1. The document presents computational methods for parameter inference from partially observed network models. In particular, it considers a duplication attachment model whose likelihood cannot be evaluated efficiently.
2. It compares importance sampling and sequential Monte Carlo methods for approximating the likelihood of the network model for a fixed parameter value. It proves that the relative variance of SMC grows polynomially rather than exponentially as with importance sampling.
3. Particle Markov chain Monte Carlo algorithms are developed to perform Bayesian parameter estimation by using the SMC algorithms within the transition dynamics, allowing inference when the exact likelihood is unknown. The approaches are numerically illustrated on small to medium sized networks.
The document describes an experiment to verify Malus' Law, which states that the intensity of light transmitted through a polarizer and analyzer varies as the cosine squared of the angle between their transmission directions. The experimental setup uses a diode laser, polarizer, rotating analyzer, and detector. Intensity readings are recorded for analyzer angles from 0-360 degrees and graphed against both the angle and cosine squared of the angle. The results show agreement with Malus' Law, verifying that intensity is directly proportional to the cosine squared of the angle.
The document discusses phasor estimation using the least squares method. It begins by introducing the problem of estimating voltage and current phasors from sampled signals for use in numerical relays. It then derives the two-sample estimation technique, noting that a minimum of two samples per cycle are needed. Noise is modeled as additive Gaussian noise, and it is shown that noise increases the standard deviation of estimates. More samples are needed to filter out noise and improve accuracy.
This document analyzes a cantilever beam with both uniformly distributed and concentrated loads. It calculates the reaction forces and bending moment using the principles of equilibrium. The bending moment is found to be 181.5 kNm. The total reaction force is 146 kN. The shear force and bending moment diagrams are drawn, with the maximum bending moment occurring at the fixed end of -42 kNm.
(1) For the balls in the semicircle to deflect by an angle of π/N in each collision, the ratio of the mass of each ball (μ) to the total mass (m) cannot be too small. The maximum deflection angle is given by sinθ = μ/m, so π/N ≤ μ/m, or π ≤ M/mN.
(2) After the first bounce, the speed of the ball (m) is approximately its initial speed (V) multiplied by 1 - μ/m. Each subsequent bounce reduces the speed by this same factor of 1 - μ/m.
(3) For the minimum mass ratio found in part
William Shakespeare was an English playwright, poet and actor born in 1564 in Stratford-upon-Avon, England. He wrote 38 plays and 154 sonnets which are considered some of the greatest works in English literature. Shakespeare wrote in Early Modern English for the Globe Theatre in London where only men and boys, whose voices had not changed, performed both male and female roles for audiences of all social classes. Some of his most famous plays include Romeo and Juliet, Hamlet, Macbeth, and A Midsummer Night's Dream.
The document discusses seven hypotheses about good language learners proposed by Naiman et al. These include being willing to guess meanings, having strong motivation to communicate, not being afraid to make mistakes, practicing language skills, monitoring their own and others' speech, and focusing on meaning.
It then discusses Oxford's classification of language learning strategies as direct or indirect. Direct strategies involve memorizing vocabulary and understanding grammar, while indirect strategies involve self-regulation like planning study time and preparation.
Oxford identifies six main categories of language learning strategies: cognitive, memory, compensation, metacognitive, affective, and social strategies. Examples are given of strategies within each category.
La epistemología es la disciplina filosófica que estudia el conocimiento científico y cómo los individuos amplían los horizontes de la ciencia. Trata los problemas planteados por la ciencia. La contabilidad cuantifica los recursos económicos y financieros de una empresa para conocer el valor relativo entre los bienes y garantizar el funcionamiento de la empresa y el estado. Los investigadores de la contabilidad establecen su campo de acción científica para incluir el balance económico general, la información financiera y el manejo de la economía de la empresa.
The document describes an experiment to verify Malus' Law, which states that the intensity of light transmitted through a polarizer and analyzer varies as the cosine squared of the angle between their transmission directions. The experimental setup uses a diode laser, polarizer, rotating analyzer, and detector. Intensity readings are recorded for analyzer angles from 0-360 degrees and graphed against both the angle and cosine squared of the angle. The results show agreement with Malus' Law, verifying that intensity is directly proportional to the cosine squared of the angle.
