Kernel Regularized Least Squares: Reducing Misspecification Bias with a Flexible and Interpretable Machine Learning Approach
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Instrumental Regression in Partially Linear Models
10-167 Research Group: Econometrics and Statistics September, 2009 Instrumental Regression in Partially Linear Models JEAN-PIERRE FLORENS, JAN JOHANNES AND SEBASTIEN VAN BELLEGEM INSTRUMENTAL REGRESSION IN PARTIALLY LINEAR MODELS∗ Jean-Pierre Florens1 Jan Johannes2 Sebastien´ Van Bellegem3 First version: January 24, 2008. This version: September 10, 2009 Abstract We consider the semiparametric regression Xtβ+φ(Z) where β and φ( ) are unknown · slope coefficient vector and function, and where the variables (X,Z) are endogeneous. We propose necessary and sufficient conditions for the identification of the parameters in the presence of instrumental variables. We also focus on the estimation of β. An incorrect parameterization of φ may generally lead to an inconsistent estimator of β, whereas even consistent nonparametric estimators for φ imply a slow rate of convergence of the estimator of β. An additional complication is that the solution of the equation necessitates the inversion of a compact operator that has to be estimated nonparametri- cally. In general this inversion is not stable, thus the estimation of β is ill-posed. In this paper, a √n-consistent estimator for β is derived under mild assumptions. One of these assumptions is given by the so-called source condition that is explicitly interprated in the paper. Finally we show that the estimator achieves the semiparametric efficiency bound, even if the model is heteroscedastic. Monte Carlo simulations demonstrate the reasonable performance of the estimation procedure on finite samples. Keywords: Partially linear model, semiparametric regression, instrumental variables, endo- geneity, ill-posed inverse problem, Tikhonov regularization, root-N consistent estimation, semiparametric efficiency bound JEL classifications: Primary C14; secondary C30 ∗We are grateful to R. -
PARAMETER DETERMINATION for TIKHONOV REGULARIZATION PROBLEMS in GENERAL FORM∗ 1. Introduction. We Are Concerned with the Solut
PARAMETER DETERMINATION FOR TIKHONOV REGULARIZATION PROBLEMS IN GENERAL FORM∗ Y. PARK† , L. REICHEL‡ , G. RODRIGUEZ§ , AND X. YU¶ Abstract. Tikhonov regularization is one of the most popular methods for computing an approx- imate solution of linear discrete ill-posed problems with error-contaminated data. A regularization parameter λ > 0 balances the influence of a fidelity term, which measures how well the data are approximated, and of a regularization term, which dampens the propagation of the data error into the computed approximate solution. The value of the regularization parameter is important for the quality of the computed solution: a too large value of λ > 0 gives an over-smoothed solution that lacks details that the desired solution may have, while a too small value yields a computed solution that is unnecessarily, and possibly severely, contaminated by propagated error. When a fairly ac- curate estimate of the norm of the error in the data is known, a suitable value of λ often can be determined with the aid of the discrepancy principle. This paper is concerned with the situation when the discrepancy principle cannot be applied. It then can be quite difficult to determine a suit- able value of λ. We consider the situation when the Tikhonov regularization problem is in general form, i.e., when the regularization term is determined by a regularization matrix different from the identity, and describe an extension of the COSE method for determining the regularization param- eter λ in this situation. This method has previously been discussed for Tikhonov regularization in standard form, i.e., for the situation when the regularization matrix is the identity. -
Least Squares Regression Principal Component Analysis a Supervised Dimensionality Reduction Method for Machine Learning in Scientific Applications
