High-dimensional Test Research Articles

Convergence and optimization of the parallel method of simultaneous directions for the solution of elliptic problems

For the solution of elliptic problems, fractional step methods and in particular alternating directions (ADI) methods are iterative methods where fractional steps are sequential. Therefore, they only accept parallelization at low level. In [T. Lu, P. Neittaanmäki, X.C. Tai, A parallel splitting-up method for partial differential equations and its applications to Navier–Stokes equations, RAIRO Modél. Math. Anal. Numér. 26 (6) (1992) 673–708], Lu et al. proposed a method where the fractional steps can be performed in parallel. We can thus speak of parallel fractional step (PFS) methods and, in particular, simultaneous directions (SDI) methods. In this paper, we perform a detailed analysis of the convergence and optimization of PFS and SDI methods, complementing what was done in [T. Lu, P. Neittaanmäki, X.C. Tai, A parallel splitting-up method for partial differential equations and its applications to Navier–Stokes equations, RAIRO Modél. Math. Anal. Numér. 26 (6) (1992) 673–708]. We describe the behavior of the method and we specify the good choice of the parameters. We also study the efficiency of the parallelization. Some 2D, 3D and high-dimensional tests confirm our results.

Some high-dimensional tests for a one-way MANOVA

A statistic is proposed for testing the equality of the mean vectors in a one-way multivariate analysis of variance. The asymptotic null distribution of this statistic, as both the sample size and the number of variables go to infinity, is shown to be normal. Thus, this test can be used when the number of variables is not small relative to the sample size. In particular, it can be used when the number of variables exceeds the degrees of freedom for error, a situation in which standard MANOVA tests are invalid. A related statistic, also having an asymptotic normal distribution, is developed for tests concerning the dimensionality of the hyperplane formed by the population mean vectors. The finite sample size performances of the normal approximations are evaluated in a simulation study.

Use of goodness-of-fit procedures in high dimensional testing

It is increasingly common in functional data analysis that inference is carried out using some form of goodness-of-fit test, such as the adaptive Neyman (AN) test of Fan [Fan, J., 1996, Journal of the American Statistical Association, 91, 674–688.], or a test incorporating any of a variety of selection diagnostics such as Akaike's information criterion (AIC) and Bayesian information criterion. These procedures as well as weighted versions of the classical χ2 test are examined by power simulation in the context of functional data analysis. It is seen that the AN test exhibits good properties consistently across a range of dimensionalities and configurations of the alternative. Certain tests based on AIC are found to exhibit comparable performance at moderate dimensionalities. Weighted χ2 tests are seen to exhibit very good properties in specific scenarios, but appear extremely sensitive to the shape of the alternative.

Micro-particle swarm optimizer for solving high dimensional optimization problems (μPSO for high dimensional optimization problems)

A simple and powerful micro-particle swarm optimizer (μPSO) is presented. For high dimensional optimization problems, the proposed μPSO needs only a small population to outperform the standard particle swarm optimizer (PSO) that uses a larger population. This superior performance is confirmed by simulation results for different high dimensional test functions.

A high-dimensional test for the equality of the smallest eigenvalues of a covariance matrix

For the test of sphericity, Ledoit and Wolf [Ann. Statist. 30 (2002) 1081–1102] proposed a statistic which is robust against high dimensionality. In this paper, we consider a natural generalization of their statistic for the test that the smallest eigenvalues of a covariance matrix are equal. Some inequalities are obtained for sums of eigenvalues and sums of squared eigenvalues. These bounds permit us to obtain the asymptotic null distribution of our statistic, as the dimensionality and sample size go to infinity together, by using distributional results obtained by Ledoit and Wolf [Ann. Statist. 30 (2002) 1081–1102]. Some empirical results comparing our test with the likelihood ratio test are also given.

Haplotype interaction analysis of unlinked regions

Genetically complex diseases are caused by interacting environmental factors and genes. As a consequence, statistical methods that consider multiple unlinked genomic regions simultaneously are desirable. Such consideration, however, may lead to a vast number of different high-dimensional tests whose appropriate analysis pose a problem. Here, we present a method to analyze case-control studies with multiple SNP data without phase information that considers gene-gene interaction effects while correcting appropriately for multiple testing. In particular, we allow for interactions of haplotypes that belong to different unlinked regions, as haplotype analysis often proves to be more powerful than single marker analysis. In addition, we consider different marker combinations at each unlinked region. The multiple testing issue is settled via the minP approach; the P value of the "best" marker/region configuration is corrected via Monte-Carlo simulations. Thus, we do not explicitly test for a specific pre-defined interaction model, but test for the global hypothesis that none of the considered haplotype interactions shows association with the disease. We carry out a simulation study for case-control data that confirms the validity of our approach. When simulating two-locus disease models, our test proves to be more powerful than association methods that analyze each linked region separately. In addition, when one of the tested regions is not involved in the etiology of the disease, only a small amount of power is lost with interaction analysis as compared to analysis without interaction. We successfully applied our method to a real case-control data set with markers from two genes controlling a common pathway. While classical analysis failed to reach significance, we obtained a significant result even after correction for multiple testing with our proposed haplotype interaction analysis. The method described here has been implemented in FAMHAP.

S36.2: High‐dimensional tests in gene‐expression analysis

S36.2: High‐dimensional tests in gene‐expression analysis

Asymptotic Results of a High Dimensional MANOVA Test and Power Comparison When the Dimension is Large Compared to the Sample Size

This paper is concerned with Dempster trace criterion for multivariate linear hypothesis which was proposed for high dimensional situation. First we derive asymptotic null and nonnull distributions of Dempster trace criterion when both the sample size and the dimension tend to infinity. Our approximations are examined through some numerical experiments. Next we compare the power of Dempster trace criterion with the ones of three classical criteria; likelihood ratio criterion, Lawley-Hotelling trace criterion, and Bartlett-Nanda-Pillai trace criterion when the dimension is large compared to the sample size.

Testing additivity by kernel-based methods - what is a reasonable test?

In the common nonparametric regression model with high dimensional predictor several tests for the hypothesis of an additive regression are investigated. The corresponding test statistics are either based on the diiferences between a fit under the assumption of additivity and a fit in the general model or based on residuals under the assumption of additivity. For all tests asymptotic normality is established under the null hypothesis of additivity and under fixed alternatives with different rates of convergence corresponding to both cases. These results are used for a comparison of the different methods. It is demonstrated that a statistic based on an empirical L1 - distance of the Nadaraya Watson and the marginal integration estimator yields the asymptotically most efficient procedure if these are compared with respect to the asymptotic behaviour under fixed alternatives.

Testing Additivity by Kernel-Based Methods: What Is a Reasonable Test?

In the common nonparametric regression model with high dimensional predictor several tests for the hypothesis of an additive regression are investigated. The corresponding test statistics are either based on the diiferences between a fit under the assumption of additivity and a fit in the general model or based on residuals under the assumption of additivity. For all tests asymptotic normality is established under the null hypothesis of additivity and under fixed alternatives with different rates of convergence corresponding to both cases. These results are used for a comparison of the different methods. It is demonstrated that a statistic based on an empirical L1 - distance of the Nadaraya Watson and the marginal integration estimator yields the asymptotically most efficient procedure if these are compared with respect to the asymptotic behaviour under fixed alternatives.