The document discusses phasor estimation using the least squares method. It begins by introducing the problem of estimating voltage and current phasors from sampled signals for use in numerical relays. It then derives the two-sample estimation technique, noting that a minimum of two samples per cycle are needed. Noise is modeled as additive Gaussian noise, and it is shown that noise increases the standard deviation of estimates. More samples are needed to filter out noise and improve accuracy.
This document analyzes a cantilever beam with both uniformly distributed and concentrated loads. It calculates the reaction forces and bending moment using the principles of equilibrium. The bending moment is found to be 181.5 kNm. The total reaction force is 146 kN. The shear force and bending moment diagrams are drawn, with the maximum bending moment occurring at the fixed end of -42 kNm.
(1) For the balls in the semicircle to deflect by an angle of π/N in each collision, the ratio of the mass of each ball (μ) to the total mass (m) cannot be too small. The maximum deflection angle is given by sinθ = μ/m, so π/N ≤ μ/m, or π ≤ M/mN.
(2) After the first bounce, the speed of the ball (m) is approximately its initial speed (V) multiplied by 1 - μ/m. Each subsequent bounce reduces the speed by this same factor of 1 - μ/m.
(3) For the minimum mass ratio found in part
William Shakespeare was an English playwright, poet and actor born in 1564 in Stratford-upon-Avon, England. He wrote 38 plays and 154 sonnets which are considered some of the greatest works in English literature. Shakespeare wrote in Early Modern English for the Globe Theatre in London where only men and boys, whose voices had not changed, performed both male and female roles for audiences of all social classes. Some of his most famous plays include Romeo and Juliet, Hamlet, Macbeth, and A Midsummer Night's Dream.
The document discusses seven hypotheses about good language learners proposed by Naiman et al. These include being willing to guess meanings, having strong motivation to communicate, not being afraid to make mistakes, practicing language skills, monitoring their own and others' speech, and focusing on meaning.
It then discusses Oxford's classification of language learning strategies as direct or indirect. Direct strategies involve memorizing vocabulary and understanding grammar, while indirect strategies involve self-regulation like planning study time and preparation.
Oxford identifies six main categories of language learning strategies: cognitive, memory, compensation, metacognitive, affective, and social strategies. Examples are given of strategies within each category.
La epistemología es la disciplina filosófica que estudia el conocimiento científico y cómo los individuos amplían los horizontes de la ciencia. Trata los problemas planteados por la ciencia. La contabilidad cuantifica los recursos económicos y financieros de una empresa para conocer el valor relativo entre los bienes y garantizar el funcionamiento de la empresa y el estado. Los investigadores de la contabilidad establecen su campo de acción científica para incluir el balance económico general, la información financiera y el manejo de la economía de la empresa.
Integrating Adaptation Mechanisms Using Control Theory Centric Architecture M...Filip Krikava
This document discusses integrating adaptation mechanisms in self-adaptive software systems using control theory models. It presents a case study of using feedback control loops and control theory models to optimize a web server's performance by self-adjusting tuning parameters. The challenges of engineering such self-adaptive systems include control challenges for control engineers and integration challenges for software engineers. The study models the web server as a multi-input multi-output system and designs a linear quadratic regulator controller to optimize performance based on CPU utilization and memory usage.
The document describes the development of an automatic MATLAB-based tool for measuring beam emittance at the Idaho Accelerator Center. An optical transition radiation screen and camera were installed to capture beam images during a quadrupole scan. MATLAB codes were developed to extract beam sizes from the images, perform a polynomial fit to determine emittance, and control the scan automatically via EPICS and MATLAB Channel Access. The tool was tested by measuring the emittance of the HRRL accelerator, reducing measurement time and error compared to manual methods.
This document describes research on developing a high-precision tsunami runup calculation method coupled with structure analysis. It discusses the need to evaluate damage from giant tsunamis considering structural destruction and debris. It proposes a 3D numerical simulator to analyze overflow, scouring, and flooding of buildings. The research aims to develop a system connecting tsunami propagation simulation with 3D structure analysis simulation. It describes a multiphysics tsunami simulator framework coupling models at different scales from the tsunami source to inundation. The framework includes STOC and CADMAS simulators connected using MPI communication. Example applications to the 2011 Tohoku tsunami demonstrate the approach.
