axiomatics for fuzzy rough sets abstract a fuzzy t rough set consists of a set x and a t similarity relation r on x where t is a lower semi continuous triangular norm we generalize the farinas prade definition for the upper approximation operator a i x i x of a fuzzy t rough set x r given originally for the special case t min to the case of arbitrary t we propose a new definition for the lower approximation operator a i x i x of x r our definition satisfies the two important identities a a a and a a a as well as a number of other interesting properties we provide axiomatics to fully characterize those upper and lower approximations

feature subset selection by gravitational search algorithm optimization a new method for feature subset selection in machine learning fss mgsa feature subset selection by modified gravitational search algorithm is presented fss mgsa is an evolutionary stochastic search algorithm based on the law of gravity and mass interactions and it can be executed when domain knowledge is not available a wrapper approach over naive bayes id3 k nearest neighbor and support vector machine learning algorithms is used to evaluate the goodness of each visited solution the key to the success of the mgsa is to utilize the piecewise linear chaotic map for increasing its diversity of species and to use sequential quadratic programming for accelerating local exploitation promising results are achieved in a variety of tasks where domain knowledge is not available the experimental results show that the proposed method has the ability of selecting the discriminating input features correctly and can achieve high accuracy of classification which is comparable to or better than well known similar classifier systems furthermore the mgsa is tested on ten functions provided by cec 2005 special session and compared with various modified gravitational search algorithm particle swarm optimization and genetic algorithm the obtained results confirm the high performance of the mgsa in solving various problems in optimization

an axiomatic approach to fuzzy cardinalities of finite fuzzy sets an axiomatic definition of fuzzy cardinalities for finite fuzzy sets defined by means of a convex fuzzy set on the natural numbers is presented in such a way that it includes the fuzzy cardinalities defined by authors like zadeh ralescu dubois wygralak and characterizes the cardinalities that fulfill the additivity property by means of the extended addition of fuzzy numbers such cardinalities result defined by two functions one nondecreasing and the other nonincreasing in a similar way to the scalar cardinality fuzzy sets and systems 110 2000 175

multiple burst correcting array codes two families of binary linear multiple burst correcting array codes are presented the codes consist of all possible n sub 1 n sub 2 arrays over gf 2 where the columns have even parity and the rows belong to any given code of length n sub 2 and minimum distance 2t it is shown that if the bits are read out diagonally instead of horizontally each diagonal followed by the preceding one viewed cyclically then the code can correct up to t bursts of length or tn sub 1 1 if each diagonal is followed by the next one the code can correct up to t bursts of length or 2t n 2 1 for t 1 some of these results are already known decoding algorithms are presented and the case t 1 is discussed in more detail

questions of cardinality of finite fuzzy sets abstract in this paper we focus our attention on finite fuzzy sets a complete simple and easily applicable cardinality theory for them is presented questions of equipotency and non classically understood cardinal numbers of finite fuzzy sets are discussed in detail also problems of arithmetical operations addition subtraction multiplication division and exponentiation on as well as ordering relation for those cardinals are carefully investigated

new array codes for multiple phased burst correction an optimal family of array codes over gf q for correcting multiple phased burst errors and erasures where each phased burst corresponds to an erroneous or erased column in a code array is introduced as for erasures these array codes have an efficient decoding algorithm which avoids multiplications or divisions over extension fields replacing these operations with cyclic shifts of vectors over gf q the erasure decoding algorithm can be adapted easily to handle single column errors as well the codes are characterized geometrically by means of parity constraints along certain diagonal lines in each code array thus generalizing a previously known construction for the special case of two erasures algebraically they can be interpreted as reed solomon codes when q is primitive in gf q the resulting codes become conventional reed solomon codes of length p over gf q sup p 1 in which case the new erasure decoding technique can be incorporated into the berlekamp massey algorithm yielding a faster way to compute the values of any prescribed number of errors

fuzzy feature selection based on min max learning rule and extension matrix in many systems such as fuzzy neural network we often adopt the language labels such as large medium small etc to split the original feature into several fuzzy features in order to reduce the computation complexity of the system after the fuzzification of features the optimal fuzzy feature subset should be selected in this paper we propose a new heuristic algorithm where the criterion is based on min max learning rule and fuzzy extension matrix is designed as the search strategy the algorithm is proved in theory and has shown its high performance over several real world benchmark data sets

a probabilistic definition of a nonconvex fuzzy cardinality the existing methods to assess the cardinality of a fuzzy set with finite support are intended to preserve the properties of classical cardinality in particular the main objective of researchers in this area has been to ensure the convexity of fuzzy cardinalities in order to preserve some properties based on the addition of cardinalities such as the additivity property we have found that in order to solve many real world problems such as the induction of fuzzy rules in data mining convex cardinalities are not always appropriate in this paper we propose a possibilistic and a probabilistic cardinality of a fuzzy set with finite support these cardinalities are not convex in general but they are most suitable for solving problems and contrary to the generalizing opinion they are found to be more intuitive for humans their suitability relies mainly on the fact that they assume dependency among objects with respect to the property to be in a fuzzy set the cardinality measures are generalized to relative ones among pairs of fuzzy sets we also introduce a definition of the entropy of a fuzzy set by using one of our probabilistic measures finally a fuzzy ranking of the cardinality of fuzzy sets is proposed and a definition of graded equipotency is introduced

variable domain fuzzy sets part ii apparatus abstract this is the second part of a two part paper on operations on fuzzy sets that may not share the same domain of definition part i described several ways of representing variable domain fuzzy sets and introduced a few basic variable domain fuzzy set operations the present largely self contained part ii further develops the theory of variable domain fuzzy sets based primarily on the representation of variable domains by a dummy membership degree that indicates the out of domain assignment error we first describe several families of algebraic operations on the extended set of membership degrees these are later employed in the investigation of basic notions of variable domain fuzzy set theory the fuzzy set operations studied in this paper include variable domain variants of unions and intersections kernels and supports heights and plinths equalities and inclusions cartesian products and fuzzy relational compositions besides the initial examination of these variable domain notions our aim is to demonstrate the viability of variable domain fuzzy set theory and highlight the similarity as well as differences between the fixed and variable domain treatments of fuzzy sets

