DOCSLIB.ORG

Algorithms for Robust Linear Regression by Exploiting the Connection to Sparse Signal Recovery

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Algorithms for Robust Linear Regression by Exploiting the Connection to Sparse Signal Recovery ALGORITHMS FOR ROBUST LINEAR REGRESSION BY EXPLOITING THE CONNECTION TO SPARSE SIGNAL RECOVERY Yuzhe Jin and Bhaskar D. Rao Department of Electrical and Computer Engineering, University of California at San Diego, La Jolla, CA 92093-0407, USA fyujin, [email protected] ABSTRACT is the fitting error. This criterion is sensitive to outliers and hence In this paper, we develop algorithms for robust linear regression by not robust. Many existing methods for robust regression follow the leveraging the connection between the problems of robust regres- idea that one should de-emphasize the impact of data with large sion and sparse signal recovery. We explicitly model the measure- deviation in order to obtain robustness. For example, the method ment noise as a combination of two terms; the first term accounts for of Least Absolute Value (LAV) [4] is probably aP well-known rep- M j j regular measurement noise modeled as zero mean Gaussian noise, resentative of this kind. This method minimizes i=1 e~i , which and the second term captures the impact of outliers. The fact that can be equivalently viewed as imposing a Laplacian distribution on the latter outlier component could indeed be a sparse vector pro- the measurement noise ei. Alternatively, the family of M-estimates vides the opportunity to leverage sparse signal reconstruction meth- [1] consider flexible weighting schemes on the fitting error e~i. ods to solve the problem of robust regression. Maximum a posteriori The weighting functions used in M-estimates aim to de-emphasize (MAP) based and empirical Bayesian inference based algorithms are samples with large deviation, and they can also be related to cer- developed for this purpose. Experimental studies on simulated and tain probability densities imposed on the measurement noise. By real data sets are presented to demonstrate the effectiveness of the assuming the measurement noise ei is drawn from the Student’s proposed algorithms. t-distribution [5], the impact of extreme errors is also effectively downscaled. Further, the model of Gaussian mixtures also has been Index Terms— robust linear regression, sparse signal recovery, employed in robust regression wherein samples of the measurement outlier detection, MAP, sparse Bayesian learning noise ei are assumed to be i.i.d. and drawn from a mixture of two Gaussians with one accounting for regular noise and the other for 1. INTRODUCTION outliers [6]. In addition, robust procedures that aim to explicitly remove the Consider the linear regression problem with the model impact of extreme errors have also been developed. For instance, | the method of Least Trimmed Squares (LTS) [7] employs the opti- yi = a xi + ei; for i = 1; 2; :::; M; (1) mization criterion that minimizes only a portion of the squared fitting L errors with smallest magnitudes. The essence of this method can be where xi 2 R is usually termed as the explanatory variable, yi is the response variable, a 2 RL is the regression coefficients, L is the viewed as roughly detecting outliers and removing their impact at the data fitting stage. This idea can be generalized to various out- model order, and ei is the measurement noise in the ith response. Note that model (1) can be compactly represented by lier diagnosis techniques [7]. For an extensive survey of previous work on robust regression and outlier detection, interested readers y = Xa + e; (2) are referred to [1][3][7] and the references therein. It is interesting to note that the probability distribution underly- | | 2 RM where X = [x1; x2; :::; xM ] , e = [e1; e2; :::; eM ] and ing various robust regression methods indeed have counterparts in | 2 RM y = [y1; y2; :::; yM ] . The goal is to determine the regres- the context of the sparse signal recovery problem which has recently sion coefficients a and this is often achieved by using a suitable opti- received much attention in many application domains [8][9]. The mization criterion. This problem has many important applications in heavy-tailed outlier-tolerating priors imposed on the measurement science and engineering. An important factor that makes this prob- noise correspond to the sparsity-inducing distributions in sparse sig- lem interesting and challenging is that the response variable y may nal recovery. The Laplacian distribution in the LAV method and its usually contain outliers. The popular ordinary Least Squares (LS) use in the corresponding `1-norm minimization based sparse signal is sensitive to outliers and hence robust regression methods are of recovery algorithms serves as an excellent example of this kind. As interest. Numerous approaches for robust