High-dimensional Data Applications Research Articles

Improving the Robustness of Variable Selection and Predictive Performance of Regularized Generalized Linear Models and Cox Proportional Hazard Models.

High-dimensional data applications often entail the use of various statistical and machine-learning algorithms to identify an optimal signature based on biomarkers and other patient characteristics that predicts the desired clinical outcome in biomedical research. Both the composition and predictive performance of such biomarker signatures are critical in various biomedical research applications. In the presence of a large number of features, however, a conventional regression analysis approach fails to yield a good prediction model. A widely used remedy is to introduce regularization in fitting the relevant regression model. In particular, a penalty on the regression coefficients is extremely useful, and very efficient numerical algorithms have been developed for fitting such models with different types of responses. This -based regularization tends to generate a parsimonious prediction model with promising prediction performance, i.e., feature selection is achieved along with construction of the prediction model. The variable selection, and hence the composition of the signature, as well as the prediction performance of the model depend on the choice of the penalty parameter used in the regularization. The penalty parameter is often chosen by K-fold cross-validation. However, such an algorithm tends to be unstable and may yield very different choices of the penalty parameter across multiple runs on the same dataset. In addition, the predictive performance estimates from the internal cross-validation procedure in this algorithm tend to be inflated. In this paper, we propose a Monte Carlo approach to improve the robustness of regularization parameter selection, along with an additional cross-validation wrapper for objectively evaluating the predictive performance of the final model. We demonstrate the improvements via simulations and illustrate the application via a real dataset.

Fully component selection: An efficient combination of feature selection and principal component analysis to increase model performance

Principal component analysis (PCA) is one the most effective and widely used dimensionality-reduction techniques which aggregates the maximal variance in the first few components. In many applications, the first few components of PCA are used in place of the original variables in statistical and machine learning models; nevertheless, such use does not guarantee the selection of the most relevant components to the target variable. This paper presents an efficient approach in which all components, rather than simply the first few, are considered as input for the random forest (RF) model, and a feature selection algorithm is integrated with the RF to select the most relevant components, hence called fully component selection (FCS). The proposed method was evaluated on spectroscopic data comprising 70 soil samples with 2050 spectral bands to estimate soil organic carbon (SOC). The FCS approach was compared to the RF model once considering the original features, then again using only the first few components, and again using all components without feature selection. The results showed that RF-FCS substantially outperformed other approaches, such that the R2 was increased between 25.2% and 55.5%. Furthermore, the findings indicated that the most relevant principal components (PCs) to the target variable were not the first few, but PC6, PC7, PC8, PC15, and PC43. Using the FCS approach increased model performance substantively, and coupling it with machine learning and statistical models is strongly recommended for high dimensional data applications.

The measure on the original space from a product measure

The measurability problems can arise in Gaussian and empirical processes underlying nonparametric and semiparametric problems critical in modern statistics and machine learning. The usual applications are high-dimensional data in random fields. To ensure the validity of recovering a true original complex object from its observed images treated in a product space that appears in existing high-dimensional data applications widely arising in physical, biological, and social sciences and engineering, we begin with the results on the σ-algebra of the original space Ω induced by a given map fT mapping from Ω to a product measure space (ΩT,FT,μT). We study the certain measurability issues on Ω with the induced measure μ from the image measure μT of the product space by establishing the outer measure ν⁎ restricted to the trace σ-algebra. We further investigate the measure μ on Ω under the map fT to obtain the conditions for the identifiability equality μT=μ∘fT−1 to hold so that a practical and rigorous foundation is provided for the nontrivial applications to work meaningfully.

