dimension reduction for nonelliptically distributed predictors sufficient dimension reduction methods often require stringent conditions on the joint distribution of the predictor or when such conditions are not satisfied rely on marginal transformation or reweighting to fulfill them approximately for example a typical dimension reduction method would require the predictor to have elliptical or even multivariate normal distribution in this paper we reformulate the commonly used dimension reduction methods via the notion of central solution space so as to circumvent the requirements of such strong assumptions while at the same time preserve the desirable properties of the classical methods such as n consistency and asymptotic normality imposing elliptical distributions or even stronger assumptions on predictors is often considered as the necessary tradeoff for overcoming the curse of dimensionality but the development of this paper shows that this need not be the case the new methods will be compared with existing methods by simulation and applied to a data set

linear systems with signed solutions linear systems ax b for which the sign patterns a and b completely determine the set of sign patterns of vectors in the solution set are studied these systems generalize the existing notion of sign solvable linear systems linear systems with signed solutions are characterized in terms of two classes of matrices the first class l matrices plays a central role in the study of sign solvable linear systems the second class is new and generalizes the class of s matrices that arise in sign solvable linear systems several constructions for the new class of matrices are given

ray solvable linear systems and ray s2ns matrices abstract ray solvable linear systems and ray s 2 ns matrices are complex generalizations of the sign solvable linear systems and s 2 ns matrices we use the determinantal ray unique matrices instead of ray nonsingular matrices as a generalization of sns matrices to generalize some fundamental results of s 2 ns matrices from the real case to complex case such as the graph theoretical characterization the inverse ray patterns and the upper bound of the number of nonzero entries of s 2 ns matrices the well known characterization of the sign solvable linear systems in terms of the l matrices and s matrices is also generalized to ray solvable linear systems and the relationships between the ray s matrices and real s matrices are investigated some examples are also given to illustrate that some results such as the characterization of the sign inconsistent linear systems do not carry over to the complex case

sublinear root detection and new hardness results for sparse polynomials over finite fields we present a deterministic 2 o t q frac t 2 t 1 o 1 algorithm to decide whether a univariate polynomial f with t monomial terms and degree q has a root in the finite field math

covariance reducing models an alternative to spectral modelling of covariance matrices we introduce covariance reducing models for studying the sample covariance matrices of a random vector observed in different populations the models are based on reducing the sample covariance matrices to an informational core that is sufficient to characterize the variance heterogeneity among the populations they possess useful equivariance properties and provide a clear alternative to spectral models for covariance matrices

aggregate inverse mean estimation for sufficient dimension reduction many well known sufficient dimension reduction methods investigate the inverse conditional moments of the predictors given the response the required linearity condition the number and ar

gate elimination circuit size lower bounds and sat upper bounds abstract most of the known lower bounds for binary boolean circuits with unrestricted depth are proved by the gate elimination method the most efficient known algorithms for the sat problem on binary boolean circuits use similar case analyses to the ones in gate elimination chen and kabanets recently showed that the known case analyses can also be used to prove average case circuit lower bounds that is lower bounds on the size of approximations of an explicit function in this paper we provide a general framework for proving worst average case lower bounds for circuits and upper bounds for sat that is built on ideas of chen and kabanets a proof in such a framework goes as follows one starts by fixing three parameters a class of circuits a circuit complexity measure and a set of allowed substitutions the main ingredient of a proof goes as follows by going through a number of cases one shows that for any circuit from the given class one can find an allowed substitution such that the given measure of the circuit reduces by a sufficient amount this case analysis immediately implies an upper bound for sat to obtain worst average case circuit complexity lower bounds one needs to present an explicit construction of a function that is a disperser extractor for the class of sources defined by the set of substitutions under consideration we show that many known proofs of circuit size lower bounds and upper bounds for sat fall into this framework using this framework we prove the following new bounds average case lower bounds of 3 24 n and 2 59 n for circuits over u 2 and b 2 respectively though the lower bound for the basis b 2 is given for a quadratic disperser whose explicit construction is not currently known and faster than 2 n sat algorithms for circuits over u 2 and b 2 of size at most 3 24 n and 2 99 n respectively here by b 2 we mean the set of all bivariate boolean functions and by u 2 the set of all bivariate boolean functions except for parity and its complement

