contour regression a general approach to dimension reduction we propose a novel approach to sufficient dimension reduction in regression based on estimating contour directions of small variation in the response these directions span the orthogonal complement of the minimal space relevant for the regression and can be extracted according to two measures of variation in the response leading to simple and general contour regression scr and gcr methodology in comparison with existing sufficient dimension reduction techniques this contour based methodology guarantees exhaustive estimation of the central subspace under ellipticity of the predictor distribution and mild additional assumptions while maintaining sqrt n consistency and computational ease moreover it proves robust to departures from ellipticity we establish population properties for both scr and gcr and asymptotic properties for scr simulations to compare performance with that of standard techniques such as ordinary least squares sliced inverse regression principal hessian directions and sliced average variance estimation confirm the advantages anticipated by the theoretical analyses we demonstrate the use of contour based methods on a data set concerning soil evaporation

on principal hessian directions for data visualization and dimension reduction another application of stein s lemma abstract modern graphical tools have enhanced our ability to learn many things from data directly with much user friendly graphical software available we are encouraged to plot a lot more often than before the benefits from direct interaction with graphics have been enormous but trailing behind these high tech advances is the issue of appropriate guidance on what to plot there are too many directions to project a high dimensional data set and unguided plotting can be time consuming and fruitless in a recent article li set up a statistical framework for study on this issue based on a notion of effective dimension reduction edr directions they are the directions to project a high dimensional input variable for the purpose of effectively viewing and studying its relationship with an output variable a methodology sliced inverse regression was introduced and shown to be useful in finding edr directions this article introduces another method for finding edr directions it begins with the observat

a decomposed equation for local entropy and entropy production in volume preserving coarse grained systems abstract in this study an equation for the local entropy is derived based on the formulation of a master equation and is applied to volume preserving maps the equation consists of the following terms unsteady convection diffusion probability weighted phase space volume expansion rate nonnegative entropy production and residuals the decomposition makes it possible to evaluate entropy production in terms of microscopic dynamics and is expected to be applicable to many coarse grained systems on the phase space when it is applied to two volume preserving multibaker chain systems it is confirmed that the summation of the nonnegative entropy production on each site numerically coincides with the entropy production introduced by gilbert et al t gilbert j r dorfman p gaspard entropy production fractals and relaxation to equilibrium phys rev lett 85 2000 1606 1609 and the phenomenological expression both in nonequilibrium steady and unsteady states the coincidence is brought about by the fact that the residual terms vanish in the thermodynamic limit when they are integrated on each site it follows that the entropy production is dominated by the nonnegative entropy production term and becomes positive in nonequilibrium states

dimension reduction and semiparametric estimation of survival models in this paper we propose a new dimension reduction method by introducing a nominal regression model with the hazard function as the conditional mean which naturally retrieves information from complete data and censored data as well moreover without requiring the linearity condition the new method can estimate the entire central subspace consistently and exhaustively the method also provides an alternative approach for the analysis of censored data assuming neither the link function nor the distribution hence it exhibits superior robustness properties numerical studies show that the method can indeed be readily used to efficiently estimate survival models explore the data structures and identify important variables

quantum chaos a decoherent definition abstract we show that the rate of increase of von neumann entropy computed from the reduced density matrix of an open quantum system is an excellent indicator of the dynamical behavior of its classical hamiltonian counterpart in decohering quantum analogs of systems which exhibit classical hamiltonian chaos entropy production rate quickly tends to a constant which is given by the sum of the positive lyapunov exponents and falls off only as the system approaches equilibrium by contrast integrable systems tend to have an entropy production rate which decreases as t 1 well before equilibrium is attained thus behavior of quantum systems in contact with the environment can be used as a test to determine the nature of their hamiltonian evolution

testing predictor contributions in sufficient dimension reduction we develop tests of the hypothesis of no effect for selected predictors in regression without assuming a model for the conditional distribution of the response given the predictors predictor effects need not be limited to the mean function and smoothing is not required the general approach is based on sufficient dimension reduction the idea being to replace the predictor vector with a lower dimensional version without loss of information on the regression methodology using sliced inverse regression is developed in detail