Prediction error after model search

Estimation of the prediction error of a linear estimation rule is difficult if the data analyst also uses data to select a set of variables and constructs the estimation rule using only the selected variables. In this work, we propose an asymptotically unbiased estimator for the prediction error after model search. Under some additional mild assumptions, we show that our estimator converges to the true prediction error in $L^{2}$ at the rate of $O(n^{-1/2})$, with $n$ being the number of data points. Our estimator applies to general selection procedures, not requiring analytical forms for the selection. The number of variables to select from can grow as an exponential factor of $n$, allowing applications in high-dimensional data. It also allows model misspecifications, not requiring linear underlying models. One application of our method is that it provides an estimator for the degrees of freedom for many discontinuous estimation rules like best subset selection or relaxed Lasso. Connection to Stein’s Unbiased Risk Estimator is discussed. We consider in-sample prediction errors in this work, with some extension to out-of-sample errors in low-dimensional, linear models. Examples such as best subset selection and relaxed Lasso are considered in simulations, where our estimator outperforms both $C_{p}$ and cross validation in various settings.

Problem formulations and solvers in linear SVM: a review

Support vector machine (SVM) is an optimal margin based classification technique in machine learning. SVM is a binary linear classifier which has been extended to non-linear data using Kernels and multi-class data using various techniques like one-versus-one, one-versus-rest, Crammer Singer SVM, Weston Watkins SVM and directed acyclic graph SVM (DAGSVM) etc. SVM with a linear Kernel is called linear SVM and one with a non-linear Kernel is called non-linear SVM. Linear SVM is an efficient technique for high dimensional data applications like document classification, word-sense disambiguation, drug design etc. because under such data applications, test accuracy of linear SVM is closer to non-linear SVM while its training is much faster than non-linear SVM. SVM is continuously evolving since its inception and researchers have proposed many problem formulations, solvers and strategies for solving SVM. Moreover, due to advancements in the technology, data has taken the form of ‘Big Data’ which have posed a challenge for Machine Learning to train a classifier on this large-scale data. In this paper, we have presented a review on evolution of linear support vector machine classification, its solvers, strategies to improve solvers, experimental results, current challenges and research directions.

High- and low-level hierarchical classification algorithm based on source separation process

High-dimensional data applications have earned great attention in recent years. We focus on remote sensing data analysis on high-dimensional space like hyperspectral data. From a methodological viewpoint, remote sensing data analysis is not a trivial task. Its complexity is caused by many factors, such as large spectral or spatial variability as well as the curse of dimensionality. The latter describes the problem of data sparseness. In this particular ill-posed problem, a reliable classification approach requires appropriate modeling of the classification process. The proposed approach is based on a hierarchical clustering algorithm in order to deal with remote sensing data in high-dimensional space. Indeed, one obvious method to perform dimensionality reduction is to use the independent component analysis process as a preprocessing step. The first particularity of our method is the special structure of its cluster tree. Most of the hierarchical algorithms associate leaves to individual clusters, and start from a large number of individual classes equal to the number of pixels; however, in our approach, leaves are associated with the most relevant sources which are represented according to mutually independent axes to specifically represent some land covers associated with a limited number of clusters. These sources contribute to the refinement of the clustering by providing complementary rather than redundant information. The second particularity of our approach is that at each level of the cluster tree, we combine both a high-level divisive clustering and a low-level agglomerative clustering. This approach reduces the computational cost since the high-level divisive clustering is controlled by a simple Boolean operator, and optimizes the clustering results since the low-level agglomerative clustering is guided by the most relevant independent sources. Then at each new step we obtain a new finer partition that will participate in the clustering process to enhance semantic capabilities and give good identification rates.

MeDiA: Mean Distance Association and Its Applications in Nonlinear Gene Set Analysis.

Probabilistic association discovery aims at identifying the association between random vectors, regardless of number of variables involved or linear/nonlinear functional forms. Recently, applications in high-dimensional data have generated rising interest in probabilistic association discovery. We developed a framework based on functions on the observation graph, named MeDiA (Mean Distance Association). We generalize its property to a group of functions on the observation graph. The group of functions encapsulates major existing methods in association discovery, e.g. mutual information and Brownian Covariance, and can be expanded to more complicated forms. We conducted numerical comparison of the statistical power of related methods under multiple scenarios. We further demonstrated the application of MeDiA as a method of gene set analysis that captures a broader range of responses than traditional gene set analysis methods.

