2. Linear Models for Continuous Data
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A Generalized Linear Model for Principal Component Analysis of Binary Data
A Generalized Linear Model for Principal Component Analysis of Binary Data Andrew I. Schein Lawrence K. Saul Lyle H. Ungar Department of Computer and Information Science University of Pennsylvania Moore School Building 200 South 33rd Street Philadelphia, PA 19104-6389 {ais,lsaul,ungar}@cis.upenn.edu Abstract they are not generally appropriate for other data types. Recently, Collins et al.[5] derived generalized criteria We investigate a generalized linear model for for dimensionality reduction by appealing to proper- dimensionality reduction of binary data. The ties of distributions in the exponential family. In their model is related to principal component anal- framework, the conventional PCA of real-valued data ysis (PCA) in the same way that logistic re- emerges naturally from assuming a Gaussian distribu- gression is related to linear regression. Thus tion over a set of observations, while generalized ver- we refer to the model as logistic PCA. In this sions of PCA for binary and nonnegative data emerge paper, we derive an alternating least squares respectively by substituting the Bernoulli and Pois- method to estimate the basis vectors and gen- son distributions for the Gaussian. For binary data, eralized linear coefficients of the logistic PCA the generalized model's relationship to PCA is anal- model. The resulting updates have a simple ogous to the relationship between logistic and linear closed form and are guaranteed at each iter- regression[12]. In particular, the model exploits the ation to improve the model's likelihood. We log-odds as the natural parameter of the Bernoulli dis- evaluate the performance of logistic PCA|as tribution and the logistic function as its canonical link. -
Understanding Linear and Logistic Regression Analyses
EDUCATION • ÉDUCATION METHODOLOGY Understanding linear and logistic regression analyses Andrew Worster, MD, MSc;*† Jerome Fan, MD;* Afisi Ismaila, MSc† SEE RELATED ARTICLE PAGE 105 egression analysis, also termed regression modeling, come). For example, a researcher could evaluate the poten- Ris an increasingly common statistical method used to tial for injury severity score (ISS) to predict ED length-of- describe and quantify the relation between a clinical out- stay by first producing a scatter plot of ISS graphed against come of interest and one or more other variables. In this ED length-of-stay to determine whether an apparent linear issue of CJEM, Cummings and Mayes used linear and lo- relation exists, and then by deriving the best fit straight line gistic regression to determine whether the type of trauma for the data set using linear regression carried out by statis- team leader (TTL) impacts emergency department (ED) tical software. The mathematical formula for this relation length-of-stay or survival.1 The purpose of this educa- would be: ED length-of-stay = k(ISS) + c. In this equation, tional primer is to provide an easily understood overview k (the slope of the line) indicates the factor by which of these methods of statistical analysis. We hope that this length-of-stay changes as ISS changes and c (the “con- primer will not only help readers interpret the Cummings stant”) is the value of length-of-stay when ISS equals zero and Mayes study, but also other research that uses similar and crosses the vertical axis.2 In this hypothetical scenario, methodology. -
Linear, Ridge Regression, and Principal Component Analysis
Linear, Ridge Regression, and Principal Component Analysis Linear, Ridge Regression, and Principal Component Analysis Jia Li Department of Statistics The Pennsylvania State University Email: [email protected] http://www.stat.psu.edu/∼jiali Jia Li http://www.stat.psu.edu/∼jiali Linear, Ridge Regression, and Principal Component Analysis Introduction to Regression I Input vector: X = (X1, X2, ..., Xp). I Output Y is real-valued. I Predict Y from X by f (X ) so that the expected loss function E(L(Y , f (X ))) is minimized. I Square loss: L(Y , f (X )) = (Y − f (X ))2 . I The optimal predictor ∗ 2 f (X ) = argminf (X )E(Y − f (X )) = E(Y | X ) . I The function E(Y | X ) is the regression function. Jia Li http://www.stat.psu.edu/∼jiali Linear, Ridge Regression, and Principal Component Analysis Example The number of active physicians in a Standard Metropolitan Statistical Area (SMSA), denoted by Y , is expected to be related to total population (X1, measured in thousands), land area (X2, measured in square miles), and total personal income (X3, measured in millions of dollars). Data are collected for 141 SMSAs, as shown in the following table. i : 1 2 3 ... 139 140 141 X1 9387 7031 7017 ... 233 232 231 X2 1348 4069 3719 ... 1011 813 654 X3 72100 52737 54542 ... 1337 1589 1148 Y 25627 15389 13326 ... 264 371 140 Goal: Predict Y from X1, X2, and X3. Jia Li http://www.stat.psu.edu/∼jiali Linear, Ridge Regression, and Principal Component Analysis Linear Methods I The linear regression model p X f (X ) = β0 + Xj βj . -
Application of General Linear Models (GLM) to Assess Nodule Abundance Based on a Photographic Survey (Case Study from IOM Area, Pacific Ocean)