on the use of reducible polynomials as random number generators the randomness properties and the hierarchy of correlation coefficients are studied of approximate maximum length sequences for which the characteristic polynomial is a product of several primitive polynomials the randomness properties are almost the same as for maximum length sequences characterized by a primitive polynomial with many terms and of the same degree reducible characteristic polynomials have acceptable figures of merit and can be of extremely high degree since they are also easily constructed and implemented reducible polynomials are strong candidates for reliable random number generation especially at the bit rates needed in large scale monte carlo simulations

dimension reduction for the conditional mean in regressions with categorical predictors consider the regression of a response y on a vector of quantitative predictors x and a categorical predictor w in this article we describe a first method for reducing the dimension of x without loss of information on the conditional mean mathrm e y x w and without requiring a prespecified parametric model the method which allows for but does not require parametric versions of the subpopulation mean functions mathrm e y x w w includes a procedure for inference about the dimension of x after reduction this work integrates previous studies on dimension reduction for the conditional mean mathrm e y x in the absence of categorical predictors and dimension reduction for the full conditional distribution of y x w the methodology we describe may be particularly useful for constructing low dimensional summary plots to aid in model building at the outset of an analysis our proposals provide an often parsimonious alternative to the standard technique of modeling with interaction terms to adapt a mean function for different subpopulations determined by the levels of w examples illustrating this and other aspects of the development are presented

combined random number generator via the generalized chinese remainder theorem the combined random number rn generator has been considered by many scholars as a good rn generator one promising type of combined rn generator recommended by l ecuyer oper res 44 1996 816 47 1999 159 is the combined multiple recursive generator mrg this paper analyzes the combined mrg via the chinese remainder theorem a new combined generator based on the generalized chinese remainder theorem and on the ore algorithm amer math monthly 59 1952 365 is presented the proposed combined generator improves the combined mrg in terms of both the suitability for various types of rn generators and the restriction on the moduli of the individual generators therefore the proposed combined generator is an ideal rn generator and is most recommended

sufficient dimension reduction for longitudinally measured predictors we propose a method to combine several predictors markers that are measured repeatedly over time into a composite marker score without assuming a model and only requiring a mild condition on the predictor distribution assuming that the first and second moments of the predictors can be decomposed into a time and a marker component via a kronecker product structure that accommodates the longitudinal nature of the predictors we develop first moment sufficient dimension reduction techniques to replace the original markers with linear transformations that contain sufficient information for the regression of the predictors on the outcome these linear combinations can then be combined into a score that has better predictive performance than a score built under a general model that ignores the longitudinal structure of the data our methods can be applied to either continuous or categorical outcome measures in simulations we focus on binary outcomes and show that our method outperforms existing alternatives by using the auc the area under the receiver operator characteristics roc curve as a summary measure of the discriminatory ability of a single continuous diagnostic marker for binary disease outcomes published 2011 this article is a us government work and is in the public domain in the usa

groupwise dimension reduction via envelope method the family of sufficient dimension reduction sdr methods that produce informative combinations of predictors or indices are particularly useful for high dimensional regression analysis in many such analyses it becomes increasingly common that there is available a priori subject knowledge of the predictors for example they belong to different groups while many recent sdr proposals have greatly expanded the scope of the methods applicability how to effectively incorporate the prior predictor structure information remains a challenge in this article we aim at dimension reduction that recovers full regression information while preserving the predictor group structure built upon a new concept of the direct sum envelope we introduce a systematic way to incorporate the group information in most existing sdr estimators as a result the reduction outcomes are much easier to interpret moreover the envelope method provides a principled way to build a variety of prior structures into dimension reduc

