Variable selection in high-dimensional space characterizes many contemporary problems in scientific discovery and decision making. Many frequently-used techniques are based on independence screening; examples include correlation ranking (Fan and Lv, 2008) or feature selection using a two-sample t-test in high-dimensional classification (Tibshirani et al., 2003). Within the context of the linear model, Fan and Lv (2008)showed that this simple correlation ranking possesses a sure independence screening property under certain conditions and that its revision, called iteratively sure independent screening (ISIS), is needed when the features are marginally unrelated but jointly related to the response variable. In this paper, we extend ISIS, without explicit definition of residuals, to a general pseudo-likelihood framework, which includes generalized linear models as a special case. Even in the least-squares setting, the new method improves ISIS by allowing variable deletion in the iterative process. Our technique allows us to select important features in high-dimensional classification where the popularly used two-sample t-method fails. A new technique is introduced to reduce the false discovery rate in the feature screening stage. Several simulated and two real data examples are presented to illustrate the methodology.
In many practices, scientists are particularly interested in detecting which of the predictors are truly associated with a multivariate response. It is more accurate to model multiple responses as one vector rather than separating each component one by one. This is particularly true for complex traits having multiple correlated components. A Bayesian multivariate variable selection (BMVS) approach is proposed to select important predictors influencing the multivariate response from a candidate pool with an ultrahigh dimension. By applying the sample-size-dependent spike and slab priors, the BM…
Modern soil mapping is characterised by the need to interpolate samples of geostatistical response observations and the availability of relatively large numbers of environmental characteristics for consideration as covariates to aid this interpolation. We demonstrate the efficiency of the Least Angle Regression algorithm for Least Absolute Shrinkage and Selection Operator (LASSO) penalized multiple linear regression at selecting covariates to aid the spatial interpolation of geostatistical soil carbon observations under an ultrahigh dimensional scenario. Where an exhaustive search of the model…
In high-dimensional model selection problems, penalized simple least-square approaches have been extensively used. This paper addresses the question of both robustness and efficiency of penalized model selection methods, and proposes a data-driven weighted linear combination of convex loss functions, together with weighted $L_1$-penalty. It is completely data-adaptive and does not require prior knowledge of the error distribution. The weighted $L_1$-penalty is used both to ensure the convexity of the penalty term and to ameliorate the bias caused by the $L_1$-penalty. In the setting with dimen…
High-dimensional variable selection is an important issue in many scientific fields, such as genomics. In this paper, we develop a sure independence feature screening pro- cedure based on kernel canonical correlation analysis (KCCA-SIS, for short). KCCA- SIS is easy to be implemented and applied. Compared to the sure independence screen- ing procedure based on the Pearson correlation (SIS, for short) developed by Fan and Lv [2008], KCCA-SIS can handle nonlinear dependencies among variables. Compared to the sure independence screening procedure based on the distance correlation (DC- SIS, for sh…
Bayesian model selection procedures based on nonlocal alternative prior densities are extended to ultrahigh dimensional settings and compared to other variable selection procedures using precision-recall curves. Variable selection procedures included in these comparisons include methods based on $g$-priors, reciprocal lasso, adaptive lasso, scad, and minimax concave penalty criteria. The use of precision-recall curves eliminates the sensitivity of our conclusions to the choice of tuning parameters. We find that Bayesian variable selection procedures based on nonlocal priors are competitive to …
Variance estimation is a fundamental problem in statistical modeling. In ultrahigh dimensional linear regressions where the dimensionality is much larger than sample size, traditional variance estimation techniques are not applicable. Recent advances on variable selection in ultrahigh dimensional linear regressions make this problem accessible. One of the major problems in ultrahigh dimensional regression is the high spurious correlation between the unobserved realized noise and some of the predictors. As a result, the realized noises are actually predicted when extra irrelevant variables are …
Quadratic regression goes beyond the linear model by simultaneously including main effects and interactions between the covariates. The problem of interaction estimation in high dimensional quadratic regression has received extensive attention in the past decade. In this article we introduce a novel method which allows us to estimate the main effects and interactions separately. Unlike existing methods for ultrahigh dimensional quadratic regressions, our proposal does not require the widely used heredity assumption. In addition, our proposed estimates have explicit formulas and obey the invari…
