On the evaluation of the joint distribution of order statistics

Dunnett and Tamhane [Dunnett, C.W., Tamhane, A.C., 1992. A step-up multiple test procedure. J. Amer. Statist. Assoc. 87, 162-170.] proposed a step-up procedure for comparing k treatments with a control and showed that the step-up procedure is more powerful than its counterpart single step and step-down procedures. Since then, several modified step-up procedures have been suggested to deal with different testing environments. In order to establish those step-up procedures, it is necessary to derive approaches for evaluating the joint distribution of the order statistics. In some cases, experimenters may have difficulty in applying those step-up procedures in multiple hypothesis testing because of the computational limitation of existing algorithms in evaluating the critical values for a large number of multiple comparisons. As a result, most procedures are only workable when the design of the experiment is balanced with k≤20 or unbalanced with k≤8. In this paper, new algorithms are proposed in order to effectively compute the joint distribution of order statistics in various situations. An extensive numerical study shows that the proposed algorithms can easily handle the testing situations with a much larger k. Examples of applying the proposed algorithms to evaluate the critical values of two existing step-up procedures are also presented.     

One-sided multiple comparisons for treatment means with a control mean

We study comparisons of several treatments with a common control when it is believed a priori that the treatment means, μ_i, are at least as large as the control mean, μ₀. In this setting, which is called a tree ordering, we study multiple comparisons that determine whether μ_i>μ₀ or μ_i=μ₀ for each treatment. The classical procedure by Dunnett (1955) and the step-down and step-up techniques by [Dunnett and Tamhane, 1991] and [Dunnett and Tamhane, 1992] are well known. The results in Marcus and Talpaz (1992) provide multiple comparisons based on the maximum likelihood estimates restricted by the tree ordering. We also study two-stage procedures that consist of the likelihood ratio test of homogeneity with the alternative constrained by the tree ordering followed by two-sample t comparisons with possibly different critical values for the two-sample comparisons. Marcus et al. (1976) discuss the use of closed tests in such situations and propose using a closed version of the restricted likelihood ratio test. We describe step-down versions of the Marcus-Talpaz, the two-stage, and the likelihood ratio procedures, as well as a closed version of the Marcus-Talpaz multiple comparison procedure. Using Monte Carlo techniques, we study the familywise errors and powers of these procedures and make some recommendations concerning techniques that perform well for all tree ordered mean vectors.     

On correlated z-values distribution in hypothesis testing

Multiple-testing problems have received much attention. Different strategies have been considered in order to deal with this problem. The false discovery rate (FDR) is, probably, the most studied criterion. On the other hand, the sequential goodness of fit (SGoF), is a recently proposed approach. Most of the developed procedures are based on the independence among the involved tests; however, in spite of being a reasonable proviso in some frameworks, independence is not realistic for a number of practical cases. Therefore, one of the main problems in order to develop appropriate methods is, precisely, the effect of the dependence among the different tests on decisions making. The consequences of the correlation on the z-values distribution in the general multitesting problem are explored. Some different algorithms are provided in order to approximate the distribution of the expected rejection proportions. The performance of the proposed methods is evaluated in a simulation study in which, for comparison purposes, the Benjamini and Hochberg method to control the FDR, the Lehmann and Romano procedure to control the tail probability of the proportion of false positives (TPPFP), and the Beta–Binomial SGoF procedure are considered. Three different dependence structures are considered. As usual, for a better understanding of the problem, several practical cases are also studied.     

Identification of k-step-ahead prediction error model and MPC control

This work studies k-step-ahead prediction error model identification and its relationship to MPC control. The use of error criteria in parameter estimation will be discussed, where the identified model is used in model predictive control (MPC). Assume that the model error is dominated by the variance part, it can be shown that a k-step-ahead prediction error model is not optimal for k-step-ahead prediction. A normal one-step-ahead prediction error criterion will be optimal for k-step-ahead prediction. Then it is argued that even when some bias exists, the result could still hold true. Therefore, for MPC identification of linear processes, one-step-ahead prediction error models fever k-step-ahead prediction models. Simulations and industrial testing data will be used to illustrate the idea.     

