hierarchical classes models for three way three mode binary data interrelations and model selection several hierarchical classes models can be considered for the modeling of three way three mode binary data including the indclas model leenen van mechelen de boeck and rosenberg 1999 the tucker3 hiclas model ceulemans van mechelen and leenen 2003 the tucker2 hiclas model ceulemans and van mechelen 2004 and the tucker1 hiclas model that is introduced in this paper two questions then may be raised 1 how are these models interrelated and 2 given a specific data set which of these models should be selected and in which rank in the present paper we deal with these questions by 1 showing that the distinct hierarchical classes models for three way three mode binary data can be organized into a partially ordered hierarchy and 2 by presenting model selection strategies based on extensions of the well known scree test and on the akaike information criterion the latter strategies are evaluated by means of an extensive simulation study and are illustrated with an application to interpersonal emotion data finally the presented hierarchy and model selection strategies are related to corresponding work by kiers 1991 for principal component models for three way three mode real valued data

the conjunctive model of hierarchical classes this paper describes the conjunctive counterpart of de boeck and rosenberg s hierarchical classes model both the original model and its conjunctive counterpart represent the set theoretical structure of a two way two mode binary matrix however unlike the original model the new model represents the row column association as a conjunctive function of a set of hypothetical binary variables the conjunctive nature of the new model further implies that it may represent some conjunctive higher order dependencies among rows and columns the substantive significance of the conjunctive model is illustrated with empirical applications finally it is shown how conjunctive and disjunctive hierarchical classes models relate to galois lattices and how hierarchical classes analysis can be useful to construct lattice models of empirical data

tabulation based 5 independent hashing with applications to linear probing and second moment estimation in the framework of wegman and carter a k independent hash function maps any k keys independently it is known that 5 independent hashing provides good expected performance in applications such as linear probing and second moment estimation for data streams the classic 5 independent hash function evaluates a degree 4 polynomial over a prime field containing the key domain n 0 ldots n 1 here we present an efficient 5 independent hash function that uses no multiplications instead for any parameter c we make 2c 1 lookups in tables of size o n 1 c in experiments on different computers our scheme gained factors of 1 8 to 10 in speed over the polynomial method we also conducted experiments on the performance of hash functions inside the above applications in particular we give realistic examples of inputs that make the most popular 2 independent hash function perform quite poorly this illustrates the advantage of using schemes with provably good expected performance for all inputs

towards more efficient spsd matrix approximation and cur matrix decomposition symmetric positive semi definite spsd matrix approximation methods have been extensively used to speed up large scale eigenvalue computation and kernel learning methods the standard sketch based method which we call the prototype model produces relatively accurate approximations but is inefficient on large square matrices the nystrom method is highly efficient but can only achieve low accuracy in this paper we propose a novel model that we call the fast spsd matrix approximation model the fast model is nearly as efficient as the nystrom method and as accurate as the prototype model we show that the fast model can potentially solve eigenvalue problems and kernel learning problems in linear time with respect to the matrix size n to achieve 1 e relative error whereas both the prototype model and the nystrom method cost at least quadratic time to attain comparable error bound empirical comparisons among the prototype model the nystrom method and our fast model demonstrate the superiority of the fast model we also contribute new understandings of the nystrom method the nystrom method is a special instance of our fast model and is approximation to the prototype model our technique can be straightforwardly applied to make the cur matrix decomposition more efficiently computed without much affecting the accuracy

indclus an individual differences generalization of the adclus model and the mapclus algorithm we present a new model and associated algorithm indclus that generalizes the shepard arabie adclus additive clustering model and the mapclus algorithm so as to represent in a clustering solution individual differences among subjects or other sources of data like mapclus the indclus generalization utilizes an alternating least squares method combined with a mathematical programming optimization procedure based on a penalty function approach to impose discrete 0 1 constraints on parameters defining cluster membership all subjects in an indclus analysis are assumed to have a common set of clusters which are differentially weighted by subjects in order to portray individual differences as such indclus provides a discrete clustering counterpart to the carroll chang indscal model for continuous spatial representations finally we consider possible generalizations of the indclus model and algorithm

