spatial data fusion for qualitative estimation of fuzzy request zones application on precision viticulture this article deals with a method used to describe and manage spatial knowledge each spatial datum is considered as an information element whose location and value are independently described the method proposed uses the information elements to infer the value of any request zone data can be either qualitative or quantitative our approach is based on a fuzzy granulation of available items of information the fusion of information elements is performed by an aggregation operator based on the choquet integral which analyzes the spatial relevancy of the information elements on the request zone we also describe an application of the method to a precision viticulture data set

normal forms in bl algebra and their contribution to universal approximation of functions this paper continues the investigation of approximating properties of generalized normal forms in fuzzy logic the problem is formalized and solved algebraically normal forms are considered in two variants infinite and finite it is proved that infinite normal forms are universal representation formulas whereas finite normal forms are universal approximation formulas for any l valued function where l is a support set of any complete bl algebra the estimation of the quality of approximation is suggested moreover functions which can be precisely represented by the discrete normal forms are considered

towards a higher degree f transform the aim of this study is to show how the f transform technique can be generalized from the case of constant components to the case of polynomial components after a general presentation of an f m transform m 0 a detailed characterization of the f 1 transform is given we apply a numeric integration technique in order to simplify the computation of f 1 transform components the inverse f m transform m 0 is defined similarly to the ordinary inverse f transform the quality of approximation using the inverse f m transform increases with an increase in m

on probability possibility transformations abstract a probability possibility transformation is a purely mechanical transformation of probabilistic support to possibilistic support and vice versa that is a conversion of the measure of support of one theory into that of the other that is independent of the problem domain in this paper it is shown that any bijective mapping between probability distributions and normalized possibility distributions that is symmetric and expansible cannot preserve the properties of projection noninteraction or conditionalization

fuzzy transform and least squares approximation analogies differences and generalizations investigating the relations between the least squares approximation techniques and the fuzzy transform in this paper we show that the discrete fuzzy transform is invariant with respect to the interpolating and least squares approximation additionally the fuzzy transform is evaluated at any point by simply resampling the continuous approximation underlying the input data using numerical linear algebra we also derive new properties e g stability to noise additivity with respect to the input data and characterizations e g radial and dual membership maps of the discrete fuzzy transform finally we define the geometry and confidence driven discrete fuzzy transforms which take into account the intrinsic geometry and the confidence weights associated to the data

on the granularity of summative kernels in this paper we propose granularity as a new index to characterize the non specificity of a summative kernel this index is intended to reflect the behavior of a kernel in the usual signal processing applications we show in different experiments that two kernels having the same granularity have very similar behavior this granularity based adaptation is compared to other adaptation methods these experiments highlight the ability of the granularity index to measure the spreading and collecting properties of a summative kernel

tight approximation for partial vertex cover with hard capacities abstract we consider the partial vertex cover problem with hard capacity constraints partial vc hc on hypergraphs in this problem we are given a hypergraph g v e with a maximum edge size f and a covering requirement r each edge is associated with a demand and each vertex is associated with a capacity and an integral available multiplicity the objective is to compute a minimum vertex multiset such that at least r units of demand from the edges are covered by the capacities of the vertices in the multiset and the multiplicity of each vertex does not exceed its available multiplicity in this paper we present an f approximation for this problem improving over a previous result of 2 f 2 1 by cheung et al to the tight extent possible our new ingredient of this work is a generalized analysis on the extreme points of the natural lp developed from previous works and a strengthened lp lower bound obtained for the optimal solutions

necessary and sufficient conditions for generalized uniform fuzzy partitions abstract the fundamental concept in the theory of fuzzy transform f transform is the fuzzy partition which is a generalization of the classical concept of partition the original definition assumes that every two normal fuzzy subsets in a partition overlap in such a way that the sum of the membership degrees at each point is equal to 1 this condition can be generalized by relaxing the assumption of normality for fuzzy sets the result is a denser fuzzy partition that may improve the approximation properties and the smoothness of the inverse f transform a fuzzy partition with this property will be referred to as general the problem is how a general fuzzy partition can be effectively constructed if we use a generating function with special properties then it is not immediately clear whether it defines a general fuzzy partition in this paper we find a necessary and sufficient condition that will enable the optimal generalized fuzzy partition to be designed more easily which is important in various practical applications of the f transform for example image processing time series analysis and solving differential equations with boundary conditions