linguistic quantifiers modeled by sugeno integrals since quantifiers have the ability of summarizing the properties of a class of objects without enumerating them linguistic quantification is a very important topic in the field of high level knowledge representation and reasoning this paper introduces a new framework for modeling quantifiers in natural languages in which each linguistic quantifier is represented by a family of fuzzy measures and the truth value of a quantified proposition is evaluated by using sugeno s integral this framework allows us to have some elegant logical properties of linguistic quantifiers we compare carefully our new model of quantification and other approaches to linguistic quantifiers a set of criteria for linguistic quantification was proposed in the previous literature the relationship between these criteria and the results obtained in the present paper is clarified some simple applications of the sugeno s integral semantics of quantifiers are presented

a rough fuzzy approach for generating classification rules the generation of effective feature pattern based classification rules is essential to the development of any intelligent classifier which is readily comprehensible to the user this paper presents an approach that integrates a potentially powerful fuzzy rule induction algorithm with a rough set assisted feature reduction method the integrated rule generation mechanism maintains the underlying semantics of the feature set through the proposed integration the original rule induction algorithm or any other similar technique that generates descriptive fuzzy rules which is sensitive to the dimensionality of the dataset becomes usable on classifying patterns composed of a moderately large number of features the resulting learned ruleset becomes manageable and may outperform rules learned using more features this as demonstrated with successful realistic applications makes the present approach effective in handling real world problems

time to depth conversion and velocity estimation by image wavefront propagation abstracta new strategy for time to depth conversion and interval velocity estimation is based entirely on image wavefront propagation without the need to follow individual image rays the procedure has three main features 1 it computes the velocity field and the traveltime directly allowing us to dispense with dynamic ray tracing 2 it requires only the knowledge of the image wavefront at the previous time step and 3 it inherently smooths the image wavefront inhibiting the formation of caustics as a consequence the method tends to be faster than the usual techniques and does not carry the constraints and limitations inherent to common ray tracing strategies synthetic tests using a gaussian velocity anomaly as well as the marmousi velocity model and two smoothed versions of it show the feasibility of the method a field data example demonstrates the use of different numerical procedures our results indicate that the present strategy can be used to construct reasonable depth velocity models t

analysis and algorithms for a regularized cauchy problem arising from a non linear elliptic pde for seismic velocity estimation in the present work we derive and study a non linear elliptic pde coming from the problem of estimation of sound speed inside the earth the physical setting of the pde allows us to pose only a cauchy problem and hence is ill posed however we are still able to solve it numerically on a long enough time interval to be of practical use we used two approaches the first approach is a finite difference time marching numerical scheme inspired by the lax friedrichs method the key features of this scheme is the lax friedrichs averaging and the wide stencil in space the second approach is a spectral chebyshev method with truncated series we show that our schemes work because of i the special input corresponding to a positive finite seismic velocity ii special initial conditions corresponding to the image rays iii the fact that our finite difference scheme contains small error terms which damp the high harmonics truncation of the chebyshev series and iv the need to compute the solution only for a short interval of time we test our numerical schemes on a collection of analytic examples and demonstrate a dramatic improvement in accuracy in the estimation of the sound speed inside the earth in comparison with the conventional dix inversion our test on the marmousi example confirms the effectiveness of the proposed approach

meta theorems on inequalities for scalar fuzzy set cardinalities we present meta theorems stating general conditions ensuring that certain inequalities for cardinalities of ordinary sets are preserved under fuzzification when adopting a scalar approach to fuzzy set cardinality the conditions pertain to the commutative conjunctor used for modelling fuzzy set intersection in particular this conjunctor should fulfil a number of bell type inequalities the advantage of these meta theorems is that repetitious calculations can be avoided this is illustrated in the demonstration of the lukasiewicz transitivity of fuzzified versions of the simple matching coefficient and the jaccard coefficient or equivalently the triangle inequality of the corresponding dissimilarity measures

on fuzzy type theory this paper is a generalization of classical simple type theory we have developed a formal system of fuzzy type theory which differs from the classical one essentially in extension of truth values from two to infinitely many the structure of truth values is assumed to be an imtl algebra residuated lattice with prelinearity and double negation extended by the baaz delta operation various properties of fuzzy type theory are proved including its completeness

fuzzy probabilistic approximation spaces and their information measures rough set theory has proven to be an efficient tool for modeling and reasoning with uncertainty information by introducing probability into fuzzy approximation space a theory about fuzzy probabilistic approximation spaces is proposed in this paper which combines three types of uncertainty probability fuzziness and roughness into a rough set model we introduce shannon s entropy to measure information quantity implied in a pawlak s approximation space and then present a novel representation of shannon s entropy with a relation matrix based on the modified formulas some generalizations of the entropy are proposed to calculate the information in a fuzzy approximation space and a fuzzy probabilistic approximation space respectively as a result uniform representations of approximation spaces and their information measures are formed with this work

iterative relief for feature weighting algorithms theories and applications relief is considered one of the most successful algorithms for assessing the quality of features in this paper we propose a set of new feature weighting algorithms that perform significantly better than relief without introducing a large increase in computational complexity our work starts from a mathematical interpretation of the seemingly heuristic relief algorithm as an online method solving a convex optimization problem with a margin based objective function this interpretation explains the success of relief in real application and enables us to identify and address its following weaknesses relief makes an implicit assumption that the nearest neighbors found in the original feature space are the ones in the weighted space and relief lacks a mechanism to deal with outlier data we propose an iterative relief i relief algorithm to alleviate the deficiencies of relief by exploring the framework of the expectation maximization algorithm we extend i relief to multiclass settings by using a new multiclass margin definition to reduce computational costs an online learning algorithm is also developed convergence analysis of the proposed algorithms is presented the results of large scale experiments on the uci and microarray data sets are reported which demonstrate the effectiveness of the proposed algorithms and verify the presented theoretical results

graph theoretical analysis of alzheimer s disease tools available in graph theory have been recently applied to signals recorded from the human brain where its cognitive functions are linked to topological properties of connectivity networks in this work we consider resting state electroencephalography eeg signals recorded from healthy subjects and patients suffering from alzheimer s disease ad in two conditions eyes open and eyes closed the eegs are used to construct functional brain networks in which the nodes are eeg sensor locations and edges represent functional connectivity between them the networks are then tested for a number of neurobiologically relevant graph theory metrics the analyses show that the network properties are stable across all conventional frequency bands ad brains in eyes closed condition show significantly reduced local efficiency and modularity measures p 0 05 wilcoxon s ranksum test we then use the network metrics as features for discriminating ad from healthy controls three feature selection methods genetic algorithms ga binary particle swarm optimization bpso and social impact theory based optimization sito are used to select the best feature set ga with support vector machines as classifier results in an accuracy of 83 in eyes close beta band the set of optimal features include edge betweenness centrality global efficiency modularity and synchronizability

