Generalized Linear Models and Generalized Additive Models
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Stochastic Process - Introduction
Stochastic Process - Introduction • Stochastic processes are processes that proceed randomly in time. • Rather than consider fixed random variables X, Y, etc. or even sequences of i.i.d random variables, we consider sequences X0, X1, X2, …. Where Xt represent some random quantity at time t. • In general, the value Xt might depend on the quantity Xt-1 at time t-1, or even the value Xs for other times s < t. • Example: simple random walk . week 2 1 Stochastic Process - Definition • A stochastic process is a family of time indexed random variables Xt where t belongs to an index set. Formal notation, { t : ∈ ItX } where I is an index set that is a subset of R. • Examples of index sets: 1) I = (-∞, ∞) or I = [0, ∞]. In this case Xt is a continuous time stochastic process. 2) I = {0, ±1, ±2, ….} or I = {0, 1, 2, …}. In this case Xt is a discrete time stochastic process. • We use uppercase letter {Xt } to describe the process. A time series, {xt } is a realization or sample function from a certain process. • We use information from a time series to estimate parameters and properties of process {Xt }. week 2 2 Probability Distribution of a Process • For any stochastic process with index set I, its probability distribution function is uniquely determined by its finite dimensional distributions. •The k dimensional distribution function of a process is defined by FXX x,..., ( 1 x,..., k ) = P( Xt ≤ 1 ,..., xt≤ X k) x t1 tk 1 k for anyt 1 ,..., t k ∈ I and any real numbers x1, …, xk . -
Generalized and Weighted Least Squares Estimation
LINEAR REGRESSION ANALYSIS MODULE – VII Lecture – 25 Generalized and Weighted Least Squares Estimation Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur 2 The usual linear regression model assumes that all the random error components are identically and independently distributed with constant variance. When this assumption is violated, then ordinary least squares estimator of regression coefficient looses its property of minimum variance in the class of linear and unbiased estimators. The violation of such assumption can arise in anyone of the following situations: 1. The variance of random error components is not constant. 2. The random error components are not independent. 3. The random error components do not have constant variance as well as they are not independent. In such cases, the covariance matrix of random error components does not remain in the form of an identity matrix but can be considered as any positive definite matrix. Under such assumption, the OLSE does not remain efficient as in the case of identity covariance matrix. The generalized or weighted least squares method is used in such situations to estimate the parameters of the model. In this method, the deviation between the observed and expected values of yi is multiplied by a weight ω i where ω i is chosen to be inversely proportional to the variance of yi. n 2 For simple linear regression model, the weighted least squares function is S(,)ββ01=∑ ωii( yx −− β0 β 1i) . ββ The least squares normal equations are obtained by differentiating S (,) ββ 01 with respect to 01 and and equating them to zero as nn n ˆˆ β01∑∑∑ ωβi+= ωiixy ω ii ii=11= i= 1 n nn ˆˆ2 βω01∑iix+= βω ∑∑ii x ω iii xy. -
Variance Function Regressions for Studying Inequality
Variance Function Regressions for Studying Inequality The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Western, Bruce and Deirdre Bloome. 2009. Variance function regressions for studying inequality. Working paper, Department of Sociology, Harvard University. Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:2645469 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Open Access Policy Articles, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#OAP Variance Function Regressions for Studying Inequality Bruce Western1 Deirdre Bloome Harvard University January 2009 1Department of Sociology, 33 Kirkland Street, Cambridge MA 02138. E-mail: [email protected]. This research was supported by a grant from the Russell Sage Foundation. Abstract Regression-based studies of inequality model only between-group differ- ences, yet often these differences are far exceeded by residual inequality. Residual inequality is usually attributed to measurement error or the in- fluence of unobserved characteristics. We present a regression that in- cludes covariates for both the mean and variance of a dependent variable. In this model, the residual variance is treated as a target for analysis. In analyses of inequality, the residual variance might be interpreted as mea- suring risk or insecurity. Variance function regressions are illustrated in an analysis of panel data on earnings among released prisoners in the Na- tional Longitudinal Survey of Youth. We extend the model to a decomposi- tion analysis, relating the change in inequality to compositional changes in the population and changes in coefficients for the mean and variance. -