a qualified description of extended fuzzy logic abstract the sense of approximate reasoning within fuzzy logic has given rise to speculation as to whether precise reasoning within classical logic can even be considered a feasible way of handling most of the world s problems now another question can be raised can fuzzy logic by itself justify handle the sort of uncertainty found in problems that belong to the class of open world problems this family of problems corresponds to the case of ambiguous mathematical theories mainly due to the fact that there is just imperfect incomplete information about them recently zadeh introduced an extension of fuzzy logic claimed to bridge the gaps in conventional approaches to the open world problems in this paper we present a basic background of extended fuzzy logic fle we address several main concepts such as f transformation and f validity enhance the definition of generalized constraints and propose that fle s rules have two layers one based on the implication model and another based on the conjunction model in addition we explore how fle can handle those conditions described in antonym pairs as well as those occurred without considering rationality in the framework both cases challenge most decision making theories we also suggest fle and second order probability are f complementary rather than f competitive the paper asserts that fle is highly capable of constructing mathematical solutions to problems including computational problems that are stated in natural language

fuzzy projection versus inverse fuzzy transform as sampling interpolation schemes in this study the inverse fuzzy transform recently introduced by perfilieva is compared to a new representation of signals that we name the fuzzy projection both representations are sampling interpolation schemes in a given fuzzy context fuzzy projection is closely related to fuzzy and inverse fuzzy transforms and is presented in this framework we deal with these two operators in the space of finite energy signals while proposing theoretical and numerical comparisons after presenting convergence results for fuzzy projection some numerical tests are performed on signals they confirm the theoretical studies and show the superiority of fuzzy projection for handling in the l 2 sense signals without too high frequencies since both representations can be regarded as low pass filtering processes they also show the robustness of fuzzy projection with respect to noise suggesting future studies in simultaneous compression and denoising

a threshold of ln n for approximating set cover given a collection f of subsets of s 1 n setcover is the problem of selecting as few as possiblesubsets from f such that their union covers s and maxk cover is the problem of selecting k subsets from f such that their union has maximum cardinality both these problems arenp hard we prove that 1 o 1 ln n is a threshold below which setcover cannot be approximated efficiently unless np has slightlysuperpolynomial time algorithms this closes the gap up to low orderterms between the ratio of approximation achievable by the greedyalogorithm which is 1 o 1 lnn and provious results of lund and yanakakis that showed hardness ofapproximation within a ratio of log 2 n 2s0 72 ln n for max k cover we show an approximationthreshold of 1 1 e up tolow order terms under assumption that p np

hukuhara difference of z numbers abstract real world problems in decision analysis economics optimization and other fields are characterized by fuzzy and partially reliable information in this regard prof zadeh suggested the concept of a z number as an ordered pair z a b of fuzzy numbers a and b the first of which describes a linguistic value and the second one is a fuzzy value of probability measure of the first one playing a role of reliability of information construction of mathematical models to handle z number based information is a new challenge which requires development of z number valued analysis including z number valued equations z number valued derivative and integral and other concepts one of the basic concepts in this realm is an inverse of addition operation in the existing literature devoted to set valued arithmetic and fuzzy arithmetic this operation is referred to as hukuhara difference for interval numbers and fuzzy numbers the existence conditions of hukuhara difference are derived unfortunately the concept of hukuhara difference for z numbers is not developed in this work for the first time we suggest a fundamental approach for development of the concept of hukuhara difference of z numbers examples are provided in the paper to show validity of the suggested approach

relational closure and partition powerset theories abstract for general powerset theories in categories new notions of relational closure or partition powerset theories in these categories are introduced these new types of powerset theories are defined as categories whose objects are the original powerset objects with relational closure or partition structures defined on these objects this construction generalizes classical constructions realized on powerset objects of all subsets of a given set or all fuzzy sets on a given set and it enables to define these structures on more general powerset objects examples of these new powerset objects are also shown

