DOCSLIB.ORG

Loss Functions for Image Restoration with Neural Networks Hang Zhao?,†, Orazio Gallo?, Iuri Frosio?, and Jan Kautz?

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Loss Functions for Image Restoration with Neural Networks Hang Zhao?,†, Orazio Gallo?, Iuri Frosio?, and Jan Kautz? 1 Loss Functions for Image Restoration with Neural Networks Hang Zhao?;y, Orazio Gallo?, Iuri Frosio?, and Jan Kautz? Abstract—Neural networks are becoming central in several popularity of `2: standard neural networks packages, such as areas of computer vision and image processing and different Caffe [11], only offer the implementation for this metric. architectures have been proposed to solve specific problems. The However, `2 suffers from well-known limitations. For in- impact of the loss layer of neural networks, however, has not ` received much attention in the context of image processing: stance, when the task at hand involves image quality, 2 the default and virtually only choice is `2. In this paper, we correlates poorly with image quality as perceived by a human bring attention to alternative choices for image restoration. In observer [12]. This is because of a number of assumptions particular, we show the importance of perceptually-motivated implicitly made when using `2. First and foremost, the use of losses when the resulting image is to be evaluated by a human `2 assumes that the impact of noise is independent of the local observer. We compare the performance of several losses, and propose a novel, differentiable error function. We show that characteristics of the image. On the contrary, the sensitivity of the quality of the results improves significantly with better loss the Human Visual System (HVS) to noise depends on local functions, even when the network architecture is left unchanged. luminance, contrast, and structure [13]. The `2 loss also works under the assumption of white Gaussian noise, which is not Index Terms—Image Processing, Image Restoration, Neural valid in general. Networks, Loss Functions. We focus on the use of neural networks for image restora- tion tasks, and we study the effect of different metrics for the network’s loss layer. We compare `2 against four error metrics I. INTRODUCTION on representative tasks: image super-resolution, JPEG artifacts OR decades, neural networks have shown various degrees removal, and joint denoising plus demosaicking. First, we test of success in several fields, ranging from robotics, to whether a different local metric such as `1 can produce better regressionF analysis, to pattern recognition. Despite the promis- results. We then evaluate the impact of perceptually-motivated ing results already produced in the 1980s on handwritten metrics. We use two state-of-the-art metrics for image quality: digit recognition [1], the popularity of neural networks in the structural similarity index (SSIM [13]) and the multi- the field of computer vision has grown exponentially only scale structural similarity index (MS-SSIM [14]). We choose recently, when deep learning boosted their performance in these among the plethora of existing indexes, because they are image recognition [2]. established measures, and because they are differentiable—a In the span of just a couple of years, neural networks have requirement for the backpropagation stage. As expected, on been employed for virtually every computer vision and image the use cases we consider, the perceptual metrics outperform processing task known to the research community. Much `2. However, and perhaps surprisingly, this is also true for `1, research has focused on the definition of new architectures that see Figure1. Inspired by this observation, we propose a novel are better suited to a specific problem [3], [4]. A large effort loss function and show its superior performance in terms of was also made to understand the inner mechanisms of neural all the metrics we consider. networks, and what their intrinsic limitations are, for instance We offer several contributions. First we bring attention to arXiv:1511.08861v3 [cs.CV] 20 Apr 2018 through the development of deconvolutional networks [5], or the importance of the error metric used to train neural networks trying to fool networks with specific inputs [6]. Other advances for image processing: despite the widely-known limitations of were made on the techniques to improve the network’s con- `2, this loss is still the de facto standard. We investigate the use vergence [7]. of three alternative error metrics (`1, SSIM, and MS-SSIM), The loss layer, despite being the effective driver of the and define a new metric that combines the advantages of `1 and network’s learning, has attracted little attention within the MS-SSIM (Section III). We also perform a thorough analysis image processing research community: the choice of the cost of their performance in terms of several image quality indexes function generally defaults to the squared `2 norm of the (SectionIV). Finally, we discuss their convergence properties. error [8], [3], [9], [10]. This is understandable, given the many We empirically show that the