Multiple Regression 1
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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. -
LECTURES 2 - 3 : Stochastic Processes, Autocorrelation Function
LECTURES 2 - 3 : Stochastic Processes, Autocorrelation function. Stationarity. Important points of Lecture 1: A time series fXtg is a series of observations taken sequentially over time: xt is an observation recorded at a specific time t. Characteristics of times series data: observations are dependent, become available at equally spaced time points and are time-ordered. This is a discrete time series. The purposes of time series analysis are to model and to predict or forecast future values of a series based on the history of that series. 2.2 Some descriptive techniques. (Based on [BD] x1.3 and x1.4) ......................................................................................... Take a step backwards: how do we describe a r.v. or a random vector? ² for a r.v. X: 2 d.f. FX (x) := P (X · x), mean ¹ = EX and variance σ = V ar(X). ² for a r.vector (X1;X2): joint d.f. FX1;X2 (x1; x2) := P (X1 · x1;X2 · x2), marginal d.f.FX1 (x1) := P (X1 · x1) ´ FX1;X2 (x1; 1) 2 2 mean vector (¹1; ¹2) = (EX1; EX2), variances σ1 = V ar(X1); σ2 = V ar(X2), and covariance Cov(X1;X2) = E(X1 ¡ ¹1)(X2 ¡ ¹2) ´ E(X1X2) ¡ ¹1¹2. Often we use correlation = normalized covariance: Cor(X1;X2) = Cov(X1;X2)=fσ1σ2g ......................................................................................... To describe a process X1;X2;::: we define (i) Def. Distribution function: (fi-di) d.f. Ft1:::tn (x1; : : : ; xn) = P (Xt1 · x1;:::;Xtn · xn); i.e. this is the joint d.f. for the vector (Xt1 ;:::;Xtn ). (ii) First- and Second-order moments. ² Mean: ¹X (t) = EXt 2 2 2 2 ² Variance: σX (t) = E(Xt ¡ ¹X (t)) ´ EXt ¡ ¹X (t) 1 ² Autocovariance function: γX (t; s) = Cov(Xt;Xs) = E[(Xt ¡ ¹X (t))(Xs ¡ ¹X (s))] ´ E(XtXs) ¡ ¹X (t)¹X (s) (Note: this is an infinite matrix). -
Spatial Autocorrelation: Covariance and Semivariance Semivariance
Spatial Autocorrelation: Covariance and Semivariancence Lily Housese P eters GEOG 593 November 10, 2009 Quantitative Terrain Descriptorsrs Covariance and Semivariogram areare numericc methods used to describe the character of the terrain (ex. Terrain roughness, irregularity) Terrain character has important implications for: 1. the type of sampling strategy chosen 2. estimating DTM accuracy (after sampling and reconstruction) Spatial Autocorrelationon The First Law of Geography ““ Everything is related to everything else, but near things are moo re related than distant things.” (Waldo Tobler) The value of a variable at one point in space is related to the value of that same variable in a nearby location Ex. Moranan ’s I, Gearyary ’s C, LISA Positive Spatial Autocorrelation (Neighbors are similar) Negative Spatial Autocorrelation (Neighbors are dissimilar) R(d) = correlation coefficient of all the points with horizontal interval (d) Covariance The degree of similarity between pairs of surface points The value of similarity is an indicator of the complexity of the terrain surface Smaller the similarity = more complex the terrain surface V = Variance calculated from all N points Cov (d) = Covariance of all points with horizontal interval d Z i = Height of point i M = average height of all points Z i+d = elevation of the point with an interval of d from i Semivariancee Expresses the degree of relationship between points on a surface Equal to half the variance of the differences between all possible points spaced a constant distance apart -
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 . -
Fast Estimation of the Median Covariation Matrix with Application to Online Robust Principal Components Analysis
Fast Estimation of the Median Covariation Matrix with Application to Online Robust Principal Components Analysis Hervé Cardot, Antoine Godichon-Baggioni Institut de Mathématiques de Bourgogne, Université de Bourgogne Franche-Comté, 9, rue Alain Savary, 21078 Dijon, France July 12, 2016 Abstract The geometric median covariation matrix is a robust multivariate indicator of dis- persion which can be extended without any difficulty to functional data. We define estimators, based on recursive algorithms, that can be simply updated at each new observation and are able to deal rapidly with large samples of high dimensional data without being obliged to store all the data in memory. Asymptotic convergence prop- erties of the recursive algorithms are studied under weak conditions. The computation of the principal components can also be performed online and this approach can be useful for online outlier detection. A simulation study clearly shows that this robust indicator is a competitive alternative to minimum covariance determinant when the dimension of the data is small and robust principal components analysis based on projection pursuit and spherical projections for high dimension data. An illustration on a large sample and high dimensional dataset consisting of individual TV audiences measured at a minute scale over a period of 24 hours confirms the interest of consider- ing the robust principal components analysis based on the median covariation matrix. All studied algorithms are available in the R package Gmedian on CRAN. Keywords. Averaging, Functional data, Geometric median, Online algorithms, Online principal components, Recursive robust estimation, Stochastic gradient, Weiszfeld’s algo- arXiv:1504.02852v5 [math.ST] 9 Jul 2016 rithm. -
Characteristics and Statistics of Digital Remote Sensing Imagery (1)
