We present a technique using data depth functions and resampling to perform best subset variable selection for a wide range of statistical models. We do this by assigning a score, called an $e$-value, to a candidate model, and use a fast bootstrap method to approximate sample versions of these $e$-values. Under general conditions, $e$-values can separate statistical models that adequately explain properties of the data from those that do not. This results in a fast algorithm that fits only a single model and evaluates $p +1$ models, $p$ being the number of predictors under consideration, as opposed to the traditional requirement of fitting and evaluating $2^{p}$ models. We illustrate in simulation experiments that our proposed method typically performs better than an array of currently used methods for variable selection in linear models and fixed effect selection in linear mixed models. As a real data application, we use our procedure to elicit climatic drivers of Indian summer monsoon precipitation.
Block-based resampling estimators have been intensively investigated for weakly dependent time processes, which has helped to inform implementation (e.g., best block sizes). However, little is known about resampling performance and block sizes under strong or long-range dependence. To establish guideposts in block selection, we consider a broad class of strongly dependent time processes, formed by a transformation of a stationary long-memory Gaussian series, and examine block-based resampling estimators for the variance of the prototypical sample mean; extensions to general statistical functio…
Bootstrap techniques (also called resampling computation techniques) have introduced new advances in modeling and model evaluation. Using resampling methods to construct a series of new samples which are based on the original data set, allows to estimate the stability of the parameters. Properties such as convergence and asymptotic normality can be checked for any particular observed data set. In most cases, the statistics computed on the generated data sets give a good idea of the confidence regions of the estimates. In this paper, we debate on the contribution of such methods for model selec…
We build penalized least-squares estimators using the slope heuristic and resampling penalties. We prove oracle inequalities for the selected estimator with leading constant asymptotically equal to 1. We compare the practical performances of these methods in a short simulation study.
The extremal index parameter theta characterizes the degree of local dependence in the extremes of a stationary time series and has important applications in a number of areas, such as hydrology, telecommunications, finance and environmental studies. In this study, a novel estimator for theta based on the asymptotic scaling of block-maxima and resampling is introduced. It is shown to be consistent and asymptotically normal for a large class of m-dependent time series. Further, a procedure for the automatic selection of its tuning parameter is developed and different types of confidence interva…
We propose a novel resampling-based method to construct an asymptotically exact test for any subset of hypotheses on coefficients in high-dimensional linear regression. It can be embedded into any multiple testing procedure to make confidence statements on relevant predictor variables. The method constructs permutation test statistics for any individual hypothesis by means of repeated splits of the data and a variable selection technique; then it defines a test for any subset by suitably aggregating its variables' test statistics. The resulting procedure is extremely flexible, as it allows dif…
Faculty of Mathematics and Physics, Charles University（查理大学数学与物理学院）；Institute for Mathematical Stochastics, Otto-von-Guericke-Universität Magdeburg（马格德堡奥托·冯·古里克大学数学随机性研究所）；Department of Statistics, TU Dortmund University（多特蒙德工业大学统计系）
Resampling techniques are widely used in statistical inference and ensemble learning, in which estimators' statistical properties are essential. However, existing methods are computationally demanding, because repetitions of estimation/learning via numerical optimization/integral for each resampled data are required. In this study, we introduce a computationally efficient method to resolve such problem: replicated vector approximate message passing. This is based on a combination of the replica method of statistical physics and an accurate approximate inference algorithm, namely the vector app…
We propose a block-resampling penalization method for marginal density estimation with nonnecessary independent observations. When the data are $β$ or $τ$-mixing, the selected estimator satisfies oracle inequalities with leading constant asymptotically equal to 1. We also prove in this setting the slope heuristic, which is a data-driven method to optimize the leading constant in the penalty.
