correspondence analysis and association models constrained by a conditional independence graph the manner in which the conditional independence graph of a multiway contingency table effects the fitting and interpretation of the goodman association model rc and of correspondence analysis ca is considered estimation of the row and column scores is presented in this context by developing a unified framework that includes both models incorporation of the conditional independence constraints inherent in the graph may lead to equal or additive scores for the corresponding marginal tables depending on the topology of the graph an example of doubly additive scores in the analysis of a burt subtable is given

log linear non uniform association models for agreement between two ratings on an ordinal scale in agreement studies when objects are rated independently by two raters or twice by the same rater an association between their ratings on two categories arises reflecting the distinguishability of these two categories for these raters when ratings are performed on an ordinal scale this association between ratings on two categories increases when the distance between these categories increases on the ordinal scale goodman s log linear models derived for the analysis of agreement between two raters on an ordinal scale assume that distinguishabilities between adjacent categories are either constant or a priori fixed log non linear models that allow variations of the distinguishabilities between adjacent categories along the scale may lead to difficulties in parameter estimation this paper describes a new class of log linear non uniform association models these models extend the log linear uniform association model by allowing variations of distinguishability between adjacent categories along the scale these new models are used to analyse ordinal agreement between dermatologists when assessing the severity of different cutaneous signs of ageing on women faces

new exact solutions to some variable coefficients problems abstract with the aid of computer symbolic computation system such as maple an extended tanh method is applied to determine the exact solutions for some nonlinear problems with variable coefficients several new soliton solutions and periodic solutions can be obtained if we taking paraments properly in these solutions the employed methods are straightforward and concise and it can also be applied to other nonlinear evolution equations in mathematical physics

new exact traveling wave solutions for the modified form of degasperis procesi equation abstract with the aid of symbolic computation a generalized auxiliary equation method is proposed to construct more general exact solutions to some types of nlpdes in this letter using the auxiliary equation method some new solitary and traveling wave solutions of such nonlinear mdp equations given by u t u xxt 4 u 2 u x 3 u x u xx uu xxx are obtained in a very general form several interesting special cases of these general solutions are also discussed

latent structure analysis of a set of multidimensional contingency tables abstract statistical methods are introduced for latent structure analysis of a set of two or more multidimensional contingency tables three basic classes of models are considered a models that assume complete homogeneity across tables b models that allow partial homogeneity across tables and c models that allow complete heterogeneity methods are presented for testing whether these models are congruent with the data in the tables and for assessing the significance of differences among the tables in the estimated parameters to illustrate the wide applicability of these models and methods we present analyses of two quite different sets of data

new solitary wave solutions to the modified forms of degasperis procesi and camassa holm equations abstract in this work we establish new solitary wave solutions to the modified forms of degasperis procesi and camassa holm equations unlike the standard degasperis procesi and camassa holm equations where multi peakon solutions arise the modified forms cause a change in the characteristic of these solutions and change it to bell shaped solitons periodic and complex solutions the extended tanh method the rational hyperbolic functions method and the rational exponential functions method are employed to reveal these new solutions

a new approach to attribute reduction of consistent and inconsistent covering decision systems with covering rough sets traditional rough set theory is mainly used to extract rules from and reduce attributes in databases in which attributes are characterized by partitions while the covering rough set theory a generalization of traditional rough set theory does the same yet characterizes attributes by covers in this paper we propose a way to reduce the attributes of covering decision systems which are databases characterized by covers first we define consistent and inconsistent covering decision systems and their attribute reductions then we state the sufficient and the necessary conditions for reduction finally we use a discernibility matrix to design algorithms that compute all the reducts of consistent and inconsistent covering decision systems numerical tests on four public data sets show that the proposed attribute reductions of covering decision systems accomplish better classification performance than those of traditional rough sets

new exact solutions for mch and mdp equations by auxiliary equation method abstract with the aid of symbolic computation auxiliary equation method is introduced to investigate modified forms of camassa holm and degasperis procesi equations a series of new exact traveling wave solutions including smooth solitary wave solution peakons singular solution periodic wave solution jacobi elliptic solution are obtained in general form these new exact solutions will enrich previous results and help us further understand the physical structures of these two nonlinear equations

exact travelling solutions for the lax s seventh order kdv equation by sech method and rational exp function method in this paper we establish exact solutions for a nonlinear evolution equation the sech method and the exp function method are used to construct the solitary travelling wave solutions of lax s seventh order kdv equation these solutions may be important of significance for the explanation of some practical physical problem

incremental rough set approach for hierarchical multicriteria classification abstract multicriteria classification refers to classify objects evaluated by a set of criteria to preference ordered decision classes dominance based rough set approach has been successfully introduced to express and reason inconsistencies with a dominance principle in multicriteria classification problems hierarchical attribute values exist extensively within many real world applications which provide a hierarchical form to organize view and analyze data from different perspectives for accommodating the preference variability in this study we consider an extension of dominance based rough set approach by applying an incremental learning technique for hierarchical multicriteria classification while attribute values dynamically vary across different levels of granulations we formalize the dynamic characteristics of knowledge granules with the cut refinement and coarsening through attribute value taxonomies in the hierarchical multicriteria decision systems in consequence incremental algorithms for computing dominance based rough approximations of preference ordered decision classes are developed by applying the resulted prior knowledge as the input and only recomputing those outputs which depend on the changed attribute values this paper presents the theoretical foundation of the proposed approach example analysis and experimental evaluation are also provided for illustration of the feasibility and efficiency