velocity independent marchenko focusing in time and depth imaging domains for media with mild lateral heterogeneity abstractthe marchenko method retrieves green s functions between the acquisition surface and any arbitrary point in the medium the process generally involves solving an inversion starting with an initial focusing function e g a direct wave green s function from the desired subsurface position typically obtained using an approximate velocity model we have formulated the marchenko method in the time imaging domain in that domain we recognize that the traveltime of the direct wave green s function is related to the cheop s traveltime pyramid commonly used in time domain processing which in turn can be readily obtained from the local slopes of the common midpoint gathers this observation allows us to substitute the velocity model based initial focusing operator with that from a data driven slope estimation process moreover we found that working in the time imaging domain allows for the specification of the desired subsurface position in terms of vertical time which is connected to the cartesian depth position via the time to depth conversion our results suggest that the prior velocity model is only required when specifying the position in depth but this requirement can be circumvented by making use of the time imaging domain within its usual assumptions e g mild lateral heterogeneity provided that those assumptions are satisfied the estimated green s functions from the proposed method have comparable quality to those obtained with the knowledge of a prior velocity model

constructive and axiomatic approaches of fuzzy approximation operators this paper presents a general framework for the study of rough set approximation operators in fuzzy environment in which both constructive and axiomatic approaches are used in constructive approach a pair of lower and upper generalized fuzzy rough and rough fuzzy respectively approximation operators is first defined the representations of both fuzzy rough approximation operators and rough fuzzy approximation operators are then presented the connections between fuzzy and crisp respectively relations and fuzzy rough and rough fuzzy respectively approximation operators are further established in axiomatic approach various classes of fuzzy approximation operators are characterized by different sets of axioms the minimal axiom sets of fuzzy approximation operators guarantee the existence of certain types of fuzzy or crisp relations producing the same operators

effect of rule weights in fuzzy rule based classification systems this paper examines the effect of rule weights in fuzzy rule based classification systems each fuzzy if then rule in our classification system has antecedent linguistic values and a single consequent class we use a fuzzy reasoning method based on a single winner rule in the classification phase the winner rule for a new pattern is the fuzzy if then rule that has the maximum compatibility grade with the new pattern when we use fuzzy if then rules with certainty grades the winner is determined as the rule with the maximum product of the compatibility grade and the certainty grade in this paper the effect of rule weights is illustrated by drawing classification boundaries using fuzzy if then rules with without certainty grades it is also shown that certainty grades play an important role when a fuzzy rule based classification system is a mixture of general rules and specific rules through computer simulations we show that comprehensible fuzzy rule based systems with high classification performance can be designed without modifying the membership functions of antecedent linguistic values when we use fuzzy if then rules with certainty grades

feature selection based on a modified fuzzy c means algorithm with supervision in this paper we propose a new approach to feature selection based on a modified fuzzy c means algorithm with supervision mfcms mfcms completes the unsupervised learning of classical fuzzy c means with labeled patterns the labeled patterns allow mfcms to accurately model the shape of each cluster and consequently to highlight the features which result to be particularly effective to characterize a cluster these features are distinguished by a low variance of their values for the patterns with a high membership degree to the cluster if with respect to these features the distance between the prototype of the cluster and the prototypes of the other clusters is high then these features have the property of discriminating between the cluster and the other clusters to take these two aspects into account for each cluster and each feature we introduce a purposely defined index the higher the value of the index the higher the discrimination capability of the feature for the cluster we execute mfcms on the training set considering all patterns as labeled then we retain the features which are associated at least for one cluster with an index larger than a threshold we applied mfcms to several real world pattern classification benchmarks we used the well known k nearest neighbors as learning algorithm we show that feature selection performed by mfcms achieved an improvement in generalization on all data sets

fuzzy cardinality and the modeling of imprecise quantification abstract the concept of cardinality of a fuzzy set has received attention from several researchers and has been defined in several apparently independent manners a systematic investigation of this notion is performed which unifies and improves previous attempts the cardinality of a fuzzy set viewed as a fuzzy integer is related to scalar cardinality indices the closely related question of the probability of a fuzzy event is dealt with lastly the usefulness of fuzzy cardinality for meaning representation of statements or queries involving fuzzy linguistic quantifiers is emphasized

membership margin based feature selection for mixed type and high dimensional data definition of membership margin based objective function weighting the antecedents of fuzzy if then rules to improve classification performance evaluate the feature importance within membership margin framework feature selection algorithm for mixed type and high dimensional data the effectiveness of the approach is proved through an extensive experimental study the present paper describes a new feature weighting method based on a membership margin distinctive properties of the proposed method include its capability to process problems characterized by mixed type data quantitative qualitative and interval as well as a huge number of features the key idea is to map simultaneously all the features of different types into a common space the membership space once all features are represented in a homogeneous space a feature weighting task can be performed in unified way this weighting approach is integrated here within a fuzzy classifier through a fuzzy rule weighted concept in order to improve its performance each antecedent fuzzy set in the fuzzy if then rule is weighted to characterize the importance of each proposition and therefore its corresponding feature weight estimation process is based on membership margin maximization to estimate a fuzzy weight of each feature in the membership space experiments on low and high dimensional real world datasets demonstrate that the proposed approach can improve significantly the performance of the fuzzy rule based as well as other state of the art classifiers and can even outperform classical feature weighting approaches in particular we show that this approach can yield meaningful results on two real world applications for cancer prognosis and industrial process diagnosis

feature subset selection in large dimensionality domains searching for an optimal feature subset from a high dimensional feature space is known to be an np complete problem we present a hybrid algorithm saga for this task saga combines the ability to avoid being trapped in a local minimum of simulated annealing with the very high rate of convergence of the crossover operator of genetic algorithms the strong local search ability of greedy algorithms and the high computational efficiency of generalized regression neural networks we compare the performance over time of saga and well known algorithms on synthetic and real datasets the results show that saga outperforms existing algorithms