Universitat Politecnica` de Catalunya Bachelor's Degree in Engineering Physics Bachelor's Thesis Least Squares Regression Principal Component Analysis A supervised dimensionality reduction method for machine learning in scientific applications Supervisor: Author: Dr. Xin Yee H´ector Pascual Herrero Dr. Joan Torras June 29, 2020 Abstract Dimension reduction is an important technique in surrogate modeling and machine learning. In this thesis, we present three existing dimension reduction methods in de- tail and then we propose a novel supervised dimension reduction method, `Least Squares Regression Principal Component Analysis" (LSR-PCA), applicable to both classifica- tion and regression dimension reduction tasks. To show the efficacy of this method, we present different examples in visualization, classification and regression problems, com- paring it to state-of-the-art dimension reduction methods. Furthermore, we present the kernel version of LSR-PCA for problems where the input are correlated non-linearly. The examples demonstrated that LSR-PCA can be a competitive dimension reduction method. 1 Acknowledgements I would like to express my gratitude to my thesis supervisor, Professor Xin Yee. I would like to thank her for giving me this wonderful opportunity and for her guidance and support during all the passing of this semester, putting herself at my disposal during the difficult times of the COVID-19 situation. Without her, the making of this thesis would not have been possible. I would like to extend my thanks to Mr. Pere Balsells, for allowing students like me to conduct their thesis abroad, as well as to the Balsells Foundation for its help and support throughout the whole stay. -
Statistical Foundations of Data Science
Statistical Foundations of Data Science Jianqing Fan Runze Li Cun-Hui Zhang Hui Zou i Preface Big data are ubiquitous. They come in varying volume, velocity, and va- riety. They have a deep impact on systems such as storages, communications and computing architectures and analysis such as statistics, computation, op- timization, and privacy. Engulfed by a multitude of applications, data science aims to address the large-scale challenges of data analysis, turning big data into smart data for decision making and knowledge discoveries. Data science integrates theories and methods from statistics, optimization, mathematical science, computer science, and information science to extract knowledge, make decisions, discover new insights, and reveal new phenomena from data. The concept of data science has appeared in the literature for several decades and has been interpreted differently by different researchers. It has nowadays be- come a multi-disciplinary field that distills knowledge in various disciplines to develop new methods, processes, algorithms and systems for knowledge dis- covery from various kinds of data, which can be either low or high dimensional, and either structured, unstructured or semi-structured. Statistical modeling plays critical roles in the analysis of complex and heterogeneous data and quantifies uncertainties of scientific hypotheses and statistical results. This book introduces commonly-used statistical models, contemporary sta- tistical machine learning techniques and algorithms, along with their mathe- matical insights and statistical theories. It aims to serve as a graduate-level textbook on the statistical foundations of data science as well as a research monograph on sparsity, covariance learning, machine learning and statistical inference. For a one-semester graduate level course, it may cover Chapters 2, 3, 9, 10, 12, 13 and some topics selected from the remaining chapters. -
Nonparametric Regression
Nonparametric Regression Statistical Machine Learning, Spring 2015 Ryan Tibshirani (with Larry Wasserman) 1 Introduction, and k-nearest-neighbors 1.1 Basic setup, random inputs Given a random pair (X; Y ) Rd R, recall that the function • 2 × f0(x) = E(Y X = x) j is called the regression function (of Y on X). The basic goal in nonparametric regression is ^ d to construct an estimate f of f0, from i.i.d. samples (x1; y1);::: (xn; yn) R R that have the same joint distribution as (X; Y ). We often call X the input, predictor,2 feature,× etc., and Y the output, outcome, response, etc. Importantly, in nonparametric regression we do not assume a certain parametric form for f0 Note for i.i.d. samples (x ; y );::: (x ; y ), we can always write • 1 1 n n yi = f0(xi) + i; i = 1; : : : n; where 1; : : : n are i.i.d. random errors, with mean zero. Therefore we can think about the sampling distribution as follows: (x1; 1);::: (xn; n) are i.i.d. draws from some common joint distribution, where E(i) = 0, and then y1; : : : yn are generated from the above model In addition, we will assume that each i is independent of xi. As discussed before, this is • actually quite a strong assumption, and you should think about it skeptically. We make this assumption for the sake of simplicity, and it should be noted that a good portion of theory that we cover (or at least, similar theory) also holds without the assumption of independence between the errors and the inputs 1.2 Basic setup, fixed inputs Another common setup in nonparametric regression is to directly assume a model • yi = f0(xi) + i; i = 1; : : : n; where now x1; : : : xn are fixed inputs, and 1; : : : are still i.i.d. -
