DOCSLIB.ORG

Linear Regression Linear Regression with Shrinkage

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Linear Regression Linear Regression with Shrinkage Linear Regression Linear Regression with Shrinkage Some slides are due to Tommi Jaakkola, MIT AI Lab Introduction The goal of regression is to make quantitative (real valued) predictions on the basis of a (vector of) features or attributes. Examples: house prices, stock values, survival time, fuel efficiency of cars, etc. Predicting vehicle fuel efficiency (mpg) from 8 attributes: A generic regression problem The input attributes are given as fixed length vectors x ∈ℜM that could come from different sources: inputs, transformation of inputs (log, square root, etc…) or basis functions. The outputs are assumed to be real valued y ∈ ℜ (with some possible restrictions). Given n iid training samples D = {( x 1 , y 1 )...( x n , y n )} from unknown distribution P(x, y), the goal is to minimize the prediction error (loss) on new examples (x, y) drawn at random from the same P(x, y). An a example of a loss function: L( f ),x( y) = ( f x)( − y)2 squared loss our prediction for x Regression Function We need to define a class of functions (types of predictions we make) . Linear prediction: f (x;w0 , w1) = w0 + w1x where w 0 , w 1 are the parameters we need estimate. Linear Regression Typically we have a set of training data ( x 1 , y 1 )...( xn , yn ) from which we estimate the parameters w 0 , w 1 . Each x i = ( x i 1 ,.., x iM ) is a vector of measurements for i th case. or h(xi ) = {h0 (xi ),..., hM (xi )} a basis expansion of xi M t y(x) ≈ f (x;w) = w0 + ∑ w j hj (x) = w h(x) j=1 (define h0 (x) = )1 Basis Functions There are many basis functions we can use e.g. Polynomial j−1 hj (x) = x 2 (x − µ ) Radial basis functions h (x) = exp − j j 2s2 x − µ Sigmoidal j hj (x) = σ s Splines, Fourier, Wavelets, etc Estimation Criterion We need a fitting/estimation criterion to select appropriate values for the parameters w 0 , w 1 based on the training set D = {( x1, y1)...( xn , yn )} For example, we can use the empirical loss: n 1 2 J n (w1, w0 ) = ∑()yi − f (xi ;w1, w0 ) n i=1 (note: the loss is the same as in evaluation) Empirical loss: motivation Ideally, we would like to find the parameters w0 , w1 , that minimize the expected loss (unlimited training data): 2 J (w0 , w1) = E(x, y ~) P (yi − f (xi ;w0 , w1)) where the expectation is over samples from P(x,y). When the number of training examples n is large: n 2 1 2 E(x,y ~) P ()yi − f (xi ;w0 , w1) ≈ ∑()yi − f (xi ;w0 , w1) n i=1 Expected loss Empirical loss Linear Regression Estimation Minimize the empirical squared loss n 1 2 J n (w0 , w1) = ∑()yi − f (xi ;w0 , w1) n i=1 n 1 2 = ∑()yi − w0 − w1xi n i=1 By setting the derivatives with respect to w 1 , w 0 to zero we get necessary conditions for the “optimal” parameter values. ∂ 2 n J n (w0 , w1) = ∑()yi − w0 − w1xi (−xi ) = 0 ∂w1 n i=1 ∂ 2 n J n (w0 , w1) = ∑()yi − w0 − w1xi (− )1 = 0 ∂w0 n i=1 Interpretation The optimality conditions 2 n ∑()yi − w0 − w1xi (−xi ) = 0 n i=1 2 n ∑()yi − w0 − w1xi (− )1 = 0 n i=1 ensure that the prediction error ε i = (y − w0 − w1xi ) is decorrelated with any linear function of the inputs. Linear Regression: Matrix Form x samples n y 1 1 w0 1 M+1 X = ... ... y = ... w = w1 1 xn y n M+1 n 1 2 t J n (w0 , w1) = ∑()()()yi − w0 − w1xi = Xw − y Xw − y n i=1 Linear Regression Solution By setting the derivatives of ( Xw − y)t (Xw − y) to zero, 2 (Xt y − Xt Xw )= 0 n we get the solution: −1 wˆ = (Xt X) Xt y The solution is a linear function of the outputs y. Statistical view of linear regression In a statistical regression model we model both the function and noise Observed output = function + noise y(x) = f (x;w) + ε where, e.g., ε ~ N(0,σ 2 ) Whatever we cannot capture with our chosen family of functions will be interpreted as noise Statistical view of linear regression f(x;w ) is trying to capture the mean of the observations y given the input x: