我们提出了一种监督主成分回归方法，用于建立函数型响应与高维预测变量之间的关联。与传统的主成分分析不同，所提出的方法建立在新定义的期望积分残差平方和之上，该平方和直接利用函数型响应与预测变量之间的关联。最小化该积分残差平方和可得到监督主成分，这等价于求解一系列非凸广义瑞利商优化问题。我们将这些非凸优化问题重新表述为带有稀疏惩罚的同步线性回归，以处理高维预测变量。在理论方面，我们证明了在特定条件下，重新表述后的回归问题能够恢复相同的监督主子空间。在统计方面，当协变量协方差具有可带状化性质时，我们为所提出的估计量建立了非渐近误差界。我们通过数值实验以及在 Human Connectome Project（人类连接组计划）fMRI 数据上的应用，证明了所提出方法的优势。
英文摘要：
We propose a supervised principal component regression method for relating functional responses with high dimensional predictors. Unlike the conventional principal component analysis, the proposed method builds on a newly defined expected integrated residual sum of squares, which directly makes use of the association between the functional response and the predictors. Minimizing the integrated residual sum of squares gives the supervised principal components, which is equivalent to solving a sequence of nonconvex generalized Rayleigh quotient optimization problems. We reformulate the nonconvex optimization problems into a simultaneous linear regression with a sparse penalty to deal with high dimensional predictors. Theoretically, we show that the reformulated regression problem can recover the same supervised principal subspace under certain conditions. Statistically, we establish non-asymptotic error bounds for the proposed estimators when the covariate covariance is bandable. We demonstrate the advantages of the proposed method through numerical experiments and an application to the Human Connectome Project fMRI data.
This paper considers linear model selection when the response is vector-valued and the predictors are randomly observed. We propose a new approach that decouples statistical inference from the selection step in a "post-inference model summarization" strategy. We study the impact of predictor uncertainty on the model selection procedure. The method is demonstrated through an application to asset pricing.
We propose a ranking and selection procedure to prioritize relevant predictors and control false discovery proportion (FDP) of variable selection. Our procedure utilizes a new ranking method built upon the de-sparsified Lasso estimator. We show that the new ranking method achieves the optimal order of minimum non-zero effects in ranking relevant predictors ahead of irrelevant ones. Adopting the new ranking method, we develop a variable selection procedure to asymptotically control FDP at a user-specified level. We show that our procedure can consistently estimate the FDP of variable selection …
Classical finite mixture regression is useful for modeling the relationship between scalar predictors and scalar responses arising from subpopulations defined by the differing associations between those predictors and responses. Here we extend the classical finite mixture regression model to incorporate functional predictors by taking a wavelet-based approach in which we represent both the functional predictors and the component-specific coefficient functions in terms of an appropriate wavelet basis. In the wavelet representation of the model, the coefficients corresponding to the functional c…
Heavy-tailed continuous shrinkage priors, such as the horseshoe prior, are widely used for sparse estimation problems. However, there is limited work extending these priors to predictors with grouping structures. Of particular interest in this article, is regression coefficient estimation where pockets of high collinearity in the covariate space are contained within known covariate groupings. To assuage variance inflation due to multicollinearity we propose the group inverse-gamma gamma (GIGG) prior, a heavy-tailed prior that can trade-off between local and group shrinkage in a data adaptive f…
In this work, we introduce a modified (rescaled) likelihood for imbalanced logistic regression. This new approach makes easier the use of exponential priors and the computation of lasso regularization path. Precisely, we study a limiting behavior for which class imbalance is artificially increased by replication of the majority class observations. If some strong overlap conditions are satisfied, the maximum likelihood estimate converges towards a finite value close to the initial one (intercept excluded) as shown by simulations with binary predictors. This solution corresponds to the extremum …
Many important modeling tasks in linear regression, including variable selection (in which slopes of some predictors are set equal to zero) and simplified models based on sums or differences of predictors (in which slopes of those predictors are set equal to each other, or the negative of each other, respectively), can be viewed as being based on imposing linear restrictions on regression parameters. In this paper, we discuss how such models can be compared using information criteria designed to estimate predictive measures like squared error and Kullback-Leibler (KL) discrepancy, in the prese…
The development and use of dimension reduction methods is prevalent in modern statistical literature. This paper reviews a class of dimension reduction techniques which aim to simultaneously select relevant predictors and find clusters within them which share a common effect on the response. Such methods have been shown to have superior performance relative to OLS estimates and the lasso [Tibshirani, 1996] especially when multicollinearity in the predictors is present. Their applications, which include genetics, epidemiology, and fMRI studies, are also discussed.