using the bootstrap to select one of a new class of dimension reduction methods dimension reduction in a regression analysis of response y given a p dimensional vector of predictors x reduces the dimension of x by replacing it with a lower dimensional linear combination x of the x s without specifying a parametric model and without loss of information about the conditional distribution of y given x we unify three existing methods sliced inverse regression sir sliced average variance estimate save and principal hessian directions phd into a larger class of methods each method estimates a particular candidate matrix essentially a matrix of parameters we introduce broad classes of dimension reduction candidate matrices and we distinguish estimators of the matrices from the matrices themselves given these classes of methods and several ways to estimate any matrix we now have the problem of selecting a particular matrix and estimation method we propose bootstrap methodology to select among candidate matrices estimators and dimension and in particular we investigate l

estimation of the mean of a multivariate normal distribution estimation of the means of independent normal random variables is considered using sum of squared errors as loss an unbiased estimate of risk is obtained for an arbitrary estimate and certain special classes of estimates are then discussed the results are applied to smoothing by use of moving averages and to trimmed analogs of the james stein estimate a suggestion is made for calculating approximate confidence sets for the mean vector centered at an arbitrary estimate

envelopes and partial least squares regression summary we build connections between envelopes a recently proposed context for efficient estimation in multivariate statistics and multivariate partial least squares pls regression in particular we establish an envelope as the nucleus of both univariate and multivariate pls which opens the door to pursuing the same goals as pls but using different envelope estimators it is argued that a likelihood based envelope estimator is less sensitive to the number of pls components that are selected and that it outperforms pls in prediction and estimation

extending sliced inverse regression sliced inverse regression sir and an associated chi squared test for dimension have been introduced as a method for reducing the dimension of regression problems whose predictor variables are normal in this article the assumptions on the predictor distribution under which the chi squared test was proved to apply are relaxed and the result is extended a general weighted chi squared test that does not require normal regressors for the dimension of a regression is given simulations show that the weighted chi squared test is more reliable than the chi squared test when the regressor distribution digresses from normality significantly and that it compares well with the chi squared test when the regressors are normal

determining the dimension in sliced inverse regression and related methods abstract sliced inverse regression sir and principal hessian directions aim to reduce the dimensionality of regression problems an important step in the method is the determination of a suitable dimension although statistical tests based on the nullity eigenvalues are usually suggested this article focuses on the quality of the estimation of the effective dimension reduction edr spaces essentially the goal is to retain only sufficiently stable subspaces the goodness of the estimation is measured by the squared trace correlation between the subspaces of the edr space and their estimates asymptotic expansions are derived and estimates deduced simulations give an insight on the behavior of the criterion and indicate how it can be used in practice

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

sequential sufficient dimension reduction for large p small n problems summary we propose a new and simple framework for dimension reduction in the large p small n setting the framework decomposes the data into pieces thereby enabling existing approaches for n p to be adapted to n p problems estimating a large covariance matrix which is a very difficult task is avoided we propose two separate paths to implement the framework our paths provide sufficient procedures for identifying informative variables via a sequential approach we illustrate the paths by using sufficient dimension reduction approaches but the paths are very general empirical evidence demonstrates the efficacy of our paths additional simulations and applications are given in an on line supplementary file

a constructive approach to the estimation of dimension reduction directions in this paper we propose two new methods to estimate the dimension reduction directions of the central subspace cs by constructing a regression model such that the directions are all captured in the regression mean compared with the inverse regression estimation methods e g j amer statist assoc 86 1991 328 332 j amer statist assoc 86 1991 316 342 j amer statist assoc 87 1992 1025 1039 the new methods require no strong assumptions on the design of covariates or the functional relation between regressors and the response variable and have better performance than the inverse regression estimation methods for finite samples compared with the direct regression estimation methods e g j amer statist assoc 84 1989 986 995 ann statist 29 2001 1537 1566 j r stat soc ser b stat methodol 64 2002 363 410 which can only estimate the directions of cs in the regression mean the new methods can detect the directions of cs exhaustively consistency of the estimators and the convergence of corresponding algorithms are proved