High-throughput DNA methylation datasets for evaluating false discovery rate methodologies

When analyzing high-throughput genomic data, the multiple comparison problem is most often addressed through estimation of the false discovery rate (FDR), using methods such as the Benjamini & Hochberg, Benjamini & Yekutieli, the q-value method, or in controlling the family-wise error rate (FWER) using Holm's step down method. To date, research studies that have compared various FDR/FWER methodologies have made use of limited simulation studies and/or have applied the methods to one or more microarray gene expression dataset(s). However, for microarray datasets the veracity of each null hypothesis tested is unknown so that an objective evaluation of performance cannot be rendered for application data. Due to the role of methylation in X-chromosome inactivation, we postulate that high-throughput methylation datasets may provide an appropriate forum for assessing the performance of commonly used FDR methodologies. These datasets preserve the complex correlation structure between probes, offering an advantage over simulated datasets. Using several methylation datasets, commonly used FDR methods including the q-value, Benjamini & Hochberg, and Benjamini & Yekutieli procedures as well as Holm's step down method were applied to identify CpG sites that are differentially methylated when comparing healthy males to healthy females. The methods were compared with respect to their ability to identify CpG sites located on sex chromosomes as significant, by reporting the sensitivity, specificity, and observed FDR. These datasets are useful for characterizing the performance of multiple comparison procedures, and may find further utility in other tasks such as comparing variable selection capabilities of classification methods and evaluating the performance of meta-analytic methods for microarray data.     

Testing equality of inverse Gaussian means under heterogeneity, based on generalized test variable

The inverse Gaussian (IG) distribution is an ideal candidate for modelling positive, right-skewed data due to the fact that its inference theory and methodology bear a striking resemblance to the normal theory and methods. For testing equality of several IG means under the assumption of equal scale parameters, there exists the ANORE (analysis of reciprocal) F test, which is analogous to the ANOVA F test for the normal distribution. In this article, the concept of generalized P-value, introduced by Tsui and Weerahandi [1989. Generalized P-values in significance testing of hypotheses in the presence of nuisance parameters. J. Amer. Statist. Assoc. 84, 602-607], is applied for testing equality of several IG means for the general cases without the assumption of homogeneity. Simulation results indicate that the proposed test has excellent type I error control under both heterogeneity and homogeneity, whereas the type I error probabilities of the ANORE test can be much larger than the nominal level under heterogeneity. The proposed procedure is illustrated using two examples.     

A finite element method with discontinuous rotations for the Mindlin–Reissner plate model

We present a continuous-discontinuous finite element method for the Mindlin–Reissner plate model based on continuous polynomials of degree k ? 2 for the transverse displacements and discontinuous polynomials of degree k ? 1 for the rotations. We prove a priori convergence estimates, uniformly in the thickness of the plate, and thus show that locking is avoided. We also derive a posteriori error estimates based on duality, together with corresponding adaptive procedures for controlling linear functionals of the error. Finally, we present some numerical results.     

Knuth-Bendix completion of theories of commuting group endomorphisms

Knuth-Bendix completions of the equational theories of k?2 commuting group endomorphisms are obtained, using automated theorem proving and modern termination checking. This improves on modern implementations of completion, where the orderings implemented cannot orient the commutation rules. The result has applications in decision procedures for automated verification.     

Estimation of the proportion of true null hypotheses in high-dimensional data under dependence

In multiple testing, a challenging issue is to provide an accurate estimation of the proportion π₀ of true null hypotheses among the whole set of tests. Besides a biological interpretation, this parameter is involved in the control of error rates such as the False Discovery Rate. Improving its estimation can result in more powerful/less conservative methods of differential analysis. Various methods for π₀ estimation have been previously developed. Most of them rely on the assumption of independent p-values distributed according to a two-component mixture model, with a uniform distribution for null p-values. In a general factor analytic framework, the impact of dependence on the properties of the estimation procedures is first investigated and exact expressions of bias and variance are provided in case of dependent data. A more accurate factor-adjusted estimator of π₀ is finally presented, which shows large improvements with respect to the standard procedures.     

