Bayesian Information Criterion The BIC Of Algebraic geometry
|
|
- Morgan Banks
- 3 years ago
- Views:
Transcription
1 Generalized BIC for Singular Models Factoring through Regular Models Shaowei Lin shaowei/ Department of Mathematics, University of California, Berkeley PhD student (Advisor: Bernd Sturmfels)
2 Abstract The Bayesian Information Criterion (BIC) is an important tool for model selection, but its use is limited only to regular models. Recently, Watanabe [5] generalized the BIC to singular models using ideas from algebraic geometry. In this paper, we compute this generalized BIC for singular models factoring through regular models, relating it to the real log canonical threshold of polynomial fiber ideals.
3 Bayesian Information Criterion The BIC is a score used for model selection. BIC = log L 0 + d 2 log N L 0 : maximum likelihood, d: dimension, N: sample size. However, it applies only for regular models.
4 Regular and Singular Models A model M(U) is regular if it is identifiable p(x u, M) = p(x u, M) x u = u and the Fisher information matrix I(u) is positive definite u. I jk (u) = log p(x u) u j log p(x u) p(x u)dx u k Otherwise, the model is singular. e.g. most exponential families are regular, while most hidden variable models are singular.
5 Factored Models M 2 (Ω) factors through M 1 (U) if for some function u(ω), p(x ω, M 2 ) = p(x u(ω), M 1 ). e.g. parametrized multivariate Gaussian models N(0, Σ(ω)).
6 Generalized BIC In 2001, Watanabe showed that for singular models M(Ω), the BIC generalizes to log L 0 + λ log N (θ 1) log log N, where λ is the smallest pole of the zeta function ζ(z) = K(ω) z ϕ(ω)dω, z C Ω and θ its multiplicity. Here, ϕ(ω) is a prior on Ω, and K(ω) the Kullback-Leibler distance to the true distribution. We compute (λ, θ) by monomializing K(ω). (a.k.a. resolution of singularities)
7 Key Idea: Fiber Ideals Even if M(Ω) is parametrized by simple polynomials, the Kullback function K(ω) is non-polynomial and difficult to monomialize. We show that if M(Ω) factors through a regular model, we can define a polynomial fiber ideal and compute (λ, θ) by monomializing this ideal.
8 Real Log Canonical Thresholds Given an ideal I = f 1,...,f r generated by polynomials, define the real log canonical threshold (λ,θ) to be the smallest pole λ and multiplicity θ of the zeta function ζ(z) = (f 1 (ω) f r (ω) 2 ) z/2 dω, z C. The RLCT is independent of the choice of generators f 1,...,f r, and can be computed by monomializing I. For monomial ideals, we find RLCTs using a geometriccombinatorial tool involving Newton polyhedra.
9 Main Result Let M 2 (Ω) factor through a regular model M 1 (U) via u(ω). Given N i.i.d. samples, let their M.L.E. in M 1 be û U. Let Z(N) be the marginal likelihood of the data given M 2. Theorem (L.) If û u(ω), then asymptotically, log Z(N) log L 0 + λ log N (θ 1) log log N where (2λ,θ) is the RLCT of the fiber ideal I = u 1 (ω) û 1,...,u d (ω) û d.
10 Example: Classical BIC Let M(U) be a regular model and û the M.L.E. of the data. Then, (2λ,θ) is the RLCT of the fiber ideal which is just (d, 1). I = u 1 û 1,...,u d û d
11 Example: Discrete Models Let M(Ω) be a model on discrete space {1, 2,...,k} with polynomial state probabilities p 1 (ω),...,p k (ω). Let q 1,...,q k be relative frequencies of the data. If q p(ω), then (2λ,θ) is the RLCT of the fiber ideal I = p 1 (ω) q 1,...,p k (ω) q k.
12 Example: Gaussian Models Let N(µ(Ω), Σ(Ω)) be a multivariate Gaussian model with sample mean ˆµ and covariance matrix ˆΣ. If ˆµ µ(ω) and ˆΣ Σ(Ω), then (2λ,θ) is the RLCT of the fiber ideal I = µ(ω) ˆµ, Σ(ω) ˆΣ.