Hyperspectral unmixing using novel conversion model.pptgrssieee
The document presents a novel hyperspectral unmixing approach called uccm-SVM that converts the abundance quantification problem into a classification problem using support vector machines. The approach is tested on both simulated and real hyperspectral images and is shown to outperform traditional mean-based techniques like FCLS in terms of accuracy while having lower computational costs for smaller training set sizes. Future work to improve the method includes enhancing performance while reducing computation for larger training sets.
Automated Generation of High-accuracy Interatomic Potentials Using Quantum Dataaimsnist
Sandia National Laboratories is developing SNAP (Spectral Neighbor Analysis Potential) potentials for molecular dynamics simulations. SNAP potentials are fitted to quantum mechanical data using bispectrum components that describe the local atomic environments. SNAP potentials have been shown to accurately reproduce properties of tantalum, including liquid structure and screw dislocation behavior not included in the training data. Work is ongoing to develop multi-element SNAP potentials, including for tungsten-beryllium alloys relevant to modeling plasma-surface interactions in nuclear fusion reactors.
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
Cross-validation is a technique used to evaluate machine learning models by reserving a portion of a dataset to test the model trained on the remaining data. There are several common cross-validation methods, including the test set method (reserving 30% of data for testing), leave-one-out cross-validation (training on all data points except one, then testing on the left out point), and k-fold cross-validation (randomly splitting data into k groups, with k-1 used for training and the remaining group for testing). The document provides an example comparing linear regression, quadratic regression, and point-to-point connection on a concrete strength dataset using k-fold cross-validation. SPSS output for the
The document discusses statistical models for networks. It covers types of inference problems in networks, simple random graphs models, and less random graph models that condition on additional network features like degree distribution or mixing patterns. It also discusses comparing two networks using quadratic assignment procedures and generating networks with appropriate degree distributions for comparison.
Efficient Implementation of Self-Organizing Map for Sparse Input Dataymelka
This document describes improvements made to the self-organizing map (SOM) algorithm to make it more efficient for sparse, high-dimensional input data. The key contributions are a sparse SOM (Sparse-Som) and sparse batch SOM (Sparse-BSom) algorithm that exploit the sparseness of the data to reduce computational complexity from O(TMD) to O(TMd), where d is the number of non-zero dimensions. Sparse-Som speeds up the BMU search and weight update phases, while Sparse-BSom further allows for efficient parallelization. Experiments show Sparse-Som and Sparse-BSom train significantly faster than standard SOM on sparse datasets, with comparable or better quality
Macromodel of High Speed Interconnect using Vector Fitting Algorithmijsrd.com
At high frequency efficient macromodeling of high speed interconnects is all time challenging task. We have presented systematic methodologies to generate rational function approximations of high-speed interconnects using vector fitting technique for any type of termination conditions and construct efficient multiport model, which is easily and directly compatible with circuit simulators.
This document discusses Matlab simulations of Markov models. It begins with an overview of Markov processes, chains, and properties. It then discusses using Matlab for simulations of hidden Markov models, including functions for decoding, generating, estimating, and training HMMs. Applications mentioned include speech recognition, gene modeling, and more. Advantages include the models' elegance, scalability, and complementarity to other techniques. Limitations include their data-intensive nature and Markov property assumptions. In conclusion, the document provides resources for learning more about Markov models and their simulation in Matlab.
Data Science Salon: MCL Clustering of Sparse GraphsFormulatedby
The increasing need for clustering in several scientific domains has inevitably driven the creation of innovative algorithms, each designed to perform more efficiently in certain applications. More specifically, in many applications the data entities involved can be portrayed effectively by a graph as a collection of nodes and edges. One of the most established algorithms for graph clustering problems is the Markov Cluster Algorithm (MCL).