poor performance of some losses desirable properties this norm possesses. There is also a less is related to local minima of the loss functions (Section V-A), well-founded, but just as relevant reason for the continued and we explain the reasons why SSIM and MS-SSIM alone do not produce the expected quality (Section V-B). For each of ?NVIDIA, yMIT Media Lab the metrics we analyze, we implement a loss layer for Caffe, The supplementary material is available at http://research.nvidia.com/ which we make available to the research community1. publication/loss-functions-image-restoration-neural-networks. The material includes additional numerical and visual comparisons, and mathematical derivations. Contact [email protected] for further questions about this work. 1https://github.com/NVlabs/PL4NN 2 1 (b) Clean (c) Noisy (d) `2 (e) `1 (f) SSIM (g) MS-SSIM (h) Mix (i) Clean (j) Noisy (k) `2 (l) `1 (m) SSIM (n) MS-SSIM (o) Mix (a) Clean image Fig. 1: Comparisons of the results of joint denoising and demosaicking performed by networks trained on different loss functions (best viewed in the electronic version by zooming in). `2, the standard loss function for neural networks for image processing, produces splotchy artifacts in flat regions (d). II. RELATED WORK independent and identically distributed Gaussian noise, to the In this paper, we target neural networks for image restora- fact that it is additive for independent noise sources. There tion, which, in our context, is the set of all the image is longer list of reasons for which we refer the reader to the processing algorithms whose goal is to output an image that work of Wang and Bovik [16]. is appealing to a human observer. To this end, we use the These properties paved the way for `2’s widespread adop- problems of super-resolution, JPEG artifacts removal, and joint tion, which was further fueled by the fact that standard demosaicking plus denoising as benchmark tests. Specifically, software packages tend to include tools to use `2, but not we show how established error measures can be adapted to many other error functions for regression. In the context of work within the loss layer of a neural network, and how this image processing, Caffe [11] actually offers only `2 as a loss 2 can positively influence the results. Here we briefly review the layer , thus discouraging researchers from testing other error existing literature on the subject of neural networks for image measures. processing, and on the subject of measures of image quality. However, it is widely accepted that `2, and consequently the Peak Signal-to-Noise Ratio, PSNR, do not correlate well with ` A. Neural networks for image restoration human’s perception of image quality [12]: 2 simply does not capture the intricate characteristics of the human visual system Following their success in several computer vision (HVS). tasks [15], [2], neural networks have received considerable There exists a rich literature of error measures, both attention in the context of image restoration. Neural networks reference-based and non reference-based, that attempt to ad- have been used for denoising [8], [3], deblurring [4], demo- dress the limitations of the simple `2 error function. For our saicking [10], and super-resolution [9] among others. To the purposes, we focus on reference-based measures. A popu- best of our knowledge, however, the work on this subject lar reference-based index is the structural similarity index has focused on tuning the architecture of the network for the (SSIM [13]). SSIM evaluates images accounting for the fact specific application; the loss layer, which effectively drives the that the HVS is sensitive to changes in local structure. Wang learning of the network to produce the desired output quality, et al. [14] extend SSIM observing that the scale at which local ` is based on 2 for all of the approaches above. structure should be analyzed is a function of factors such We show that a better choice for the error measure has as image-to-observer distance. To account for these factors, a strong impact on the quality of the results. Moreover, we they propose MS-SSIM, a multi-scale version of SSIM that ` show that even when 2 is the appropriate loss, alternating the weighs SSIM computed at different scales according to the ` training loss function with a related loss, such as 1, can lead sensitivity of the HVS. Experimental results have shown the to finding a better solution for `2. superiority of SSIM-based indexes over `2. As a consequence, SSIM has been widely employed as a metric to evaluate image B. Evaluating image quality processing algorithms. Moreover, given that it can be used The mean squared error, `2, is arguably the dominant error as a differentiable cost function, SSIM has also been used measure across very diverse fields, from regression problems, in iterative algorithms designed for image compression [16], to pattern recognition, to signal and image processing. Among image reconstruction [17], denoising and super-resolution [18], the main reasons for its popularity is the fact that it is convex and even downscaling [19].