13 Acknowledgements Many thanks go to Mathias Drton and Sumio Watanabe for enlightening discussions and key ideas.
14 References 1. V. I. Arnol d, S. M. Guseĭn-Zade and A. N. Varchenko: Singularities of Differentiable Maps, Vol. II, Birkhäuser, Boston, H. Hironaka: Resolution of singularities of an algebraic variety over a field of characteristic zero I, II, Ann. of Math. (2) 79 (1964) S. Lin: Asymptotic Approximation of Marginal Likelihood Integrals, preprint arxiv: (2010). 4. S. Watanabe: Algebraic analysis for nonidentifiable learning machines, Neural Computation 13 (2001) S. Watanabe: Algebraic Geometry and Statistical Learning Theory, Cambridge Monographs on Applied and Computational Mathematics 25, Cambridge University Press, Cambridge, G. Schwarz: Estimating the Dimension of a Model, Annals of Statistics 6 (1978)
Linear Classification. Volker Tresp Summer 2015
Linear Classification Volker Tresp Summer 2015 1 Classification Classification is the central task of pattern recognition Sensors supply information about an object: to which class do the object belong
More informationBasics of Statistical Machine Learning
CS761 Spring 2013 Advanced Machine Learning Basics of Statistical Machine Learning Lecturer: Xiaojin Zhu jerryzhu@cs.wisc.edu Modern machine learning is rooted in statistics. You will find many familiar
More informationClass #6: Non-linear classification. ML4Bio 2012 February 17 th, 2012 Quaid Morris
Class #6: Non-linear classification ML4Bio 2012 February 17 th, 2012 Quaid Morris 1 Module #: Title of Module 2 Review Overview Linear separability Non-linear classification Linear Support Vector Machines
More informationStatistical Machine Learning
Statistical Machine Learning UoC Stats 37700, Winter quarter Lecture 4: classical linear and quadratic discriminants. 1 / 25 Linear separation For two classes in R d : simple idea: separate the classes
More informationInterpreting Kullback-Leibler Divergence with the Neyman-Pearson Lemma
Interpreting Kullback-Leibler Divergence with the Neyman-Pearson Lemma Shinto Eguchi a, and John Copas b a Institute of Statistical Mathematics and Graduate University of Advanced Studies, Minami-azabu
More informationPackage MixGHD. June 26, 2015
Type Package Package MixGHD June 26, 2015 Title Model Based Clustering, Classification and Discriminant Analysis Using the Mixture of Generalized Hyperbolic Distributions Version 1.7 Date 2015-6-15 Author
More informationSTA 4273H: Statistical Machine Learning
STA 4273H: Statistical Machine Learning Russ Salakhutdinov Department of Statistics! rsalakhu@utstat.toronto.edu! http://www.cs.toronto.edu/~rsalakhu/ Lecture 6 Three Approaches to Classification Construct
More informationMaximum Likelihood Estimation
Math 541: Statistical Theory II Lecturer: Songfeng Zheng Maximum Likelihood Estimation 1 Maximum Likelihood Estimation Maximum likelihood is a relatively simple method of constructing an estimator for
More informationCPM High Schools California Standards Test (CST) Results for 2004-2010
CPM High California Standards Test (CST) Results for 2004-2010 The tables below show a comparison between CPM high schools and all high schools in California based on the percentage of students who scored
More informationStatistical Machine Learning from Data
Samy Bengio Statistical Machine Learning from Data 1 Statistical Machine Learning from Data Gaussian Mixture Models Samy Bengio IDIAP Research Institute, Martigny, Switzerland, and Ecole Polytechnique
More informationAlgebraic Computation Models. Algebraic Computation Models
Algebraic Computation Models Ζυγομήτρος Ευάγγελος ΜΠΛΑ 201118 Φεβρουάριος, 2013 Reasons for using algebraic models of computation The Turing machine model captures computations on bits. Many algorithms
More informationMATH 590: Meshfree Methods
MATH 590: Meshfree Methods Chapter 7: Conditionally Positive Definite Functions Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2010 fasshauer@iit.edu MATH 590 Chapter
More informationStatistics Graduate Courses
Statistics Graduate Courses STAT 7002--Topics in Statistics-Biological/Physical/Mathematics (cr.arr.).organized study of selected topics. Subjects and earnable credit may vary from semester to semester.