Next DSS MIA Event - https://datascience.salon/miami/
When dealing with large and complex datasets, the underlying graphs can easily reach proportions that independent computing systems are inadequate to deal with. Additionally, the graphs encountered are typically sparse: the number of edges is far smaller than might be possible in a fully-connected graph. Consequently, there is a concrete need for algorithms that are designed to handle sparse graph clustering utilizing distributed computing resources.
Our motivation was the development of a distributed architecture, able to accommodate large and sparse graphs, to actualize the MCL and R-MCL algorithm. The Apache Spark framework was chosen due to its ability to utilize distributed resources and its proven track record. Although Spark is a framework capable of handling massive datasets, it currently does not provide rich support for computation with sparse matrices and sparse graphs. Hence, methods have been implemented to enable the exploitation of sparse adjacency matrices in distributed sparse matrix multiplication, a critical component of MCL. The proposed solution can handle arbitrarily large inputs, provide almost linear speed-up with the addition of computational resources and output results directly comparable to the non-distributed reference MCL implementation.
Approaches to formal verification of ams designAmbuj Mishra
Masters thesis on Approaches to formal verification of analog and mixed signal designs presented in June 2016 at International Institute of Information Technology, Bangalore (IIITB).
The document summarizes the Monte Carlo method for simulating complex physical systems. It describes how Monte Carlo simulations can provide approximate solutions to problems that are difficult to solve analytically. Such simulations are important for understanding systems with many interacting particles, like spin glasses, and for problems in fields like condensed matter physics, particle physics, and quantum gravity. Markov chain Monte Carlo methods are surveyed as a way to realize the importance sampling idea in computer simulations of complex systems.
The document describes Bayesian model updating research using adaptive Bayesian filters and data-centric approaches. It outlines previous contributions, future research plans, and short-term objectives. The focus is on Bayesian updating with MCMC and TMCMC approaches to more accurately and efficiently update model parameters. Model reduction techniques are proposed in the frequency domain and time domain to address incomplete measured responses. Numerical studies on a shear building model demonstrate that the Bayesian updating algorithm can estimate parameters well when using 45 data sets and hyperparameters of 0.001, 0.001, with a maximum error of 2.5%.
1) Randomized numerical linear algebra (RandNLA) algorithms can be used to solve large-scale least-squares problems by computing a randomized sketch of the design matrix in two steps and then obtaining approximate solutions.
2) The document implements and evaluates these RandNLA algorithms in Apache Spark on datasets up to terabytes in size, finding that Spark is well-suited due to the algorithms' parallelism and Spark's ability to cache data in memory.
3) The evaluation compares the performance of low-precision solvers that directly use the sketch and high-precision solvers that employ the sketch as a preconditioner, finding that both approaches can efficiently solve least-squares problems on large datasets.
This document presents a new method for simultaneously coordinating the design of STATCOM controllers and power system stabilizers (PSS) using a cultural algorithm. The cultural algorithm optimizes the parameters of the STATCOM and PSS to improve power system stability. Simulation results on single machine infinite bus and multi-machine test systems showed the proposed coordinated design technique effectively enhances system dynamics and performance compared to other optimization methods. The cultural algorithm provides a wide search region and high convergence speed for coordinating STATCOM and PSS controllers to improve power system stability.
This document summarizes a study of built-in self-test (BIST) approaches for detecting single stuck-at faults in combinational logic circuits. Pseudorandom test patterns generated by a linear feedback shift register (LFSR) were applied in parallel and serially to benchmark circuits. Applying patterns in parallel via test-per-clock achieved high fault coverage but required a large LFSR for circuits with many inputs. Reseeding the LFSR improved coverage when an initial seed was ineffective. Seed selection and minimum LFSR size for different application methods were evaluated to optimize BIST fault detection.
Integrating Adaptation Mechanisms Using Control Theory Centric Architecture M...Filip Krikava
This document discusses integrating adaptation mechanisms in self-adaptive software systems using control theory models. It presents a case study of using feedback control loops and control theory models to optimize a web server's performance by self-adjusting tuning parameters. The challenges of engineering such self-adaptive systems include control challenges for control engineers and integration challenges for software engineers. The study models the web server as a multi-input multi-output system and designs a linear quadratic regulator controller to optimize performance based on CPU utilization and memory usage.
