a generalizable formulation of conditional logit with diagnostics abstract the conditional logit model is a multinomial logit model that permits the inclusion of choice specific attributes this article shows that the conditional logit model will maximize entropy given a set of attribute value preserving constraints a correspondence between the maximum entropy me and maximum likelihood ml estimates for logit probabilities is established some easily computable and useful diagnostics for logit analysis are provided and it is shown that an evaluation of the relative importance of attributes can be made using the me formulation the me formulation is also generalized to accommodate initial choice probabilities into the logit model an example is given key words choice models entropy kullback leibler discrimination information function relative importance

strong data processing inequalities for input constrained additive noise channels this paper quantifies the intuitive observation that adding noise reduces available information by means of nonlinear strong data processing inequalities consider the random variables w to x to y forming a markov chain where y x z with x and z real valued independent and x bounded in l p norm it is shown that i w y le f i i w x with f i t whenever t 0 if and only if z has a density whose support is not disjoint from any translate of itself a related question is to characterize for what couplings w x the mutual information i w y is close to maximum possible to that end we show that in order to saturate the channel i e for i w y to approach capacity it is mandatory that i w x to infty under suitable conditions on the channel a key ingredient for this result is a deconvolution lemma which shows that postconvolution total variation distance bounds the preconvolution kolmogorov smirnov distance explicit bounds are provided for the special case of the additive gaussian noise channel with quadratic cost constraint these bounds are shown to be order optimal for this case simplified proofs are provided leveraging gaussian specific tools such as the connection between information and estimation i mmse and talagrand s information transportation inequality

approximate non dominated sorting for evolutionary many objective optimization non dominated sorting has widely been adopted and shown to be very effective in dominance based evolutionary multi objective optimization where the number of objectives is two or three in dealing with many objective optimization problems where the number of objectives is more than three however the effectiveness of dominance based evolutionary algorithms will seriously degrade and the computational complexity of non dominated sorting will considerably increase in this paper we propose an approximate non dominated sorting algorithm for many objective optimization where the dominance relationship between two solutions is determined by a maximum of three objective comparisons on top of a sorted population according to one of the objectives the time complexity of the proposed approximate sorting is independent of the number of objectives which significantly enhances the time efficiency in particular when the number of objectives is large to examine the influence of inaccurate sorting on the search performance the proposed approximate sorting method is embedded in three popular dominance based evolutionary many objective optimization algorithms our experimental results on 16 widely used many objective optimization benchmark problems suggest that the approximate non dominated sorting method is not only computationally much more efficient than the state of the art accurate non dominated sorting methods but also leads to better search performance on most test problems used in this study

an approach to efficient planning with numerical fluents and multi criteria plan quality dealing with numerical information is practically important in many real world planning domains where the executability of an action can depend on certain numerical conditions and the action effects can consume or renew some critical continuous resources which in pddl can be represented by numerical fluents when a planning problem involves numerical fluents the quality of the solutions can be expressed by an objective function that can take different plan quality criteria into account we propose an incremental approach to automated planning with numerical fluents and multi criteria objective functions for pddl numerical planning problems the techniques in this paper significantly extend the framework of planning with action graphs and local search implemented in the lpg planner we define the numerical action graph na graph representation for numerical plans and we propose some new local search techniques using this representation including a heuristic search neighborhood for na graphs a heuristic evaluation function based on relaxed numerical plans and an incremental method for plan quality optimization based on particular search restarts moreover we analyze our approach through an extensive experimental study aimed at evaluating the importance of some specific techniques for the performance of the approach and at analyzing its effectiveness in terms of fast computation of a valid plan and quality of the best plan that can be generated within a given cpu time limit overall the results show that our planner performs quite well compared to other state of the art planners handling numerical fluents

blind separation of instantaneous mixture of sources via an independent component analysis in this paper we introduce a procedure for separating a multivariate distribution into nearly independent components based on minimizing a criterion defined in terms of the kullback leibner distance by replacing the unknown density with a kernel estimate we derive useful forms of this criterion when only a sample from that distribution is available we also compute the gradient and hessian of our criteria for use in an iterative minimization setting this gradient to zero yields a set of separating functions similar to the ones considered in the source separation problem except that here these functions are adapted to the observed data finally some simulations are given illustrating the good performance of the method

