The document discusses univariate and multivariate extreme value theory. It introduces limit probabilities for maxima and maximum domains of attraction in univariate extreme value theory. It also discusses limit distributions of multivariate maxima and the multivariate domain of attraction. The outline previews that the document will cover introduction to limits of maxima, univariate extreme value theory, and multivariate extreme value theory.
This document discusses key concepts in probability theory, including:
1) Markov's inequality and Chebyshev's inequality, which relate the probability that a random variable exceeds a value to its expected value and variance.
2) The weak law of large numbers and central limit theorem, which describe how the means of independent random variables converge to the expected value and follow a normal distribution as the number of variables increases.
3) Stochastic processes, which are collections of random variables indexed by time or another parameter and can model evolving systems. Examples of stochastic processes and their properties are provided.
1. The document discusses maximum likelihood estimation and Bayesian parameter estimation for machine learning problems involving parametric densities like the Gaussian.
2. Maximum likelihood estimation finds the parameter values that maximize the probability of obtaining the observed training data. For Gaussian distributions with unknown mean and variance, MLE returns the sample mean and variance.
3. Bayesian parameter estimation treats the parameters as random variables and uses prior distributions and observed data to obtain posterior distributions over the parameters. This allows incorporation of prior knowledge with the training data.
There are various reasons why we would want to find the extreme (maximum and minimum values) of a function. Fermat's Theorem tells us we can find local extreme points by looking at critical points. This process is known as the Closed Interval Method.
This document defines and provides examples of expectation, or the average value, of random variables. It discusses properties of expectations including how the expectation of a function of a random variable is calculated. It also defines and gives properties of variance, covariance, conditional expectation, and conditional variance. Examples are provided throughout to illustrate key concepts.
This document provides an overview of probability theory concepts related to random variables. It defines random variables and their probability mass functions and cumulative distribution functions. It describes different types of random variables including discrete, continuous, Bernoulli, binomial, geometric, Poisson, uniform, exponential, gamma, and normal random variables. It also covers concepts of joint and marginal distributions as well as independent and conditional random variables. The document uses mathematical notation to formally define these concepts.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
This document discusses several topics related to Fourier transforms including:
1) Representing polynomials in value representation by evaluating them at roots of unity allows for faster multiplication using the Discrete Fourier Transform (DFT).
2) The DFT reduces the complexity of the Discrete Fourier Transform (DFT) from O(n2) to O(n log n) by formulating it recursively.
3) Converting images from the spatial to frequency domain using techniques like the Discrete Cosine Transform (DCT) allows for image compression by retaining only low frequency components with large coefficients.
Lesson 20: Derivatives and the Shapes of Curves (slides)Matthew Leingang
This document contains lecture notes on derivatives and the shapes of curves from a Calculus I class taught by Professor Matthew Leingang at New York University. The notes cover using derivatives to determine the intervals where a function is increasing or decreasing, classifying critical points as maxima or minima, using the second derivative to determine concavity, and applying the first and second derivative tests. Examples are provided to illustrate finding intervals of monotonicity for various functions.
This document discusses key concepts in probability theory, including:
1) Markov's inequality and Chebyshev's inequality, which relate the probability that a random variable exceeds a value to its expected value and variance.
2) The weak law of large numbers and central limit theorem, which describe how the means of independent random variables converge to the expected value and follow a normal distribution as the number of variables increases.
3) Stochastic processes, which are collections of random variables indexed by time or another parameter and can model evolving systems. Examples of stochastic processes and their properties are provided.
1. The document discusses maximum likelihood estimation and Bayesian parameter estimation for machine learning problems involving parametric densities like the Gaussian.
2. Maximum likelihood estimation finds the parameter values that maximize the probability of obtaining the observed training data. For Gaussian distributions with unknown mean and variance, MLE returns the sample mean and variance.
3. Bayesian parameter estimation treats the parameters as random variables and uses prior distributions and observed data to obtain posterior distributions over the parameters. This allows incorporation of prior knowledge with the training data.
There are various reasons why we would want to find the extreme (maximum and minimum values) of a function. Fermat's Theorem tells us we can find local extreme points by looking at critical points. This process is known as the Closed Interval Method.
This document defines and provides examples of expectation, or the average value, of random variables. It discusses properties of expectations including how the expectation of a function of a random variable is calculated. It also defines and gives properties of variance, covariance, conditional expectation, and conditional variance. Examples are provided throughout to illustrate key concepts.
This document provides an overview of probability theory concepts related to random variables. It defines random variables and their probability mass functions and cumulative distribution functions. It describes different types of random variables including discrete, continuous, Bernoulli, binomial, geometric, Poisson, uniform, exponential, gamma, and normal random variables. It also covers concepts of joint and marginal distributions as well as independent and conditional random variables. The document uses mathematical notation to formally define these concepts.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
This document discusses several topics related to Fourier transforms including:
1) Representing polynomials in value representation by evaluating them at roots of unity allows for faster multiplication using the Discrete Fourier Transform (DFT).
2) The DFT reduces the complexity of the Discrete Fourier Transform (DFT) from O(n2) to O(n log n) by formulating it recursively.
3) Converting images from the spatial to frequency domain using techniques like the Discrete Cosine Transform (DCT) allows for image compression by retaining only low frequency components with large coefficients.
Lesson 20: Derivatives and the Shapes of Curves (slides)Matthew Leingang
This document contains lecture notes on derivatives and the shapes of curves from a Calculus I class taught by Professor Matthew Leingang at New York University. The notes cover using derivatives to determine the intervals where a function is increasing or decreasing, classifying critical points as maxima or minima, using the second derivative to determine concavity, and applying the first and second derivative tests. Examples are provided to illustrate finding intervals of monotonicity for various functions.
The document discusses evaluating definite integrals. It begins by reviewing the definition of the definite integral as a limit and properties of integrals such as additivity. It then covers estimating integrals using the Midpoint Rule and properties for comparing integrals. Examples are provided of evaluating definite integrals using known formulas or the Midpoint Rule. The integral is discussed as computing the total change, and an outline of future topics like indefinite integrals and computing area is presented.
