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Here are some other potential applications of EM: - EM can be used for parameter estimation in hidden Markov models (HMMs). The hidden states are the latent variables estimated using EM. - EM can be used for topic modeling using latent Dirichlet allocation (LDA). The topics are the latent variables estimated from documents. - As mentioned in the document, EM can also be used for Gaussian mixture models (GMMs) for clustering and density estimation. The cluster assignments are latent. - EM can be used for missing data problems, where the missing values are treated as latent variables estimated each iteration. - Bayesian networks and directed graphical models more generally can also be estimated using EM by treating the conditional probabilities as latent

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Lecture 18: Gaussian Mixture Models and Expectation Maximization

This document discusses Gaussian mixture models (GMMs) and the expectation-maximization (EM) algorithm. GMMs model data as coming from a mixture of Gaussian distributions, with each data point assigned soft responsibilities to the different components. EM is used to estimate the parameters of GMMs and other latent variable models. It iterates between an E-step, where responsibilities are computed based on current parameters, and an M-step, where new parameters are estimated to maximize the expected complete-data log-likelihood given the responsibilities. EM converges to a local optimum for fitting GMMs to data.

GMM

This document discusses Gaussian mixture models (GMMs) and their use in applications like speaker recognition and language identification. GMMs represent a probability density function as a weighted sum of Gaussian distributions. GMM parameters are estimated from training data using Expectation-Maximization or Maximum A Posteriori estimation. GMMs are computationally inexpensive and well-suited for text-independent tasks without strong prior knowledge of content.

Mc culloch pitts neuron

Here is a MATLAB program to implement logic functions using a McCulloch-Pitts neuron:
% McCulloch-Pitts neuron for logic functions
% Inputs
x1 = 1;
x2 = 0;
% Weights
w1 = 1;
w2 = 1;
% Threshold
theta = 2;
% Net input
net = x1*w1 + x2*w2;
% Activation function
if net >= theta
y = 1;
else
y = 0;
end
% Output
disp(y)
This implements a basic AND logic gate using a McCulloch-Pitts neuron.

Neural Networks: Rosenblatt's Perceptron

Lecture slides on Rosenblatt's Perceptron as a part of a course on Neural Networks based on Haykin's Book.

Speaker Recognition using Gaussian Mixture Model

This presentation slide contains, Introduction to Gaussian mixture model and its application in identifying speaker.

K-means and GMM

This document discusses clustering methods using the EM algorithm. It begins with an overview of machine learning and unsupervised learning. It then describes clustering, k-means clustering, and how k-means can be formulated as an optimization of a biconvex objective function solved via an iterative EM algorithm. The document goes on to describe mixture models and how the EM algorithm can be used to estimate the parameters of a Gaussian mixture model (GMM) via maximum likelihood.

Clustering：k-means, expect-maximization and gaussian mixture model

This document discusses K-means clustering, Expectation Maximization (EM), and Gaussian mixture models (GMM). It begins with an overview of unsupervised learning and introduces K-means as a simple clustering algorithm. It then describes EM as a general algorithm for maximum likelihood estimation that can be applied to problems like GMM. GMM is presented as a density estimation technique that models data using a weighted sum of Gaussian distributions. EM is described as a method for estimating the parameters of a GMM from data.

Evolutionary Computing

This document describes how a genetic algorithm can be used to select the best cricket team from a pool of players without human intervention. It discusses representing the team as a chromosome with 11 genes corresponding to each player position. A fitness function calculates the total performance score for each team. Selection is done with tournament selection to choose teams for reproduction. Crossover and mutation operators combine genes to generate new teams, while invalid teams are removed. The process runs until convergence on the highest scoring team.

Lecture 18: Gaussian Mixture Models and Expectation Maximization

This document discusses Gaussian mixture models (GMMs) and the expectation-maximization (EM) algorithm. GMMs model data as coming from a mixture of Gaussian distributions, with each data point assigned soft responsibilities to the different components. EM is used to estimate the parameters of GMMs and other latent variable models. It iterates between an E-step, where responsibilities are computed based on current parameters, and an M-step, where new parameters are estimated to maximize the expected complete-data log-likelihood given the responsibilities. EM converges to a local optimum for fitting GMMs to data.

GMM

This document discusses Gaussian mixture models (GMMs) and their use in applications like speaker recognition and language identification. GMMs represent a probability density function as a weighted sum of Gaussian distributions. GMM parameters are estimated from training data using Expectation-Maximization or Maximum A Posteriori estimation. GMMs are computationally inexpensive and well-suited for text-independent tasks without strong prior knowledge of content.

Mc culloch pitts neuron

Here is a MATLAB program to implement logic functions using a McCulloch-Pitts neuron:
% McCulloch-Pitts neuron for logic functions
% Inputs
x1 = 1;
x2 = 0;
% Weights
w1 = 1;
w2 = 1;
% Threshold
theta = 2;
% Net input
net = x1*w1 + x2*w2;
% Activation function
if net >= theta
y = 1;
else
y = 0;
end
% Output
disp(y)
This implements a basic AND logic gate using a McCulloch-Pitts neuron.

Neural Networks: Rosenblatt's Perceptron

Lecture slides on Rosenblatt's Perceptron as a part of a course on Neural Networks based on Haykin's Book.

Speaker Recognition using Gaussian Mixture Model

This presentation slide contains, Introduction to Gaussian mixture model and its application in identifying speaker.

K-means and GMM

This document discusses clustering methods using the EM algorithm. It begins with an overview of machine learning and unsupervised learning. It then describes clustering, k-means clustering, and how k-means can be formulated as an optimization of a biconvex objective function solved via an iterative EM algorithm. The document goes on to describe mixture models and how the EM algorithm can be used to estimate the parameters of a Gaussian mixture model (GMM) via maximum likelihood.

Clustering：k-means, expect-maximization and gaussian mixture model

This document discusses K-means clustering, Expectation Maximization (EM), and Gaussian mixture models (GMM). It begins with an overview of unsupervised learning and introduces K-means as a simple clustering algorithm. It then describes EM as a general algorithm for maximum likelihood estimation that can be applied to problems like GMM. GMM is presented as a density estimation technique that models data using a weighted sum of Gaussian distributions. EM is described as a method for estimating the parameters of a GMM from data.

Evolutionary Computing

This document describes how a genetic algorithm can be used to select the best cricket team from a pool of players without human intervention. It discusses representing the team as a chromosome with 11 genes corresponding to each player position. A fitness function calculates the total performance score for each team. Selection is done with tournament selection to choose teams for reproduction. Crossover and mutation operators combine genes to generate new teams, while invalid teams are removed. The process runs until convergence on the highest scoring team.