finding the number of clusters in a dataset one of the most difficult problems in cluster analysis is identifying the number of groups in a dataset most previously suggested approaches to this problem are either somewhat ad hoc or require parametric assumptions and complicated calculations in this article we develop a simple yet powerful nonparametric method for choosing the number of clusters based on distortion a quantity that measures the average distance per dimension between each observation and its closest cluster center our technique is computationally efficient and straightforward to implement we demonstrate empirically its effectiveness not only for choosing the number of clusters but also for identifying underlying structure on a wide range of simulated and real world datasets in addition we give a rigorous theoretical justification for the method based on information theoretic ideas specifically results from the subfield of electrical engineering known as rate distortion theory allow us to describe the behavior of the dist

some inequalities for information divergence and related measures of discrimination inequalities which connect information divergence with other measures of discrimination or distance between probability distributions are used in information theory and its applications to mathematical statistics ergodic theory and other scientific fields we suggest new inequalities of this type often based on underlying identities as a consequence we obtain certain improvements of the well known pinsker inequality our study depends on two measures of discrimination called capacitory discrimination and triangular discrimination the discussion contains references to related research and comparison with other measures of discrimination e g ali silvey csiszar 1996 1966 divergences and in particular the hellinger distance

probability distributions conditioned by the available information gamma distribution and moments given a gamma probability distribution g as the observed distribution and the information available on moments of the random variable the probability distribution is derived such that the g 2 distance between and g is minimum the explicit expressions for the density function minimum g 2 measure and moments are given since in actual applications the available information is in the form of observed frequency distribution and sometimes on mean geometric mean and or variance of random variable these cases are considered in detail followed by a numerical illustration

contraction coefficients for noisy quantum channels generalized relative entropy monotone riemannian metrics geodesic distance and trace distance are all known to decrease under the action of quantum channels we give some new bounds on and relationships between the maximal contraction for these quantities

sms emoa multiobjective selection based on dominated hypervolume abstract the hypervolume measure or s metric is a frequently applied quality measure for comparing the results of evolutionary multiobjective optimisation algorithms emoa the new idea is to aim explicitly for the maximisation of the dominated hypervolume within the optimisation process a steady state emoa is proposed that features a selection operator based on the hypervolume measure combined with the concept of non dominated sorting the algorithm s population evolves to a well distributed set of solutions thereby focussing on interesting regions of the pareto front the performance of the devised s metric selection emoa sms emoa is compared to state of the art methods on two and three objective benchmark suites as well as on aeronautical real world applications

sufficient subalgebras and the relative entropy of states of a von neumann algebra a subalgebram 0 of a von neumann algebram is called weakly sufficient with respect to a pair of states if the relative entropy of and coincides with the relative entropy of their restrictions tom 0 the main result says thatm 0 is weakly sufficient for if and only ifm 0 contains the radon nikodym cocycle d d t other conditions are formulated in terms of generalized conditional expectations and the relative hamiltonian

monotone riemannian metrics and relative entropy on noncommutative probability spaces we use the relative modular operator to define a generalized relative entropy for any convex operator function g on 0 satisfying g 1 0 we show that these convex operator functions can be partitioned into convex subsets each of which defines a unique symmetrized relative entropy a unique family parametrized by density matrices of continuous monotone riemannian metrics a unique geodesic distance on the space of density matrices and a unique monotone operator function satisfying certain symmetry and normalization conditions we describe these objects explicitly in several important special cases including g w log w which yields the familiar logarithmic relative entropy the relative entropies riemannian metrics and geodesic distances obtained by our procedure all contract under completely positive trace preserving maps we then define and study the maximal contraction associated with these quantities

the dobrushin ergodicity coefficient and the ergodicity of noncommutative markov chains abstract in this paper we study the notion of the dobrushin ergodicity coefficient for positive contractions defined on the predual m of von neumann algebra m moreover in terms of this coefficient we prove ergodic type theorems for nonhomogeneous markov chains on m

information indices unification and applications the unified framework of information theoretic statistics was established by kullback 1959 since then numerous information indices have been developed in various contexts this paper represents many of these indices in a unified context the unification thread is the discrimination information function information indices are all logarithmic measures of discrepancy between two probability distributions first we present a summary of informational aspects of the basic information functions a unification of various information theoretic modeling approaches and some explication in terms of traditional measures we then tabulate a unified representation of assortments of information indices developed in the literature for maximum entropy modeling covariate information and influence diagnostics the subjects of these indices include parametric model fitting nonparametric entropy estimation categorical data analysis the linear and exponential family regression and time series the coverage however is not exhaustive the tabulation includes sampling theory and bayesian indices but the focus is on interpretation as descriptive measures and inferential properties are noted tangentially finally applications of some information indices are illustrated through modeling duration data for sprint s churned customer and choice of long distance provider