1. The progress report outlines work on a discriminatively trained, multiscale, deformable part model for object detection.
2. Modifications are proposed to optimize the model's function, use lower dimensional but more informative features, and predict bounding boxes.
3. Adding contextual information is also discussed to help rescore detections using surrounding detections from other models.
Lesson 15: Exponential Growth and Decay (slides)Matthew Leingang
Many problems in nature are expressible in terms of a certain differential equation that has a solution in terms of exponential functions. We look at the equation in general and some fun applications, including radioactivity, cooling, and interest.
The document discusses the principle of maximum entropy. It explains that maximum entropy is an approach for making probability assignments where the assigned probability distribution should have the largest entropy or uncertainty possible, subject to whatever information is known. It describes applications of maximum entropy modeling such as part-of-speech tagging and logistic regression. Maximum entropy and maximum likelihood methods are related as they both aim to make distributions as uniform as possible based on available information.
The document introduces a new distance measure between probability density functions called the Laplacian PDF distance. This distance measure has a connection to kernel-based learning theory via the Parzen window technique for density estimation. In a kernel feature space defined by the eigenspectrum of the Laplacian data matrix, the Laplacian PDF distance is shown to measure the cosine of the angle between cluster mean vectors. The Laplacian data matrix and its eigenspectrum can be obtained automatically based on the data, allowing the feature space mapping to be determined in an unsupervised manner.
The document provides a review outline for Midterm I in Math 1a. It includes the following topics:
- The Intermediate Value Theorem
- Limits (concept, computation, limits involving infinity)
- Continuity (concept, examples)
- Derivatives (concept, interpretations, implications, computation)
- It also provides learning objectives and outlines for each topic.
Image sciences, image processing, image restoration, photo manipulation. Image and videos representation. Digital versus analog imagery. Quantization and sampling. Sources and models of noises in digital CCD imagery: photon, thermal and readout noises. Sources and models of blurs. Convolutions and point spread functions. Overview of other standard models, problems and tasks: salt-and-pepper and impulse noises, half toning, inpainting, super-resolution, compressed sensing, high dynamic range imagery, demosaicing. Short introduction to other types of imagery: SAR, Sonar, ultrasound, CT and MRI. Linear and ill-posed restoration problems.
The document discusses the concept of PAC (Probably Approximately Correct) learning. It begins by describing a learning scenario where a hidden hypothesis is chosen by nature, and a learner tries to approximate this hypothesis based on randomly generated training data. It then defines what it means for a learned hypothesis to be "bad" or have high test error, and shows that by choosing a large enough random training set, the probability of learning a bad hypothesis can be bounded. Finally, it provides the formula for calculating the minimum size of the random training set needed to guarantee this probability bound.
The document discusses probability distributions and their natural parameters. It provides examples of several common distributions including the Bernoulli, multinomial, Gaussian, and gamma distributions. For each distribution, it derives the natural parameter representation and shows how to write the distribution in the form p(x|η) = h(x)g(η)exp{η^T μ(x)}. Maximum likelihood estimation for these distributions is also briefly discussed.
The closed interval method tells us how to find the extreme values of a continuous function defined on a closed, bounded interval: we check the end points and the critical points.
Higher-order (F, α, β, ρ, d)-convexity is considered. A multiobjective programming problem (MP) is considered. Mond-Weir and Wolfe type duals are considered for multiobjective programming problem. Duality results are established for multiobjective programming problem under higher-order (F, α, β, ρ, d)- convexity assumptions. The results are also applied for multiobjective fractional programming problem.
This document discusses likelihood methods for continuous-time models in finance. It describes approximating the transition density function pX of a continuous-time process through a series of transformations to get closer to a normal distribution. This allows representing pX as a series expansion involving Hermite polynomials. Computing the expansion coefficients allows obtaining an explicit closed-form approximation to pX. Maximizing the approximate likelihood results in an estimator that converges to the true MLE as the number of terms increases.
This document presents a modified Mann iteration method for finding common fixed points of a countable family of multivalued mappings in Banach spaces. It introduces using the best approximation operator PTn to define the iteration (1.2), where xn+1 is defined as a convex combination of the previous iterate xn and an element in PTn xn . The paper aims to establish weak and strong convergence theorems for this iterative method to find a common fixed point for a countable family of nonexpansive multivalued mappings. It provides relevant background on fixed points, nonexpansive mappings, and best approximation operators in Banach and Hilbert spaces.
The document discusses using the derivative to determine whether a function is increasing or decreasing over an interval. It provides examples of using the sign of the derivative to determine if a function is increasing or decreasing. It also discusses using the second derivative test to determine if a stationary point is a relative maximum or minimum. Specifically:
- The sign of the derivative indicates whether the function is increasing or decreasing over an interval. Positive derivative means increasing, negative means decreasing.
- Stationary points where the derivative is zero require the second derivative test to determine if it is a relative maximum or minimum. Positive second derivative means a relative minimum, negative means a maximum.
- Examples demonstrate finding stationary points, using the first and second derivative
This document summarizes results on analyzing stochastic gradient descent (SGD) algorithms for minimizing convex functions. It shows that a continuous-time version of SGD (SGD-c) can strongly approximate the discrete-time version (SGD-d) under certain conditions. It also establishes that SGD achieves the minimax optimal convergence rate of O(t^-1/2) for α=1/2 by using an "averaging from the past" procedure, closing the gap between previous lower and upper bound results.
Global illumination techniques for the computation of hight quality images in...Frederic Perez
This document appears to be a dissertation defense presentation summarizing work on rendering high quality images accounting for global illumination in general environments, including participating media. The presentation covers global illumination fundamentals, resolution methods for participating media, two first pass methods for solving global illumination, link probabilities for importance sampling, and progressive radiance computation methods. It aims to render high quality images for general scenes potentially including participating media and general optical properties.