HOPFIELD NETWORK

This document presents information on Hopfield networks through a slideshow presentation. It begins with an introduction to Hopfield networks, describing them as fully connected, single layer neural networks that can perform pattern recognition. It then discusses the properties of Hopfield networks, including their symmetric weights and binary neuron outputs. The document proceeds to provide derivations of the Hopfield network model based on an additive neuron model. It concludes by discussing applications of Hopfield networks.

Graph Based Clustering

The document discusses graph-based clustering methods. It describes how graphs can be used to represent real-world networks from domains like biology, technology, social networks, and economics. It introduces the idea of using minimal spanning trees and hierarchical clustering to identify clusters in graph data. Two common algorithms for finding minimal spanning trees are described: Prim's algorithm and Kruskal's algorithm. Different strategies for iteratively deleting branches from the minimal spanning tree are also summarized to form clusters, such as deleting the branch with the maximum weight or inconsistent branches based on a reference value.

Neural Networks: Radial Bases Functions (RBF)

This document discusses kernel methods and radial basis function (RBF) networks. It begins with an introduction and overview of Cover's theory of separability of patterns. It then revisits the XOR problem and shows how it can be solved using Gaussian hidden functions. The interpolation problem is explained and how RBF networks can perform strict interpolation through a set of training data points. Radial basis functions that satisfy Micchelli's theorem allowing for a nonsingular interpolation matrix are presented. Finally, the structure and training of RBF networks using k-means clustering and recursive least squares estimation is covered.

Support vector machines (svm)

A Support Vector Machine (SVM) is a discriminative classifier formally defined by a separating hyperplane. In other words, given labeled training data (supervised learning), the algorithm outputs an optimal hyperplane which categorizes new examples. In two dimentional space this hyperplane is a line dividing a plane in two parts where in each class lay in either side.

Hebbian Learning

The document discusses various types of Hebbian learning including:
1) Unsupervised Hebbian learning where weights are strengthened based on actual neural responses to stimuli without a target output.
2) Supervised Hebbian learning where weights are strengthened based on the desired neural response rather than the actual response to better approximate a target output.
3) Recognition networks like the instar rule which only updates weights when a neuron's output is active to recognize specific input patterns.

Point processing

This document discusses various point processing and gray level transformation techniques used in image enhancement. It describes point processing as operating directly on pixel intensity values individually to alter them using transformation functions. The document outlines several basic gray level transformations including linear, logarithmic and power law. It also discusses piecewise linear transformations such as contrast stretching, intensity level slicing, and bit plane slicing. These transformations are used to enhance images by modifying their brightness, contrast and emphasis on certain gray levels.

Perceptron (neural network)

i. Perceptron
Representation & Issues
Classification
learning
ii. linear Separability

Intro to Model Selection

This document provides an introduction to statistical model selection. It discusses various approaches to model selection including predictive risk, Bayesian methods, information theoretic measures like AIC and MDL, and adaptive methods. The key goals of model selection are to understand the bias-variance tradeoff and select models that offer the best guaranteed predictive performance on new data. Model selection aims to find the right level of complexity to explain patterns in available data while avoiding overfitting.

Principles of soft computing-Associative memory networks

The document discusses various types of associative memory networks including auto-associative, hetero-associative, bidirectional associative memory (BAM), and Hopfield networks. It describes the architecture, training algorithms, and testing procedures for each type of network. The key points are: Auto-associative networks store and recall patterns using the same input and output vectors, while hetero-associative networks use different input and output vectors. BAM networks perform bidirectional retrieval of patterns. Hopfield networks are auto-associative single-layer recurrent networks that can converge to stable states representing stored patterns. Hebbian learning and energy functions are important concepts in analyzing the storage and recall capabilities of these associative memory networks.

Lecture 4 Decision Trees (2): Entropy, Information Gain, Gain Ratio

attribute selection, constructing decision trees, decision trees, divide and conquer, entropy, gain ratio, information gain, machine leaning, pruning, rules, suprisal

Defuzzification

Defuzzification is the process of producing a quantifiable result in Crisp logic, given fuzzy sets and corresponding membership degrees. It is the process that maps a fuzzy set to a crisp set. It is typically needed in fuzzy control systems.

Adaptive Resonance Theory (ART)

Introduction to Adaptive Resonance Theory (ART) neural networks including:
Introduction (Stability-Plasticity Dilemma)
ART Network
ART Types
Basic ART network Architecture
ART Algorithm and Learning
ART Computational Example
ART Application
Conclusion
Main References

Regularization in deep learning

Presentation in Vietnam Japan AI Community in 2019-05-26.
The presentation summarizes what I've learned about Regularization in Deep Learning.
Disclaimer: The presentation is given in a community event, so it wasn't thoroughly reviewed or revised.

Artificial Neural Networks Lect3: Neural Network Learning rules

The document discusses various neural network learning rules:
1. Error correction learning rule (delta rule) adapts weights based on the error between the actual and desired output.
2. Memory-based learning stores all training examples and classifies new inputs based on similarity to nearby examples (e.g. k-nearest neighbors).
3. Hebbian learning increases weights of simultaneously active neuron connections and decreases others, allowing patterns to emerge from correlations in inputs over time.
4. Competitive learning (winner-take-all) adapts the weights of the neuron most active for a given input, allowing unsupervised clustering of similar inputs across neurons.

Hopfield Networks

The document discusses Hopfield networks, which are neural networks with fixed weights and adaptive activations. It describes two types - discrete and continuous Hopfield nets. Discrete Hopfield nets use binary activations that are updated asynchronously, allowing an energy function to be defined. They can serve as associative memory. Continuous Hopfield nets have real-valued activations and can solve optimization problems like the travelling salesman problem. The document provides details on the architecture, energy functions, algorithms, and applications of both network types.

Fuzzy inference systems

Fuzzy inference systems use fuzzy logic to map inputs to outputs. There are two main types:
Mamdani systems use fuzzy outputs and are well-suited for problems involving human expert knowledge. Sugeno systems have faster computation using linear or constant outputs.
The fuzzy inference process involves fuzzifying inputs, applying fuzzy logic operators, and using if-then rules. Outputs are determined through implication, aggregation, and defuzzification. Mamdani systems find the centroid of fuzzy outputs while Sugeno uses weighted averages, making it more efficient.

Decision Tree - C4.5&CART

This document discusses decision tree algorithms C4.5 and CART. It explains that ID3 has limitations in dealing with continuous data and noisy data, which C4.5 aims to address through techniques like post-pruning trees to avoid overfitting. CART uses binary splits and measures like Gini index or entropy to produce classification trees, and sum of squared errors to produce regression trees. It also performs cost-complexity pruning to find an optimal trade-off between accuracy and model complexity.