comparison of channels criteria for domination by a symmetric channel this paper studies the basic question of whether a given channel v can be dominated in the precise sense of being more noisy by a q ary symmetric channel the concept of less noisy relation between channels originated in network information theory broadcast channels and is defined in terms of mutual information or kullback leibler divergence we provide an equivalent characterization in terms of chi 2 divergence furthermore we develop a simple criterion for domination by a q ary symmetric channel in terms of the minimum entry of the stochastic matrix defining the channel v the criterion is strengthened for the special case of additive noise channels over finite abelian groups finally it is shown that domination by a symmetric channel implies via comparison of dirichlet forms a logarithmic sobolev inequality for the original channel

relative entropy under mappings by stochastic matrices the relative g entropy of two finite discrete probability distributions x x1 xn and y y1 yn is defined as hg x y kxkg yk kk 1 where g 1 r is convex and g 0 0 when g t log 1 t then hg x y kxklog xk yk the usual relative entropy let pn x rn ixi 1 xi 0 i our major results is that for any m n column stochastic matrix a the contraction coefficient defined as g a sup hg ax ay hg x y x y pn x y satisfies g a 1 a where a minj k i min aij aik is dobrushin s coefficient of ergodicity consequently g a 1 if and only if a is scrambling upper and lower bounds on g a are established analogous results hold for markov chains in continuous time

a note on approximation of uniform distributions from variable to fixed length codes in this correspondence we prove that the probability distribution induced on the leaves of a tunstall parse tree for a given source is a unique lower bound in the partially ordered set of the probability distributions induced by all possible parse trees with a same number of leaves and ordered according to the majorization partial order we apply this result to the problem of optimally approximating a uniform distribution with flips of a biased coin

estimating divergence functionals and the likelihood ratio by convex risk minimization we develop and analyze m estimation methods for divergence functionals and the likelihood ratios of two probability distributions our method is based on a nonasymptotic variational characterization of f divergences which allows the problem of estimating divergences to be tackled via convex empirical risk optimization the resulting estimators are simple to implement requiring only the solution of standard convex programs we present an analysis of consistency and convergence for these estimators given conditions only on the ratios of densities we show that our estimators can achieve optimal minimax rates for the likelihood ratio and the divergence functionals in certain regimes we derive an efficient optimization algorithm for computing our estimates and illustrate their convergence behavior and practical viability by simulations

assessing granger non causality using nonparametric measure of conditional independence in recent years granger causality has become a popular method in a variety of research areas including engineering neuroscience and economics however despite its simplicity and wide applicability the linear granger causality is an insufficient tool for analyzing exotic stochastic processes such as processes involving non linear dynamics or processes involving causality in higher order statistics in order to analyze such processes more reliably a different approach toward granger causality has become increasingly popular this new approach employs conditional independence as a tool to discover granger non causality without any assumption on the underlying stochastic process this paper discusses the concept of discovering granger non causality using measures of conditional independence and proposes a novel measure of conditional independence in brief the proposed approach estimates the conditional distribution function through a kernel based least square regression approach this paper also explores the strengths and weaknesses of the proposed method compared to other available methods and provides a detailed comparison of these methods using a variety of synthetic data sets

asymptotic distribution theory for nonparametric entropy measures of serial dependence entropy is a classical statistical concept with appealing properties establishing asymptotic distribution theory for smoothed nonparametric entropy measures of dependence has so far proved challenging in this paper we develop an asymptotic theory for a class of kernel based smoothed nonparametric entropy measures of serial dependence in a time series context we use this theory to derive the limiting distribution of granger and lin s 1994 normalized entropy measure of serial dependence which was previously not available in the literature we also apply our theory to construct a new entropy based test for serial dependence providing an alternative to robinson s 1991 approach to obtain accurate inferences we propose and justify a consistent smoothed bootstrap procedure the naive bootstrap is not consistent for our test our test is useful in for example testing the random walk hypothesis evaluating density forecasts and identifying important lags of a time series it is asymptotically locally more powerful than robinson s 1991 test as is confirmed in our simulation an application to the daily s p 500 stock price index illustrates our approach