partial inverse regression in regression with a vector of quantitative predictors sufficient dimension reduction methods can effectively reduce the predictor dimension while preserving full regression information and assuming no parametric model however all current reduction methods require the sample size n to be greater than the number of predictors p it is well known that partial least squares can deal with problems with n p we first establish a link between partial least squares and sufficient dimension reduction motivated by this link we then propose a new dimension reduction method entitled partial inverse regression we show that its sample estimator is consistent and that its performance is similar to or superior to partial least squares when n p especially when the regression model is nonlinear or heteroscedastic an example involving the spectroscopy analysis of biscuit dough is also given

iterative sliced inverse regression for segmentation of ultrasound and mr images in this study we propose an integrated approach based on iterative sliced inverse regression isir for the segmentation of ultrasound and magnetic resonance mr images the approach integrates two stages the first is the unsupervised clustering which combines multidimensional scaling mds with k means the dimension reduction based on mds is employed to obtain fewer representative variates as input variables for k means this step intends to generate the initial group labels of the training data for the second stage of supervised segmentation we then combine the sir with the nearest mean classifier nmc or the support vector machine svm to iteratively update the group labels for supervised segmentation the method of sir is introduced by li sliced inverse regression for dimension reduction j am stat assoc 86 1991 316 342 to explore the effective dimension reduction e d r directions from the training data embedded in high dimensional space the test data are then projected onto these directions and the classifiers are further applied to classify the test data the integrated approach based on isir is evaluated on simulated and clinical images which include ultrasound and mr images the evaluation results indicate that this approach provides an improvement of image segmentation over the methods to be compared without dimension reduction

an adaptive estimation of dimension reduction space summary searching for an effective dimension reduction space is an important problem in regression especially for high dimensional data we propose an adaptive approach based on semiparametric models which we call the conditional minimum average variance estimation mave method within quite a general setting the mave method has the following advantages most existing methods must undersmooth the nonparametric link function estimator to achieve a faster rate of consistency for the estimator of the parameters than for that of the nonparametric function in contrast a faster consistency rate can be achieved by the mave method even without undersmoothing the nonparametric link function estimator the mave method is applicable to a wide range of models with fewer restrictions on the distribution of the covariates to the extent that even time series can be included because of the faster rate of consistency for the parameter estimators it is possible for us to estimate the dimension of the space consistently the relationship of the mave method with other methods is also investigated in particular a simple outer product gradient estimator is proposed as an initial estimator in addition to theoretical results we demonstrate the efficacy of the mave method for high dimensional data sets through simulation two real data sets are analysed by using the mave approach

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

determining the dimensionality in sliced inverse regression abstract a general regression problem is one in which a response variable can be expressed as some function of one or more different linear combinations of a set of explanatory variables as well as a random error term sliced inverse regression is a method for determining these linear combinations in this article we address the problem of determining how many linear combinations are involved procedures based on conditional means and conditional covariance matrices as well as a procedure combining the two approaches are considered in each case we develop a test that has an asymptotic chi squared distribution when the vector of explanatory variables is sampled from an elliptically symmetric distribution

dimension reduction in regression without matrix inversion summary regressions in which the fixed number of predictors p exceeds the number of independent observational units n occur in a variety of scientific fields sufficient dimension reduction provides a promising approach to such problems by restricting attention to d n linear combinations of the original p predictors however standard methods of sufficient dimension reduction require inversion of the sample predictor covariance matrix we propose a method for estimating the central subspace that eliminates the need for such inversion and is applicable regardless of the n p relationship simulations show that our method compares favourably with standard large sample techniques when the latter are applicable we illustrate our method with a genomics application

on sliced inverse regression with high dimensional covariates sliced inverse regression is a promising method for the estimation of the central dimension reduction subspace cdr space in semiparametric regression models it is particularly useful in tackling cases with high dimensional covariates in this article we study the asymptotic behavior of the estimate of the cdr space with high dimensional covariates that is when the dimension of the covariates goes to infinity as the sample size goes to infinity strong and weak convergence are obtained we also suggest an estimation procedure of the bayes information criterion type to ascertain the dimension of the cdr space and derive the consistency a simulation study is conducted