The document describes the development of an automatic MATLAB-based tool for measuring beam emittance at the Idaho Accelerator Center. An optical transition radiation screen and camera were installed to capture beam images during a quadrupole scan. MATLAB codes were developed to extract beam sizes from the images, perform a polynomial fit to determine emittance, and control the scan automatically via EPICS and MATLAB Channel Access. The tool was tested by measuring the emittance of the HRRL accelerator, reducing measurement time and error compared to manual methods.
This document describes research on developing a high-precision tsunami runup calculation method coupled with structure analysis. It discusses the need to evaluate damage from giant tsunamis considering structural destruction and debris. It proposes a 3D numerical simulator to analyze overflow, scouring, and flooding of buildings. The research aims to develop a system connecting tsunami propagation simulation with 3D structure analysis simulation. It describes a multiphysics tsunami simulator framework coupling models at different scales from the tsunami source to inundation. The framework includes STOC and CADMAS simulators connected using MPI communication. Example applications to the 2011 Tohoku tsunami demonstrate the approach.
Hyperspectral unmixing using novel conversion model.pptgrssieee
The document presents a novel hyperspectral unmixing approach called uccm-SVM that converts the abundance quantification problem into a classification problem using support vector machines. The approach is tested on both simulated and real hyperspectral images and is shown to outperform traditional mean-based techniques like FCLS in terms of accuracy while having lower computational costs for smaller training set sizes. Future work to improve the method includes enhancing performance while reducing computation for larger training sets.
Automated Generation of High-accuracy Interatomic Potentials Using Quantum Dataaimsnist
Sandia National Laboratories is developing SNAP (Spectral Neighbor Analysis Potential) potentials for molecular dynamics simulations. SNAP potentials are fitted to quantum mechanical data using bispectrum components that describe the local atomic environments. SNAP potentials have been shown to accurately reproduce properties of tantalum, including liquid structure and screw dislocation behavior not included in the training data. Work is ongoing to develop multi-element SNAP potentials, including for tungsten-beryllium alloys relevant to modeling plasma-surface interactions in nuclear fusion reactors.
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
Cross-validation is a technique used to evaluate machine learning models by reserving a portion of a dataset to test the model trained on the remaining data. There are several common cross-validation methods, including the test set method (reserving 30% of data for testing), leave-one-out cross-validation (training on all data points except one, then testing on the left out point), and k-fold cross-validation (randomly splitting data into k groups, with k-1 used for training and the remaining group for testing). The document provides an example comparing linear regression, quadratic regression, and point-to-point connection on a concrete strength dataset using k-fold cross-validation. SPSS output for the
The document discusses statistical models for networks. It covers types of inference problems in networks, simple random graphs models, and less random graph models that condition on additional network features like degree distribution or mixing patterns. It also discusses comparing two networks using quadratic assignment procedures and generating networks with appropriate degree distributions for comparison.
Efficient Implementation of Self-Organizing Map for Sparse Input Dataymelka
This document describes improvements made to the self-organizing map (SOM) algorithm to make it more efficient for sparse, high-dimensional input data. The key contributions are a sparse SOM (Sparse-Som) and sparse batch SOM (Sparse-BSom) algorithm that exploit the sparseness of the data to reduce computational complexity from O(TMD) to O(TMd), where d is the number of non-zero dimensions. Sparse-Som speeds up the BMU search and weight update phases, while Sparse-BSom further allows for efficient parallelization. Experiments show Sparse-Som and Sparse-BSom train significantly faster than standard SOM on sparse datasets, with comparable or better quality
Macromodel of High Speed Interconnect using Vector Fitting Algorithmijsrd.com
At high frequency efficient macromodeling of high speed interconnects is all time challenging task. We have presented systematic methodologies to generate rational function approximations of high-speed interconnects using vector fitting technique for any type of termination conditions and construct efficient multiport model, which is easily and directly compatible with circuit simulators.