the set of fuzzy rational numbers and flexible querying a fuzzy bag is a bag in which each occurrence of an element is associated with a grade of membership this notion can be viewed as a generalization of the concepts of set fuzzy set and bag the set of fuzzy integers n f provides a general characterization in which all these different concepts are treated in a uniform way and can then be composed in the field of databases the use of fuzzy bags is motivated by their ability to manage both quantities and preferences however n f becomes too restricted a framework when dealing with queries based on difference or division operations so a more general structure based on the set of fuzzy relative integers z f in which exact differences can be performed has been first developed in this paper we carry on with this approach and we extend z f to the set of fuzzy rational numbers q f this context leads to define a closed system of multiplicative operations and allows to perform exact divisions applied to flexible querying of databases q f and the notion of division on fuzzy numbers allow to generalize the relational division they define a sound basis for queries involving ratios between quantities

rule weight specification in fuzzy rule based classification systems this paper shows how the rule weight of each fuzzy rule can be specified in fuzzy rule based classification systems first we propose two heuristic methods for rule weight specification next the proposed methods are compared with existing ones through computer simulations on artificial numerical examples and real world pattern classification problems simulation results show that the proposed methods outperform the existing ones in the case of multiclass pattern classification problems with many classes

distributed representation of fuzzy rules and its application to pattern classification abstract this paper introduces the concept of distributed representation of fuzzy rules and applies it to classification problems distributed representation is implemented by superimposing many fuzzy rules corresponding to different fuzzy partitions of a pattern space this means that we simulatenously employ many fuzzy rule tables corresponding to different fuzzy partitions in fuzzy inference in order to apply distributed representation of fuzzy rules to pattern classification problems we first propose an algorithm to generate fuzzy rules from numerical data next we propose a fuzzy inference method using the generated fuzzy rules the classification power of distributed representation is compared with that of ordinary fuzzy rules which can be viewed as local representation

uncertainty measure on fuzzy partitions in this paper we face the problem of constructing the uncertainty or perhaps expected information of a fuzzy partition in terms of the crisp measures of its cuts we solve the problem in the cases in which the measure is either compositive or branching

a fast feature weighting algorithm of data gravitation classification the data gravitation classification dgc model is a new classification method that has gained great research interest in recent years feature weights are the key parameters in dgc models because the classification performances of these models are very sensitive to such feature weights the available dgc models use wrapper like algorithms to obtain their optimised feature weights although such algorithms produce high classification accuracies but they contribute to the high computational complexities of the dgc models in this study we propose a fast feature weight algorithm for dgc models called ffw dgc we use the concepts of feature discrimination and redundancy to measure the importance of a feature after which two fuzzy subsets are constructed to respectively represent these concepts next we combine the two fuzzy subsets to compute the feature weights used in gravitational computing we conduct our experiments on 25 standard data sets and 22 imbalance data sets and compare ffw dgc with 11 kinds of classifiers including the swarm intelligence based dgc pso dgc model competitive results of ffw dgc demonstrate that it can obtain high classification accuracies but also hundreds of times of speedup ratios compared with pso dgc

rough sets decision algorithms and bayes theorem abstract rough set based data analysis starts from a data table called an information system the information system contains data about objects of interest characterized in terms of some attributes often we distinguish in the information system condition and decision attributes such information system is called a decision table the decision table describes decisions in terms of conditions that must be satisfied in order to carry out the decision specified in the decision table with every decision table a set of decision rules called a decision algorithm can be associated it is shown that every decision algorithm reveals some well known probabilistic properties in particular it satisfies the total probability theorem and bayes theorem these properties give a new method of drawing conclusions from data without referring to prior and posterior probabilities inherently associated with bayesian reasoning

generalized cardinal numbers and operations on them abstract in this paper we present a cardinality theory for so called vaguely defined objects which are mild generalizations of fuzzy sets obtained by introducing lower and upper approximations of the membership functions the theory makes use of the sentential calculus in the infinite valued ukasiewicz logic and can be applied to fuzzy sets with arbitrary supports some kinds of twofold fuzzy sets partial sets etc the notion of equipotency of vaguely defined objects is studied in detail the resulting generalized cardinal numbers are convenient tools for describing the powers of vaguely defined objects in the second part of the paper basic operations on the generalized cardinals are defined and carefully investigated similarities and anomalies in comparison with the classical arithmetic of the usual cardinals are indicated

unsupervised feature selection using feature similarity in this article we describe an unsupervised feature selection algorithm suitable for data sets large in both dimension and size the method is based on measuring similarity between features whereby redundancy therein is removed this does not need any search and therefore is fast a new feature similarity measure called maximum information compression index is introduced the algorithm is generic in nature and has the capability of multiscale representation of data sets the superiority of the algorithm in terms of speed and performance is established extensively over various real life data sets of different sizes and dimensions it is also demonstrated how redundancy and information loss in feature selection can be quantified with an entropy measure

an axiomatic approach to scalar cardinalities of fuzzy sets abstract we present an axiomatic approach to scalar cardinalities of fuzzy sets which is based on a system of three simple postulates a characterization theorem for those cardinalities is given the infinite family of possible scalar cardinalities of a fuzzy set generated by the postulates contains as particular cases all standard concepts of scalar cardinality e g the sigma count of a fuzzy set the cardinality of its core or support and the cardinality of its t level set

a graded approach to cardinal theory of finite fuzzy sets part i in this article we propose a fuzzy class relation assigning to each pair of finite fuzzy sets a degree to which they are equipollent which indicates that they have the same number of elements the concepts of fuzzy sets and fuzzy classes in the class of all sets in zfc are introduced and several standard relations and constructions such as the fuzzy power set and exponentiation are defined a functional approach to the cardinal theory of finite fuzzy sets based on graded equipollence is shown and a relation to generalized cardinals and wygralak s cardinal theory of finite fuzzy sets defined over triangular norms is demonstrated

on the entropy of fuzzy measures fuzzy measures provides a structure for modeling the knowledge available about variables whose values are unknown and uncertain a large class of different types of uncertainty can be represented in this framework in this work we provide a measure of entropy that can be used to calculate the amount of uncertainty associated with a fuzzy measure

a reformulation of entropy in the presence of indistinguishability operators this paper deals with the measurement of entropy when an indistinguishability relation on the set of events has been defined our approach states that entropy could be measured in terms of the observed distinguishability of the set of events in this sense the observer paradigm is introduced and definitions for joint and conditional entropy under this paradigm are given we also present some interesting properties and relationships with shannon s entropy measure