"PAC Learning - a discussion on the original paper by Valiant" presentation @...Adrian Florea
This document discusses PAC (probably approximately correct) learning, which was introduced in Valiant's 1984 paper. It defines key concepts in PAC learning like concepts, concept classes, learning algorithms, hypothesis spaces, and error rates. It also proves theorems like the theorem of ε-exhausting the version space, which shows that the number of training examples needed is logarithmic in the size of the hypothesis space. As an example, it shows that learning conjunctions of Boolean literals is PAC learnable, while learning all concepts is not PAC learnable.
This document contains notes from a Stat310 class on moments. It discusses the definition of moments, including that the mean is the first moment and variance is the second central moment. It provides formulas for the mean, variance, skewness, and kurtosis. Examples of different distributions are shown graphically and their properties discussed, including the binomial and Poisson distributions. The moment generating function is introduced and how it can be used to find the mean and variance. Students are asked to practice computing these properties for different distributions.
O documento discute os desafios da educação baseada em leitura e escrita em contraste com as novas mídias digitais. Também alerta sobre os riscos do plágio na internet e da utilização indevida da Wikipedia em trabalhos acadêmicos. Finalmente, pede aos estudantes para refletirem sobre ganhos e perdas da leitura e escrita digital.
The document discusses evaluating definite integrals. It begins by reviewing the definition of the definite integral as a limit and properties of integrals such as additivity. It then covers estimating integrals using the Midpoint Rule and properties for comparing integrals. Examples are provided of evaluating definite integrals using known formulas or the Midpoint Rule. The integral is discussed as computing the total change, and an outline of future topics like indefinite integrals and computing area is presented.
1. The progress report outlines work on a discriminatively trained, multiscale, deformable part model for object detection.
2. Modifications are proposed to optimize the model's function, use lower dimensional but more informative features, and predict bounding boxes.
3. Adding contextual information is also discussed to help rescore detections using surrounding detections from other models.
Lesson 15: Exponential Growth and Decay (slides)Matthew Leingang
Many problems in nature are expressible in terms of a certain differential equation that has a solution in terms of exponential functions. We look at the equation in general and some fun applications, including radioactivity, cooling, and interest.
The document discusses the principle of maximum entropy. It explains that maximum entropy is an approach for making probability assignments where the assigned probability distribution should have the largest entropy or uncertainty possible, subject to whatever information is known. It describes applications of maximum entropy modeling such as part-of-speech tagging and logistic regression. Maximum entropy and maximum likelihood methods are related as they both aim to make distributions as uniform as possible based on available information.
The document introduces a new distance measure between probability density functions called the Laplacian PDF distance. This distance measure has a connection to kernel-based learning theory via the Parzen window technique for density estimation. In a kernel feature space defined by the eigenspectrum of the Laplacian data matrix, the Laplacian PDF distance is shown to measure the cosine of the angle between cluster mean vectors. The Laplacian data matrix and its eigenspectrum can be obtained automatically based on the data, allowing the feature space mapping to be determined in an unsupervised manner.
The document provides a review outline for Midterm I in Math 1a. It includes the following topics:
- The Intermediate Value Theorem
- Limits (concept, computation, limits involving infinity)
- Continuity (concept, examples)
- Derivatives (concept, interpretations, implications, computation)
- It also provides learning objectives and outlines for each topic.
Image sciences, image processing, image restoration, photo manipulation. Image and videos representation. Digital versus analog imagery. Quantization and sampling. Sources and models of noises in digital CCD imagery: photon, thermal and readout noises. Sources and models of blurs. Convolutions and point spread functions. Overview of other standard models, problems and tasks: salt-and-pepper and impulse noises, half toning, inpainting, super-resolution, compressed sensing, high dynamic range imagery, demosaicing. Short introduction to other types of imagery: SAR, Sonar, ultrasound, CT and MRI. Linear and ill-posed restoration problems.
The document discusses the concept of PAC (Probably Approximately Correct) learning. It begins by describing a learning scenario where a hidden hypothesis is chosen by nature, and a learner tries to approximate this hypothesis based on randomly generated training data. It then defines what it means for a learned hypothesis to be "bad" or have high test error, and shows that by choosing a large enough random training set, the probability of learning a bad hypothesis can be bounded. Finally, it provides the formula for calculating the minimum size of the random training set needed to guarantee this probability bound.
The document discusses probability distributions and their natural parameters. It provides examples of several common distributions including the Bernoulli, multinomial, Gaussian, and gamma distributions. For each distribution, it derives the natural parameter representation and shows how to write the distribution in the form p(x|η) = h(x)g(η)exp{η^T μ(x)}. Maximum likelihood estimation for these distributions is also briefly discussed.
The closed interval method tells us how to find the extreme values of a continuous function defined on a closed, bounded interval: we check the end points and the critical points.
Higher-order (F, α, β, ρ, d)-convexity is considered. A multiobjective programming problem (MP) is considered. Mond-Weir and Wolfe type duals are considered for multiobjective programming problem. Duality results are established for multiobjective programming problem under higher-order (F, α, β, ρ, d)- convexity assumptions. The results are also applied for multiobjective fractional programming problem.
This document discusses likelihood methods for continuous-time models in finance. It describes approximating the transition density function pX of a continuous-time process through a series of transformations to get closer to a normal distribution. This allows representing pX as a series expansion involving Hermite polynomials. Computing the expansion coefficients allows obtaining an explicit closed-form approximation to pX. Maximizing the approximate likelihood results in an estimator that converges to the true MLE as the number of terms increases.
This document presents a modified Mann iteration method for finding common fixed points of a countable family of multivalued mappings in Banach spaces. It introduces using the best approximation operator PTn to define the iteration (1.2), where xn+1 is defined as a convex combination of the previous iterate xn and an element in PTn xn . The paper aims to establish weak and strong convergence theorems for this iterative method to find a common fixed point for a countable family of nonexpansive multivalued mappings. It provides relevant background on fixed points, nonexpansive mappings, and best approximation operators in Banach and Hilbert spaces.