More informationLinear Threshold Units
Linear Threshold Units w x hx (... w n x n w We assume that each feature x j and each weight w j is a real number (we will relax this later) We will study three different algorithms for learning linear
More informationMATHEMATICS. Administered by the Department of Mathematical and Computing Sciences within the College of Arts and Sciences. Degree Requirements
MATHEMATICS Administered by the Department of Mathematical and Computing Sciences within the College of Arts and Sciences. Paul Feit, PhD Dr. Paul Feit is Professor of Mathematics and Coordinator for Mathematics.
More informationTHE MULTIVARIATE ANALYSIS RESEARCH GROUP. Carles M Cuadras Departament d Estadística Facultat de Biologia Universitat de Barcelona
THE MULTIVARIATE ANALYSIS RESEARCH GROUP Carles M Cuadras Departament d Estadística Facultat de Biologia Universitat de Barcelona The set of statistical methods known as Multivariate Analysis covers a
More informationTwo Topics in Parametric Integration Applied to Stochastic Simulation in Industrial Engineering
Two Topics in Parametric Integration Applied to Stochastic Simulation in Industrial Engineering Department of Industrial Engineering and Management Sciences Northwestern University September 15th, 2014
More informationCCNY. BME I5100: Biomedical Signal Processing. Linear Discrimination. Lucas C. Parra Biomedical Engineering Department City College of New York
BME I5100: Biomedical Signal Processing Linear Discrimination Lucas C. Parra Biomedical Engineering Department CCNY 1 Schedule Week 1: Introduction Linear, stationary, normal - the stuff biology is not
More informationSection 5. Stan for Big Data. Bob Carpenter. Columbia University
Section 5. Stan for Big Data Bob Carpenter Columbia University Part I Overview Scaling and Evaluation data size (bytes) 1e18 1e15 1e12 1e9 1e6 Big Model and Big Data approach state of the art big model
More informationSTATISTICA Formula Guide: Logistic Regression. Table of Contents
: Table of Contents... 1 Overview of Model... 1 Dispersion... 2 Parameterization... 3 Sigma-Restricted Model... 3 Overparameterized Model... 4 Reference Coding... 4 Model Summary (Summary Tab)... 5 Summary
More informationMessage-passing sequential detection of multiple change points in networks
Message-passing sequential detection of multiple change points in networks Long Nguyen, Arash Amini Ram Rajagopal University of Michigan Stanford University ISIT, Boston, July 2012 Nguyen/Amini/Rajagopal
More informationLogistic Regression. Vibhav Gogate The University of Texas at Dallas. Some Slides from Carlos Guestrin, Luke Zettlemoyer and Dan Weld.
Logistic Regression Vibhav Gogate The University of Texas at Dallas Some Slides from Carlos Guestrin, Luke Zettlemoyer and Dan Weld. Generative vs. Discriminative Classifiers Want to Learn: h:x Y X features
More informationAcknowledgments. Data Mining with Regression. Data Mining Context. Overview. Colleagues
Data Mining with Regression Teaching an old dog some new tricks Acknowledgments Colleagues Dean Foster in Statistics Lyle Ungar in Computer Science Bob Stine Department of Statistics The School of the
More informationComputational algebraic geometry
Computational algebraic geometry Learning coefficients via symbolic and numerical methods Anton Leykin Georgia Tech AIM, Palo Alto, December 2011 Let k be a field (R or C). Ideals varieties Ideal in R
More informationON THE DEGREES OF FREEDOM IN RICHLY PARAMETERISED MODELS
COMPSTAT 2004 Symposium c Physica-Verlag/Springer 2004 ON THE DEGREES OF FREEDOM IN RICHLY PARAMETERISED MODELS Salvatore Ingrassia and Isabella Morlini Key words: Richly parameterised models, small data
More informationLecture 3: Linear methods for classification
Lecture 3: Linear methods for classification Rafael A. Irizarry and Hector Corrada Bravo February, 2010 Today we describe four specific algorithms useful for classification problems: linear regression,
More informationMATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors. Jordan canonical form (continued).