regression have been de- another example, the LTS method exhibits very similar ingredient veloped [1][2][3] with the goal of extracting the model parameters to the thresholding method that is used for finding sparse solutions. reliably in the presence of outliers. Our work examines this connection more deeply. Intuitively, this connection is made possible by the fact that outliers are events that 1.1. Background occur infrequently, and thus sparse. To make the connection more explicit, in Section 2.1 we develop a two component model for the As a popular technique, ordinaryP LS estimation determines the M | additive noise and reformulate the regression problem such that the model parameters a by minimizing e~2; where e~ = y − a x i=1 i i i i usefulness of sparse recovery methods is evident. Using this new This research was supported by NSF Grants IIS-0613595 and CCF- formulation and connection, in Section 2.2 we develop algorithms 0830612. for the robust regression which are based on sparse signal recovery methods. They are then evaluated in Section 3 and demonstrated to denoising [8]) and `1-regularization problem [11]. By estimating w, be effective in combating the negative impact of outliers. these algorithms determine how each observation is contaminated. In contrast, the LAVmethod, which can be obtained by letting ζ ! 0 2. TWO COMPONENT MODEL AND SPARSE SIGNAL in (6), assumes a Laplacian prior on the total noise and minimizes RECOVERY ALGORITHMS FOR ROBUST REGRESSION the sum of `1-norm of the fitting errors. As a result, it is not able to clarify the underlying mechanism of noise contamination. 2.1. The two component model To solve the above optimization problems, (5) and (6) will be- come convex optimization problems when p = 1 and (5) will be We leverage the fact that outliers occur infrequently and hence are considered in the simulation study. Motivated by the analysis in sparse. Unfortunately, the linear model (1) leaves us little opportu- [8, 5.2]p and our experience, we choose the regularization parameter nity to take advantage of this observation, since a single measure- σ~ 2 log M 1 λ = 3 , where σ~ is a proper estimation of scale. Procedures ment noise term ei deals with both the impact of outlier and regular can be developed for other choices of p [12]. noise. To explicitly make use of the sparsity of outliers, we suggest an alternative model by splitting e into two independent additive i 2.2.2. Empirical Bayesian inference based robust regression components, namely wi and ϵi, as follows, | This method adopts the empirical Bayesian approach for robust yi = a xi + wi + ϵi; for i = 1; 2; :::; M: (3) regression. In particular, we utilize the Sparse Bayesian learning methodology developed in [13][14]. To this end, it is assumed that The interpretations of xi; yi and a are carried over from (1). If a wi is a random variable with prior distribution wi ∼ N (0; γi), response yi is not an outlier, then the corresponding wi is assumed where γi is the hyperparameter that controls the variance of each wi to be zero. If yi is an outlier, then wi can be viewed as the anomalous and has to be learnt. If γi = 0, it means the corresponding wi will be error in yi such that (yi −wi) appears to be a response contaminated zero, resulting in no anomalous error being added into observation only by regular noise. The term ϵi, on the other hand, contains the yi. If γi > 0, an anomalous noise whose magnitude depends on γi regular measurement noise in response yi, and it is modeled as i.i.d. 2 will contaminate yi, and it results in an outlier in the measurement. zero mean Gaussian noise, i.e. ϵi ∼ N (0; σ ). Compactly, model (3) can be represented by To estimate the regression coefficients, we jointly find [ ] a^; γ^; σ^2 = arg max P (yjX; a; γ; σ2); (7) a 2 y = Xa + w + ϵ = [X; I] + ϵ; (4) a;γ,σ w where γ , fγ1; γ2; :::; γM g. Then w can be estimated by the pos- | where, in addition to (2), w = [w1; w2; :::; wM ] , and ϵ = terior mean, i.e. [ϵ ; ϵ ; :::; ϵ ]|. By definition, w is a sparse vector, which means 1 2 M 2 the number of nonzero entries of w is (much) smaller than the w^ = E[wjX; y; a^; γ^; σ^ ]: (8) length of w. As we shall see, this model (4) enables the opportunity Note that the essence of this method is that the robust regres- to adapt sparse signal recovery methods to robust regression. sion problem is cast into the framework of Sparse Bayesian learning (SBL) with appropriate modifications, and this is made possible by 2.2. Sparse signal recovery algorithms for robust linear regres- our proposed two component noise modeling technique. The algo- sion rithm development, analysis and experimental study of the original Since w is sparse, one can utilize ideas from sparse signal recovery SBL for sparse signal recovery have been extensively discussed in to develop robust linear regression methods. In this work, we con- [13][14][15].