Within the statistical and machine learning literature, regularization techniques are often used to construct sparse (predictive) models. Most regularization strategies only work for data where all predictors are treated identically, such as Lasso regression for (continuous) predictors treated as linear effects. However, many predictive problems involve different types of predictors and require a tailored regularization term. We propose a multi-type Lasso penalty that acts on the objective function as a sum of subpenalties, one for each type of predictor. As such, we allow for predictor select…
Variable selection is an important statistical problem. This problem becomes more challenging when the candidate predictors are of mixed type (e.g. continuous and binary) and impact the response variable in nonlinear and/or non-additive ways. In this paper, we review existing variable selection approaches for the Bayesian additive regression trees (BART) model, a nonparametric regression model, which is flexible enough to capture the interactions between predictors and nonlinear relationships with the response. An emphasis of this review is on the capability of identifying relevant predictors.…
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…
This work studies finite-sample properties of the risk of the minimum-norm interpolating predictor in high-dimensional regression models. If the effective rank of the covariance matrix $Σ$ of the $p$ regression features is much larger than the sample size $n$, we show that the min-norm interpolating predictor is not desirable, as its risk approaches the risk of trivially predicting the response by 0. However, our detailed finite-sample analysis reveals, surprisingly, that this behavior is not present when the regression response and the features are {\it jointly} low-dimensional, following a w…
In this paper, we propose methods for functional predictor selection and the estimation of smooth functional coefficients simultaneously in a scalar-on-function regression problem under high-dimensional multivariate functional data setting. In particular, we develop two methods for functional group-sparse regression under a generic Hilbert space of infinite dimension. We show the convergence of algorithms and the consistency of the estimation and the selection (oracle property) under infinite-dimensional Hilbert spaces. Simulation studies show the effectiveness of the methods in both the selec…
Principal component regression (PCR) is a two-stage procedure: the first stage performs principal component analysis (PCA) and the second stage constructs a regression model whose explanatory variables are replaced by principal components obtained by the first stage. Since PCA is performed by using only explanatory variables, the principal components have no information about the response variable. To address the problem, we propose a one-stage procedure for PCR in terms of singular value decomposition approach. Our approach is based upon two loss functions, a regression loss and a PCA loss, w…
The model implied by factor score predictors does not reproduce the non-diagonal elements of the observed covariance matrix as well as the factor loadings. It is therefore investigated whether it is possible to estimate factor loadings for which the model implied by the factor score predictors optimally reproduces the non-diagonal elements of the observed covariance matrix. Accordingly, loading estimates are proposed for which the model implied by the factor score predictors allows for a least-squares approximation of the non-diagonal elements of the observed covariance matrix. This estimation…
Principal loading analysis is a dimension reduction method that discards variables which have only a small distorting effect on the covariance matrix. As a special case, principal loading analysis discards variables that are not correlated with the remaining ones. In multivariate linear regression on the other hand, predictors that are neither correlated with both the remaining predictors nor with the dependent variables have a regression coefficients equal to zero. Hence, if the goal is to select a number of predictors, variables that do not correlate are discarded as it is also done in princ…
We develop a novel Bayesian method to select important predictors in regression models with multiple responses of diverse types. A sparse Gaussian copula regression model is used to account for the multivariate dependencies between any combination of discrete and/or continuous responses and their association with a set of predictors. We utilize the parameter expansion for data augmentation strategy to construct a Markov chain Monte Carlo algorithm for the estimation of the parameters and the latent variables of the model. Based on a centered parametrization of the Gaussian latent variables, we…
We propose a new method for supervised learning, especially suited to wide data where the number of features is much greater than the number of observations. The method combines the lasso ($\ell_1$) sparsity penalty with a quadratic penalty that shrinks the coefficient vector toward the leading principal components of the feature matrix. We call the proposed method the "principal components lasso" ("pcLasso"). The method can be especially powerful if the features are pre-assigned to groups (such as cell-pathways, assays or protein interaction networks). In that case, pcLasso shrinks each group…
In scientific applications, multivariate observations often come in tandem with temporal or spatial covariates, with which the underlying signals vary smoothly. The standard approaches such as principal component analysis and factor analysis neglect the smoothness of the data, while multivariate linear or nonparametric regression fail to leverage the correlation information among multivariate response variables. We propose a novel approach named nonparametric principal subspace regression to overcome these issues. By decoupling the model discrepancy, a simple and general two-step framework is …
We analyse the prediction error of principal component regression (PCR) and prove non-asymptotic upper bounds for the corresponding squared risk. Under mild assumptions, we show that PCR performs as well as the oracle method obtained by replacing empirical principal components by their population counterparts. Our approach relies on upper bounds for the excess risk of principal component analysis.
