, 𝒚𝑠 =
𝑦𝑠
1
𝑦𝑠
2
⋮
𝑦𝑠
𝑁𝑠
(1.8)
14/12/2022 Two-step Autoregressive Model - Loc Nguyen 8
9. 1. Two-step autoregressive model
Let X and Y be the sets of all site dataset Xs and Ys, respectively. Let α and σ2 be the sets of all αs and all σs
2,
respectively. The joint probability of the global dataset (X, Y) is:
𝑓 𝑿, 𝒀 𝛼, 𝜎2 =
𝑠=1
𝑆
𝑓 𝑌𝑠 𝑋𝑠, 𝑋𝑠∗, 𝛼𝑠, 𝜎𝑠
2
=
𝑠=1
𝑆
2𝜋𝜎𝑠
2 −
𝑁𝑠
2 ∗
𝑠=1
𝑆
exp −
1
2𝜎𝑠
2
𝑖=1
𝑁𝑠
𝑦𝑠
𝑖
− 𝛼𝑠
𝑇𝒙𝑠
𝑖
+ 𝜌
𝑠∗∈nb 𝑠
𝑎𝑠𝑠∗ 𝑦𝑠∗
𝑖
− 𝛼𝑠∗
𝑇
𝒙𝑠∗
𝑖
2
The log-likelihood function is logarithm function of such joint probability:
𝑙 𝛼, 𝜎2
= −
1
2
𝑠=1
𝑆
𝑁𝑠log 2𝜋𝜎𝑠
2
−
1
2
𝑠=1
𝑆
1
𝜎𝑠
2
𝑖=1
𝑁𝑠
𝑦𝑠
𝑖
− 𝛼𝑠
𝑇
𝒙𝑠
𝑖
+ 𝜌
𝑠∗∈nb 𝑠
𝑎𝑠𝑠∗ 𝑦𝑠∗
𝑖
− 𝛼𝑠∗
𝑇
𝒙𝑠∗
𝑖
2
Let αs
* = (αs1
*, αs2
*,…, αsk
*) be the optimal estimate of regressive coefficients and thus, αs
* is a maximizer of
l(α, σ2).
𝛼𝑠
∗ = argmax
𝛼1,𝛼2,…,𝛼𝑠,…,𝛼 𝑆
𝑙 𝛼, 𝜎2
14/12/2022 Two-step Autoregressive Model - Loc Nguyen 9
10. 1. Two-step autoregressive model
By taking first-order partial derivatives of l(α, σ2) with regard to αs, we obtain:
𝜕𝑙 𝛼, 𝜎2
𝜕𝛼𝑠
=
1
𝜎𝑠
2
𝑖=1
𝑁𝑠
𝑦𝑠
𝑖
− 𝛼𝑠
𝑇𝒙𝑠
𝑖
+ 𝜌
𝑠∗∈nb 𝑠
𝑎𝑠𝑠∗ 𝑦𝑠∗
𝑖
− 𝛼𝑠∗
𝑇
𝒙𝑠∗
𝑖
𝒙𝑠
𝑖
𝑇
When the first-order partial derivatives of l(α, σ2) with regard to αs are equal to zero, it gets locally maximal. In other
words, αs
* is solution of the following equation resulted from setting such derivatives to be zero.
