multi adjoint t concept lattices the t concept lattice is introduced as a set of triples associated to graded tabular information interpreted in a non commutative fuzzy logic following the general techniques of formal concept analysis and based on the works by georgescu and popescu given a non commutative conjunctor it is possible to provide generalizations of the mappings for the intension and the extension in two different ways and this generates a pair of concept lattices in this paper we show that the information common to both concept lattices can be seen as a sublattice of the cartesian product of both concept lattices the multi adjoint framework can be applied to this general t concept lattice and its usefulness is illustrated by a working example

statistical mechanics and phase transitions in clustering a new approach to clustering based on statistical physics is presented the problem is formulated as fuzzy clustering and the association probability distribution is obtained by maximizing the entropy at a given average variance the corresponding lagrange multiplier is related to the temperature and motivates a deterministic annealing process where the free energy is minimized at each temperature critical temperatures are derived for phase transitions when existing clusters split it is a hierarchical clustering estimating the most probable cluster parameters at various average variances

variable threshold concept lattices in this paper the definition of a variable threshold concept lattice is introduced based on a galois connection three kinds of variable threshold concept lattices in which diverse requirements of knowledge discovery can be satisfied by adjusting a threshold are defined the number of formal concepts in a variable threshold concept lattice is far less than that in a fuzzy concept lattice the three kinds of variable threshold concept lattices are constructed between two crisp sets between a crisp set and a fuzzy set and between a fuzzy set and a crisp set their properties are analogous to that of the classical concept lattices and can be induced by the fuzzy concept lattice

set approximations in fuzzy formal concept analysis formal concept analysis and rough set theory are two important tools in knowledge representation and knowledge discovery in relational information systems the purpose of this paper is to study rough set approximations within formal concept analysis in fuzzy environment properties of existent fuzzy concept lattices derived from an adjoint pair of operations are first reviewed and examined based on both lattice theoretic and fuzzy set theoretic operators two new pairs of rough fuzzy set approximations within fuzzy formal contexts are then defined finally properties of the rough fuzzy set approximation operators are presented in detail

notes on bell sejnowski pdf matching neuron this article investigates the behavior of a single input single unit neuron model of the bell sejnowski class which learn through the maximum entropy principle in order to understand its probability density function matching ability

ica using spacings estimates of entropy this paper presents a new algorithm for the independent components analysis ica problem based on an efficient entropy estimator like many previous methods this algorithm directly minimizes the measure of departure from independence according to the estimated kullback leibler divergence between the joint distribution and the product of the marginal distributions we pair this approach with efficient entropy estimators from the statistics literature in particular the entropy estimator we use is consistent and exhibits rapid convergence the algorithm based on this estimator is simple computationally efficient intuitively appealing and outperforms other well known algorithms in addition the estimator s relative insensitivity to outliers translates into superior performance by our ica algorithm on outlier tests we present favorable comparisons to the kernel ica fast ica jade and extended infomax algorithms in extensive simulations we also provide public domain source code for our algorithms

concept lattices with negative information classical formal concept analysis fca extracts represents and manages knowledge from positive information i e its fundamental data model is a binary relation between a set of objects and attributes indicating the presence of a property in an object however some applications require to treat the absence of some property in an object as a negative information to be explicitly represented and managed too although mixed positive and negative information has been addressed in the past in fca such approaches maintain the standard framework which hides the specific semantics and avoids the further use of direct techniques and methods for mixed information in this work the foundations of fca are extended and in particular mixed concept lattices are studied in depth the main result of this work is a characterization theorem specifying in lattice theoretic terms which lattices are isomorphic to a mixed concept lattice

solution algorithms for nonlinear transient heat conduction analysis employing element by element iterative strategies abstract despite continuing advancements in computer technology there are many problems of engineering interest that exceed the combined capabilities of today s numerical algorithms and computational hardware the resources required by traditional finite element algorithms tend to grow geometrically as the problem size is increased thus for the forseeable future there will be problems of interest which cannot be adequately modeled using currently available algorithms for this reason we have undertaken the development of algorithms whose resource needs grow only linearly with problem size in addition these new algorithms will fully exploit the parallel processing capability available in the new generation of multi processor computers the approach taken in the element by element solution procedures is to approximate the global implicit operator by a product of lower order operators this type of product approximation originated with adi techniques and was further refined into the method of fraction steps the current effort involves the use of a more natural operator split for finite element analysis based on element operators this choice of operator splitting based on element operators has several advantages first it fits easily within the architecture of current finite element programs second it allows the development of parallel algorithms finally the computational expense varies only linearly with the number of elements the particular problems considered arise from nonlinear transient heat conduction the nonlinearity enters through both material temperature dependence and radiation boundary conditions the latter condition typically introduces a stiff component in the resultant matrix odes which precludes effective use of explicit solution techniques on the other hand typical implicit techniques involving direct equation solving can be prohibitively expensive instead the matrix equations are solved by combining a modified newton raphson iteration scheme for the nonlinear algebraic problem with an element by element preconditioned conjugate gradient procedure for the linear problems an adaptive intercommunicative strategy is developed for setting termination criteria for the nonlinear and linear equation solving algorithms errors and computational efficiency in the transient solution are controlled by an automatic time step selection strategy the resultant procedure has proven to be both accurate and reliable in the solution of medium size test problems