models for the analysis of association in multivariate contingency tables abstract the present work describes a family of models that can be used for the analysis of association in multiway cross classifications the models are complementary to the usual i e the so called hierarchical log linear models an advantage of this family of models is that it is frequently possible to give a more parsimonious description of the association than is possible with the usual log linear models approach the models will be especially useful when some or all of the variables are ordinal the models are derived from a reparameterization of the saturated model based on singular value decompositions of interaction terms in the log linear parameterization multivariate association models given in goodman 1979 1986 clogg 1982a b agresti and kezouh 1983 gilula and haberman 1988 and becker and clogg 1989 are obtained as special cases the utility of the models developed in this article is demonstrated with the analysis of two examples the first example a cross classification of

reducts within the variable precision rough sets model a further investigation abstract one fundamental aspect of the variable precision rough sets vprs model involves a search for subsets of condition attributes which provide the same information for classification purposes as the full set of available attributes such subsets are labelled approximate reducts or reducts being defined for a specified classification error denoted by this paper undertakes a further investigation of the criteria for a reduct within vprs certain anomalies and interesting implications are identified an additional condition is suggested for finding reducts which assures a more general level knowledge equivalent to that of the full set of attributes

fixed distance latent class models for the analysis of sets of two way contingency tables this paper develops and describes the application of modified latent class models for analyzing sets of two way contingency tables the proposed fixed distance models differ from traditional latent class models in that the assumption of local stochastic independence is superseded by allowing interactions of the manifest variables within each class which can be represented by a single association parameter as an example two data sets on eye color by hair color collected in caithness n1 5 387 and aberdeen n2 22 361 and fixed distance models with up to six classes three classes per data set are considered finally leading to satisfactory fit and rather simple interpretation

the analysis of three way contingency tables by three mode association models therc m association model goodman 1979 1985 1986 1991 is useful for analyzing the relationship between the variables of a 2 way cross classification the models presented here are generalizations of therc m association model for 3 way tables the family of models proposed here 3 mode association models use tucker s 3 mode components model tucker 1964 1966 kroonenberg 1983 to represent either the three factor interaction or the combined effects of two and three factor interactions an example from a study in developmental psychology kramer gottman 1992 is provided to illustrate the usefulness of the proposed models

quasisymmetric models for the analysis of square contingency tables summary a parametric model is presented for the analysis of square contingency tables where there is a one to one correspondence between the categories of the row variable and the categories of the column variable the model is applicable to cross classifications where the categories are unordered partially ordered or completely ordered but the interpretation of the model and associated parameters is most straightforward when the categories are at least partially ordered connections with the mover stayer model and the usual model of quasisymmetry are described two data sets are used to illustrate the utility of the model

a quasi nonmetric method for multidimensional scaling via an extended euclidean model an extended two way euclidean multidimensional scaling mds model which assumes both common and specific dimensions is described and contrasted with the standard two way mds model in this extended two way euclidean model then stimuli or other objects are assumed to be characterized by coordinates onr common dimensions in addition each stimulus is assumed to have a dimension or dimensions specific to it alone the overall distance between objecti and objectj then is defined as the square root of the ordinary squared euclidean distance plus terms denoting the specificity of each object the specificity s j can be thought of as the sum of squares of coordinates on those dimensions specific to objecti all of which have nonzero coordinatesonly for objecti in practice we may think of there being just one such specific dimension for each object as this situation is mathematically indistinguishable from the case in which there are more than one we further assume that ij f d ij e ij where ij is the proximity value e g similarity or dissimilarity of objectsi andj d ij is the extended euclidean distance defined above whilee ij is an error term assumed i i d n 0 2 f is assumed either a linear function in the metric case or a monotone spline of specified form in the quasi nonmetric case a numerical procedure alternating a modified newton raphson algorithm with an algorithm for fitting an optimal monotone spline or linear function is used to secure maximum likelihood estimates of the paramstatistics can be used to test hypotheses about the number of common dimensions and or the existence of specific in addition tor common dimensions this approach is illustrated with applications to both artificial data and real data on judged similarity of nations

minimum cost attribute reduction in decision theoretic rough set models in classical rough set models attribute reduction generally keeps the positive or non negative regions unchanged as these regions do not decrease with the addition of attributes however the monotonicity property in decision theoretic rough set models does not hold this is partly due to the fact that all regions are determined according to the bayesian decision procedure consequently it is difficult to evaluate and interpret region preservation attribute reduction in decision theoretic rough set models this paper provides a new definition of attribute reduct for decision theoretic rough set models the new attribute reduction is formulated as an optimization problem the objective is to minimize the cost of decisions theoretical analysis shows the meaning of the optimization problem both the problem definition and the objective function have good interpretation a heuristic approach a genetic approach and a simulated annealing approach to the new problem are proposed experimental results on several data sets indicate the efficiency of these approaches