𝑖=1
𝑁𝑠
𝑦𝑠
𝑖
− 𝛼𝑠
𝑇𝒙𝑠
𝑖
+ 𝜌
𝑠∗∈nb 𝑠
𝑎𝑠𝑠∗ 𝑦𝑠∗
𝑖
− 𝛼𝑠∗
𝑇
𝒙𝑠∗
𝑖
𝒙𝑠
𝑖
𝑇
= 𝟎𝑇
As a convention, ass* = 0 if ys*
(i) or xs*
(i) does not exist. The equation above is re-written as follows:
𝒚𝑠
𝑇
𝑿𝑠 − 𝛼𝑠
𝑇
𝑿𝑠
𝑇
𝑿𝑠 − 𝜌
𝑠∗∈nb 𝑠
𝑎𝑠𝑠∗ 𝒚𝑠∗
𝑇
𝑿𝑠 − 𝛼𝑠∗
𝑇
𝑿𝑠
𝑇
𝑿𝑠
= 𝒚𝑠
𝑇 − 𝜌
𝑠∗∈nb 𝑠
𝑎𝑠𝑠∗𝒚𝑠∗
𝑇
𝑿𝑠 − 𝛼𝑠
𝑇 − 𝜌
𝑠∗∈nb 𝑠
𝑎𝑠𝑠∗𝛼𝑠∗
𝑇
𝑿𝑠
𝑇𝑿𝑠 = 𝟎𝑇
This implies
𝛼𝑠 − 𝜌
𝑠∗∈nb 𝑠
𝑎𝑠𝑠∗𝛼𝑠∗ = 𝑿𝑠
𝑇𝑿𝑠
−1𝑿𝑠
𝑇 𝒚𝑠 − 𝜌
𝑠∗∈nb 𝑠
𝑎𝑠𝑠∗𝒚𝑠∗
14/12/2022 Two-step Autoregressive Model - Loc Nguyen 10
11. 1. Two-step autoregressive model
In general, the estimates αs
* for all regressive coefficient vectors αs where s from
1 to |S| is solution of the linear equation system consisting of |S| equations as
follows:
𝛼𝑠 − 𝜌
𝑠∗∈𝑛𝑏 𝑠
𝑎𝑠𝑠∗𝛼𝑠∗ = 𝑿𝑠
𝑇
𝑿𝑠
−1
𝑿𝑠
𝑇
𝒚𝑠 − 𝜌
𝑠∗∈𝑛𝑏 𝑠
𝑎𝑠𝑠∗𝒚𝑠∗
𝑠=1, 𝑆
(1.9)
It is not difficult to solve the linear equation system above. As a result, the
spatial autoregressive model is re-written as follows:
𝑌𝑠 = 𝛼𝑠
∗ 𝑇
𝑋𝑠 + 𝜌
𝑠∗∈nb 𝑠
𝑎𝑠𝑠∗ 𝑌𝑠∗ − 𝛼𝑠∗
∗ 𝑇
𝑋𝑠∗ (1.10)
14/12/2022 Two-step Autoregressive Model - Loc Nguyen 11
12. 1. Two-step autoregressive model
In the second step, temporal autoregressive model is estimated. For simplicity, we follow the Markov
condition in which, future cases only depend on current cases. When building up spatial autoregressive
model, we ignore environmental features Xs in dataset or we focus on a given site s. Suppose the current
number of cases Yst at site s was determined previously by spatial autoregressive model. As a convention,
the “s” index is removed from the response variable and so we have Yt+1 = Ys(t+1) and Yt = Yst. We will
forecast next (unknown) Yt+1 based on current (known) Yt. The temporal autoregressive model is defined as
follows (Eshel, 2003, p. 1):
𝑌𝑡+1 = 𝛽𝑌𝑡 + 𝜀 (1.11)
Where ε is the error random variable and β is temporal coefficient with note that mean of ε is 0 such that
𝐸 𝑌𝑡+1 = 𝛽𝑌𝑡
Let yt be the value of Yt at timepoint t. Suppose there are T timepoints and we have t = 1, 2,…, T. The sum
of squared errors is defined as follows:
𝑆 𝛽 =
𝑡=1
𝑇−1
𝑦𝑡+1 − 𝛽𝑦𝑡
2
Let β* be the estimate of β. The least squares method sets β* as the minimizer of S(β).
𝛽∗
= argmin
𝛽
𝑆 𝛽
14/12/2022 Two-step Autoregressive Model - Loc Nguyen 12
13. 1. Two-step autoregressive model
By taking first-order derivative of S(β) with regard to β, we obtain (Eshel, 2003, p. 1):
𝑑𝑆 𝛽
𝑑𝛽
= −2
𝑡=1
𝑇−1
𝑦𝑡+1 − 𝛽𝑦𝑡 𝑦𝑡
The temporal coefficient β* is solution of the following equation resulted from setting such
derivative to be zero.
