rotation to simple loadings using component loss functions the oblique case component loss functions clfs similar to those used in orthogonal rotation are introduced to define criteria for oblique rotation in factor analysis it is shown how the shape of the clf affects the performance of the criterion it defines for example it is shown that monotone concave clfs give criteria that are minimized by loadings with perfect simple structure when such loadings exist moreover if the clfs are strictly concave minimizing must produce perfect simple structure whenever it exists examples show that methods defined by concave clfs perform well much more generally while it appears important to use a concave clf the specific clf used is less important for example the very simple linear clf gives a rotation method that can easily outperform the most popular oblique rotation methods promax and quartimin and is competitive with the more complex simplimax and geomin methods

simplimax oblique rotation to an optimal target with simple structure factor analysis and principal component analysis are usually followed by simple structure rotations of the loadings these rotations optimize a certain criterion e g varimax oblimin designed to measure the degree of simple structure of the pattern matrix simple structure can be considered optimal if a usually large number of pattern elements is exactly zero in the present paper a class of oblique rotation procedures is proposed to rotate a pattern matrix such that it optimally resembles a matrix which has an exact simple pattern it is demonstrated that this method can recover relatively complex simple structures where other well known simple structure rotation techniques fail

a phase transition behavior for brownian motions interacting through their ranks consider a time varying collection of n points on the positive real axis modeled as exponentials of n brownian motions whose drift vector at every time point is determined by the relative ranks of the coordinate processes at that time if at each time point we divide the points by their sum under suitable assumptions the rescaled point process converges to a stationary distribution depending on n and the vector of drifts as time goes to infinity this stationary distribution can be exactly computed using a recent result of pal and pitman the model and the rescaled point process are both central objects of study in models of equity markets introduced by banner fernholz and karatzas in this paper we look at the behavior of this point process under the stationary measure as n tends to infinity under a certain continuity at the edge condition on the drifts we show that one of the following must happen either i all points converge to 0 or ii the maximum goes to 1 and the rest go to 0 or iii the processes converge in law to a non trivial poisson dirichlet distribution the underlying idea of the proof is inspired by talagrand s analysis of the low temperature phase of derrida s random energy model of spin glasses the main result establishes a universality property for the bfk models and aids in explicit asymptotic computations using known results about the poisson dirichlet law

stochastic differential equations with reflecting boundary conditions on resout des equations differentielles stochastiques a conditions aux limites reflechissantes par une approche directe basee sur le probleme de skorokhod

multidimensional reflected brownian motions having exponential stationary distributions on considere la distribution stationnaire d un mouvement brownien reflechi dans un domaine d dimensionnel g on etudie deux classes de processus de ce type

on the mathematical model for linear elastic systems with analytic damping in the present paper we investigate linear elastic systems with damping ddot y b dot y ay 0 in hilbert spaces where a is a positive definite unbounded linear operator and b is a closed linear operator related in various ways to a alpha frac 1 2 leqq alpha leqq 1 we discuss the spectral property of these systems and obtain some fundamental results for the holomorphic property and the exponential stability of the semigroups associated with these systems

approximated penalized maximum likelihood for exploratory factor analysis an orthogonal case the problem of penalized maximum likelihood pml for an exploratory factor analysis efa model is studied in this paper an efa model is typically estimated using maximum likelihood and then the estimated loading matrix is rotated to obtain a sparse representation penalized maximum likelihood simultaneously fits the efa model and produces a sparse loading matrix to overcome some of the computational drawbacks of pml an approximation to pml is proposed in this paper it is further applied to an empirical dataset for illustration a simulation study shows that the approximation naturally produces a sparse loading matrix and more accurately estimates the factor loadings and the covariance matrix in the sense of having a lower mean squared error than factor rotations under various conditions

least controls for a class of constrained linear stochastic systems this paper introduces a class of constrained linear stochastic systems the main objective is to investigate whether there exists a least control in a sense that it exerts least effort in controlling the system to within a predetermined region our approach of finding a least control is a to characterize a class of functional polyhedral sets of functions which have least elements and b to construct a least control process through these least elements existence of a least control is established and the connection between a least control and a solution of the dynamic complementarity problem dcp is also discussed