strong data processing inequalities and phi sobolev inequalities for discrete channels the noisiness of a channel can be measured by comparing suitable functionals of the input and output distributions for instance the worst case ratio of output relative entropy to input relative entropy for all possible pairs of input distributions is bounded from above by unity by the data processing theorem however for a fixed reference input distribution this quantity may be strictly smaller than one giving the so called strong data processing inequalities sdpis the same considerations apply to an arbitrary phi divergence this paper presents a systematic study of optimal constants in the sdpis for discrete channels including their variational characterizations upper and lower bounds structural results for channels on product probability spaces and the relationship between the sdpis and the so called phi sobolev inequalities another class of inequalities that can be used to quantify the noisiness of a channel by controlling entropy like functionals of the input distribution by suitable measures of input output correlation several applications to information theory discrete probability and statistical physics are discussed

on the relative entropy fora any subset of the bounded operators on a hilbert space containing the unit and and restrictions of states on toa ent a the entropy of relative to given the information ina is defined and given an axiomatic characterisation it is compared with ent the relative entropy introduced by umegaki and generalised by various authors which is defined only fora an algebra it is proved that ent and ent s agree on pairs of normal states on an injective von neumann algebra it is also proved that ent always has all the most important properties known for ent s monotonicity concavity w upper semicontinuity etc

the relative information generating function abstract the aim of the paper is to introduce the relative information generating function whose derivatives give well known statistical indices such as the kullback leibler divergence between two probability distributions and watanabe s measure of interdependence it contains golomb s generating function as a particular case and takes simple forms for the binomial and the poisson distributions neither of which fits into golomb s model besides its unifying role the relative information generating function suggests new information indices as for instance the standard deviation of the variation of information

on the selection of measures of distance between probability distributions abstract a notion of relative strength and weakness is defined for the comparison of measures of distance between probability distributions and is shown to be appropriate for use in the selection of distances as quasioptimal design criteria in problems such as signal selection and detector design when the preferred criteria probability of error and asymptotic relative efficiency are intractable csiszar s work on the topological properties of divergences a class of distances also investigated by ali and silvey enables one to identify strong distances and provides a justification for the experimental findings of previous investigators

a non parametric independence test using permutation entropy in the present paper we construct a new simple consistent and powerful test for independence by using symbolic dynamics and permutation entropy as a measure of serial dependence we also give a standard asymptotic distribution of an affine transformation of the permutation entropy under the null hypothesis of independence the test statistic and its standard limit distribution are invariant to any monotonic transformation the test applies to time series with discrete or continuous distributions eventhough the test is based on entropy measures it avoids smoothed non parametric estimation an application to several daily financial time series illustrates our approach

maxallent maximizers of all entropies and uncertainty of uncertainty a b s t r a c t the entropy maximum approach maxent was developed as a minimization of the subjective uncertainty measured by the boltzmann gibbs shannon entropy many new entropies have been invented in the second half of the 20th century now there exists a rich choice of entropies for fitting needs this diversity of entropies gave rise to a maxent anarchism the maxent approach is now the conditional maximization of an appropriate entropy for the evaluation of the probability distribution when our information is partial and incomplete the rich choice of non classical entropies causes a new problem which entropy is better for a given class of applications we understand entropy as a measure of uncertainty which increases in markov processes in this work we describe the most general ordering of the distribution space with respect to which all continuous time markov processes are monotonic the markov order for inference this approach results in a set of conditionally most random distributions each distribution from this set is a maximizer of its own entropy this uncertainty of uncertainty is unavoidable in the analysis of nonequilibrium systems surprisingly the constructive description of this set of maximizers is possible two decomposition theorems for markov processes provide a tool for this description

equivalence of certain entropy contraction coefficients abstract the entropy contraction coefficient associated with the convex function g and column stochastic matrix a is defined as g a sup x y h g ax ay h g x y where h g x y denotes the relative g entropy of the probability vectors x and y we show that for each fixed a all entropy contraction coefficients associated with convex operator functions are equivalent in particular log a w 2 a a where the subscripts log and w 2 refer respectively to the usual logarithmic relative entropy and to the relative g entropy for the function g w w 2 we also give a useful new characterization of w 2 a

random unistochastic matrices an ensemble of random unistochastic orthostochastic matrices is defined by taking squared moduli of elements of random unitary orthogonal matrices distributed according to the haar measure on u n or o n an ensemble of symmetric unistochastic matrices is obtained with use of unitary symmetric matrices pertaining to the circular orthogonal ensemble we study the distribution of complex eigenvalues of bistochastic unistochastic and orthostochastic matrices in the complex plane we compute averages entropy traces over the ensembles of unistochastic matrices and present inequalities concerning the entropies of products of bistochastic matrices