locality in distributed graph algorithms this paper concerns a number of algorithmic problems on graphs and how they may be solved in a distributed fashion the computational model is such that each node of the graph is occupied by a processor which has its own id processors are restricted to collecting data from others which are at a distance at most t away from them in t time units but are otherwise computationally unbounded this model focuses on the issue of locality in distributed processing namely to what extent a global solution to a computational problem can be obtained from locally available data three results are proved within this model a 3 coloring of an n cycle requires time omega log n this bound is tight by previous work of cole and vishkin any algorithm for coloring the d regular tree of radius r which runs for time at most 2r 3 requires at least omega sqrt d colors in an n vertex graph of largest degree delta an o delta 2 coloring may be found in time o log n

generalized uniform fuzzy partition in this note we propose a solution to an open problem which was presented by mesiar and stup anova we give two counterexamples to show that the hypothesis is not sufficient to get the result and we propose that the sufficient condition of the hypothesis is held by modifying the symmetry condition

a segmentation method for images compressed by fuzzy transforms in this paper we describe a segmentation method applied to images which are compressed by using fuzzy transforms the segmentation of the images is realized via the fgfcm fast generalized fuzzy c means clustering algorithm which is robust to noise and outliers the optimal number of clusters is determined via the pcaes partition coefficient and exponential separation validity index we use a similarity measure defined via lukasiewicz t norm for comparison between the original image and the reconstructed images the best results are obtained if this similarity measure overcomes a threshold value experimentally determined from the analysis of the trend of it with respect to the psnr peak signal to noise ratio

lattice valued f transforms and similarity relations abstract the relationships between lattice valued f transforms and similarity relations are investigated several similarity relations are naturally associated with f transforms three types of such similarity relations related to lattice valued f transforms are investigated namely similarity relationships that are related to spaces with fuzzy partitions as ground structures of the f transform relationships between extensional hulls with respect to similarity relations and f transforms and relationships between similarities of pairs of functions on the one hand and similarities of the f transforms of these functions on the other some examples of possible applications are also presented

on the hardness of approximating minimization problems we prove results indicating that it is hard to compute efficiently good approximate solutions to the graph coloring set covering and other related minimization problems specifically there is an e 0 such that graph coloring cannot be approximated with ratio n e unless p np set covering cannot be approximated with ratio c log n for any c 1 4 unless np is contained in dtime n poly log n similar results follow for related problems such as clique cover fractional chromatic number dominating set and others

linguistic characterization of time series the goal of this paper is to provide an overview of applications of special soft computing theories the fuzzy transform and fuzzy natural logic to analysis forecasting and mining information from time series the focus is especially placed on the ability of methods of fuzzy natural logic to provide information in sentences of natural language our approach is based on the decomposition of time series into three components the trend cycle seasonal component and noise the trend cycle is extracted using the f transform and its course is characterized by automatically generated linguistic description the latter is then used to forecast the trend cycle the trend cycle can be furthermore decomposed into trend general tendency and cycle the former is computed again using the f transform moreover the f1 transform makes it possible to estimate the direction of the trend which can then be characterized by expressions of natural language stagnating slightly increasing sharply decreasing etc finally we focus on selected problems of mining information from time series first we suggest an algorithm for finding intervals of monotonous behavior and then show how the theory of intermediate quantifiers a constituent of fuzzy natural logic and generalized aristotle s syllogisms can be applied to automatic summarization of information on time series

a smoothing filter based on fuzzy transform this paper is devoted to the smoothing of discrete functions using the fuzzy transform introduced by perfilieva we generalize a smoothing filter based on the fuzzy transform recently proposed by us to obtain a better control on the smoothed functions for this purpose a generalization of the concept of fuzzy partition is suggested and the smoothing filter is defined as a combination of the direct discrete fuzzy transform and a slightly modified inverse continuous fuzzy transform an approximation behavior total variation of smoothed functions and statistical properties including the description of the white noise reduction and the asymptotic expression of bias and variance are investigated and discussed the proposed filter is compared with the nadaraya watson estimator and the results are illustrated assuming financial data