The document discusses using the derivative to determine whether a function is increasing or decreasing over an interval. It provides examples of using the sign of the derivative to determine if a function is increasing or decreasing. It also discusses using the second derivative test to determine if a stationary point is a relative maximum or minimum. Specifically:
- The sign of the derivative indicates whether the function is increasing or decreasing over an interval. Positive derivative means increasing, negative means decreasing.
- Stationary points where the derivative is zero require the second derivative test to determine if it is a relative maximum or minimum. Positive second derivative means a relative minimum, negative means a maximum.
- Examples demonstrate finding stationary points, using the first and second derivative
This document summarizes results on analyzing stochastic gradient descent (SGD) algorithms for minimizing convex functions. It shows that a continuous-time version of SGD (SGD-c) can strongly approximate the discrete-time version (SGD-d) under certain conditions. It also establishes that SGD achieves the minimax optimal convergence rate of O(t^-1/2) for α=1/2 by using an "averaging from the past" procedure, closing the gap between previous lower and upper bound results.
Global illumination techniques for the computation of hight quality images in...Frederic Perez
This document appears to be a dissertation defense presentation summarizing work on rendering high quality images accounting for global illumination in general environments, including participating media. The presentation covers global illumination fundamentals, resolution methods for participating media, two first pass methods for solving global illumination, link probabilities for importance sampling, and progressive radiance computation methods. It aims to render high quality images for general scenes potentially including participating media and general optical properties.
"PAC Learning - a discussion on the original paper by Valiant" presentation @...Adrian Florea
This document discusses PAC (probably approximately correct) learning, which was introduced in Valiant's 1984 paper. It defines key concepts in PAC learning like concepts, concept classes, learning algorithms, hypothesis spaces, and error rates. It also proves theorems like the theorem of ε-exhausting the version space, which shows that the number of training examples needed is logarithmic in the size of the hypothesis space. As an example, it shows that learning conjunctions of Boolean literals is PAC learnable, while learning all concepts is not PAC learnable.
This document contains notes from a Stat310 class on moments. It discusses the definition of moments, including that the mean is the first moment and variance is the second central moment. It provides formulas for the mean, variance, skewness, and kurtosis. Examples of different distributions are shown graphically and their properties discussed, including the binomial and Poisson distributions. The moment generating function is introduced and how it can be used to find the mean and variance. Students are asked to practice computing these properties for different distributions.
O documento discute os desafios da educação baseada em leitura e escrita em contraste com as novas mídias digitais. Também alerta sobre os riscos do plágio na internet e da utilização indevida da Wikipedia em trabalhos acadêmicos. Finalmente, pede aos estudantes para refletirem sobre ganhos e perdas da leitura e escrita digital.
This document summarizes key concepts from Chapter 4 of the textbook, which introduces multivariate probability and related topics. It begins by explaining that multivariate probability generalizes the treatment of random variables from a single variable X to multiple variables X1, X2, ..., Xm. It then discusses multivariate probability density functions and how to reduce the dimensionality through marginal and conditional probability density functions. Finally, it covers topics like moments, independence, correlation, and the multivariate uniform distribution.
The document discusses using Taylor series approximations to evaluate expressions. It finds the maximum error of approximating e-0.5 as 1 - 0.5 + 0.5^2, determines the value of x for which this approximation is accurate to within 0.01, and establishes that an 8th degree Taylor polynomial is needed to approximate e^x in [-1,1] with an error less than 0.001.
This document discusses moment generating functions (MGFs), which are defined as the expectation of e^tx, where x is a random variable and t is a real number for which the expectation is finite. The MGF completely determines the distribution of a random variable. Higher moments describe properties like symmetry, peakedness, and kurtosis of a probability distribution. The MGF can be used to find moments of a random variable.
The document discusses arc length, curvature, and formulas for calculating them for curves in 2D and 3D. It provides the definitions of arc length, unit tangent vector, and curvature. Examples are given of finding the curvature of various curves, such as circles, helices, and parabolas, at given points using parametrizations of the curves. In general, the curvature of a plane curve f(x) = y is k(x) = f''(x) / [1+(f'(x))^2]^(3/2).
Guillaume Théaudière, Stratège Médias Universal Mc Cann (www.umww.com) présente Wave 5 : The socialisation of brands, une étude sur les usages des médias sociaux dans le monde avec un focus sur les marques et la manière dont elles peuvent les utiliser.
Why socialising is amazing, how we are able to design for socilisation, and what psychological and anthropological tools are at our disposal, followed by a demo of CrowdScanner which is designed for socialisation with strangers.
There are various reasons why we would want to find the extreme (maximum and minimum values) of a function. Fermat's Theorem tells us we can find local extreme points by looking at critical points. This process is known as the Closed Interval Method.
The document discusses statistical models and exponential families. It states that for most of the course, data is assumed to be a random sample from a distribution F. Repetition of observations via the law of large numbers and central limit theorem increases information about F. Exponential families are a class of parametric distributions with convenient analytic properties, where the density can be written as a function of natural parameters in an exponential form. Examples of exponential families include the binomial and normal distributions.
This document summarizes multivariate extreme value theory and methods for analyzing the joint behavior of extremes from multiple variables. It discusses three main approaches:
1) Limit theorems for multivariate sample maxima, which characterize the limiting distribution of component-wise maxima.
2) Alternative formulations by Ledford-Tawn and Heffernan-Tawn that allow for more flexible dependence structures between variables.
3) Max-stable processes, which generalize univariate extreme value distributions to the multivariate case through the use of exponent measures.
Estimation of multivariate extreme value models poses challenges due to their nonregular behavior and potential for high dimensionality. Most methods transform to unit Fréchet margins before modeling dependence structure.