geometric representation of association between categories categories can be counted rated or ranked but they cannot be measured likewise persons or individuals can be counted rated or ranked but they cannot be measured either nevertheless psychology has realized early on that it can take an indirect road to measurement what can be measured is the strength of association between categories in samples or populations and what can be quantitatively compared are counts ratings or rankings made under different circumstances or originating from different persons the strong demand for quantitative analysis of categorical data has thus created a variety of statistical methods with substantial contributions from psychometrics and sociometrics what is the common basis of these methods dealing with categories the basic element they share is that the sample space has a special geometry in which categories or persons are point masses forming a simplex while distributions of counts or profiles of ratings are centers of gravity which are also point masses rankings form a discrete subset in the interior of the simplex known as the permutation polytope and paired comparisons form another subset on the edges of the simplex distances between point masses form the basic tool of analysis the paper gives some history of major concepts which naturally leads to a new concept the shadow point it is then shown how loglinear models luce and rasch models unfolding models correspondence analysis and homogeneity analysis forced classification and classification trees as well as other models and methods fit into this particular geometrical framework

analysis of sets of two way contingency tables using association models abstract a class of models is introduced for the analysis of group differences in the association between two discrete variables the rc m association model for two way tables is reviewed and alternative weighting systems for identifying interaction parameters are presented this model is generalized for the setting where a two way contingency table is available for two or more groups various restricted models can be used to examine possible sources of intergroup heterogeneity in the association these sources pertain to heterogeneity in the intrinsic association and or in the scores for the row and column variables the importance of weights used to identify the row and column scores is emphasized a classical set of data previously analyzed by many authors is used to illustrate the advantages of the models and methods developed here

multigranulation rough sets from partition to covering the classical multigranulation rough set mgrs theory offers a formal theoretical framework for solving the complex problem under multigranulation environment however it is noticeable that mgrs theory cannot be applied in multi source information systems with a covering environment in the real world to address this issue we firstly present in this paper three types of covering based multigranulation rough sets in which set approximations are defined by different covering approximation operators then by using two different approximation strategies i e seeking common reserving difference and seeking common rejecting difference two kinds of covering based multigranulation rough set are presented namely a covering based optimistic multigranulation rough set and a covering based pessimistic multigranulation rough sets finally we develop some properties and several uncertainty measures of the covering based multigranulation rough sets these results will enrich the mgrs theory and enlarge its application scope

a note on exact travelling wave solutions for the modified camassa holm and degasperis procesi equations abstract exact solutions for the modified camassa holm and degasperis procesi equations by liu et al 2010 y f liu x y zhu j x he factorization technique and new exact solutions for the modified camassa holm and degasperis procesi equations appl math comput 217 2010 1658 1665 are investigated liu et al has used the factorization technique to reduce the modified camassa holm and degasperis procesi equations to first order ordinary differential equations and then derived some exact travelling wave solutions by direct integral method in this note we will explain that the implementation of the so called factorization technique is completely unnecessary moreover based on the method of complete discrimination system for polynomial we shall demonstrate that the general explicit exact solution and its classification for the above two types of equations can be obtained directly and many exact solutions by liu et al are our special cases besides some known results in previously relevant literatures are extended and some simple remarks are also made

explicit peakon and solitary wave solutions for the modified fornberg whitham equation in this paper the modified fornberg whitham equation is studied by using the bifurcation theory and the method of phase portraits analysis in some parametric conditions some peakons and solitary waves are found and their exact parametric representations in explicit form are obtained

toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic there are three basic concepts that underlie human cognition granulation organization and causation informally granulation involves decomposition of whole into parts organization involves integration of parts into whole and causation involves association of causes with effects granulation of an object a leads to a collection of granules of a with a granule being a clump of points objects drawn together by indistinguishability similarity proximity or functionality for example the granules of a human head are the forehead nose cheeks ears eyes etc in general granulation is hierarchical in nature a familiar example is the granulation of time into years months days hours minutes etc modes of information granulation ig in which the granules are crisp c granular play important roles in a wide variety of methods approaches and techniques crisp ig however does not reflect the fact that in almost all of human reasoning and concept formation the granules are fuzzy f granular the granules of a human head for example are fuzzy in the sense that the boundaries between cheeks nose forehead ears etc are not sharply defined furthermore the attributes of fuzzy granules e g length of nose are fuzzy as are their values long short very long etc the fuzziness of granules their attributes and their values is characteristic of ways in which humans granulate and manipulate information

exact travelling wave solutions of a generalized camassa holm equation using the integral bifurcation method abstract in this paper a generalized camassa holm equation is studied by using the integral bifurcation method many travelling waves such as peaked compacton compacton peaked solitary wave solitary wave and kink like wave are found in some parameter conditions exact parametric representations of these travelling waves in explicit form and implicit form are obtained

nonlinear variants of the improved boussinesq equation with compact and noncompact structures variants of the improved boussinesq equation with positive and negative exponents are investigated it is formally shown that these nonlinear variants give rise to compact and non compact physical structures where compactons solitons solitary patterns and periodic solutions are developed the presented sine cosine ansatz is reliable

observations on the tanh coth expansion method for finding solutions to nonlinear evolution equations abstract the tanh coth expansion method for finding solitary travelling wave solutions to nonlinear evolution equations has been used extensively in the literature it is a natural extension to the basic tanh function expansion method which was developed in the 1990s it usually delivers three types of solution namely a tanh function expansion a coth function expansion and a tanh coth expansion it is known that for every tanh function expansion solution there is a corresponding coth function expansion solution it is shown that there is a tanh coth expansion solution that is merely a disguised version of the coth solution in many papers such tanh coth solutions are erroneously claimed to be new however other tanh coth solutions may be delivered that are genuinely new in the sense that they would not be delivered via the basic tanh function method similar remarks apply to tan cot and tan cot expansion solutions