Load more

Recommended publications
  • Robust Statistics Part 3: Regression Analysis
    Robust Statistics Part 3: Regression analysis Peter Rousseeuw LARS-IASC School, May 2019 Peter Rousseeuw Robust Statistics, Part 3: Regression LARS-IASC School, May 2019 p. 1 Linear regression Linear regression: Outline 1 Classical regression estimators 2 Classical outlier diagnostics 3 Regression M-estimators 4 The LTS estimator 5 Outlier detection 6 Regression S-estimators and MM-estimators 7 Regression with categorical predictors 8 Software Peter Rousseeuw Robust Statistics, Part 3: Regression LARS-IASC School, May 2019 p. 2 Linear regression Classical estimators The linear regression model The linear regression model says: yi = β0 + β1xi1 + ... + βpxip + εi ′ = xiβ + εi 2 ′ ′ with i.i.d. errors εi ∼ N(0,σ ), xi = (1,xi1,...,xip) and β =(β0,β1,...,βp) . ′ Denote the n × (p + 1) matrix containing the predictors xi as X =(x1,..., xn) , ′ ′ the vector of responses y =(y1,...,yn) and the error vector ε =(ε1,...,εn) . Then: y = Xβ + ε Any regression estimate βˆ yields fitted values yˆ = Xβˆ and residuals ri = ri(βˆ)= yi − yˆi . Peter Rousseeuw Robust Statistics, Part 3: Regression LARS-IASC School, May 2019 p. 3 Linear regression Classical estimators The least squares estimator Least squares estimator n ˆ 2 βLS = argmin ri (β) β i=1 X If X has full rank, then the solution is unique and given by ˆ ′ −1 ′ βLS =(X X) X y The usual unbiased estimator of the error variance is n 1 σˆ2 = r2(βˆ ) LS n − p − 1 i LS i=1 X Peter Rousseeuw Robust Statistics, Part 3: Regression LARS-IASC School, May 2019 p. 4 Linear regression Classical estimators Outliers in regression Different types of outliers: vertical outlier good leverage point • • y • • • regular data • ••• • •• ••• • • • • • • • • • bad leverage point • • •• • x Peter Rousseeuw Robust Statistics, Part 3: Regression LARS-IASC School, May 2019 p.
    [Show full text]
  • Quantile Regression for Overdispersed Count Data: a Hierarchical Method Peter Congdon
    Congdon Journal of Statistical Distributions and Applications (2017) 4:18 DOI 10.1186/s40488-017-0073-4 RESEARCH Open Access Quantile regression for overdispersed count data: a hierarchical method Peter Congdon Correspondence: [email protected] Abstract Queen Mary University of London, London, UK Generalized Poisson regression is commonly applied to overdispersed count data, and focused on modelling the conditional mean of the response. However, conditional mean regression models may be sensitive to response outliers and provide no information on other conditional distribution features of the response. We consider instead a hierarchical approach to quantile regression of overdispersed count data. This approach has the benefits of effective outlier detection and robust estimation in the presence of outliers, and in health applications, that quantile estimates can reflect risk factors. The technique is first illustrated with simulated overdispersed counts subject to contamination, such that estimates from conditional mean regression are adversely affected. A real application involves ambulatory care sensitive emergency admissions across 7518 English patient general practitioner (GP) practices. Predictors are GP practice deprivation, patient satisfaction with care and opening hours, and region. Impacts of deprivation are particularly important in policy terms as indicating effectiveness of efforts to reduce inequalities in care sensitive admissions. Hierarchical quantile count regression is used to develop profiles of central and extreme quantiles according to specified predictor combinations. Keywords: Quantile regression, Overdispersion, Poisson, Count data, Ambulatory sensitive, Median regression, Deprivation, Outliers 1. Background Extensions of Poisson regression are commonly applied to overdispersed count data, focused on modelling the conditional mean of the response. However, conditional mean regression models may be sensitive to response outliers.