minerals Article Application of General Linear Models (GLM) to Assess Nodule Abundance Based on a Photographic Survey (Case Study from IOM Area, Pacific Ocean) Monika Wasilewska-Błaszczyk * and Jacek Mucha Department of Geology of Mineral Deposits and Mining Geology, Faculty of Geology, Geophysics and Environmental Protection, AGH University of Science and Technology, 30-059 Cracow, Poland; [email protected] * Correspondence: [email protected] Abstract: The success of the future exploitation of the Pacific polymetallic nodule deposits depends on an accurate estimation of their resources, especially in small batches, scheduled for extraction in the short term. The estimation based only on the results of direct seafloor sampling using box corers is burdened with a large error due to the long sampling interval and high variability of the nodule abundance. Therefore, estimations should take into account the results of bottom photograph analyses performed systematically and in large numbers along the course of a research vessel. For photographs taken at the direct sampling sites, the relationship linking the nodule abundance with the independent variables (the percentage of seafloor nodule coverage, the genetic types of nodules in the context of their fraction distribution, and the degree of sediment coverage of nodules) was determined using the general linear model (GLM). Compared to the estimates obtained with a simple Citation: Wasilewska-Błaszczyk, M.; linear model linking this parameter only with the seafloor nodule coverage, a significant decrease Mucha, J. Application of General in the standard prediction error, from 4.2 to 2.5 kg/m2, was found. The use of the GLM for the Linear Models (GLM) to Assess assessment of nodule abundance in individual sites covered by bottom photographs, outside of Nodule Abundance Based on a direct sampling sites, should contribute to a significant increase in the accuracy of the estimation of Photographic Survey (Case Study nodule resources. -
The Simple Linear Regression Model
The Simple Linear Regression Model Suppose we have a data set consisting of n bivariate observations {(x1, y1),..., (xn, yn)}. Response variable y and predictor variable x satisfy the simple linear model if they obey the model yi = β0 + β1xi + ǫi, i = 1,...,n, (1) where the intercept and slope coefficients β0 and β1 are unknown constants and the random errors {ǫi} satisfy the following conditions: 1. The errors ǫ1, . , ǫn all have mean 0, i.e., µǫi = 0 for all i. 2 2 2 2. The errors ǫ1, . , ǫn all have the same variance σ , i.e., σǫi = σ for all i. 3. The errors ǫ1, . , ǫn are independent random variables. 4. The errors ǫ1, . , ǫn are normally distributed. Note that the author provides these assumptions on page 564 BUT ORDERS THEM DIFFERENTLY. Fitting the Simple Linear Model: Estimating β0 and β1 Suppose we believe our data obey the simple linear model. The next step is to fit the model by estimating the unknown intercept and slope coefficients β0 and β1. There are various ways of estimating these from the data but we will use the Least Squares Criterion invented by Gauss. The least squares estimates of β0 and β1, which we will denote by βˆ0 and βˆ1 respectively, are the values of β0 and β1 which minimize the sum of errors squared S(β0, β1): n 2 S(β0, β1) = X ei i=1 n 2 = X[yi − yˆi] i=1 n 2 = X[yi − (β0 + β1xi)] i=1 where the ith modeling error ei is simply the difference between the ith value of the response variable yi and the fitted/predicted valuey ˆi. -
Principal Component Analysis (PCA) As a Statistical Tool for Identifying Key Indicators of Nuclear Power Plant Cable Insulation
Iowa State University Capstones, Theses and Graduate Theses and Dissertations Dissertations 2017 Principal component analysis (PCA) as a statistical tool for identifying key indicators of nuclear power plant cable insulation degradation Chamila Chandima De Silva Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/etd Part of the Materials Science and Engineering Commons, Mechanics of Materials Commons, and the Statistics and Probability Commons Recommended Citation De Silva, Chamila Chandima, "Principal component analysis (PCA) as a statistical tool for identifying key indicators of nuclear power plant cable insulation degradation" (2017). Graduate Theses and Dissertations. 16120. https://lib.dr.iastate.edu/etd/16120 This Thesis is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Principal component analysis (PCA) as a statistical tool for identifying key indicators of nuclear power plant cable insulation degradation by Chamila C. De Silva A thesis submitted to the graduate faculty in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Major: Materials Science and Engineering Program of Study Committee: Nicola Bowler, Major Professor Richard LeSar Steve Martin The student author and the program of study committee are solely responsible for the content of this thesis. The Graduate College will ensure this thesis is globally accessible and will not permit alterations after a degree is conferred. -
Generalized Linear Models