Network models are applied in numerous domains where data can be represented as a system of interactions among pairs of actors. While both statistical and mechanistic network models are increasingly capable of capturing various dependencies amongst these actors, these dependencies imply the lack of independence. This poses statistical challenges for analyzing such data, especially when there is only a single observed network, and often leads to intractable likelihoods regardless of the modeling paradigm, which limit the application of existing statistical methods for networks. We explore a sub…
With recent advances in high throughput technology, researchers often find themselves running a large number of hypothesis tests (thousands+) and esti- mating a large number of effect-sizes. Generally there is particular interest in those effects estimated to be most extreme. Unfortunately naive estimates of these effect-sizes (even after potentially accounting for multiplicity in a testing procedure) can be severely biased. In this manuscript we explore this bias from a frequentist perspective: we give a formal definition, and show that an oracle estimator using this bias dominates the naive …
Weak signal identification and inference are very important in the area of penalized model selection, yet they are under-developed and not well-studied. Existing inference procedures for penalized estimators are mainly focused on strong signals. In this paper, we propose an identification procedure for weak signals in finite samples, and pro- vide a transition phase in-between noise and strong signal strengths. We also introduce a new two-step inferential method to construct better confidence intervals for the identified weak signals. Our theory development assumes that variables are orthogona…
Stability of a Generalized Debiased Lasso with Applications to Resampling-Based Variable Selection
Department of Statistics, University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校统计系）；Department of Electrical and Computer Engineering, the Grainger College of Engineering（格拉inger工程学院电子与计算机工程系）
It is common to show the confidence intervals or $p$-values of selected features, or predictor variables in regression, but they often involve selection bias. The selective inference approach solves this bias by conditioning on the selection event. Most existing studies of selective inference consider a specific algorithm, such as Lasso, for feature selection, and thus they have difficulties in handling more complicated algorithms. Moreover, existing studies often consider unnecessarily restrictive events, leading to over-conditioning and lower statistical power. Our novel and widely-applicabl…
Skolkovo Institute of Science and Technology（斯科尔科沃科学技术研究所）；Institute for Information Transmission Problems（信息传输问题研究所）
One possible approach to tackle the class imbalance in classification tasks is to resample a training dataset, i.e., to drop some of its elements or to synthesize new ones. There exist several widely-used resampling methods. Recent research showed that the choice of resampling method significantly affects the quality of classification, which raises resampling selection problem. Exhaustive search for optimal resampling is time-consuming and hence it is of limited use. In this paper, we describe an alternative approach to the resampling selection. We follow the meta-learning concept to build res…
In this paper we consider the problem of constructing confidence intervals for coefficients of martingale regression models (in particular, time series models) after variable selection. Although constructing confidence intervals are common practice in statistical analysis, it is challenging in our framework due to the data-dependence of the selected model and the correlation among the variables being selected and not selected. We first introduce estimators for the selected coefficients and show that it is consistent under martingale regression model, in which the observations can be dependent …
An approximate method for conducting resampling in Lasso, the $\ell_1$ penalized linear regression, in a semi-analytic manner is developed, whereby the average over the resampled datasets is directly computed without repeated numerical sampling, thus enabling an inference free of the statistical fluctuations due to sampling finiteness, as well as a significant reduction of computational time. The proposed method is based on a message passing type algorithm, and its fast convergence is guaranteed by the state evolution analysis, when covariates are provided as zero-mean independently and identi…
In statistical exercises where there are several candidate models, the traditional approach is to select one model using some data driven criterion and use that model for estimation, testing and other purposes, ignoring the variability of the model selection process. We discuss some problems associated with this approach. An alternative scheme is to use a model-averaged estimator, that is, a weighted average of estimators obtained under different models, as an estimator of a parameter. We show that the risk associated with a Bayesian model-averaged estimator is bounded as a function of the sam…
RobustiPy: An efficient next generation multiversal library with model selection, averaging, resampling, and explainable artificial intelligence
University of Oxford（牛津大学）；Leverhulme Centre for Demographic Science, University of Oxford（牛津大学勒弗休姆人口科学中心）；ESRC Centre for Care, University of Oxford（牛津大学经济与社会研究委员会照护研究中心）