    [Show full text]
  • Robust Linear Regression: a Review and Comparison Arxiv:1404.6274
    Robust Linear Regression: A Review and Comparison Chun Yu1, Weixin Yao1, and Xue Bai1 1Department of Statistics, Kansas State University, Manhattan, Kansas, USA 66506-0802. Abstract Ordinary least-squares (OLS) estimators for a linear model are very sensitive to unusual values in the design space or outliers among y values. Even one single atypical value may have a large effect on the parameter estimates. This article aims to review and describe some available and popular robust techniques, including some recent developed ones, and compare them in terms of breakdown point and efficiency. In addition, we also use a simulation study and a real data application to compare the performance of existing robust methods under different scenarios. arXiv:1404.6274v1 [stat.ME] 24 Apr 2014 Key words: Breakdown point; Robust estimate; Linear Regression. 1 Introduction Linear regression has been one of the most important statistical data analysis tools. Given the independent and identically distributed (iid) observations (xi; yi), i = 1; : : : ; n, 1 in order to understand how the response yis are related to the covariates xis, we tradi- tionally assume the following linear regression model T yi = xi β + "i; (1.1) where β is an unknown p × 1 vector, and the "is are i.i.d. and independent of xi with E("i j xi) = 0. The most commonly used estimate for β is the ordinary least square (OLS) estimate which minimizes the sum of squared residuals n X T 2 (yi − xi β) : (1.2) i=1 However, it is well known that the OLS estimate is extremely sensitive to the outliers.
    [Show full text]
  • A Guide to Robust Statistical Methods in Neuroscience
    A GUIDE TO ROBUST STATISTICAL METHODS IN NEUROSCIENCE Authors: Rand R. Wilcox1∗, Guillaume A. Rousselet2 1. Dept. of Psychology, University of Southern California, Los Angeles, CA 90089-1061, USA 2. Institute of Neuroscience and Psychology, College of Medical, Veterinary and Life Sciences, University of Glasgow, 58 Hillhead Street, G12 8QB, Glasgow, UK ∗ Corresponding author: [email protected] ABSTRACT There is a vast array of new and improved methods for comparing groups and studying associations that offer the potential for substantially increasing power, providing improved control over the probability of a Type I error, and yielding a deeper and more nuanced understanding of data. These new techniques effectively deal with four insights into when and why conventional methods can be unsatisfactory. But for the non-statistician, the vast array of new and improved techniques for comparing groups and studying associations can seem daunting, simply because there are so many new methods that are now available. The paper briefly reviews when and why conventional methods can have relatively low power and yield misleading results. The main goal is to suggest some general guidelines regarding when, how and why certain modern techniques might be used. Keywords: Non-normality, heteroscedasticity, skewed distributions, outliers, curvature. 1 1 Introduction The typical introductory statistics course covers classic methods for comparing groups (e.g., Student's t-test, the ANOVA F test and the Wilcoxon{Mann{Whitney test) and studying associations (e.g., Pearson's correlation and least squares regression). The two-sample Stu- dent's t-test and the ANOVA F test assume that sampling is from normal distributions and that the population variances are identical, which is generally known as the homoscedastic- ity assumption.
    [Show full text]
  • Robust Bayesian General Linear Models ⁎ W.D
    www.elsevier.com/locate/ynimg NeuroImage 36 (2007) 661–671 Robust Bayesian general linear models ⁎ W.D. Penny, J. Kilner, and F. Blankenburg Wellcome Department of Imaging Neuroscience, University College London, 12 Queen Square, London WC1N 3BG, UK Received 22 September 2006; revised 20 November 2006; accepted 25 January 2007 Available online 7 May 2007 We describe a Bayesian learning algorithm for Robust General Linear them from the data (Jung et al., 1999). This is, however, a non- Models (RGLMs). The noise is modeled as a Mixture of Gaussians automatic process and will typically require user intervention to rather than the usual single Gaussian. This allows different data points disambiguate the discovered components. In fMRI, autoregressive to be associated with different noise levels and effectively provides a (AR) modeling can be used to downweight the impact of periodic robust estimation of regression coefficients. A variational inference respiratory or cardiac noise sources (Penny et al., 2003). More framework is used to prevent overfitting and provides a model order recently, a number of approaches based on robust regression have selection criterion for noise model order. This allows the RGLM to default to the usual GLM when robustness is not required. The method been applied to imaging data (Wager et al., 2005; Diedrichsen and is compared to other robust regression methods and applied to Shadmehr, 2005). These approaches relax the assumption under- synthetic data and fMRI. lying ordinary regression that the errors be normally (Wager et al., © 2007 Elsevier Inc. All rights reserved. 2005) or identically (Diedrichsen and Shadmehr, 2005) distributed. In Wager et al.