ordinal association in contingency tables some interpretive aspects abstract two families of models for ordered contingency tables goodman s association models and canonical correlation models are investigated and compared with respect to the interpretation of their parameters we show that the two families of models actually refer to different kinds of ordinal association stochastic order extremity for correlation models and stochastic order entropy for association models this difference is related to the way the two models scale interaction the scale difference is proven to be of substantial consequence especially under strong association

modelling ordered categorical data recent advances and future challenges this article summarizes recent advances in the modelling of ordered categorical ordinal response variables we begin by reviewing some models for ordinal data introduced in the literature in the past 25 years we then survey recent extensions of these models and related methodology for special types of applications such as for repeated measurement and other forms of clustering we also survey other aspects of ordinal modelling such as small sample analyses power and sample size considerations and availability of software throughout we suggest problem areas for future research and we highlight challenges for statisticians who deal with ordinal data

rough set models in multigranulation spaces there exist several approaches to rough set approximations in a multigranulation space namely a family of equivalence relations in this paper we propose a unified framework to classify and compare existing studies an underlying principle is to explain rough sets in a multigranulation space through rough sets derived by using individual equivalence relations two basic models are suggested one model is based on a combination of a family of equivalence relations into an equivalence relation and the construction of approximations with respect to the combined relation by combining equivalence relations through set intersection and union respectively we construct two sub models the other model is based on the construction of a family of approximations from a set of equivalence relations and a combination of the family of approximations by using set intersection and union to combine a family of approximations respectively we again build two sub models as a result we have a total of four models we examine these models and give conditions under which some of them become the same

solitons solutions for some nonlinear evolution equations abstract in this paper we establish new solitary wave solutions to the modified kawahara equation by the sine cosine method moreover the periodic solutions and bell shaped solitons solutions to the generalized fifth order kdv equation are obtained the tanh method is used to handle the double sine gordon equation and the double sinh gordon equation families of exact travelling wave solutions are formally derived the rational triangle sine cosine method is introduced and to be constructed complex solutions to the modified degasperis procesi dp equation and the modified camassa holm ch equation

symbolic analysis and exact travelling wave solutions to a new modified novikov equation in this paper a new modified novikov equation is introduced multiple exact travelling wave solutions are obtained by using the extended tanh function method furthermore based on the homogeneous balance method an auto backlund transformation is derived and some solitary wave solutions to this new equation are established

graphical representations and odds ratios in a distance association model for the analysis of cross classified data although rc m association models have become a generally useful tool for the analysis of cross classified data the graphical representation resulting from such an analysis can at times be misleading the relationships present between row category points and column category points cannot be interpreted by inter point distances but only through projection in order to avoid incorrect interpretation by a distance rule joint plots should be made that either represent the row categories or the column categories as vectors in contrast the present study proposes models in which the distances between row and column points can be interpreted directly with a large small distance corresponding to a small large value for the association the models provide expressions for the odds ratios in terms of distances which is a feature that makes the proposed models attractive reparametrizations to the usual rc m parametrization comparisons to existing data analysis techniques plus an overview of related models and their connections are also provided

association models and canonical correlation in the analysis of cross classifications having ordered categories abstract the association models considered in goodman 1979a for the analysis of cross classifications having ordered categories are presented in a somewhat different form in the present article to facilitate comparison of the results obtained using these models with those obtained using the earlier canonical correlation approach both the association models and the canonical correlation approach can provide meaningful scores for the row and column categories and these scores can be used to partition into relevant components the usual chi squared statistic for testing the null hypothesis of statistical independence between the row classification and column classification however while the usual procedure for testing the statistical significance of chi squared components is invalid when these components are based on the canonical correlations a corresponding procedure is valid when these components are obtained with the association models the components of association obtained with the association mo

canonical analysis of contingency tables by maximum likelihood abstract canonical analysis has often been employed instead of log linear models to analyze the relationship of two polytomous random variables however until the last few years analysis has been informal in this article models are examined that place nontrivial restrictions on the values of the canonical parameters so that a parsimonious description of association is obtained maximum likelihood is used to obtain parameter estimates for these restricted models approximate confidence intervals are derived for parameters and chi squared tests are used to check adequacy of models the resulting models may be used to determine the appropriateness of latent class analysis or to determine whether a set of canonical scores has specified patterns results are illustrated through analysis of two tables previously analyzed in the statistical literature comparisons are made with alternate methods of analysis based on a log linear parameterization of cell probabilities it is shown that canonical analysis wh

a class of nonlinear fourth order variant of a generalized camassa holm equation with compact and noncompact solutions abstract in this paper we study a class of nonlinear fourth order analogue of a generalized camassa holm equation it is shown that this class gives compactons conventional solitons solitary patterns and periodic solutions it is also found that the qualitative change in the physical structure of solutions depends mainly on the exponent of the wave function u x t positive or negative and on the coefficient of u n as well

new peakon solitary wave and periodic wave solutions for the modified camassa holm equation abstract in this paper an independent variable transformation is introduced to solve the modified camassa holm equation using the bifurcation theory and the method of phase portrait analysis some peakons solitary waves and periodic waves are found and their exact parametric representations in explicit form and in implicit form are obtained