MATH 423 Linear Algebra II Lecture 38: Generalized eigenvectors Jordan canonical form (continued) Jordan canonical form A Jordan block is a square matrix of the form λ 1 0 0 0 0 λ 1 0 0 0 0 λ 0 0 J = 0
More informationLinear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices
MATH 30 Differential Equations Spring 006 Linear algebra and the geometry of quadratic equations Similarity transformations and orthogonal matrices First, some things to recall from linear algebra Two
More informationBindel, Spring 2012 Intro to Scientific Computing (CS 3220) Week 3: Wednesday, Feb 8
Spaces and bases Week 3: Wednesday, Feb 8 I have two favorite vector spaces 1 : R n and the space P d of polynomials of degree at most d. For R n, we have a canonical basis: R n = span{e 1, e 2,..., e
More informationBayesian Classifier for a Gaussian Distribution, Decision Surface Equation, with Application
Iraqi Journal of Statistical Science (18) 2010 p.p. [35-58] Bayesian Classifier for a Gaussian Distribution, Decision Surface Equation, with Application ABSTRACT Nawzad. M. Ahmad * Bayesian decision theory
More informationGLM, insurance pricing & big data: paying attention to convergence issues.
GLM, insurance pricing & big data: paying attention to convergence issues. Michaël NOACK - michael.noack@addactis.com Senior consultant & Manager of ADDACTIS Pricing Copyright 2014 ADDACTIS Worldwide.
More informationBayesian logistic betting strategy against probability forecasting. Akimichi Takemura, Univ. Tokyo. November 12, 2012
Bayesian logistic betting strategy against probability forecasting Akimichi Takemura, Univ. Tokyo (joint with Masayuki Kumon, Jing Li and Kei Takeuchi) November 12, 2012 arxiv:1204.3496. To appear in Stochastic
More informationCOURSE SYLLABUS Pre-Calculus A/B Last Modified: April 2015
COURSE SYLLABUS Pre-Calculus A/B Last Modified: April 2015 Course Description: In this year-long Pre-Calculus course, students will cover topics over a two semester period (as designated by A and B sections).
More informationIntroduction to General and Generalized Linear Models
Introduction to General and Generalized Linear Models General Linear Models - part I Henrik Madsen Poul Thyregod Informatics and Mathematical Modelling Technical University of Denmark DK-2800 Kgs. Lyngby
More informationPATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 4: LINEAR MODELS FOR CLASSIFICATION
PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 4: LINEAR MODELS FOR CLASSIFICATION Introduction In the previous chapter, we explored a class of regression models having particularly simple analytical
More informationFactoring Cubic Polynomials
Factoring Cubic Polynomials Robert G. Underwood 1. Introduction There are at least two ways in which using the famous Cardano formulas (1545) to factor cubic polynomials present more difficulties than
More informationi=1 In practice, the natural logarithm of the likelihood function, called the log-likelihood function and denoted by
Statistics 580 Maximum Likelihood Estimation Introduction Let y (y 1, y 2,..., y n be a vector of iid, random variables from one of a family of distributions on R n and indexed by a p-dimensional parameter
More informationExact Inference for Gaussian Process Regression in case of Big Data with the Cartesian Product Structure
Exact Inference for Gaussian Process Regression in case of Big Data with the Cartesian Product Structure Belyaev Mikhail 1,2,3, Burnaev Evgeny 1,2,3, Kapushev Yermek 1,2 1 Institute for Information Transmission
More informationMath 143 - College Algebra (Online)
Math 143 - College Algebra (Online) 3 Credits Term: Instructor: Spring 2010 Ken Floyd (208) 732-6583 Office Location: Office Hours: Shields 206E 1-2 PM, M - F E-mail: kfloyd@csi.edu 10 AM - Noon Saturday
More informationNEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS
NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document
More informationby the matrix A results in a vector which is a reflection of the given
Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the y-axis We observe that