fuzzy cardinals based on the generalized equality of fuzzy subsets the purpose of this paper is to present a new concept of the cardinality of finite fuzzy subsets which is based on the notion of the generalized equality the proposed idea will be compared with several approaches which were earlier offered in fuzzy literature using the new definition of the fuzzy cardinality we next introduce a notion of fuzzy semicardinals i e we construct fuzzy natural numbers defining degrees to which a finite fuzzy subset has respectively at most least and less than more than k elements various properties of different fuzzy cardinals and semicardinals as well as some interdependences between them are also given

on triangular norm based generalized cardinals and singular fuzzy sets there are three basic types of triangular norm based generalized cardinals of fuzzy sets namely generalized fgcounts flcounts and fecounts all of them are convex fuzzy sets of usual cardinal numbers our attention will be focused on generalized fecounts if nonstrict archimedean triangular norms are involved generalized fecounts of many fuzzy sets become the zero function those singular fuzzy sets being totally dissimilar to any set of any cardinality and their properties are the subject of this paper a relationship between singularity and fuzziness measures suited to fuzzy sets with complements induced by a strict negation will also be discussed

an axiomatic characterization of a fuzzy generalization of rough sets in rough set theory the lower and upper approximation operators defined by a fixed binary relation satisfy many interesting properties several authors have proposed various fuzzy generalizations of rough approximations in this paper we introduce the definitions for generalized fuzzy lower and upper approximation operators determined by a residual implication then we find the assumptions which permit a given fuzzy set theoretic operator to represent a upper or lower approximation derived from a special fuzzy relation different classes of fuzzy rough set algebras are obtained from different types of fuzzy relations and different sets of axioms of fuzzy set theoretic operator guarantee the existence of different types of fuzzy relations which produce the same operator finally we study the composition of two approximation spaces it is proved that the approximation operators in the composition space are just the composition of the approximation operators in the two fuzzy approximation spaces

measuring properties of fuzzy sets a general technique and its use in fuzzy query evaluation abstract a fuzzy set can be viewed as a weighted max combination or as a convex combination of a family of nested ordinary sets the latter point of view allows the interpretation of a fuzzy set as a consonant random set then any numerical set function applied to a fuzzy set leads to a result which is a random number the expectation attached to this random number provides a scalar evaluation of the fuzzy set with respect to the considered set function this approach is very general and can be used for various kinds of measurements cardinality confidence measures attached to the occurrence of events diameter or perimeter of a fuzzy region distance between two sets etc also the fuzziness of the set to evaluate naturally induces a fuzzy set of possible evaluations and expected values of the lower and upper bounds of this fuzzy set of numbers can be computed the weighted max combination view which leads to another type of scalar evaluation is also briefly discussed in this paper the above approach is more particularly applied to the treatment of queries asking for some global evaluation of a set of items specified by means of a fuzzy property this kind of query can be managed by languages like sql in case of a crisp set of items when this set is fuzzily specified a scalar answer can still be provided as well as a fuzzy valued or interval valued one if we want to keep track of the effect of the fuzziness of the query the case where the data base contains precise information but also the more general case of fuzzy information data bases are considered

using simulated annealing to optimize the feature selection problem in marketing applications the feature selection also specification problem is concerned with finding the most influential subset of predictors in predictive modeling from a much larger set of potential predictors that can contain hundreds of predictors the problem belongs to the realm of combinatorial optimization where the objective is to find the subset of variables that optimize the value of some goodness of fit function due to the dimensionality of the problem the feature selection problem belongs to the group of np hard problems most of the available predictors are noisy or redundant and add very little if any to the prediction power of the model using all the predictors in the model often results in strong over fitting and very poor predictions constructing a prediction model by checking out all possible subsets is impractical due to computational volume looking on the contribution of each predictor separately is not accurate because it ignores the inter correlations between predictors as a result no analytic solution is available for the feature selection problem requiring that one resorts to heuristics in this paper we employ the simulated annealing sa approach which is one of the leading stochastic search methods for specifying a large scale linear regression model the sa results are compared to the results of the more common stepwise regression swr approach for model specification the models are applied on realistic data sets in database marketing we also use simulated data sets to investigate what data characteristics make the swr approach equivalent to the supposedly more superior sa approach

aggregation of subjective evaluations based on discrete fuzzy numbers in this paper aggregation functions defined on the set of all discrete fuzzy numbers whose support is a subset of consecutive natural numbers are introduced and the particular cases of uninorms and nullnorms are studied in detail these aggregation functions are constructed from discrete aggregation functions defined on a finite chain and they are applied to the aggregation of subjective evaluations

cardinality quantifiers and the aggregation of fuzzy criteria abstract in this paper we are interested in counting the number of elements of a fuzzy set we study two concepts of fuzzy cardinality one which gives the answer as a fuzzy set or fuzzy number and the other which gives an ordinary integer relationships between these concepts and others which have been studied in the literature e g sigma count are also investigated some properties of the new concepts are studied such as fuzzy convexity monotonicity and additivity the concepts of fuzzy cardinality are important for a better understanding of fuzzy quantified rules of the form qx s are a or qa s are b where q is a fuzzy quantifier and a b are fuzzy sets we study such rules as well as aggregation rules of the form q of a1 ap contain x

a multi objective evolutionary algorithm for feature selection based on mutual information with a new redundancy measure a new feature redundancy measurement based on mutual information was proposed a multi objective evolutionary algorithm for feature selection was presented pareto optimality was used to evaluate candidate feature subsets and find compact feature subsets experiments showed that our algorithm could select compact and discriminative feature subsets feature selection is an important task in data mining and pattern recognition especially for high dimensional data it aims to select a compact feature subset with the maximal discriminative capability the discriminability of a feature subset requires that selected features have a high relevance to class labels whereas the compactness demands a low redundancy within the selected feature subset this paper defines a new feature redundancy measurement capable of accurately estimating mutual information between features with respect to the target class mifs cr based on a relevance measure and this new redundancy measure a multi objective evolutionary algorithm with class dependent redundancy for feature selection mecy fs is presented the mecy fs algorithm employs the pareto optimality to evaluate candidate feature subsets and finds compact feature subsets with both the maximal relevance and the minimal redundancy experiments on benchmark datasets are conducted to validate the effectiveness of the new redundancy measure and the mecy fs algorithm is verified to be able to generate compact feature subsets with a high predictive capability