a single general method for the analysis of cross classified data reconciliation and synthesis of some methods of pearson yule and fisher and also some methods of correspondence analysis and association analysis 1 2 3 4 5 6 7 8 9 10 contents introduction and summary a general measure of nonindependence for the cells in the table a general method of nonindependence analysis some simple examples some special cases of generalized nonindependence analysis standardized generalized nonindependence analysis overall indices and graphical displays of generalized nonindependence saturated and unsaturated models symmetric nonindependence and generalized quasi symmetry closing comments on 2 x 2 table measures of nonindependence and some brief historical notes

new periodic and soliton solutions of nonlinear evolution equations abstract in this paper the tanh and sine cosine methods are used to construct exact periodic and soliton solutions of nonlinear evolution equations arising in mathematical physics many new families of exact travelling wave solutions of the generalized hirota satsuma coupled kdv system generalized zakharov equations and 2 1 dimensional broer kaup kupershmidt system are successfully obtained the obtained solutions include solitons kinks and plane periodic solutions these solutions may be important of significance for the explanation of some practical physical problems

factorization technique and new exact solutions for the modified camassa holm and degasperis procesi equations abstract in this paper the recent factorization technique is applied to the modified camassa holm and degasperis procesi equations and two first order ordinary differential equations are obtained respectively subsequently some new exact solitary wave solutions for the two equations are proposed the figures for the bell type and peakon type solutions of the modified camassa holm are plotted to describe the properties of the solutions

variants of the two dimensional boussinesq equation with compactons solitons and periodic solutions variants of the two dimensional boussinesq equation with positive and negative exponents are studied the sine cosine ansatz is fruitfully used to carry out the analysis exact solutions of different physical structures compactons solitary patterns solitons and periodic solutions are obtained the quantitative change in the physical structure of the solutions is shown to depend mainly on the exponent of the wave function u x t and on the ratio a b of the derivatives of u x t

the tanh method exact solutions of the sine gordon and the sinh gordon equations in this work we use the tanh method for traveling wave solutions of the sine gordon and the sinh gordon equations several exact solutions of distinct physical structures are obtained the method is powerful with minimal algebra work and is demonstrated for four models

theory and applications of granular labelled partitions in multi scale decision tables abstract granular computing and acquisition of if then rules are two basic issues in knowledge representation and data mining a formal approach to granular computing with multi scale data measured at different levels of granulations is proposed in this paper the concept of labelled blocks determined by a surjective function is first introduced lower and upper label block approximations of sets are then defined multi scale granular labelled partitions and multi scale decision granular labelled partitions as well as their derived rough set approximations are further formulated to analyze hierarchically structured data finally the concept of multi scale information tables in the context of rough set is proposed and the unravelling of decision rules at different scales in multi scale decision tables is discussed

a new index of fit based on mixture methods for the analysis of contingency tables summary a framework based on mixture methods is proposed for evaluating goodness of fit in the analysis of contingency tables for a given model h applied to a contingency table p we consider the two point mixture p 1 r i1 ri2 with 7r the mixing proportion o r 1 and hi and i2 the tables of probabilities for each latent class or component in the unstructured approach recommended here the mixture model applies h to hi but does not impose any restrictions on i2 a contingency table p can generally be represented as such a two point mixture for an interval of 7r values we define our index of lack of fit ir to be the smallest such 7r i e 7r is the fraction of the population that cannot be described by model h this approach can be contrasted with the structured approach that applies model h to both h and i2 and leads to conventional latent class models when h is the hypothesis of independence the case where h is the hypothesis of row column independence and p is a two way contingency table is covered in detail but the procedure is quite general

the tanh method for traveling wave solutions of nonlinear equations in this work we employ the tanh method for traveling wave solutions of nonlinear equations the study is extended to equations that do not have tanh polynomial solutions the efficiency of the method is demonstrated by applying it for a variety of selected equations

a new approach of optimal scale selection to multi scale decision tables as a special case of information table multi scale decision table can usually be observed in real life world in such table objects may take different values under the same attribute measured at different scales based on inclusion relation of subsets of attributes and coarse relation of scales of attributes multi layered granulations and stratified rough set approximations in multi scale decision tables are shown from the perspective of granular computing compared with a special case studied by wu and leung the multi scale decision tables of diverse attributes with different numbers of levels of scales are studied in this paper furthermore complement model and lattice model are proposed to analyze the optimal scale selection for multi scale decision tables correspondingly algorithms of the two models are designed and some experiments are performed to testify feasibilities of these proposed algorithms and to make comparisons of the models

ideal point discriminant analysis revisited with a special emphasis on visualization ideal point discriminant analysis is a classification tool which uses highly intuitive multidimensional scaling procedures however in the last paper takane wrote about it he concludes that the interpretation is rather intricate and calls that a weakness of the model we summarize the conditions that provide an easy interpretation and show that in maximum dimensionality they can be obtained without any loss for reduced dimensionality it is conjectured that loss is minor which is examined using several data sets

