controlling correlations in latin hypercube samples abstract monte carlo integration is competitive for high dimensional integrands latin hypercube sampling is a stratification technique that reduces the variance of the integral previous work has shown that the additive part of the integrand is integrated with error op n 1 2 under latin hypercube sampling with n integrand evaluations a bilinear part of the integrand is more accurately estimated if the sample correlations among input variables are negligible other authors have proposed an algorithm for controlling these correlations we show that their method reduces the correlations by roughly a factor of 3 for 10 n 500 we propose a method that based on simulations appears to produce correlations of order op n 3 2 an analysis of the algorithm indicates that it cannot be expected to do better than n 3 2

a comment on some recent results concerning the reverse order law for 1 3 4 inverses abstract this paper has been motivated by the one of liu and yang d liu h yang the reverse order law for 1 3 4 inverse of the product of two matrices appl math comp 215 12 2010 4293 4303 in which the authors consider separately the cases when ab 1 3 4 b 1 3 4 a 1 3 4 and ab 1 3 4 b 1 3 4 a 1 3 4 where a c n m and b c m n here we prove that ab 1 3 4 b 1 3 4 a 1 3 4 is actually equivalent to ab 1 3 4 b 1 3 4 a 1 3 4 we show that ab 1 3 4 b 1 3 4 a 1 3 4 can only be possible if n m and in this case we present purely algebraic necessary and sufficient conditions for this inclusion to hold also we give some new characterizations of b 1 3 4 a 1 3 4 ab 1 3 4

on parallel sum of operators abstract the basic properties of the parallel sum were developed for nonnegative matrices in finite dimensional spaces however without suitable restrictions very little of the preceding theories will hold for bounded linear operators a and b acting in hilbert space in this paper generalization to non selfadjoint operators is considered and various properties of parallel sum a b a a b b are given meanwhile the common upper and lower bounds of positive operators by using the parallel sum are given and some relationships between the parallel sum of projection operators are obtained

on design orthogonality maximin distance and projection uniformity for computer experiments space filling designs are widely used in both computer and physical experiments column orthogonality maximin distance and projection uniformity are three basic and popular space filling criteria

a general rotation method for orthogonal latin hypercubes summaryorthogonal latin hypercubes provide a class of useful designs for computer experiments among the available methods for constructing such designs the method of rotation is particularly prominent due to its theoretical appeal as well as its space filling properties this paper presents a general method of rotation for constructing orthogonal latin hypercubes making the rotation idea applicable to many more situations than the original method allows in addition to general theoretical results many new orthogonal latin hypercubes are obtained and tabulated

a new and flexible method for constructing designs for computer experiments we develop a new method for constructing good designs for computer experiments the method derives its power from its basic structure that builds large designs using small designs we specialize the method for the construc tion of orthogonal latin hypercubes and obtain many results along the way in terms of run sizes the existence problem of orthogonal latin hypercubes is completely solved we also present an explicit result showing how large or thogonal latin hypercubes can be constructed using small orthogonal latin hypercubes another appealing feature of our method is that it can easily be adapted to construct other designs we examine how to make use of the method to construct nearly orthogonal and cascading latin hypercubes

strong orthogonal arrays and associated latin hypercubes for computer experiments this paper introduces constructs and studies a new class of arrays called strong orthogonal arrays as suitable designs for computer experiments a strong orthogonal array of strength t enjoys better space filling properties than a comparable orthogonal array in all dimensions lower than t while retaining the space filling properties of the latter in t dimensions latin hypercubes based on strong orthogonal arrays of strength t are more space filling than comparable orthogonal array based latin hypercubes in all g dimensions for any 2 g t 1 copyright 2013 oxford university press

a general approach to analyzing epidemiologic data that contain misclassification errors misclassification is a common source of bias and reduced efficiency in the analysis of discrete data several methods have been proposed to adjust for misclassification using information on error rates i gathered by resampling the study population ii gathered by sampling a separate population or iii assumed a priori we present unified methods for incorporating these types of information into analyses based on log linear models and maximum likelihood estimation general variance expressions are developed examples from epidemiologic studies are used to demonstrate the proposed methodology