a multi level conceptual data reduction approach based on the lukasiewicz implication starting from fuzzy binary data represented as tables in the fuzzy relational database in this paper we use fuzzy formal concept analysis to reduce the tables size to only keep the minimal rows in each table without losing knowledge i e association rules extracted from reduced databases are identical at given precision level more specifically we develop a fuzzy extension of a previously proposed algorithm for crisp data reduction without loss of knowledge the fuzzy galois connection based on the lukasiewicz implication is mainly used in the definition of the closure operator according to a precision level which makes data reduction sensitive to the variation of this precision level

code vector density in topographic mappings scalar case the author derives some new results that build on his earlier work 1989 of combining vector quantization vq theory and topographic mapping tm theory a vq model with a noisy transmission medium is used to model the processes that occur in tms which leads to the standard tm training algorithm albeit with a slight modification to the encoding process to emphasize this difference the model is called a topographic vector quantizer tvq in the continuum limit of the one dimensional scalar tvq it is found that the density of code vectors is proportional to p x sup a alpha 1 3 assuming that the transmission medium introduces additive noise with a zero mean symmetric monotically decreasing probability density this result is dramatically different from the result that is predicted when the standard tm training algorithm is used with a uniform symmetric neighborhood n n and it is noted that this difference arises entirely from using minimum distortion rather than nearest neighbor encoding

probabilistic self organizing maps for continuous data the original self organizing feature map did not define any probability distribution on the input space however the advantages of introducing probabilistic methodologies into self organizing map models were soon evident this has led to a wide range of proposals which reflect the current emergence of probabilistic approaches to computational intelligence the underlying estimation theories behind them derive from two main lines of thought the expectation maximization methodology and stochastic approximation methods here we present a comprehensive view of the state of the art with a unifying perspective of the involved theoretical frameworks in particular we examine the most commonly used continuous probability distributions self organization mechanisms and learning schemes special emphasis is given to the connections among them and their relative advantages depending on the characteristics of the problem at hand furthermore we evaluate their performance in two typical applications of self organizing maps classification and visualization

consistent sparse factorization abstract efficient numerical solution of large elliptic systems is often facilitated with an approximate space factorization as a preconditioner for some base iteration a new procedure for generating sparse factors is described this preconditioner captures low modes of the actual system by maintaining the same order local truncation error numerical results supporting theoretical expectations are given

reduction method for concept lattices based on rough set theory and its application rough set theory and formal concept analysis are two complementary mathematical tools for data analysis in this paper we study the reduction of the concept lattices based on rough set theory and propose two kinds of reduction methods for the above concept lattices first we present the sufficient and necessary conditions for justifying whether an attribute and an object are dispensable or indispensable in the above concept lattices based on the above justifying conditions we propose a kind of multi step attribute reduction method and object reduction method for the concept lattices respectively then on the basis of the defined discernibility functions of the concept lattices we propose a kind of single step reduction method for the concept lattices additionally the relations between the attribute reduction of the concept lattices in fca and the attribute reduction of the information system in rough set theory are discussed in detail at last we apply the above multi step attribute reduction method for the concept lattices based on rough set theory to the reduction of the redundant premises of the multiple rules used in the job shop scheduling problem the numerical computational results show that the reduction method for the concept lattices is effective in the reduction of the multiple rules

entropy based kernel mixture modeling for topographic map formation a new information theoretic learning algorithm for kernel based topographic map formation is introduced in the one dimensional case the algorithm is aimed at uniformizing the cumulative distribution of the kernel mixture densities by maximizing its differential entropy a nonparametric differential entropy estimator is used on which normalized gradient ascent is performed both differentiable and nondifferentiable kernels are in principle supported such as gaussian and rectangular on off kernels the relation is shown with joint entropy maximization of the kernel outputs the learning algorithm s performance is assessed and compared with the theoretically optimal performance a fixed point rule is derived for the case of heterogeneous kernel mixtures finally an extension of the algorithm to the multidimensional case is suggested