Flexible Signal Denoising Via Flexible Empirical Bayes Shrinkage
Journal of Machine Learning Research 22 (2021) 1-28 Submitted 1/19; Revised 9/20; Published 5/21 Flexible Signal Denoising via Flexible Empirical Bayes Shrinkage Zhengrong Xing [email protected] Department of Statistics University of Chicago Chicago, IL 60637, USA Peter Carbonetto [email protected] Research Computing Center and Department of Human Genetics University of Chicago Chicago, IL 60637, USA Matthew Stephens [email protected] Department of Statistics and Department of Human Genetics University of Chicago Chicago, IL 60637, USA Editor: Edo Airoldi Abstract Signal denoising—also known as non-parametric regression—is often performed through shrinkage estima- tion in a transformed (e.g., wavelet) domain; shrinkage in the transformed domain corresponds to smoothing in the original domain. A key question in such applications is how much to shrink, or, equivalently, how much to smooth. Empirical Bayes shrinkage methods provide an attractive solution to this problem; they use the data to estimate a distribution of underlying “effects,” hence automatically select an appropriate amount of shrinkage. However, most existing implementations of empirical Bayes shrinkage are less flexible than they could be—both in their assumptions on the underlying distribution of effects, and in their ability to han- dle heteroskedasticity—which limits their signal denoising applications. Here we address this by adopting a particularly flexible, stable and computationally convenient empirical Bayes shrinkage method and apply- ing it to several signal denoising problems. These applications include smoothing of Poisson data and het- eroskedastic Gaussian data. We show through empirical comparisons that the results are competitive with other methods, including both simple thresholding rules and purpose-built empirical Bayes procedures. -
18.650 (F16) Lecture 10: Generalized Linear Models (Glms)
Statistics for Applications Chapter 10: Generalized Linear Models (GLMs) 1/52 Linear model A linear model assumes Y X (µ(X), σ2I), | ∼ N And ⊤ IE(Y X) = µ(X) = X β, | 2/52 Components of a linear model The two components (that we are going to relax) are 1. Random component: the response variable Y X is continuous | and normally distributed with mean µ = µ(X) = IE(Y X). | 2. Link: between the random and covariates X = (X(1),X(2), ,X(p))⊤: µ(X) = X⊤β. · · · 3/52 Generalization A generalized linear model (GLM) generalizes normal linear regression models in the following directions. 1. Random component: Y some exponential family distribution ∼ 2. Link: between the random and covariates: ⊤ g µ(X) = X β where g called link function� and� µ = IE(Y X). | 4/52 Example 1: Disease Occuring Rate In the early stages of a disease epidemic, the rate at which new cases occur can often increase exponentially through time. Hence, if µi is the expected number of new cases on day ti, a model of the form µi = γ exp(δti) seems appropriate. ◮ Such a model can be turned into GLM form, by using a log link so that log(µi) = log(γ) + δti = β0 + β1ti. ◮ Since this is a count, the Poisson distribution (with expected value µi) is probably a reasonable distribution to try. 5/52 Example 2: Prey Capture Rate(1) The rate of capture of preys, yi, by a hunting animal, tends to increase with increasing density of prey, xi, but to eventually level off, when the predator is catching as much as it can cope with. -
Variance Function Estimation in Multivariate Nonparametric Regression by T
Variance Function Estimation in Multivariate Nonparametric Regression by T. Tony Cai, Lie Wang University of Pennsylvania Michael Levine Purdue University Technical Report #06-09 Department of Statistics Purdue University West Lafayette, IN USA October 2006 Variance Function Estimation in Multivariate Nonparametric Regression T. Tony Cail, Michael Levine* Lie Wangl October 3, 2006 Abstract Variance function estimation in multivariate nonparametric regression is consid- ered and the minimax rate of convergence is established. Our work uses the approach that generalizes the one used in Munk et al (2005) for the constant variance case. As is the case when the number of dimensions d = 1, and very much contrary to the common practice, it