a neural network approach to the fuzzy transform the paper deals with the f transform technique which was introduced as a method for an approximate representation of continuous functions the same task is solved by many methods from different areas neural networks also belong to techniques which have been proved to be powerful for approximation objectives they provide us with many advantages especially incremental way of learning parameters the paper inherits neural approaches to the f transform method and presents experiments justifying the proposed approach incremental way of determination of certain parameters of the f transform method which were up to now given by batch formula enriches possible areas of application of the method by fast on line processes moreover the neural approach is demonstrated to be an appropriate one for finding such fuzzy partition of a domain which respects better a given set of measured samples which are to be approximated by a continuous function with no predetermined shape

a color image reduction based on fuzzy transforms we present a new method for color image reduction based on the concept of fuzzy transform any image in a single band can be considered as a fuzzy matrix which is subdivided into submatrices called blocks each block is compressed with various compression rates by means of a fuzzy transform in two variables we compare our method with recent three algorithms due to g beliakov h bustince and d paternain based on the minimizing penalty functions defined over a discrete lattice the quality of the reduced image is measured by the mean square error mse and penalty function pen obtained by comparing both magnified and original images we also point out a threshold of the compression rate beyond which the mse follows a linear trend and the corresponding loss of information is still acceptable

b spline signal processing ii efficiency design and applications for pt i see ibid vol 41 no 2 p 821 33 1993 a class of recursive filtering algorithms for the efficient implementation of b spline interpolation and approximation techniques is described in terms of simplicity of realization and reduction of computational complexity these algorithms compare favorably with conventional matrix approaches a filtering interpretation low pass filter followed by an exact polynomial spline interpolator of smoothing spline and least squares approximation methods is proposed these techniques are applied to the design of digital filters for cubic spline signal processing an efficient implementation of a smoothing spline edge detector is proposed it is also shown how to construct a cubic spline image pyramid that minimizes the loss of information in passage from one resolution level to the next in terms of common measures of fidelity this data structure appears to be superior to the gaussian laplacian pyramid

logical structure of fuzzy if then rules abstract this paper provides a logical basis for manipulation with fuzzy if then rules our theory is wide enough and it encompasses not only finding a conclusion by means of the compositional rule of inference due to lotfi a zadeh but also other kinds of approximate reasoning methods e g perception based deduction provided that there exists a possibility to characterize them within a formal logical system in contrast with other approaches employing variants of multiple valued first order logic the approach presented here employs fuzzy type theory of v novak which has sufficient expressive power to present the essential concepts and results in a compact elegant and justifiable form within the effectively formalized representation developed here based on a complete logical system it is possible to reconstruct numerous well known properties of cri related fuzzy inference methods albeit not from the analytic point of view as usually presented but as formal derivations of the logical system employed the authors are confident that eventually all relevant knowledge about fuzzy inference methods based on fuzzy if then rule bases will be represented formalized and backed up by proof within the well founded logical representation presented here an immediate positive consequence of this approach is that suddenly all elements of a fuzzy inference method based on fuzzy if then rules are first class citizens of the representation there are clear logically founded definitions for fuzzy if then rule bases to be consistent complete or independent

improved approximation algorithms for the vertex cover problem in graphs and hypergraphs we obtain improved algorithms for finding small vertex covers in bounded degree graphs and hypergraphs we use semidefinite programming to relax the problems and introduce new rounding techniques for these relaxations on graphs with maximum degree at most delta the algorithm achieves a performance ratio of 2 1 o 1 frac 2 ln ln delta ln delta for large delta which improves the previously known ratio of 2 frac log delta o 1 delta obtained by halld orsson and radhakrishnan using similar techniques we also present improved approximations for the vertex cover problem in hypergraphs for k uniform hypergraphs with n vertices we achieve a ratio of k 1 o 1 frac k ln ln n ln n for large n and for k uniform hypergraphs with maximum degree at most delta the algorithm achieves a ratio of k 1 o 1 frac k k 1 ln ln delta ln delta for large delta these results considerably improve the previous best ratio of k 1 c delta frac 1 k 1 for bounded degree k uniform hypergraphs and k 1 c n frac k 1 k for general k uniform hypergraphs both obtained by krivelevich using similar techniques we also obtain an approximation algorithm for the weighted independent set problem matching a recent result of halldorsson