    [Show full text]
  • Sketching for M-Estimators: a Unified Approach to Robust Regression
    Sketching for M-Estimators: A Unified Approach to Robust Regression Kenneth L. Clarkson∗ David P. Woodruffy Abstract ing algorithm often results, since a single pass over A We give algorithms for the M-estimators minx kAx − bkG, suffices to compute SA. n×d n n where A 2 R and b 2 R , and kykG for y 2 R is specified An important property of many of these sketching ≥0 P by a cost function G : R 7! R , with kykG ≡ i G(yi). constructions is that S is a subspace embedding, meaning d The M-estimators generalize `p regression, for which G(x) = that for all x 2 R , kSAxk ≈ kAxk. (Here the vector jxjp. We first show that the Huber measure can be computed norm is generally `p for some p.) For the regression up to relative error in O(nnz(A) log n + poly(d(log n)=")) d time, where nnz(A) denotes the number of non-zero entries problem of minimizing kAx − bk with respect to x 2 R , of the matrix A. Huber is arguably the most widely used for inputs A 2 Rn×d and b 2 Rn, a minor extension of M-estimator, enjoying the robustness properties of `1 as well the embedding condition implies S preserves the norm as the smoothness properties of ` . 2 of the residual vector Ax − b, that is kS(Ax − b)k ≈ We next develop algorithms for general M-estimators. kAx − bk, so that a vector x that makes kS(Ax − b)k We analyze the M-sketch, which is a variation of a sketch small will also make kAx − bk small.
    [Show full text]
  • On Robust Regression with High-Dimensional Predictors
    On robust regression with high-dimensional predictors Noureddine El Karoui ∗, Derek Bean ∗ , Peter Bickel ∗ , Chinghway Lim y, and Bin Yu ∗ ∗University of California, Berkeley, and yNational University of Singapore Submitted to Proceedings of the National Academy of Sciences of the United States of America We study regression M-estimates in the setting where p, the num- pointed out a surprising feature of the regime, p=n ! κ > 0 ber of covariates, and n, the number of observations, are both large for LSE; fitted values were not asymptotically Gaussian. He but p ≤ n. We find an exact stochastic representation for the dis- was unable to deal with this regime otherwise, see the discus- Pn 0 tribution of β = argmin p ρ(Y − X β) at fixed p and n b β2R i=1 i i sion on p.802 of [6]. under various assumptions on the objective function ρ and our sta- In this paper we intend to, in part heuristically and with tistical model. A scalar random variable whose deterministic limit \computer validation", analyze fully what happens in robust rρ(κ) can be studied when p=n ! κ > 0 plays a central role in this regression when p=n ! κ < 1. We do limit ourselves to Gaus- representation. Furthermore, we discover a non-linear system of two deterministic sian covariates but present grounds that the behavior holds equations that characterizes rρ(κ). Interestingly, the system shows much more generally. We also investigate the sensitivity of that rρ(κ) depends on ρ through proximal mappings of ρ as well as our results to the geometry of the design matrix.