Fuzzy partition based soft subspace clustering and its applications in high dimensional data

As one of the most popular clustering techniques for high dimensional data, soft subspace clustering (SSC) algorithms have been receiving a great deal of attention in recent years. Unfortunately, most existing works do not cluster high dimensional sparse data and noisy data in an effective manner. In this study, a novel soft subspace clustering algorithm called PI-SSC is proposed. By introducing a partition index (PI) into the objective function, a novel soft subspace clustering algorithm that combines the concepts of hard and fuzzy clustering is proposed. Furthermore, the robust property of PI-SSC is analyzed from the viewpoint of ε-insensitive distance. A convergence theorem for PI-SSC is also established by applying Zangwill’s convergence theorem. The results of the experiment demonstrate the effectiveness of the proposed algorithm in high dimensional sparse text data and noisy texture data.

Minimax estimation of large covariance matrices under L1-norm

Driven by a wide range of applications in high-dimensional data analy- sis, there has been significant recent interest in the estimation of large covariance matrices. In this paper, we consider optimal estimation of a covariance matrix as well as its inverse over several commonly used parameter spaces under the matrix l1 norm. Both minimax lower and upper bounds are derived. The lower bounds are established by using hypothesis testing arguments, where at the core are a novel construction of collections of least favorable multivariate nor- mal distributions and the bounding of the affinities between mixture distributions. The lower bound analysis also provides insight into where the difficulties of the covariance matrix estimation problem arise. A specific thresholding estimator and tapering estimator are constructed and shown to be minimax rate optimal. The optimal rates of convergence established in the paper can serve as a benchmark for the performance of covariance matrix estimation methods.

Using cross-validation to evaluate predictive accuracy of survival risk classifiers based on high-dimensional data

Developments in whole genome biotechnology have stimulated statistical focus on prediction methods. We review here methodology for classifying patients into survival risk groups and for using cross-validation to evaluate such classifications. Measures of discrimination for survival risk models include separation of survival curves, time-dependent ROC curves and Harrell's concordance index. For high-dimensional data applications, however, computing these measures as re-substitution statistics on the same data used for model development results in highly biased estimates. Most developments in methodology for survival risk modeling with high-dimensional data have utilized separate test data sets for model evaluation. Cross-validation has sometimes been used for optimization of tuning parameters. In many applications, however, the data available are too limited for effective division into training and test sets and consequently authors have often either reported re-substitution statistics or analyzed their data using binary classification methods in order to utilize familiar cross-validation. In this article we have tried to indicate how to utilize cross-validation for the evaluation of survival risk models; specifically how to compute cross-validated estimates of survival distributions for predicted risk groups and how to compute cross-validated time-dependent ROC curves. We have also discussed evaluation of the statistical significance of a survival risk model and evaluation of whether high-dimensional genomic data adds predictive accuracy to a model based on standard covariates alone.

Super-sparse principal component analyses for high-throughput genomic data

BackgroundPrincipal component analysis (PCA) has gained popularity as a method for the analysis of high-dimensional genomic data. However, it is often difficult to interpret the results because the principal components are linear combinations of all variables, and the coefficients (loadings) are typically nonzero. These nonzero values also reflect poor estimation of the true vector loadings; for example, for gene expression data, biologically we expect only a portion of the genes to be expressed in any tissue, and an even smaller fraction to be involved in a particular process. Sparse PCA methods have recently been introduced for reducing the number of nonzero coefficients, but these existing methods are not satisfactory for high-dimensional data applications because they still give too many nonzero coefficients.ResultsHere we propose a new PCA method that uses two innovations to produce an extremely sparse loading vector: (i) a random-effect model on the loadings that leads to an unbounded penalty at the origin and (ii) shrinkage of the singular values obtained from the singular value decomposition of the data matrix. We develop a stable computing algorithm by modifying nonlinear iterative partial least square (NIPALS) algorithm, and illustrate the method with an analysis of the NCI cancer dataset that contains 21,225 genes.ConclusionsThe new method has better performance than several existing methods, particularly in the estimation of the loading vectors.