Deep Learning Frameworks slides

The document discusses deep learning concepts and frameworks. It provides an overview of deep learning concepts such as neural networks, layers, nodes, weights, activation functions, and optimization techniques. It also discusses specific deep learning frameworks including TensorFlow, Torch, and Theano. These frameworks can be compared based on factors like speed, ease of use, programming languages, hardware support, community size, and algorithms supported.

Max net

Self-organizing networks can perform unsupervised clustering by mapping high-dimensional input patterns into a smaller number of clusters in output space through competitive learning. Fixed weight competitive networks like Maxnet, Mexican Hat net, and Hamming net use competitive learning with fixed weights. Maxnet uses winner-take-all competition to select the neuron whose weights best match the input. Mexican Hat net has both excitatory and inhibitory connections between neurons to enhance contrast. Hamming net determines which exemplar vector most closely matches the input using the Hamming distance measure.

Independent Component Analysis

This document discusses independent component analysis (ICA) for blind source separation. ICA is a method to estimate original signals from observed signals consisting of mixed original signals and noise. It introduces the ICA model and approach, including whitening, maximizing non-Gaussianity using kurtosis and negentropy, and fast ICA algorithms. The document provides examples applying ICA to separate images and discusses approaches to improve ICA, including using differential filtering. ICA is an important technique for blind source separation and independent component estimation from observed signals.

The Inverse Smoluchowski Problem, Particles In Turbulence 2011, Potsdam, Marc...

This document summarizes Colm Connaughton's presentation on solving the inverse Smoluchowski problem to determine particle collision kernels from observed cluster size distributions. It describes how the forward problem maps kernels to distributions but the inverse problem is ill-posed. Tikhonov regularization is used to obtain approximate kernel reconstructions from numerical solutions with known test kernels, demonstrating partial success in reconstructing kernel features despite ill-posedness. Future work aims to address limitations and applicability to real problems.

Manuscript 1334

1) The document discusses travelling wave solutions for pulse propagation in negative index materials (NIMs) in the presence of an external source.
2) It obtains fractional-type solutions containing trigonometric and hyperbolic functions by using a fractional transform to map the governing equation to an elliptic equation.
3) Specific solutions include periodic solutions and bright/dark solitary wave solutions, with the intensity profiles of the bright solitary wave shown.

HOPFIELD NETWORK

This document presents information on Hopfield networks through a slideshow presentation. It begins with an introduction to Hopfield networks, describing them as fully connected, single layer neural networks that can perform pattern recognition. It then discusses the properties of Hopfield networks, including their symmetric weights and binary neuron outputs. The document proceeds to provide derivations of the Hopfield network model based on an additive neuron model. It concludes by discussing applications of Hopfield networks.

Graph Based Clustering

The document discusses graph-based clustering methods. It describes how graphs can be used to represent real-world networks from domains like biology, technology, social networks, and economics. It introduces the idea of using minimal spanning trees and hierarchical clustering to identify clusters in graph data. Two common algorithms for finding minimal spanning trees are described: Prim's algorithm and Kruskal's algorithm. Different strategies for iteratively deleting branches from the minimal spanning tree are also summarized to form clusters, such as deleting the branch with the maximum weight or inconsistent branches based on a reference value.

Neural Networks: Radial Bases Functions (RBF)

This document discusses kernel methods and radial basis function (RBF) networks. It begins with an introduction and overview of Cover's theory of separability of patterns. It then revisits the XOR problem and shows how it can be solved using Gaussian hidden functions. The interpolation problem is explained and how RBF networks can perform strict interpolation through a set of training data points. Radial basis functions that satisfy Micchelli's theorem allowing for a nonsingular interpolation matrix are presented. Finally, the structure and training of RBF networks using k-means clustering and recursive least squares estimation is covered.

Support vector machines (svm)

A Support Vector Machine (SVM) is a discriminative classifier formally defined by a separating hyperplane. In other words, given labeled training data (supervised learning), the algorithm outputs an optimal hyperplane which categorizes new examples. In two dimentional space this hyperplane is a line dividing a plane in two parts where in each class lay in either side.

Hebbian Learning

The document discusses various types of Hebbian learning including:
1) Unsupervised Hebbian learning where weights are strengthened based on actual neural responses to stimuli without a target output.
2) Supervised Hebbian learning where weights are strengthened based on the desired neural response rather than the actual response to better approximate a target output.
3) Recognition networks like the instar rule which only updates weights when a neuron's output is active to recognize specific input patterns.

Point processing

This document discusses various point processing and gray level transformation techniques used in image enhancement. It describes point processing as operating directly on pixel intensity values individually to alter them using transformation functions. The document outlines several basic gray level transformations including linear, logarithmic and power law. It also discusses piecewise linear transformations such as contrast stretching, intensity level slicing, and bit plane slicing. These transformations are used to enhance images by modifying their brightness, contrast and emphasis on certain gray levels.

Perceptron (neural network)

i. Perceptron
Representation & Issues
Classification
learning
ii. linear Separability

Intro to Model Selection

This document provides an introduction to statistical model selection. It discusses various approaches to model selection including predictive risk, Bayesian methods, information theoretic measures like AIC and MDL, and adaptive methods. The key goals of model selection are to understand the bias-variance tradeoff and select models that offer the best guaranteed predictive performance on new data. Model selection aims to find the right level of complexity to explain patterns in available data while avoiding overfitting.

Principles of soft computing-Associative memory networks

The document discusses various types of associative memory networks including auto-associative, hetero-associative, bidirectional associative memory (BAM), and Hopfield networks. It describes the architecture, training algorithms, and testing procedures for each type of network. The key points are: Auto-associative networks store and recall patterns using the same input and output vectors, while hetero-associative networks use different input and output vectors. BAM networks perform bidirectional retrieval of patterns. Hopfield networks are auto-associative single-layer recurrent networks that can converge to stable states representing stored patterns. Hebbian learning and energy functions are important concepts in analyzing the storage and recall capabilities of these associative memory networks.

Lecture 4 Decision Trees (2): Entropy, Information Gain, Gain Ratio

attribute selection, constructing decision trees, decision trees, divide and conquer, entropy, gain ratio, information gain, machine leaning, pruning, rules, suprisal

Defuzzification

Defuzzification is the process of producing a quantifiable result in Crisp logic, given fuzzy sets and corresponding membership degrees. It is the process that maps a fuzzy set to a crisp set. It is typically needed in fuzzy control systems.