regression analysis for categorical variables with outcome subject to nonignorable nonresponse abstract we develop a log linear model for categorical response subject to nonignorable nonresponse the paper differs from fay 1986 in its focus on estimation and hypothesis testing in a regression setting as opposed to imputation in a multivariate setting we present several new results concerning the existence of solutions on the boundary of the parameter space and the construction of confidence intervals for estimates we illustrate the method by estimating the proportion of voters preferring truman in a 1948 preelection poll mosteller hyman mccarthy marks and truman 1949 results may depend strongly on the model assumed for nonresponse goodness of fit tests are available for comparing alternative models

space filling fractional factorial designs fractional factorial designs are widely used in various scientific investigations and industrial applications level permutation of factors could alter their geometrical structures and statistical properties this article studies space filling properties of fractional factorial designs under two commonly used space filling measures discrepancy and maximin distance when all possible level permutations are considered the average discrepancy is expressed as a linear combination of generalized word length pattern for fractional factorial designs with any number of levels and any discrepancy defined by a reproducing kernel generalized minimum aberration designs are shown to have good space filling properties on average in terms of both discrepancy and distance several novel relationships between distance distribution and generalized word length pattern are derived it is also shown that level permutations can improve space filling properties for many existing saturated designs a two step construction pro

further results on the reverse order law for the moore penrose inverse in rings with involution abstract we present some equivalent conditions of the reverse order law for the moore penrose inverse in rings with involution extending some well known results to more general settings then we apply this result to obtain a set of equivalent conditions to the reverse order rule for the weighted moore penrose inverse in c algebras

reverse order law for the group inverses this paper is to present some equivalent conditions concerning the reverse order law ab b a for the group invertible operators a b on a hilbert space h

space filling properties of good lattice point sets we study space filling properties of good lattice point sets and obtain some general theoretical results we show that linear level permutation does not decrease the minimum distance for good lattice point sets and we identify several classes of such sets with large minimum distance based on good lattice point sets some maximin distance designs are also constructed

analysis of multiple partially ordered responses to belief items with don t know option understanding beliefs values and preferences of patients is a tenet of contemporary health sciences this application was motivated by the analysis of multiple partially ordered set poset responses from an inventory on layman beliefs about diabetes the partially ordered set arises because of two features in the data first the response options contain a don t know dk option and second there were two consecutive occasions of measurement as predicted by the common sense model of illness beliefs about diabetes were not necessarily stable across the two measurement occasions instead of analyzing the two occasions separately we studied the joint responses across the occasions as a poset response few analytic methods exist for data structures other than ordered or nominal categories poset responses are routinely collapsed and then analyzed as either rank ordered or nominal data leading to the loss of nuanced information that might be present within poset categories in this paper we developed a general class of item response models for analyzing the poset data collected from the common sense model of diabetes inventory the inferential object of interest is the latent trait that indicates congruence of belief with the biomedical model to apply an item response model to the poset diabetes inventory we proved that a simple coding algorithm circumvents the requirement of writing new codes such that standard irt software could be directly used for the purpose of item estimation and individual scoring simulation experiments were used to examine parameter recovery for the proposed poset model

uniform projection designs efficient designs are in high demand in practice for both computer and physical experiments existing designs such as maximin distance designs and uniform designs may have bad low dimensional projections which is undesirable when only a few factors are active we propose a new design criterion called uniform projection criterion by focusing on projection uniformity uniform projection designs generated under the new criterion scatter points uniformly in all dimensions and have good space filling properties in terms of distance uniformity and orthogonality we show that the new criterion is a function of the pairwise l 1 distances between the rows so that the new criterion can be computed at no more cost than a design criterion that ignores projection properties we develop some theoretical results and show that maximin l 1 equidistant designs are uniform projection designs in addition a class of asymptotically optimal uniform projection designs based on good lattice point sets are constructed we further illustrate an application of uniform projection designs via a multidrug combination experiment