E[y | x] = E[ f (x; w) + ε | x] = f (x;w) where E[ y| x ] is the conditional expectation of y given x, evaluated according to the model (not according to the underlying distribution of X) Statistical view of linear regression According to our statistical model y(x) = f (x;w) + ε, ε ~ N ,0( σ 2 ) the outputs y given x are normally distributed with mean f (x;w) and variance σ2: 1 1 p(y | x, w,σ 2 ) = exp − (y − f (x;w)) 2 2 2πσ 2 2σ (we model the uncertainty in the predictions, not just the mean) Maximum likelihood estimation Given observations D = { ( x 1 , y 1 ),..., ( x n , y n ) } we find the parameters w that maximize the likelihood of the outputs: n 2 2 L(w,σ ) = ∏ p(yi | xi , w,σ ) i=1 n 1 1 n = exp − ()y − f ()x ;w 2 2 2 ∑ k k 2πσ 2σ i=1 Maximize log-likelihood n 1 1 n log L(w,σ 2 ) = log − ()y − f ()x ;w 2 2 2 ∑ k k 2πσ 2σ i=1 minimize Maximum likelihood estimation Thus n 2 wMLE = arg min ∑()yi − f ()xi ;w w i=1 But the empirical squared loss is n 1 2 J n (w) = ∑()yi − f (xi ; w) n i=1 Least-squares Linear Regression is MLE for Gaussian noise !!! Linear Regression (is it “good”?) Simple model Straightforward solution BUT • MLS is not a good estimator for prediction error • The matrix XTX could be ill conditioned Inputs are correlated Input dimension is large Training set is smal l Linear Regression - Example In this example the output is generated by the model (where ε is a small white Gaussian noise): y = 8.0 x1 + 2.0 x2 + ε Three training sets with different correlations between the two inputs were randomly chosen, and the linear regression solution was applied. Linear Regression – Example And the results are… RSS=0.29RSS =0.0901465 RSS=0.204RSS =0.104955 RSS=0.47RSS =0.0725469 w=0.82,b=80.807276 0.209 , 0.209322 < w=0.42,b=80.434832 0.56 , 0.568331 < w=30.36,b=830.2363, -29.21-29.2196 < 2 1.5 3 3 1 2 2 0.5 1 2 2 2 1 x x x 0 0 0 -0.5 -1 -1 -1 -1.5 -2 -2 -2 -1 0 1 2 3 -2 -1 0 1 2 3 -2 -1 0 1 2 3 x1 x1 x1 x ,x correlated x ,x strongly x1,x 2 uncorrelated 1 2 1 2 correlated Strong correlation can cause the coefficients to be very large, and they will cancel each other to achieve a good RSS . Linear Regression What can be done? Shrinkage methods Ridge Regression Lasso PCR (Principal Components Regression) PLS (Partial Least Squares) Shrinkage methods Before we proceed: Since the following methods are not invariant under input scale, we will assume the input is normalized (mean 0, variance 1): xij′ xij′ ← xij − x j xij ← σ j The offset w0 is always estimated as w0 = (∑ yi /) N i and we will work with centered y (meaning y y-w0) Ridge Regression Ridge regression shrinks the regression coefficients by imposing a penalty on their size ( also called weight decay) In ridge regression, we add a quadratic penalty on the weights: 2 N M M J (w) = y − w − x w + λ w2 ∑ i 0 ∑ ij j ∑ j i=1 j=1 j=1 where λ ≥ 0 is a tuning parameter that controls the amount of shrinkage. The size constraint prevents the phenomenon of wildly large coefficients that cancel each other from occurring. Ridge Regression Solution Ridge regression in matrix form: J (w) = (y − Xw )t (y − Xw )+ λwt w The solution is ridge t −1 t LS t −1 t wˆ = (X X + λI M ) X y wˆ = (X X) X y The solution adds a positive constant to the diagonal of XTX before inversion. This makes the problem non singular even if X does not have full column rank. For orthogonal inputs the ridge estimates are the scaled version of least squares estimates: wˆ ridge = γ wˆ LS 0 ≤ γ ≤1 Ridge Regression (insights) The matrix X can be represented by it’s SVD: X = UDV T • U is a N*M matrix, it’s columns span the column space of X • D is a M*M diagonal matrix of singular values • V is a M*M matrix, it’s columns span the row space of X Lets see how this method looks in the Principal Components coordinates of X Ridge Regression (insights) X '= XV Xw = ()XV ()V T w = X 'w' −1 The least squares solution is given by: wˆ ls = (Xt X) Xt y −1 wˆ'ls = V T wˆ ls = V T (VDU TUDV T ) VDU T y = D−1U T y The Ridge Regression is similarly given by: −1 wˆ'ridge = V T wˆ ridge = V T (VDU TUDV T + λI ) VDU T y = 2 −1 T 2 −1 2 ls = (D + λI ) DU y = (D + λI ) D wˆ' Diagonal Ridge Regression (insights) 2 ridge d j ls wˆ' j = 2 wˆ' j d j + λ In the PCA axes, the ridge coefficients are just scaled LS coefficients! The coefficients that correspond to smaller input variance directions are scaled down more.