Adaptive Resonance Theory (ART)

Introduction to Adaptive Resonance Theory (ART) neural networks including:
Introduction (Stability-Plasticity Dilemma)
ART Network
ART Types
Basic ART network Architecture
ART Algorithm and Learning
ART Computational Example
ART Application
Conclusion
Main References

Regularization in deep learning

Presentation in Vietnam Japan AI Community in 2019-05-26.
The presentation summarizes what I've learned about Regularization in Deep Learning.
Disclaimer: The presentation is given in a community event, so it wasn't thoroughly reviewed or revised.

Artificial Neural Networks Lect3: Neural Network Learning rules

The document discusses various neural network learning rules:
1. Error correction learning rule (delta rule) adapts weights based on the error between the actual and desired output.
2. Memory-based learning stores all training examples and classifies new inputs based on similarity to nearby examples (e.g. k-nearest neighbors).
3. Hebbian learning increases weights of simultaneously active neuron connections and decreases others, allowing patterns to emerge from correlations in inputs over time.
4. Competitive learning (winner-take-all) adapts the weights of the neuron most active for a given input, allowing unsupervised clustering of similar inputs across neurons.

Hopfield Networks

The document discusses Hopfield networks, which are neural networks with fixed weights and adaptive activations. It describes two types - discrete and continuous Hopfield nets. Discrete Hopfield nets use binary activations that are updated asynchronously, allowing an energy function to be defined. They can serve as associative memory. Continuous Hopfield nets have real-valued activations and can solve optimization problems like the travelling salesman problem. The document provides details on the architecture, energy functions, algorithms, and applications of both network types.

Fuzzy inference systems

Fuzzy inference systems use fuzzy logic to map inputs to outputs. There are two main types:
Mamdani systems use fuzzy outputs and are well-suited for problems involving human expert knowledge. Sugeno systems have faster computation using linear or constant outputs.
The fuzzy inference process involves fuzzifying inputs, applying fuzzy logic operators, and using if-then rules. Outputs are determined through implication, aggregation, and defuzzification. Mamdani systems find the centroid of fuzzy outputs while Sugeno uses weighted averages, making it more efficient.

Decision Tree - C4.5&CART

This document discusses decision tree algorithms C4.5 and CART. It explains that ID3 has limitations in dealing with continuous data and noisy data, which C4.5 aims to address through techniques like post-pruning trees to avoid overfitting. CART uses binary splits and measures like Gini index or entropy to produce classification trees, and sum of squared errors to produce regression trees. It also performs cost-complexity pruning to find an optimal trade-off between accuracy and model complexity.

Deep Learning Frameworks slides

The document discusses deep learning concepts and frameworks. It provides an overview of deep learning concepts such as neural networks, layers, nodes, weights, activation functions, and optimization techniques. It also discusses specific deep learning frameworks including TensorFlow, Torch, and Theano. These frameworks can be compared based on factors like speed, ease of use, programming languages, hardware support, community size, and algorithms supported.

Max net

Self-organizing networks can perform unsupervised clustering by mapping high-dimensional input patterns into a smaller number of clusters in output space through competitive learning. Fixed weight competitive networks like Maxnet, Mexican Hat net, and Hamming net use competitive learning with fixed weights. Maxnet uses winner-take-all competition to select the neuron whose weights best match the input. Mexican Hat net has both excitatory and inhibitory connections between neurons to enhance contrast. Hamming net determines which exemplar vector most closely matches the input using the Hamming distance measure.

Independent Component Analysis

This document discusses independent component analysis (ICA) for blind source separation. ICA is a method to estimate original signals from observed signals consisting of mixed original signals and noise. It introduces the ICA model and approach, including whitening, maximizing non-Gaussianity using kurtosis and negentropy, and fast ICA algorithms. The document provides examples applying ICA to separate images and discusses approaches to improve ICA, including using differential filtering. ICA is an important technique for blind source separation and independent component estimation from observed signals.

HOPFIELD NETWORK

HOPFIELD NETWORK

Graph Based Clustering

Graph Based Clustering

Neural Networks: Radial Bases Functions (RBF)

Neural Networks: Radial Bases Functions (RBF)

Support vector machines (svm)

Support vector machines (svm)

Hebbian Learning

Hebbian Learning

Point processing

Point processing

Perceptron (neural network)

Perceptron (neural network)

Intro to Model Selection

Intro to Model Selection

Principles of soft computing-Associative memory networks

Principles of soft computing-Associative memory networks

Lecture 4 Decision Trees (2): Entropy, Information Gain, Gain Ratio

Lecture 4 Decision Trees (2): Entropy, Information Gain, Gain Ratio

Defuzzification

Defuzzification

Adaptive Resonance Theory (ART)

Adaptive Resonance Theory (ART)

Regularization in deep learning

Regularization in deep learning

Artificial Neural Networks Lect3: Neural Network Learning rules

Artificial Neural Networks Lect3: Neural Network Learning rules

Hopfield Networks

Hopfield Networks

Fuzzy inference systems

Fuzzy inference systems

Decision Tree - C4.5&CART

Decision Tree - C4.5&CART

Deep Learning Frameworks slides

Deep Learning Frameworks slides

Max net

Max net

Independent Component Analysis

Independent Component Analysis

The Inverse Smoluchowski Problem, Particles In Turbulence 2011, Potsdam, Marc...

This document summarizes Colm Connaughton's presentation on solving the inverse Smoluchowski problem to determine particle collision kernels from observed cluster size distributions. It describes how the forward problem maps kernels to distributions but the inverse problem is ill-posed. Tikhonov regularization is used to obtain approximate kernel reconstructions from numerical solutions with known test kernels, demonstrating partial success in reconstructing kernel features despite ill-posedness. Future work aims to address limitations and applicability to real problems.

Manuscript 1334

1) The document discusses travelling wave solutions for pulse propagation in negative index materials (NIMs) in the presence of an external source.
2) It obtains fractional-type solutions containing trigonometric and hyperbolic functions by using a fractional transform to map the governing equation to an elliptic equation.
3) Specific solutions include periodic solutions and bright/dark solitary wave solutions, with the intensity profiles of the bright solitary wave shown.

Manuscript 1334-1

1) The document discusses travelling wave solutions for pulse propagation in negative index materials (NIMs) in the presence of an external source.
2) It obtains fractional-type solutions containing trigonometric and hyperbolic functions by using a fractional transform to map the governing equation to an elliptic equation.
3) Specific solutions include dark/bright solitary waves described by a sech-squared profile, as well as periodic solutions.

2012 mdsp pr12 k means mixture of gaussian

The document provides the course calendar and lecture plan for a machine learning course. The course calendar lists the class dates and topics to be covered from September to January, including Bayes estimation, Kalman filters, particle filters, hidden Markov models, Bayesian decision theory, principal component analysis, and clustering algorithms. The lecture plan focuses on clustering methods, including k-means clustering, mixtures of Gaussians models, and using the expectation-maximization (EM) algorithm to estimate the parameters of Gaussian mixture models.