reverse order laws for least squares g inverses and minimum norm g inverses of products of two matrices in this paper we derive equivalent conditions for reverse order laws of least squares g inverses and minimum norm g inverses of product of two matrices a and b by applying the product singular value decomposition p svd

the absorption laws for the generalized inverses abstract in this paper we give necessary and sufficient conditions for the absorption laws in terms of 1 1 2 1 3 and 1 4 inverses also we consider the various types of mixed absorption law for the generalized inverses

miscellanea construction of orthogonal latin hypercube designs summary latin hypercube designs have found wide application such designs guarantee uniform samples for the marginal distribution of each input variable we propose a method for constructing orthogonal latin hypercube designs in which all the linear terms are orthogonal not only to each other but also to the quadratic terms this construction method is convenient and flexible and the resulting designs can accommodate many more factors than can existing ones

uniform design theory and application a uniform design ud seeks design points that are uniformly scattered on the domain it has been popular since 1980 a survey of ud is given in the first portion the fundamental idea and construction method are presented and discussed and examples are given for illustration it is shown that ud s have many desirable properties for a wide variety of applications furthermore we use the global optimization algorithm threshold accepting to generate ud s with low discrepancy the relationship between uniformity and orthogonality is investigated it turns out that most ud s obtained here are indeed orthogonal

two stage sensitivity based group screening in computer experiments sophisticated computer codes that implement mathematical models of physical processes can involve large numbers of inputs and screening to determine the most active inputs is critical for understanding the input output relationship this article presents a new two stage group screening methodology for identifying active inputs in stage 1 groups of inputs showing low activity are screened out in stage 2 individual inputs from the active groups are identified inputs are evaluated through their estimated total effect sensitivity indices tsis which are compared with a benchmark null tsi distribution created from added low noise inputs examples show that compared with other procedures the proposed method provides more consistent and accurate results for high dimensional screening additional details and computer code are provided in supplementary materials available online

a general approach to categorical data analysis with missing data using generalized linear models with composite links a general approach for analyzing categorical data when there are missing data is described and illustrated the method is based on generalized linear models with composite links the approach can be used among other applications to fill in contingency tables with supplementary margins fit loglinear models when data are missing fit latent class models without or with missing data on observed variables fit models with fused cells including many models from genetics and to fill in tables or fit models to data when variables are more finely categorized for some cases than others both newton like and em methods are easy to implement for parameter estimation

noncollapsing space filling designs for bounded nonrectangular regions many researchers use computer simulators as experimental tools especially when physical experiments are infeasible when computer codes are computationally intensive nonparametric predictors can be fitted to training data for detailed exploration of the input output relationship the accuracy of such flexible predictors is enhanced by taking training inputs to be space filling if there are inputs that have little or no effect on the response it is desirable that the design be noncollapsing in the sense of having space filling lower dimensional projections this article describes an algorithm for constructing noncollapsing space filling designs for bounded input regions that are of possibly high dimension online supplementary materials provide the code for the algorithm examples of its use and show its performance in multiple settings

system identification via cur factored hankel approximation subspace based system identification for dynamical systems is a sound system theoretic way to obtain linear time invariant system models from data the interplay of data and systems theory is ref

maximum projection designs for computer experiments summary space filling properties are important in designing computer experiments the traditional maximinandminimaxdistancedesignsconsideronlyspace fillinginthefull dimensionalspace thiscanresultinpoorprojectionsontolower dimensionalspaces whichisundesirablewhenonly af ew factors are active restricting maximin distance design to the class of latin hypercubes can improveone dimensionalprojectionsbutcannotguaranteegoodspace fillingpropertiesinlarger subspaces we propose designs that maximize space filling properties on projections to all subsets of factors we call our designs maximum projection designs our design criterion can be computed at no more cost than a design criterion that ignores projection properties