indclas a three way hierarchical classes model a three way three mode extension of de boeck and rosenberg s 1988 two way two mode hierarchical classes model is presented for the analysis of individual differences in binary object attribute arrays in line with the two way hierarchical classes model the three way extension represents both the association relation among the three modes and the set theoretical relations among the elements of each model an algorithm for fitting the model is presented and evaluated in a simulation study the model is illustrated with data on psychiatric diagnosis finally the relation between the model and extant models for three way data is discussed

spectral analysis of fokker planck and related operators arising from linear stochastic differential equations we study spectral properties of certain families of linear second order differential operators arising from linear stochastic differential equations we construct a basis in the hilbert space of square integrable functions using modified hermite polynomials and obtain a representation for these operators from which their eigenvalues and eigenfunctions can be computed in particular we completely describe the spectrum of the fokker planck operator on an appropriate invariant subspace of rapidly decaying functions the eigenvalues of the fokker planck operator provide information about the speed of convergence of the corresponding probability distribution to steady state which is important for stochastic estimation and control applications we show that the operator families under consideration can be realized as solutions of differential equations in the double bracket form on an operator lie algebra which leads to a simple expression for the flow of their eigenfunctions

tucker3 hierarchical classes analysis this paper presents a new model for binary three way three mode data called tucker3 hierarchical classes model tucker3 hiclas this new model generalizes leenen van mechelen de boeck and rosenberg s 1999 individual differences hierarchical classes model indclas like the indclas model the tucker3 hiclas model includes a hierarchical classification of the elements of each mode and a linking structure among the three hierarchies unlike indclas tucker3 hiclas a does not restrict the hierarchical classifications of the three modes to have the same rank and b allows for more complex linking structures among the three hierarchies an algorithm to fit the tucker3 hiclas model is described and evaluated in an extensive simulation study an application of the model to hostility data is discussed

an lqr lqg theory for systems with saturating actuators an extension of the lqr lqg methodology to systems with saturating actuators referred to as slqr slqg where s stands for saturating is obtained the development is based on the method of stochastic linearization whereby the saturation is replaced by a gain calculated as a function of the variance of the signal at its input using the stochastically linearized system and the lagrange multiplier technique solutions of the slqr slqg problems are derived these solutions are given by standard riccati and lyapunov equations coupled with two transcendental equations which characterize both the variance of the signal at the saturation input and the lagrange multiplier associated with the constrained minimization problem it is shown that under standard stabilizability and detectability conditions these equations have a unique solution which can be found by a simple bisection algorithm when the level of saturation tends to infinity these equations reduce to their standard lqr lqg counterparts in addition the paper investigates the properties of closed loop systems with slqr slqg controllers and saturating actuators in this regard it is shown that slqr slqg controllers ensure semi global stability by appropriate choice of a parameter in the performance criterion finally the paper illustrates the techniques developed by a ship roll damping problem

tucker2 hierarchical classes analysis this paper presents a new hierarchical classes model called tucker2 hiclas for binary three way three mode data as any three way hierarchical classes model the tucker2 hiclas model includes a representation of the association relation among the three modes and a hierarchical classification of the elements of each mode a distinctive feature of the tucker2 hiclas model being closely related to the tucker3 hiclas model ceulemans van mechelen leenen 2003 is that one of the three modes is minimally reduced and hence that the differences among the association patterns of the elements of this mode are maximally retained in the model moreover as compared to tucker3 hiclas tucker2 hiclas implies three rather than four different types of parameters and as such is simpler to interpret two types of tucker2 hiclas models are distinguished a disjunctive and a conjunctive type an algorithm for fitting the tucker2 hiclas model is described and evaluated in a simulation study the model is illustrated with longitudinal data on interpersonal emotions