Load more

Recommended publications
  • Shrinkage Priors for Bayesian Penalized Regression
    Shrinkage Priors for Bayesian Penalized Regression. Sara van Erp1, Daniel L. Oberski2, and Joris Mulder1 1Tilburg University 2Utrecht University This manuscript has been published in the Journal of Mathematical Psychology. Please cite the published version: Van Erp, S., Oberski, D. L., & Mulder, J. (2019). Shrinkage priors for Bayesian penalized regression. Journal of Mathematical Psychology, 89, 31–50. doi:10.1016/j.jmp.2018.12.004 Abstract In linear regression problems with many predictors, penalized regression tech- niques are often used to guard against overfitting and to select variables rel- evant for predicting an outcome variable. Recently, Bayesian penalization is becoming increasingly popular in which the prior distribution performs a function similar to that of the penalty term in classical penalization. Specif- ically, the so-called shrinkage priors in Bayesian penalization aim to shrink small effects to zero while maintaining true large effects. Compared toclas- sical penalization techniques, Bayesian penalization techniques perform sim- ilarly or sometimes even better, and they offer additional advantages such as readily available uncertainty estimates, automatic estimation of the penalty parameter, and more flexibility in terms of penalties that can be considered. However, many different shrinkage priors exist and the available, often quite technical, literature primarily focuses on presenting one shrinkage prior and often provides comparisons with only one or two other shrinkage priors. This can make it difficult for researchers to navigate through the many prior op- tions and choose a shrinkage prior for the problem at hand. Therefore, the aim of this paper is to provide a comprehensive overview of the literature on Bayesian penalization.
    [Show full text]
  • Shrinkage Estimation of Rate Statistics Arxiv:1810.07654V1 [Stat.AP] 17 Oct
    Published, CS-BIGS 7(1):14-25 http://www.csbigs.fr Shrinkage estimation of rate statistics Einar Holsbø Department of Computer Science, UiT — The Arctic University of Norway Vittorio Perduca Laboratory of Applied Mathematics MAP5, Université Paris Descartes This paper presents a simple shrinkage estimator of rates based on Bayesian methods. Our focus is on crime rates as a motivating example. The estimator shrinks each town’s observed crime rate toward the country-wide average crime rate according to town size. By realistic simulations we confirm that the proposed estimator outperforms the maximum likelihood estimator in terms of global risk. We also show that it has better coverage properties. Keywords : Official statistics, crime rates, inference, Bayes, shrinkage, James-Stein estimator, Monte-Carlo simulations. 1. Introduction that most of the best schools—according to a variety of performance measures—were small. 1.1. Two counterintuitive random phenomena As it turns out, there is nothing special about small schools except that they are small: their It is a classic result in statistics that the smaller over-representation among the best schools is the sample, the more variable the sample mean. a consequence of their more variable perfor- The result is due to Abraham de Moivre and it mance, which is counterbalanced by their over- tells us that the standard deviation of the mean p representation among the worst schools. The is sx¯ = s/ n, where n is the sample size and s observed superiority of small schools was sim- the standard deviation of the random variable ply a statistical fluke. of interest.