Self-checking circuits and decoding algorithms for binary hamming and BCH codes and Reed-Solomon codes over GF(2 m )

We consider problems of detecting errors in combinational circuits and algorithms for the decoding of linear codes. We show that a totally self-checking combinatorial circuit for the decoding of a binary Hamming [n, k] code can be constructed if and only if n = 2 ^r ? 1, r = n?k. We introduce the notion of a totally self-checking combinational circuit detecting error clusters of size at most µ; for shortened Hamming [n,k] codes, we construct totally self-checking decoding combinational circuits detecting error clusters of size at most µ, 2 ≤ µ < n?k. We describe single-error protected and self-checking algorithms: the extended Euclidean algorithm and decoding algorithms for binary BCH codes and Reed-Solomon codes over GF(2 ^m ).     

Improving the bit-parallel NFA of Baeza-Yates and Navarro for approximate string matching

We propose a new variant of the bit-parallel NFA of Baeza-Yates and Navarro (BPD) for approximate string matching [R. Baeza-Yates, G. Navarro, Faster approximate string matching, Algorithmica 23 (1999) 127-158]. BPD is one of the most practical approximate string matching algorithms under moderate pattern lengths and error levels [G. Myers, A fast bit-vector algorithm for approximate string matching based on dynamic programming, J. ACM 46 (3) 1989 395-415; G. Navarro, M. Raffinot, Flexible Pattern Matching in Strings—Practical On-line Search Algorithms for Texts and Biological Sequences, Cambridge University Press, Cambridge, UK, 2002]. Given a length-m pattern and an error threshold k, the original BPD requires (m−k)(k+2) bits of space to represent an NFA with (m−k)(k+1) states. In this paper we remove redundancy from the original NFA representation. Our variant requires (m−k)(k+1) bits of space, which is optimal in the sense that exactly one bit per state is used. The space efficiency is achieved by using an alternative, but equally or even more efficient, simulation algorithm for the bit-parallel NFA. We also present experimental results to compare our modified NFA against the original BPD and its main competitors. Our new variant is more efficient than the original BPD, and it hence takes over/extends the role of the original BPD as one of the most practical approximate string matching algorithms under moderate values of k and m.     

Performance of Balanced Two-Stage Empirical Predictors of Realized Cluster Latent Values from Finite Populations: A Simulation Study

Predictors of random effects are usually based on the popular mixed effects (ME) model developed under the assumption that the sample is obtained from a conceptual infinite population; such predictors are employed even when the actual population is finite. Two alternatives that incorporate the finite nature of the population are obtained from the superpopulation model proposed by Scott and Smith (1969. Estimation in multi-stage surveys. J. Amer. Statist. Assoc. 64, 830-840) or from the finite population mixed model recently proposed by Stanek and Singer (2004. Predicting random effects from finite population clustered samples with response error. J. Amer. Statist. Assoc. 99, 1119-1130). Predictors derived under the latter model with the additional assumptions that all variance components are known and that within-cluster variances are equal have smaller mean squared error (MSE) than the competitors based on either the ME or Scott and Smith's models. As population variances are rarely known, we propose method of moment estimators to obtain empirical predictors and conduct a simulation study to evaluate their performance. The results suggest that the finite population mixed model empirical predictor is more stable than its competitors since, in terms of MSE, it is either the best or the second best and when second best, its performance lies within acceptable limits. When both cluster and unit intra-class correlation coefficients are very high (e.g., 0.95 or more), the performance of the empirical predictors derived under the three models is similar.     

A Mixed DG Method for Linearized Incompressible Magnetohydrodynamics

We introduce and analyze a discontinuous Galerkin method for the numerical discretization of a stationary incompressible magnetohydrodynamics model problem. The fluid unknowns are discretized with inf-sup stable discontinuous ? _k ³ ?? _k?1 elements whereas the magnetic part of the equations is approximated by discontinuous ? _k ³ ?? _k+1 elements. We carry out a complete a-priori error analysis of the method and prove that the energy norm error is convergent of order k in the mesh size. These results are verified in a series of numerical experiments.     