This document discusses Matlab simulations of Markov models. It begins with an overview of Markov processes, chains, and properties. It then discusses using Matlab for simulations of hidden Markov models, including functions for decoding, generating, estimating, and training HMMs. Applications mentioned include speech recognition, gene modeling, and more. Advantages include the models' elegance, scalability, and complementarity to other techniques. Limitations include their data-intensive nature and Markov property assumptions. In conclusion, the document provides resources for learning more about Markov models and their simulation in Matlab.
Data Science Salon: MCL Clustering of Sparse GraphsFormulatedby
The increasing need for clustering in several scientific domains has inevitably driven the creation of innovative algorithms, each designed to perform more efficiently in certain applications. More specifically, in many applications the data entities involved can be portrayed effectively by a graph as a collection of nodes and edges. One of the most established algorithms for graph clustering problems is the Markov Cluster Algorithm (MCL).
Next DSS MIA Event - https://datascience.salon/miami/
When dealing with large and complex datasets, the underlying graphs can easily reach proportions that independent computing systems are inadequate to deal with. Additionally, the graphs encountered are typically sparse: the number of edges is far smaller than might be possible in a fully-connected graph. Consequently, there is a concrete need for algorithms that are designed to handle sparse graph clustering utilizing distributed computing resources.
Our motivation was the development of a distributed architecture, able to accommodate large and sparse graphs, to actualize the MCL and R-MCL algorithm. The Apache Spark framework was chosen due to its ability to utilize distributed resources and its proven track record. Although Spark is a framework capable of handling massive datasets, it currently does not provide rich support for computation with sparse matrices and sparse graphs. Hence, methods have been implemented to enable the exploitation of sparse adjacency matrices in distributed sparse matrix multiplication, a critical component of MCL. The proposed solution can handle arbitrarily large inputs, provide almost linear speed-up with the addition of computational resources and output results directly comparable to the non-distributed reference MCL implementation.
Approaches to formal verification of ams designAmbuj Mishra
Masters thesis on Approaches to formal verification of analog and mixed signal designs presented in June 2016 at International Institute of Information Technology, Bangalore (IIITB).
The document summarizes the Monte Carlo method for simulating complex physical systems. It describes how Monte Carlo simulations can provide approximate solutions to problems that are difficult to solve analytically. Such simulations are important for understanding systems with many interacting particles, like spin glasses, and for problems in fields like condensed matter physics, particle physics, and quantum gravity. Markov chain Monte Carlo methods are surveyed as a way to realize the importance sampling idea in computer simulations of complex systems.
The document describes Bayesian model updating research using adaptive Bayesian filters and data-centric approaches. It outlines previous contributions, future research plans, and short-term objectives. The focus is on Bayesian updating with MCMC and TMCMC approaches to more accurately and efficiently update model parameters. Model reduction techniques are proposed in the frequency domain and time domain to address incomplete measured responses. Numerical studies on a shear building model demonstrate that the Bayesian updating algorithm can estimate parameters well when using 45 data sets and hyperparameters of 0.001, 0.001, with a maximum error of 2.5%.
1) Randomized numerical linear algebra (RandNLA) algorithms can be used to solve large-scale least-squares problems by computing a randomized sketch of the design matrix in two steps and then obtaining approximate solutions.
2) The document implements and evaluates these RandNLA algorithms in Apache Spark on datasets up to terabytes in size, finding that Spark is well-suited due to the algorithms' parallelism and Spark's ability to cache data in memory.
3) The evaluation compares the performance of low-precision solvers that directly use the sketch and high-precision solvers that employ the sketch as a preconditioner, finding that both approaches can efficiently solve least-squares problems on large datasets.
This document presents a new method for simultaneously coordinating the design of STATCOM controllers and power system stabilizers (PSS) using a cultural algorithm. The cultural algorithm optimizes the parameters of the STATCOM and PSS to improve power system stability. Simulation results on single machine infinite bus and multi-machine test systems showed the proposed coordinated design technique effectively enhances system dynamics and performance compared to other optimization methods. The cultural algorithm provides a wide search region and high convergence speed for coordinating STATCOM and PSS controllers to improve power system stability.