on distribution weighted partial least squares with diverging number of highly correlated predictors summary because highly correlated data arise from many scientific fields we investigate parameter estimation in a semiparametric regression model with diverging number of predictors that are highly correlated for this we first develop a distribution weighted least squares estimator that can recover directions in the central subspace then use the distribution weighted least squares estimator as a seed vector and project it onto a krylov space by partial least squares to avoid computing the inverse of the covariance of predictors thus distrbution weighted partial least squares can handle the cases with high dimensional and highly correlated predictors furthermore we also suggest an iterative algorithm for obtaining a better initial value before implementing partial least squares for theoretical investigation we obtain strong consistency and asymptotic normality when the dimension p of predictors is of convergence rate o n1 2 log n and o n1 3 respectively where n is the sample size when there are no other constraints on the covariance of predictors the rates n1 2 and n1 3 are optimal we also propose a bayesian information criterion type of criterion to estimate the dimension of the krylov space in the partial least squares procedure illustrative examples with a real data set and comprehensive simulations demonstrate that the method is robust to non ellipticity and works well even in small n large p problems

contour projected dimension reduction in regression analysis we employ contour projection cp to develop a new dimension reduction theory accordingly we introduce the notions of the central contour subspace and generalized contour subspace we show that both of their structural dimensions are no larger than that of the central subspace cook regression graphics 1998b wiley furthermore we employ cp sliced inverse regression cp sliced average variance estimation and cp directional regression to estimate the generalized contour subspace and we subsequently obtain their theoretical properties monte carlo studies demonstrate that the three cp based dimension reduction methods out perform their corresponding non cp approaches when the predictors have heavy tailed elliptical distributions an empirical example is also presented to illustrate the usefulness of the cp method

partial envelopes for efficient estimation in multivariate linear regression we introduce the partial envelope model which leads to a parsimonious method for multivariate linear regression when some of the predictors are of special interest it has the potential to achieve massive efficiency gains compared with the standard model in the estimation of the coefficients for the selected predictors the partial envelope model is a variation on the envelope model proposed by cook et al 2010 but as it focuses on part of the predictors it has looser restrictions and can further improve the efficiency we develop maximum likelihood estimation for the partial envelope model and discuss applications of the bootstrap an example is provided to illustrate some of its operating characteristics

gradient based kernel dimension reduction for regression this article proposes a novel approach to linear dimension reduction for regression using nonparametric estimation with positive definite kernels or reproducing kernel hilbert spaces rkhss the purpose of the dimension reduction is to find such directions in the explanatory variables that explain the response sufficiently this is called sufficient dimension reduction the proposed method is based on an estimator for the gradient of the regression function considered for the feature vectors mapped into rkhss it is proved that the method is able to estimate the directions that achieve sufficient dimension reduction in comparison with other existing methods the proposed one has wide applicability without strong assumptions on the distributions or the type of variables and needs only eigendecomposition for estimating the projection matrix the theoretical analysis shows that the estimator is consistent with certain rate under some conditions the experimental results demonstrate that the proposed metho

on almost linearity of low dimensional projections from high dimensional data this paper studies the shapes of low dimensional projections from high dimensional data after standardization let mathbf x be a p dimensional random variable with mean zero and identity covariance for a projection beta mathbf x beta 1 find another direction b so that the regression curve of b mathbf x against beta mathbf x is as nonlinear as possible we show that when the dimension of mathbf x is large for most directions beta even the most nonlinear regression is still nearly linear our method depends on the construction of a pair of p dimensional random variables mathbf w 1 mathbf w 2 called the rotational twin and its density function with respect to the standard normal density with this we are able to obtain closed form expressions for measuring deviation from normality and deviation from linearity in a suitable sense of average as an interesting by product from a given set of data we can find simple unbiased estimates of e f beta mathbf x t phi 1 t 1 2 and e lbrack e mathbf x mid beta beta mathbf x t 2 t 2 f 2 beta mathbf x t phi 2 1 t rbrack where phi 1 is the standard normal density f beta mathbf x is the density for beta mathbf x and the e is taken with respect to the uniformly distributed beta this is achieved without any smoothing and without resorting to any laborious projection procedures such as grand tours our result is related to the work of diaconis and freedman the impact of our result on several fronts of data analysis is discussed for example it helps establish the validity of regression analysis when the link function of the regression model may be grossly wrong a further generalization which replaces beta mathbf x by b mathbf x with b beta 1 ldots beta k for k randomly selected orthonormal vectors beta i i 1 ldots k helps broaden the scope of application of sliced inverse regression sir

