Ultrahigh Dimensionality Research Articles

GWASinlps: non-local prior based iterative SNP selection tool for genome-wide association studies.

Multiple marker analysis of the genome-wide association study (GWAS) data has gained ample attention in recent years. However, because of the ultra high-dimensionality of GWAS data, such analysis is challenging. Frequently used penalized regression methods often lead to large number of false positives, whereas Bayesian methods are computationally very expensive. Motivated to ameliorate these issues simultaneously, we consider the novel approach of using non-local priors in an iterative variable selection framework. We develop a variable selection method, named, iterative non-local prior based selection for GWAS, or GWASinlps, that combines, in an iterative variable selection framework, the computational efficiency of the screen-and-select approach based on some association learning and the parsimonious uncertainty quantification provided by the use of non-local priors. The hallmark of our method is the introduction of 'structured screen-and-select' strategy, that considers hierarchical screening, which is not only based on response-predictor associations, but also based on response-response associations and concatenates variable selection within that hierarchy. Extensive simulation studies with single nucleotide polymorphisms having realistic linkage disequilibrium structures demonstrate the advantages of our computationally efficient method compared to several frequentist and Bayesian variable selection methods, in terms of true positive rate, false discovery rate, mean squared error and effect size estimation error. Further, we provide empirical power analysis useful for study design. Finally, a real GWAS data application was considered with human height as phenotype. An R-package for implementing the GWASinlps method is available at https://cran.r-project.org/web/packages/GWASinlps/index.html. Supplementary data are available at Bioinformatics online.

Sure independence screening for real medical Poisson data

ABSTRACTThe statistical modeling of big data bases constitutes one of the most challenging issues, especially nowadays. The issue is even more critical in case of a complicated correlation structure. Variable selection plays a vital role in statistical analysis of large data bases and many methods have been proposed so far to deal with the aforementioned problem. One of such methods is the Sure Independence Screening which has been introduced to reduce dimensionality to a relatively smaller scale. This method, though simple, produces remarkable results even under both ultra high dimensionality and big scale in terms of sample size problems. In this paper we dealt with the analysis of a big real medical data set assuming a Poisson regression model. We support the analysis by conducting simulated experiments taking into consideration the correlation structure of the design matrix.

Tucker Tensor Regression and Neuroimaging Analysis.

Neuroimaging data often take the form of high dimensional arrays, also known as tensors. Addressing scientific questions arising from such data demands new regression models that take multidimensional arrays as covariates. Simply turning an image array into a vector would both cause extremely high dimensionality and destroy the inherent spatial structure of the array. In a recent work, Zhou et al. (2013) proposed a family of generalized linear tensor regression models based upon the CP (CANDECOMP/PARAFAC) decomposition of regression coefficient array. Low rank approximation brings the ultrahigh dimensionality to a manageable level and leads to efficient estimation. In this article, we propose a tensor regression model based on the more flexible Tucker decomposition. Compared to the CP model, Tucker regression model allows different number of factors along each mode. Such flexibility leads to several advantages that are particularly suited to neuroimaging analysis, including further reduction of the number of free parameters, accommodation of images with skewed dimensions, explicit modeling of interactions, and a principled way of image downsizing. We also compare the Tucker model with CP numerically on both simulated data and a real magnetic resonance imaging data, and demonstrate its effectiveness in finite sample performance.

Two-stage identification of SNP effects on dynamic poplar growth.

This project proposes an approach to identify significant single nucleotide polymorphism (SNP) effects, both additive and dominant, on the dynamic growth of poplar in diameter and height. The annual changes in yearly phenotypes based on regular observation periods are considered to represent multiple responses. In total 156,362 candidate SNPs are studied, and the phenotypes of 64 poplar trees are recorded. To address this ultrahigh dimensionality issue, this paper adopts a two-stage approach. First, the conventional genome-wide association studies (GWAS) and the distance correlation sure independence screening (DC-SIS) methods (Li etal., 2012) were combined to reduce the model dimensions at the sample size; second, a grouped penalized regression was applied to further refine the model and choose the final sparse SNPs. The multiple response issue was also carefully addressed. The SNP effects on the dynamic diameter and height growth patterns of poplar were systematically analyzed. In addition, a series of intensive simulation studies was performed to validate the proposed approach.

Feature Screening for Nonparametric and Semiparametric Models with Ultrahigh-Dimensional Covariates

This paper considers the feature screening and variable selection for ultrahigh dimensional covariates. The new feature screening procedure base on conditional expectation which is used to differentiate whether an explanatory variable contributes to a response variable or not, without requiring a specific parametric form of the underlying data model. The authors estimate the marginal conditional expectation by kernel regression estimator. The proposed method is showed to have sure screen property. The authors propose an iterative kernel estimator algorithm to reduce the ultrahigh dimensionality to an appropriate scale. Simulation results and real data analysis demonstrate the proposed method works well and performs better than competing methods.