majorization randomness and dependence for multivariate distributions the preorder relation of hardy littlewood and polya 1929 day 1973 and chong 1974 1976 is applied to multivariate probability densities this preorder which is called majorization here can be interpreted as an ordering of randomness when used to compare multivariate densities with the same marginal densities it can be interpreted as an ordering of dependence or conditional dependence results in hickey 1983 1984 and joe 1985 are generalized a relative entropy function is proposed as a measure of dependence or conditional dependence for multivariate densities with the same marginals

an exact expression for the gap in the data processing inequality for f divergences when two input discrete distributions pass through a memoryless channel the data processing inequality expresses the fact that the f divergence decreases as is well known in the case of the kullback leibler divergence the chain rule makes it possible to obtain an exact expression for the gap in the data processing inequality as a conditional divergence in this paper we provide a general expression for the gap valid for any convex function f which generalizes the well known formula for the kullback leibler case the result is a sum of bregman divergences which shows non negativity we then show how this identity may be used to give simple proofs of two classical results in linear algebra

on contraction properties of markov kernels we study lipschitz contraction properties of general markov kernels seen as operators on spaces of probability measures equipped with entropy like distances universal quantitative bounds on the associated ergodic constants are deduced from dobrushin s ergodic coefficient strong contraction properties in orlicz spaces for relative densities are proved under more restrictive mixing assumptions we also describe contraction bounds in the entropy sense around arbitrary probability measures by introducing a suitable dirichlet form and the corresponding modified logarithmic sobolev constants the interest in these bounds is illustrated on the example of inhomogeneous gaussian chains in particular the existence of an invariant measure is not required in general

dissipation of information in channels with input constraints one of the basic tenets in information theory the data processing inequality states that the output divergence does not exceed the input divergence for any channel for channels without input constraints various estimates on the amount of such contraction are known dobrushin s coefficient for the total variation being perhaps the most well known this paper investigates channels with an average input cost constraint it is found that while the contraction coefficient typically equals one no contraction the information nevertheless dissipates a certain nonlinear function the dobrushin curve of the channel is proposed to quantify the amount of dissipation tools for evaluating the dobrushin curve of additive noise channels are developed based on coupling arguments some basic applications in stochastic control uniqueness of gibbs measures and fundamental limits of noisy circuits are discussed as an application it is shown that in the chain of n power constrained relays and gaussian channels the end to end mutual information and maximal squared correlation decay as theta log log n log n which is in stark contrast with the exponential decay in chains of discrete channels similarly the behavior of noisy circuits composed of gates with bounded fan in and broadcasting of information on trees of bounded degree does not experience threshold behavior in the signal to noise ratio snr namely unlike the case of discrete channels the probability of bit error stays bounded away from 1 2 regardless of the snr

bi goal evolution for many objective optimization problems this paper presents a meta objective optimization approach called bi goal evolution bige to deal with multi objective optimization problems with many objectives in multi objective optimization it is generally observed that 1 the conflict between the proximity and diversity requirements is aggravated with the increase of the number of objectives and 2 the pareto dominance loses its effectiveness for a high dimensional space but works well on a low dimensional space inspired by these two observations bige converts a given multi objective optimization problem into a bi goal objective optimization problem regarding proximity and diversity and then handles it using the pareto dominance relation in this bi goal domain implemented with estimation methods of individuals performance and the classic pareto nondominated sorting procedure bige divides individuals into different nondominated layers and attempts to put well converged and well distributed individuals into the first few layers from a series of extensive experiments on four groups of well defined continuous and combinatorial optimization problems with 5 10 and 15 objectives bige has been found to be very competitive against five state of the art algorithms in balancing proximity and diversity the proposed approach is the first step towards a new way of addressing many objective problems as well as indicating several important issues for future development of this type of algorithms

approximation theoretic derivation of logarithmic entropy principles for inverse problems and unique extension of the maximum entropy method to incorporate prior knowledge the minimum distance approach for reconstructing a positive function based on knowledge of finitely many linear functional values is examined two important classes of directed distances for signal processing and statistical inference are discussed by imposing conditions analogous to those satisfied by linear projections in hilbert space two logarithmic entropy principles are derived one of these involves the itakura saito distortion measure of communication theory and uniquely extends burg s maximum entropy method to incorporate prior knowledge the other uses the kullback leibler distance of statistics