CHAPTER 6 Generalized linear models 6.1 Introduction Generalized linear modeling is a framework for statistical analysis that includes linear and logistic regression as special cases. Linear regression directly predicts continuous data y from a linear predictor Xβ = β0 + X1β1 + + Xkβk.Logistic regression predicts Pr(y =1)forbinarydatafromalinearpredictorwithaninverse-··· logit transformation. A generalized linear model involves: 1. A data vector y =(y1,...,yn) 2. Predictors X and coefficients β,formingalinearpredictorXβ 1 3. A link function g,yieldingavectoroftransformeddataˆy = g− (Xβ)thatare used to model the data 4. A data distribution, p(y yˆ) | 5. Possibly other parameters, such as variances, overdispersions, and cutpoints, involved in the predictors, link function, and data distribution. The options in a generalized linear model are the transformation g and the data distribution p. In linear regression,thetransformationistheidentity(thatis,g(u) u)and • the data distribution is normal, with standard deviation σ estimated from≡ data. 1 1 In logistic regression,thetransformationistheinverse-logit,g− (u)=logit− (u) • (see Figure 5.2a on page 80) and the data distribution is defined by the proba- bility for binary data: Pr(y =1)=y ˆ. This chapter discusses several other classes of generalized linear model, which we list here for convenience: The Poisson model (Section 6.2) is used for count data; that is, where each • data point yi can equal 0, 1, 2, ....Theusualtransformationg used here is the logarithmic, so that g(u)=exp(u)transformsacontinuouslinearpredictorXiβ to a positivey ˆi.ThedatadistributionisPoisson. It is usually a good idea to add a parameter to this model to capture overdis- persion,thatis,variationinthedatabeyondwhatwouldbepredictedfromthe Poisson distribution alone. -
Time-Series Regression and Generalized Least Squares in R*
Time-Series Regression and Generalized Least Squares in R* An Appendix to An R Companion to Applied Regression, third edition John Fox & Sanford Weisberg last revision: 2018-09-26 Abstract Generalized least-squares (GLS) regression extends ordinary least-squares (OLS) estimation of the normal linear model by providing for possibly unequal error variances and for correlations between different errors. A common application of GLS estimation is to time-series regression, in which it is generally implausible to assume that errors are independent. This appendix to Fox and Weisberg (2019) briefly reviews GLS estimation and demonstrates its application to time-series data using the gls() function in the nlme package, which is part of the standard R distribution. 1 Generalized Least Squares In the standard linear model (for example, in Chapter 4 of the R Companion), E(yjX) = Xβ or, equivalently y = Xβ + " where y is the n×1 response vector; X is an n×k +1 model matrix, typically with an initial column of 1s for the regression constant; β is a k + 1 ×1 vector of regression coefficients to estimate; and " is 2 an n×1 vector of errors. Assuming that " ∼ Nn(0; σ In), or at least that the errors are uncorrelated and equally variable, leads to the familiar ordinary-least-squares (OLS) estimator of β, 0 −1 0 bOLS = (X X) X y with covariance matrix 2 0 −1 Var(bOLS) = σ (X X) More generally, we can assume that " ∼ Nn(0; Σ), where the error covariance matrix Σ is sym- metric and positive-definite. Different diagonal entries in Σ error variances that are not necessarily all equal, while nonzero off-diagonal entries correspond to correlated errors. -
Simple Linear Regression: Straight Line Regression Between an Outcome Variable (Y ) and a Single Explanatory Or Predictor Variable (X)
1 Introduction to Regression \Regression" is a generic term for statistical methods that attempt to fit a model to data, in order to quantify the relationship between the dependent (outcome) variable and the predictor (independent) variable(s). Assuming it fits the data reasonable well, the estimated model may then be used either to merely describe the relationship between the two groups of variables (explanatory), or to predict new values (prediction). There are many types of regression models, here are a few most common to epidemiology: Simple Linear Regression: Straight line regression between an outcome variable (Y ) and a single explanatory or predictor variable (X). E(Y ) = α + β × X Multiple Linear Regression: Same as Simple Linear Regression, but now with possibly multiple explanatory or predictor variables. E(Y ) = α + β1 × X1 + β2 × X2 + β3 × X3 + ::: A special case is polynomial regression. 2 3 E(Y ) = α + β1 × X + β2 × X + β3 × X + ::: Generalized Linear Model: Same as Multiple Linear Regression, but with a possibly transformed Y variable. This introduces considerable flexibil- ity, as non-linear and non-normal situations can be easily handled. G(E(Y )) = α + β1 × X1 + β2 × X2 + β3 × X3 + ::: In general, the transformation function G(Y ) can take any form, but a few forms are especially common: • Taking G(Y ) = logit(Y ) describes a logistic regression model: E(Y ) log( ) = α + β × X + β × X + β × X + ::: 1 − E(Y ) 1 1 2 2 3 3 2 • Taking G(Y ) = log(Y ) is also very common, leading to Poisson regression for count data, and other so called \log-linear" models. -