High-dimensional tests for functional networks of brain anatomic regions.

Exploring resting-state brain functional connectivity of autism spectrum disorders (ASD) using functional magnetic resonance imaging (fMRI) data has become a popular topic over the past few years. The data in a standard brain template consist of over 170,000 voxel specific points in time for each human subject. Such an ultra-high dimensionality makes the voxel-level functional connectivity analysis (involving four billion voxel pairs) both statistically and computationally inefficient. In this work, we introduce a new framework to identify the functional brain network at the anatomical region level for each individual. We propose two pairwise tests to detect region dependence, and one multiple testing procedure to identify global structures of the network. The limiting null distribution of each test statistic is derived. It is also shown that the tests are rate optimal when the alternative block networks are sparse. The numerical studies show that the proposed tests are valid and powerful. We apply our method to a resting-state fMRI study on autism and identify patient-unique and control-unique hub regions. These findings are biologically meaningful and consistent with the existing literature.

Trace Pursuit: A General Framework for Model-Free Variable Selection

ABSTRACTWe 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. 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 trace pursuit and other model-free variable selection methods are compared through numerical studies. Supplementary materials for this article are available online.

Regularized Quantile Regression and Robust Feature Screening for Single Index Models.

We propose both a penalized quantile regression and an independence screening procedure to identify important covariates and to exclude unimportant ones for a general class of ultrahigh dimensional single-index models, in which the conditional distribution of the response depends on the covariates via a single-index structure. We observe that the linear quantile regression yields a consistent estimator of the direction of the index parameter in the single-index model. Such an observation dramatically reduces computational complexity in selecting important covariates in the single-index model. We establish an oracle property for the penalized quantile regression estimator when the covariate dimension increases at an exponential rate of the sample size. From a practical perspective, however, when the covariate dimension is extremely large, the penalized quantile regression may suffer from at least two drawbacks: computational expediency and algorithmic stability. To address these issues, we propose an independence screening procedure which is robust to model misspecification, and has reliable performance when the distribution of the response variable is heavily tailed or response realizations contain extreme values. The new independence screening procedure offers a useful complement to the penalized quantile regression since it helps to reduce the covariate dimension from ultrahigh dimensionality to a moderate scale. Based on the reduced model, the penalized linear quantile regression further refines selection of important covariates at different quantile levels. We examine the finite sample performance of the newly proposed procedure by Monte Carlo simulations and demonstrate the proposed methodology by an empirical analysis of a real data set.

Exploiting Linkage Disequilibrium for Ultrahigh-Dimensional Genome-Wide Data with an Integrated Statistical Approach.

Genome-wide data with millions of single-nucleotide polymorphisms (SNPs) can be highly correlated due to linkage disequilibrium (LD). The ultrahigh dimensionality of big data brings unprecedented challenges to statistical modeling such as noise accumulation, the curse of dimensionality, computational burden, spurious correlations, and a processing and storing bottleneck. The traditional statistical approaches lose their power due to [Formula: see text] (n is the number of observations and p is the number of SNPs) and the complex correlation structure among SNPs. In this article, we propose an integrated distance correlation ridge regression (DCRR) approach to accommodate the ultrahigh dimensionality, joint polygenic effects of multiple loci, and the complex LD structures. Initially, a distance correlation (DC) screening approach is used to extensively remove noise, after which LD structure is addressed using a ridge penalized multiple logistic regression (LRR) model. The false discovery rate, true positive discovery rate, and computational cost were simultaneously assessed through a large number of simulations. A binary trait of Arabidopsis thaliana, the hypersensitive response to the bacterial elicitor AvrRpm1, was analyzed in 84 inbred lines (28 susceptibilities and 56 resistances) with 216,130 SNPs. Compared to previous SNP discovery methods implemented on the same data set, the DCRR approach successfully detected the causative SNP while dramatically reducing spurious associations and computational time.

Type 2 Diabetes Biomarkers of Human Gut Microbiota Selected via Iterative Sure Independent Screening Method.

