Maximum Likelihood Estimation
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An Introduction to Psychometric Theory with Applications in R
What is psychometrics? What is R? Where did it come from, why use it? Basic statistics and graphics TOD An introduction to Psychometric Theory with applications in R William Revelle Department of Psychology Northwestern University Evanston, Illinois USA February, 2013 1 / 71 What is psychometrics? What is R? Where did it come from, why use it? Basic statistics and graphics TOD Overview 1 Overview Psychometrics and R What is Psychometrics What is R 2 Part I: an introduction to R What is R A brief example Basic steps and graphics 3 Day 1: Theory of Data, Issues in Scaling 4 Day 2: More than you ever wanted to know about correlation 5 Day 3: Dimension reduction through factor analysis, principal components analyze and cluster analysis 6 Day 4: Classical Test Theory and Item Response Theory 7 Day 5: Structural Equation Modeling and applied scale construction 2 / 71 What is psychometrics? What is R? Where did it come from, why use it? Basic statistics and graphics TOD Outline of Day 1/part 1 1 What is psychometrics? Conceptual overview Theory: the organization of Observed and Latent variables A latent variable approach to measurement Data and scaling Structural Equation Models 2 What is R? Where did it come from, why use it? Installing R on your computer and adding packages Installing and using packages Implementations of R Basic R capabilities: Calculation, Statistical tables, Graphics Data sets 3 Basic statistics and graphics 4 steps: read, explore, test, graph Basic descriptive and inferential statistics 4 TOD 3 / 71 What is psychometrics? What is R? Where did it come from, why use it? Basic statistics and graphics TOD What is psychometrics? In physical science a first essential step in the direction of learning any subject is to find principles of numerical reckoning and methods for practicably measuring some quality connected with it. -
The Gompertz Distribution and Maximum Likelihood Estimation of Its Parameters - a Revision
Max-Planck-Institut für demografi sche Forschung Max Planck Institute for Demographic Research Konrad-Zuse-Strasse 1 · D-18057 Rostock · GERMANY Tel +49 (0) 3 81 20 81 - 0; Fax +49 (0) 3 81 20 81 - 202; http://www.demogr.mpg.de MPIDR WORKING PAPER WP 2012-008 FEBRUARY 2012 The Gompertz distribution and Maximum Likelihood Estimation of its parameters - a revision Adam Lenart ([email protected]) This working paper has been approved for release by: James W. Vaupel ([email protected]), Head of the Laboratory of Survival and Longevity and Head of the Laboratory of Evolutionary Biodemography. © Copyright is held by the authors. Working papers of the Max Planck Institute for Demographic Research receive only limited review. Views or opinions expressed in working papers are attributable to the authors and do not necessarily refl ect those of the Institute. The Gompertz distribution and Maximum Likelihood Estimation of its parameters - a revision Adam Lenart November 28, 2011 Abstract The Gompertz distribution is widely used to describe the distribution of adult deaths. Previous works concentrated on formulating approximate relationships to char- acterize it. However, using the generalized integro-exponential function Milgram (1985) exact formulas can be derived for its moment-generating function and central moments. Based on the exact central moments, higher accuracy approximations can be defined for them. In demographic or actuarial applications, maximum-likelihood estimation is often used to determine the parameters of the Gompertz distribution. By solving the maximum-likelihood estimates analytically, the dimension of the optimization problem can be reduced to one both in the case of discrete and continuous data. -
Cluster Analysis for Gene Expression Data: a Survey
Cluster Analysis for Gene Expression Data: A Survey Daxin Jiang Chun Tang Aidong Zhang Department of Computer Science and Engineering State University of New York at Buffalo Email: djiang3, chuntang, azhang @cse.buffalo.edu Abstract DNA microarray technology has now made it possible to simultaneously monitor the expres- sion levels of thousands of genes during important biological processes and across collections of related samples. Elucidating the patterns hidden in gene expression data offers a tremen- dous opportunity for an enhanced understanding of functional genomics. However, the large number of genes and the complexity of biological networks greatly increase the challenges of comprehending and interpreting the resulting mass of data, which often consists of millions of measurements. A first step toward addressing this challenge is the use of clustering techniques, which is essential in the data mining process to reveal natural structures and identify interesting patterns in the underlying data. Cluster analysis seeks to partition a given data set into groups based on specified features so that the data points within a group are more similar to each other than the points in different groups. A very rich literature on cluster analysis has developed over the past three decades. Many conventional clustering algorithms have been adapted or directly applied to gene expres- sion data, and also new algorithms have recently been proposed specifically aiming at gene ex- pression data. These clustering algorithms have been proven useful for identifying biologically relevant groups of genes and samples. In this paper, we first briefly introduce the concepts of microarray technology and discuss the basic elements of clustering on gene expression data. -