We propose trace pursuit for model-free variable selection under the sufficient dimension reduction paradigm. Two distinct algorithms are proposed: stepwise trace pursuit and forward trace pursuit. Stepwise trace pursuit achieves selection consistency with fixed p, and is readily applicable in the challenging setting with p>n. Forward trace pursuit can serve as an initial screening step to speed up the computation in the case of ultrahigh dimensionality. The screening consistency property of forward trace pursuit based on sliced inverse regression is established. Finite sample performances of …
High-dimensional data are commonly seen in modern statistical applications, variable selection methods play indispensable roles in identifying the critical features for scientific discoveries. Traditional best subset selection methods are computationally intractable with a large number of features, while regularization methods such as Lasso, SCAD and their variants perform poorly in ultrahigh-dimensional data due to low computational efficiency and unstable algorithm. Sure screening methods have become popular alternatives by first rapidly reducing the dimension using simple measures such as m…
In ultrahigh dimensional setting, independence screening has been both theoretically and empirically proved a useful variable selection framework with low computation cost. In this work, we propose a two-step framework by using marginal information in a different perspective from independence screening. In particular, we retain significant variables rather than screening out irrelevant ones. The new method is shown to be model selection consistent in the ultrahigh dimensional linear regression model. To improve the finite sample performance, we then introduce a three-step version and character…
When fitting statistical models, some predictors are often found to be correlated with each other, and functioning together. Many group variable selection methods are developed to select the groups of predictors that are closely related to the continuous or categorical response. These existing methods usually assume the group structures are well known. For example, variables with similar practical meaning, or dummy variables created by categorical data. However, in practice, it is impractical to know the exact group structure, especially when the variable dimensional is large. As a result, the…
In recent years many sparse linear discriminant analysis methods have been proposed for high-dimensional classification and variable selection. However, most of these proposals focus on binary classification and they are not directly applicable to multiclass classification problems. There are two sparse discriminant analysis methods that can handle multiclass classification problems, but their theoretical justifications remain unknown. In this paper, we propose a new multiclass sparse discriminant analysis method that estimates all discriminant directions simultaneously. We show that when appl…
This paper studies macroeconomic forecasting and variable selection using a folded-concave penalized regression with a very large number of predictors. The penalized regression approach leads to sparse estimates of the regression coefficients, and is applicable even if the dimensionality of the model is much larger than the sample size. The first half of the paper discusses the theoretical aspects of a folded-concave penalized regression when the model exhibits time series dependence. Specifically, we show the oracle inequality and the oracle property for ultrahigh-dimensional time-dependent r…
In this paper we focus on the empirical variable-selection peformance of subsample-ordered least angle regression (Solar) -- a novel ultrahigh dimensional redesign of lasso -- on the empirical data with complicated dependence structures and, hence, severe multicollinearity and grouping effect issues. Previous researches show that Solar largely alleviates several known high-dimensional issues with least-angle regression and $\mathcal{L}_1$ shrinkage. Also, With the same computation load, solar yields substantiali mprovements over two lasso solvers (least-angle regression for lasso and coordinat…
We explore the theoretical and numerical property of a fully Bayesian model selection method in sparse ultrahigh-dimensional settings, i.e., $p\gg n$, where $p$ is the number of covariates and $n$ is the sample size. Our method consists of (1) a hierarchical Bayesian model with a novel prior placed over the model space which includes a hyperparameter $t_n$ controlling the model size, and (2) an efficient MCMC algorithm for automatic and stochastic search of the models. Our theory shows that, when specifying $t_n$ correctly, the proposed method yields selection consistency, i.e., the posterior …
Numerous variable selection methods rely on a two-stage procedure, where a sparsity-inducing penalty is used in the first stage to predict the support, which is then conveyed to the second stage for estimation or inference purposes. In this framework, the first stage screens variables to find a set of possibly relevant variables and the second stage operates on this set of candidate variables, to improve estimation accuracy or to assess the uncertainty associated to the selection of variables. We advocate that more information can be conveyed from the first stage to the second one: we use the …