A new fuzzy logic control scheme for enhanced performance: simulation and experimental studies

In most fuzzy logic control schemes, the control signal is determined by using the error signal, e(.), and its first derivative, e̊ (.), as the input signals. In this paper, these inputs signals e(.) and e̊ (.) are modified respectively to e_m (.) and e̊_m (.), where e_m (.) is given by e(.) + k_me̊ (.) and k_m is a constant. This modification enables the performance of the overall system to be further improved through a proper choice of k_m . Moreover, the proposed scheme is simple and easy to implement as the number of signal to be measured remains unchanged after the modification. A non-linear power system is first chosen to demonstrate the effectiveness of the proposed scheme through simulations. The proposed scheme is specifically applied to enhance the damping of the mechanical mode oscillation of this non-linear power system. A real-time experimental study which involves the control of a servomotor is then performed.     

Randomized Priority Queues for Fast Parallel Access

We present simple randomized algorithms for parallel priority queues on distributed memory machines. Inserting O(n) elements or deleting the O(n) out ofmsmallest elements usingnprocessors requires O(T_coll+ log(m/n)) amortized time with high probability whereT_collbounds the time for performing prefix sums and randomized routing. The memory requirement is bounded by (m/n)(1 +o(1)) + O(logn) whp. These bounds are an improvement over the best previously known algorithms for many interconnection networks and even matches the speed of the best known PRAM algorithms. Generalizations for accessing thek⪢nsmallest elements are even more efficient. A portable implementation using MPI demonstrates that our approach is already useful for medium scale parallelism. Two parallel selection algorithms for randomly placed data are a spin-off. One runs in time O(T_coll) with high probability, beating a lower bound for the worst case. The other requires only a single reduction operation.     

Numerical resolution of discontinuous Galerkin methods for time dependent wave equations

The discontinuous Galerkin (DG) method is known to provide good wave resolution properties, especially for long time simulation. In this paper, using Fourier analysis, we provide a quantitative error analysis for the semi-discrete DG method applied to time dependent linear convection equations with periodic boundary conditions. We apply the same technique to show that the error is of order k + 2 superconvergent at Radau points on each element and of order 2k + 1 superconvergent at the downwind point of each element, when using piecewise polynomials of degree k. An analysis of the fully discretized approximation is also provided. We compute the number of points per wavelength required to obtain a fixed error for several fully discrete schemes. Numerical results are provided to verify our error analysis.     

A tabu search algorithm for the min–max k-Chinese postman problem

In this paper we present a tabu search algorithm for the min–max k-Chinese postman problem (MM k-CPP). Given an undirected edge-weighted graph and a distinguished depot node, the MM k  -CPP consists of finding k>1$k>1$ tours (starting and ending at the depot node) such that each edge is traversed by at least one tour and the length of the longest tour is minimized. A special emphasis is put on investigating the trade-off between running time effort and solution quality when applying different improvement procedures in the course of the neighborhood construction. Furthermore, different neighborhoods are analyzed. Extensive computational results show that the tabu search algorithm outperforms all known heuristics and improvement procedures.     

Model-based mean square error estimators for k-nearest neighbour predictions and applications using remotely sensed data for forest inventories

New model-based estimators of the uncertainty of pixel-level and areal k-nearest neighbour (k_nn) predictions of attribute Y from remotely-sensed ancillary data X are presented. Non-parametric functions predict Y from scalar ‘Single Index Model’ transformations of X. Variance functions generated estimates of the variance of Y. Three case studies, with data from the Forest Inventory and Analysis program of the U.S. Forest Service, the Finnish National Forest Inventory, and Landsat ETM+ ancillary data, demonstrate applications of the proposed estimators. Nearly unbiased k_nn predictions of three forest attributes were obtained. Estimates of mean square error indicate that k_nn is an attractive technique for integrating remotely-sensed and ground data for the provision of forest attribute maps and areal predictions.