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

on sign inconsistent linear systems abstract a linear system of equations ax b is called sign inconsistent if for each matrix a b with the same sign pattern as a b the corresponding linear system a x b is not solvable sign inconsistent linear systems have close relationships with sign solvable and conditionally sign solvable linear systems in this paper we study various properties of sign inconsistent linear systems we give a complete characterization of sign inconsistent linear systems in terms of l matrices we also obtain complete characterizations of the minimally sign inconsistent linear systems in terms of the l canonical forms and their corresponding canonical digraphs of the barely l matrices

some comments on cp i study the typical configuration of a cp plot when the number of variables in the regression problem is large and there are many weak effects i show that a particular configuration that is very commonly seen can arise in a simple way i give a formula by means of which the risk incurred by the minimum cp rule can be estimated

sliced regression for dimension reduction a new dimension reduction method involving slicing the region of the response and applying local kernel regression to each slice is proposed compared with the traditional inverse regression methods e g sliced inverse regression sir the new method is free of the linearity condition and has much better estimation accuracy compared with the direct estimation methods e g mave the new method is much more robust against extreme values and can capture the entire central subspace cs exhaustively to determine the cs dimension a consistent cross validation criterion is developed extensive numerical studies including a real example confirm our theoretical findings

efficient integration of sufficient dimension reduction and prediction in discriminant analysis abstractsufficient dimension reduction sdr methods are popular model free tools for preprocessing and data visualization in regression problems where the number of variables is large unfortunate

some hypothesis tests for the covariance matrix when the dimension is large compared to the sample size this paper analyzes whether standard covariance matrix tests work when dimensionality is large and in particular larger than sample size in the latter case the singularity of the sample covariance matrix makes likelihood ratio tests degenerate but other tests based on quadratic forms of sample covariance matrix eigenvalues remain well defined we study the consistency property and limiting distribution of these tests as dimensionality and sample size go to infinity together with their ratio converging to a finite nonzero limit we find that the existing test for sphericity is robust against high dimensionality but not the test for equality of the covariance matrix to a given matrix for the latter test we develop a new correction to the existing test statistic that makes it robust against high dimensionality

asymptotic inference for eigenvectors abstract asymptotic procedures are given for testing certain hypotheses concerning eigenvectors and for constructing confidence regions for eigenvectors these asymptotic procedures are derived under fairly general conditions on the estimates of the matrix whose eigenvectors are of interest applications of the general results to principal components analysis and canonical variate analysis are given author

dimension reduction in regressions through cumulative slicing estimation in this paper we offer a complete methodology of cumulative slicing estimation to sufficient dimension reduction in parallel to the classical slicing estimation we develop three methods that are termed respectively as cumulative mean estimation cumulative variance estimation and cumulative directional regression the strong consistency for p o n1 2 log n and the asymptotic normality for p o n1 2 are established where p is the dimension of the predictors and n is sample size such asymptotic results improve the rate p o n1 3 in many existing contexts of semiparametric modeling in addition we propose a modified bic type criterion to estimate the structural dimension of the central subspace its consistency is established when p o n1 2 extensive simulations are carried out for comparison with existing methods and a real data example is presented for illustration

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

sufficient dimension reduction through discretization expectation estimation in the context of sufficient dimension reduction the goal is to parsimoniously recover the central subspace of a regression model many inverse regression methods use slicing estimation to recover the central subspace the efficacy of slicing estimation depends heavily upon the number of slices however the selection of the number of slices is an open and long standing problem in this paper we propose a discretization expectation estimation method which avoids selecting the number of slices while preserving the integrity of the central subspace this generic method assures root n consistency and asymptotic normality of slicing estimators for many inverse regression methods and can be applied to regressions with multivariate responses a bic type criterion for the dimension of the central subspace is proposed comprehensive simulations and an illustrative application show that our method compares favourably with existing estimators