is often not desirable to base the estimator of the variance func- tion on the residuals from an optimal estimator of the mean. Instead it is desirable to use estimators of the mean with minimal bias. Another important conclusion is that the first order difference-based estimator that achieves minimax rate of convergence in one-dimensional case does not do the same in the high dimensional case. Instead, the optimal order of differences depends on the number of dimensions. Keywords: Minimax estimation, nonparametric regression, variance estimation. AMS 2000 Subject Classification: Primary: 62G08, 62G20. Department of Statistics, The Wharton School, University of Pennsylvania, Philadelphia, PA 19104. The research of Tony Cai was supported in part by NSF Grant DMS-0306576. 'Corresponding author. Address: 250 N. University Street, Purdue University, West Lafayette, IN 47907. Email: [email protected]. Phone: 765-496-7571. Fax: 765-494-0558 1 Introduction We consider the multivariate nonparametric regression problem where yi E R, xi E S = [0, Ild C Rd while a, are iid random variables with zero mean and unit variance and have bounded absolute fourth moments: E lail 5 p4 < m. -
Variance Function Program
Variance Function Program Version 15.0 (for Windows XP and later) July 2018 W.A. Sadler 71B Middleton Road Christchurch 8041 New Zealand Ph: +64 3 343 3808 e-mail: [email protected] (formerly at Nuclear Medicine Department, Christchurch Hospital) Contents Page Variance Function Program 15.0 1 Windows Vista through Windows 10 Issues 1 Program Help 1 Gestures 1 Program Structure 2 Data Type 3 Automation 3 Program Data Area 3 User Preferences File 4 Auto-processing File 4 Graph Templates 4 Decimal Separator 4 Screen Settings 4 Scrolling through Summaries and Displays 4 The Variance Function 5 Variance Function Output Examples 8 Variance Stabilisation 11 Histogram, Error and Biological Variation Plots 12 Regression Analysis 13 Limit of Blank (LoB) and Limit of Detection (LoD) 14 Bland-Altman Analysis 14 Appendix A (Program Delivery) 16 Appendix B (Program Installation) 16 Appendix C (Program Removal) 16 Appendix D (Example Data, SD and Regression Files) 16 Appendix E (Auto-processing Example Files) 17 Appendix F (History) 17 Appendix G (Changes: Version 14.0 to Version 15.0) 18 Appendix H (Version 14.0 Bug Fixes) 19 References 20 Variance Function Program 15.0 1 Variance Function Program 15.0 The variance function (the relationship between variance and the mean) has several applications in statistical analysis of medical laboratory data (eg. Ref. 1), but the single most important use is probably the construction of (im)precision profile plots (2). This program (VFP.exe) was initially aimed at immunoassay data which exhibit relatively extreme variance relationships. It was based around the “standard” 3-parameter variance function, 2 J σ = (β1 + β2U) 2 where σ denotes variance, U denotes the mean, and β1, β2 and J are the parameters (3, 4). -
4.3 Least Squares Approximations
218 Chapter 4. Orthogonality 4.3 Least Squares Approximations It often happens that Ax D b has no solution. The usual reason is: too many equations. The matrix has more rows than columns. There are more equations than unknowns (m is greater than n). The n columns span a small part of m-dimensional space. Unless all measurements are perfect, b is outside that column space. Elimination reaches an impossible equation and stops. But we can’t stop just because measurements include noise. To repeat: We cannot always get the error e D b Ax down to zero. When e is zero, x is an exact solution to Ax D b. When the length of e is as small as possible, bx is a least squares solution. Our goal in this section is to compute bx and use it. These are real problems and they need an answer. The previous section emphasized p (the projection). This section emphasizes bx (the least squares solution). They are connected by p D Abx. The fundamental equation is still ATAbx D ATb. Here is a short unofficial way to reach this equation: When Ax D b has no solution, multiply by AT and solve ATAbx D ATb: Example 1 A crucial application of least squares is fitting a straight line to m points. Start with three points: Find the closest line to the points .0; 6/; .1; 0/, and .2; 0/. No straight line b D C C Dt goes through those three points. We are asking for two numbers C and D that satisfy three equations. -
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. -
Statistical Properties of Least Squares Estimates