eigenvalue inequalities for products of matrix exponentials motivated by models from stochastic population biology and statistical mechanics we prove new inequalities of the form cp eae 2 cp eatr where a and b are n x n complex matrices 1 n 00 and cp is a real valued continuous function of the eigenvalues of its matrix argument for example if a is essentially nonnegative b is diagonal real and cp is the spectral radius then holds if in addition a is irreducible and b has at least two different diagonal elements then the inequality is strict the proof uses kingman s theorem on the logconvexity of the spectral radius lie s product formula and perturbation theory we conclude with conjectures

the diamagnetic coulomb problem at high field strength asymptotic analysis the paper deals with asymptotic expansions in cylindrical coordinates for the schrodinger equation of the diamagnetic coulomb problem with infinite nuclear mass the basis functions introduced by liu and starace are analysed analytical asymptotic expansions are given for the basis functions and eigenvalues belonging to them using these analytical asymptotic expansions are obtained for the coupling coefficients and solutions of the system of second order ordinary differential equations which arise if the wavefunction is expanded in terms of the liu starace basis functions the role of the asymptotic expansions is elucidated for the numerical solution of the non adiabatic approximation and for finding non trivial auto ionizing states

topology selection for particle swarm optimization particle swarm optimization pso uses a social topology for particles to share information among neighbors during optimization a large number of existing literatures have shown that the topology affects the performance of pso and an optimal topology is problem dependent but currently there is a lack of study on this issue in this paper we first analyze a class of deterministic regular topologies with regard to what affect the optimality of algorithmic parameters e g the number of particles and the topological degree so as to provide a guide to topology selections for pso both theoretical analysis and numerical experiments are performed and reported in detail the theoretical analysis reveals that the optimality of algorithmic parameters is dependent on the computational budget available in particular the optimal number of particles increases unstrictly as the computational budget increases while for any fixed number of particles the optimal degree decreases unstrictly as computational budget increases the only condition is that the computational budget cannot exceed a constant measuring the hardness of the benchmark function set with a total of 198 regular topologies and 9 different numbers of particles tested on 90 benchmark functions using a recently reported data profiling technique numerical experiments verify the theoretical derivations based on these results two formulas are developed to help choose optimal topology parameters for increased ease and applicability of pso to real world problems

information distinguishability with application to analysis of failure data abstract in maximum entropy me modeling the information discrepancy between two distributions is measured in terms of their entropy difference in discrimination information statistics the information discrepancy between two distributions is measured in terms of the kullback leibler function i e relative entropy or cross entropy this article presents an equivalence between kullback leibler functions and entropy differences involving an me distribution based on this equivalence the concept of information discrimination id distinguishability is introduced as a unifying framework for the two methods of measuring information discrepancy between distributions applications of id distinguishability as diagnostics for examining robustness of parametric procedures and sensitivity of nonparametric statistics across parametric families of distributions is proposed the equivalence result facilitates estimation of kullback leibler functions in terms of entropy estimates application of the id distinguish

on a class of non separable quantum mechanical eigenvalue problems an idea of born is reviewed and elaborated to non separable quantum mechanical eigenvalue problems in which the schrodinger equation can be solved exactly for a subconfiguration by subconfiguration we mean a subsystem in which one dynamic variable of the whole system is considered as parameter derivations with respect to this variable are omitted the eigenfunctions in the subconfiguration e g the eigenfunctions of a born oppenheimer approximation are used as a basis to expand the eigenfunction of the complete problem by analytical methods it is shown how to construct the complete ensemble of solutions which can be systematically mapped and classified by their analytical behaviour in one of the singularities in a regular singularity a modification of the numerov procedure is given to the numerical solution of the coupled second order ordinary differential equations which arise from our treatment the analytical asymptotic solutions are used to bridge over the asymptotic regions in which the error of the numerov procedure is large as a concrete example the comprehensive asymptotic analysis of the schrodinger equation of a hydrogen like ion in strong homogeneous magnetic field is presented practical methods and computational aspects are discussed and finally a few actual numerical results are reported some energy levels are given as a function of field strength