Machine Learning

The document outlines machine learning topics including k-means clustering and mixtures of Gaussians. It introduces k-means clustering as a method to partition data into K clusters by minimizing distances between points and cluster centers. It also describes mixtures of Gaussians models as a combination of Gaussian distributions that can model complex data distributions by adjusting means, covariances, and mixing coefficients of the Gaussian components. The Expectation-Maximization (EM) algorithm is introduced as a way to estimate parameters in mixtures of Gaussians models.

Monte Caro Simualtions, Sampling and Markov Chain Monte Carlo

Pseudorandom
Pseudorandom The document discusses Monte Carlo methods and Markov chain Monte Carlo (MCMC). It provides examples of using Monte Carlo simulations to estimate pi and solve Buffon's needle problem. It also discusses random walks in Markov chains, the PageRank algorithm used by Google, and challenges with high-dimensional integrals and distributions that do not have a closed-form inverse. MCMC methods are presented as a way to address these challenges.

Ordinary abelian varieties having small embedding degree

International Workshop on Pairings in Cryptography 12-15 June 2005, Dublin, Ireland and
`Mathematical Problems and Techniques in Cryptology' workshop, Barcelona, June 2005
Slides for the 2005 paper: S. D. Galbraith, J. McKee and P. Valenca, "Ordinary abelian varieties having small embedding degree"

How to design a linear control system

How to design a linear control system?
in this article you can learn designing of a linear control system.

The Gaussian Hardy-Littlewood Maximal Function

This document presents a summary of a talk on building a harmonic analytic theory for the Gaussian measure and the Ornstein-Uhlenbeck operator. It discusses how the Gaussian measure is non-doubling but satisfies a local doubling property. It introduces Gaussian cones and shows how they allow proving maximal function estimates for the Ornstein-Uhlenbeck semigroup in a similar way as for the heat semigroup. The talk outlines estimates for the Mehler kernel of the Ornstein-Uhlenbeck semigroup and combines them to obtain boundedness of the maximal function.

Cluster-cluster aggregation with (complete) collisional fragmentation

This document summarizes a presentation on cluster-cluster aggregation models with collisional fragmentation. It discusses mean-field theories of aggregation with a source of monomers and collision-induced fragmentation. Stationary solutions to the Smoluchowski equation are presented for both local and nonlocal aggregation kernels. While stationary nonlocal solutions exist, they are dynamically unstable. Simplified models with complete fragmentation and a source/sink of monomers produce exact solutions analogous to previous work. Nonlocality and the instability of stationary states require further study.

Color Coding-Related Techniques

Narrow sieves, representative sets and divide-and-color are three breakthrough techniques related to color coding, which led to the design of extremely fast parameterized algorithms. In this talk, I will discuss the power and limitations of these techniques. I will also briefly address some recent developments related to these techniques, including general schemes for mixing them.

MLHEP 2015: Introductory Lecture #4

* tuning gradient boosting over decision trees (GBDT)
* speeding up predictions for online triggers: lookup tables
* PCA, autoencoder, manifold learning
* structural learning: Markov chain, LDA
* remarks on collaborative research

Stochastic Approximation and Simulated Annealing

AACIMP 2010 Summer School lecture by Leonidas Sakalauskas. "Applied Mathematics" stream. "Stochastic Programming and Applications" course. Part 8.
More info at http://summerschool.ssa.org.ua

Quantization

The document discusses scalar quantization and the Lloyd-Max algorithm. It provides examples of using the Lloyd-Max algorithm to design scalar quantizers for Gaussian and Laplacian distributed signals. The algorithm works by iteratively calculating decision thresholds and representative levels to minimize mean squared error. At high rates, the distortion-rate function of a Lloyd-Max quantizer is approximated. The document also discusses entropy-constrained scalar quantization and an iterative algorithm to design those quantizers.

Cluster aggregation with complete collisional fragmentation

The document summarizes research on cluster-cluster aggregation (CCA) models where particles stick together upon contact. It discusses mean-field kinetic equations to model CCA with sources and sinks of particles. For the case of complete fragmentation, it presents an exact solution to the kinetic equations. It finds that nonlocal cascades where larger clusters interact mostly with smaller ones can be unstable, leading to oscillatory behavior over time rather than a stationary state. The document outlines approaches to model the nonlocal case using approximations to the Smoluchowski kinetic equation.

Diffraction,unit 2

This document discusses the phenomenon of diffraction - how light bends or spreads when encountering an obstacle or opening. It provides details on diffraction patterns created by single slits, edges, and gratings. Key points covered include the characteristics of diffraction patterns such as bright and dark bands, as well as the differences between Fresnel and Fraunhofer diffraction based on the distances between the light source, obstacle, and viewing screen. Equations for determining the positions of maxima and minima in diffraction patterns are also presented.

The Inverse Smoluchowski Problem, Particles In Turbulence 2011, Potsdam, Marc...

The Inverse Smoluchowski Problem, Particles In Turbulence 2011, Potsdam, Marc...

Manuscript 1334

Manuscript 1334

Manuscript 1334-1

Manuscript 1334-1

2012 mdsp pr12 k means mixture of gaussian

2012 mdsp pr12 k means mixture of gaussian

Machine Learning

Machine Learning

Monte Caro Simualtions, Sampling and Markov Chain Monte Carlo

Monte Caro Simualtions, Sampling and Markov Chain Monte Carlo

Ordinary abelian varieties having small embedding degree

Ordinary abelian varieties having small embedding degree

How to design a linear control system

How to design a linear control system

The Gaussian Hardy-Littlewood Maximal Function

The Gaussian Hardy-Littlewood Maximal Function

Cluster-cluster aggregation with (complete) collisional fragmentation

Cluster-cluster aggregation with (complete) collisional fragmentation

Color Coding-Related Techniques

Color Coding-Related Techniques

MLHEP 2015: Introductory Lecture #4

MLHEP 2015: Introductory Lecture #4

Stochastic Approximation and Simulated Annealing

Stochastic Approximation and Simulated Annealing

Quantization

Quantization

Cluster aggregation with complete collisional fragmentation

Cluster aggregation with complete collisional fragmentation

Diffraction,unit 2

Diffraction,unit 2

GraphRAG for Life Science to increase LLM accuracy

GraphRAG for life science domain, where you retriever information from biomedical knowledge graphs using LLMs to increase the accuracy and performance of generated answers

Taking AI to the Next Level in Manufacturing.pdf

Read Taking AI to the Next Level in Manufacturing to gain insights on AI adoption in the manufacturing industry, such as:
1. How quickly AI is being implemented in manufacturing.
2. Which barriers stand in the way of AI adoption.
3. How data quality and governance form the backbone of AI.
4. Organizational processes and structures that may inhibit effective AI adoption.
6. Ideas and approaches to help build your organization's AI strategy.