entropy production from open volume preserving to dissipative systems we generalize gaspard s method for computing the e entropy production rate in hamiltonian systems to dissipative systems with attractors considered earlier by tel vollmer and breymann this approach leads to a natural definition of a coarse grained gibbs entropy which is extensive and which can be expressed in terms of the srb measures and volumes of the coarse graining sets which cover the attractor one can also study the entropy and entropy production as functions of the degree of resolution of the coarse graining process and examine the limit as the coarse graining size approaches zero we show that this definition of the gibbs entropy leads to a positive rate of irreversible entropy production for reversible dissipative systems we apply the method to the case of a two dimensional map based upon a model considered by vollmer tel and breymann that is a deterministic version of a biased random walk we treat both volume preserving and dissipative versions of the basic map and make a comparison between the two cases we discuss the e entropy production rate as a function of the size of the coarse graining cells for these biased random walks and for an open system with flux boundary conditions show regions of exponential growth and decay of the rate of entropy production as the size of the cells decreases this work describes in some detail the relation between the results of gaspard those of of tel vollmer and breymann and those of ruelle on entropy production in various systems described by anosov or anosov like maps

iterative simulation methods abstract the standard methods of generating sample from univariate distributions often become hopelessly inefficient when applied to realizations of stochastic processes iterative methods are not widely known amongst statisticians but some are standard practice in statistical physics and chemistry the methods are surveyed and compared with particular reference to their convergence properties

initializing generalized feedback shift register pseudorandom number generators the generalized feedback shift register pseudorandom number generators proposed by lewis and payne provide a very attractive method of random number generation unfortunately the published initialization procedure can be extremely time consuming this paper considers an alternative method of initialization based on a natural polynomial representation for the terms of a feedback shift register sequence that results in substantial improvements in the initialization process

sliced inverse regression for dimension reduction abstract modern advances in computing power have greatly widened scientists scope in gathering and investigating information from many variables information which might have been ignored in the past yet to effectively scan a large pool of variables is not an easy task although our ability to interact with data has been much enhanced by recent innovations in dynamic graphics in this article we propose a novel data analytic tool sliced inverse regression sir for reducing the dimension of the input variable x without going through any parametric or nonparametric model fitting process this method explores the simplicity of the inverse view of regression that is instead of regressing the univariate output variable y against the multivariate x we regress x against y forward regression and inverse regression are connected by a theorem that motivates this method the theoretical properties of sir are investigated under a model of the form y f 1 x k x e where the k s are the unknown

martingale difference correlation and its use in high dimensional variable screening in this article we propose a new metric the so called martingale difference correlation to measure the departure of conditional mean independence between a scalar response variable v and a vector predictor variable u our metric is a natural extension of distance correlation proposed by szekely rizzo and bahirov which is used to measure the dependence between v and u the martingale difference correlation and its empirical counterpart inherit a number of desirable features of distance correlation and sample distance correlation such as algebraic simplicity and elegant theoretical properties we further use martingale difference correlation as a marginal utility to do high dimensional variable screening to screen out variables that do not contribute to conditional mean of the response given the covariates further extension to conditional quantile screening is also described in detail and sure screening properties are rigorously justified both simulation results and real data illustrations demonstr

optimal sufficient dimension reduction for the conditional mean in multivariate regression summary the aim of this article is to develop optimal sufficient dimension reduction methodology for the conditional mean in multivariate regression the context is roughly the same as that of a related method by cook setodji 2003 but the new method has several advantages it is asymptotically optimal in the sense described herein and its test statistic for dimension always has a chi squared distribution asymptotically under the null hypothesis additionally the optimal method allows tests of predictor effects a comparison of the two methods is provided