More informationISU Department of Mathematics. Graduate Examination Policies and Procedures
ISU Department of Mathematics Graduate Examination Policies and Procedures There are four primary criteria to be used in evaluating competence on written or oral exams. 1. Knowledge Has the student demonstrated
More informationMultivariate Normal Distribution
Multivariate Normal Distribution Lecture 4 July 21, 2011 Advanced Multivariate Statistical Methods ICPSR Summer Session #2 Lecture #4-7/21/2011 Slide 1 of 41 Last Time Matrices and vectors Eigenvalues
More informationUnderstanding Poles and Zeros
MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.14 Analysis and Design of Feedback Control Systems Understanding Poles and Zeros 1 System Poles and Zeros The transfer function
More informationCONTROL SYSTEMS, ROBOTICS, AND AUTOMATION - Vol. V - Relations Between Time Domain and Frequency Domain Prediction Error Methods - Tomas McKelvey
COTROL SYSTEMS, ROBOTICS, AD AUTOMATIO - Vol. V - Relations Between Time Domain and Frequency Domain RELATIOS BETWEE TIME DOMAI AD FREQUECY DOMAI PREDICTIO ERROR METHODS Tomas McKelvey Signal Processing,
More informationExam C, Fall 2006 PRELIMINARY ANSWER KEY
Exam C, Fall 2006 PRELIMINARY ANSWER KEY Question # Answer Question # Answer 1 E 19 B 2 D 20 D 3 B 21 A 4 C 22 A 5 A 23 E 6 D 24 E 7 B 25 D 8 C 26 A 9 E 27 C 10 D 28 C 11 E 29 C 12 B 30 B 13 C 31 C 14
More informationA HYBRID GENETIC ALGORITHM FOR THE MAXIMUM LIKELIHOOD ESTIMATION OF MODELS WITH MULTIPLE EQUILIBRIA: A FIRST REPORT
New Mathematics and Natural Computation Vol. 1, No. 2 (2005) 295 303 c World Scientific Publishing Company A HYBRID GENETIC ALGORITHM FOR THE MAXIMUM LIKELIHOOD ESTIMATION OF MODELS WITH MULTIPLE EQUILIBRIA:
More information171:290 Model Selection Lecture II: The Akaike Information Criterion
171:290 Model Selection Lecture II: The Akaike Information Criterion Department of Biostatistics Department of Statistics and Actuarial Science August 28, 2012 Introduction AIC, the Akaike Information
More informationINDIRECT INFERENCE (prepared for: The New Palgrave Dictionary of Economics, Second Edition)
INDIRECT INFERENCE (prepared for: The New Palgrave Dictionary of Economics, Second Edition) Abstract Indirect inference is a simulation-based method for estimating the parameters of economic models. Its
More informationRESONANCES AND BALLS IN OBSTACLE SCATTERING WITH NEUMANN BOUNDARY CONDITIONS
RESONANCES AND BALLS IN OBSTACLE SCATTERING WITH NEUMANN BOUNDARY CONDITIONS T. J. CHRISTIANSEN Abstract. We consider scattering by an obstacle in R d, d 3 odd. We show that for the Neumann Laplacian if
More informationFactorial experimental designs and generalized linear models
Statistics & Operations Research Transactions SORT 29 (2) July-December 2005, 249-268 ISSN: 1696-2281 www.idescat.net/sort Statistics & Operations Research c Institut d Estadística de Transactions Catalunya
More informationMATHEMATICAL METHODS OF STATISTICS
MATHEMATICAL METHODS OF STATISTICS By HARALD CRAMER TROFESSOK IN THE UNIVERSITY OF STOCKHOLM Princeton PRINCETON UNIVERSITY PRESS 1946 TABLE OF CONTENTS. First Part. MATHEMATICAL INTRODUCTION. CHAPTERS
More informationOn The So-Called Huber Sandwich Estimator and Robust Standard Errors by David A. Freedman
On The So-Called Huber Sandwich Estimator and Robust Standard Errors by David A. Freedman Abstract The Huber Sandwich Estimator can be used to estimate the variance of the MLE when the underlying model
More informationBig Data, Statistics, and the Internet
Big Data, Statistics, and the Internet Steven L. Scott April, 4 Steve Scott (Google) Big Data, Statistics, and the Internet April, 4 / 39 Summary Big data live on more than one machine. Computing takes
More informationPa8ern Recogni6on. and Machine Learning. Chapter 4: Linear Models for Classifica6on
Pa8ern Recogni6on and Machine Learning Chapter 4: Linear Models for Classifica6on Represen'ng the target values for classifica'on If there are only two classes, we typically use a single real valued output