We propose a new two stage algorithm LING for large scale regression problems. LING has the same risk as the well known Ridge Regression under the fixed design setting and can be computed much faster. Our experiments have shown that LING performs well in terms of both prediction accuracy and computational efficiency compared with other large scale regression algorithms like Gradient Descent, Stochastic Gradient Descent and Principal Component Regression on both simulated and real datasets.
We study a dimensionality reduction technique for finite mixtures of high-dimensional multivariate response regression models. Both the dimension of the response and the number of predictors are allowed to exceed the sample size. We consider predictor selection and rank reduction to obtain lower-dimensional approximations. A class of estimators with a fast rate of convergence is introduced. We apply this result to a specific procedure, introduced in [11], where the relevant predictors are selected by the Group-Lasso.
Identifying homogeneous subgroups of variables can be challenging in high dimensional data analysis with highly correlated predictors. We propose a new method called Hexagonal Operator for Regression with Shrinkage and Equality Selection, HORSES for short, that simultaneously selects positively correlated variables and identifies them as predictive clusters. This is achieved via a constrained least-squares problem with regularization that consists of a linear combination of an L_1 penalty for the coefficients and another L_1 penalty for pairwise differences of the coefficients. This specificat…
Multi-task learning is increasingly used to investigate the association structure between multiple responses and a single set of predictor variables in many applications. In the era of big data, the coexistence of incomplete outcomes, large number of responses, and high dimensionality in predictors poses unprecedented challenges in estimation, prediction, and computation. In this paper, we propose a scalable and computationally efficient procedure, called PEER, for large-scale multi-response regression with incomplete outcomes, where both the numbers of responses and predictors can be high-dim…
The dual problem of testing the predictive significance of a particular covariate, and identification of the set of relevant covariates is common in applied research and methodological investigations. To study this problem in the context of functional linear regression models with predictor variables observed over a grid and a scalar response, we consider basis expansions of the functional covariates and apply the likelihood ratio test. Based on p-values from testing each predictor, we propose a new variable selection method, which is consistent in selecting the relevant predictors from set of…
Penalized regression is an attractive framework for variable selection problems. Often, variables possess a grouping structure, and the relevant selection problem is that of selecting groups, not individual variables. The group lasso has been proposed as a way of extending the ideas of the lasso to the problem of group selection. Nonconvex penalties such as SCAD and MCP have been proposed and shown to have several advantages over the lasso; these penalties may also be extended to the group selection problem, giving rise to group SCAD and group MCP methods. Here, we describe algorithms for fitt…
This paper studies the related problems of prediction, covariance estimation, and principal component analysis for the spiked covariance model with heteroscedastic noise. We consider an estimator of the principal components based on whitening the noise, and we derive optimal singular value and eigenvalue shrinkers for use with these estimated principal components. Underlying these methods are new asymptotic results for the high-dimensional spiked model with heteroscedastic noise, and consistent estimators for the relevant population parameters. We extend previous analysis on out-of-sample pred…
In areas of application, including actuarial science and demography, it is increasingly common to consider a time series of curves; an example of this is age-specific mortality rates observed over a period of years. Given that age can be treated as a discrete or continuous variable, a dimension reduction technique, such as principal component analysis, is often implemented. However, in the presence of moderate to strong temporal dependence, static principal component analysis commonly used for analyzing independent and identically distributed data may not be adequate. As an alternative, we con…
Stanford University（斯坦福大学）；MIT（麻省理工学院）；Microsoft Research, New England（新英格兰微软研究院）
We show how to efficiently project a vector onto the top principal components of a matrix, without explicitly computing these components. Specifically, we introduce an iterative algorithm that provably computes the projection using few calls to any black-box routine for ridge regression. By avoiding explicit principal component analysis (PCA), our algorithm is the first with no runtime dependence on the number of top principal components. We show that it can be used to give a fast iterative method for the popular principal component regression problem, giving the first major runtime improvemen…
École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）
Data analyses based on linear methods constitute the simplest, most robust, and transparent approaches to the automatic processing of large amounts of data for building supervised or unsupervised machine learning models. Principal covariates regression (PCovR) is an underappreciated method that interpolates between principal component analysis and linear regression, and can be used to conveniently reveal structure-property relations in terms of simple-to-interpret, low-dimensional maps. Here we provide a pedagogic overview of these data analysis schemes, including the use of the kernel trick t…
We tackle modelling and inference for variable selection in regression problems with many predictors and many responses. We focus on detecting hotspots, i.e., predictors associated with several responses. Such a task is critical in statistical genetics, as hotspot genetic variants shape the architecture of the genome by controlling the expression of many genes and may initiate decisive functional mechanisms underlying disease endpoints. Existing hierarchical regression approaches designed to model hotspots suffer from two limitations: their discrimination of hotspots is sensitive to the choice…
We address the problem of multicollinearity in a function-on-scalar regression model by using a prior which simultaneously selects, clusters, and smooths functional effects. Our methodology groups effects of highly correlated predictors, performing dimension reduction without dropping relevant predictors from the model. We validate our approach via a simulation study, showing superior performance relative to existing dimension reduction approaches in the function-on-scalar literature. We also demonstrate the use of our model on a data set of age specific fertility rates from the United Nations…
Whereas translating the coding of predictor variables does not change the fit of a polynomial least squares regression, penalized polynomial regressions are potentially affected. A result on which terms can be penalized to maintain the invariance to translations of the coding has earlier been published. A generalization of a corresponding proposition, which requires a more precise mathematical framework, is presented in this short note.
