Narrow Results

This paper discusses the multivariate Behrens–Fisher problem, which tests the equality of two population mean vectors under heteroscedasticity. The classical Wald-type test is not applicable in high dimensions owing to the singularity of sample covariance matrices. A ridgelized Wald-type (RIWT) test is then proposed. Using the exact four-moment theorem, its asymptotic null distribution is derived under moment conditions. Thus, our test can accommodate non-Gaussian distributions with general covariance matrices. Simulation results demonstrate the good performance of the proposed test.

The prediction of future values of time series is a challenging task in many fields. In particular, making prediction based on short-term data is believed to be difficult. Here, we propose a method to predict systems' low-dimensional dynamics from high-dimensional but short-term data. Intuitively, it can be considered as a transformation from the inter-variable information of the observed high-dimensional data into the corresponding low-dimensional but long-term data, thereby equivalent to prediction of time series data. Technically, this method can be viewed as an inverse implementation of delayed embedding reconstruction. Both methods and algorithms are developed. To demonstrate the effectiveness of the theoretical result, benchmark examples and real-world problems from various fields are studied.

Driven by mutual benefits, there is a demand for transactional data sharing among organizations or parties for research or business analysis purpose. It becomes an essential concern to provide privacy-preserving data sharing and meanwhile maintain data utility, due to the fact that transactional data may contain sensitive personal information. Existing privacy-preserving methods, such as k-anonymity and l-diversity, cannot handle high-dimensional sparse data well, since they would bring about much data distortion in the anonymization process. In this paper, we use bipartite graphs with node attributes to model high-dimensional sparse data, and then propose a privacy-preserving approach for sharing transactional data in a new vision, in which the bipartite graph is anonymized into a weighted bipartite graph by clustering node attributes. Our approach can maintain privacy of the associations between entities and resist certain attackers with knowledge of partial items. Experiments have been performed on real-life data sets to measure the information loss and the accuracy of answering aggregate queries. Experimental results show that the approach improves the balance of performance between privacy protection and data utility.

The ad hoc nature of the clustering methods makes simulated data paramount in assessing the performance of clustering methods. Real datasets could be used in the evaluation of clustering methods with the major drawback of missing the assessment of many test scenarios. In this paper, we propose a formal quantification of component overlap. This quantification is derived from a set of theorems which allow us to derive an automatic method for artificial data generation. We also derive a method to estimate parameters of existing models and to evaluate the results of other approaches. Automatic estimation of the overlap rate can also be used as an unsupervised learning approach in data mining to determine the parameters of mixture models from actual observations.

The high-throughput correlated DNA methylation (DNAmeth) dataset generated from Illumina Infinium Human Methylation 27 (IIHM 27K) BeadChip assay. In the DNAmeth data, there are several CpG sites for every gene, and these grouped CpG sites are highly correlated. Most of the current filtering-based ranking (FBR) methods do not consider the group correlation structures. Obtaining the significant features with the FBR methods and applying these features to the classifiers to attain the best classification accuracy in highly correlated DNAmeth data is a challenging task. In this research, we introduce a resampling of group least absolute shrinkage and selection operator (glasso) FBR method capable of ignoring the unrelated features in the data considering the group correlation among the features. The various classifiers, such as random forests (RF), Naive Bayes (NB), and support vector machines (SVM) with the significant CpGs obtained from the proposed resampling of group lasso-based ranking (RGLR) method helped to boost the classification accuracy. Through simulated and experimental prostate DNAmeth data, we showed that higher performance of accuracy, sensitivity, specificity, and geometric mean is achieved by ignoring the unimportant CpG sites through the RGLR method.

Subspace clustering is a challenging high-dimensional data mining task. There have been several approaches proposed in the literature to identify clusters in subspaces, however their performance and quality is highly affected by input parameters. A little research is done so far on identifying proper parameter values automatically. Other observed drawbacks are requirement of multiple database scans resulting into increased demand for computing resources and generation of many redundant clusters. Here, we propose a parameter light subspace clustering method for numerical data hereafter referred to as CLUSLINK. The algorithm is based on single linkage clustering method and works in bottom up, greedy fashion. The only input user has to provide is how coarse or fine the resulting clusters should be, and if not given, the algorithm operates with default values. The empirical results obtained over synthetic and real benchmark datasets show significant improvement in terms of accuracy and execution time.

High-dimensional data that often lie in low-dimensional subspaces are ubiquitous in many fields of signal and image processing, pattern recognition, machine learning, etc. Finding sparse representations of data points in a dictionary built by using the collection of data helps to study low-dimensional subspaces. In this paper, we consider the problem of subspace-sparse representation for data composed of two distinct features. Applying different measures to the coefficients of the two distinct features under different dictionaries makes the model used in this paper more general. We present theoretical guarantee for subspace-sparse representation under conditions on the subspaces and data. The program used in this paper can handle data with structured corruption, missing entries, sparse outliers and random bounded noise well.

Random forest is an ensemble classification algorithm. It performs well when most predictive variables are noisy and can be used when the number of variables is much larger than the number of observations. The use of bootstrap samples and restricted subsets of attributes makes it more powerful than simple ensembles of trees. The main advantage of a random forest classifier is its explanatory power: it measures variable importance or impact of each factor on a predicted class label. These characteristics make the algorithm ideal for microarray data. It was shown to build models with high accuracy when tested on high-dimensional microarray datasets. Current implementations of random forest in the machine learning and statistics community, however, limit its usability for mining over large datasets, as they require that the entire dataset remains permanently in memory. We propose a new framework, an optimized implementation of a random forest classifier, which addresses specific properties of microarray data, takes computational complexity of a decision tree algorithm into consideration, and shows excellent computing performance while preserving predictive accuracy. The implementation is based on reducing overlapping computations and eliminating dependency on the size of main memory. The implementation's excellent computational performance makes the algorithm useful for interactive data analyses and data mining.