two dimensional half rate codes using two dimensional finite field filter banks this correspondence introduces two dimensional 2 d filter bank codes tdfbcs the synthesis bank of a two channel two dimensional filter bank over a finite field is used to design the 2 d code and the corresponding analysis bank is used to generate the syndrome of the code first we study the encoder of half rate tdfbcs and show that these linear codes are lattice cyclic it is proven that any 2 d lattice cyclic code can also be generated by a 2 d filter bank second we introduce a methodology to design tdfbcs over binary erasure channels these codes have a simple and efficient maximum likelihood ml decoding algorithm for burst erasures we show that half rate tdfbcs of dimensions n1 x n2 can recover burst erasures of size up to n1 x n2 2 and n1 2 x n2 using the proposed decoding technique finally we present examples of tdfbcs that satisfy the reiger bound with equality i e they are capable of correcting any burst of size n1 x n2 2

fuzzy quantification a state of the art abstract quantified sentences are a very powerful notion for modelling statements in natural language nl but in practice they have been used to solve several problems this paper is intended to offer a global view of the development on this branch until now focusing in the different approaches dealing with quantification specially those involving imprecision called fuzzy quantification we put attention to the different mechanisms for defining them the evaluation methods for measuring their fulfilment as well as the properties they should satisfy

entropy of fuzzy partitions a general model abstract the entropy of fuzzy partitions with regard to a fuzzy measure is analyzed a general model based on pseudo arithmetical operations and g functions will be created the entropy of fuzzy partitions introduced by ruspini inform and control 15 1969 22 and dumitrescu j math anal appl 176 1993 359 is shown to be reasonable only up to an isomorphism

time to depth conversion and seismic velocity estimation using time migration velocity the objective was to build an efficient algorithm 1 to estimate seismic velocity from time migration velocity and 2 to convert time migrated images to depth we established theoretical relations between the time migration velocity and seismic velocity in two and three dimensions using paraxial ray tracing theory the relation in two dimensions implies that the conventional dix velocity is the ratio of the interval seismic velocity and the geometric spreading of image rays we formulated an inverse problem of finding seismic velocity from the dix velocity and developed a numerical procedure for solving it the procedure consists of two steps 1 computation of the geometric spreading of image rays and the true seismic velocity in time domain coordinates from the dix velocity 2 conversion of the true seismic velocity from the time domain to the depth domain and computation of the transition matrices from time domain coordinates todepth for step 1 we derived a partial differential equation pde i

fuzzy sets with triangular norms and their cardinality theory the most advanced and adequate approach to the question of cardinality of a fuzzy set seems to be that offering a fuzzy perception of cardinality the resulting convex fuzzy sets of usual cardinals of nonnegative integers in the finite case are then called generalized cardinal numbers three types of them are of special interest and importance namely fgcounts flcounts and fecounts in this paper first we show that their original forms are suitable only for fuzzy sets with the classical min and max operations second we propose an appropriate generalization to fuzzy sets with triangular norms and conorms further we investigate the resulting generalized fgcounts flcounts and fecounts from the viewpoint of the corresponding equipotency relations inequalities and arithmetical operations

feature selection with svd entropy some modification and extension many approaches have been developed for dimensionality reduction these approaches can broadly be categorized into supervised and unsupervised methods in case of supervised dimensionality reduction for any input vector the target value is known which can be a class label also in a supervised approach our objective is to select a subset of features that has adequate discriminating power to predict the target value this target value for an input vector is absent in case of an unsupervised approach in an unsupervised scheme we mainly try to find a subset that can capture the inherent structure of the data such as the neighborhood relation or the cluster structure in this work we first study a singular value decomposition svd based unsupervised feature selection approach proposed by varshavsky et al then we propose a modification of this method to improve its performance an svd entropy based supervised feature selection algorithm is also developed in this paper performance evaluation of the algorithms is done on altogether 13 benchmark and one synthetic data sets the quality of the selected features is assessed using three indices sammon s error se cluster preservation index cpi and misclassification error mce using a 1 nearest neighbor 1 nn classifier besides showing the improvement of the modified unsupervised scheme over the existing one we have also made a comparative study of the modified unsupervised and the proposed supervised algorithms with one well known unsupervised and two popular supervised feature selection methods respectively our results reveal the effectiveness of the proposed algorithms in selecting relevant features

fuzzy relations an fuzzy functions abstract in this paper we apply the standard element free mathematical definitions of equivalence relation and function as relations to the case of fuzzy relations one of the surprising results of this approach is that fuzzy functions take their values from the complemented elements of a brouwerian lattice thus in the common case of the values being taken from the unit interval the only fuzzy functions of this type are the ordinary functions this might suggest that fuzzy set theory is more than just a generalization of ordinary set theory

a discussion on fuzzy cardinality and quantification some applications in image processing in this paper we discuss on some different representations of the cardinality of a fuzzy set and their use in fuzzy quantification we have considered the widely employed sigma count fuzzy numbers and gradual numbers gradual numbers assign numbers to values of a relevance scale typically 0 1 contrary to sigma count and fuzzy numbers they provide a precise representation of the cardinality of a fuzzy set we illustrate our claims by calculating the cardinality of the fuzzy set of pixels that match a certain fuzzy color in an image for that purpose we consider fuzzy color spaces previously defined by the authors consisting of a collection of fuzzy sets providing a suitable conceptual quantization with soft boundaries of crisp color spaces finally we show the suitability of our approaches to fuzzy quantification for different applications in image processing first the calculation of histograms second the definition of the notion of dominant fuzzy color and the calculation of the degree to which we can say that a certain color is dominant in an image

a proposed method for learning rule weights in fuzzy rule based classification systems in fuzzy rule based classification systems frbcss rule weighting has often been used as a simple mechanism to tune the classifier in past research a number of heuristic rule weight specification methods have been proposed for this purpose a learning algorithm based on reward and punishment has also been proposed to adjust the weights of each fuzzy rule in the rule base in this paper a new method of learning rule weight in frbcss is proposed the method can be used when single winner or weighted vote methods of reasoning is used compared with reward and punishment scheme the proposed method is much faster and more effective another advantage of the proposed method is that during the learning process redundant rules are removed i e by setting their weights to zero the final rule base usually contains much fewer rules than the initial one this feature is very useful since a compact rule base is usually desired from the point of efficiency and interpretability a number of uci data sets are used to assess the performance of the proposed method in comparison with reward and punishment scheme

a framework for fuzzy quantification models analysis a framework for description of fuzzy quantification models is presented within this framework the fuzzy quantified statements evaluation problem is described as the compatibility between the fuzzy quantifier and a fuzzy cardinality or a fuzzy aggregation measure a list of desirable properties for quantification models is presented and those models that fit the framework are confronted with it