Salesforce Integration for Bonterra Impact Management (fka Social Solutions A...

Sidekick Solutions uses Bonterra Impact Management (fka Social Solutions Apricot) and automation solutions to integrate data for business workflows.
We believe integration and automation are essential to user experience and the promise of efficient work through technology. Automation is the critical ingredient to realizing that full vision. We develop integration products and services for Bonterra Case Management software to support the deployment of automations for a variety of use cases.
This video focuses on integration of Salesforce with Bonterra Impact Management.
Interested in deploying an integration with Salesforce for Bonterra Impact Management? Contact us at sales@sidekicksolutionsllc.com to discuss next steps.

June Patch Tuesday

Ivanti’s Patch Tuesday breakdown goes beyond patching your applications and brings you the intelligence and guidance needed to prioritize where to focus your attention first. Catch early analysis on our Ivanti blog, then join industry expert Chris Goettl for the Patch Tuesday Webinar Event. There we’ll do a deep dive into each of the bulletins and give guidance on the risks associated with the newly-identified vulnerabilities.

JavaLand 2024: Application Development Green Masterplan

My presentation slides I used at JavaLand 2024

GNSS spoofing via SDR (Criptored Talks 2024)

In the realm of cybersecurity, offensive security practices act as a critical shield. By simulating real-world attacks in a controlled environment, these techniques expose vulnerabilities before malicious actors can exploit them. This proactive approach allows manufacturers to identify and fix weaknesses, significantly enhancing system security.
This presentation delves into the development of a system designed to mimic Galileo's Open Service signal using software-defined radio (SDR) technology. We'll begin with a foundational overview of both Global Navigation Satellite Systems (GNSS) and the intricacies of digital signal processing.
The presentation culminates in a live demonstration. We'll showcase the manipulation of Galileo's Open Service pilot signal, simulating an attack on various software and hardware systems. This practical demonstration serves to highlight the potential consequences of unaddressed vulnerabilities, emphasizing the importance of offensive security practices in safeguarding critical infrastructure.

leewayhertz.com-AI in predictive maintenance Use cases technologies benefits ...

Predictive maintenance is a proactive approach that anticipates equipment failures before they happen. At the forefront of this innovative strategy is Artificial Intelligence (AI), which brings unprecedented precision and efficiency. AI in predictive maintenance is transforming industries by reducing downtime, minimizing costs, and enhancing productivity.

A Comprehensive Guide to DeFi Development Services in 2024

DeFi represents a paradigm shift in the financial industry. Instead of relying on traditional, centralized institutions like banks, DeFi leverages blockchain technology to create a decentralized network of financial services. This means that financial transactions can occur directly between parties, without intermediaries, using smart contracts on platforms like Ethereum.
In 2024, we are witnessing an explosion of new DeFi projects and protocols, each pushing the boundaries of what’s possible in finance.
In summary, DeFi in 2024 is not just a trend; it’s a revolution that democratizes finance, enhances security and transparency, and fosters continuous innovation. As we proceed through this presentation, we'll explore the various components and services of DeFi in detail, shedding light on how they are transforming the financial landscape.
At Intelisync, we specialize in providing comprehensive DeFi development services tailored to meet the unique needs of our clients. From smart contract development to dApp creation and security audits, we ensure that your DeFi project is built with innovation, security, and scalability in mind. Trust Intelisync to guide you through the intricate landscape of decentralized finance and unlock the full potential of blockchain technology.
Ready to take your DeFi project to the next level? Partner with Intelisync for expert DeFi development services today!

Introduction of Cybersecurity with OSS at Code Europe 2024

I develop the Ruby programming language, RubyGems, and Bundler, which are package managers for Ruby. Today, I will introduce how to enhance the security of your application using open-source software (OSS) examples from Ruby and RubyGems.
The first topic is CVE (Common Vulnerabilities and Exposures). I have published CVEs many times. But what exactly is a CVE? I'll provide a basic understanding of CVEs and explain how to detect and handle vulnerabilities in OSS.
Next, let's discuss package managers. Package managers play a critical role in the OSS ecosystem. I'll explain how to manage library dependencies in your application.
I'll share insights into how the Ruby and RubyGems core team works to keep our ecosystem safe. By the end of this talk, you'll have a better understanding of how to safeguard your code.

Driving Business Innovation: Latest Generative AI Advancements & Success Story

Are you ready to revolutionize how you handle data? Join us for a webinar where we’ll bring you up to speed with the latest advancements in Generative AI technology and discover how leveraging FME with tools from giants like Google Gemini, Amazon, and Microsoft OpenAI can supercharge your workflow efficiency.
During the hour, we’ll take you through:
Guest Speaker Segment with Hannah Barrington: Dive into the world of dynamic real estate marketing with Hannah, the Marketing Manager at Workspace Group. Hear firsthand how their team generates engaging descriptions for thousands of office units by integrating diverse data sources—from PDF floorplans to web pages—using FME transformers, like OpenAIVisionConnector and AnthropicVisionConnector. This use case will show you how GenAI can streamline content creation for marketing across the board.
Ollama Use Case: Learn how Scenario Specialist Dmitri Bagh has utilized Ollama within FME to input data, create custom models, and enhance security protocols. This segment will include demos to illustrate the full capabilities of FME in AI-driven processes.
Custom AI Models: Discover how to leverage FME to build personalized AI models using your data. Whether it’s populating a model with local data for added security or integrating public AI tools, find out how FME facilitates a versatile and secure approach to AI.
We’ll wrap up with a live Q&A session where you can engage with our experts on your specific use cases, and learn more about optimizing your data workflows with AI.
This webinar is ideal for professionals seeking to harness the power of AI within their data management systems while ensuring high levels of customization and security. Whether you're a novice or an expert, gain actionable insights and strategies to elevate your data processes. Join us to see how FME and AI can revolutionize how you work with data!