This document summarizes a study of built-in self-test (BIST) approaches for detecting single stuck-at faults in combinational logic circuits. Pseudorandom test patterns generated by a linear feedback shift register (LFSR) were applied in parallel and serially to benchmark circuits. Applying patterns in parallel via test-per-clock achieved high fault coverage but required a large LFSR for circuits with many inputs. Reseeding the LFSR improved coverage when an initial seed was ineffective. Seed selection and minimum LFSR size for different application methods were evaluated to optimize BIST fault detection.
1. EMAILS:
• JUNSHAN WANG: wangjunshan@nus.edu.sg
• AJAY JASRA: staja@nus.edu.sg
• MARIA DE IORIO: m.deiorio@ucl.ac.uk
In the following article we provide an exposition of exact computational methods to
perform parameter inference from partially observed network models. In particular,
we consider the duplication attachment (DA) model which has a likelihood function
that typically cannot be evaluated in any reasonable computational time. We
consider a number of importance sampling (IS) and sequential Monte Carlo (SMC)
methods for approximating the likelihood of the network model for a fixed
parameter value. It is well-known that for IS, the relative variance of the likelihood
estimate typically grows at an exponential rate in the time parameter (here this is
associated to the size of the network): we prove that, under assumptions, the SMC
method will have relative variance which can grow only polynomially. In order to
perform parameter estimation, we develop particle Markov chain Monte Carlo
(PMCMC) algorithms to perform Bayesian inference. Such algorithms use the afore-
mentioned SMC algorithms within the transition dynamics. The approaches are
illustrated numerically.
!
ABSTRACT
OBJECTIVES
NUMERICAL
ILLUSTRATION
(CONTINUED)
DPF (N=100) DPF (N=1000) DPF (N=10000)
Relative variance CPU time
2. Parameter estimation
• Auto-correlation plots
Marginal MCMC PMCMC with SMC PMCMC with DPF
• Density plots
IID sampling Marginal MCMC PMCMC with SMC PMCMC with DPF
CONCLUSION
ACKNOWLEGEMENTS
1
Department
of
Sta.s.cs
&
Applied
Probability,
Na.onal
University
of
Singapore,
Singapore,
117546,
SG.
2
Department
of
Sta.s.cal
Science,
University
College,
London,
WC1E
6BT,
UK.
JUNSHAN
WANG1
&
AJAY
JASRA1
&
MARIA
DE
IORIO2
Computa.onal
Methods
for
a
Class
of
Network
Models
COMPUTATIONAL
METHODS
NUMERICAL
ILLUSTRATIONS
1. Likelihood approximation comparison.
IS (N=1000) IS (N=10000) ESS of IS (N=100,1000,10000)
SMC (N=1000) SMC (N=10000) ESS of SMC (N=100,1000,10000)
0.05 0.25 0.45 0.65 0.85
−1
0
1
2
3
4
5
6
7
x 10
−11
Parameter p
Likelihood
True Estimate Upper&Lower
0.05 0.25 0.45 0.65 0.85
−1
0
1
2
3
4
5
6
7
x 10
−11
Parameter p
Likelihood
True Estimate Upper&Lower
0.05 0.25 0.45 0.65 0.85
0
10
20
30
Parameter p, N=100
ESS
0.05 0.25 0.45 0.65 0.85
0
20
40
60
80
Parameter p, N=1000
ESS
0.05 0.25 0.45 0.65 0.85
0
200
400
600
Parameter p, N=10000
ESS
1 2 3 4 5 6 7 8 9
0
50
100
Time, N=100
ESS&UN
ESS UN
1 2 3 4 5 6 7 8 9
0
500
1000
Time, N=1000
ESS&UN
1 2 3 4 5 6 7 8 9
0
5000
10000
Time, N=10000
ESS&UN
0.05 0.25 0.45 0.65 0.85
−1
0
1
2
3
4
5
6
7
x 10
−11
Parameter p
Likelihood
True Estimate Upper&Lower
0.05 0.25 0.45 0.65 0.85
−1
0
1
2
3
4
5
6
7
x 10
−11
Parameter p
Likelihood
True Estimate Upper&Lower
0.05 0.25 0.45 0.65 0.85
−1
0