fuzzy criteria for feature selection the presence of less relevant or highly correlated features often decrease classification accuracy feature selection in which most informative variables are selected for model generation is an important step in data driven modeling in feature selection one often tries to satisfy multiple criteria such as feature discriminating power model performance or subset cardinality therefore a multi objective formulation of the feature selection problem is more appropriate in this paper we propose to use fuzzy criteria in feature selection by using a fuzzy decision making framework this formulation allows for a more flexible definition of the goals in feature selection and avoids the problem of weighting different goals is classical multi objective optimization the optimization problem is solved using an ant colony optimization algorithm proposed in our previous work we illustrate the added value of the approach by applying our proposed fuzzy feature selection algorithm to eight benchmark problems

rough sets methodology for sorting problems in presence of multiple attributes and criteria abstract we consider a sorting classification problem in the presence of multiple attributes and criteria called the mac however the domains of attributes are not preference ordered while the domains of criteria scales are totally ordered by preference relations among the attributes we distinguish between qualitative attributes and quantitative attributes in order to construct a comprehensive preference model that could be used to support the sorting task we consider preferential information of the decision maker dm in the form of assignment examples i e exemplary assignments of some reference actions to the decision classes the preference model inferred from these examples is a set of if then decision rules the rules are derived from rough approximations of decision classes made up of reference actions they satisfy conditions of completeness and dominance and manage with possible ambiguity inconsistencies in the set of examples our idea of rough approximations involves three relations together indiscernibility similarity and dominance defined on qualitative and quantitative attributes and on criteria respectively the usefulness of this approach is illustrated by an example

fuzzy class theory the paper introduces a simple yet powerful axiomatization of zadeh s notion of fuzzy set based on formal fuzzy logic the presented formalism is strong enough to serve as foundations of a large part of fuzzy mathematics its essence is elementary fuzzy set theory cast as two sorted first order theory over fuzzy logic which is generalized to simple type theory we show a reduction of the elementary fuzzy set theory to fuzzy propositional calculus and a general method of fuzzification of classical mathematical theories within this formalism in this paper we restrict ourselves to set relations and operations that are definable without any structure on the universe of objects presupposed however we also demonstrate how to add structure to the universe of discourse within our framework

constructive granular systems with universal approximation and fast knowledge discovery conventional gradient descent learning algorithms for soft computing systems have the learning speed bottleneck problem and the local minima problem to effectively solve the two problems the n variable constructive granular system with high speed granular constructive learning is proposed based on granular computing and soft computing and proved to be a universal approximator the fast granular constructive learning algorithm can highly speed up granular knowledge discovery by directly calculating all parameters of the n variable constructive granular system using training data and then construct the n variable constructive granular system with any required accuracy using a small number of granular rules predictive granular knowledge discovery simulation results indicate that the direct calculation based granular constructive algorithm is better than the conventional gradient descent learning algorithm in terms of learning speed learning error and prediction error

velocity continuation and the anatomy of residual prestack time migration velocity continuation is an imaginary continuous process of seismic image transformation in the postmigration domain it generalizes the concepts of residual and cascaded migrations understanding the laws of velocity continuation is crucially important for a successful application of time migration velocity analysis these laws predict the changes in the geometry and intensity of reflection events on migrated images with the change of the migration velocity in this paper i derive kinematic and dynamic laws for the case of prestack residual migration from simple geometric principles the main theoretical result is a decomposition of prestack velocity continuation into three different components corresponding to residual normal moveout residual dip moveout and residual zero offset migration i analyze the contribution and properties of each of the three components separately this theory forms the basis for constructing efficient finite difference and spectral algorithms for time migration velocity ana

a class of array codes correcting multiple column erasures a family of binary array codes of size p 1 spl times n with p a prime correcting multiple column erasures is proposed the codes coincide with a subclass of shortened reed solomon codes and achieve the maximum possible correcting capability complexity of encoding and decoding is proportional to rnp where r is the number of correctable erasures i e is simpler than the forney decoding algorithm the length n of the codes is at most 2p 1 that is twice as big as the length of the blaum roth codes having comparable decoding complexity

retrieving images in fuzzy object relational databases using dominant color descriptors in this paper a fuzzy approach for image retrieval on the basis of color features is presented the proposal deals with vagueness in the color description and introduces the use of fuzzy database models to store and retrieve imprecise data to face the color description the concept of dominant fuzzy color is proposed using linguistic labels for representing the color information in terms of hue saturation and intensity to deal with fuzzy data in our database model we use a general approach which can support the manipulation of fuzzy objects in an object relational database system this allows the retrieval of images by performing flexible queries on the database

a graded approach to cardinal theory of finite fuzzy sets part ii fuzzy cardinality measures and their relationship to graded equipollence abstract in this article we propose an axiomatic system for fuzzy cardinality measures referred to as fuzzy c measures for short assigning to each finite fuzzy set a generalized cardinal that expresses the number of elements that the fuzzy set contains the system generalizes an axiomatic system introduced by j casasnovas and j torrens 2003 we show that each fuzzy c measure is determined by two appropriate homomorphisms between the reducts of residuated dually residuated rdr lattices for linearly ordered rdr lattices we prove that each fuzzy c measure is a product of a non decreasing and a non increasing fuzzy c measure which indicates that there is a close relation between fuzzy c measures and fgcount flcount and fecount provided by l a zadeh 1983 and generalized by m wygralak 2001 finally the relationship of fuzzy c measures to graded equipollence introduced in the first part of this contribution is analyzed

the cardinality of fuzzy sets and the continuum hypothesis abstract up to now the research of the cardinality of fuzzy sets hardly advanced since it is difficult to give it an appropriate definition in this paper we give a general definition of the cardinality of fuzzy sets based on the fuzzy mappings and not only obtain a large part of the results with respect to the cardinality in cantor s set theory but also get many new properties of the cardinality of fuzzy sets especially from the cardinality of fuzzy sets we may gain some new ideas on the continuum hypothesis which is a well known problem