Characteristics and statistics of digital remote sensing imagery (1) Digital Images: 1 Digital Image • With raster data structure, each image is treated as an array of values of the pixels. • Image data is organized as rows and columns (or lines and pixels) start from the upper left corner of the image. • Each pixel (picture element) is treated as a separate unite. Statistics of Digital Images Help: • Look at the frequency of occurrence of individual brightness values in the image displayed • View individual pixel brightness values at specific locations or within a geographic area; • Compute univariate descriptive statistics to determine if there are unusual anomalies in the image data; and • Compute multivariate statistics to determine the amount of between-band correlation (e.g., to identify redundancy). 2 Statistics of Digital Images It is necessary to calculate fundamental univariate and multivariate statistics of the multispectral remote sensor data. This involves identification and calculation of – maximum and minimum value –the range, mean, standard deviation – between-band variance-covariance matrix – correlation matrix, and – frequencies of brightness values The results of the above can be used to produce histograms. Such statistics provide information necessary for processing and analyzing remote sensing data. A “population” is an infinite or finite set of elements. A “sample” is a subset of the elements taken from a population used to make inferences about certain characteristics of the population. (e.g., training signatures) 3 Large samples drawn randomly from natural populations usually produce a symmetrical frequency distribution. Most values are clustered around the central value, and the frequency of occurrence declines away from this central point. -
Covariance of Cross-Correlations: Towards Efficient Measures for Large-Scale Structure
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by RERO DOC Digital Library Mon. Not. R. Astron. Soc. 400, 851–865 (2009) doi:10.1111/j.1365-2966.2009.15490.x Covariance of cross-correlations: towards efficient measures for large-scale structure Robert E. Smith Institute for Theoretical Physics, University of Zurich, Zurich CH 8037, Switzerland Accepted 2009 August 4. Received 2009 July 17; in original form 2009 June 13 ABSTRACT We study the covariance of the cross-power spectrum of different tracers for the large-scale structure. We develop the counts-in-cells framework for the multitracer approach, and use this to derive expressions for the full non-Gaussian covariance matrix. We show that for the usual autopower statistic, besides the off-diagonal covariance generated through gravitational mode- coupling, the discreteness of the tracers and their associated sampling distribution can generate strong off-diagonal covariance, and that this becomes the dominant source of covariance as spatial frequencies become larger than the fundamental mode of the survey volume. On comparison with the derived expressions for the cross-power covariance, we show that the off-diagonal terms can be suppressed, if one cross-correlates a high tracer-density sample with a low one. Taking the effective estimator efficiency to be proportional to the signal-to-noise ratio (S/N), we show that, to probe clustering as a function of physical properties of the sample, i.e. cluster mass or galaxy luminosity, the cross-power approach can outperform the autopower one by factors of a few. -
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. -
On the Use of the Autocorrelation and Covariance Methods for Feedforward Control of Transverse Angle and Position Jitter in Linear Particle Beam Accelerators*
'^C 7 ON THE USE OF THE AUTOCORRELATION AND COVARIANCE METHODS FOR FEEDFORWARD CONTROL OF TRANSVERSE ANGLE AND POSITION JITTER IN LINEAR PARTICLE BEAM ACCELERATORS* Dean S. Ban- Advanced Photon Source, Argonne National Laboratory, 9700 S. Cass Ave., Argonne, IL 60439 ABSTRACT It is desired to design a predictive feedforward transverse jitter control system to control both angle and position jitter m pulsed linear accelerators. Such a system will increase the accuracy and bandwidth of correction over that of currently available feedback correction systems. Intrapulse correction is performed. An offline process actually "learns" the properties of the jitter, and uses these properties to apply correction to the beam. The correction weights calculated offline are downloaded to a real-time analog correction system between macropulses. Jitter data were taken at the Los Alamos National Laboratory (LANL) Ground Test Accelerator (GTA) telescope experiment at Argonne National Laboratory (ANL). The experiment consisted of the LANL telescope connected to the ANL ZGS proton source and linac. A simulation of the correction system using this data was shown to decrease the average rms jitter by a factor of two over that of a comparable standard feedback correction system. The system also improved the correction bandwidth. INTRODUCTION Figure 1 shows the standard setup for a feedforward transverse jitter control system. Note that one pickup #1 and two kickers are needed to correct beam position jitter, while two pickup #l's and one kicker are needed to correct beam trajectory-angle jitter. pickup #1 kicker pickup #2 Beam Fast loop Slow loop Processor Figure 1. Feedforward Transverse Jitter Control System It is assumed that the beam is fast enough to beat the correction signal (through the fast loop) to the kicker. -
Lecture 4 Multivariate Normal Distribution and Multivariate CLT
Lecture 4 Multivariate normal distribution and multivariate CLT. T We start with several simple observations. If X = (x1; : : : ; xk) is a k 1 random vector then its expectation is × T EX = (Ex1; : : : ; Exk) and its covariance matrix is Cov(X) = E(X EX)(X EX)T : − − Notice that a covariance matrix is always symmetric Cov(X)T = Cov(X) and nonnegative definite, i.e. for any k 1 vector a, × a T Cov(X)a = Ea T (X EX)(X EX)T a T = E a T (X EX) 2 0: − − j − j � We will often use that for any vector X its squared length can be written as X 2 = XT X: If we multiply a random k 1 vector X by a n k matrix A then the covariancej j of Y = AX is a n n matrix × × × Cov(Y ) = EA(X EX)(X EX)T AT = ACov(X)AT : − − T Multivariate normal distribution. Let us consider a k 1 vector g = (g1; : : : ; gk) of i.i.d. standard normal random variables. The covariance of g is,× obviously, a k k identity × matrix, Cov(g) = I: Given a n k matrix A, the covariance of Ag is a n n matrix × × � := Cov(Ag) = AIAT = AAT : Definition. The distribution of a vector Ag is called a (multivariate) normal distribution with covariance � and is denoted N(0; �): One can also shift this disrtibution, the distribution of Ag + a is called a normal distri bution with mean a and covariance � and is denoted N(a; �): There is one potential problem 23 with the above definition - we assume that the distribution depends only on covariance ma trix � and does not depend on the construction, i.e. -
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.