−2
𝑡=1
𝑇−1
𝑦𝑡+1 − 𝛽𝑦𝑡 𝑦𝑡 = 0
It is easy to determine the formula for calculating β* (Eshel, 2003, p. 1).
𝛽∗ =
𝑡=1
𝑇−1
𝑦𝑡𝑦𝑡+1
𝑡=1
𝑇−1
𝑦𝑡
2 (1.12)
Please pay attention that values yt are calculated by CAR model estimated in the first step. The
temporal autoregressive model is re-written as follows:
𝑌𝑠 𝑡+1 = 𝛽∗𝑌𝑠𝑡 (1.13)
Equation 1.12 to calculate β* is simplest form of the well-known Yule-Walker equation.
14/12/2022 Two-step Autoregressive Model - Loc Nguyen 13
14. 2. Integrated with logistic function
In normal CAR model, the responsor Ys which is the outcome number of cases at site s is real variable but in the situation
that Ys is category variable whose G ordered values are v1, v2,…, vG, new modification of CAR model is required. The
multinomial logistic function of Ys with regard to value vg at site s, which is associated spatial autoregression, is proposed
as follows:
𝑓𝑔 𝑌𝑠 𝑋𝑠, 𝑋𝑠∗, 𝛼𝑠𝑔 =
1
1 + exp − 𝛼𝑠𝑔
𝑇
𝑋𝑠 + 𝜌 𝑠∗∈nb 𝑠 𝑎𝑠𝑠∗ 𝛼𝑠𝑔
𝑇
𝑋𝑠∗ − 𝛼𝑠∗𝑔
𝑇
𝑋𝑠∗
(2.1)
Note that αsg = (αsg1, αsg2,…, αsgk) which is site regressive coefficient vector with regard to value yg is the main parameter
which is estimated here. Let pg be the probability that Ys receives value yg, where g is from 1 to G. If G values v1, v2,…, vG
are not ordered, pg is the same to the proposed multinomial logistic function of Ys.
𝑝 𝑋𝑠 𝛼𝑠𝑔 = 𝑃 𝑌𝑠 = 𝑣𝑔 𝑋𝑠, 𝑋𝑠∗, 𝛼𝑠𝑔 = 𝑓𝑔 𝑌𝑠 𝑋𝑠, 𝑋𝑠∗, 𝛼𝑠𝑔
=
1
1 + exp − 𝛼𝑠𝑔
𝑇
𝑋𝑠 + 𝜌 𝑠∗∈nb 𝑠 𝑎𝑠𝑠∗ 𝛼𝑠𝑔
𝑇
𝑋𝑠∗ − 𝛼𝑠∗𝑔
𝑇
𝑋𝑠∗
(2.2)
If G values v1, v2,…, vG are ordered such that v1 ≤ v2 ≤ … ≤ vG then, according to Bender and Grouven (Bender & Grouven,
1997, p. 547), pg becomes the cumulative probability such that Ys ≤ vg as follows:
𝑝 𝑋𝑠 𝛼𝑠𝑔 = 𝑃 𝑌𝑠 ≤ 𝑣𝑔 𝑋𝑠, 𝑋𝑠∗, 𝛼𝑠𝑔 =
𝑖=1
𝑔
𝑓𝑖 𝑌𝑠 𝑋𝑠, 𝑋𝑠∗, 𝛼𝑠𝑖 (2.3)
Obviously, p(Xs|αsg) is the CAR model integrated with logistic function, which is called LCAR model.