quantitative information architecture granular computing and rough set models in the double quantitative approximation space of precision and grade because precision and grade act as fundamental quantitative information in approximation space they are used in relative and absolute quantifications respectively at present the double quantification regarding precision and grade is a novel and valuable subject but quantitative information fusion has become a key problem thus this paper constructs the double quantitative approximation space of precision and grade pg approx space and tackles the fusion problem using normal logical operations it further conducts double quantification studies on granular computing and rough set models 1 first for quantitative information organization and storage we construct space and plane forms of pg approx space using the cartesian product and for quantitative information extraction and fusion we establish semantics construction and semantics granules of pg approx space 2 second by granular computing we investigate three primary granular issues quantitative semantics complete system and optimal calculation accordingly six types of fundamental granules are proposed based on the semantic microscopic and macroscopic descriptions their semantics forms structures calculations and relationships are studied and the granular hierarchical structure is achieved 3 finally we investigate rough set models in pg approx space accordingly model regions are proposed by developing the classical regions model expansion is systematically analyzed some models are constructed as their structures are obtained and a concrete model is provided based on the quantitative information architecture this paper systematically conducts and investigates double quantification and establishes a fundamental and general exploration framework

invariance and factorial models summary two factors having the same set of levels are said to be homologous this paper aims to extend the domain of factorial models to designs that include homologous factors in doing so it is necessary first to identify the characteristic property of those vector spaces that constitute the standard factorial models we argue here that essentially every interesting statistical model specified by a vector space is necessarily a representation of some algebraic category logical consistency of the sort associated with the standard marginality conditions is guaranteed by category representations but not by group representations marginality is thus interpreted as invariance under selection of factor levels z representations and invariance under replication of levels s representations for designs in which each factor occurs once the representations of the product category coincide with the standard factorial models for designs that include homologous factors the set of srepresentations is a subset of the z representations it is shown that symmetry and quasi symmetry are representations in both senses but that not all representations include the constant functions intercept the beginnings of an extended algebra for constructing general z representations is described and illustrated by a diallel cross design

a systematic study on attribute reduction with rough sets based on general binary relations attribute reduction is considered as an important preprocessing step for pattern recognition machine learning and data mining this paper provides a systematic study on attribute reduction with rough sets based on general binary relations we define a relation information system a consistent relation decision system and a relation decision system and their attribute reductions furthermore we present a judgment theorem and a discernibility matrix associated with attribute reduction in each type of system based on the discernibility matrix we can compute all the reducts finally the experimental results with uci data sets show that the proposed reduction methods are an effective technique to deal with complex data sets

knowledge reduction in random information systems via dempster shafer theory of evidence knowledge reduction is one of the main problems in the study of rough set theory this paper deals with knowledge reduction in random information systems based on dempster shafer theory of evidence the concepts of belief and plausibility reducts in random information systems are first introduced it is proved that both of belief reduct and plausibility reduct are equivalent to classical reduct in random information systems the relative belief and plausibility reducts in consistent and inconsistent random decision systems are then proposed and compared to the relative reduct and relationships between the new reducts and some existing ones are examined

computation of maximum likelihood estimates in association models abstract in association models for cross classified data computation of maximum likelihood estimates mle s is relatively difficult due to the nonlinear constraints on the parameters currently available procedures based on the scoring algorithm for constrained maximum likelihood are relatively unreliable and other cyclic procedures are relatively slow and do not provide estimated asymptotic standard deviations as by products of calculations to facilitate computations it is noted that in standard association models removal of constraints results in underidentification of parameters but does not affect the model itself so that the mle s of cell probabilities and conditional probabilities are unaffected given this observation maximum likelihood estimation may be accomplished by unconstrained maximization of an objective function with two components a log likelihood ratio and a sum of squares representing deviations of parameters from their constraints the objective function is then maximized by u

properties of ideal point classification models for bivariate binary data the ideal point classification ipc model was originally proposed for analysing multinomial data in the presence of predictors in this paper we studied properties of the ipc model for analysing bivariate binary data with a specific focus on three quantities 1 the marginal probabilities 2 the association structure between the two binary responses and 3 the joint probabilities we found that the ipc model with a specific class point configuration represents either the marginal probabilities or the association structure however the ipc model is not able to represent both quantities at the same time we then derived a new parametrization of the model the bivariate ipc bipc model which is able to represent both the marginal probabilities and the association structure like the standard ipc model the results of the bipc model can be displayed in a biplot from which the effects of predictors on the binary responses and on their association can be read we will illustrate our findings with a psychological example relating personality traits to depression and anxiety disorders

using association models to analyse agreement data two examples two examples demonstrate how one can use association models to analyse agreement data the first example concerns intra rater variability in the classification of sputum cytology slides and the second deals with variability associated with the reporting of passive smoking histories the paper emphasizes models in which one estimates category scores from the data that is models that are not in the log linear family of models such models have use in assessment of category distinguishability and provide insights not easily obtained with log linear models

mgrs a multi granulation rough set the original rough set model was developed by pawlak which is mainly concerned with the approximation of sets described by a single binary relation on the universe in the view of granular computing the classical rough set theory is established through a single granulation this paper extends pawlak s rough set model to a multi granulation rough set model mgrs where the set approximations are defined by using multi equivalence relations on the universe a number of important properties of mgrs are obtained it is shown that some of the properties of pawlak s rough set theory are special instances of those of mgrs moreover several important measures such as accuracy email protected quality of email protected and precision of email protected are presented which are re interpreted in terms of a classic measure based on sets the marczewski steinhaus metric and the inclusion degree measure a concept of approximation reduct is introduced to describe the smallest attribute subset that preserves the lower approximation and upper approximation of all decision classes in mgrs as well finally we discuss how to extract decision rules using mgrs unlike the decision rules and rules from pawlak s rough set model the form of decision rules in mgrs is or several pivotal algorithms are also designed which are helpful for applying this theory to practical issues the multi granulation rough set model provides an effective approach for problem solving in the context of multi granulations