Regularized Radial Basis Function Models for Stochastic Simulation
Proceedings of the 2014 Winter Simulation Conference A. Tolk, S. Y. Diallo, I. O. Ryzhov, L. Yilmaz, S. Buckley, and J. A. Miller, eds. REGULARIZED RADIAL BASIS FUNCTION MODELS FOR STOCHASTIC SIMULATION Yibo Ji Sujin Kim Department of Industrial and Systems Engineering National University of Singapore 119260, Singapore ABSTRACT We propose a new radial basis function (RBF) model for stochastic simulation, called regularized RBF (R-RBF). We construct the R-RBF model by minimizing a regularized loss over a reproducing kernel Hilbert space (RKHS) associated with RBFs. The model can flexibly incorporate various types of RBFs including those with conditionally positive definite basis. To estimate the model prediction error, we first represent the RKHS as a stochastic process associated to the RBFs. We then show that the prediction model obtained from the stochastic process is equivalent to the R-RBF model and derive the associated mean squared error. We propose a new criterion for efficient parameter estimation based on the closed form of the leave-one-out cross validation error for R-RBF models. Numerical results show that R-RBF models are more robust, and yet fairly accurate compared to stochastic kriging models. 1 INTRODUCTION In this paper, we aim to develop a radial basis function (RBF) model to approximate a real-valued smooth function using noisy observations obtained via black-box simulations. The observations are assumed to capture the information on the true function f through the commonly-used additive error model, d y = f (x) + e(x); x 2 X ⊂ R ; 2 where the random noise is characterized by a normal random variable e(x) ∼ N 0;se (x) , whose variance is determined by a continuously differentiable variance function se : X ! R. -
Regularization Tools
Regularization Tools A Matlab Package for Analysis and Solution of Discrete Ill-Posed Problems Version 4.1 for Matlab 7.3 Per Christian Hansen Informatics and Mathematical Modelling Building 321, Technical University of Denmark DK-2800 Lyngby, Denmark [email protected] http://www.imm.dtu.dk/~pch March 2008 The software described in this report was originally published in Numerical Algorithms 6 (1994), pp. 1{35. The current version is published in Numer. Algo. 46 (2007), pp. 189{194, and it is available from www.netlib.org/numeralgo and www.mathworks.com/matlabcentral/fileexchange Contents Changes from Earlier Versions 3 1 Introduction 5 2 Discrete Ill-Posed Problems and their Regularization 9 2.1 Discrete Ill-Posed Problems . 9 2.2 Regularization Methods . 11 2.3 SVD and Generalized SVD . 13 2.3.1 The Singular Value Decomposition . 13 2.3.2 The Generalized Singular Value Decomposition . 14 2.4 The Discrete Picard Condition and Filter Factors . 16 2.5 The L-Curve . 18 2.6 Transformation to Standard Form . 21 2.6.1 Transformation for Direct Methods . 21 2.6.2 Transformation for Iterative Methods . 22 2.6.3 Norm Relations etc. 24 2.7 Direct Regularization Methods . 25 2.7.1 Tikhonov Regularization . 25 2.7.2 Least Squares with a Quadratic Constraint . 25 2.7.3 TSVD, MTSVD, and TGSVD . 26 2.7.4 Damped SVD/GSVD . 27 2.7.5 Maximum Entropy Regularization . 28 2.7.6 Truncated Total Least Squares . 29 2.8 Iterative Regularization Methods . 29 2.8.1 Conjugate Gradients and LSQR . -
Enabling Deeper Learning on Big Data for Materials Informatics Applications