Vitaly Vanchurin "General relativity from non-equilibrium thermodynamics of q...SEENET-MTP
1) The document proposes that general relativity can emerge from quantum mechanics in the limit of many degrees of freedom, similar to how thermodynamics emerges from classical mechanics with many particles.
2) It suggests defining statistical ensembles over wave functions using an "infoton field" to obtain a spatially covariant description of quantum information, represented by an information tensor.
3) A dual theory description of computational complexity is developed using the infoton field, arriving at a Klein-Gordon theory with an inverse metric related to computational parameters like the number of qubits. This provides a space-time covariant description of quantum computation.
A brief introduction to Hartree-Fock and TDDFTJiahao Chen
The document provides an overview of time-dependent density functional theory (TDDFT) for computing molecular excited states. It begins with an introduction to the Born-Oppenheimer approximation and variational principle. It then discusses the Hartree-Fock and Kohn-Sham equations as self-consistent field methods for calculating ground states, and linear response theory for calculating excited states within TDDFT. The contents section outlines the topics to be covered, including basis functions, Hartree-Fock theory, density functional theory, and time-dependent DFT.
A common fixed point theorem in cone metric spacesAlexander Decker
This academic article summarizes a common fixed point theorem for continuous and asymptotically regular self-mappings on complete cone metric spaces. The theorem extends previous results to cone metric spaces, which generalize metric spaces by replacing real numbers with an ordered Banach space. It proves that under certain contractive conditions, the self-mapping has a unique fixed point. The proof constructs a Cauchy sequence that converges to the fixed point.
Homomorphism and Anti-homomorphism of Multi-Fuzzy Ideal and Multi-Anti Fuzzy ...iosrjce
IOSR Journal of Mathematics(IOSR-JM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
The document discusses multiobjective optimization and evolutionary algorithms. It defines multiobjective optimization problems as having multiple objective functions to minimize subject to constraints. Pareto optimal solutions are those that are not dominated by any other solutions in terms of all objectives. Evolutionary algorithms are used to approximate the Pareto front and find Pareto optimal solutions. Non-dominated sorting and crowding distance are used to select the next population in NSGA-II. The hypervolume indicator measures the size of the space covered by the Pareto front approximations.
This document introduces the finite element method for solving partial differential equations. It discusses using a "master element" to perform calculations that then get transformed to individual mesh elements. The method is described for a general diffusion equation, integrating it by parts and discretizing it using basis functions defined on mesh elements. This leads to a system of equations relating the unknown values at different nodes in the mesh at each time step. Transforming between the master element coordinates and the actual mesh coordinates completes the description of how the finite element method sets up and solves the discrete system of equations approximating the original PDE.
This document contains lecture notes on machine learning and deep learning. It discusses regression, classification, and neural networks. For regression and classification, it presents the optimal functions that minimize error and relates them to conditional expectations. It also provides bounds on the generalization error of functions learned through empirical risk minimization. For neural networks, it discusses their ability to approximate functions and bounds the VC-dimension of neural networks with multiple hidden layers.
A tutorial on the Frobenious Theorem, one of the most important results in differential geometry, with emphasis in its use in nonlinear control theory. All results are accompanied by proofs, but for a more thorough and detailed presentation refer to the book of A. Isidori.
In topological inference, the goal is to extract information about a shape, given only a sample of points from it. There are many approaches to this problem, but the one we focus on is persistent homology. We get a view of the data at different scales by imagining the points are balls and consider different radii. The shape information we want comes in the form of a persistence diagram, which describes the components, cycles, bubbles, etc in the space that persist over a range of different scales.
To actually compute a persistence diagram in the geometric setting, previous work required complexes of size n^O(d). We reduce this complexity to O(n) (hiding some large constants depending on d) by using ideas from mesh generation.
This talk will not assume any knowledge of topology. This is joint work with Gary Miller, Benoit Hudson, and Steve Oudot.
This document summarizes a lecture on modified gravity theories. It discusses scalar-tensor theories as a framework for modifying general relativity, and mentions specific theories like DGP and f(R) gravity. It outlines two screening mechanisms - chameleon screening, where the scalar field acquires an environment-dependent mass, and Vainshtein screening, where derivative interactions lead to a scale-dependent suppression of the scalar force. Tests of modified gravity theories are discussed at both large cosmological and small solar system scales.
First-order logic (FOL) is a formal system used in mathematics, philosophy, linguistics, and computer science to represent knowledge about domains involving objects and relations. FOL extends propositional logic with quantifiers and predicates to describe properties of and relations between objects. Well-formed formulas in FOL involve constants, variables, functions, predicates, quantifiers, and logical connectives. The meaning and truth of FOL statements is determined with respect to a structure called a model that specifies a domain of objects and interpretations of symbols. FOL can be used to represent knowledge about many different domains and perform logical inference.
This document provides an overview of Dirichlet processes and their applications. It begins with background on probability mass functions and density functions. It then discusses the probability simplex and the Dirichlet distribution. The Dirichlet process is defined as a distribution over distributions that allows modeling probability distributions over infinite sample spaces. An example application involves using Dirichlet processes to learn hierarchical morphology paradigms by modeling stems and suffixes as being generated independently from Dirichlet processes. References for further reading are also provided.
Those are the slides for my Master course on Monte Carlo Statistical Methods given in conjunction with the Monte Carlo Statistical Methods book with George Casella.
The document discusses cumulative distribution functions (CDFs) and probability density functions (PDFs) for continuous random variables. It provides definitions and properties of CDFs and PDFs. For CDFs, it describes how they give the probability that a random variable is less than or equal to a value. For PDFs, it explains how they provide the probability of a random variable taking on a particular value. The document also gives examples of CDFs and PDFs for exponential and uniform random variables.
The document discusses numerical methods for solving nonlinear equations, including root finding and systems of nonlinear equations. It covers the basics of nonlinear solvers like bisection, Newton's method, and fixed-point iteration. For one-dimensional root finding, it analyzes the convergence properties and order of convergence for these methods. It then extends the discussion to systems of nonlinear equations and shows how Newton's method can be applied by taking derivatives to form the Jacobian matrix.