Reliability Engineering: Today and Beyond
Reliability Engineering: Today and Beyond Keynote Talk at the 6th Annual Conference of the Institute for Quality and Reliability Tsinghua University People's Republic of China by Professor Mohammad Modarres Director, Center for Risk and Reliability Department of Mechanical Engineering Outline – A New Era in Reliability Engineering – Reliability Engineering Timeline and Research Frontiers – Prognostics and Health Management – Physics of Failure – Data-driven Approaches in PHM – Hybrid Methods – Conclusions New Era in Reliability Sciences and Engineering • Started as an afterthought analysis – In enduing years dismissed as a legitimate field of science and engineering – Worked with small data • Three advances transformed reliability into a legitimate science: – 1. Availability of inexpensive sensors and information systems – 2. Ability to better described physics of damage, degradation, and failure time using empirical and theoretical sciences – 3. Access to big data and PHM techniques for diagnosing faults and incipient failures • Today we can predict abnormalities, offer just-in-time remedies to avert failures, and making systems robust and resilient to failures Seventy Years of Reliability Engineering – Reliability Engineering Initiatives in 1950’s • Weakest link • Exponential life model • Reliability Block Diagrams (RBDs) – Beyond Exp. Dist. & Birth of System Reliability in 1960’s • Birth of Physics of Failure (POF) • Uses of more proper distributions (Weibull, etc.) • Reliability growth • Life testing • Failure Mode and Effect Analysis -
The Likelihood Principle
1 01/28/99 ãMarc Nerlove 1999 Chapter 1: The Likelihood Principle "What has now appeared is that the mathematical concept of probability is ... inadequate to express our mental confidence or diffidence in making ... inferences, and that the mathematical quantity which usually appears to be appropriate for measuring our order of preference among different possible populations does not in fact obey the laws of probability. To distinguish it from probability, I have used the term 'Likelihood' to designate this quantity; since both the words 'likelihood' and 'probability' are loosely used in common speech to cover both kinds of relationship." R. A. Fisher, Statistical Methods for Research Workers, 1925. "What we can find from a sample is the likelihood of any particular value of r [a parameter], if we define the likelihood as a quantity proportional to the probability that, from a particular population having that particular value of r, a sample having the observed value r [a statistic] should be obtained. So defined, probability and likelihood are quantities of an entirely different nature." R. A. Fisher, "On the 'Probable Error' of a Coefficient of Correlation Deduced from a Small Sample," Metron, 1:3-32, 1921. Introduction The likelihood principle as stated by Edwards (1972, p. 30) is that Within the framework of a statistical model, all the information which the data provide concerning the relative merits of two hypotheses is contained in the likelihood ratio of those hypotheses on the data. ...For a continuum of hypotheses, this principle -
Biostatistics (BIOSTAT) 1
Biostatistics (BIOSTAT) 1 This course covers practical aspects of conducting a population- BIOSTATISTICS (BIOSTAT) based research study. Concepts include determining a study budget, setting a timeline, identifying study team members, setting a strategy BIOSTAT 301-0 Introduction to Epidemiology (1 Unit) for recruitment and retention, developing a data collection protocol This course introduces epidemiology and its uses for population health and monitoring data collection to ensure quality control and quality research. Concepts include measures of disease occurrence, common assurance. Students will demonstrate these skills by engaging in a sources and types of data, important study designs, sources of error in quarter-long group project to draft a Manual of Operations for a new epidemiologic studies and epidemiologic methods. "mock" population study. BIOSTAT 302-0 Introduction to Biostatistics (1 Unit) BIOSTAT 429-0 Systematic Review and Meta-Analysis in the Medical This course introduces principles of biostatistics and applications Sciences (1 Unit) of statistical methods in health and medical research. Concepts This course covers statistical methods for meta-analysis. Concepts include descriptive statistics, basic probability, probability distributions, include fixed-effects and random-effects models, measures of estimation, hypothesis testing, correlation and simple linear regression. heterogeneity, prediction intervals, meta regression, power assessment, BIOSTAT 303-0 Probability (1 Unit) subgroup analysis and assessment of publication -
A Family of Skew-Normal Distributions for Modeling Proportions and Rates with Zeros/Ones Excess