Load more

Recommended publications
  • A Simple Censored Median Regression Estimator
    Statistica Sinica 16(2006), 1043-1058 A SIMPLE CENSORED MEDIAN REGRESSION ESTIMATOR Lingzhi Zhou The Hong Kong University of Science and Technology Abstract: Ying, Jung and Wei (1995) proposed an estimation procedure for the censored median regression model that regresses the median of the survival time, or its transform, on the covariates. The procedure requires solving complicated nonlinear equations and thus can be very difficult to implement in practice, es- pecially when there are multiple covariates. Moreover, the asymptotic covariance matrix of the estimator involves the density of the errors that cannot be esti- mated reliably. In this paper, we propose a new estimator for the censored median regression model. Our estimation procedure involves solving some convex min- imization problems and can be easily implemented through linear programming (Koenker and D'Orey (1987)). In addition, a resampling method is presented for estimating the covariance matrix of the new estimator. Numerical studies indi- cate the superiority of the finite sample performance of our estimator over that in Ying, Jung and Wei (1995). Key words and phrases: Censoring, convexity, LAD, resampling. 1. Introduction The accelerated failure time (AFT) model, which relates the logarithm of the survival time to covariates, is an attractive alternative to the popular Cox (1972) proportional hazards model due to its ease of interpretation. The model assumes that the failure time T , or some monotonic transformation of it, is linearly related to the covariate vector Z 0 Ti = β0Zi + "i; i = 1; : : : ; n: (1.1) Under censoring, we only observe Yi = min(Ti; Ci), where Ci are censoring times, and Ti and Ci are independent conditional on Zi.
    [Show full text]
  • Cluster Analysis, a Powerful Tool for Data Analysis in Education
    International Statistical Institute, 56th Session, 2007: Rita Vasconcelos, Mßrcia Baptista Cluster Analysis, a powerful tool for data analysis in Education Vasconcelos, Rita Universidade da Madeira, Department of Mathematics and Engeneering Caminho da Penteada 9000-390 Funchal, Portugal E-mail: [email protected] Baptista, Márcia Direcção Regional de Saúde Pública Rua das Pretas 9000 Funchal, Portugal E-mail: [email protected] 1. Introduction A database was created after an inquiry to 14-15 - year old students, which was developed with the purpose of identifying the factors that could socially and pedagogically frame the results in Mathematics. The data was collected in eight schools in Funchal (Madeira Island), and we performed a Cluster Analysis as a first multivariate statistical approach to this database. We also developed a logistic regression analysis, as the study was carried out as a contribution to explain the success/failure in Mathematics. As a final step, the responses of both statistical analysis were studied. 2. Cluster Analysis approach The questions that arise when we try to frame socially and pedagogically the results in Mathematics of 14-15 - year old students, are concerned with the types of decisive factors in those results. It is somehow underlying our objectives to classify the students according to the factors understood by us as being decisive in students’ results. This is exactly the aim of Cluster Analysis. The hierarchical solution that can be observed in the dendogram presented in the next page, suggests that we should consider the 3 following clusters, since the distances increase substantially after it: Variables in Cluster1: mother qualifications; father qualifications; student’s results in Mathematics as classified by the school teacher; student’s results in the exam of Mathematics; time spent studying.