qualitative and quantitative analysis of stochastic processes based on measured data i theory and applications to synthetic data analysis of stochastic processes governed by the langevin equation is discussed the analysis is based on a general method for non parametric estimation of deterministic and random terms of the langevin equation directly from given data separate estimation of the terms corresponds to decomposition of process dynamics into deterministic and random components such decomposition provides a basis for qualitative and quantitative analysis of process dynamics in part i the following analysis possibilities are described and illustrated using various synthetic datasets 1 qualitative inspection of the estimated terms presented as fields 2 reconstruction of the deterministic and stochastic evolution of the process and 3 approximation of the deterministic term by an analytical function and quantitative treatment of the equations obtained in part ii these analysis possibilities are applied to experimental datasets from metal cutting and laser beam welding

on the structural properties of f m transform with applications abstract the aim of this study is to obtain some direct formulas for f m transforms of derivative integral and multiplication of functions in order to extend the application and flexibility of f m transform method using the derived explicit formula one can use f m transform method straightforwardly to solve various linear and nonlinear problems including differential and integral equations in this study the general form of the higher order fuzzy transforms are constructed using any arbitrary basis functions instead of the classical polynomial ones

polynomial alias higher degree fuzzy transform of complex valued functions abstract in this article we propose a general approach to the computation of components of the direct higher degree fuzzy transform apart from the orthogonal bases of the subspaces of polynomials of weighted hilbert spaces with respect to a generalized uniform fuzzy partition which are used in all papers on fuzzy transform of higher degree we admit also the non orthogonal bases an advantage of using non orthogonal bases consists in the possibility of replacing orthogonal polynomials derivation of which by the gram schmidt orthogonalization process can be questionable difficult or imprecise by suitable non orthogonal polynomials of much simpler form we present a simple matrix calculus and show how it can be used to introduce the components of the direct higher degree fuzzy transform with the help of the monomial basis we prove a convergence theorem and an approximation theorem for the higher degree fuzzy transform the results are illustrated by examples including a comparison with standard methods

operations fitting triangular norm based biresiduation abstract given a t norm t and the corresponding biresiduation t an operation k fits the latter or shortly is fitting if t is respected by k in all its arguments in this paper the structure of fitting operations is investigated special attention is paid to the case of basic t norms the connection between the fitting property and the lipschitz property is stressed some examples are given

construction of image reduction operators using averaging aggregation functions in this work we present an image reduction algorithm based on averaging aggregation functions we axiomatically define the concepts of image reduction operator and local reduction operator we study the construction of the latter by means of averaging functions and we propose an image reduction algorithm image reduction operator we analyze the properties of several averaging functions and their effect on the image reduction algorithm finally we present experimental results where we apply our algorithm in two different applications analyzing the best operators for each concrete application

supporting trading strategies by inverse fuzzy transform trading in finance requires to define strategies able to identify early trading buy sell signals i e conditions which suggest to enter or exit a position in order to exploit early mover advantage trading signals are generally identified by looking at the time series of prices and volumes several technical analysis indicators and strategies have been proposed and are commonly in use in this paper we propose inverse fuzzy transform as a means for building a new class of technical indicators experimental results show that this approach outperforms simple and exponential moving average when embedded in common strategies

fuzzy transforms for compression and decompression of color videos we use fuzzy transforms for coding decoding color frames of videos and we compare the results with the same frames reconstructed with the standard jpeg method we classify all the frames in intra frames and predictive frames by adopting a similarity measure based on the lukasiewicz t norm and a pre processing phase which determines the best similarity threshold value the compression is made on particular frames called d frames obtained from a suitable difference defined on the values of the pixels of an intra frame and a predictive frame under high compression rates we see that the peak signal to noise ratio of the frames obtained with the fuzzy transforms is averagely close to the psnr obtained with the jpeg method we use the videos at url sampl eng ohiostate edu sampl database htm