orthogonal and nearly orthogonal designs for computer experiments we introduce a method for constructing a rich class of designs that are suitable for use in computer experiments the designs include latin hypercube designs and two level fractional factorial designs as special cases and fill the vast vacuum between these two familiar classes of designs the basic construction method is simple building a series of larger designs based on a given small design if the base design is orthogonal the resulting designs are orthogonal likewise if the base design is nearly orthogonal the resulting designs are nearly orthogonal we present two generalizations of our basic construction method the first generalization improves the projection properties of the basic method the second generalization gives rise to designs that have smaller correlations sample constructions are presented and properties of these designs are discussed

moore penrose inverse in rings with involution abstract we study the moore penrose inverse mp inverse in the setting of rings with involution the results include the relation between regular mp invertible and well supported elements we present an algebraic proof of the reverse order rule for the mp inverse valid under certain conditions on mp invertible elements applications to c algebras are given

when is b a a generalized inverse of ab in practice factorizations of a generalized inverse often arise from factorizations of the matrix which is to be inverted in addition to full rank factorizations normal factorizations and singular value decompositions svd there are other factorizations of particular matrices that are natural to certain problems for instance in statistics answers to questions like when is b a a generalized inverse of ab may thus be a computational tool besides they are of significant interest because they provide us with insights into the g inversion of matrix products

strong orthogonal arrays of strength two plus strong orthogonal arrays were recently introduced and studied in he and tang biometrika100 2013 254 260 as a class of space filling designs for computer experiments to enjoy the benefits of better space filling properties when compared to ordinary orthogonal arrays strong orthogonal arrays need to have strength three or higher which may require run sizes that are too large for experimenters to afford to address this problem we introduce a new class of arrays called strong orthogonal arrays of strength two plus these arrays while being more economical than strong orthogonal arrays of strength three still enjoy the better two dimensional space filling property of the latter among the many results we have obtained on the characterizations and constructions of strong orthogonal arrays of strength two plus worth special mention is their intimate connection with second order saturated designs

reverse order law for the moore penrose inverse abstract in this paper we present new results related to the reverse order law for the moore penrose inverse of operators on hilbert spaces some finite dimensional results are extended to infinite dimensional settings

a method of constructing space filling orthogonal designs abstractthis article presents a method of constructing a rich class of orthogonal designs that include orthogonal latin hypercubes as special cases two prominent features of the method are its simplicity and generality in addition to orthogonality the resulting designs enjoy some attractive space filling properties making them very suitable for computer experiments

using rank formulas to characterize equalities for moore penrose inverses of matrix products a group of rank formulas related to the moore penrose inverses ab and b a are established various interesting consequences of these rank formulas are obtained the extensions of these results to the weighted moore penrose inverses ab m n and b p na m p are considered

the reverse order law revisited abstract necessary and sufficient conditions are given for the triple reverse order law abc c b a to hold some special cases are considered

construction of maximin distance latin squares and related latin hypercube designs summarymaximin distance latin hypercube designs are widely used in computer experiments yet their construction is challenging based on number theory and finite fields we propose three algebraic methods to construct maximin distance latin squares as special latin hypercube designs we develop lower bounds on their minimum distances the resulting latin squares and related latin hypercube designs have larger minimum distances than existing ones and are especially appealing for high dimensional applications

orthogonal array based latin hypercubes abstract in this article we use orthogonal arrays oa s to construct latin hypercubes besides preserving the univariate stratification properties of latin hypercubes these strength r oa based latin hypercubes also stratify each r dimensional margin therefore such oa based latin hypercubes provide more suitable designs for computer experiments and numerical integration than do general latin hypercubes we prove that when used for integration the sampling scheme with oa based latin hypercubes offers a substantial improvement over latin hypercube sampling