1 STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS Recall: Assumption: E(Y|x) = η0 + η1x (linear conditional mean function) Data: (x1, y1), (x2, y2), … , (xn, yn) ˆ Least squares estimator: E (Y|x) = "ˆ 0 +"ˆ 1 x, where SXY "ˆ = "ˆ = y -"ˆ x 1 SXX 0 1 2 SXX = ∑ ( xi - x ) = ∑ xi( xi - x ) SXY = ∑ ( xi - x ) (yi - y ) = ∑ ( xi - x ) yi Comments: 1. So far we haven’t used any assumptions about conditional variance. 2. If our data were the entire population, we could also use the same least squares procedure to fit an approximate line to the conditional sample means. 3. Or, if we just had data, we could fit a line to the data, but nothing could be inferred beyond the data. 4. (Assuming again that we have a simple random sample from the population.) If we also assume e|x (equivalently, Y|x) is normal with constant variance, then the least squares estimates are the same as the maximum likelihood estimates of η0 and η1. Properties of "ˆ 0 and "ˆ 1 : n #(xi " x )yi n n SXY i=1 (xi " x ) 1) "ˆ 1 = = = # yi = "ci yi SXX SXX i=1 SXX i=1 (x " x ) where c = i i SXX Thus: If the xi's are fixed (as in the blood lactic acid example), then "ˆ 1 is a linear ! ! ! combination of the yi's. Note: Here we want to think of each y as a random variable with distribution Y|x . Thus, ! ! ! ! ! i i ! if the yi’s are independent and each Y|xi is normal, then "ˆ 1 is also normal. -
Examining Residuals
Stat 565 Examining Residuals Jan 14 2016 Charlotte Wickham stat565.cwick.co.nz So far... xt = mt + st + zt Variable Trend Seasonality Noise measured at time t Estimate and{ subtract off Should be left with this, stationary but probably has serial correlation Residuals in Corvallis temperature series Temp - loess smooth on day of year - loess smooth on date Your turn Now I have residuals, how could I check the variance doesn't change through time (i.e. is stationary)? Is the variance stationary? Same checks as for mean except using squared residuals or absolute value of residuals. Why? var(x) = 1/n ∑ ( x - μ)2 Converts a visual comparison of spread to a visual comparison of mean. Plot squared residuals against time qplot(date, residual^2, data = corv) + geom_smooth(method = "loess") Plot squared residuals against season qplot(yday, residual^2, data = corv) + geom_smooth(method = "loess", size = 1) Fitted values against residuals Looking for mean-variance relationship qplot(temp - residual, residual^2, data = corv) + geom_smooth(method = "loess", size = 1) Non-stationary variance Just like the mean you can attempt to remove the non-stationarity in variance. However, to remove non-stationary variance you divide by an estimate of the standard deviation. Your turn For the temperature series, serial dependence (a.k.a autocorrelation) means that today's residual is dependent on yesterday's residual. Any ideas of how we could check that? Is there autocorrelation in the residuals? > corv$residual_lag1 <- c(NA, corv$residual[-nrow(corv)]) > head(corv) . residual residual_lag1 . 1.5856663 NA xt-1 = lag 1 of xt . -0.4928295 1.5856663 . -
Models Where the Least Trimmed Squares and Least Median of Squares Estimators Are Maximum Likelihood
Models where the Least Trimmed Squares and Least Median of Squares estimators are maximum likelihood Vanessa Berenguer-Rico, Søren Johanseny& Bent Nielsenz 1 September 2019 Abstract The Least Trimmed Squares (LTS) and Least Median of Squares (LMS) estimators are popular robust regression estimators. The idea behind the estimators is to …nd, for a given h; a sub-sample of h ‘good’observations among n observations and esti- mate the regression on that sub-sample. We …nd models, based on the normal or the uniform distribution respectively, in which these estimators are maximum likelihood. We provide an asymptotic theory for the location-scale case in those models. The LTS estimator is found to be h1=2 consistent and asymptotically standard normal. The LMS estimator is found to be h consistent and asymptotically Laplace. Keywords: Chebychev estimator, LMS, Uniform distribution, Least squares esti- mator, LTS, Normal distribution, Regression, Robust statistics. 1 Introduction The Least Trimmed Squares (LTS) and the Least Median of Squares (LMS) estimators sug- gested by Rousseeuw (1984) are popular robust regression estimators. They are de…ned as follows. Consider a sample with n observations, where some are ‘good’and some are ‘out- liers’. The user chooses a number h and searches for a sub-sample of h ‘good’observations. The idea is to …nd the sub-sample with the smallest residual sum of squares –for LTS –or the least maximal squared residual –for LMS. In this paper, we …nd models in which these estimators are maximum likelihood. In these models, we …rst draw h ‘good’regression errors from a normal distribution, for LTS, or a uniform distribution, for LMS.