an overview of fuzzy quantifiers i interpretations quantification is an important topic in fuzzy theory and its applications an overview is presented for quantification in fuzzy theory after a brief review of quantifiers in first order logic two approaches of generalizing quantifiers are given the algebraic method and the substitution method by distinguishing the fuzziness of predicates and quantifiers various approaches to quantification in fuzzy logic can be organized quantifiers in first order logic can be generalized in crisp sense and these generalized quantifiers can also be applied to fuzzy sets moreover quantifiers themselves can be fuzzy i e they can only be represented by a fuzzy set these different kinds of quantifications are identified quantifiers relate close to the concept of the cardinality of a fuzzy set which is summarized before investigating fuzzy quantifications different to classical logic various semantics of propositions in fuzzy logic fall into different frameworks which are known as the possibility distribution based reasoning system and the many valued fuzzy logics accordingly numerical and possibilistic interpretation explored in literature are reviewed conforming to these two frameworks

the three semantics of fuzzy sets abstract three main semantics for membership functions seem to exist in the literature similarity preference and uncertainty each semantics underlies a particular class of applications similarity notions are exploited in clustering analysis and fuzzy controllers uncertainty is captured by fuzzy sets in the framework of possibility theory the membership function of a fuzzy set is also sometimes a kind of utility function that represents flexible constraints in decision problems this paper advocates the claim that progress in operational semantics of membership functions presupposes that these distinct semantics be acknowledged and related to more basic measurement issues in terms of distance cost and frequency on which scientific traditions exist

fuzzy color histogram and its use in color image retrieval a conventional color histogram cch considers neither the color similarity across different bins nor the color dissimilarity in the same bin therefore it is sensitive to noisy interference such as illumination changes and quantization errors furthermore cchs large dimension or histogram bins requires large computation on histogram comparison to address these concerns this paper presents a new color histogram representation called fuzzy color histogram fch by considering the color similarity of each pixel s color associated to all the histogram bins through fuzzy set membership function a novel and fast approach for computing the membership values based on fuzzy c means algorithm is introduced the proposed fch is further exploited in the application of image indexing and retrieval experimental results clearly show that fch yields better retrieval results than cch such computing methodology is fairly desirable for image retrieval over large image databases

extension of discrete t norms and t conorms to discrete fuzzy numbers the aim of this paper is to extend a monotone operation defined on a discrete setting to a closed binary operation on the partially ordered set a 1 of discrete fuzzy numbers whose support is a subset of natural numbers moreover we study several properties like the monotonicity commutativity associativity or the boundary condition when the extended operation is a discrete t norm or a discrete t conorm defined on a finite chain l 0 1 n n also we study the frank s equation for t norms and t conorms defined on a special subset of a 1 finally we propose an application to obtain consensus opinion based on discrete fuzzy weighted normed operators

weighted fuzzy pattern matching the fuzzy pattern matching technique has been developed in the framework of fuzzy set and possibility theory in order to take into account the imprecision and the uncertainty pervading values which have to be compared in a matching process this technique has proved very useful for implementing patterns of approximate reasoning in expert system inference engines and for designing retrieval systems capable of managing incomplete and fuzzy information data bases and vague queries in this paper the fuzzy pattern matching procedure is improved by introducing weights assessing the relative importance of atoms in the pattern matching indices are obtained using weighted versions of the minimum and maximum operations of fuzzy set theory this approach is extended to the case of variable weights and is related to aggregation schemes proposed by yager when modeling soft quantified statements such as most of the criteria must be met a notion of soft partial matching is thus modelled

codes correcting phased burst erasures we introduce a family of binary array codes of size t spl times n correcting multiple phased burst erasures of size t the codes achieve maximal correcting capability i e being considered as codes over gf 2 sup t they are mds the length of the codes is n spl sigma sub l 1 sup l sub l sup t where l is a constant or is slowly growing in t the complexity of encoding and decoding is proportional to rnml where r is the number of correctable erasures and m is the smallest number such that 2 sup t 1 modulo m this compares favorably with the complexity of decoding codes obtained from the shortened reed solomon codes having the same parameters

fuzzy rough sets assisted attribute selection attribute selection as refers to the problem of selecting those input attributes or features that are most predictive of a given outcome a problem encountered in many areas such as machine learning pattern recognition and signal processing unlike other dimensionality reduction methods attribute selectors preserve the original meaning of the attributes after reduction this has found application in tasks that involve datasets containing huge numbers of attributes in the order of tens of thousands which for some learning algorithms might be impossible to process further recent examples include text processing and web content classification as techniques have also been applied to small and medium sized datasets in order to locate the most informative attributes for later use one of the many successful applications of rough set theory has been to this area the rough set ideology of using only the supplied data and no other information has many benefits in as where most other methods require supplementary knowledge however the main limitation of rough set based attribute selection in the literature is the restrictive requirement that all data is discrete in classical rough set theory it is not possible to consider real valued or noisy data this paper investigates a novel approach based on fuzzy rough sets fuzzy rough feature selection frfs that addresses these problems and retains dataset semantics frfs is applied to two challenging domains where a feature reducing step is important namely web content classification and complex systems monitoring the utility of this approach is demonstrated and is compared empirically with several dimensionality reducers in the experimental studies frfs is shown to equal or improve classification accuracy when compared to the results from unreduced data classifiers that use a lower dimensional set of attributes which are retained by fuzzy rough reduction outperform those that employ more attributes returned by the existing crisp rough reduction method in addition it is shown that frfs is more powerful than the other as techniques in the comparative study

time migration velocity analysis by velocity continuation time migration velocity analysis can be performed by velocity continuation an incremental process that transforms migrated seismic sections according to changes in the migration velocity velocity continuation enhances residual normal moveout correction by properly taking into account both vertical and lateral movements of events on seismic images finite difference and spectral algorithms provide efficient practical implementations for velocity continuation synthetic and field data examples demonstrate the performance of the method and confirm theoretical expectations

fast time to depth conversion and interval velocity estimation in the case of weak lateral variations time domain processing has a long history in seismic imaging and has always been a powerful workhorse that is routinely used it generally leads to an expeditious construction of the subsurface velocity model in time which can later be expressed in the cartesian depth coordinates via a subsequent time to depth conversion the conventional practice of such a conversion is done using dix inversion which is exact in the case of laterally homogeneous media for other media with lateral heterogeneity the time to depth conversion involves solving a more complex system of partial differential equations pdes we have developed an efficient alternative for time to depth conversion and interval velocity estimation based on the assumption of weak lateral velocity variations by considering only first order perturbative effects from lateral variations the exact system of pdes required to accomplish the exact conversion reduces to a simpler system that can be solved efficiently in a layer stripping downward stepping fashion numerical synthetic and field data examples show that our method can achieve reasonable accuracy and is significantly more efficient than previously proposed methods with a speedup by an order of magnitude