optimal sliced latin hypercube designs sliced latin hypercube designs slhds have important applications in designing computer experiments with continuous and categorical factors however a randomly generated slhd can be poor in terms of space filling and based on the existing construction method that generates the slhd column by column using sliced permutation matrices it is also difficult to search for the optimal slhd in this article we develop a new construction approach that first generates the small latin hypercube design in each slice and then arranges them together to form the slhd the new approach is intuitive and can be easily adapted to generate orthogonal slhds and orthogonal array based slhds more importantly it enables us to develop general algorithms that can search for the optimal slhd efficiently

reverse order laws for the generalized inverses of multiple matrix products by using the ranks of matrices this article gives necessary and sufficient conditions for the reverse product a n a 2a 1of the moore penrose inverses of a1 a2 an to a 1 inverse inner inverse 1 2 inverse reflexive inner inverse 1 3 inverse 1 4 inverse 1 2 3 inverse 1 2 4 inverse or moore penrose inverse of matrix product a a1a2 an

on the connection between maximin distance designs and orthogonal designs summarymaximin distance designs and orthogonal designs are widely used in computer and physical experiments we characterize a broad class of maximin distance designs by establishing new bounds on the minimum intersite distance for mirror symmetric and general u type designs we show that maximin distance designs and orthogonal designs are closely related and coincide under some conditions

orthogonal column latin hypercubes and their application in computer experiments abstract latin hypercubes have been frequently used in conducting computer experiments in this paper a class of orthogonal latin hypercubes that preserves orthogonality among columns is proposed applying an orthogonal latin hypercube design to a computer experiment benefits the data analysis in two ways first it retains the orthogonality of traditional experimental designs the estimates of linear effects of all factors are uncorrelated not only with each other but also with the estimates of all quadratic effects and bilinear interactions second it can facilitate nonparametric fitting procedures because one can select good space filling designs within the class of orthogonal latin hypercubes according to selection criteria

log linear models for doubly sampled categorical data fitted by the em algorithm abstract double sampling experiments can be expressed as incomplete multiway contingency tables and analyzed by using techniques appropriate for fitting log linear models the framework described can be applied to experiments with any number of factors and measurement devices maximum likelihood estimation via the em algorithm leads to straightforward expressions for covariances of estimates of the parameters and functions of the parameters

reverse order law in c algebras we study equivalent conditions for the reverse order law a1a2 an an a1 a1a2 anan a1 in c algebras as corollaries we obtain some recent and special results

identities concerning the reverse order law for the moore penrose inverse we prove some identities related to the reverse order law for the moore penrose inverse of operators on hilbert spaces extending some results from y tian s cheng linear multilinear algebra 52 2004 and r e cline siam rev vol 6 no 1 1964 to infinite dimensional settings

reverse order laws for the drazin inverses abstract this paper is to present some equivalent conditions concerning the reverse order law p q d q d p d for the drazin invertible operators p and q with partial commutative properties p p q 0 or q p q 0 or p p q q d 0 or q p p d q 0

multilevel and latent variable modeling with composite links and exploded likelihoods composite links and exploded likelihoods are powerful yet simple tools for specifying a wide range of latent variable models applications considered include survival or duration models models for rankings small area estimation with census information models for ordinal responses item response models with guessing randomized response models unfolding models latent class models with random effects multilevel latent class models models with log normal latent variables and zero inflated poisson models with random effects some of the ideas are illustrated by estimating an unfolding model for attitudes to female work participation

analysis of longitudinal binary data from multiphase sampling the efficient use of surrogate or auxiliary information has been investigated within both model based and design based approaches to data analysis particularly in the context of missing data here we consider the use of such data in epidemiological studies of disease incidence in which surrogate measures of disease status are available for all subjects at two time points but definitive diagnoses are available only in stratified subsamples we briefly review methods for the analysis of two phase studies of disease prevalence at a single time point and we discuss the extension of four of these methods to the analysis of incidence studies their performance is compared with special reference to a study of the incidence of senile dementia