    [Show full text]
  • Linear Methods for Regression and Shrinkage Methods
    Linear Methods for Regression and Shrinkage Methods Reference: The Elements of Statistical Learning, by T. Hastie, R. Tibshirani, J. Friedman, Springer 1 Linear Regression Models Least Squares • Input vectors • is an attribute / feature / predictor (independent variable) • The linear regression model: • The output is called response (dependent variable) • ’s are unknown parameters (coefficients) 2 Linear Regression Models Least Squares • A set of training data • Each corresponding to attributes • Each is a class attribute value / a label • Wish to estimate the parameters 3 Linear Regression Models Least Squares • One common approach ‐ the method of least squares: • Pick the coefficients to minimize the residual sum of squares: 4 Linear Regression Models Least Squares • This criterion is reasonable if the training observations represent independent draws. • Even if the ’s were not drawn randomly, the criterion is still valid if the ’s are conditionally independent given the inputs . 5 Linear Regression Models Least Squares • Make no assumption about the validity of the model • Simply finds the best linear fit to the data 6 Linear Regression Models Finding Residual Sum of Squares • Denote by the matrix with each row an input vector (with a 1 in the first position) • Let be the N‐vector of outputs in the training set • Quadratic function in the parameters: 7 Linear Regression Models Finding Residual Sum of Squares • Set the first derivation to zero: • Obtain the unique solution: 8 Linear Regression
    [Show full text]
  • Linear, Ridge Regression, and Principal Component Analysis
    Linear, Ridge Regression, and Principal Component Analysis Linear, Ridge Regression, and Principal Component Analysis Jia Li Department of Statistics The Pennsylvania State University Email: [email protected] http://www.stat.psu.edu/∼jiali Jia Li http://www.stat.psu.edu/∼jiali Linear, Ridge Regression, and Principal Component Analysis Introduction to Regression I Input vector: X = (X1, X2, ..., Xp). I Output Y is real-valued. I Predict Y from X by f (X ) so that the expected loss function E(L(Y , f (X ))) is minimized. I Square loss: L(Y , f (X )) = (Y − f (X ))2 . I The optimal predictor ∗ 2 f (X ) = argminf (X )E(Y − f (X )) = E(Y | X ) . I The function E(Y | X ) is the regression function. Jia Li http://www.stat.psu.edu/∼jiali Linear, Ridge Regression, and Principal Component Analysis Example The number of active physicians in a Standard Metropolitan Statistical Area (SMSA), denoted by Y , is expected to be related to total population (X1, measured in thousands), land area (X2, measured in square miles), and total personal income (X3, measured in millions of dollars). Data are collected for 141 SMSAs, as shown in the following table. i : 1 2 3 ... 139 140 141 X1 9387 7031 7017 ... 233 232 231 X2 1348 4069 3719 ... 1011 813 654 X3 72100 52737 54542 ... 1337 1589 1148 Y 25627 15389 13326 ... 264 371 140 Goal: Predict Y from X1, X2, and X3. Jia Li http://www.stat.psu.edu/∼jiali Linear, Ridge Regression, and Principal Component Analysis Linear Methods I The linear regression model p X f (X ) = β0 + Xj βj .