    [Show full text]
  • The Detection of Heteroscedasticity in Regression Models for Psychological Data
    Psychological Test and Assessment Modeling, Volume 58, 2016 (4), 567-592 The detection of heteroscedasticity in regression models for psychological data Andreas G. Klein1, Carla Gerhard2, Rebecca D. Büchner2, Stefan Diestel3 & Karin Schermelleh-Engel2 Abstract One assumption of multiple regression analysis is homoscedasticity of errors. Heteroscedasticity, as often found in psychological or behavioral data, may result from misspecification due to overlooked nonlinear predictor terms or to unobserved predictors not included in the model. Although methods exist to test for heteroscedasticity, they require a parametric model for specifying the structure of heteroscedasticity. The aim of this article is to propose a simple measure of heteroscedasticity, which does not need a parametric model and is able to detect omitted nonlinear terms. This measure utilizes the dispersion of the squared regression residuals. Simulation studies show that the measure performs satisfactorily with regard to Type I error rates and power when sample size and effect size are large enough. It outperforms the Breusch-Pagan test when a nonlinear term is omitted in the analysis model. We also demonstrate the performance of the measure using a data set from industrial psychology. Keywords: Heteroscedasticity, Monte Carlo study, regression, interaction effect, quadratic effect 1Correspondence concerning this article should be addressed to: Prof. Dr. Andreas G. Klein, Department of Psychology, Goethe University Frankfurt, Theodor-W.-Adorno-Platz 6, 60629 Frankfurt; email: [email protected] 2Goethe University Frankfurt 3International School of Management Dortmund Leipniz-Research Centre for Working Environment and Human Factors 568 A. G. Klein, C. Gerhard, R. D. Büchner, S. Diestel & K. Schermelleh-Engel Introduction One of the standard assumptions underlying a linear model is that the errors are inde- pendently identically distributed (i.i.d.).
    [Show full text]
  • Testing Hypotheses for Differences Between Linear Regression Lines
    United States Department of Testing Hypotheses for Differences Agriculture Between Linear Regression Lines Forest Service Stanley J. Zarnoch Southern Research Station e-Research Note SRS–17 May 2009 Abstract The general linear model (GLM) is often used to test for differences between means, as for example when the Five hypotheses are identified for testing differences between simple linear analysis of variance is used in data analysis for designed regression lines. The distinctions between these hypotheses are based on a priori assumptions and illustrated with full and reduced models. The studies. However, the GLM has a much wider application contrast approach is presented as an easy and complete method for testing and can also be used to compare regression lines. The for overall differences between the regressions and for making pairwise GLM allows for the use of both qualitative and quantitative comparisons. Use of the Bonferroni adjustment to ensure a desired predictor variables. Quantitative variables are continuous experimentwise type I error rate is emphasized. SAS software is used to illustrate application of these concepts to an artificial simulated dataset. The values such as diameter at breast height, tree height, SAS code is provided for each of the five hypotheses and for the contrasts age, or temperature. However, many predictor variables for the general test and all possible specific tests. in the natural resources field are qualitative. Examples of qualitative variables include tree class (dominant, Keywords: Contrasts, dummy variables, F-test, intercept, SAS, slope, test of conditional error. codominant); species (loblolly, slash, longleaf); and cover type (clearcut, shelterwood, seed tree). Fraedrich and others (1994) compared linear regressions of slash pine cone specific gravity and moisture content Introduction on collection date between families of slash pine grown in a seed orchard.
    [Show full text]
  • 5. Dummy-Variable Regression and Analysis of Variance