    [Show full text]
  • Robust Regression in Stata
    The Stata Journal (yyyy) vv, Number ii, pp. 1–23 Robust Regression in Stata Vincenzo Verardi University of Namur (CRED) and Universit´eLibre de Bruxelles (ECARES and CKE) Rempart de la Vierge, 8, B-5000 Namur. E-mail: [email protected]. Vincenzo Verardi is Associated Researcher of the FNRS and gratefully acknowledges their finacial support Christophe Croux K.U.Leuven, Faculty of Business and Economics Naamsestraat 69. B-3000, Leuven E-mail: [email protected]. Abstract. In regression analysis, the presence of outliers in the data set can strongly distort the classical least squares estimator and lead to un- reliable results. To deal with this, several robust-to-outliers methods have been proposed in the statistical literature. In Stata, some of these methods are available through the commands rreg and qreg. Unfortunately, these methods only resist to some specific types of outliers and turn out to be ineffective under alternative scenarios. In this paper we present more ef- fective robust estimators that we implemented in Stata. We also present a graphical tool that allows recognizing the type of detected outliers. KEYWORDS: S-estimators, MM-estimators, Outliers, Robustness JEL CLASSIFICATION: C12, C21, C87 c yyyy StataCorp LP st0001 2 Robust Regression in Stata 1 Introduction The objective of linear regression analysis is to study how a dependent variable is linearly related to a set of regressors. In matrix notation, the linear regression model is given by: y = Xθ + ε (1) where, for a sample of size n, y is the (n × 1) vector containing the values for the dependent variable, X is the (n × p) matrix containing the values for the p regressors and ε is the (n × 1) vector containing the error terms.
    [Show full text]
  • Robust Fitting of Parametric Models Based on M-Estimation
    Robust Fitting of Parametric Models Based on M-Estimation Andreas Ruckstuhl ∗ IDP Institute of Data Analysis and Process Design ZHAW Zurich University of Applied Sciences in Winterthur Version 2018† ∗Email Address: [email protected]; Internet: http://www.idp.zhaw.ch †The author thanks Amanda Strong for their help in translating the original German version into English. Contents 1. Basic Concepts 1 1.1. The Regression Model and the Outlier Problem . ..... 1 1.2. MeasuringRobustness . 3 1.3. M-Estimation................................. 7 1.4. InferencewithM-Estimators . 10 2. Linear Regression 12 2.1. RegressionM-Estimation . 12 2.2. Example from Molecular Spectroscopy . 13 2.3. General Regression M-Estimation . 17 2.4. Robust Regression MM-Estimation . 19 2.5. Robust Inference and Variable Selection . ...... 23 3. Generalized Linear Models 29 3.1. UnifiedModelFormulation . 29 3.2. RobustEstimatingProcedure . 30 4. Multivariate Analysis 34 4.1. Robust Estimation of the Covariance Matrix . ..... 34 4.2. PrincipalComponentAnalysis . 39 4.3. Linear Discriminant Analysis . 42 5. Baseline Removal Using Robust Local Regression 45 5.1. AMotivatingExample.. .. .. .. .. .. .. .. 45 5.2. LocalRegression ............................... 45 5.3. BaselineRemoval............................... 49 6. Some Closing Comments 51 6.1. General .................................... 51 6.2. StatisticsPrograms. 51 6.3. Literature ................................... 52 A. Appendix 53 A.1. Some Thoughts on the Location Model . 53 A.2. Calculation of Regression M-Estimations . ....... 54 A.3. More Regression Estimators with High Breakdown Points ........ 56 Bibliography 58 Objectives The block course Robust Statistics in the post-graduated course (Weiterbildungslehrgang WBL) in applied statistics at the ETH Zürich should 1. introduce problems where robust procedures are advantageous, 2. explain the basic idea of robust methods for linear models, and 3.