    [Show full text]
  • Kernel Mean Shrinkage Estimators
    Journal of Machine Learning Research 17 (2016) 1-41 Submitted 5/14; Revised 9/15; Published 4/16 Kernel Mean Shrinkage Estimators Krikamol Muandet∗ [email protected] Empirical Inference Department, Max Planck Institute for Intelligent Systems Spemannstraße 38, T¨ubingen72076, Germany Bharath Sriperumbudur∗ [email protected] Department of Statistics, Pennsylvania State University University Park, PA 16802, USA Kenji Fukumizu [email protected] The Institute of Statistical Mathematics 10-3 Midoricho, Tachikawa, Tokyo 190-8562 Japan Arthur Gretton [email protected] Gatsby Computational Neuroscience Unit, CSML, University College London Alexandra House, 17 Queen Square, London - WC1N 3AR, United Kingdom ORCID 0000-0003-3169-7624 Bernhard Sch¨olkopf [email protected] Empirical Inference Department, Max Planck Institute for Intelligent Systems Spemannstraße 38, T¨ubingen72076, Germany Editor: Ingo Steinwart Abstract A mean function in a reproducing kernel Hilbert space (RKHS), or a kernel mean, is central to kernel methods in that it is used by many classical algorithms such as kernel principal component analysis, and it also forms the core inference step of modern kernel methods that rely on embedding probability distributions in RKHSs. Given a finite sample, an empirical average has been used commonly as a standard estimator of the true kernel mean. Despite a widespread use of this estimator, we show that it can be improved thanks to the well- known Stein phenomenon. We propose a new family of estimators called kernel mean shrinkage estimators (KMSEs), which benefit from both theoretical justifications and good empirical performance. The results demonstrate that the proposed estimators outperform the standard one, especially in a \large d, small n" paradigm.
    [Show full text]
  • Generalized Shrinkage Methods for Forecasting Using Many Predictors
    Generalized Shrinkage Methods for Forecasting using Many Predictors This revision: February 2011 James H. Stock Department of Economics, Harvard University and the National Bureau of Economic Research and Mark W. Watson Woodrow Wilson School and Department of Economics, Princeton University and the National Bureau of Economic Research AUTHOR’S FOOTNOTE James H. Stock is the Harold Hitchings Burbank Professor of Economics, Harvard University (e-mail: [email protected]). Mark W. Watson is the Howard Harrison and Gabrielle Snyder Beck Professor of Economics and Public Affairs, Princeton University ([email protected]). The authors thank Jean Boivin, Domenico Giannone, Lutz Kilian, Serena Ng, Lucrezia Reichlin, Mark Steel, and Jonathan Wright for helpful discussions, Anna Mikusheva for research assistance, and the referees for helpful suggestions. An earlier version of the theoretical results in this paper was circulated earlier under the title “An Empirical Comparison of Methods for Forecasting Using Many Predictors.” Replication files for the results in this paper can be downloaded from http://www.princeton.edu/~mwatson. This research was funded in part by NSF grants SBR-0214131 and SBR-0617811. 0 Generalized Shrinkage Methods for Forecasting using Many Predictors JBES09-255 (first revision) This revision: February 2011 [BLINDED COVER PAGE] ABSTRACT This paper provides a simple shrinkage representation that describes the operational characteristics of various forecasting methods designed for a large number of orthogonal predictors (such as principal components). These methods include pretest methods, Bayesian model averaging, empirical Bayes, and bagging. We compare empirically forecasts from these methods to dynamic factor model (DFM) forecasts using a U.S. macroeconomic data set with 143 quarterly variables spanning 1960-2008.
    [Show full text]
  • Cross-Validation, Shrinkage and Variable Selection in Linear Regression Revisited
    Open Journal of Statistics, 2013, 3, 79-102 http://dx.doi.org/10.4236/ojs.2013.32011 Published Online April 2013 (http://www.scirp.org/journal/ojs) Cross-Validation, Shrinkage and Variable Selection in Linear Regression Revisited Hans C. van Houwelingen1, Willi Sauerbrei2 1Department of Medical Statistics and Bioinformatics, Leiden University Medical Center, Leiden, The Netherlands 2Institut fuer Medizinische Biometrie und Medizinische Informatik, Universitaetsklinikum Freiburg, Freiburg, Germany Email: [email protected] Received December 8, 2012; revised January 10, 2013; accepted January 26, 2013 Copyright © 2013 Hans C. van Houwelingen, Willi Sauerbrei. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT In deriving a regression model analysts often have to use variable selection, despite of problems introduced by data- dependent model building. Resampling approaches are proposed to handle some of the critical issues. In order to assess and compare several strategies, we will conduct a simulation study with 15 predictors and a complex correlation struc- ture in the linear regression model. Using sample sizes of 100 and 400 and estimates of the residual variance corre- sponding to R2 of 0.50 and 0.71, we consider 4 scenarios with varying amount of information. We also consider two examples with 24 and 13 predictors, respectively. We will discuss the value of cross-validation, shrinkage and back- ward elimination (BE) with varying significance level. We will assess whether 2-step approaches using global or pa- rameterwise shrinkage (PWSF) can improve selected models and will compare results to models derived with the LASSO procedure.