1
2
3
4
5
6
7
x 10
−11
Parameter p
Likelihood
True Estimate Upper&Lower
0.05 0.25 0.45 0.65 0.85
−1
0
1
2
3
4
5
6
7
x 10
−11
Parameter p
Likelihood
True Estimate Upper&Lower
0.05 0.25 0.45 0.65 0.85
−1
0
1
2
3
4
5
6
7
x 10
−11
Parameter p
Likelihood
True Estimate Upper&Lower
0.05 0.25 0.45 0.65 0.85
−1
0
1
2
3
4
5
6
7
x 10
−11
Parameter p
Likelihood
True
SMC
IS
DPF
Upper of SMC
Lower of SMC
Upper of IS
Lower of IS
Upper of DPF
Lower of DPF
size IS STRA DPF
5 0.0003 0.0002 0.0000
6 0.0027 0.0030 0.0000
7 0.0043 0.0064 0.0000
8 0.0158 0.0142 0.0000
9 0.0149 0.0136 0.0010
10 0.0419 0.0128 0.0036
11 0.1512 0.0364 0.0084
12 0.5659 0.1115 0.0079
13 1.4224 0.3022 0.0657
−0.2 0 0.2 0.4 0.6 0.8 1
0
20
40
60
80
100
120
140
160
Parameter p
Frequency
−0.2 0 0.2 0.4 0.6 0.8 1
0
20
40
60
80
100
120
140
160
Parameter p
Frequency
−0.2 0 0.2 0.4 0.6 0.8 1
0
20
40
60
80
100
120
140
160
Parameter p
Frequency
−0.2 0 0.2 0.4 0.6 0.8 1
0
20
40
60
80
100
120
140
160
Parameter p
Frequency
0 2100 4200 6300
−0.05
0
0.05
Lag k
Auto−correlation
0 2100 4200 6300
−0.05
0
0.05
Lag k
Auto−correlation
0 2100 4200 6300
−0.05
0
0.05
Lag k
Auto−correlation
CONTACT
1. Approximate the likelihood of the network model.
• Given a reducible graph G! and a fixed parameter value θ, the recursive manner
of the likelihood is:
L! G! =
1
t
ω!(v, G!)!L! δ(G!, v)
!∈!(!!)
with L! G!!
= 1, ω! v, G! = Ρ!(G!|δ(G!, v)) is the transition probability and
R(G!) is the collection of removable vertices of G!.
2. Perform parameter estimation.
• We will follow a Bayesian procedure and place a prior probability distribution !(!)
on the parameter; we will then seek to sample from the associated posterior
distribution !(!) ∝ L! G! !!(!) using MCMC.
1. Likelihood approximation.#
• Importance Sampling (IS)!
Advantage: run-time savings.
Disadvantage: the relative variance is !(ϰ!!!!) for some ϰ > 1.
• Sequential Monte Carlo (SMC)!
Advantage: the relative variance is no worse than !((! − !!)!).
Disadvantage: evolve on a finite state-space.
• Discrete Particle Filter (DPF)!
Advantage: explore the whole state-space.
Disadvantage: only excellent for small to medium size networks.
2. Parameter estimation.#
• Particle Markov Chain Monte Carlo (PMCMC)!
Advantage: applicable when the exact likelihood is unknown.
Disadvantage: scalability restriction due to both memory and computational demands.
!
1. The relative variance of the SMC method will only grow at a polynomial
rate in the number removable nodes. Whilst the relative variance of the
IS estimate of the likelihood typically grows at an exponential rate in the
number of removable nodes.
2. For small to medium sized networks, the DPF and DPF inside MCMC
seemed to perform better versus the SMC based versions. In general,
however, the computational time was much higher and this value was
quite high for each of our algorithms.
3. The two PMCMC algorithms perform similarly to the marginal MCMC. In
addition, they produce solutions consistent with i.i.d. sampling, which
means such methodology can be useful for network models.
!
• The second author was supported by an MOE Singapore grant.
• Special thanks to Prof. Ajay Jasra for his assistance and cooperation in
accomplishing this paper.
• This paper is about to appear on the Journal of Computational Biology and
able to be downloaded at
http://www.stat.nus.edu.sg/~staja/smc_network2.pdf.
!