Type 2 diabetes, which is a complex metabolic disease influenced by genetic and environment, has become a worldwide problem. Previous published results focused on genetic components through genome-wide association studies that just interpret this disease to some extent. Recently, two research groups published metagenome-wide association studies (MGWAS) result that found meta-biomarkers related with type 2 diabetes. However, One key problem of analyzing genomic data is that how to deal with the ultra-high dimensionality of features. From a statistical viewpoint it is challenging to filter true factors in high dimensional data. Various methods and techniques have been proposed on this issue, which can only achieve limited prediction performance and poor interpretability. New statistical procedure with higher performance and clear interpretability is appealing in analyzing high dimensional data. To address this problem, we apply an excellent statistical variable selection procedure called iterative sure independence screening to gene profiles that obtained from metagenome sequencing, and 48/24 meta-markers were selected in Chinese/European cohorts as predictors with 0.97/0.99 accuracy in AUC (area under the curve), which showed a better performance than other model selection methods, respectively. These results demonstrate the power and utility of data mining technologies within the large-scale and ultra-high dimensional genomic-related dataset for diagnostic and predictive markers identifying.

Demand forecasting with high dimensional data: The case of SKU retail sales forecasting with intra- and inter-category promotional information

In marketing analytics applications in OR, the modeler often faces the problem of selecting key variables from a large number of possibilities. For example, SKU level retail store sales are affected by inter and intra category effects which potentially need to be considered when deciding on promotional strategy and producing operational forecasts. But no research has yet put this well accepted concept into forecasting practice: an obvious obstacle is the ultra-high dimensionality of the variable space. This paper develops a four steps methodological framework to overcome the problem. It is illustrated by investigating the value of both intra- and inter-category SKU level promotional information in improving forecast accuracy. The method consists of the identification of potentially influential categories, the building of the explanatory variable space, variable selection and model estimation by a multistage LASSO regression, and the use of a rolling scheme to generate forecasts. The success of this new method for dealing with high dimensionality is demonstrated by improvements in forecasting accuracy compared to alternative methods of simplifying the variable space. The empirical results show that models integrating more information perform significantly better than the baseline model when using the proposed methodology framework. In general, we can improve the forecasting accuracy by 12.6 percent over the model using only the SKU's own predictors. But of the improvements achieved, 95 percent of it comes from the intra-category information, and only 5 percent from the inter-category information. The substantive marketing results also have implications for promotional category management.

Robust sure independence screening for ultrahigh dimensional non-normal data

Sure independence screening (SIS) has been proposed to reduce the ultrahigh dimensionality down to a moderate scale and proved to enjoy the sure screening property under Gaussian linear models. However, the observed response is often skewed or heavy-tailed with extreme values in practice, which may dramatically deteriorate the performance of SIS. To this end, we propose a new robust sure independence screening (RoSIS) via considering the correlation between each predictor and the distribution function of the response. The proposed approach contributes to the literature in the following three folds: First, it is able to reduce ultrahigh dimensionality effectively. Second, it is robust to heavy tails or extreme values in the response. Third, it possesses both sure screening property and ranking consistency property under milder conditions. Furthermore, we demonstrate its excellent finite sample performance through numerical simulations and a real data example.

Latent Feature Decompositions for Integrative Analysis of Multi-Platform Genomic Data

Increased availability of multi-platform genomics data on matched samples has sparked research efforts to discover how diverse molecular features interact both within and between platforms. In addition, simultaneous measurements of genetic and epigenetic characteristics illuminate the roles their complex relationships play in disease progression and outcomes. However, integrative methods for diverse genomics data are faced with the challenges of ultra-high dimensionality and the existence of complex interactions both within and between platforms. We propose a novel modeling framework for integrative analysis based on decompositions of the large number of platform-specific features into a smaller number of latent features. Subsequently we build a predictive model for clinical outcomes accounting for both within- and between-platform interactions based on Bayesian model averaging procedures. Principal components, partial least squares and non-negative matrix factorization as well as sparse counterparts of each are used to define the latent features, and the performance of these decompositions is compared both on real and simulated data. The latent feature interactions are shown to preserve interactions between the original features and not only aid prediction but also allow explicit selection of outcome-related features. The methods are motivated by and applied to a glioblastoma multiforme data set from The Cancer Genome Atlas to predict patient survival times integrating gene expression, microRNA, copy number and methylation data. For the glioblastoma data, we find a high concordance between our selected prognostic genes and genes with known associations with glioblastoma. In addition, our model discovers several relevant cross-platform interactions such as copy number variation associated gene dosing and epigenetic regulation through promoter methylation. On simulated data, we show that our proposed method successfully incorporates interactions within and between genomic platforms to aid accurate prediction and variable selection. Our methods perform best when principal components are used to define the latent features.

Adaptive Lasso estimators for ultrahigh dimensional generalized linear models

Generalized linear models have been more widely used than linear models which exclude categorical variables. The penalized method becomes an effective tool to study ultrahigh dimensional generalized linear models. In this paper, we study theoretical results of the adaptive Lasso for generalized linear models in terms of diverging number of parameters and ultrahigh dimensionality. The asymptotic results are examined by several simulation studies.