generalized extensionality of fuzzy relations in this work we investigate a relationship between extensionality and continuity of fuzzy relations we found conditions when extensionality implies continuity and vice versa

chatter onset in non regenerative cutting a numerical study a non linear model of non regenerative cutting is analyzed when damped the model has a stable fixed point for small values of chip thickness as chip thickness is increased the fixed point becomes unstable via a sub critical hopf bifurcation this transition corresponds to the onset of chatter when dynamic noise is added to the system chatter onset can occur in the sub critical parameter region although chatter free cutting is predicted by the stability analysis of the fixed point a noisy bifurcation diagram is compared qualitatively to a diagram obtained experimentally in regenerative cutting similar features regarding the onset of chatter are observed in both diagrams

on approximations of zadeh s extension principle in this paper we provide several results concerning the approximation of zadeh s extension of a given function we provide several simple observations due to which we provide a general procedure allowing to approximate zadeh s extension of any continuous map among other things we demonstrated that the technique of f transform can also be used to approximate this extension further we studied how the choice of the metric on the space of fuzzy sets resp fuzzy numbers can affect the quality of the approximation and we clarified that in comparison to recent habits the choice of the levelwise metric need not be the most appropriate one and that the approximation process could be simplified our results are general in the sense that we mostly require nothing but the continuity of the original map

non additive interval valued f transform this article proposes a new interval valued fuzzy transform its construction is based on a possibilistic interpretation of the partition on which the fuzzy transform is built the main advantage of this approach is that it provides specific interval valued functions whose interpretation is straightforward this interpretation relates to a traditional sampling reconstruction framework where little is known about the sampling and or reconstructing kernels numerous properties of the proposed approach are proved that could be useful for function analysis and comparison in the experimental section we illustrate some properties of the proposed transform while highlighting interesting features of the obtained framework

f transform with parametric generalized fuzzy partitions the fuzzy transform f transform has some interesting properties that allow its use in curve fitting and smoothing we suggest the following a a generalization of fuzzy partitions for the f transform to control and improve the smoothing effect and b a parametrization of the fuzzy numbers constituting the basic functions related to the fuzzy partitions an optimization criterion is then suggested to obtain the f transform with best approximation properties by estimating the basic functions from a family of parametric fuzzy numbers the generalized fuzzy partitions are characterized by the number of subintervals the bandwidth of the partition forming the core of each basic function and we suggest two procedures for bandwidth selection one based on taut string variation and a second based on generalized cross validation some illustrative examples are included

multi dimensional fuzzy transform and projection fuzzy and inverse fuzzy transforms introduced by i perfilieva is an important tool for signal and image fuzzy representation it led to several variants as the natural least squares minimization named fuzzy projection we deal with this transform proposing first to show its simplicity and its easy numerical implementation in any dimension of space secondly we point out that a key parameter permits to control its numerical robustness thirdly we show that this parameter also governs the stability of the representation by fuzzy projection when choosing different partitions of the space we conclude in discussing linear representation especially triangular representation for signals and pyramidal representation for images

interval valued probability density estimation based on quasi continuous histograms proof of the conjecture the sensitivity of histogram computation to the choice of a reference interval and number of bins can be attenuated by replacing the crisp partition on which the histogram is built by a fuzzy partition this involves replacing the crisp counting process by a distributed weighted voting process the counterpart to this low sensitivity is some confusion in the count values a value of 10 in the accumulator associated with a bin can mean 10 observations in the bin or 40 observations near the bin this confusion can bias the statistical decision process based on such a histogram in a recent paper we proposed a method that links the probability measure associated with any subset of the reference interval with the accumulator values of a fuzzy partition based histogram the method consists of transferring counts associated with each bin proportionally to its interaction with the considered subset two methods have been proposed which are called precise and imprecise pignistic transfer imprecise pignistic transfer accounts for the interactivity of two consecutive cells in order to propagate in the estimated probability measure counting confusion due to fuzzy granulation imprecise pignistic transfer has been conjectured to include precise pignistic transfer the present article proposes a proof of this conjecture