resolvent positive operators resolvent positive operators on an ordered banach space with generating and normal positive cone are by definition linear possibly unbounded operators whose resolvent exists and is positive on a right half line even though these operators are defined by a simple purely algebraic condition analogues of the basic results of the theory of q semigroups can be proved for them in fact if a is resolvent positive and has a dense domain then the cauchy problem associated with a has a unique solution for every initial value in the domain of a and the solution is positive if the initial value is positive also the converse is true if we assume that a has a non empty resolvent set and d a n e is dense in moreover every positive resolvent is a laplace stieltjes transform of a so called integrated semigroup and conversely every such increasing non degenerate integrated semigroup defines a unique resolvent positive operator

on the diffusion approximation to two parallel queues with processor sharing a two dimensional queuing model is considered in which the server in the second queue helps the server in the first queue during periods when the second queue is empty the system is analyzed in the heavy traffic limit and explicit approximate solutions are obtained to the resulting diffusion equations using singular perturbation methods approximate asymptotic formulas are obtained for the stationary distribution of the number of customers as well as for some first passage time problems associated with the busy period it is shown that these formulas reduce to the asymptotic expansions of the exact solutions when the latter are available

brownian motion in a wedge with oblique reflection this work is concerned with the existence and uniqueness of a strong markov process that has continuous sample paths and the following additional properties i the state space is an infinite two dimensional wedge and the process behaves in the interior of the wedge like an ordinary brownian motion ii the process reflects instantaneously at the boundary of the wedge the angle of reflection being constant along each side iii the amount of time that the process spends at the comer of the wedge is zero i e the set of times for which the process is at the comer has lebesgue measure zero hereafter let be the angle of the wedge 0 2 let 1 and 2 be the angles of reflection on the two sides of the wedge measured from the inward normals the positive angles being toward the corner 1 2 and set 1 2 the question of existence and uniqueness is recast as a submartingale problem in the style used by stroock and varadhan diffusion processes with boundary conditions comm pure appl math 24 1971 pp 147 225 for diffusions on smooth domains with smooth boundary conditions it is shown that no solution exists if 2 in this case there is a unique continuous strong markov process satisfying i ii above it reaches the corner of the wedge almost surely and it remains there if 2 however then there is a unique continuous strong markov process statisfying i iii it is shown that starting away from the corner this process does not reach the corner of the wedge if 0 and does reach the corner if 0 2 the general theory of multi dimensional diffusions does not apply to the above problem because in general the boundary of the state space is not smooth and there is a discontinuity in the direction of reflection at the corner for some values of the process arises from diffusion approximations to storage systems and queueing networks i the state space is an infinite two dimensional wedge and the process behaves in the interior of the wedge like an ordinary brownian motion ii the process reflects instantaneously at the boundary of the wedge and the angle of reflection being constant along each side iii the amount of time that the process spends at the corner of the wedge is zero i e the set of times for which the process is at the corner has lebesgue measure zero

quartic rotation criteria and algorithms most of the currently used analytic rotation criteria for simple structure in factor analysis are summarized and identified as members of a general symmetric family of quartic criteria a unified development of algorithms for orthogonal and direct oblique rotation using arbitrary criteria from this family is given these algorithms represent fairly straightforward extensions of present methodology and appear to be the best methods currently available

strong solutions of stochastic equations with rank based coefficients we study finite and countably infinite systems of stochastic differential equations in which the drift and diffusion coefficients of each component particle are determined by its rank in the vector of all components of the solution we show that strong existence and uniqueness hold until the first time three particles collide motivated by this result we improve significantly the existing conditions for the absence of such triple collisions in the case of finite dimensional systems and provide the first condition of this type for systems with a countable infinity of particles