www.nature.com/scientificreports OPEN Enabling deeper learning on big data for materials informatics applications Dipendra Jha1, Vishu Gupta1, Logan Ward2,3, Zijiang Yang1, Christopher Wolverton4, Ian Foster2,3, Wei‑keng Liao1, Alok Choudhary1 & Ankit Agrawal1* The application of machine learning (ML) techniques in materials science has attracted signifcant attention in recent years, due to their impressive ability to efciently extract data‑driven linkages from various input materials representations to their output properties. While the application of traditional ML techniques has become quite ubiquitous, there have been limited applications of more advanced deep learning (DL) techniques, primarily because big materials datasets are relatively rare. Given the demonstrated potential and advantages of DL and the increasing availability of big materials datasets, it is attractive to go for deeper neural networks in a bid to boost model performance, but in reality, it leads to performance degradation due to the vanishing gradient problem. In this paper, we address the question of how to enable deeper learning for cases where big materials data is available. Here, we present a general deep learning framework based on Individual Residual learning (IRNet) composed of very deep neural networks that can work with any vector‑ based materials representation as input to build accurate property prediction models. We fnd that the proposed IRNet models can not only successfully alleviate the vanishing gradient problem and enable deeper learning, but also lead to signifcantly (up to 47%) better model accuracy as compared to plain deep neural networks and traditional ML techniques for a given input materials representation in the presence of big data. -
On the Determination of Proper Regularization Parameter
Universität Stuttgart Geodätisches Institut On the determination of proper regularization parameter α-weighted BLE via A-optimal design and its comparison with the results derived by numerical methods and ridge regression Bachelorarbeit im Studiengang Geodäsie und Geoinformatik an der Universität Stuttgart Kun Qian Stuttgart, Februar 2017 Betreuer: Dr.-Ing. Jianqing Cai Universität Stuttgart Prof. Dr.-Ing. Nico Sneeuw Universität Stuttgart Erklärung der Urheberschaft Ich erkläre hiermit an Eides statt, dass ich die vorliegende Arbeit ohne Hilfe Dritter und ohne Benutzung anderer als der angegebenen Hilfsmittel angefertigt habe; die aus fremden Quellen direkt oder indirekt übernommenen Gedanken sind als solche kenntlich gemacht. Die Arbeit wurde bisher in gleicher oder ähnlicher Form in keiner anderen Prüfungsbehörde vorgelegt und auch noch nicht veröffentlicht. Ort, Datum Unterschrift III Abstract In this thesis, several numerical regularization methods and the ridge regression, which can help improve improper conditions and solve ill-posed problems, are reviewed. The deter- mination of the optimal regularization parameter via A-optimal design, the optimal uniform Tikhonov-Phillips regularization (a-weighted biased linear estimation), which minimizes the trace of the mean square error matrix MSE(ˆx), is also introduced. Moreover, the comparison of the results derived by A-optimal design and results derived by numerical heuristic methods, such as L-curve, Generalized Cross Validation and the method of dichotomy is demonstrated. According to the comparison, the A-optimal design regularization parameter has been shown to have minimum trace of MSE(ˆx) and its determination has better efficiency. VII Contents List of Figures IX List of Tables XI 1 Introduction 1 2 Least Squares Method and Ill-Posed Problems 3 2.1 The Special Gauss-Markov medel . -
The L-Curve and Its Use in the Numerical Treatment of Inverse Problems 1 Introduction
The L-curve and its use in the numerical treatment of inverse problems P. C. Hansen Department of Mathematical Modelling, Technical University of Denmark, DK-2800 Lyngby, Denmark Abstract The L-curve is a log-log plot of the norm of a regularized solution versus the norm of the corresponding residual norm. It is a convenient graphical tool for displaying the trade-off between the size of a regularized solution and its fit to the given data, as the regularization parameter varies. The L-curve thus gives insight into the regularizing properties of the underlying regularization method, and it is an aid in choosing an appropriate regu- larization parameter for the given data. In this chapter we summarize the main properties of the L-curve, and demonstrate by examples its usefulness and its limitations both as an analysis tool and as a method for choosing the regularization parameter. 1 Introduction Practically all regularization methods for computing stable solutions to inverse problems involve a trade-off between the “size” of the reg- ularized solution and the quality of the fit that it provides to the given data. What distinguishes the various regularization methods is how they measure these quantities, and how they decide on the optimal trade-off between the two quantities. For example, given the discrete linear least-squares problem min kA x¡bk2 (which specializes to A x = b if A is square), the classical regularization method devel- oped independently by Phillips [31] and Tikhonov [35] (but usually referred to as Tikhonov regularization) amounts — in its most general 1 form — to solving the minimization problem n o 2 2 2 x¸ = arg min kA x ¡ bk2 + ¸ kL (x ¡ x0)k2 ; (1) where ¸ is a real regularization parameter that must be chosen by the user. -