Chapter 2 Simple Linear Regression Analysis the Simple
Chapter 2 Simple Linear Regression Analysis The simple linear regression model We consider the modelling between the dependent and one independent variable. When there is only one independent variable in the linear regression model, the model is generally termed as a simple linear regression model. When there are more than one independent variables in the model, then the linear model is termed as the multiple linear regression model. The linear model Consider a simple linear regression model yX01 where y is termed as the dependent or study variable and X is termed as the independent or explanatory variable. The terms 0 and 1 are the parameters of the model. The parameter 0 is termed as an intercept term, and the parameter 1 is termed as the slope parameter. These parameters are usually called as regression coefficients. The unobservable error component accounts for the failure of data to lie on the straight line and represents the difference between the true and observed realization of y . There can be several reasons for such difference, e.g., the effect of all deleted variables in the model, variables may be qualitative, inherent randomness in the observations etc. We assume that is observed as independent and identically distributed random variable with mean zero and constant variance 2 . Later, we will additionally assume that is normally distributed. The independent variables are viewed as controlled by the experimenter, so it is considered as non-stochastic whereas y is viewed as a random variable with Ey()01 X and Var() y 2 . Sometimes X can also be a random variable. -
Week 7: Multiple Regression
Week 7: Multiple Regression Brandon Stewart1 Princeton October 24, 26, 2016 1These slides are heavily influenced by Matt Blackwell, Adam Glynn, Jens Hainmueller and Danny Hidalgo. Stewart (Princeton) Week 7: Multiple Regression October 24, 26, 2016 1 / 145 Where We've Been and Where We're Going... Last Week I regression with two variables I omitted variables, multicollinearity, interactions This Week I Monday: F matrix form of linear regression I Wednesday: F hypothesis tests Next Week I break! I then ::: regression in social science Long Run I probability ! inference ! regression Questions? Stewart (Princeton) Week 7: Multiple Regression October 24, 26, 2016 2 / 145 1 Matrix Algebra Refresher 2 OLS in matrix form 3 OLS inference in matrix form 4 Inference via the Bootstrap 5 Some Technical Details 6 Fun With Weights 7 Appendix 8 Testing Hypotheses about Individual Coefficients 9 Testing Linear Hypotheses: A Simple Case 10 Testing Joint Significance 11 Testing Linear Hypotheses: The General Case 12 Fun With(out) Weights Stewart (Princeton) Week 7: Multiple Regression October 24, 26, 2016 3 / 145 Why Matrices and Vectors? Here's one way to write the full multiple regression model: yi = β0 + xi1β1 + xi2β2 + ··· + xiK βK + ui Notation is going to get needlessly messy as we add variables Matrices are clean, but they are like a foreign language You need to build intuitions over a long period of time (and they will return in Soc504) Reminder of Parameter Interpretation: β1 is the effect of a one-unit change in xi1 conditional on all other xik . We are going to review the key points quite quickly just to refresh the basics. -
The Conspiracy of Random Predictors and Model Violations Against Classical Inference in Regression 1 Introduction
The Conspiracy of Random Predictors and Model Violations against Classical Inference in Regression A. Buja, R. Berk, L. Brown, E. George, E. Pitkin, M. Traskin, K. Zhang, L. Zhao March 10, 2014 Dedicated to Halbert White (y2012) Abstract We review the early insights of Halbert White who over thirty years ago inaugurated a form of statistical inference for regression models that is asymptotically correct even under \model misspecification.” This form of inference, which is pervasive in econometrics, relies on the \sandwich estimator" of standard error. Whereas the classical theory of linear models in statistics assumes models to be correct and predictors to be fixed, White permits models to be \misspecified” and predictors to be random. Careful reading of his theory shows that it is a synergistic effect | a \conspiracy" | of nonlinearity and randomness of the predictors that has the deepest consequences for statistical inference. It will be seen that the synonym \heteroskedasticity-consistent estimator" for the sandwich estimator is misleading because nonlinearity is a more consequential form of model deviation than heteroskedasticity, and both forms are handled asymptotically correctly by the sandwich estimator. The same analysis shows that a valid alternative to the sandwich estimator is given by the \pairs bootstrap" for which we establish a direct connection to the sandwich estimator. We continue with an asymptotic comparison of the sandwich estimator and the standard error estimator from classical linear models theory. The comparison shows that when standard errors from linear models theory deviate from their sandwich analogs, they are usually too liberal, but occasionally they can be too conservative as well.