This document discusses various methods for estimating normalizing constants that arise when evaluating integrals numerically. It begins by noting there are many computational methods for approximating normalizing constants across different communities. It then lists the topics that will be covered in the upcoming workshop, including discussions on estimating constants using Monte Carlo methods and Bayesian versus frequentist approaches. The document provides examples of estimating normalizing constants using Monte Carlo integration, reverse logistic regression, and Xiao-Li Meng's maximum likelihood estimation approach. It concludes by discussing some of the challenges in bringing a statistical framework to constant estimation problems.
1. Introduction
Univariate Extreme Value Theory
Multivariate Extreme Value Theory
Extreme Values and Probability Distribution
Functions on Finite Dimensional Spaces
Do Dai Chi
Thesis advisor: Assoc.Prof.Dr. Ho Dang Phuc
K53 - Undergraduate Program in Mathematics
Viet Nam National University - University of Science
December 7, 2012
Do Dai Chi EVT and Probability D.Fs on F.D.S
2. Introduction
Univariate Extreme Value Theory
Multivariate Extreme Value Theory
Outline
1 Introduction
Limit Probabilities for Maxima
Maximum Domains of Attraction
2 Univariate Extreme Value Theory
Max-Stable Distributions
Extremal Value Distributions
Domain of Attration Condition
3 Multivariate Extreme Value Theory
Limit Distributions of Multivariate Maxima
Multivariate Domain of Atrraction
Do Dai Chi EVT and Probability D.Fs on F.D.S
3. Introduction
Univariate Extreme Value Theory
Multivariate Extreme Value Theory
Outline
1 Introduction
Limit Probabilities for Maxima
Maximum Domains of Attraction
2 Univariate Extreme Value Theory
Max-Stable Distributions
Extremal Value Distributions
Domain of Attration Condition
3 Multivariate Extreme Value Theory
Limit Distributions of Multivariate Maxima
Multivariate Domain of Atrraction
Do Dai Chi EVT and Probability D.Fs on F.D.S
4. Introduction
Univariate Extreme Value Theory
Multivariate Extreme Value Theory
Outline
1 Introduction
Limit Probabilities for Maxima
Maximum Domains of Attraction
2 Univariate Extreme Value Theory
Max-Stable Distributions
Extremal Value Distributions
Domain of Attration Condition
3 Multivariate Extreme Value Theory
Limit Distributions of Multivariate Maxima
Multivariate Domain of Atrraction
Do Dai Chi EVT and Probability D.Fs on F.D.S
5. Introduction
Limit Probabilities for Maxima
Univariate Extreme Value Theory
Maximum Domains of Attraction
Multivariate Extreme Value Theory
Motivation
Extreme value theory developed from an interest in studying
the behavior of the extremes of i.i.d random variables.
Historically, the study of extremes can be dated back to
Nicholas Bernoulli who studied the mean largest distance from
the origin to n points scattered randomly on a straight line of
some fixed length.
Our focus is on probabilistic aspects of univariate modelling
and of the behaviour of extremes.
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6. Introduction
Limit Probabilities for Maxima
Univariate Extreme Value Theory
Maximum Domains of Attraction
Multivariate Extreme Value Theory
Limit Probabilities for Maxima
Sample maxima:
Mn = max(X1 , . . . , Xn ), n ≥ 1. (1)
P(Mn ≤ x) = F n (x). (2)
Renormalization :
∗ Mn − bn
Mn = (3)
an
for {an > 0} and {bn } ∈ R.
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7. Introduction
Limit Probabilities for Maxima
Univariate Extreme Value Theory
Maximum Domains of Attraction
Multivariate Extreme Value Theory
Limit Probabilities for Maxima
Definition
A univariate distribution function F , belong to the maximum
domain of attraction of a distribution function G if
1 G is non-degenerate distribution.
2 There exist real valued sequence an > 0, bn ∈ R, such that
Mn − bn d
P ≤x = F n (an x + bn ) → G (x). (4)
an
Extremal Limit Problem : Finding the limit distribution G (x).
Domain of Attraction Problem: Finding the F (x) (F ∈ D(G )).
Mn −bn
P an ≤ x = P(Mn ≤ un ) where un = an x + bn .
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8. Introduction
Limit Probabilities for Maxima
Univariate Extreme Value Theory
Maximum Domains of Attraction
Multivariate Extreme Value Theory
Limit Probabilities for Maxima
Example (standard exponential distribution)
FX (x) = 1 − e −x , x > 0. (5)
Taking an = 1 and bn = log n, we have
Mn − bn
P ≤x = F n (x + log n) = [1 − e −(x+log n) ]n
an
= [1 − n−1 e −x ]n → exp(−e −x ) (6)
=: Λ(x), x ∈ R.
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9. Introduction
Limit Probabilities for Maxima
Univariate Extreme Value Theory
Maximum Domains of Attraction
Multivariate Extreme Value Theory
Limit Probabilities for Maxima
Remark
min(X1 , . . . , Xn ) = − max(−X1 , . . . , −Xn ). (7)
Now we are faced with certain questions:
1 Given any F , does there exist G such that F ∈ D(G ) ?
2 Given any F , if G exist, is it unique ?
3 Can we characterize the class of all possible limits G
according to definition definition #1 ?
4 Given a limit G , what properties should F have so that
F ∈ D(G ) ?
5 How can we compute an , bn ?
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10. Introduction
Limit Probabilities for Maxima
Univariate Extreme Value Theory
Maximum Domains of Attraction
Multivariate Extreme Value Theory
Maximum Domains of Attraction
Theorem (Poisson approximation)
For given τ ∈ [0, ∞] and a sequence {un } of real numbers, the
following two conditions are equivalent for F = 1 − F
1 nF (un ) → τ as n → ∞,
2 P(Mn ≤ un ) → e −τ as n → ∞.
We denote f (x−) = limy ↑x f (y )
Theorem
Let F be a d.f. with right endpoint xF ≤ ∞ and let τ ∈ (0, ∞).