More informationDynamic Eigenvalues for Scalar Linear Time-Varying Systems
Dynamic Eigenvalues for Scalar Linear Time-Varying Systems P. van der Kloet and F.L. Neerhoff Department of Electrical Engineering Delft University of Technology Mekelweg 4 2628 CD Delft The Netherlands
More informationRow Ideals and Fibers of Morphisms
Michigan Math. J. 57 (2008) Row Ideals and Fibers of Morphisms David Eisenbud & Bernd Ulrich Affectionately dedicated to Mel Hochster, who has been an inspiration to us for many years, on the occasion
More informationItem selection by latent class-based methods: an application to nursing homes evaluation
Item selection by latent class-based methods: an application to nursing homes evaluation Francesco Bartolucci, Giorgio E. Montanari, Silvia Pandolfi 1 Department of Economics, Finance and Statistics University
More informationClassification by Pairwise Coupling
Classification by Pairwise Coupling TREVOR HASTIE * Stanford University and ROBERT TIBSHIRANI t University of Toronto Abstract We discuss a strategy for polychotomous classification that involves estimating
More informationDegree Bounds for Gröbner Bases in Algebras of Solvable Type
Degree Bounds for Gröbner Bases in Algebras of Solvable Type Matthias Aschenbrenner University of California, Los Angeles (joint with Anton Leykin) Algebras of Solvable Type... introduced by Kandri-Rody
More informationarxiv:hep-th/0507236v1 25 Jul 2005
Non perturbative series for the calculation of one loop integrals at finite temperature Paolo Amore arxiv:hep-th/050736v 5 Jul 005 Facultad de Ciencias, Universidad de Colima, Bernal Diaz del Castillo
More informationResearch Article The General Traveling Wave Solutions of the Fisher Equation with Degree Three
Advances in Mathematical Physics Volume 203, Article ID 65798, 5 pages http://dx.doi.org/0.55/203/65798 Research Article The General Traveling Wave Solutions of the Fisher Equation with Degree Three Wenjun
More informationThese slides follow closely the (English) course textbook Pattern Recognition and Machine Learning by Christopher Bishop
Music and Machine Learning (IFT6080 Winter 08) Prof. Douglas Eck, Université de Montréal These slides follow closely the (English) course textbook Pattern Recognition and Machine Learning by Christopher
More informationz-scores AND THE NORMAL CURVE MODEL
z-scores AND THE NORMAL CURVE MODEL 1 Understanding z-scores 2 z-scores A z-score is a location on the distribution. A z- score also automatically communicates the raw score s distance from the mean A
More informationAnomaly detection for Big Data, networks and cyber-security
Anomaly detection for Big Data, networks and cyber-security Patrick Rubin-Delanchy University of Bristol & Heilbronn Institute for Mathematical Research Joint work with Nick Heard (Imperial College London),
More informationExtreme Value Modeling for Detection and Attribution of Climate Extremes
Extreme Value Modeling for Detection and Attribution of Climate Extremes Jun Yan, Yujing Jiang Joint work with Zhuo Wang, Xuebin Zhang Department of Statistics, University of Connecticut February 2, 2016
More informationA hidden Markov model for criminal behaviour classification
RSS2004 p.1/19 A hidden Markov model for criminal behaviour classification Francesco Bartolucci, Institute of economic sciences, Urbino University, Italy. Fulvia Pennoni, Department of Statistics, University
More informationMax-Min Representation of Piecewise Linear Functions
Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 43 (2002), No. 1, 297-302. Max-Min Representation of Piecewise Linear Functions Sergei Ovchinnikov Mathematics Department,
More informationCalculation of Minimum Distances. Minimum Distance to Means. Σi i = 1
Minimum Distance to Means Similar to Parallelepiped classifier, but instead of bounding areas, the user supplies spectral class means in n-dimensional space and the algorithm calculates the distance between
More informationREVIEW EXERCISES DAVID J LOWRY