rotation to simple loadings using component loss functions the orthogonal case component loss functions clfs are used to generalize the quartimax criterion for orthogonal rotation in factor analysis these replace the fourth powers of the factor loadings by an arbitrary function of the second powers criteria of this form were introduced by a number of authors primarily katz and rohlf 1974 and rozeboom 1991 but there has been essentially no follow up to this work a method so simple natural and general deserves to be investigated more completely a number of theoretical results are derived including the fact that any method using a concave clf will recover perfect simple structure whenever it exists and there are methods that will recover thurstone simple structure whenever it exists specific clfs are identified and it is shown how to compare these using standardized plots numerical examples are used to illustrate and compare clf and other methods sorted absolute loading plots are introduced to aid in comparing results and setting parameters for methods that require them

the reflection map with discontinuities we study the multidimensional reflection map on the spacesd 0 t r k andd 0 8 r k of right continuous r k valued functions on 0 t or 0 8 with left limits endowed with variants of the skorohod 1956 m1 topology the reflection map was used with the continuous mapping theorem by harrison and reiman 1981 and reiman 1984 to establish heavy traffic limit theorems with reflected brownian motion limit processes for vector valued queue length waiting time and workload stochastic processes in single class open queueing networks since brownian motion and reflected brownian motion have continuous sample paths the topology of uniform convergence over bounded intervals could be used for those results variants of them1 topologies are needed to obtain alternative discontinuous limits approached gradually by the converging processes as occurs in stochastic fluid networks with bursty exogenous input processes e g with on off sources having heavy tailed on periods or off periods having infinite variance we show that the reflection map is continuous at limits without simultaneous jumps of opposite sign in the coordinate functions provided that the productm1 topology is used as a consequence the reflection map is continuous with the productm1 topology at all functions that have discontinuities in only one coordinate at a time that continuity property also holds for more general reflection maps and is sufficient to support limit theorems for stochastic processes in most applications we apply the continuity of the reflection map to obtain limits for buffer content stochastic processes in stochastic fluid networks

integrated semigroups and integrated solutions to abstract cauchy problems abstract concepts which have been developed in the theory of weakly continuous semigroups on dual banach spaces are extended to integrated semigroups the relation to integrated solutions of homogeneous and inhomogeneous abstract cauchy problems is established the results are illustrated for the wave equation in l2 rm

newton algorithms for analytic rotation an implicit function approach in this paper implicit function based parameterizations for orthogonal and oblique rotation matrices are proposed the parameterizations are used to construct newton algorithms for minimizing differentiable rotation criteria applied to m factors and p variables the speed of the new algorithms is compared to that of existing algorithms and to that of newton algorithms based on alternative parameterizations several rotation criteria were examined and the algorithms were evaluated over a range of values for m initial guesses for newton algorithms were improved by subconvergence iterations of the gradient projection algorithm simulation results suggest that no one algorithm is fastest for minimizing all criteria for all values of m among competing algorithms the gradient projection algorithm alone was faster than the implicit function algorithm for minimizing a quartic criterion over oblique rotation matrices when m is large in all other conditions however the implicit function algorithms were competitive with or faster than the fastest existing algorithms the new algorithms showed the greatest advantage over other algorithms when minimizing a nonquartic component loss criterion

the skorohod oblique reflection problem in domains with corners and application to stochastic differential equations the skorohod oblique reflection problem for d w d a general domain in d x x d a convex cone of directions of reflection w a function ind d is considered it is first proved under a condition on d corresponding to x not being simultaneously too large and too much skewed with respect to d that given a sequence wn of functions converging in the skorohod topology tow any sequence xn n of solutions to the skorohod problem for d wn is relatively compact and any of its limit points is a solution to the skorohod problem for d w next it is shown that if d satisfies the uniform exterior sphere condition and another requirement then solutions to the skorohod problem for d w exist for everyw d d with small enough jump size the requirement is met in the case when d is piecewisecb1 is generated by continuous vector fields on the faces ofd and x makes and angle in a suitable sense of less than 2 with the cone of inward normals atd for everyx d existence of obliquely reflecting brownian motion and of weak solutions to stochastic differential equations with oblique reflection boundary conditions is derived