The Cox proportional hazards model has been widely used in cancer genomic research that aims to identify genes from high-dimensional gene expression space associated with the survival time of patients. With the increase in expertly curated biological pathways, it is challenging to incorporate such complex networks in fitting a high-dimensional Cox model. This paper considers a Bayesian framework that employs the Ising prior to capturing relations among genes represented by graphs. A spike-and-slab prior is also assigned to each of the coefficients for the purpose of variable selection. The iterated conditional modes/medians (ICM/M) algorithm is proposed for the implementation for Cox models. The ICM/M estimates hyperparameters using conditional modes and obtains coefficients through conditional medians. This procedure produces some coefficients that are exactly zero, making the model more interpretable. Comparisons of the ICM/M and other regularized Cox models were carried out with both simulated and real data. Compared to lasso, adaptive lasso, elastic net, and DegreeCox, the ICM/M yielded more parsimonious models with consistent variable selection. The ICM/M model also provided a smaller number of false positives than the other methods and showed promising results in terms of predictive accuracy. In terms of computing times among the network-aware methods, the ICM/M algorithm is substantially faster than DegreeCox even when incorporating a large complex network. The implementation of the ICM/M algorithm for Cox regression model is provided in R package icmm, available on the Comprehensive R Archive Network (CRAN).

Let $N_p(\mu ,\mathrm{\Sigma })$ be a $p$-dimensional normal distribution. Testing $\mathrm{\Sigma }$ equal to a given matrix or $(\mu ,\mathrm{\Sigma })$ equal to a given pair through the likelihood ratio test (LRT) is a classical problem in the multivariate analysis. When the population dimension $p$ is fixed, it is known that the LRT statistics go to ${\chi }^2$-distributions. When $p$ is large, simulation shows that the approximations are far from accurate. For the two LRT statistics, in the high-dimensional cases, we obtain their central limit theorems under a big class of alternative hypotheses. In particular, the alternative hypotheses are not local ones. We do not need the assumption that $p$ and $n$ are proportional to each other. The condition $n-1>p\to \infty$ suffices in our results.

This paper considers empirical likelihood inference for a high-dimensional partially functional linear model. An empirical log-likelihood ratio statistic is constructed for the regression coefficients of non-functional predictors and proved to be asymptotically normally distributed under some regularity conditions. Moreover, maximum empirical likelihood estimators of the regression coefficients of non-functional predictors are proposed and their asymptotic properties are obtained. Simulation studies are conducted to demonstrate the performance of the proposed procedure and a real data set is analyzed for illustration.

Independent component model (ICM) is a widely-used population distribution in high-dimensional data analysis and random matrix theory. In this work, we study the kurtosis of the ICM, which is an important parameter in asymptotic distributions of many commonly used statistics and is also an important criterion for measuring the heavy-tailed nature of the data. Based on U-statistics, we develop an estimation method. Theoretically, we show that the proposed estimator is consistent under regular conditions, especially we relax the restriction that the data dimension and the sample size are of the same order. Furthermore, we derive the asymptotic normality of the estimator, which allows us to construct confidence intervals and hypothesis testing statistics. Computationally, we provide a fast and efficient algorithm by leveraging the matrix structure, where the computational complexity is essentially equivalent to computing the sample covariance matrix and the Gram matrix. Finally, the effectiveness of the proposed method is demonstrated through simulated data and real-world data.

This chapter aims to summarize the theories, models, methods and tools of modern econometrics which we have covered in the previous chapters. We first review the classical assumptions of the linear regression model and discuss the historical development of modern econometrics by various relaxations of the classical assumptions. We also discuss the challenges and opportunities for econometrics in the Big data era and point out some important directions for the future development of econometrics.

In bioinformatics with special emphasis on computational biology, genomic science, polygenic models, and computational sequence analysis, principles of molecular genetics (biology) provide room for stochastics to comprehend the basic differences between mathematical exactness and biological diversity. With a large number of sites or loci having mostly categorical (qualitative) responses and imprecise dependence patterns, standard (discrete or continuous) multivariate statistical modeling and analysis may encounter roadblocks of various kinds. Limitations of likelihoods and their variants are appraised and contrasted with the inadequacy of the knowledge discovery and data mining approach that exploits mainly computational algorithms. Alternative approaches that take into account underlying biological implications to a greater (and parametrics to a lesser) extent are appraised in the light of validity and robustness considerations.

We present a new method for exploring cancer gene expression data based on tools from algebraic topology. Our method selects a small relevant subset from tens of thousands of genes while simultaneously identifying nontrivial higher order topological features, i.e., holes, in the data. We first circumvent the problem of high dimensionality by dualizing the data, i.e., by studying genes as points in the sample space. Then we select a small subset of the genes as landmarks to construct topological structures that capture persistent, i.e., topologically significant, features of the data set in its first homology group. Furthermore, we demonstrate that many members of these loops have been implicated for cancer biogenesis in scientific literature. We illustrate our method on five different data sets belonging to brain, breast, leukemia, and ovarian cancers.