S S symmetry Article A Family of Skew-Normal Distributions for Modeling Proportions and Rates with Zeros/Ones Excess Guillermo Martínez-Flórez 1, Víctor Leiva 2,* , Emilio Gómez-Déniz 3 and Carolina Marchant 4 1 Departamento de Matemáticas y Estadística, Facultad de Ciencias Básicas, Universidad de Córdoba, Montería 14014, Colombia; [email protected] 2 Escuela de Ingeniería Industrial, Pontificia Universidad Católica de Valparaíso, 2362807 Valparaíso, Chile 3 Facultad de Economía, Empresa y Turismo, Universidad de Las Palmas de Gran Canaria and TIDES Institute, 35001 Canarias, Spain; [email protected] 4 Facultad de Ciencias Básicas, Universidad Católica del Maule, 3466706 Talca, Chile; [email protected] * Correspondence: [email protected] or [email protected] Received: 30 June 2020; Accepted: 19 August 2020; Published: 1 September 2020 Abstract: In this paper, we consider skew-normal distributions for constructing new a distribution which allows us to model proportions and rates with zero/one inflation as an alternative to the inflated beta distributions. The new distribution is a mixture between a Bernoulli distribution for explaining the zero/one excess and a censored skew-normal distribution for the continuous variable. The maximum likelihood method is used for parameter estimation. Observed and expected Fisher information matrices are derived to conduct likelihood-based inference in this new type skew-normal distribution. Given the flexibility of the new distributions, we are able to show, in real data scenarios, the good performance of our proposal. Keywords: beta distribution; centered skew-normal distribution; maximum-likelihood methods; Monte Carlo simulations; proportions; R software; rates; zero/one inflated data 1. -
Big Data for Reliability Engineering: Threat and Opportunity
Reliability, February 2016 Big Data for Reliability Engineering: Threat and Opportunity Vitali Volovoi Independent Consultant [email protected] more recently, analytics). It shares with the rest of the fields Abstract - The confluence of several technologies promises under this umbrella the need to abstract away most stormy waters ahead for reliability engineering. News reports domain-specific information, and to use tools that are mainly are full of buzzwords relevant to the future of the field—Big domain-independent1. As a result, it increasingly shares the Data, the Internet of Things, predictive and prescriptive lingua franca of modern systems engineering—probability and analytics—the sexier sisters of reliability engineering, both statistics that are required to balance the otherwise orderly and exciting and threatening. Can we reliability engineers join the deterministic engineering world. party and suddenly become popular (and better paid), or are And yet, reliability engineering does not wear the fancy we at risk of being superseded and driven into obsolescence? clothes of its sisters. There is nothing privileged about it. It is This article argues that“big-picture” thinking, which is at the rarely studied in engineering schools, and it is definitely not core of the concept of the System of Systems, is key for a studied in business schools! Instead, it is perceived as a bright future for reliability engineering. necessary evil (especially if the reliability issues in question are safety-related). The community of reliability engineers Keywords - System of Systems, complex systems, Big Data, consists of engineers from other fields who were mainly Internet of Things, industrial internet, predictive analytics, trained on the job (instead of receiving formal degrees in the prescriptive analytics field). -
Interactive Statistical Graphics/ When Charts Come to Life
Titel Event, Date Author Affiliation Interactive Statistical Graphics When Charts come to Life [email protected] www.theusRus.de Telefónica Germany Interactive Statistical Graphics – When Charts come to Life PSI Graphics One Day Meeting Martin Theus 2 www.theusRus.de What I do not talk about … Interactive Statistical Graphics – When Charts come to Life PSI Graphics One Day Meeting Martin Theus 3 www.theusRus.de … still not what I mean. Interactive Statistical Graphics – When Charts come to Life PSI Graphics One Day Meeting Martin Theus 4 www.theusRus.de Interactive Graphics ≠ Dynamic Graphics • Interactive Graphics … uses various interactions with the plots to change selections and parameters quickly. Interactive Statistical Graphics – When Charts come to Life PSI Graphics One Day Meeting Martin Theus 4 www.theusRus.de Interactive Graphics ≠ Dynamic Graphics • Interactive Graphics … uses various interactions with the plots to change selections and parameters quickly. • Dynamic Graphics … uses animated / rotating plots to visualize high dimensional (continuous) data. Interactive Statistical Graphics – When Charts come to Life PSI Graphics One Day Meeting Martin Theus 4 www.theusRus.de Interactive Graphics ≠ Dynamic Graphics • Interactive Graphics … uses various interactions with the plots to change selections and parameters quickly. • Dynamic Graphics … uses animated / rotating plots to visualize high dimensional (continuous) data. 1973 PRIM-9 Tukey et al. Interactive Statistical Graphics – When Charts come to Life PSI Graphics One Day Meeting Martin Theus 4 www.theusRus.de Interactive Graphics ≠ Dynamic Graphics • Interactive Graphics … uses various interactions with the plots to change selections and parameters quickly. • Dynamic Graphics … uses animated / rotating plots to visualize high dimensional (continuous) data. -