newton noda iteration for finding the perron pair of a weakly irreducible nonnegative tensor we present a newton noda iteration nni for computing the perron pair of a weakly irreducible nonnegative mth order tensor mathscr a a by combining the idea of newton s method with the idea of the noda iteration the method requires the selection of a positive parameter theta k k in the kth iteration and produces a scalar sequence approximating the spectral radius of mathscr a a and a positive vector sequence approximating the perron vector we propose a halving procedure to determine the parameters theta k k starting with theta k 1 k 1 for each k such that the scalar sequence is monotonically decreasing convergence of this sequence to the spectral radius of mathscr a a and convergence of the vector sequence to the perron vector is guaranteed for any initial positive unit vector as long as the sequence theta k k so chosen is bounded below by a positive constant in this case we always have theta k 1 k 1 near convergence and the convergence is quadratic very often the halving procedure will return theta k 1 k 1 i e no halving is actually used for each k if the tensor is semisymmetric m ge 4 m 4 and theta k 1 k 1 then the computational work in the kth iteration of nni is roughly the same as that for one iteration of the ng qi zhou algorithm which is linearly convergent for the smaller class of weakly primitive tensors

monkey tests for random number generators this article describes some very simple as well as some quite sophisticated tests that shed light on the suitability of certain random number generators

some comments on computer implementation of random number generators abstract several workers in simulation studies are concerned with the correct and portable computer implementation of random number generators since approximate calculations may lead to accumulated errors in various machines all implementations must be exact in a previous work gentle this journal 1990 presented several implementations for multiplicative congruential generators the aim of this paper is to describe some more methods especially the implementation using the decomposition of the modulus is very efficient

dimension reduction in binary response regression abstract the idea of dimension reduction without loss of information can be quite helpful for guiding the construction of summary plots in regression without requiring a prespecified model focusing on the central subspace we investigate such sufficient dimension reduction in regressions with a binary response three existing methods sliced inverse regression principal hessian direction and sliced average variance estimation and one new method difference of covariances are studied for their ability to estimate the central subspace and produce sufficient summary plots combining these numerical methods with the graphical methods proposed earlier by cook leads to a novel paradigm for the analysis of binary response regressions

random number generation with the recursion x t x t 3p x t 3q abstract a generalized feedback shift register gfsr algorithm proposed by lewis and payne 1973 uses a primitive trinomial to generate a sequence of pseudorandom numbers we propose a similar algorithm which uses a primitive polynomial with many nonzero terms but generates a number as fast as the original gfsr algorithm our sequence is guaranteed to be equidistributed in higher dimensions and to have a good autocorrelation property extensive statistical tests have been performed on the sequences generated by our algorithm and the results were quite satisfactory

compound random number generators abstract recently marsaglia 1984 proceedings of the sixteenth symposium on the interface noted a trend toward increasingly more demanding use of random number generators after observing that for current uses generators must pass increasingly more sophisticated tests for randomness in order to provide acceptable results marsaglia described a class of combination generators and recommended two specific combination generators that passed an extensive battery of stringent tests for randomness the present article describes an alternative class of generators which produce random values by compounding or intermixing several different sequences of random values these compound generators are easier to implement especially on smaller computers than are marsaglia s recommended generators five specific compound generators two standard generators and marsaglia s two combination generators were subjected to several tests for randomness including marsaglia s stringent tests the standard generator

on estimation efficiency of the central mean subspace summary we investigate the estimation efficiency of the central mean subspace in the framework of sufficient dimension reduction we derive the semiparametric efficient score and study its practical applicability despite the difficulty caused by the potential high dimension issue in the variance component we show that locally efficient estimators can be constructed in practice we conduct simulation studies and a real data analysis to demonstrate the finite sample performance and gain in efficiency of the proposed estimators in comparison with several existing methods

revaluation of the first order upwind difference scheme to solve coarse grained master equations this study addresses the initial boundary value problem of coarse grained probability measure on the state space in which a differentiable vector field v is given and as a consequence the differenced continuity equation using the first order upwind difference scheme uds based on the finite volume method appears as the physical substance on the coarse grained dynamics at first the uds is theoretically shown to be equivalent to a class of coarse grained master equations cgme brought by a principle that we cannot distinguish state points in the same partition with each other the principle is based on the formulation of non equilibrium statistical mechanics to resolve the macroscopic irreversibility moreover the entropy production evaluated by the uds is also shown to be in accord with the average volume contraction rate in the steady state this is essential for the non equilibrium statistical dynamics and was numerically confirmed under the principle of coarse graining the uds is very superior to the conventional monte carlo method in computer time and storage and is very useful to solve the cgme

