Dimension reduction is an important tool for analyzing high-dimensional data. The predictor envelope is a method of dimension reduction for regression that assumes certain linear combinations of the predictors are immaterial to the regression. The method can result in substantial gains in estimation efficiency and prediction accuracy over traditional maximum likelihood and least squares estimates. While predictor envelopes have been developed and studied for independent data, no work has been done adapting predictor envelopes to spatial data. In this work, the predictor envelope is adapted to a popular spatial model to form the spatial predictor envelope (SPE). Maximum likelihood estimates for the SPE are derived, along with asymptotic distributions for the estimates given certain assumptions, showing the SPE estimates to be asymptotically more efficient than estimates of the original spatial model. The effectiveness of the proposed model is illustrated through simulation studies and the analysis of a geo-chemical data set.
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 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…
In this paper, we propose a Spatial Robust Mixture Regression model to investigate the relationship between a response variable and a set of explanatory variables over the spatial domain, assuming that the relationships may exhibit complex spatially dynamic patterns that cannot be captured by constant regression coefficients. Our method integrates the robust finite mixture Gaussian regression model with spatial constraints, to simultaneously handle the spatial nonstationarity, local homogeneity, and outlier contaminations. Compared with existing spatial regression models, our proposed model as…
We derive the properties and demonstrate the desirability of a model-based method for estimating the spatially-varying effects of covariates on the quantile function. By modeling the quantile function as a combination of I-spline basis functions and Pareto tail distributions, we allow for flexible parametric modeling of the extremes while preserving non-parametric flexibility in the center of the distribution. We further establish that the model guarantees the desired degree of differentiability in the density function and enables the estimation of non-stationary covariance functions dependent…
University of Chicago（芝加哥大学）；Stanford University（斯坦福大学）；Harvard University（哈佛大学）；University of British Columbia（不列颠哥伦比亚大学）；Amii（阿尔伯塔机器智能研究所）；Toyota Technological Institute at Chicago（芝加哥丰田技术研究所）
We prove a new generalization bound that shows for any class of linear predictors in Gaussian space, the Rademacher complexity of the class and the training error under any continuous loss $\ell$ can control the test error under all Moreau envelopes of the loss $\ell$. We use our finite-sample bound to directly recover the "optimistic rate" of Zhou et al. (2021) for linear regression with the square loss, which is known to be tight for minimal $\ell_2$-norm interpolation, but we also handle more general settings where the label is generated by a potentially misspecified multi-index model. The …
Soil Organic Carbon (SOC) estimation is crucial to manage both natural and anthropic ecosystems and has recently been put under the magnifying glass after the Paris agreement 2016 due to its relationship with greenhouse gas. Statistical applications have dominated the SOC stock mapping at regional scale so far. However, the community has hardly ever attempted to implement Quantile Regression (QR) to spatially predict the SOC distribution. In this contribution, we test QR to estimate SOC stock (0-30 $cm$ depth) in the agricultural areas of a highly variable semi-arid region (Sicily, Italy, arou…
Modern technology often generates data with complex structures in which both response and explanatory variables are matrix-valued. Existing methods in the literature are able to tackle matrix-valued predictors but are rather limited for matrix-valued responses. In this article, we study matrix-variate regressions for such data, where the response Y on each experimental unit is a random matrix and the predictor X can be either a scalar, a vector, or a matrix, treated as non-stochastic in terms of the conditional distribution Y|X. We propose models for matrix-variate regressions and then develop…
Envelope methods offer targeted dimension reduction for various models. The overarching goal is to improve efficiency in multivariate parameter estimation by projecting the data onto a lower-dimensional subspace known as the envelope. Envelope approaches have advantages in analyzing data with highly correlated variables, but their iterative Grassmannian optimization algorithms do not scale very well with ultra high-dimensional data. While the connections between envelopes and partial least squares in multivariate linear regression have promoted recent progress in high-dimensional studies of en…
The $α$--regression for compositional data: a unified framework for standard, temporal and spatial regression models including compositional predictors
University of Crete（克里特大学）；King Saud bin Abdulaziz University for Health Sciences（国王萨勒曼·本·阿卜杜勒阿齐兹健康科学大学）；King Abdullah International Medical Research Center（阿卜杜拉国王国际医学研究中心）；Northern Border University（北部边境大学）
Envelope model also known as multivariate regression model was proposed to solve the multiple response regression problems. It measures the linear association between predictors and multiple responses by using the minimal reducing subspace of the covariance matrix that accommodates the mean function. However, in many real applications, data may consist many unknown confounding factors or they just come from different resources. Thus, there might be some heterogeneous dependency across the whole population and divide them into different groups. For example, there exists several subtypes across …
Mapping Drivers of Greenness: Spatial Variable Selection for MODIS Vegetation Indices
Department of Mathematical and Statistical Sciences, Marquette University（数学与统计学系，马歇尔大学）；Department of Computer Science, Georgia Institute of Technology（计算机科学系，佐治亚理工学院）；Department of Applied Mathematics and Statistics, Stony Brook University（应用数学与统计学系，石溪大学）；Department of Statistics, Texas A&M University（统计学系，德克萨斯阿姆大学）
Envelope method was recently proposed as a method to reduce the dimension of responses in multivariate regressions. However, when there exists missing data, the envelope method using the complete case observations may lead to biased and inefficient results. In this paper, we generalize the envelope estimation when the predictors and/or the responses are missing at random. Specifically, we incorporate the envelope structure in the expectation-maximization (EM) algorithm. As the parameters under the envelope method are not pointwise identifiable, the EM algorithm for the envelope method was not …
Spatially continuous modelling of aggregated outcome data
Imperial College London（伦敦帝国理工学院）；Lancaster University（兰卡斯特大学）；University of Edinburgh（爱丁堡大学）；King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）；Northumbria University（诺森比亚大学）
We tackle the problem of multiscale regression for predictors that are spatially or temporally indexed, or with a pre-specified multiscale structure, with a Bayesian modular approach. The regression function at the finest scale is expressed as an additive expansion of coarse to fine step functions. Our Modular and Multiscale (M&M) methodology provides multiscale decomposition of high-dimensional data arising from very fine measurements. Unlike more complex methods for functional predictors, our approach provides easy interpretation of the results. Additionally, it provides a quantification of …
The spatial panel regression model has shown great success in modelling econometric and other types of data that are observed both spatially and temporally with associated predictor variables. However, model checking via testing for spatial correlations in spatial-temporal residuals is still lacking. We propose a general methodology for fast permutation testing of local and global indicators of spatial association. This methodology extends past statistics for univariate spatial data that can be written as a gamma index for matrix similarity to the multivariate and panel data settings. This inc…
Small Area Estimation under Spatial Regimes: Spatially Clustered Fay-Herriot Models for Agricultural Indicators
University of Milano-Bicocca（米兰比可卡大学）；Fondazione Eni Enrico Mattei (FEEM)（埃尼·恩里科·马泰伊基金会）；University of Campania "Luigi Vanvitelli"（坎帕尼亚路易吉·范维特利大学）；Keio University（庆应义塾大学）
East China Normal University（华东师范大学）；King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）；North Carolina State University（北卡罗来纳州立大学）
In spatial statistics, a common objective is to predict values of a spatial process at unobserved locations by exploiting spatial dependence. Kriging provides the best linear unbiased predictor using covariance functions and is often associated with Gaussian processes. However, when considering non-linear prediction for non-Gaussian and categorical data, the Kriging prediction is no longer optimal, and the associated variance is often overly optimistic. Although deep neural networks (DNNs) are widely used for general classification and prediction, they have not been studied thoroughly for data…