open queueing networks in heavy traffic this paper presents heavy traffic limit theorems for the queue length and sojourn time processes associated with open queueing networks these limit theorems state that properly normalized sequences of queue length and sojourn time processes converge weakly to a certain diffusion as the network traffic intensity converges to unity the limit diffusion is reflected brownian motion on the nonnegative orthant this process behaves like brownian motion on the interior of its state space and reflects instantaneously on the boundaries the reflection direction is a constant for each boundary hyperplane

convergence rates for rank based models with applications to portfolio theory we determine rates of convergence of rank based interacting diffusions and semimartingale reflecting brownian motions to equilibrium bounds on fluctuations of additive functionals are obtained using transportation cost information inequalities for markov processes we work out various applications to the rank based abstract equity markets used in stochastic portfolio theory for example we produce quantitative bounds including constants for fluctuations of market weights and occupation times of various ranks for individual coordinates another important application is the comparison of performance between symmetric functionally generated portfolios and the market portfolio this produces estimates of probabilities of beating the market

complete second order linear differential equations with almost periodic solutions abstract this paper characterizes equations of the type u t bu t au t 0 in a banach space where a b are densely defined closed linear operators and whose solutions are all almost periodic

reflected brownian motion with skew symmetric data in a polyhedral domain this paper is concerned with the characterization and invariant measures of certain reflected brownian motions rbm s in polyhedral domains the kind of rbm studied here behaves like d dimensional brownian motion with constant drift in the interior of a simple polyhedron and is instantaneously reflected at the boundary in directions that depend on the face that is hit under the assumption that the directions of reflection satisfy a certain skew symmetry condition first introduced in harrison williams 9 it is shown that such an rbm can be characterized in terms of a family of submartingales and that it reaches non smooth parts of the boundary with probability zero in 9 a purely analytic problem associated with such an rbm was solved here the exponential form solution obtained in 9 is shown to be the density of an invariant measure for the rbm furthermore if the density is integrable over the polyhedral state space then it yields the unique stationary distribution for the rbm in the proofs of these results a key role is played by a dual process for the rbm and by results in 9 for reflected brownian motions on smooth approximating domains

laplace transforms and integrated regularized semigroups in locally convex spaces abstract in this paper we consider the vector valued laplace transforms r times r 0 integrated semigroups and regularized semigroups in the context of sequentially complete locally convex spaces our theorems develop the corresponding results in 1 11 including the well known integrated version of the classical widder s representation theorem of laplace transforms for functions taking values in banach spaces moreover we study a class of differential operators on certain function spaces optimal conditions making them the generators of integrated or regularized semigroups are obtained we finally show some applications to abstract cauchy problems

on the diffusion approximation to a fork and join queueing model the use of singular perturbation methods to obtain approximate analytic solutions to the types of diffusion equations that are used to model queueing networks is shown as an example a two dimensional fork and join model is considered approximations to the stationary distribution of the number of customers in the system as well as the residual busy period are computed the formulas obtained clearly show the dependence of the solution on the variables and the parameters of the model

the weighted oblimin rotation cureton mulaik 1975 proposed the weighted varimax rotation so that varimax kaiser 1958 could reach simple solutions when the complexities of the variables in the solution are larger than one in the present paper the weighting procedure proposed by cureton mulaik 1975 is applied to direct oblimin clarkson jennrich 1988 and the rotation method obtained is called weighted oblimin it has been tested on artificial complex data and real data and the results seem to indicate that even though direct oblimin rotation fails when applied to complex data weighted oblimin gives good results if a variable with complexity one can be found for each factor in the pattern although the weighting procedure proposed by cureton mulaik is based on landahl s 1938 expression for orthogonal factors weighted oblimin seems to be adequate even with highly oblique factors the new rotation method was compared to other rotation methods based on the same weighting procedure and whenever a variable with complexity one could be found for each factor in the pattern weighted oblimin gave the best results when rotating a simple empirical loading matrix weighted oblimin seemed to slightly increase the performance of direct oblimin