Groupwise Predictor Envelope Models for Multivariate Linear Regression
The Institute of Statistical Mathematics（统计数学研究所）；Graduate School of Mathematics, Kyushu University（九州大学大学院数理学府）；Faculty of Mathematics, Kyushu University（九州大学数学部）
We provide a remedy for two concerns that have dogged the use of principal components in regression: (i) principal components are computed from the predictors alone and do not make apparent use of the response, and (ii) principal components are not invariant or equivariant under full rank linear transformation of the predictors. The development begins with principal fitted components [Cook, R. D. (2007). Fisher lecture: Dimension reduction in regression (with discussion). Statist. Sci. 22 1--26] and uses normal models for the inverse regression of the predictors on the response to gain reducti…
We propose a principal components regression method based on maximizing a joint pseudo-likelihood for responses and predictors. Our method uses both responses and predictors to select linear combinations of the predictors relevant for the regression, thereby addressing an oft-cited deficiency of conventional principal components regression. The proposed estimator is shown to be consistent in a wide range of settings, including ones with non-normal and dependent observations; conditions on the first and second moments suffice if the number of predictors ($p$) is fixed and the number of observat…
Deciding which predictors to use plays an integral role in deriving statistical models in a wide range of applications. Motivated by the challenges of predicting events across a telecommunications network, we propose a semi-automated, joint model-fitting and predictor selection procedure for linear regression models. Our approach can model and account for serial correlation in the regression residuals, produces sparse and interpretable models and can be used to jointly select models for a group of related responses. This is achieved through fitting linear models under constraints on the number…
In prediction problems with more predictors than observations, it can sometimes be helpful to use a joint probability model, $π(Y,X)$, rather than a purely conditional model, $π(Y \mid X)$, where $Y$ is a scalar response variable and $X$ is a vector of predictors. This approach is motivated by the fact that in many situations the marginal predictor distribution $π(X)$ can provide useful information about the parameter values governing the conditional regression. However, under very mild misspecification, this marginal distribution can also lead conditional inferences astray. Here, we explore t…
A regression model is proposed for the analysis of an ordinal response variable depending on a set of multiple covariates containing ordinal and potentially other variables. The proportional odds model (McCullagh (1980)) is used for the ordinal response, and constrained maximum likelihood estimation is used to account for the ordinality of covariates. Ordinal predictors are coded by dummy variables. The parameters associated to the categories of the ordinal predictor(s) are constrained, enforcing them to be monotonic (isotonic or antitonic). A decision rule is introduced for classifying the or…
Traditionally, the least squares regression is mainly concerned with studying the effects of individual predictor variables, but strongly correlated variables generate multicollinearity which makes it difficult to study their effects. Existing methods for handling multicollinearity such as ridge regression are complicated. To resolve the multicollinearity issue without abandoning the simple least squares regression, for situations where predictor variables are in groups with strong within-group correlations but weak between-group correlations, we propose to study the effects of the groups with…
Compositional data are common in many fields, both as outcomes and predictor variables. The inventory of models for the case when both the outcome and predictor variables are compositional is limited and the existing models are difficult to interpret, due to their use of complex log-ratio transformations. We develop a transformation-free linear regression model where the expected value of the compositional outcome is expressed as a single Markov transition from the compositional predictor. Our approach is based on generalized method of moments thereby not requiring complete specification of da…
This article proposes a Bayesian approach to regression with a continuous scalar response and an undirected network predictor. Undirected network predictors are often expressed in terms of symmetric adjacency matrices, with rows and columns of the matrix representing the nodes, and zero entries signifying no association between two corresponding nodes. Network predictor matrices are typically vectorized prior to any analysis, thus failing to account for the important structural information in the network. This results in poor inferential and predictive performance in presence of small sample s…