REVIEW EXERCISES DAVID J LOWRY Contents 1. Introduction 1 2. Elementary Functions 1 2.1. Factoring and Solving Quadratics 1 2.2. Polynomial Inequalities 3 2.3. Rational Functions 4 2.4. Exponentials and
More informationPCHS ALGEBRA PLACEMENT TEST
MATHEMATICS Students must pass all math courses with a C or better to advance to the next math level. Only classes passed with a C or better will count towards meeting college entrance requirements. If
More informationMethod of Stationary phase. Reference: Hormander vol I. Steve Zelditch Department of Mathematics Northwestern University
Method of Stationary phase Reference: Hormander vol I Steve Zelditch Department of Mathematics Northwestern University 1 Method of Stationary Phase We now describe the method of stationary phase, which
More informationMCS 563 Spring 2014 Analytic Symbolic Computation Wednesday 9 April. Hilbert Polynomials
Hilbert Polynomials For a monomial ideal, we derive the dimension counting the monomials in the complement, arriving at the notion of the Hilbert polynomial. The first half of the note is derived from
More informationProf. Dr. Raymond Hemmecke
Prof. Dr. Raymond Hemmecke Combinatorial Optimization Zentrum Mathematik Technische Universität Munich 85747 Garching Germany Tel.: +49-89-289 16864 Fax: +49-89-289 16859 email: hemmecke@ma.tum.de Curriculum
More informationComponent Ordering in Independent Component Analysis Based on Data Power
Component Ordering in Independent Component Analysis Based on Data Power Anne Hendrikse Raymond Veldhuis University of Twente University of Twente Fac. EEMCS, Signals and Systems Group Fac. EEMCS, Signals
More informationGaussian Processes to Speed up Hamiltonian Monte Carlo
Gaussian Processes to Speed up Hamiltonian Monte Carlo Matthieu Lê Murray, Iain http://videolectures.net/mlss09uk_murray_mcmc/ Rasmussen, Carl Edward. "Gaussian processes to speed up hybrid Monte Carlo
More informationHow To Test Granger Causality Between Time Series
A general statistical framework for assessing Granger causality The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation As Published
More informationLarge Sample Covariance Matrices and High-Dimensional Data Analysis
Large Sample Covariance Matrices and High-Dimensional Data Analysis High-dimensional data appear in many fields, and their analysis has become increasingly important in modern statistics. However, it has
More information30. Bode Plots. Introduction
0. Bode Plots Introduction Each of the circuits in this problem set is represented by a magnitude Bode plot. The network function provides a connection between the Bode plot and the circuit. To solve these
More informationPenalized Splines - A statistical Idea with numerous Applications...
Penalized Splines - A statistical Idea with numerous Applications... Göran Kauermann Ludwig-Maximilians-University Munich Graz 7. September 2011 1 Penalized Splines - A statistical Idea with numerous Applications...
More informationProbabilistic properties and statistical analysis of network traffic models: research project
Probabilistic properties and statistical analysis of network traffic models: research project The problem: It is commonly accepted that teletraffic data exhibits self-similarity over a certain range of
More informationUNDERSTANDING INFORMATION CRITERIA FOR SELECTION AMONG CAPTURE-RECAPTURE OR RING RECOVERY MODELS
UNDERSTANDING INFORMATION CRITERIA FOR SELECTION AMONG CAPTURE-RECAPTURE OR RING RECOVERY MODELS David R. Anderson and Kenneth P. Burnham Colorado Cooperative Fish and Wildlife Research Unit Room 201 Wagar
More informationUnsupervised Learning
Unsupervised Learning Zoubin Ghahramani Gatsby Computational Neuroscience Unit University College London, UK zoubin@gatsby.ucl.ac.uk http://www.gatsby.ucl.ac.uk/~zoubin September 16, 2004 Abstract We give
More informationThe CUSUM algorithm a small review. Pierre Granjon
The CUSUM algorithm a small review Pierre Granjon June, 1 Contents 1 The CUSUM algorithm 1.1 Algorithm............................... 1.1.1 The problem......................... 1.1. The different steps......................