bayesian exploratory factor analysis this paper develops and applies a bayesian approach to exploratory factor analysis that improves on ad hoc classical approaches our framework relies on dedicated factor models and simultaneously determines the number of factors the allocation of each measurement to a unique factor and the corresponding factor loadings classical identification criteria are applied and integrated into our bayesian procedure to generate models that are stable and clearly interpretable a monte carlo study confirms the validity of the approach the method is used to produce interpretable low dimensional aggregates from a high dimensional set of psychological measurements

derivative free gradient projection algorithms for rotation a simple modification substantially simplifies the use of the gradient projection gp rotation algorithms of jennrich 2001 2002 these algorithms require subroutines to compute the value and gradient of any specific rotation criterion of interest the gradient can be difficult to derive and program it is shown that using numerical gradients gives almost precisely the same results as using exact gradients the resulting algorithm is very easy to use because the only problem specific code required is that needed to define the rotation criterion the computing time is increased when using numerical gradients but it is still very modest for most purposes while used extensively elsewhere numerical derivatives seem to be underutilized in statistics

spectra of pseudodifferential operators generating integrated semigroups consider the lp realization opp a of a pseudodifferential operator with symbol a e sp70 having constant coefficients we show that for a certain class of symbols the spectrum of opp a is independent of p this implies that opp a generates an n times integrated semigroup on lp i rn for a certain n if and only if p opp a 8 0 and the numerical range of a is contained in a left half plane our method allows us also to construct examples of operators generating integrated semigroups on lp rn if and only if p is sufficiently close to 2

entire solutions of higher order abstract cauchy problems abstract we consider the abstract cauchy problem for the higher order differential equation u n t i 0 n 1 a i u i t 0 t 0 u j 0 u j 0 j n 1 wherea0 an 1are closed linear operators in a complex banach spacee it is shown that if there exist 0 2 r 0 and a positive integerhsuch that nr and hakr 0 k n 1 are bounded in c arg e i 2 r wherer n n 1i 0 iai 1 then there exists a dense subsetgofensuch that for every initial value u0 un 1 g has a unique entire solution that is a solution which can be extended analytically to the whole complex plane as a corollary we get that the cauchy problems for the parabolic higher order linear differential equations have a unique entire solution for a dense set of initial data

on the convergence and approximation of integrated semigroups abstract we extend the well known theorem on convergence and approximation of c 0 semigroups due to h f trotter and t kato to the context of integrated semigroups the result is obtained as a consequence of an integrated version of the trotter kato theorem based on the laplace transform an application to the complete second order equation is given

a simple general method for oblique rotation a simple and very general algorithm for oblique rotation is identified while motivated by the rotation problem in factor analysis it may be used to minimize almost any function of a not necessarily square matrix whose columns are restricted to have unit length the algorithm has two steps the first is to compute the gradient of the rotation criterion and the second is to project this onto a manifold of matrices with unit length columns for this reason it is called a gradient projection algorithm because the projection step is very simple implementation of the algorithm involves little more than computing the gradient of the rotation criterion which for many applications is very simple it is proven that the algorithm is strictly monotone that is as long as it is not already at a stationary point each step will decrease the value of the criterion examples from a variety of areas are used to demonstrate the algorithm including oblimin rotation target rotation simplimax rotation and rotation to similarity and simplicity simultaneously while it may be the algorithm is not intended for use as a standard algorithm for well established problems but rather as a tool for investigating new methods where its generality and simplicity may save an investigator substantial effort

sparse versus simple structure loadings the component loadings are interpreted by considering their magnitudes which indicates how strongly each of the original variables relates to the corresponding principal component the usual ad hoc practice in the interpretation process is to ignore the variables with small absolute loadings or set to zero loadings smaller than some threshold value this in fact makes the component loadings sparse in an artificial and a subjective way we propose a new alternative approach which produces sparse loadings in an optimal way the introduced approach is illustrated on two well known data sets and compared to the existing rotation methods