best asymptotic normality of the kernel density entropy estimator for smooth densities in the random sampling setting we estimate the entropy of a probability density distribution by the entropy of a kernel density estimator using the double exponential kernel under mild smoothness and moment conditions we show that the entropy of the kernel density estimator equals a sum of independent and identically distributed i i d random variables plus a perturbation which is asymptotically negligible compared to the parametric rate n sup 1 2 an essential part in the proof is obtained by exhibiting almost sure bounds for the kullback leibler divergence between the kernel density estimator and its expected value the basic technical tools are doob s submartingale inequality and convexity jensen s inequality

relating attribute reduction in formal object oriented and property oriented concept lattices attribute reduction is an important step in reducing computational complexity in order to extract information from relational systems three of these systems are the formal object oriented and property oriented concept lattices attribute reduction in the last two concept lattices has recently been studied the relation with the first concept lattice is very important since two important independent tools to extract information from databases the formal concept analysis and rough set theory will be related this paper studies attribute reduction in these three frameworks the main results are that the classification of each attribute into absolutely necessary relatively necessary and absolutely unnecessary attributes is independent of the framework considered and that an attribute reduct in one of these relational systems is also an attribute reduct in the others

on attribute reduction in concept lattices the polynomial time discernibility matrix based method becomes the cr method abstract we eliminate unnecessary computations of attribute discernibility sets in discernibility matrix based methods dm methods of attribute reduction in concept lattices we obtain a polynomial time algorithm we call a skim dm method the skim dm method however is merely slightly different from the standard clarification and reduction cr method this result enables us to express a relationship of the cr method and the dm methods the dm methods are basically the cr method with up to an exponential amount of unnecessary computations additionally we experimentally evaluate the skim dm method in comparison with the cr method

attribute reduction in generalized one sided formal contexts in this paper we present a new pair of adjoint mappings between a power set and the direct product of complete lattices the proposed pair of adjoint mappings form a galois connection and the corresponding concept lattice is constructed from a generalized one sided formal context we also propose a lattice keep based attribute reduction approach for generalized one sided formal contexts specifically we present concrete judgment theorems and an algorithm to calculate the attribute reducts in generalized one sided formal contexts furthermore according to the importance of attributes we discuss attributive characteristics for the proposed generalized one sided concept lattice

how to generate ordered maps by maximizing the mutual information between input and output signals a learning rule that performs gradient ascent in the average mutual information between input and an output signal is derived for a system having feedforward and lateral interactions several processes emerge as components of this learning rule hebb like modification and cooperation and competition among processing nodes topographic map formation is demonstrated using the learning rule an analytic expression relating the average mutual information to the response properties of nodes and their geometric arrangement is derived in certain cases this yields a relation between the local map magnification factor and the probability distribution in the input space the results provide new links between unsupervised learning and information theoretic optimization in a system whose properties are biologically motivated

on attribute reduction in concept lattices methods based on discernibility matrix are outperformed by basic clarification and reduction abstract several recent papers have studied the problem of attribute reduction in concept lattices by using a discernibility matrix in various extensions of formal concept analysis we recall the clarification and reduction method by ganter wille and show that it is strictly superior to the methods based on discernibility matrix consequently we show how to adapt the clarification and reduction method to the extensions of formal concept analysis considered in the recent papers

self organizing mixture networks for probability density estimation a self organizing mixture network somn is derived for learning arbitrary density functions the network minimizes the kullback leibler information metric by means of stochastic approximation methods the density functions are modeled as mixtures of parametric distributions a mixture needs not to be homogenous i e it can have different density profiles the first layer of the network is similar to kohonen s self organizing map som but with the parameters of the component densities as the learning weights the winning mechanism is based on maximum posterior probability and updating of the weights is limited to a small neighborhood around the winner the second layer accumulates the responses of these local nodes weighted by the learned mixing parameters the network possesses a simple structure and computational form yet yields fast and robust convergence the network has a generalization ability due to the relative entropy criterion used applications to density profile estimation and pattern classification are presented the somn can also provide an insight to the role of neighborhood function used in the som

sums of functions of nearest neighbor distances moment bounds limit theorems and a goodness of fit test abstract the limiting behavior of sums of functions of nearest neighbor distances is studied for an m dimensional sample a central limit theorem and moment bounds for such sums and an invariance principle for the empirical process of nearest neighbor distances are both established as a consequence the asymptotic behavior of a practicable goodness of fit test is obtained based on nearest neighbor distances