More informationBasic Math Course Map through algebra and calculus
Basic Math Course Map through algebra and calculus This map shows the most common and recommended transitions between courses. A grade of C or higher is required to move from one course to the next. For
More informationA REVIEW OF CONSISTENCY AND CONVERGENCE OF POSTERIOR DISTRIBUTION
A REVIEW OF CONSISTENCY AND CONVERGENCE OF POSTERIOR DISTRIBUTION By Subhashis Ghosal Indian Statistical Institute, Calcutta Abstract In this article, we review two important issues, namely consistency
More informationOrthogonal Diagonalization of Symmetric Matrices
MATH10212 Linear Algebra Brief lecture notes 57 Gram Schmidt Process enables us to find an orthogonal basis of a subspace. Let u 1,..., u k be a basis of a subspace V of R n. We begin the process of finding
More informationLinear Discrimination. Linear Discrimination. Linear Discrimination. Linearly Separable Systems Pairwise Separation. Steven J Zeil.
Steven J Zeil Old Dominion Univ. Fall 200 Discriminant-Based Classification Linearly Separable Systems Pairwise Separation 2 Posteriors 3 Logistic Discrimination 2 Discriminant-Based Classification Likelihood-based:
More informationBayesX - Software for Bayesian Inference in Structured Additive Regression
BayesX - Software for Bayesian Inference in Structured Additive Regression Thomas Kneib Faculty of Mathematics and Economics, University of Ulm Department of Statistics, Ludwig-Maximilians-University Munich
More informationClovis Community College Core Competencies Assessment 2014 2015 Area II: Mathematics Algebra
Core Assessment 2014 2015 Area II: Mathematics Algebra Class: Math 110 College Algebra Faculty: Erin Akhtar (Learning Outcomes Being Measured) 1. Students will construct and analyze graphs and/or data
More informationAlgebra II. Weeks 1-3 TEKS
Algebra II Pacing Guide Weeks 1-3: Equations and Inequalities: Solve Linear Equations, Solve Linear Inequalities, Solve Absolute Value Equations and Inequalities. Weeks 4-6: Linear Equations and Functions:
More informationMATH. ALGEBRA I HONORS 9 th Grade 12003200 ALGEBRA I HONORS
* Students who scored a Level 3 or above on the Florida Assessment Test Math Florida Standards (FSA-MAFS) are strongly encouraged to make Advanced Placement and/or dual enrollment courses their first choices
More informationAn Internal Model for Operational Risk Computation
An Internal Model for Operational Risk Computation Seminarios de Matemática Financiera Instituto MEFF-RiskLab, Madrid http://www.risklab-madrid.uam.es/ Nicolas Baud, Antoine Frachot & Thierry Roncalli
More informationNew Methods Providing High Degree Polynomials with Small Mahler Measure
New Methods Providing High Degree Polynomials with Small Mahler Measure G. Rhin and J.-M. Sac-Épée CONTENTS 1. Introduction 2. A Statistical Method 3. A Minimization Method 4. Conclusion and Prospects
More informationCorrected Diffusion Approximations for the Maximum of Heavy-Tailed Random Walk
Corrected Diffusion Approximations for the Maximum of Heavy-Tailed Random Walk Jose Blanchet and Peter Glynn December, 2003. Let (X n : n 1) be a sequence of independent and identically distributed random
More informationProbabilistic Linear Classification: Logistic Regression. Piyush Rai IIT Kanpur
Probabilistic Linear Classification: Logistic Regression Piyush Rai IIT Kanpur Probabilistic Machine Learning (CS772A) Jan 18, 2016 Probabilistic Machine Learning (CS772A) Probabilistic Linear Classification:
More informationExtrinsic geometric flows
On joint work with Vladimir Rovenski from Haifa Paweł Walczak Uniwersytet Łódzki CRM, Bellaterra, July 16, 2010 Setting Throughout this talk: (M, F, g 0 ) is a (compact, complete, any) foliated, Riemannian
More informationModel-Based Recursive Partitioning for Detecting Interaction Effects in Subgroups
Model-Based Recursive Partitioning for Detecting Interaction Effects in Subgroups Achim Zeileis, Torsten Hothorn, Kurt Hornik http://eeecon.uibk.ac.at/~zeileis/ Overview Motivation: Trees, leaves, and
More information