some models for the analysis of association in multiway cross classifications having ordered categories abstract goodman recently presented a class of models for the analysis of association between two discrete ordinal variables the association was measured in terms of the odds ratios in 2 2 subtables formed from adjacent rows and adjacent columns of the cross classification and models were devised that allowed the odds ratios to depend on an overall effect on row effects on column effects and on other effects this article presents some generalizations of this approach appropriate for multiway cross classifications including a models for the analysis of conditional association b models for the analysis of partial association and c models for the analysis of symmetric association three cross classifications are analyzed with these models and methods and rather simple interpretations of the association in each are provided

on the bayesian analysis of population size summary we consider the problem of estimating the total size of a population from a series of incomplete census data we observe that inference is typically highly sensitive to the choice of model and we demonstrate how bayesian model averaging techniques easily overcome this problem we combine and extend the work of madigan york 1997 and dellaportas forster 1999 using reversible jump markov chain monte carlo simulation to calculate posterior model probabilities which can then be used to estimate modelaveraged statistics of interest we provide a detailed description of the simulation procedures involved and consider a wide variety of modelling issues such as the range of models considered their parameterisation both prior choice and sensitivity and computational efficiency we consider a detailed example concerning adolescent injuries in pennsylvania on the basis of medical school and survey data in the context of this example we discuss the relationship between posterior model probabilities and the associated information criteria values for model selection we also discuss cost efficiency issues with particular reference to inclusion and exclusion of sources on the grounds of cost we consider a decision theoretic approach which balances the cost and accuracy of different combinations of data sources to guide future decisions on data collection

modeling rough granular computing based on approximation spaces the results reported in this paper create a step toward the rough set based foundations of data mining and machine learning the approach is based on calculi of approximation spaces in this paper we present the summarization and extension of our results obtained since 2003 when we started investigations on foundations of approximation of partially defined concepts see e g 2 3 7 37 20 21 5 42 39 38 40 we discuss some important issues for modeling granular computations aimed at inducing compound granules relevant for solving problems such as approximation of complex concepts or selecting relevant actions plans for reaching target goals the problems discussed in this article are crucial for building computer systems that assist researchers in scientific discoveries in many areas such as biology in this paper we present foundations for modeling of granular computations inside of system that is based on granules called approximation spaces our approach is based on the rough set approach introduced by pawlak 24 25 approximation spaces are fundamental granules used in searching for relevant complex granules called as data models e g approximations of complex concepts functions or relations in particular we discuss some issues that are related to generalizations of the approximation space introduced in 33 34 we present examples of rough set based strategies for the extension of approximation spaces from samples of objects onto a whole universe of objects this makes it possible to present foundations for inducing data models such as approximations of concepts or classifications analogous to the approaches for inducing different types of classifiers known in machine learning and data mining searching for relevant approximation spaces and data models are formulated as complex optimization problems the proposed interactive granular computing systems should be equipped with efficient heuristics that support searching for semi optimal granules

approximate distribution reducts in inconsistent interval valued ordered decision tables abstract many methods based on the rough set theory to deal with information systems have been proposed in recent decades in practice some information systems are based on dominance relations and may be inconsistent because of various factors moreover taking the imprecise evaluations and assignments in the description of objects into account single valued information systems have been generalized to interval valued information systems in this paper by introducing a dominance relation to interval valued ordered information systems we establish a dominance based rough set approach which is mainly based on substitution of the indiscernibility relation by the dominance relation to extract the minimal decision rules approximate distribution reducts are proposed in inconsistent interval valued ordered decision tables this paper presents a theoretical method based on the discernibility matrix to enumerate all reducts and a practical approach on the basis of significance to find one reduct and two equivalent definitions of approximate distribution reducts are also introduced in addition numerical examples are employed to examine the validity of the approaches proposed in this paper

from fuzzy cognitive maps to granular cognitive maps fuzzy cognitive maps fcms form a class of graph oriented fuzzy models describing causal relationships among concepts in this study we augment these models by introducing their generalization coming in the form of granular fcms in contrast with fcms in the granular fcms the connections between the nodes states are described in the form of information granules especially intervals and fuzzy sets key scenarios in which granular models and granular fcms arise are presented in order to offer a compelling rationale behind the formation of such models in the context of system modeling we show that information granularity emerges as an important design asset we discuss detailed schemes of allocation of information granularity and quantify a performance of the resulting granular fcm in terms of a coverage criterion for illustrative purposes the detailed studies are completed for granular fcms with interval valued connections

gee for multinomial responses using a local odds ratios parameterization in this article we propose a generalized estimating equations gee approach for correlated ordinal or nominal multinomial responses using a local odds ratios parameterization our motivation lies upon observing that i modeling the dependence between correlated multinomial responses via the local odds ratios is meaningful both for ordinal and nominal response scales and ii ordinary gee methods might not ensure the joint existence of the estimates of the marginal regression parameters and of the dependence structure to avoid ii we treat the so called working association vector as a nuisance parameter vector that defines the local odds ratios structure at the marginalized contingency tables after tabulating the responses without a covariate adjustment at each time pair to estimate and simultaneously approximate adequately possible underlying dependence structures we employ the family of association models proposed by goodman in simulations the parameter estimators with the proposed gee method for a marginal cumulative probit model appear to be less biased and more efficient than those with the independence working model especially for studies having time varying covariates and strong correlation