There exists a sequence (un ) satisfying nF (un ) → τ if and only if
F (x)
lim =1 (8)
x↑xF F (x−)
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11. Introduction
Limit Probabilities for Maxima
Univariate Extreme Value Theory
Maximum Domains of Attraction
Multivariate Extreme Value Theory
Example (Geometric distribution)
P(X = k) = p(1 − p)k−1 , 0 < p < 1, k ∈ N. (9)
For this distribution, we have
∞ −1
F (k) k−1 r −1
= 1 − (1 − p) (1 − p)
F (k − 1) r =k
= 1 − p ∈ (0, 1). (10)
No limit P(Mn ≤ un ) → ρ exists except for ρ = 0 or 1, that
implies there is no non-degenerate limit distribution for the
maxima in the geometric distribution case.
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12. Introduction
Limit Probabilities for Maxima
Univariate Extreme Value Theory
Maximum Domains of Attraction
Multivariate Extreme Value Theory
Maximum Domains of Attraction
Definition
Distribution functions U(x) and V (x) are of the same type if for
some A > 0, B ∈ R
V (x) = U(Ax + B) (11)
d X −B
Y = (12)
A
Example (Normal distribution function)
x −µ
N(µ, σ 2 , x) = N(0, 1, ) for σ > 0, µ ∈ R. (13)
σ
d
Xµ,σ = σX0,1 + µ. (14)
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13. Introduction
Limit Probabilities for Maxima
Univariate Extreme Value Theory
Maximum Domains of Attraction
Multivariate Extreme Value Theory
Convergence to types theorem
Theorem (Convergence to types theorem)
Suppose U(x) and V (x) are two non-degenerate d.f.’s . Suppose
for n ≥ 1, Fn is a distribution, an ≥ 0, αn > 0, bn , βn ∈ R and
d d
Fn (an x + bn ) → U(x), Fn (αn x + βn ) → V (x). (15)
Then as n → ∞
αn βn − bn
→ A > 0, → B ∈ R, (16)
an an
and
V (x) = U(Ax + B) (17)
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14. Introduction Max-Stable Distributions
Univariate Extreme Value Theory Extremal Value Distributions
Multivariate Extreme Value Theory Domain of Attration Condition
Max-Stable Distributions
What are the possible (non-degenerate) limit laws for the
maxima Mn when properly normalised and centred?
Definition
A non-degenerate random d.f. F is max-stable if for X1 , X2 , . . . , Xn
i.i.d. F there exist an > 0, bn ∈ R such that
d
Mn = an X1 + bn . (18)
Theorem (Limit property of max-stable laws)
The class of all max-stable d.f.’s coincide with the class of all limit
laws G for maxima of i.i.d. random variables.
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15. Introduction Max-Stable Distributions
Univariate Extreme Value Theory Extremal Value Distributions
Multivariate Extreme Value Theory Domain of Attration Condition
Extremal Value Distributions
Theorem (Extremal types theorem)
Suppose there exist sequence {an > 0} and {bn ∈ R}, such that
Mn − bn d
→G
an
where G is non-degenerate, then G is of one the following three
types:
1 Type I, Gumbel : Λ(x) = exp{−e −x }, x ∈ R.
0 if x < 0
2 Type II, Fr´chet :
e Φα (x) =
exp{−x −α } if x ≥ 0
for some α > 0.
exp{−(−x)α } if x < 0
3 Type III, Weibull : Ψα (x) =
1 if x ≥ 0
for some α > 0
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16. Introduction Max-Stable Distributions
Univariate Extreme Value Theory Extremal Value Distributions
Multivariate Extreme Value Theory Domain of Attration Condition
Extremal Value Distributions
Remark
1 Suppose X > 0, then
1
X ∼ Ψα ⇔ − ∼ Ψα ⇔ log X α ∼ Λ (19)
X
2 Class of Extreme Value distributions = Max-stable
distributions = Distributions appearing as limits in Definition
definition #1
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17. Introduction Max-Stable Distributions
Univariate Extreme Value Theory Extremal Value Distributions
Multivariate Extreme Value Theory Domain of Attration Condition
Extremal Value Distributions
Example (standard Fr´chet distribution)
e
1
F (x) = exp(− ), x > 0. (20)
x
For an = n and bn = 0.
M n − bn 1 n
P ≤x = F n (nx) = [exp{− }]
an nx
n
= exp(− ) = F (x) (21)
nx
Because of the max-stability of F - is also the standard
Fr´chet distribution.
e
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18. Introduction Max-Stable Distributions
Univariate Extreme Value Theory Extremal Value Distributions
Multivariate Extreme Value Theory Domain of Attration Condition
Extremal Value Distributions
Example (Uniform distribution)
F (x) = x for 0 ≤ x ≤ 1.
1
For fixed x < 0, suppose n > −x and let an = n and bn = 1.
Mn − bn
P ≤x = F n (n−1 x + 1)
an
x n
= 1+ → ex (22)
n
The limit distribution is of Weibull type, that means Weibull
distribution are max-stable.
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19. Introduction Max-Stable Distributions
Univariate Extreme Value Theory Extremal Value Distributions
Multivariate Extreme Value Theory Domain of Attration Condition
Generalized Extreme Value Distributions
Definition (Generalized Extreme Value Distributions)
For any γ ∈ R, defined the distribution
1
exp(−(1 + γx) γ ), if 1 + γx > 0;
Gγ (x) = (23)
− exp{−e −x } if γ = 0.
is an extreme value distribution. The parameter γ is called the
extreme value index.