    Sociology 740 John Fox Lecture Notes 5. Dummy-Variable Regression and Analysis of Variance Copyright © 2014 by John Fox Dummy-Variable Regression and Analysis of Variance 1 1. Introduction I One of the limitations of multiple-regression analysis is that it accommo- dates only quantitative explanatory variables. I Dummy-variable regressors can be used to incorporate qualitative explanatory variables into a linear model, substantially expanding the range of application of regression analysis. c 2014 by John Fox Sociology 740 ° Dummy-Variable Regression and Analysis of Variance 2 2. Goals: I To show how dummy regessors can be used to represent the categories of a qualitative explanatory variable in a regression model. I To introduce the concept of interaction between explanatory variables, and to show how interactions can be incorporated into a regression model by forming interaction regressors. I To introduce the principle of marginality, which serves as a guide to constructing and testing terms in complex linear models. I To show how incremental I -testsareemployedtotesttermsindummy regression models. I To show how analysis-of-variance models can be handled using dummy variables. c 2014 by John Fox Sociology 740 ° Dummy-Variable Regression and Analysis of Variance 3 3. A Dichotomous Explanatory Variable I The simplest case: one dichotomous and one quantitative explanatory variable. I Assumptions: Relationships are additive — the partial effect of each explanatory • variable is the same regardless of the specific value at which the other explanatory variable is held constant. The other assumptions of the regression model hold. • I The motivation for including a qualitative explanatory variable is the same as for including an additional quantitative explanatory variable: to account more fully for the response variable, by making the errors • smaller; and to avoid a biased assessment of the impact of an explanatory variable, • as a consequence of omitting another explanatory variables that is relatedtoit.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Pubh 7405: REGRESSION ANALYSIS
    PubH 7405: REGRESSION ANALYSIS Review #2: Simple Correlation & Regression COURSE INFORMATION • Course Information are at address: www.biostat.umn.edu/~chap/pubh7405 • On each class web page, there is a brief version of the lecture for the day – the part with “formulas”; you can review, preview, or both – and as often as you like. • Follow Reading & Homework assignments at the end of the page when & if applicable. OFFICE HOURS • Instructor’s scheduled office hours: 1:15 to 2:15 Monday & Wednesday, in A441 Mayo Building • Other times are available by appointment • When really needed, can just drop in and interrupt me; could call before coming – making sure that I’m in. • I’m at a research facility on Fridays. Variables • A variable represents a characteristic or a class of measurement. It takes on different values on different subjects/persons. Examples include weight, height, race, sex, SBP, etc. The observed values, also called “observations,” form items of a data set. • Depending on the scale of measurement, we have different types of data. There are “observed variables” (Height, Weight, etc… each takes on different values on different subjects/person) and there are “calculated variables” (Sample Mean, Sample Proportion, etc… each is a “statistic” and each takes on different values on different samples). The Standard Deviation of a calculated variable is called the Standard Error of that variable/statistic. A “variable” – sample mean, sample standard deviation, etc… included – is like a “function”; when you apply it to a target element in its domain, the result is a “number”. For example, “height” is a variable and “the height of Mrs.
    [Show full text]
  • Tests of Hypotheses Using Statistics
    Tests of Hypotheses Using Statistics Adam Massey¤and Steven J. Millery Mathematics Department Brown University Providence, RI 02912 Abstract We present the various methods of hypothesis testing that one typically encounters in a mathematical statistics course. The focus will be on conditions for using each test, the hypothesis tested by each test, and the appropriate (and inappropriate) ways of using each test. We conclude by summarizing the di®erent tests (what conditions must be met to use them, what the test statistic is, and what the critical region is). Contents 1 Types of Hypotheses and Test Statistics 2 1.1 Introduction . 2 1.2 Types of Hypotheses . 3 1.3 Types of Statistics . 3 2 z-Tests and t-Tests 5 2.1 Testing Means I: Large Sample Size or Known Variance . 5 2.2 Testing Means II: Small Sample Size and Unknown Variance . 9 3 Testing the Variance 12 4 Testing Proportions 13 4.1 Testing Proportions I: One Proportion . 13 4.2 Testing Proportions II: K Proportions . 15 4.3 Testing r £ c Contingency Tables . 17 4.4 Incomplete r £ c Contingency Tables Tables . 18 5 Normal Regression Analysis 19 6 Non-parametric Tests 21 6.1 Tests of Signs . 21 6.2 Tests of Ranked Signs . 22 6.3 Tests Based on Runs . 23 ¤E-mail: [email protected] yE-mail: [email protected] 1 7 Summary 26 7.1 z-tests . 26 7.2 t-tests . 27 7.3 Tests comparing means . 27 7.4 Variance Test . 28 7.5 Proportions . 28 7.6 Contingency Tables .