14/12/2022 Two-step Autoregressive Model - Loc Nguyen 14
15. 2. Integrated with logistic function
Essentially, LCAR is determined by estimating parameters αsg. As a convention, let
𝑝𝑠𝑔 = 𝑝 𝑋𝑠 𝛼𝑠𝑔
𝑔=1
𝐺
𝑝𝑠𝑔 = 1
(2.4)
The logarithm of psg which is called logit of psg is defined as follows:
logit 𝑝𝑠𝑔 = log
𝑝𝑠𝑔
1 − 𝑝𝑠𝑔
= 𝛼𝑠𝑔
𝑇
𝑋𝑠 + 𝜌
𝑠∗∈nb 𝑠
𝑎𝑠𝑠∗ 𝛼𝑠𝑔
𝑇
𝑋𝑠∗ − 𝛼𝑠∗𝑔
𝑇
𝑋𝑠∗ (2.5)
Suppose Ys obeys multinomial distribution so that the parameters αsg at site s are determined by maximum likelihood
estimation (MLE) method (Czepiel, 2002). The joint probability of the global dataset (X, Y) in reduced form is:
𝑓 𝑿, 𝒀 𝛼 =
𝑠=1
𝑆
𝑔=1
𝐺
𝑖=1,𝑦𝑠
𝑖
=𝑣𝑔
𝑁𝑠
𝑝𝑠𝑔
𝑖
Where psg
(i) is the evaluation of psg with the instance Xs = xs
(i). The log-likelihood function is logarithm function of such
joint probability:
𝑙 𝛼 = −
𝑠=1
𝑆
𝑔=1
𝐺
𝑖=1,𝑦𝑠
𝑖
=𝑣𝑔
𝑁𝑠
log 1 + exp − 𝛼𝑠𝑔
𝑇
𝒙𝑠
𝑖
+ 𝜌
𝑠∗∈nb 𝑠
𝑎𝑠𝑠∗ 𝛼𝑠𝑔
𝑇
𝒙𝑠∗
𝑖
− 𝛼𝑠∗𝑔
𝑇
𝒙𝑠∗
𝑖
14/12/2022 Two-step Autoregressive Model - Loc Nguyen 15
16. 2. Integrated with logistic function
Let αsg
* = (αsg1
*, αsg2
*,…, αsgk
*) be the optimal estimate of regressive coefficients which is a maximizer of l(α)
with the constraint 𝑔=1
𝐺
𝑝𝑠𝑔 = 1 which is equivalent to:
log
𝑔=1
𝐺
𝑝𝑠𝑔 = 0
Let L0(α, λ) be Lagrange function defined as follows:
𝐿0 𝛼, 𝜆 = 𝑙 𝛼 +
𝑠=1
𝑆
𝑔=1
𝐺
𝑖=1,𝑦𝑠
𝑖
=𝑣𝑔
𝑁𝑠
𝜆𝑖
𝑠𝑔
log 𝑝𝑠𝑔
𝑖
Where λi are Lagrange multipliers such that λi ≥ 0 with note that each λi
(sg) implies λi at site s given value vg.
Because logarithm is concave function, we apply Jessen inequation to approximate the Lagrange function as
follows:
𝐿0 𝛼, 𝜆 ≤ 𝐿 𝛼, 𝜆 = 𝑙 𝛼 +
𝑠=1
𝑆
𝑔=1
𝐺
𝑖=1,𝑦𝑠
𝑖
=𝑣𝑔
𝑁𝑠
𝜆𝑖
𝑠𝑔
log 𝑝𝑠𝑔
𝑖
We now focus on maximizing L(α, λ).