1 For γ > 0, we have Fr´chet class of distributions.
e
2 For γ = 0, we have Gumbel class of distributions.
3 For γ < 0, we have Weibull class of distributions.
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20. Introduction Max-Stable Distributions
Univariate Extreme Value Theory Extremal Value Distributions
Multivariate Extreme Value Theory Domain of Attration Condition
Domain of Attration Condition
Theorem (von Mises’condition)
Let F be a distribution function. Suppose F ”(x) exists and F (x)
is positive for all x in some left neighborhood of xF . If
1 − F (t)
lim (t) =γ (24)
t↑xF F
or equivalently
(1 − F (t))F (t)
lim = −γ − 1 (25)
t↑xF (F (t))2
then F is in the domain of attraction of Gγ (F ∈ D(Gγ )).
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21. Introduction Max-Stable Distributions
Univariate Extreme Value Theory Extremal Value Distributions
Multivariate Extreme Value Theory Domain of Attration Condition
Domain of Attration Condition
Example (standard normal distribution)
Let F (x) = N(x). We have
1 2
F (x) = n(x) = √ e −x /2 (26)
2π
1 2
F (x) = − √ xe −x /2 = −xn(x) (27)
2π
Using Mills’ ratio, we have 1 − N(x) ∼ x −1 n(x).
(1 − F (x))F (x) −x −1 n(x)xn(x)
lim = lim = −1. (28)
x→∞ (F (x))2 x→∞ (n(x))2
Then γ = 0 and F ∈ D(Λ) - Gumbel distribution.
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22. Introduction
Limit Distributions of Multivariate Maxima
Univariate Extreme Value Theory
Multivariate Domain of Atrraction
Multivariate Extreme Value Theory
Limit Distributions of Multivariate Maxima
For d-dimensional vectors x = (x (1) , . . . , x (d) ).
Marginal ordering: x ≤ y means x (j) ≤ y (j) , j = 1, . . . , d.
Component-wise maximum:
x ∨ y := (x (1) ∨ y (1) , . . . , x (d) ∨ y (d) ) (29)
Our approach for extreme value analysis will be based on the
Componentwise maxima depending on Marginal ordering.
(1) (d)
If Xn = (Xn , . . . , Xn ), then
n n
(1) (d) (1) (d)
Mn = ( Xi , . . . , Xi ) = (Mn , . . . , Mn ) (30)
i=1 i=1
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23. Introduction
Limit Distributions of Multivariate Maxima
Univariate Extreme Value Theory
Multivariate Domain of Atrraction
Multivariate Extreme Value Theory
Max-infinitely Divisible Distributions
Definition
The d.f. F on Rd is max-infinitely divisible or max-id if for every n
there exists a distribution Fn on Rd such that
n
F = Fn . (31)
Theorem
Suppose that for n ≥ 0, Fn are probability distribution functions on
n d
Rd . If Fn → F0 then F0 is max-id. Consequently,
1 F is max-id if and only if F t is a d.f. for all t > 0.
2 The class of max-id distributions is closed under weak
d
convergence: If Gn are max-id and Gn → G0 , then G0 is
max-id.
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24. Introduction
Limit Distributions of Multivariate Maxima
Univariate Extreme Value Theory
Multivariate Domain of Atrraction
Multivariate Extreme Value Theory
Multivariate Domain of Atrraction
Definition
A multivariate distribution function F is said to be in the domain
of attraction of a multivariate distribution function G if
1 G has non-degenerate marginal distributions Gi , i = 1, . . . , d.
(i) (i)
2 There exist sequence an > 0 and bn ∈ R, such that
(i) (i)
Mn − bn
P (i)
≤ x (i) = F n (an x (1) + bn , . . . , an x (d) + bn )
(1) (1) (d) (d)
an
d
→ G (x) (32)
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25. Introduction
Limit Distributions of Multivariate Maxima
Univariate Extreme Value Theory
Multivariate Domain of Atrraction
Multivariate Extreme Value Theory
Max-stability
Definition (Max-stable distribution)
A distribution G (x) is max-stable if for i = 1, . . . , d and every
t > 0, there exist functions α(i) (t) > 0 , β (i) (t) such that
G t (x) = G (α(1) (t)x (1) + β (1) (t), . . . , α(d) (t)x (d) + β (d) (t)). (33)
Every max-stable distribution is max-id.
Theorem
The class of multivariate extreme value distributions is precisely
the class of max-stable d.f.’s with non-degenerate marginals.
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26. Introduction
Limit Distributions of Multivariate Maxima
Univariate Extreme Value Theory
Multivariate Domain of Atrraction
Multivariate Extreme Value Theory
Conclusion
Extreme value theory is concerned with distributional properties of
the maximum Mn of n i.i.d. random variables.
1 Extremal Types Theorem, which exhibits the possible limiting
forms for the distribution of Mn under linear normalizations.
2 A simple necessary and sufficient condition under which
P{Mn ≤ un } converges, for a given sequence of constants
{un }.
The maximum of n multivariate observations is defined by the
vector of componentwise maxima.
The structure of the family of limiting distributions can be
studied in terms of max-stable distributions. We discuss
characterizations of the limiting multivariate extreme value
distributions.
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27. Introduction
Limit Distributions of Multivariate Maxima
Univariate Extreme Value Theory
Multivariate Domain of Atrraction
Multivariate Extreme Value Theory
Bibliography
[S. Resnick]
Extreme Values, Regular Variation, and Point Processes
(Springer, 1987)
[de Haan, Laurens and Ferreira, Ana]
Extreme Value Theory: An Introduction (Springer, 2006)
[Leadbetter, M. R. and Lindgren, G. and Rootz´n, H. ]
e
Extremes and Related Properties of Random Sequences and
Processes (Springer-Verlag, 1983)
[Bikramjit Dass]
A course in Multivariate Extremes (Spring-2010)
Do Dai Chi EVT and Probability D.Fs on F.D.S
28. Introduction
Limit Distributions of Multivariate Maxima
Univariate Extreme Value Theory
Multivariate Domain of Atrraction
Multivariate Extreme Value Theory
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