on universal classes of extremely random constant time hash functions a family of functions f that map 0 m 1 into 0 n 1 is said to be h wise independent if any tuple of h distinct points in 0 m 1 have a corresponding image for a randomly selected f in f that is uniformly distributed in 0 n 1 h this paper shows that for suitably fixed epsilon 1 and any h m epsilon there are families of h wise independent functions that can be evaluated in constant time for the standard random access model of computation it is also proven that any such family requires a storage array of m delta random seeds for a suitable delta 1 these seeds can be pseudorandom values precomputed from an initial o h random seeds a simple adaptation yields n epsilon wise independent functions that require n delta storage in many cases where m gg n lower bounds are presented to show that neither storage requirement can be materially reduced previous constructions of random functions having constant evaluation time and sublinear storage exhibited only a constant degree of independence unfortunately the explicit randomized constructions while requiring a constant number of operations are far too slow for any practical application however nonconstructive existence arguments are given which suggest that this factor might be eliminated the problem of eliminating this factor is shown to be equivalent to a fundamental open question in graph theory as a consequence of these constructions many probabilistic algorithms from traditional hashing to ranade s emulation of common pram algorithms can for the first time be shown to achieve up to constant factors their expected asymptotic performance for a programmable albeit formal and currently impractical model of computation and a research direction is now available that may eventually lead to implementations that are fast and provably sound

linear probing with constant independence hashing with linear probing dates back to the 1950s and is among the most widely studied algorithms in recent years it has become one of the most important hash table organizations because it uses the cache of modern computers very well unfortunately previous analyses relied either on complicated and space consuming hash functions or on the unrealistic assumption of free access to a hash function with random and independent function values carter and wegman in their seminal paper on universal hashing have already raised the question of extending their analysis to linear probing however we show in this paper that linear probing using a pairwise independent family may have expected logarithmic cost per operation on the positive side we show that 5 wise independence is enough to ensure constant expected time per operation this resolves the question of finding a space and time efficient hash function that provably ensures good performance for linear probing

sparser johnson lindenstrauss transforms we give two different and simple constructions for dimensionality reduction in e2 via linear mappings that are sparse only an o e fraction of entries in each column of our embedding matrices are non zero to achieve distortion 1 e with high probability while still achieving the asymptotically optimal number of rows these are the first constructions to provide subconstant sparsity for all values of parameters improving upon previous works of achlioptas 2003 and dasgupta et al 2010 such distributions can be used to speed up applications where e2 dimensionality reduction is used

nonlinear feedback systems perturbed by noise steady state probability distributions and optimal control we describe a class of nonlinear feedback systems perturbed by white noise for which explicit formulas for steady state probability densities can be found we show that this class includes what has been called monotemperaturic systems in earlier work and establish relationships with lyapunov functions for the corresponding deterministic systems we also treat a number of stochastic optimal control problems in the case of quantized feedback with performance criteria formulated in terms of the steady state probability density

hierarchical classes model and data analysis a discrete categorical model and a corresponding data analysis method are presented for two way two mode objects attributes data arrays with 0 1 entries the model contains the following two basic components a set theoretical formulation of the relations among objects and attributes a boolean decomposition of the matrix the set theoretical formulation defines a subset of the possible decompositions as consistent with it a general method for graphically representing the set theoretical decomposition is described the data analysis algorithm dubbed hiclas aims at recovering the underlying structure in a data matrix by minimizing the discrepancies between the data and the recovered structure hiclas is evaluated with a simulation study and two empirical applications

asymptotics of the spectral gap with applications to the theory of simulated annealing soit m une variete de riemann compacte connexe a n dimensions de produit interieur et soit u m r 1 une fonction lisse donnee pour 0 on definit 7b l c m c m par 7b l o exp u exp u o ou et sont le gradient et la divergence sur m soit la taille de l ecart entre 0 et le reste du spectre de 7b l on etudie le comportement asymptotique de