Independence screening is a powerful method for variable selection for `Big Data' when the number of variables is massive. Commonly used independence screening methods are based on marginal correlations or variations of it. In many applications, researchers often have some prior knowledge that a certain set of variables is related to the response. In such a situation, a natural assessment on the relative importance of the other predictors is the conditional contributions of the individual predictors in presence of the known set of variables. This results in conditional sure independence screen…
Sure screening technique has been considered as a powerful tool to handle the ultrahigh dimensional variable selection problems, where the dimensionality p and the sample size n can satisfy the NP dimensionality log p=O(n^a) for some a>0 (Fan & Lv 2008). The current paper aims to simultaneously tackle the "universality" and "effectiveness" of sure screening procedures. For the "universality", we develop a general and unified framework for nonparametric screening methods from a loss function perspective. Consider a loss function to measure the divergence of the response variable and the underly…
In recent years the ultrahigh dimensional linear regression problem has attracted enormous attentions from the research community. Under the sparsity assumption most of the published work is devoted to the selection and estimation of the significant predictor variables. This paper studies a different but fundamentally important aspect of this problem: uncertainty quantification for parameter estimates and model choices. To be more specific, this paper proposes methods for deriving a probability density function on the set of all possible models, and also for constructing confidence intervals f…
High-dimensional data subject to heavy-tailed phenomena and heterogeneity are commonly encountered in various scientific fields and bring new challenges to the classical statistical methods. In this paper, we combine the asymmetric square loss and huber-type robust technique to develop the robust expectile regression for ultrahigh dimensional heavy-tailed heterogeneous data. Different from the classical huber method, we introduce two different tuning parameters on both sides to account for possibly asymmetry and allow them to diverge to reduce bias induced by the robust approximation. In the r…
Chain event graphs are a family of probabilistic graphical models that generalise Bayesian networks and have been successfully applied to a wide range of domains. Unlike Bayesian networks, these models can encode context-specific conditional independencies as well as asymmetric developments within the evolution of a process. More recently, new model classes belonging to the chain event graph family have been developed for modelling time-to-event data to study the temporal dynamics of a process. However, existing model selection algorithms for chain event graphs and its variants rely on all par…
Beyond Cross-Validation: Adaptive Parameter Selection for Kernel-Based Gradient Descents
Center for Intelligent Decision-Making and Machine Learning, School of Management, Xi’an Jiaotong University, Xi’an, China（智能决策与机器学习中心，管理学院，西安交通大学，西安，中国）；Department of Mathematics, The University of Hong Kong, Hongkong, China（数学系，香港大学，香港，中国）
Independence screening methods such as the two sample $t$-test and the marginal correlation based ranking are among the most widely used techniques for variable selection in ultrahigh dimensional data sets. In this short note, simple examples are used to demonstrate potential problems with the independence screening methods in the presence of correlated predictors. Also, an example is considered where all important variables are independent among themselves and all but one important variables are independent with the unimportant variables. Furthermore, a real data example from a genome wide as…
The expanding number of assets offers more opportunities for investors but poses new challenges for modern portfolio management (PM). As a central plank of PM, portfolio selection by expected utility maximization (EUM) faces uncontrollable estimation and optimization errors in ultrahigh-dimensional scenarios. Past strategies for high-dimensional PM mainly concern only large-cap companies and select many stocks, making PM impractical. We propose a sample-average approximation-based portfolio strategy to tackle the difficulties above with cardinality constraints. Our strategy bypasses the estima…
A fully Bayesian approach is proposed for ultrahigh-dimensional nonparametric additive models in which the number of additive components may be larger than the sample size, though ideally the true model is believed to include only a small number of components. Bayesian approaches can conduct stochastic model search and fulfill flexible parameter estimation by stochastic draws. The theory shows that the proposed model selection method has satisfactory properties. For instance, when the hyperparameter associated with the model prior is correctly specified, the true model has posterior probabilit…
Bayesian variable selection often assumes normality, but the effects of model misspecification are not sufficiently understood. There are sound reasons behind this assumption, particularly for large $p$: ease of interpretation, analytical and computational convenience. More flexible frameworks exist, including semi- or non-parametric models, often at the cost of some tractability. We propose a simple extension of the Normal model that allows for skewness and thicker-than-normal tails but preserves tractability. It leads to easy interpretation and a log-concave likelihood that facilitates optim…