Feature Selection for Varying Coefficient Models With Ultrahigh-Dimensional Covariates

This article is concerned with feature screening and variable selection for varying coefficient models with ultrahigh-dimensional covariates. We propose a new feature screening procedure for these models based on conditional correlation coefficient. We systematically study the theoretical properties of the proposed procedure, and establish their sure screening property and the ranking consistency. To enhance the finite sample performance of the proposed procedure, we further develop an iterative feature screening procedure. Monte Carlo simulation studies were conducted to examine the performance of the proposed procedures. In practice, we advocate a two-stage approach for varying coefficient models. The two-stage approach consists of (a) reducing the ultrahigh dimensionality by using the proposed procedure and (b) applying regularization methods for dimension-reduced varying coefficient models to make statistical inferences on the coefficient functions. We illustrate the proposed two-stage approach by a real data example. Supplementary materials for this article are available online.

A note on the one-step estimator for ultrahigh dimensionality

The one-step estimator, covering various penalty functions, enjoys the oracle property with a good initial estimator. The initial estimator can be chosen as the least squares estimator or maximum likelihood estimator in low-dimensional settings. However, it is not available in ultrahigh dimensionality. In this paper, we study the one-step estimator with the initial estimator being marginal ordinary least squares estimates in the ultrahigh linear model. Under some appropriate conditions, we show that the one-step estimator is selection consistent. Finite sample performance of the proposed procedure is assessed by Monte Carlo simulation studies.

Regularized Matrix Regression

Modern technologies are producing a wealth of data with complex structures. For instance, in two-dimensional digital imaging, flow cytometry and electroencephalography, matrix-type covariates frequently arise when measurements are obtained for each combination of two underlying variables. To address scientific questions arising from those data, new regression methods that take matrices as covariates are needed, and sparsity or other forms of regularization are crucial owing to the ultrahigh dimensionality and complex structure of the matrix data. The popular lasso and related regularization methods hinge on the sparsity of the true signal in terms of the number of its non-zero coefficients. However, for the matrix data, the true signal is often of, or can be well approximated by, a low rank structure. As such, the sparsity is frequently in the form of low rank of the matrix parameters, which may seriously violate the assumption of the classical lasso. We propose a class of regularized matrix regression methods based on spectral regularization. A highly efficient and scalable estimation algorithm is developed, and a degrees-of-freedom formula is derived to facilitate model selection along the regularization path. Superior performance of the method proposed is demonstrated on both synthetic and real examples.

Tensor Regression with Applications in Neuroimaging Data Analysis

Classical regression methods treat covariates as a vector and estimate a corresponding vector of regression coefficients. Modern applications in medical imaging generate covariates of more complex form such as multidimensional arrays (tensors). Traditional statistical and computational methods are proving insufficient for analysis of these high-throughput data due to their ultrahigh dimensionality as well as complex structure. In this article, we propose a new family of tensor regression models that efficiently exploit the special structure of tensor covariates. Under this framework, ultrahigh dimensionality is reduced to a manageable level, resulting in efficient estimation and prediction. A fast and highly scalable estimation algorithm is proposed for maximum likelihood estimation and its associated asymptotic properties are studied. Effectiveness of the new methods is demonstrated on both synthetic and real MRI imaging data. Supplementary materials for this article are available online.

Testing the Statistical Significance of an Ultra-High-Dimensional Naive Bayes Classifier

The naive Bayes approach is one of the most popular methods used for classification. Nevertheless, how to test its statistical significance under an ultra-high-dimensional (UHD) setup is not well understood. To fill this important theoretical gap, we propose a novel testing statistic with a standard normal asymptotic null distribution, even if the predictor dimension is considerably larger than the sample size. This makes the proposed method useful for UHD data analysis. Simulation studies are presented to demonstrate its finite sample performance and a text classification example is described for illustration.

Estimating Networks: Lasso for Spatial Weights

In applied work, often the spatial neighbouring matrix is assumed known and available to the researcher. This paper provides a method for estimating it within longitudinal data. The salient feature is the ultra-high dimensionality of the problem, which is addressed with an adaptation of the Least Absolute Shrinkage and Selection Operator (Lasso). The main result is that, under identification and sparsity conditions, the estimator is consistent for the true neighbouring matrix, both under L1 and prediction norms. The proposed model nests a graphical model and suggests several economic applications. Most notably, it provides a framework for estimating networks based on observable cohorts, as opposed to assuming prior knowledge. Finally, Monte Carlo evidence is presented, along with an application for contagion of government bond yields in the wake of the recent European crisis.