inference on the primary parameter of interest with the aid of dimension reduction estimation summary as high dimensional data become routinely available in applied sciences sufficient dimension reduction has been widely employed and its research has received considerable attention however with the majority of sufficient dimension reduction methodology focusing on the dimension reduction step complete analysis and inference after dimension reduction have yet to receive much attention we couple the strategy of sufficient dimension reduction with a flexible semiparametric model we concentrate on inference with respect to the primary variables of interest and we employ sufficient dimension reduction to bring down the dimension of the regression effectively extensive simulations demonstrate the efficacy of the method proposed and a real data analysis is presented for illustration

simulated division with approximate factoring for the multiple recursive generator with both unrestricted multiplier and non mersenne prime modulus abstract this paper focuses on devising a general and efficient way of generating random numbers for the multiple recursive generator with both unrestricted multiplier and non mersenne prime modulus we propose a new algorithm that embeds the technique of approximate factoring into the simulated division method the proposed new algorithm improves the decomposition method in terms of both the suitability for various word sizes of the computers and the efficiency characteristics such as the number of arithmetic operations required and the computational time empirical simulations are conducted to compare and evaluate the computational time of this algorithm with the decomposition method for various computers

principal hessian directions revisited abstract li has suggested the method of principal hessian directions for estimating plotting directions that capture curvature in regression functions this article revisits li s proposal offering a number of suggestions for improved application

dimension reduction for conditional mean in regression in many situations regression analysis is mostly concerned with inferring about the conditional mean of the response given the predictors and less concerned with the other aspects of the conditional distribution in this paper we develop dimension reduction methods that incorporate this consideration we introduce the notion of the central mean subspace cms a natural inferential object for dimension reduction when the mean function is of interest we study properties of the cms and develop methods to estimate it these methods include a new class of estimators which requires fewer conditions than phd and which displays a clear advantage when one of the conditions for phd is violated cms also reveals a transparent distinction among the existing methods for dimension reduction ols phd sir and save we apply the new methods to a data set involving recumbent cows

determining the dimension of iterative hessian transformation the central mean subspace cms and iterative hessian transformation iht have been introduced recently for dimension reduction when the conditional mean is of interest suppose that x is a vector valued predictor and y is a scalar response the basic problem is to find a lower dimensional predictor t x such that e y x e y t x the cms defines the inferential object for this problem and iht provides an estimating procedure compared with other methods iht requires fewer assumptions and has been shown to perform well when the additional assumptions required by those methods fail in this paper we give an asymptotic analysis of iht and provide stepwise asymptotic hypothesis tests to determine the dimension of the cms as estimated by iht here the original iht method has been modified to be invariant under location and scale transformations to provide empirical support for our asymptotic results we will present a series of simulation studies these agree well with the theory the method is applied to analyze an ozone data set

effective and efficient restriction on producing the multipliers for the multiple recursive random number generator abstract the most efficient and easiest way for implementing a multiple recursive random number generator is the approximate factoring af method in the literature the af method can be applied in the three types of restriction on multiplier this paper considers the effectiveness and efficiency of these restrictions from the theoretical and empirical points of view the most effective and efficient restriction is presented the number of arithmetic operations required by the af method is also discussed

optimal sufficient dimension reduction in regressions with categorical predictors though partial sliced inverse regression partial sir chiaromonte et al 2002 sufficient dimension reduction in regressions with categorical predictors ann statist 30 475 497 extended the scope of sufficient dimension reduction to regressions with both continuous and categorical predictors its requirement of homogeneous predictor covariances across the subpopulations restricts its application in practice when this condition fails partial sir may provide misleading results in this article we propose a new estimation method via a minimum discrepancy approach without this restriction our method is optimal in terms of asymptotic efficiency and its test statistic for testing the dimension of the partial central subspace always has an asymptotic chi squared distribution it also gives us the ability to test predictor effects an asymptotic chi squared test of the conditional independence hypothesis that the response is independent of a selected subset of the continuous predictors given the remaining predictors is obtained