a construction of compact and noncompact solutions for nonlinear dispersive equations of even order a construction of compact and noncompact solutions for nonlinear dispersive partial differential equations of even order is considered two model equations of even orders k 4 are adopted to carry out this study the analysis reveals the presence of distinct roots of different indices that play a significant role in the solutions

analyzing rectangular tables by joint and constrained multidimensional scaling abstract this paper selectively discusses multidimensional scaling methods addressing both theory and applications most of the discussion is focussed upon the analysis of rectangular tables the term joint mds is introduced for an analysis which gives a meaningful joint representation of the set of row and the set of column objects of such a table so that the dissimilarities either within sets or between sets are optimally approximated from this perspective it is possible to give an unified account of the relationships between classical multidimensional scaling principal components analysis and correspondence analysis by imposing constraints the general model can be structurally changed in a number of ways for example it can be made discrete that is in principle it subsumes subset models and quadratic assignment as special cases it can also be made more flexible in the sense that external information may be used in order to tune the analysis to specific hypotheses finally an elegant joint mds model called multidimensional unfolding is discussed as a technique unfolding unfortunately suffers from being extremely sensitive to ill conditioned data a new way of coping with this problem is indicated and an application is given in which the bootstrap technique succesfully establishes the stability of the results

correspondence analysis of incidence and abundance data properties in terms of a unimodal response model correspondence analysis is commonly used by ecologists to analyze data on the incidence or abundance of species in samples the first few axes are interpreted as latent variables and are presumed to relate to underlying environmental variables in this paper correspondence analysis is shown to approximate the maximum likelihood solution of explicit unimodal response models in one latent variable these models are logistic linear for presence absence data and loglinear for poisson counts with predictors that are quadratic in the latent variable the approximation is best when the maxima and tolerances widths of the response curves are equal and the species optima and the sample values of the latent variable are equally spaced it is still fairly good for uniformly distributed optima and sample values as shown by simulation for the models extended to two latent variables the approximation is often bad because of the horseshoe effect in correspondence analysis but improves considerably in the simulations when this effect is removed as it is in detrended correspondence analysis

analysis of contingency tables by ideal point discriminant analysis cross classified data are frequently encountered in behavioral and social science research the loglinear model and dual scaling correspondence analysis are two representative methods of analyzing such data an alternative method based on ideal point discriminant analysis da is proposed for analysis of contingency tables which in a certain sense encompasses the two existing methods a variety of interesting structures can be imposed on rows and columns of the tables through manipulations of predictor variables and or as direct constraints on model parameters this along with maximum likelihood estimation of the model parameters allows interesting model comparisons this is illustrated by the analysis of several data sets

the analysis of multivariate contingency tables by restricted canonical and restricted association models abstract restricted canonical models and restricted association models are proposed and applied to multiway contingency tables these models have been previously applied to two way contingency tables however multivariate generalization has been impeded in the past since canonical and association models both depend on singular value decompositions that apply only to two way arrays in this article this restriction to two way arrays is overcome by division of the cross classified variables into explanatory and response variables the explanatory variables are treated as a single polytomous variable and the response variables are treated as a second single polytomous variable in this fashion the multiway table is reduced to a two way array to which traditional canonical and association models may be applied use of linear restrictions on parameters in canonical and association models is especially important in multiway tables if useful models are to be constructed the class of models considered in th

a study on nonlinear dispersive partial differential equations of compact and noncompact solutions in this paper we study two genuinely nonlinear dispersive partial differential equations in one and higher dimensional spaces we show that the focusing branch and the defocusing branch of these equations each leads to a different physical structure in addition we develop general solutions of compact and noncompact support which are not obtainable through classic techniques

analytic study on two nonlinear evolution equations by using the g g expansion method abstract the validity and reliability of the so called g g expansion method is tested by applying it to two nonlinear evolutionary equations solutions in more general forms are obtained when the parameters are taken as special values it is observed that the previously known solutions can be recovered new rational function solutions are also presented being concise and less restrictive the method can be applied to many nonlinear partial differential equations

an information fusion approach by combining multigranulation rough sets and evidence theory multigranulation rough set mgrs theory provides two kinds of qualitative combination rules that are generated by optimistic and pessimistic multigranulation fusion functions they are used to aggregate multiple granular structures from a set theoretic standpoint however the two combination rules seem to lack robustness because one is too relaxed and the other too restrictive to solve some practical problems dempster s combination rule in the evidence theory has been employed to aggregate information coming from multiple sources however it fails to deal with conflict evidence to overcome these limitations we focus on the combination of granular structures with both reliability and conflict from multiple sources which has been a challenging task in the field of granular computing we first address the connection between multigranulation rough set theory and the evidence theory then a two grade fusion approach involved in the evidence theory and multigranulation rough set theory is proposed which is based on a well defined distance function among granulation structures finally an illustrative example is given to show the effectiveness of the proposed fusion method the results of this study will be useful for pooling the uncertain data from different sources and significant for establishing a new direction of granular computing