Cluster Analysis Or Clustering Is a Common Technique for Statistical
IOSR Journal of Engineering Apr. 2012, Vol. 2(4) pp: 719-725 AN OVERVIEW ON CLUSTERING METHODS T. Soni Madhulatha Associate Professor, Alluri Institute of Management Sciences, Warangal. ABSTRACT Clustering is a common technique for statistical data analysis, which is used in many fields, including machine learning, data mining, pattern recognition, image analysis and bioinformatics. Clustering is the process of grouping similar objects into different groups, or more precisely, the partitioning of a data set into subsets, so that the data in each subset according to some defined distance measure. This paper covers about clustering algorithms, benefits and its applications. Paper concludes by discussing some limitations. Keywords: Clustering, hierarchical algorithm, partitional algorithm, distance measure, I. INTRODUCTION finding the length of the hypotenuse in a triangle; that is, it Clustering can be considered the most important is the distance "as the crow flies." A review of cluster unsupervised learning problem; so, as every other problem analysis in health psychology research found that the most of this kind, it deals with finding a structure in a collection common distance measure in published studies in that of unlabeled data. A cluster is therefore a collection of research area is the Euclidean distance or the squared objects which are “similar” between them and are Euclidean distance. “dissimilar” to the objects belonging to other clusters. Besides the term data clustering as synonyms like cluster The Manhattan distance function computes the analysis, automatic classification, numerical taxonomy, distance that would be traveled to get from one data point to botrology and typological analysis. the other if a grid-like path is followed. -
Cluster Analysis: What It Is and How to Use It Alyssa Wittle and Michael Stackhouse, Covance, Inc
PharmaSUG 2019 - Paper ST-183 Cluster Analysis: What It Is and How to Use It Alyssa Wittle and Michael Stackhouse, Covance, Inc. ABSTRACT A Cluster Analysis is a great way of looking across several related data points to find possible relationships within your data which you may not have expected. The basic approach of a cluster analysis is to do the following: transform the results of a series of related variables into a standardized value such as Z-scores, then combine these values and determine if there are trends across the data which may lend the data to divide into separate, distinct groups, or "clusters". A cluster is assigned at a subject level, to be used as a grouping variable or even as a response variable. Once these clusters have been determined and assigned, they can be used in your analysis model to observe if there is a significant difference between the results of these clusters within various parameters. For example, is a certain age group more likely to give more positive answers across all questionnaires in a study or integration? Cluster analysis can also be a good way of determining exploratory endpoints or focusing an analysis on a certain number of categories for a set of variables. This paper will instruct on approaches to a clustering analysis, how the results can be interpreted, and how clusters can be determined and analyzed using several programming methods and languages, including SAS, Python and R. Examples of clustering analyses and their interpretations will also be provided. INTRODUCTION A cluster analysis is a multivariate data exploration method gaining popularity in the industry. -
11. Parameter Estimation
11. Parameter Estimation Chris Piech and Mehran Sahami May 2017 We have learned many different distributions for random variables and all of those distributions had parame- ters: the numbers that you provide as input when you define a random variable. So far when we were working with random variables, we either were explicitly told the values of the parameters, or, we could divine the values by understanding the process that was generating the random variables. What if we don’t know the values of the parameters and we can’t estimate them from our own expert knowl- edge? What if instead of knowing the random variables, we have a lot of examples of data generated with the same underlying distribution? In this chapter we are going to learn formal ways of estimating parameters from data. These ideas are critical for artificial intelligence. Almost all modern machine learning algorithms work like this: (1) specify a probabilistic model that has parameters. (2) Learn the value of those parameters from data. Parameters Before we dive into parameter estimation, first let’s revisit the concept of parameters. Given a model, the parameters are the numbers that yield the actual distribution. In the case of a Bernoulli random variable, the single parameter was the value p. In the case of a Uniform random variable, the parameters are the a and b values that define the min and max value. Here is a list of random variables and the corresponding parameters. From now on, we are going to use the notation q to be a vector of all the parameters: Distribution Parameters Bernoulli(p) q = p Poisson(l) q = l Uniform(a,b) q = (a;b) Normal(m;s 2) q = (m;s 2) Y = mX + b q = (m;b) In the real world often you don’t know the “true” parameters, but you get to observe data.