a model free test for reduced rank in multivariate regression we propose a test of dimension in multivariate regression this test is in the spirit of tests on the rank of the coefficient matrix in a multivariate linear model but it does not require a prespecified model the test may be particularly useful at the outset of an analysis before a multivariate model is posited because it can lead to low dimensional summary plots that are inferred to contain all of the sample information on the multivariate mean function

a positivity preserving inexact noda iteration for computing the smallest eigenpair of a large irreducible m m matrix in this paper based on the noda iteration we present inexact noda iterations ini to find the smallest eigenvalue and the associated positive eigenvector of a large irreducible nonsingular m m matrix the positivity of approximations is critical in applications and if the approximations lose the positivity then they may be meaningless and could not be interpreted we propose two different inner tolerance strategies for solving the inner linear systems involved and prove that the convergence of resulting ini algorithms is globally linear and superlinear with the convergence order frac 1 sqrt 5 2 1 52 respectively the proposed ini algorithms are structure preserving and maintains the positivity of approximate eigenvectors we also revisit the exact noda iteration and establish a new quadratic convergence result all the above is first done for the problem of computing the perron root and the positive perron vector of an irreducible nonnegative matrix and is then adapted to computing the smallest eigenpair of the irreducible nonsingular m m matrix numerical examples illustrate that the proposed ini algorithms are practical and they always preserve the positivity of approximate eigenvectors we compare them with the jacobi davidson method the implicitly restarted arnoldi method and the explicitly restarted krylov schur method all of which cannot guarantee the positivity of approximate eigenvectors and illustrate that the overall efficiency of the ini algorithms is competitive with and can be considerably higher than the latter three methods

entropy production in open volume preserving systems we describe a mechanism leading to positive entropy production in volume preserving systems under nonequilibrium conditions we consider volume preserving systems sustaining a diffusion process like the multibaker map or the lorentz gas a continuous flux of particles is imposed across the system resulting in a steady gradient of concentration in the limit where such flux boundary conditions are imposed at arbitrarily separated boundaries for a fixed gradient the invariant measure becomes singular for instance in the multibaker map the limit invariant measure has a cumulative function given in terms of the nondifferentiable takagi function because of this singularity of the invariant measure the entropy must be defined as a coarse grained entropy instead of the fined grained gibbs entropy which would require the existence of a regular measure with a density the coarse grained entropy production is then shown to be asymptotically positive and moreover given by the entropy production expected from irreversible thermodynamics

diagnostic studies in sufficient dimension reduction sufficient dimension reduction in regression aims to reduce the predictor dimension by replacing the original predictors with some set of linear combinations of them without loss of information numerous dimension reduction methods have been developed based on this paradigm however little effort has been devoted to diagnostic studies within the context of dimension reduction in this paper we introduce methods to check goodness of fit for a given dimension reduction subspace the key idea is to extend the so called distance correlation to measure the conditional dependence relationship between the covariates and the response given a reduction subspace our methods require minimal assumptions which are usually much less restrictive than the conditions needed to justify the original methods asymptotic properties of the test statistic are studied numerical examples demonstrate the effectiveness of the proposed approach

sparse serial tests of uniformity for random number generators different versions of the serial test for testing the uniformity and independence of vectors of successive values produced by a pseudo random number generator are studied these tests partition the t dimensional unit hypercube into k cubic cells of equal volume generate n points vectors in this hypercube count how many points fall in each cell and compute a test statistic defined as the sum of values of some univariate function f applied to these k individual counters both overlapping and nonoverlapping vectors are considered for different families of generators such as linear congruential tausworthe nonlinear inversive etc different ways of choosing these functions and of choosing k are compared and formulas are obtained for the estimated sample size required to reject the null hypothesis of independent uniform random variables as a function of the period length of the generator for the classes of alternatives that correspond to linear generators the most efficient tests turn out to have k gg n in contrast to what is usually done or recommended in simulation books and to use overlapping vectors