nonuniform acc circuit lower bounds the class acc consists of circuit families with constant depth over unbounded fan in and or not and modm gates where m 1 is an arbitrary constant we prove the following nexp the class of languages accepted in nondeterministic exponential time does not have nonuniform acc circuits of polynomial size the size lower bound can be slightly strengthened to quasipolynomials and other less natural functions enp the class of languages recognized in 2o n time with an np oracle doesn t have nonuniform acc circuits of 2no 1 size the lower bound gives an exponential size depth tradeoff for every d m there is a 0 such that enp doesn t have depth d acc circuits of size 2n with modm gates previously it was not known whether expnp had depth 3 polynomial size circuits made out of only mod6 gates the high level strategy is to design faster algorithms for the circuit satisfiability problem over acc circuits then prove that such algorithms entail these lower bounds the algorithms combine known properties of acc with fast rectangular matrix multiplication and dynamic programming while the second step requires a strengthening of the author s prior work

slicing regression a link free regression method slicing regression a link free regression m e t h o d author s naihua duan and k e r c h a u l i s o u r c e the annals of statistics v o l 19 n o 2 t u n 1991 p p 5 0 5 5 3 0 p u b l i s h e d b y institute of mathematical statistics s t a b l e u r l http www jstor org stable 2242072 accessed 16 05 2011 your use of the jstor archive indicates your acceptance of jstor s terms and conditions of use available at http www jstor org page info about policies terms jsp jstor s terms and conditions of use provides in part that unless you have obtained prior permission you may not download an entire issue of a journal or multiple copies of articles and you may use content in the jstor archive only for your personal non commercial use please contact the publisher regarding any further use of this work publisher contact information may be obtained at http www jstor org action showpublisher publishercode ims each copy of any part of a jstor transmission must contain the same copyright notice that appears on the screen or printed page of such transmission jstor is a not for profit service that helps scholars researchers and students discover use and build upon a wide range of content in a trusted digital archive we use information technology and tools to increase productivity and facilitate new forms of scholarship for more information about jstor please contact support jstor org http www jstor org

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

fused estimators of the central subspace in sufficient dimension reduction when studying the regression of a univariate variable y on a vector x of predictors most existing sufficient dimension reduction sdr methods require the construction of slices of y to estimate moments of the conditional distribution of x given y but there is no widely accepted method for choosing the number of slices while a poorly chosen slicing scheme may produce miserable results we propose a novel and easily implemented fusing method that can mitigate the problem of choosing a slicing scheme and improve estimation efficiency at the same time we develop two fused estimators called fire and dire based on an optimal inverse regression estimator the asymptotic variance of fire is no larger than that of the original methods regardless of the choice of slicing scheme while dire is less computational intense and more robust simulation studies show that the fused estimators perform effectively the same as or substantially better than the parent methods fused estimators based on other methods can be

improving exhaustive search implies superpolynomial lower bounds the p vs np problem arose from the question of whether exhaustive search is necessary for problems with short verifiable solutions we do not know if even a slight algorithmic improvement over exhaustive search is universally possible for all np problems and to date no major consequences have been derived from the assumption that an improvement exists we show that there are natural np and bpp problems for which minor algorithmic improvements over the trivial deterministic simulation already entail lower bounds such as sf nexp not subseteq sf p rm poly and sf logspace neq sf np these results are especially interesting given that similar improvements have been found for many other hard problems optimistically one might hope our results suggest a new path to lower bounds pessimistically they show that carrying out the seemingly modest program of finding slightly better algorithms for all search problems may be extremely difficult if not impossible we also prove unconditional s

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

estimating sufficient reductions of the predictors in abundant high dimensional regressions we study the asymptotic behavior of a class of methods for sufficient dimension reduction in high dimension regressions as the sample size and number of predictors grow in various alignments it is demonstrated that these methods are consistent in a variety of settings particularly in abundant regressions where most predictors contribute some information on the response and oracle rates are possible simulation results are presented to support the theoretical conclusion

using intraslice covariances for improved estimation of the central subspace in regression summary popular methods for estimating the central subspace in regression require slicing a continuous response however slicing can result in loss of information and in some cases that loss can be substantial we use intraslice covariances to construct improved inference methods for the central subspace these methods are optimal within a class of quadratic inference functions and permit chi squared tests of conditional independence hypotheses involving the predictors our experience gained through simulation is that the new method is never worse than existing methods and can be substantially better