the least weighted deviation abstract the probability distribution p is found that minimizes the weighted deviation from a given probability distribution q when the range and the mean value of a random variable are known a particular weighted deviation is taken into account only to the extent to which the optimization problem has a unique and calculable solution the commonest known measures of derivation pearson s chi square the kullback leibler divergence etc are reobtained as functional variants of the deviations used in a euclidean normed space an adequate choice of the weights may improve the accuracy in predicting the unknown probability distribution in the 19th century r wolf tossed an unfair die 20 000 times and obtained a probability distribution yielding a mean value of the number of pips equal to 3 5983 knowing can we predict wolf s distribution after tossing his die only 1 000 or fewer times the choice of the weights in the weighted deviation used in prediction proves to be essential even in the case when these weights depend only on the given probability distribution q

nonparametric bayes inference on conditional independence in many application areas a primary focus is on assessing evidence in the data refuting the assumption of independence of y and x conditionally on z with y response variables x predictors of interest and z covariates ideally one would have methods available that avoid parametric assumptions allow y x z to be random variables on arbitrary spaces with arbitrary dimension and accommodate rapid consideration of different candidate predictors as a formal decision theoretic approach has clear disadvantages in this context we instead rely on an encompassing nonparametric bayes model for the joint distribution of y x and z with conditional mutual information used as a summary of the strength of conditional dependence we construct a functional of the encompassing model and empirical measure for estimation of conditional mutual information the implementation relies on a single markov chain monte carlo run under the encompassing model with conditional mutual information for candidate models calculated as a byproduct we provide an asymptotic theory supporting the approach and apply the method to variable selection the methods are illustrated through simulations and criminology applications

the diamagnetic coulomb problem an eigenvalue problem with two singularities by using the angular oblate spheroidal functions as basis functions a bounded wavefunction is constructed in the singularities of the schrodinger equation of the diamagnetic coulomb problem with infinite nuclear mass the expansion in terms of these functions is a model to resolve singularities in an eigenvalue problem of non separable partial differential equations of non relativistic quantum mechanics a comprehensive asymptotic analysis reveals the complete set of asymptotic solutions makes possible a uniform numerical treatment of the bound autoionizing continuum and continuum levels and indicates how to find hitherto unknown low lying stationary levels an example the splitting of the ground level has been found numerically by an iterative shooting method

speeding up many objective optimization by monte carlo approximations many state of the art evolutionary vector optimization algorithms compute the contributing hypervolume for ranking candidate solutions however with an increasing number of objectives calculating the volumes becomes intractable therefore although hypervolume based algorithms are often the method of choice for bi criteria optimization they are regarded as not suitable for many objective optimization recently monte carlo methods have been derived and analyzed for approximating the contributing hypervolume turning theory into practice we employ these results in the ranking procedure of the multi objective covariance matrix adaptation evolution strategy mo cma es as an example of a state of the art method for vector optimization it is empirically shown that the approximation does not impair the quality of the obtained solutions given a budget of objective function evaluations while considerably reducing the computation time in the case of multiple objectives these results are obtained on common benchmark functions as well as on two design optimization tasks thus employing monte carlo approximations makes hypervolume based algorithms applicable to many objective optimization

approximation quality of the hypervolume indicator in order to allow a comparison of otherwise incomparable sets many evolutionary multi objective optimizers use indicator functions to guide the search and to evaluate the performance of search algorithms the most widely used indicator is the hypervolume indicator it measures the volume of the dominated portion of the objective space bounded from below by a reference point though the hypervolume indicator is very popular it has not been shown that maximizing the hypervolume indicator of sets of bounded size is indeed equivalent to the overall objective of finding a good approximation of the pareto front to address this question we compare the optimal approximation ratio with the approximation ratio achieved by two dimensional sets maximizing the hypervolume indicator we bound the optimal multiplicative approximation ratio of n points by 1 q 1 n for arbitrary pareto fronts furthermore we prove that the same asymptotic approximation ratio is achieved by sets of n points that maximize the hypervolume indicator however there is a provable gap between the two approximation ratios which is even exponential in the ratio between the largest and the smallest value of the front we also examine the additive approximation ratio of the hypervolume indicator in two dimensions and prove that it achieves the optimal additive approximation ratio apart from a small ratio