COVID-19 and Machine Learning: Investigation and Prediction Team #5
COVID-19 and Machine Learning: Investigation and Prediction Team #5: Kiran Brar Olivia Alexander Kimberly Segura 1 Abstract The COVID-19 virus has affected over four million people in the world. In the U.S. alone, the number of positive cases have exceeded one million, making it the most affected country. There is clear urgency to predict and ultimately decrease the spread of this infectious disease. Therefore, this project was motivated to test and determine various machine learning models that can accurately predict the number of confirmed COVID-19 cases in the U.S. using available time-series data. COVID-19 data was coupled with state demographic data to investigate the distribution of cases and potential correlations between demographic features. Concerning the four machine learning models tested, it was hypothesized that LSTM and XGBoost would result in the lowest errors due to the complexity and power of these models, followed by SVR and linear regression. However, linear regression and SVR had the best performance in this study which demonstrates the importance of testing simpler models and only adding complexity if the data requires it. We note that LSTM’s low performance was most likely due to the size of the training dataset available at the time of this research as deep learning requires a vast amount of data. Additionally, each model’s accuracy improved after implementing time-series preprocessing techniques of power transformations, normalization, and the overall restructuring of the time-series problem to a supervised machine learning problem using lagged values. This research can be furthered by predicting the number of deaths and recoveries as well as extending the models by integrating healthcare capacity and social restrictions in order to increase accuracy or to forecast infection, death, and recovery rates for future dates. -
Arxiv:1707.06852V1 [Stat.ME] 21 Jul 2017 Dynamical System Dx T = G(T, X , Θ), (1.1) Dt T Where G Is a Known Function and Θ Is a Parameter
A Statistical Perspective on Inverse and Inverse Regression Problems Debashis Chatterjeea, Sourabh Bhattacharyaa,∗ aInterdisciplinary Statistical Research Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata - 700108, India Abstract Inverse problems, where in broad sense the task is to learn from the noisy response about some unknown function, usually represented as the argument of some known functional form, has received wide attention in the general scientific disciplines. How- ever, in mainstream statistics such inverse problem paradigm does not seem to be as popular. In this article we provide a brief overview of such problems from a statistical, particularly Bayesian, perspective. We also compare and contrast the above class of problems with the perhaps more statistically familiar inverse regression problems, arguing that this class of problems contains the traditional class of inverse problems. In course of our review we point out that the statistical literature is very scarce with respect to both the inverse paradigms, and substantial research work is still necessary to develop the fields. Keywords: Bayesian analysis, Inverse problems, Inverse regression problems, Regularization, Reproducing Kernel Hilbert Space (RKHS), Palaeoclimate reconstruction 1. Introduction The similarities and dissimilarities between inverse problems and the more tradi- tional forward problems are usually not clearly explained in the literature, and often “ill-posed” is the term used to loosely characterize inverse problems. We point out that these two problems may have the same goal or different goal, while both consider the same model given the data. We first elucidate using the traditional case of deterministic differential equations, that the goals of the two problems may be the same.