    [Show full text]
  • Correlation and Regression Analysis
    OIC ACCREDITATION CERTIFICATION PROGRAMME FOR OFFICIAL STATISTICS Correlation and Regression Analysis TEXTBOOK ORGANISATION OF ISLAMIC COOPERATION STATISTICAL ECONOMIC AND SOCIAL RESEARCH AND TRAINING CENTRE FOR ISLAMIC COUNTRIES OIC ACCREDITATION CERTIFICATION PROGRAMME FOR OFFICIAL STATISTICS Correlation and Regression Analysis TEXTBOOK {{Dr. Mohamed Ahmed Zaid}} ORGANISATION OF ISLAMIC COOPERATION STATISTICAL ECONOMIC AND SOCIAL RESEARCH AND TRAINING CENTRE FOR ISLAMIC COUNTRIES © 2015 The Statistical, Economic and Social Research and Training Centre for Islamic Countries (SESRIC) Kudüs Cad. No: 9, Diplomatik Site, 06450 Oran, Ankara – Turkey Telephone +90 – 312 – 468 6172 Internet www.sesric.org E-mail [email protected] The material presented in this publication is copyrighted. The authors give the permission to view, copy download, and print the material presented that these materials are not going to be reused, on whatsoever condition, for commercial purposes. For permission to reproduce or reprint any part of this publication, please send a request with complete information to the Publication Department of SESRIC. All queries on rights and licenses should be addressed to the Statistics Department, SESRIC, at the aforementioned address. DISCLAIMER: Any views or opinions presented in this document are solely those of the author(s) and do not reflect the views of SESRIC. ISBN: xxx-xxx-xxxx-xx-x Cover design by Publication Department, SESRIC. For additional information, contact Statistics Department, SESRIC. i CONTENTS Acronyms
    [Show full text]
  • Profiling Heteroscedasticity in Linear Regression Models
    358 The Canadian Journal of Statistics Vol. 43, No. 3, 2015, Pages 358–377 La revue canadienne de statistique Profiling heteroscedasticity in linear regression models Qian M. ZHOU1*, Peter X.-K. SONG2 and Mary E. THOMPSON3 1Department of Statistics and Actuarial Science, Simon Fraser University, Burnaby, British Columbia, Canada 2Department of Biostatistics, University of Michigan, Ann Arbor, Michigan, U.S.A. 3Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario, Canada Key words and phrases: Heteroscedasticity; hybrid test; information ratio; linear regression models; model- based estimators; sandwich estimators; screening; weighted least squares. MSC 2010: Primary 62J05; secondary 62J20 Abstract: Diagnostics for heteroscedasticity in linear regression models have been intensively investigated in the literature. However, limited attention has been paid on how to identify covariates associated with heteroscedastic error variances. This problem is critical in correctly modelling the variance structure in weighted least squares estimation, which leads to improved estimation efficiency. We propose covariate- specific statistics based on information ratios formed as comparisons between the model-based and sandwich variance estimators. A two-step diagnostic procedure is established, first to detect heteroscedasticity in error variances, and then to identify covariates the error variance structure might depend on. This proposed method is generalized to accommodate practical complications, such as when covariates associated with the heteroscedastic variances might not be associated with the mean structure of the response variable, or when strong correlation is present amongst covariates. The performance of the proposed method is assessed via a simulation study and is illustrated through a data analysis in which we show the importance of correct identification of covariates associated with the variance structure in estimation and inference.
    [Show full text]
  • Regression Analysis
    CALIFORNIA STATE UNIVERSITY, SACRAMENTO College of Business Administration NOTES FOR DATA ANALYSIS [Eleventh Edition] Manfred W. Hopfe, Ph.D. Stanley A. Taylor, Ph.D. Last Revised July 2008 by Min Li, Ph.D. 1 NOTES FOR DATA ANALYSIS [Tenth Edition] As stated in previous editions, the topics presented in this publication, which we have produced to assist our students, have been heavily influenced by the Making Statistics More Effective in Schools of Business Conferences held throughout the United States. The first conference was held at the University of Chicago in 1986. The School of Business Administration at California State University, Sacramento, hosted the tenth annual conference June 15-17, 1995. Most recent conferences were held at Babson College, (June 1999) and Syracuse University (June 2000). As with any publication in its developmental stages, there will be errors. If you find any errors, we ask for your feedback since this is a dynamic publication we continually revise. Throughout the semester you will be provided additional handouts to supplement the material in this book. Manfred W. Hopfe, Ph.D. Stanley A. Taylor, Ph.D. Carmichael, California August 2000 2 TABLE OF CONTENTS REVIEW................................................................................................................................................................ 5 Statistics vs. Parameters.................................................................................................................................... 5 Mean and Variance ..........................................................................................................................................
    [Show full text]