projection of a binary criterion into a model of hierarchical classes a formal analysis is made of how to project an attribute criterion into the hierarchical classes model for object by attribute data proposed by de boeck and rosenberg the projection is conceptualized as the prediction of the attribute criterion by means of a logical rule defined on the basis of attribute combinations from the model eliminative and constructive strategies are proposed to find logical rules with maximal predictive power and minimal formula complexity logical analyses of a real data set are reported and compared with a logistic regression to demonstrate the usefulness of the logical strategies and to show the complementarity of logical and probabilistic approaches

models for ordinal hierarchical classes analysis this paper proposes an ordinal generalization of the hierarchical classes model originally proposed by de boeck and rosenberg 1998 any hierarchical classes model implies a decomposition of a two way two mode binary arraym into two component matrices called bundle matrices which represent the association relation and the set theoretical relations among the elements of both modes inm whereas the original model restricts the bundle matrices to be binary the ordinal hierarchical classes model assumes that the bundles are ordinal variables with a prespecified number of values this generalization results in a classification model with classes ordered along ordinal dimensions the ordinal hierarchical classes model is shown to subsume coombs and kao s 1955 model for nonmetric factor analysis an algorithm is described to fit the model to a given data set and is subsequently evaluated in an extensive simulation study an application of the model to student housing data is discussed

hierarchical classes modeling of rating data hierarchical classes hiclas models constitute a distinct family of structural models for n way n mode data all members of the family include n simultaneous and linked classifications of the elements of the n modes implied by the data those classifications are organized in terms of hierarchical if then type relations moreover the models are accompanied by comprehensive insightful graphical representations up to now the hierarchical classes family has been limited to dichotomous or dichotomized data in the present paper we propose a novel extension of the family to two way two mode rating data hiclas r the hiclas r model preserves the representation of simultaneous and linked classifications as well as of generalized if then type relations and keeps being accompanied by a comprehensive graphical representation it is shown to bear interesting relationships with classical real valued two way component analysis and with methods of optimal scaling

the chic model a global model for coupled binary data often problems result in the collection of coupled data which consist of different n way n mode data blocks that have one or more modes in common to reveal the structure underlying such data an integrated modeling strategy with a single set of parameters for the common mode s that is estimated based on the information in all data blocks may be most appropriate such a strategy implies a global model consisting of different n way n mode submodels and a global loss function that is a weighted sum of the partial loss functions associated with the different submodels in this paper such a global model for an integrated analysis of a three way three mode binary data array and a two way two mode binary data matrix that have one mode in common is presented a simulated annealing algorithm to estimate the model parameters is described and evaluated in a simulation study an application of the model to real psychological data is discussed

stochastic algorithms for robustness of control performances in recent years there has been a growing interest in developing statistical learning methods to provide approximate solutions to difficult control problems in particular randomized algorithms have become a very popular tool used for stability and performance analysis as well as for design of control systems however as randomized algorithms provide an efficient solution procedure to the intractable problems stochastic methods bring closer to understanding the properties of the real systems the topic of this paper is the use of stochastic methods in order to solve the problem of control robustness the case of parametric stochastic uncertainty is considered necessary concepts regarding stochastic control theory and stochastic differential equations are introduced then a convergence analysis is provided by means of the chernoff bounds which guarantees robustness in mean and in probability as an illustration the robustness of control performances of example control systems is computed

the local minima problem in hierarchical classes analysis an evaluation of a simulated annealing algorithm and various multistart procedures hierarchical classes models are quasi order retaining boolean decomposition models for n way n mode binary data to fit these models to data rationally started alternating least squares or equivalently alternating least absolute deviations algorithms have been proposed extensive simulation studies showed that these algorithms succeed quite well in recovering the underlying truth but frequently end in a local minimum in this paper we evaluate whether or not this local minimum problem can be mitigated by means of two common strategies for avoiding local minima in combinatorial data analysis simulated annealing sa and use of a multistart procedure in particular we propose a generic sa algorithm for hierarchical classes analysis and three different types of random starts the effectiveness of the sa algorithm and the random starts is evaluated by reanalyzing data sets of previous simulation studies the reported results support the use of the proposed sa algorithm in combination with a random multistart procedure regardless of the properties of the data set under study