    [Show full text]
  • Robust Regression Examples
    Chapter 9 Robust Regression Examples Chapter Table of Contents OVERVIEW ...................................177 FlowChartforLMS,LTS,andMVE......................179 EXAMPLES USING LMS AND LTS REGRESSION ............180 Example 9.1 LMS and LTS with Substantial Leverage Points: Hertzsprung- Russell Star Data . ......................180 Example9.2LMSandLTS:StacklossData..................184 Example 9.3 LMS and LTS Univariate (Location) Problem: Barnett and LewisData...........................193 EXAMPLES USING MVE REGRESSION ..................196 Example9.4BrainlogData...........................196 Example9.5MVE:StacklossData.......................201 EXAMPLES COMBINING ROBUST RESIDUALS AND ROBUST DIS- TANCES ................................208 Example9.6Hawkins-Bradu-KassData....................209 Example9.7StacklossData..........................218 REFERENCES ..................................220 176 Chapter 9. Robust Regression Examples SAS OnlineDoc: Version 8 Chapter 9 Robust Regression Examples Overview SAS/IML has three subroutines that can be used for outlier detection and robust re- gression. The Least Median of Squares (LMS) and Least Trimmed Squares (LTS) subroutines perform robust regression (sometimes called resistant regression). These subroutines are able to detect outliers and perform a least-squares regression on the remaining observations. The Minimum Volume Ellipsoid Estimation (MVE) sub- routine can be used to find the minimum volume ellipsoid estimator, which is the location and robust covariance matrix that can be used
    [Show full text]
  • Robust Regression Analysis of Ochroma Pyramidale (Balsa-Tree)
    Article Improving the Modeling of the Height–Diameter Relationship of Tree Species with High Growth Variability: Robust Regression Analysis of Ochroma pyramidale (Balsa-Tree) Jorge Danilo Zea-Camaño 1,*, José R. Soto 2, Julio Eduardo Arce 1, Allan Libanio Pelissari 1 , Alexandre Behling 1, Gabriel Agostini Orso 1, Marcelino Santiago Guachambala 3 and Rozane de Loyola Eisfeld 1 1 Department of Forest Sciences, Federal University of Parana, Campus III, 3400 Lothário Meissner Av., Curitiba 80000-000, Paraná State, Brazil; [email protected] (J.E.A.); [email protected] (A.L.P.); [email protected] (A.B.); [email protected] (G.A.O.); [email protected] (R.d.L.E.) 2 School of Natural Resources and the Environmental, The University of Arizona, 1064 E. Lowell St., Tucson, AZ 85721, USA; [email protected] 3 PLANTABAL S.A.—3ACOREMATERIALS, Ofic. 102 Mz. CC3 SL. 120, Complejo Industrial Quevedo, 1 Km 4 2 vía a Valencia, Quevedo 120501, Ecuador; [email protected] * Correspondence: [email protected]; Tel.: +55-83-99926-8487 Received: 23 January 2020; Accepted: 9 March 2020; Published: 12 March 2020 Abstract: Ochroma pyramidale (Cav. ex. Lam.) Urb. (balsa-tree) is a commercially important tree species that ranges from Mexico to northern Brazil. Due to its low weight and mechanical endurance, the wood is particularly well-suited for wind turbine blades, sporting equipment, boats and aircrafts; as such, it is in high market demand and plays an important role in many regional economies. This tree species is also well-known to exhibit a high degree of variation in growth.
    [Show full text]
  • Experimental Design and Robust Regression
    Rochester Institute of Technology RIT Scholar Works Theses 12-15-2017 Experimental Design and Robust Regression Pranay Kumar [email protected] Follow this and additional works at: https://scholarworks.rit.edu/theses Recommended Citation Kumar, Pranay, "Experimental Design and Robust Regression" (2017). Thesis. Rochester Institute of Technology. Accessed from This Thesis is brought to you for free and open access by RIT Scholar Works. It has been accepted for inclusion in Theses by an authorized administrator of RIT Scholar Works. For more information, please contact [email protected]. Rochester Institute of Technology Experimental Design and Robust Regression A Thesis Submitted in partial fulfillment of the requirements for the degree of Master of Science in Industrial Engineering in the Department of Industrial & Systems Engineering Kate Gleason College of Engineering by Pranay Kumar December 15, 2017 DEPARTMENT OF INDUSTRIAL AND SYSTEMS ENGINEERING KATE GLEASON COLLEGE OF ENGINEERING ROCHESTER INSTITUTE OF TECHNOLOGY ROCHESTER, NEW YORK CERTIFICATE OF APPROVAL M.S. DEGREE THESIS The M.S. Degree Thesis of Pranay Kumar has been examined and approved by the thesis committee as satisfactory for the thesis requirement for the Master of Science degree Approved by: ____________________________________ Dr. Rachel Silvestrini, Thesis Advisor ____________________________________ Dr. Brian Thorn Experimental Design and Robust Regression Abstract: Design of Experiments (DOE) is a very powerful statistical methodology, especially when used with linear regression analysis. The use of ordinary least squares (OLS) estimation of linear regression parameters requires the errors to have a normal distribution. However, there are numerous situations when the error distribution is non- normal and using OLS can result in inaccurate parameter estimates.
    [Show full text]