Monitoring and Managing Anomaly Detection on OpenShift.pdf

Monitoring and Managing Anomaly Detection on OpenShift
Overview
Dive into the world of anomaly detection on edge devices with our comprehensive hands-on tutorial. This SlideShare presentation will guide you through the entire process, from data collection and model training to edge deployment and real-time monitoring. Perfect for those looking to implement robust anomaly detection systems on resource-constrained IoT/edge devices.
Key Topics Covered
1. Introduction to Anomaly Detection
- Understand the fundamentals of anomaly detection and its importance in identifying unusual behavior or failures in systems.
2. Understanding Edge (IoT)
- Learn about edge computing and IoT, and how they enable real-time data processing and decision-making at the source.
3. What is ArgoCD?
- Discover ArgoCD, a declarative, GitOps continuous delivery tool for Kubernetes, and its role in deploying applications on edge devices.
4. Deployment Using ArgoCD for Edge Devices
- Step-by-step guide on deploying anomaly detection models on edge devices using ArgoCD.
5. Introduction to Apache Kafka and S3
- Explore Apache Kafka for real-time data streaming and Amazon S3 for scalable storage solutions.
6. Viewing Kafka Messages in the Data Lake
- Learn how to view and analyze Kafka messages stored in a data lake for better insights.
7. What is Prometheus?
- Get to know Prometheus, an open-source monitoring and alerting toolkit, and its application in monitoring edge devices.
8. Monitoring Application Metrics with Prometheus
- Detailed instructions on setting up Prometheus to monitor the performance and health of your anomaly detection system.
9. What is Camel K?
- Introduction to Camel K, a lightweight integration framework built on Apache Camel, designed for Kubernetes.
10. Configuring Camel K Integrations for Data Pipelines
- Learn how to configure Camel K for seamless data pipeline integrations in your anomaly detection workflow.
11. What is a Jupyter Notebook?
- Overview of Jupyter Notebooks, an open-source web application for creating and sharing documents with live code, equations, visualizations, and narrative text.
12. Jupyter Notebooks with Code Examples
- Hands-on examples and code snippets in Jupyter Notebooks to help you implement and test anomaly detection models.

Skybuffer SAM4U tool for SAP license adoption

Manage and optimize your license adoption and consumption with SAM4U, an SAP free customer software asset management tool.
SAM4U, an SAP complimentary software asset management tool for customers, delivers a detailed and well-structured overview of license inventory and usage with a user-friendly interface. We offer a hosted, cost-effective, and performance-optimized SAM4U setup in the Skybuffer Cloud environment. You retain ownership of the system and data, while we manage the ABAP 7.58 infrastructure, ensuring fixed Total Cost of Ownership (TCO) and exceptional services through the SAP Fiori interface.

Nordic Marketo Engage User Group_June 13_ 2024.pptx

Slides from event

Presentation of the OECD Artificial Intelligence Review of Germany

Consult the full report at https://www.oecd.org/digital/oecd-artificial-intelligence-review-of-germany-609808d6-en.htm

Best 20 SEO Techniques To Improve Website Visibility In SERP

Boost your website's visibility with proven SEO techniques! Our latest blog dives into essential strategies to enhance your online presence, increase traffic, and rank higher on search engines. From keyword optimization to quality content creation, learn how to make your site stand out in the crowded digital landscape. Discover actionable tips and expert insights to elevate your SEO game.

How to Interpret Trends in the Kalyan Rajdhani Mix Chart.pdf

A Mix Chart displays historical data of numbers in a graphical or tabular form. The Kalyan Rajdhani Mix Chart specifically shows the results of a sequence of numbers over different periods.

5th LF Energy Power Grid Model Meet-up Slides

5th Power Grid Model Meet-up
It is with great pleasure that we extend to you an invitation to the 5th Power Grid Model Meet-up, scheduled for 6th June 2024. This event will adopt a hybrid format, allowing participants to join us either through an online Mircosoft Teams session or in person at TU/e located at Den Dolech 2, Eindhoven, Netherlands. The meet-up will be hosted by Eindhoven University of Technology (TU/e), a research university specializing in engineering science & technology.
Power Grid Model
The global energy transition is placing new and unprecedented demands on Distribution System Operators (DSOs). Alongside upgrades to grid capacity, processes such as digitization, capacity optimization, and congestion management are becoming vital for delivering reliable services.
Power Grid Model is an open source project from Linux Foundation Energy and provides a calculation engine that is increasingly essential for DSOs. It offers a standards-based foundation enabling real-time power systems analysis, simulations of electrical power grids, and sophisticated what-if analysis. In addition, it enables in-depth studies and analysis of the electrical power grid’s behavior and performance. This comprehensive model incorporates essential factors such as power generation capacity, electrical losses, voltage levels, power flows, and system stability.
Power Grid Model is currently being applied in a wide variety of use cases, including grid planning, expansion, reliability, and congestion studies. It can also help in analyzing the impact of renewable energy integration, assessing the effects of disturbances or faults, and developing strategies for grid control and optimization.
What to expect
For the upcoming meetup we are organizing, we have an exciting lineup of activities planned:
-Insightful presentations covering two practical applications of the Power Grid Model.
-An update on the latest advancements in Power Grid -Model technology during the first and second quarters of 2024.
-An interactive brainstorming session to discuss and propose new feature requests.
-An opportunity to connect with fellow Power Grid Model enthusiasts and users.

WeTestAthens: Postman's AI & Automation Techniques

Postman's AI and Automation Techniques

GraphRAG for Life Science to increase LLM accuracy

GraphRAG for Life Science to increase LLM accuracy

Taking AI to the Next Level in Manufacturing.pdf

Taking AI to the Next Level in Manufacturing.pdf

Overcoming the PLG Trap: Lessons from Canva's Head of Sales & Head of EMEA Da...

Overcoming the PLG Trap: Lessons from Canva's Head of Sales & Head of EMEA Da...

Salesforce Integration for Bonterra Impact Management (fka Social Solutions A...

Salesforce Integration for Bonterra Impact Management (fka Social Solutions A...

June Patch Tuesday

June Patch Tuesday

JavaLand 2024: Application Development Green Masterplan

JavaLand 2024: Application Development Green Masterplan

GNSS spoofing via SDR (Criptored Talks 2024)

GNSS spoofing via SDR (Criptored Talks 2024)

leewayhertz.com-AI in predictive maintenance Use cases technologies benefits ...

leewayhertz.com-AI in predictive maintenance Use cases technologies benefits ...

A Comprehensive Guide to DeFi Development Services in 2024

A Comprehensive Guide to DeFi Development Services in 2024

Deep Dive: AI-Powered Marketing to Get More Leads and Customers with HyperGro...

Deep Dive: AI-Powered Marketing to Get More Leads and Customers with HyperGro...