close point spatial tests and their application to random number generators we study statistical tests of uniformity based on thel p distances between them nearest pairs of points forn points generated uniformly over thek dimensional unit hypercube or unit torus the number of distinct pairs at distance no more thant fort 0 is a stochastic process whose initial part after an appropriate transformation and asn 8 is asymptotically a poisson process with unit rate convergence to this asymptotic is slow in the hypercube as soon ask exceeds 2 or 3 due to edge effects but is reasonably fast in the torus we look at the quality of approximation of the exact distributions of the tests statistics by their asymptotic distributions discuss computational issues and apply the tests to random number generators linear congruential generators fail decisively certain variants of the tests as soon asn approaches the square root of the period length

on partial sufficient dimension reduction with applications to partially linear multi index models partial dimension reduction is a general method to seek informative convex combinations of predictors of primary interest which includes dimension reduction as its special case when the predictors in the remaining part are constants in this article we propose a novel method to conduct partial dimension reduction estimation for predictors of primary interest without assuming that the remaining predictors are categorical to this end we first take the dichotomization step such that any existing approach for partial dimension reduction estimation can be employed then we take the expectation step to integrate over all the dichotomic predictors to identify the partial central subspace as an example we use the partially linear multi index model to illustrate its applications for semiparametric modeling simulations and real data examples are given to illustrate our methodology

thoughts on pseudorandom number generators abstract much of the informal discussion at the workshop concerned the merits of different pseudorandom number generators here we record some comments based on comparing generators across a wide range of machines

estimating the structural dimension of regressions via parametric inverse regression a new estimation method for the dimension of a regression at the outset of an analysis is proposed a linear subspace spanned by projections of the regressor vector x which contains part or all of the modelling information for the regression of a vector y on x and its dimension are estimated via the means of parametric inverse regression smooth parametric curves are fitted to the p inverse regressions via a multivariate linear model no restrictions are placed on the distribution of the regressors the estimate of the dimension of the regression is based on optimal estimation procedures a simulation study shows the method to be more powerful than sliced inverse regression in some situations

sufficient dimension reduction via inverse regression a family of dimension reduction methods the inverse regression ir family is developed by minimizing a quadratic objective function an optimal member of this family the inverse regression estimator ire is proposed along with inference methods and a computational algorithm the ire has at least three desirable properties 1 its estimated basis of the central dimension reduction subspace is asymptotically efficient 2 its test statistic for dimension has an asymptotic chi squared distribution and 3 it provides a chi squared test of the conditional independence hypothesis that the response is independent of a selected subset of predictors given the remaining predictors current methods like sliced inverse regression belong to a suboptimal class of the ir family comparisons of these methods are reported through simulation studies the approach developed here also allows a relatively straightforward derivation of the asymptotic null distribution of the test statistic for dimension used in slic

groupwise dimension reduction in many regression applications the predictors fall naturally into a number of groups or domains and it is often desirable to establish a domain specific relation between the predictors and the response in this article we consider dimension reduction that incorporates such domain knowledge the proposed method is based on the derivative of the conditional mean where the differential operator is constrained to the form of a direct sum this formulation also accommodates the situations where dimension reduction is focused only on part of the predictors as such it extends partial dimension reduction to cases where the blocked predictors are continuous through simulation and real data analyses we show that the proposed method achieves greater accuracy and interpretability than the dimension reduction methods that ignore group information furthermore the new method does not require the stringent conditions on the predictor distribution that are required by existing methods

matrices and the structure of random number sequences abstract this note provides a short self contained treatment using linear algebra and matrix theory for establishing maximal periods underlying structure and choice of starting values for shift register and lagged fibonacci random number generators