    [Show full text]
  • Mining Big Data Using Parsimonious Factor, Machine Learning, Variable Selection and Shrinkage Methods Hyun Hak Kim1 and Norman R
    Mining Big Data Using Parsimonious Factor, Machine Learning, Variable Selection and Shrinkage Methods Hyun Hak Kim1 and Norman R. Swanson2 1Bank of Korea and 2Rutgers University February 2016 Abstract A number of recent studies in the economics literature have focused on the usefulness of factor models in the context of prediction using “big data” (see Bai and Ng (2008), Dufour and Stevanovic (2010), Forni et al. (2000, 2005), Kim and Swanson (2014a), Stock and Watson (2002b, 2006, 2012), and the references cited therein). We add to this literature by analyzing whether “big data” are useful for modelling low frequency macroeconomic variables such as unemployment, in‡ation and GDP. In particular, we analyze the predictive bene…ts associated with the use of principal component analysis (PCA), independent component analysis (ICA), and sparse principal component analysis (SPCA). We also evaluate machine learning, variable selection and shrinkage methods, including bagging, boosting, ridge regression, least angle regression, the elastic net, and the non-negative garotte. Our approach is to carry out a forecasting “horse-race” using prediction models constructed using a variety of model speci…cation approaches, factor estimation methods, and data windowing methods, in the context of the prediction of 11 macroeconomic variables relevant for monetary policy assessment. In many instances, we …nd that various of our benchmark models, including autoregressive (AR) models, AR models with exogenous variables, and (Bayesian) model averaging, do not dominate speci…cations based on factor-type dimension reduction combined with various machine learning, variable selection, and shrinkage methods (called “combination”models). We …nd that forecast combination methods are mean square forecast error (MSFE) “best” for only 3 of 11 variables when the forecast horizon, h = 1, and for 4 variables when h = 3 or 12.
    [Show full text]
  • Shrinkage Improves Estimation of Microbial Associations Under Di↵Erent Normalization Methods 1 2 1,3,4, 5,6,7, Michelle Badri , Zachary D
    bioRxiv preprint doi: https://doi.org/10.1101/406264; this version posted April 4, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC 4.0 International license. Shrinkage improves estimation of microbial associations under di↵erent normalization methods 1 2 1,3,4, 5,6,7, Michelle Badri , Zachary D. Kurtz , Richard Bonneau ⇤, and Christian L. Muller¨ ⇤ 1Department of Biology, New York University, New York, 10012 NY, USA 2Lodo Therapeutics, New York, 10016 NY, USA 3Center for Computational Biology, Flatiron Institute, Simons Foundation, New York, 10010 NY, USA 4Computer Science Department, Courant Institute, New York, 10012 NY, USA 5Center for Computational Mathematics, Flatiron Institute, Simons Foundation, New York, 10010 NY, USA 6Institute of Computational Biology, Helmholtz Zentrum M¨unchen, Neuherberg, Germany 7Department of Statistics, Ludwig-Maximilians-Universit¨at M¨unchen, Munich, Germany ⇤correspondence to: [email protected], cmueller@flatironinstitute.org ABSTRACT INTRODUCTION Consistent estimation of associations in microbial genomic Recent advances in microbial amplicon and metagenomic survey count data is fundamental to microbiome research. sequencing as well as large-scale data collection efforts Technical limitations, including compositionality, low sample provide samples across different microbial habitats that are sizes, and technical variability, obstruct standard
    [Show full text]
  • Shrinkage Estimation for Functional Principal Component Scores, with Application to the Population Kinetics of Plasma Folate
    Shrinkage Estimation for Functional Principal Component Scores, with Application to the Population Kinetics of Plasma Folate Fang Yao1, Hans-Georg M¨uller1; 2, Andrew J. Clifford3, Steven R. Dueker3, Jennifer Follett3, Yumei Lin3, Bruce A. Buchholz4 and John S. Vogel4 February 16, 2003 1Department of Statistics, University of California, One Shields Ave., Davis, CA 95616 2Corresponding Author, e-mail: [email protected] 3Department of Nutrition, University of California, One Shields Ave., Davis, CA 95616 4Center for Accelerator Mass Spectrometry, Lawrence Livermore National Laboratory, Livermore, CA 94551 1 ABSTRACT We present the application of a nonparametric method to perform functional principal com- ponents analysis for functional curve data that consist of measurements of a random trajectory for a sample of subjects. This design typically consists of an irregular grid of time points on which repeated measurements are taken for a number of subjects. We introduce shrinkage estimates for the functional principal component scores that serve as the random effects in the model. Scatter- plot smoothing methods are used to estimate the mean function and covariance surface of this model. We propose improved estimation in the neighborhood of and at the diagonal of the covari- ance surface, where the measurement errors are reflected. The presence of additive measurement errors motivates shrinkage estimates for the functional principal components scores. Shrinkage estimates are developed through best linear prediction and in a generalized version, aiming at min- imizing one-curve-leave-out prediction error. The estimation of individual trajectories combines data obtained from that individual as well as all other individuals. We apply our methods to new data regarding the analysis of the level of 14C-folate in plasma as a function of time since dosing healthy adults with a small tracer dose of 14C-folic acid.