a simple general procedure for orthogonal rotation a very general algorithm for orthogonal rotation is identified it is shown that when an algorithm parameter is sufficiently large the algorithm converges monotonically to a stationary point of the rotation criterion from any starting value because a sufficiently large is in general hard to find a modification that does not require it is introduced without this requirement the modified algorithm is not only very general but also very simple its implementation involves little more than computing the gradient of the rotation criterion while the modified algorithm converges monotonically from any starting value it is not guaranteed to converge to a stationary point it however does so in all of our examples while motivated by the rotation problem in factor analysis the algorithms discussed may be used to optimize almost any function of a not necessarily square column wise orthonormal matrix a number of these more general applications are considered empirical examples show that the modified algorithm can be reasonably fast but its purpose is to save an investigator s effort rather than that of his or her computer this makes it more appropriate as a research tool than as an algorithm for established methods

norm continuity fort 0 of propagators of arbitrary order abstract differential equations in hilbert spaces let acpn u n t n 1k 0aku k t 0 t 0 u k 0 uk 0 k n 1 be strongly wellposed where theak 0 k n 1 are densely defined closed linear operators in a hilbert space letsk t 0 k n 1 be the propagators of acpn andp ni n 1k 0 kak it is shown thats k k t 0 k n 1 is norm continuous fort 0 if and only if lim omega to infty left tau 0 i omega n 1 p tau 0 i omega 1 right 0 lim omega to infty left tau 0 i omega k 1 overline p tau 0 i omega 1 a k right 0 qquad 1 le k n le n 1 for a suitable real number 0

reflected brownian motion on an orthant we consider a k dimensional diffusion process z whose state space is the nonnegative orthant on the interior of the orthant z behaves like a kdimensional brownian motion with arbitrary covariance matrix and drift vector at each of the k 1 dimensional hyperplanes that form the boundary of the orthant z reflects instantaneously in a direction that is constant over that hyperplane there is no extant theory of multidimensional diffusion that applies to this process because the boundary of its state space is not smooth we adopt an approach that requires a restriction on the directions of reflection but reiman has shown that this restriction is met by all diffusions z arising as heavy traffic limits in open k station queuing networks our process z is defined as a path to path mapping of k dimensional brownian motion from this construction it follows that z is a continuous markov process and a semimartingale using the latter property we obtain a change of variable formula from which one can develop a complete analytical theory for the process z

semigroups arising from elastic systems with dissipation abstract in this paper we investigate the properties norm continuity differentiability analyticity and exponential stability of semigroups arising from elastic systems with dissipation the resulting theorems are complete in a sense extensions of 1 theorem 1 1 and 2 theorem 1 1

brownian motion with polar drift on cherche a approfondir la connaissance du comportement des diffusions multidimensionnelles pres des points singuliers

a factor simplicity index we propose an index for assessing the degree of factor simplicity in the context of principal components and exploratory factor analysis the new index which is called loading simplicity is based on the idea that the communality of each variable should be related to few components or factors so that the loadings in each variable are either zero or as far from zero as possible this index does not depend on the scale of the factors and its maximum and minimum are only related to the degree of simplicity in the loading matrix the aim of the index is to enable the degree of simplicity in loading matrices to be compared

integrated semigroups and higher order abstract equations abstract let e be a banach space b 0 b n 1 closed linear operators on e and f 0 e this paper is devoted to the study of well posedness of the cauchy problem for higher order abstract differential equations formula using the theory of integrated semigroups we obtain some concise and useful criteria

brownian motion in a wedge with variable skew reflection does planar brownian motion confined to a wedge by skew reflection on the sides approach the vertex of the wedge this question has been answered by varadhan and williams in the case where the direction of reflection is constant on each of the sides but here we address the question when the direction reflected is allowed to vary a necessary condition and a sufficient condition are obtained for the vertex to be reached the conditions are of a geometric nature and the gap between them is quite small

a characterization of norm continuity of propagators for second order abstract differential equations abstract in this paper we obtain a concise characterization of norm continuity for t 0 of propagators for the complete second order abstract differential equation on a banach space e u t bu t au t 0 t 0 where b l e as a consequence we discover that a strongly continuous cosine operator function or operator group is norm continuous for t 0 if and only if its generator is bounded

a multidimensional process involving local time on presente une approche probabiliste a un probleme pose par mckean 1967 pour la construction d un systeme de n particules sur r avec un generateur formel