a note on variable threshold concept lattices threshold based operators are reducible to classical concept forming operators the present paper deals with formal concept analysis of data with fuzzy attributes we clarify several points of a new approach of s q fan w x zhang variable threshold concept lattice inf sci accepted for publication which is based on using thresholds in concept forming operators we show that the extent and intent forming operators from s q fan w x zhang inf sci accepted for publication can be defined in terms of basic fuzzy set operations and the original operators as introduced and studied e g in r belohlavek fuzzy galois connections math logic quarterly 45 4 1999 497 504 r belohlavek concept lattices and order in fuzzy logic ann pure appl logic 128 2004 277 298 s pollandt fuzzy begriffe springer verlag berlin heidelberg 1997 as a consequence main properties of the new operators from s q fan w x zhang inf sci accepted for publication including the properties studied in s q fan w x zhang inf sci accepted for publication can be obtained as consequences of the original operators from r belohlavek 1999 r belohlavek 2004 s pollandt 1997

joint entropy maximization in kernel based topographic maps a new learning algorithm for kernel based topographic map formation is introduced the kernel parameters are adjusted individually so as to maximize the joint entropy of the kernel outputs this is done by maximizing the differential entropies of the individual kernel outputs given that the map s output redundancy due to the kernel overlap needs to be minimized the latter is achieved by minimizing the mutual information between the kernel outputs as a kernel the radial incomplete gamma distribution is taken since for a gaussian input density the differential entropy of the kernel output will be maximal since the theoretically optimal joint entropy performance can be derived for the case of nonoverlapping gaussian mixture densities a new clustering algorithm is suggested that uses this optimum as its null distribution finally it is shown that the learning algorithm is similar to one that performs stochastic gradient descent on the kullback leibler divergence for a heteroskedastic gaussian mixture density model

concept analysis via rough set and afs algebra formal concept analysis fca was originally proposed by wille 1982 which is an important theory for data analysis and knowledge discovery afs axiomatic fuzzy set algebra was proposed by liu x liu the fuzzy theory based on afs algebras and afs structure journal of mathematical analysis and applications 217 1998 459 478 x liu the topology on afs algebra and afs structure journal of mathematical analysis and applications 217 1998 479 489 which is a semantic methodology relating to the fuzzy theory combining above two theories we propose afs formal concept which can be viewed as the generalization and development of monotone concept proposed by deogun and saquer 2003 moreover we show that the set of all afs formal concepts forms a complete lattice afs formal concept can be applied to represent the logic operations of queries in information retrieval furthermore we give an approach to find the afs formal concepts whose intents extents approximate any element of afs algebra by virtue of rough set theory

formal concept analysis based on fuzzy granularity base for different granulations this paper introduces granular computing grc into formal concept analysis fca it provides a unified model for concept lattice building and rule extraction on a fuzzy granularity base for different granulations one of the strengths of grc is that larger granulations help to hide some specific details whereas fca in a grc context can prevent losses due to concept lattice complexity however the number of superfluous rules increases exponentially with the scale of the decision context to overcome this we present some inference rules and maximal rules and prove that the set of all these maximal rules is complete and nonredundant thus users who want to obtain decision rules should generate maximal rules examples demonstrate that application of the method is valid and practicable in summary this approach utilizes fca in a grc context and provides a practical basis for data analysis and processing

an information maximization approach to blind separation and blind deconvolution we derive a new self organizing learning algorithm that maximizes the information transferred in a network of nonlinear units the algorithm does not assume any knowledge of the input distributions and is defined here for the zero noise limit under these conditions information maximization has extra properties not found in the linear case linsker 1989 the nonlinearities in the transfer function are able to pick up higher order moments of the input distributions and perform something akin to true redundancy reduction between units in the output representation this enables the network to separate statistically independent components in the inputs a higher order generalization of principal components analysis we apply the network to the source separation or cocktail party problem successfully separating unknown mixtures of up to 10 speakers we also show that a variant on the network architecture is able to perform blind deconvolution cancellation of unknown echoes and reverberation in a speech signal finally we derive dependencies of information transfer on time delays we suggest that information maximization provides a unifying framework for problems in blind signal processing

cooperative information maximization with gaussian activation functions for self organizing maps in this paper we propose a new information theoretic method to produce explicit self organizing maps soms competition is realized by maximizing mutual information between input patterns and competitive units competitive unit outputs are computed by the gaussian function of distance between input patterns and competitive units a property of this gaussian function is that as distance becomes smaller a neuron tends to fire strongly cooperation processes are realized by taking into account the firing rates of neighboring neurons we applied our method to uniform distribution learning chemical compound classification and road classification experimental results confirmed that cooperation processes could significantly increase information content in input patterns when cooperative operations are not effective in increasing information mutual information as well as entropy maximization is used to increase information experimental results showed that entropy maximization could be used to increase information and to obtain clearer soms because competitive units are forced to be equally used on average