regularity properties and pathologies of position space renormalization group transformations scope and limitations of gibbsian theory we reconsider the conceptual foundations of the renormalization group rg formalism and prove some rigorous theorems on the regularity properties and possible pathologies of the rg map our main results apply to local in position space rg maps acting on systems of bounded spins compact single spin space regarding regularity we show that the rg map defined on a suitable space of interactions formal hamiltonians is always single valued and lipschitz continuous on its domain of definition this rules out a recently proposed scenario for the rg description of first order phase transitions on the pathological side we make rigorous some arguments of griffiths pearce and israel and prove in several cases that the renormalized measure is not a gibbs measure for any reasonable interaction this means that the rg map is ill defined and that the conventional rg description of first order phase transitions is not universally valid for decimation or kadanoff transformations applied to the ising model in dimension d 3 these pathologies occur in a full neighborhood 0 h e of the low temperature part of the first order phase transition surface for block averaging transformations applied to the ising model in dimension d 2 the pathologies occur at low temperatures for arbitrary magnetic field strength pathologies may also occur in the critical region for ising models in dimension d 4 we discuss the heuristic and numerical evidence on rg pathologies in the light of our rigorous theorems in addition we discuss critically the concept of gibbs measure which is at the heart of present day classical statistical mechanics we provide a careful and we hope pedagogical overview of the theory of gibbsian measures as well as the less familiar non gibbsian measures emphasizing the distinction between these two objects and the possible occurrence of the latter in different physical situations we give a rather complete catalogue of the known examples of such occurrences the main message of this paper is that despite a well established tradition gibbsiannessshould not be taken for granted

multi objective evolutionary algorithms based on the summation of normalized objectives and diversified selection most multi objective evolutionary algorithms moeas use the concept of dominance in the search process to select the top solutions as parents in an elitist manner however as moeas are probabilistic search methods some useful information may be wasted if the dominated solutions are completely disregarded in addition the diversity may be lost during the early stages of the search process leading to a locally optimal or partial pareto front beside this the non domination sorting process is complex and time consuming to overcome these problems this paper proposes multi objective evolutionary algorithms based on summation of normalized objective values and diversified selection snov ds the performance of this algorithm is tested on a set of benchmark problems using both multi objective evolutionary programming moep and multi objective differential evolution mode with the proposed method the performance metric has improved significantly and the speed of the parent selection process has also increased when compared with the non domination sorting in addition the proposed algorithm also outperforms ten other algorithms

the diamagnetic coulomb problem at low field strength i analysis of the spherical basis using legendre polynomials spherical basis asymptotic expansions are given in the singularities r 0 of the schrodinger equation of the diamagnetic coulomb problem with infinite nuclear mass it is pointed out that the truncated expansion converges to an unbounded solution at r the complete expansion has either a trivial or a divergent solution only

the diamagnetic coulomb problem at high field strength numerical analysis with the help of the basis proposed by liu and starace the solution of the schrodinger equation with a strong magnetic field is investigated in this paper besides the analytical characterization this basis allows an easy way for numerical treatment of the problem we present an efficient numerical method for determining both the eigenvalues of the basis equation and those of the accompanying coupled equations without the computation of the eigenfunctions the theoretically stable behaviour of the method is confirmed by numerical evaluations for the parameter values which were involved into computations the results are obtained with satisfactory accuracy

relative entropy measures of multivariate dependence abstract there has been a lot of work on measures of dependence or association for bivariate probability distributions or bivariate data these measures usually assume that the variables are both continuous or both categorical in comparison there is very little work on multivariate or conditional measures of dependence the purpose of this article is to discuss measures of multivariate dependence and measures of conditional dependence based on relative entropies these measures are conceptually very general as they can be used for a set of variables that can be a mixture of continuous ordinal categorical and nominal categorical variables for continuous or ordinal categorical variables a certain transformation of relative entropy to the interval 0 1 leads to generalizations of the correlation multiple correlation and partial correlation coefficients if all variables are nominal categorical the relative entropies are standardized to take a maximum of 1 and then transformed so that in the bivar

a novel approach to probability distribution aggregation today s business world is highly competitive and unpredictable so effective decision making is of primary importance however it is difficult to make effective decisions when sufficient information is not available and decision making in such situations involves a high risk of error conventional statistics based approaches to such problems are not effective because in such situations decision making is usually in the hands of a small panel of experts however the expert opinions can be represented by probability distribution functions thus such a problem reduces to the aggregation of a set of probability distribution functions to an aggregated or consensus distribution in this paper we propose a new approach to address this problem the novelties of the proposed approach include 1 the problem is formulated as an optimization problem and 2 the overlapping area between an individual expert s distribution and an aggregated distribution is taken to measure the expertise level of that expert and subsequently to determine the weight of the expert the proposed approach in this paper is illustrated by an example reported in literature handled with the delphi method which also shows the effectiveness of our approach