14/12/2022 Two-step Autoregressive Model - Loc Nguyen 16
17. 2. Integrated with logistic function
We now focus on maximizing L(α, λ). The first-order partial derivative of L(α, λ) with regard to αsg is:
𝜕𝐿 𝛼, 𝜆
𝜕𝛼𝑠𝑔
=
𝑖=1,𝑦𝑠
𝑖
=𝑣𝑔
𝑁𝑠
𝜆𝑖 + 1 1 − 𝑝𝑠𝑔
𝑖
𝒙𝑠
𝑖
𝑇
+ 𝜌
𝑠∗∈nb 𝑠
𝑎𝑠𝑠∗ 𝒙𝑠∗
𝑖
𝑇
Note, ass* = 0 if ys*
(i) or xs*
(i) does not exist. Let
𝑍𝑠
𝑖
= 𝒙𝑠
𝑖
+ 𝜌
𝑠∗∈nb 𝑠
𝑎𝑠𝑠∗𝒙𝑠∗
𝑖
(2.6)
We have:
𝜕𝐿 𝛼, 𝜆
𝜕𝛼𝑠𝑔
=
𝑖=1,𝑦𝑠
𝑖
=𝑣𝑔
𝑁𝑠
𝜆𝑖 + 1 1 − 𝑝𝑠𝑔
𝑖
𝑍𝑠
𝑖
𝑇
Setting the first-order partial derivative of L(α, λ) with regard to λi to be 0, we obtain:
𝜕𝐿 𝛼, 𝜆
𝜕𝜆𝑖
= 𝜆𝑖
𝑔=1
𝐺
log 𝑝𝑠𝑔
𝑖
= 0 ⇒ 𝜆𝑖 = 0
Substituting all λi = 0 into the derivative
𝜕𝐿 𝛼,𝜆
𝜕𝛼𝑠𝑔
, we have:
𝜕𝐿 𝛼, 𝜆
𝜕𝛼𝑠𝑔
=
𝑖=1,𝑦𝑠
𝑖
=𝑣𝑔
𝑁𝑠
1 − 𝑝𝑠𝑔
𝑖
𝑍𝑠
𝑖
𝑇
14/12/2022 Two-step Autoregressive Model - Loc Nguyen 17
18. 2. Integrated with logistic function
By setting such derivative
𝜕𝐿 𝛼,𝜆
𝜕𝛼𝑠𝑔
to be zero, we obtain:
𝑖=1,𝑦𝑠
𝑖
=𝑣𝑔
𝑁𝑠
𝑝𝑠𝑔
𝑖
𝑍𝑠
𝑖
=
𝑖=1,𝑦𝑠
𝑖
=𝑣𝑔
𝑁𝑠
𝑍𝑠
𝑖
As a result, the estimates αsg
* for all regressive coefficient vectors αsg over all sites and all
values v1, v2,…, vG is solution of the linear equation system consisting of |S|G equations as
follows:
𝑖=1,𝑦𝑠
𝑖
=𝑣𝑔
𝑁𝑠
𝑝𝑠𝑔
𝑖
𝑍𝑠
𝑖
=
𝑖=1,𝑦𝑠
𝑖
=𝑣𝑔
𝑁𝑠
𝑍𝑠
𝑖
𝑠=1, 𝑆 ,𝑔=1,𝐺
(2.7)
In general, the equation 2.7 is used to estimate LCAR model, which can be solved by
numerical methods like Newton-Raphson method for nonlinear equation system (Burden &
Faires, 2011, pp. 638-642).
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19. 3. Conclusions
In this research, I proposed a simple integration model for spatial-
temporal autoregression which consists of two consequent and
separated steps. Moreover, I also proposed the integration of
logistic model and autoregressive model called LCAR which is
slightly different from traditional CAR model. However, the
traditional AR model known as temporal autoregressive model is
still simple with assumption of first-order Markov property. In the
future trend, I will improve AR model with autoregressive moving
average (ARMA) and autoregressive integrated moving average
(ARIMA).
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20. References
1. Bender, R., & Grouven, U. (1997, September 5). Ordinal logistic regression in medical
research. Journal of the Royal College of Physicians of London, 31(5), 546-551.
Retrieved from https://pubmed.ncbi.nlm.nih.gov/9429194/
2. Burden, R. L., & Faires, D. J. (2011). Numerical Analysis (9th Edition ed.). (M. Julet,
Ed.) Brooks/Cole Cengage Learning.
3. Czepiel, S. A. (2002). Maximum Likelihood Estimation of Logistic Regression Models:
Theory and Implementation. Czepiel's website http://czep.net.
4. Eshel, G. (2003). The Yule Walker Equations for the AR Coefficients. University of
Wharton, Statistics Department. Michael Steele Homepage. Retrieved 11 24, 2018,
from http://www-
stat.wharton.upenn.edu/~steele/Courses/956/ResourceDetails/YWSourceFiles/YW-
Eshel.pdf
5. Mariella, L., & Tarantino, M. (2010). Spatial Temporal Conditional Auto-Regressive
Model: A New Autoregressive Matrix. Austrian Journal of Statistics, 39(3), 223-244.
doi:10.17713/ajs.v39i3.246
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21. Thank you for listening
21
Two-step Autoregressive Model - Loc Nguyen
14/12/2022