While many statistical models and methods are now available for network analysis, resampling network data remains a challenging problem. Cross-validation is a useful general tool for model selection and parameter tuning, but is not directly applicable to networks since splitting network nodes into groups requires deleting edges and destroys some of the network structure. Here we propose a new network resampling strategy based on splitting node pairs rather than nodes applicable to cross-validation for a wide range of network model selection tasks. We provide a theoretical justification for our…
Barber and Candes recently introduced a feature selection method called knockoff+ that controls the false discovery rate (FDR) among the selected features in the classical linear regression problem. Knockoff+ uses the competition between the original features and artificially created knockoff features to control the FDR [1]. We generalize Barber and Candes' knockoff construction to generate multiple knockoffs and use those in conjunction with a recently developed general framework for multiple competition-based FDR control [9]. We prove that using our initial multiple-knockoff construction the…
Contamination can severely distort an estimator unless the estimation procedure is suitably robust. This is a well-known issue and has been addressed in Robust Statistics, however, the relation of contamination and distorted variable selection has been rarely considered in literature. As for variable selection, many methods for sparse model selection have been proposed, including the Stability Selection which is a meta-algorithm based on some variable selection algorithm in order to immunize against particular data configurations. We introduce the variable selection breakdown point that quanti…
Estimating the number of clusters (K) is a critical and often difficult task in cluster analysis. Many methods have been proposed to estimate K, including some top performers using resampling approach. When performing cluster analysis in high-dimensional data, simultaneous clustering and feature selection is needed for improved interpretation and performance. To our knowledge, none has investigated simultaneous estimation of K and feature selection in an exploratory cluster analysis. In this paper, we propose a resampling method to meet this gap and evaluate its performance under the sparse K-…
In this work, we show that under specific choices of the copula, the lasso, elastic net, and $g$-prior are particular cases of `copula prior,' for regularization and variable selection method. We present `lasso with Gauss copula prior' and `lasso with t-copula prior.' The simulation study and real-world data for regression, classification, and large time-series data show that the `copula prior' often outperforms the lasso and elastic net while having a comparable sparsity of representation. Also, the copula prior encourages a grouping effect. The strongly correlated predictors tend to be in or…
Cluster Analysis with Resampling for Validation and Exploration (CARVE)
Department of Statistics, Columbia University, New York, NY, USA（哥伦比亚大学统计学系）；Center for Theoretical Neuroscience, Zuckerman Mind Brain Behavior Institute, Columbia University, New York, NY, USA（哥伦比亚大学理论神经科学中心、Zuckerman思维-大脑-行为研究所）；Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN, USA（诺丁汉大学应用与计算数学与统计学系）；School of Data and Information Sciences, University of North Carolina at Chapel Hill, Chapel Hill, NC, USA（北卡罗来纳大学夏洛特分校数据与信息科学学院）；Irving Institute for Cancer Dynamics, Columbia University, New York, NY, USA（哥伦比亚大学癌症动力学伊万·里弗斯研究所）
High-dimensional predictive models, those with more measurements than observations, require regularization to be well defined, perform well empirically, and possess theoretical guarantees. The amount of regularization, often determined by tuning parameters, is integral to achieving good performance. One can choose the tuning parameter in a variety of ways, such as through resampling methods or generalized information criteria. However, the theory supporting many regularized procedures relies on an estimate for the variance parameter, which is complicated in high dimensions. We develop a suite …
University of Minnesota Twin Cities（明尼苏达大学双城分校）；Splunk（Splunk公司）
In the context of supervised parametric models, we introduce the concept of e-values. An e-value is a scalar quantity that represents the proximity of the sampling distribution of parameter estimates in a model trained on a subset of features to that of the model trained on all features (i.e. the full model). Under general conditions, a rank ordering of e-values separates models that contain all essential features from those that do not. The e-values are applicable to a wide range of parametric models. We use data depths and a fast resampling-based algorithm to implement a feature selection pr…
Variable selection for recovering sparsity in nonadditive nonparametric models has been challenging. This problem becomes even more difficult due to complications in modeling unknown interaction terms among high dimensional variables. There is currently no variable selection method to overcome these limitations. Hence, in this paper we propose a variable selection approach that is developed by connecting a kernel machine with the nonparametric multiple regression model. The advantages of our approach are that it can: (1) recover the sparsity, (2) automatically model unknown and complicated int…