    [Show full text]
  • Image Denoising Via Residual Kurtosis Minimization
    Image Denoising via Residual Kurtosis Minimization Tristan A. Hearn∗ Lothar Reichely Abstract A new algorithm for the removal of additive uncorrelated Gaussian noise from a digital image is presented. The algorithm is based on a data driven methodology for the adaptive thresholding of wavelet coefficients. This methodology is derived from higher order statistics of the residual image, and requires no a priori estimate of the level of noise contamination of an image. 1 Introduction With the mass production of digital images of many kinds, the need for efficient methods to restore corrupted images has become immediate. A major source of image contamination is noise. Image noise may arise from quantization of the image data, transmission errors, electronic interference from the imaging hardware, as well as from other sources [1]. Many methods have been proposed for denoising of digital images. Some are based on finite-impulse response (FIR) filtering, where the noisy image is convolved with a smoothing kernel to reduce the visible effect of noise. This may introduce undesirable artificial blur to an image [16]. Some methods cannot reliably distinguish between noise and edge information in an image; they may require a priori knowledge of the noise level in the contaminated image, or might introduce waveform artifacts. A recent discussion of many available methods can be found in [2]. Denoising methods based on the discrete wavelet transform are popular. These methods are similar to spectral filtering methods, but can take ad- vantage of the partial localization in space and frequency offered by wavelet transforms. This allows efficient separation of noise from impulsive signal ∗Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA.
    [Show full text]
  • An Empirical Bayes Approach to Shrinkage Estimation on the Manifold of Symmetric Positive-Definite Matrices∗†
    An Empirical Bayes Approach to Shrinkage Estimation on the Manifold of Symmetric Positive-Definite Matrices∗† Chun-Hao Yang1, Hani Doss1 and Baba C. Vemuri2 1Department of Statistics 2Department of Computer Information Science & Engineering University of Florida October 28, 2020 Abstract The James-Stein estimator is an estimator of the multivariate normal mean and dominates the maximum likelihood estimator (MLE) under squared error loss. The original work inspired great interest in developing shrinkage estimators for a variety of problems. Nonetheless, research on shrinkage estimation for manifold-valued data is scarce. In this paper, we propose shrinkage estimators for the parameters of the Log-Normal distribution defined on the manifold of N × N symmetric positive-definite matrices. For this manifold, we choose the Log-Euclidean metric as its Riemannian metric since it is easy to compute and is widely used in applications. By using the Log-Euclidean distance in the loss function, we derive a shrinkage estimator in an ana- lytic form and show that it is asymptotically optimal within a large class of estimators including the MLE, which is the sample Fr´echet mean of the data. We demonstrate the performance of the proposed shrinkage estimator via several simulated data exper- iments. Furthermore, we apply the shrinkage estimator to perform statistical inference in diffusion magnetic resonance imaging problems. arXiv:2007.02153v3 [math.ST] 27 Oct 2020 Keywords: Stein's unbiased risk estimate, Fr´echet mean, Tweedie's estimator ∗This research was in part funded by the NSF grants IIS-1525431 and IIS-1724174 to Vemuri. †This manuscript has been submitted to a journal.
    [Show full text]