graphics for regressions with a binary response abstract central dimension reduction subspaces which characterize the dependence of a response variable on one or more predictors are developed and then used to guide the construction and interpretation of graphics for regression problems with a binary response variable graphical methods requiring neither a link function nor residuals are suggested for both development and criticism of model components implied by the central dimension reduction subspace

structured ordinary least squares a sufficient dimension reduction approach for regressions with partitioned predictors and heterogeneous units summary in many scientific and engineering fields advanced experimental and computing technologies are producing data that are not just high dimensional but also internally structured for instance statistical units may have heterogeneous origins from distinct studies or subpopulations and features may be naturally partitioned based on experimental platforms generating them or on information available about their roles in a given phenomenon in a regression analysis exploiting this known structure in the predictor dimension reduction stage that precedes modeling can be an effective way to integrate diverse data to pursue this we propose a novel sufficient dimension reduction sdr approach that we call structured ordinary least squares sols this combines ideas from existing sdr literature to merge reductions performed within groups of samples and or predictors in particular it leads to a version of ols for grouped predictors that requires far less computation than recently proposed groupwise sdr procedures and provides an informal yet effective variable selection tool in these settings we demonstrate the performance of sols by simulation and present a first application to genomic data the r package ssdr publicly available on cran includes all procedures necessary to implement the sols approach

simulation of a directed random walk model the effect of pseudo random number correlations abstract we investigate the mechanism that leads to systematic deviations in cluster monte carlo simulations when correlated pseudo random numbers are used we present a simple model which enables an analysis of the effects due to correlations in several types of pseudo random number sequences this model provides qualitative understanding of the bias mechanism in a class of cluster monte carlo algorithms

accurate computation of the smallest eigenvalue of a diagonally dominant m matrix if each off diagonal entry and the sum of each row of a diagonally dominant m matrix are known to certain relative accuracy then its smallest eigenvalue and the entries of its inverse are known to the same order relative accuracy independent of any condition numbers in this paper we devise algorithms that compute these quantities with relative errors in the magnitude of the machine precision rounding error analysis and numerical examples are presented to demonstrate the numerical behaviour of the algorithms

inexact generalized noda iterations for generalized eigenproblems abstract in this paper we present an inexact generalized noda iteration igni for finding the perron pair of the generalized eigenproblem arising from economic models we prove that the approximate eigenvalue sequence generated by igni converges globally linearly furthermore we also propose an enhanced inexact generalized noda iteration eigni and prove that the approximate eigenvector sequence generated by eigni algorithms converges superlinearly numerical examples illustrate that the proposed igni and eigni algorithms are efficient

random number generations generators on vector supercomputers and other advanced architectures this paper is divided into three parts the first part surveys the more common techniques for generating and testing uniformly distributed random numbers the second part considers the efficient ge

on the periods of generalized fibonacci recurrences we give a simple condition for a linear recurrence mod 2w of degree r to have the maximal possible period 2m 1 2 1 it follows that the period is maximal in the cases of interest for pseudorandom number gen eration i e for three term linear recurrences defined by trinomials which are primitive mod 2 and of degree r 2 we consider the enumeration of cer tain exceptional polynomials which do not give maximal period and list all such polynomials of degree less than 15

computer implementation of random number generators abstract good theoretical work on algorithms is often defeated by a poor computer implementation this is particularly true of random number generators for two reasons the overwhelming concern for speed and the inherent ill conditioning of the problem the emphasis on speed has often resulted in shortcuts see carta 1990 these shortcuts often accept approximations which can be disastrous in random number generation other shortcuts are very machine specific and then often get used in the wrong environment the problem of random number generation using a deterministic recursion is ill conditioned in the sense that small deviations relative errors of the order of 10 10 can cause complete loss of accuracy this is why any approximation is disastrous in this paper we will concentrate on the coding considerations for a special case of a good algorithm namely a multiplicative congruential generator with properly chosen multiplier and modulus the discussion will use fortran constructs but it is applicable to any programming language we will also address implementation of recursive generators on vector processors