Introduction of Cybersecurity with OSS at Code Europe 2024

Introduction of Cybersecurity with OSS at Code Europe 2024

Driving Business Innovation: Latest Generative AI Advancements & Success Story

Driving Business Innovation: Latest Generative AI Advancements & Success Story

Monitoring and Managing Anomaly Detection on OpenShift.pdf

Monitoring and Managing Anomaly Detection on OpenShift.pdf

Skybuffer SAM4U tool for SAP license adoption

Skybuffer SAM4U tool for SAP license adoption

Nordic Marketo Engage User Group_June 13_ 2024.pptx

Nordic Marketo Engage User Group_June 13_ 2024.pptx

Presentation of the OECD Artificial Intelligence Review of Germany

Presentation of the OECD Artificial Intelligence Review of Germany

Best 20 SEO Techniques To Improve Website Visibility In SERP

Best 20 SEO Techniques To Improve Website Visibility In SERP

How to Interpret Trends in the Kalyan Rajdhani Mix Chart.pdf

How to Interpret Trends in the Kalyan Rajdhani Mix Chart.pdf

5th LF Energy Power Grid Model Meet-up Slides

5th LF Energy Power Grid Model Meet-up Slides

WeTestAthens: Postman's AI & Automation Techniques

WeTestAthens: Postman's AI & Automation Techniques

- 1. Expectation Maximization and Mixture of Gaussians 1
- 2. (bpm 125) Recommend me Bpm some music! 90! Discover groups of similar songs… Only my railgun (bpm Bach Sonata 120) #1 (bpm 60) My Music Collection 2
- 3. (bpm 125) Recommend me some music! bpm Discover groups 120 of similar songs… Only my railgun (bpm Bach Sonata 120) #1 (bpm 60) My Music Collection bpm 60 3
- 4. An unsupervised classifying method 4
- 5. 1. Initialize K “means” µk , one for each class µ1 Eg. Use random starting points, or € choose k random € µ2 points from the set €K=2 5
- 6. 1 0 2. Phase 1: Assign each point to closest mean µk 3. Phase 2: Update means of the new clusters € 6
- 7. 2. Phase 1: Assign each point to closest mean µk 3. Phase 2: Update means of the new clusters € 0 1 7
- 8. 2. Phase 1: Assign each point to closest mean 3. Phase 2: Update means of the new clusters 8
- 9. 2. Phase 1: Assign each point to closest mean 3. Phase 2: Update means of the new clusters 9
- 10. 2. Phase 1: Assign each point to closest mean 3. Phase 2: Update means of the new clusters 10
- 11. 0 1 2. Phase 1: Assign each point to closest mean µk 3. Phase 2: Update means of the new clusters € 11
- 12. 2. Phase 1: Assign each point to closest mean 3. Phase 2: Update means of the new clusters 12
- 13. 2. Phase 1: Assign each point to closest mean µk 3. Phase 2: Update means of the new clusters € 13
- 14. 2. Phase 1: Assign each point to closest mean 3. Phase 2: Update means of the new clusters 14
- 15. 4. When means do not change anymore clustering DONE. 15
- 16. InK-means, a point can only have 1 class But what about points that lie in between groups? eg. Jazz + Classical 16
- 17. The Famous “GMM”: Gaussian Mixture Model 17
- 18. Mean p(X) = N(X | µ,Σ) Variance Gaussian == “Normal” distribution 18
- 19. p(X) = N(X | µ,Σ) + N(X | µ,Σ) 19
- 20. p(X) = N(X | µ1,Σ1 ) + N(X | µ2 ,Σ 2 ) Example: Variance 20
- 21. p(X) = π 1N(X | µ1,Σ1 ) + π 2 N(X | µ2 ,Σ 2 ) k Example: Mixing Coefficient ∑π k =1 k=1 € π 1 = 0.7 π 2 = 0.3 21
- 22. K p(X) = ∑ π k N(X | µk ,Σ k ) k=1 Example: K =2 € € 22
- 23. K-means is a Mixture of classifier Gaussians is a probability model We can USE it as a “soft” classifier 23
- 24. K-means is a Mixture of classifier Gaussians is a probability model We can USE it as a “soft” classifier 24
- 25. K-means is a Mixture of classifier Gaussians is a probability model We can USE it as a “soft” classifier Parameter to fit to data: Parameters to fit to data: • Mean µk • Mean µk • Covariance Σ k • Mixing coefficient π k € € 25 €
- 26. EM for GMM 26
- 27. 1. Initialize means µk 1 0 2. E Step: Assign each point to a cluster 3. M Step: Given clusters, refine mean µk of each cluster k 4. Stop when change in means is small € € 27
- 28. 1. Initialize Gaussian* parameters: means µk , covariances Σ k and mixing coefficients π k 2. E Step: Assign each point Xn an assignment score γ (znk ) for each cluster k 0.5 0.5 3. M Step: Given scores, adjust µk ,€ k ,Σ k π for€each cluster k € 4. Evaluate € likelihood. If likelihood or parameters converge, stop. € € € *There are k Gaussians 28
- 29. 1. Initialize µk , Σk π k , one for each Gaussian k € π2 Σ2 Tip! Use K-means € € result to initialize: µ2 µk ← µk Σk ← cov(cluster(K)) € € π k ← Number of pointspoints in k € Total number of 29 €
- 30. Latent variable 2. E Step: For each .7 .3 point Xn, determine its assignment score to each Gaussian k: is called a “responsibility”: how much is this Gaussian k γ (znk ) responsible for this point Xn? 30
- 31. 3. M Step: For each Gaussian k, update parameters using new γ (znk ) Responsibility for this Xn Mean of Gaussian k € Find the mean that “fits” the assignment scores best 31
- 32. 3. M Step: For each Gaussian k, update parameters using new γ (znk ) Covariance matrix € of Gaussian k Just calculated this! 32
- 33. 3. M Step: For each Gaussian k, update parameters using new γ (znk ) Mixing Coefficient € eg. 105.6/200 for Gaussian k Total # of points 33
- 34. 4. Evaluate log likelihood. If likelihood or parameters converge, stop. Else go to Step 2 (E step). Likelihood is the probability that the data X was generated by the parameters you found. ie. Correctness! 34
- 35. 35
- 36. old Hidden 1. Initialize parameters θ variables old 2. E Step: Evaluate p(Z | X,θ ) 3. M Step: Evaluate Observed variables € € Likelihood where 4. Evaluate log likelihood. If likelihood or parameters converge, stop. Else θ old ← θ new and go to E Step. 36
- 37. K-means can be formulated as EM EM for Gaussian Mixtures EM for Bernoulli Mixtures EM for Bayesian Linear Regression 37
- 38. “Expectation” Calculated the fixed, data-dependent parameters of the function Q. “Maximization” Once the parameters of Q are known, it is fully determined, so now we can maximize Q. 38
- 39. We learned how to cluster data in an unsupervised manner Gaussian Mixture Models are useful for modeling data with “soft” cluster assignments Expectation Maximization is a method used when we have a model with latent variables (values we don’t know, but estimate with each step) 0.5 0.5 39
- 40. Myquestion: What other applications could use EM? How about EM of GMMs? 40