differentiation by the f transform and application to edge detection in this contribution we consider integrable functions of many in particular two variables and extend for them the technique of f transform to f1 transform we prove that linear coefficients in the representation of the f1 transform component approximate the first partial derivatives we give estimates for this approximation finally we discuss the application to the edge detection problem we propose a new algorithm where the technique of f1 transform is combined with the canny detector a theoretical justification and advantages of the proposed approach are emphasized and illustrative examples are presented

general approximation of fuzzy numbers by f transform in this paper we will prove that in most of the cases the extended inverse fuzzy transform preserves the quasi concavity of a fuzzy number and hence it can be used to generate fuzzy numbers by approximating the restriction of the membership function to its support in the case of continuous fuzzy numbers with cores containing more than one element the rate of uniform convergence is of linear type and the same holds when we approximate the important characteristics of a fuzzy number such as the value or the ambiguity moreover we have the preservation of the support and the convergence of the core which in addition can be determined precisely in the case of continuous fuzzy numbers with one element core it is in general necessary to normalize the approximation but the support is preserved again and the core can be determined exactly in this case too moreover the approximations have again linear rate of uniform convergence

axiomatic of lattice valued f transform abstract axioms for upper and lower fuzzy transformation systems are introduced and it is proved that a transformation function satisfies axioms for upper or lower respectively transformation systems if and only if it is an upper or lower respectively f transform categories of upper lower respectively transformation systems are introduced and it is proved that both categories are isomorphic to the category of spaces with fuzzy partitions we prove that lower and upper f transforms preserve extensional fuzzy sets associated with spaces with fuzzy partitions

a distributed 2 e approximation for vertex cover in o log δ e log log δ rounds we present a simple deterministic distributed 2 e approximation algorithm for minimum weight vertex cover which completes in o log elog log rounds where is the maximum degree in the graph for any e 0 that is at most o 1 for a constant e this implies a constant approximation in o log log log rounds which contradicts the lower bound of kmw10

a new reconstruction from the f transform components a new notion of adjoint fuzzy partition is introduced and the reconstruction of a function from its f transform components is analyzed an analogy with the nyquistshannonkotelnikov sampling theorem is discussed special attention is paid to the case where sample values of a function come with noise we show that the double reconstruction through the f transform components significantly reduces the influence of noise

b spline signal processing i theory the use of continuous b spline representations for signal processing applications such as interpolation differentiation filtering noise reduction and data compressions is considered the b spline coefficients are obtained through a linear transformation which unlike other commonly used transforms is space invariant and can be implemented efficiently by linear filtering the same property also applies for the indirect b spline transform as well as for the evaluation of approximating representations using smoothing or least squares splines the filters associated with these operations are fully characterized by explicitly evaluating their transfer functions for b splines of any order applications to differentiation filtering smoothing and least squares approximation are examined the extension of such operators for higher dimensional signals such as digital images is considered

development of autofocusing algorithm based on fuzzy transforms in this paper we propose a new passive auto focusing algorithm to acquire images with enhanced quality through a digital camera the focus measure operations play a key role in a passive auto focusing method the proposed focus measure operation is based on a localized version of image variance method the localization of the image variance focus measure operation can be implemented with the use of fuzzy partition and fuzzy transform in order to validate our approach experimental comparative studies are conducted on test images with the use of different focus measure operators the selected operators have been chosen from an extensive review of the state of the art approaches a comparative analysis demonstrates that the proposed algorithm is superior to the conventional methods

qualitative and quantitative analysis of stochastic processes based on measured data ii applications to experimental data analysis of stochastic processes governed by the langevin equation is discussed the analysis is based on a general method for non parametric estimation of deterministic and random terms of the langevin equation directly from given data separate estimation of the terms corresponds to the decomposition of process dynamics into deterministic and random components part i of the paper presented several possibilities for qualitative and quantitative analysis of process dynamics based on such decomposition in part ii some of these analysis possibilities are applied to experimental datasets from metal cutting and laser beam welding

fuzzy transforms of monotone functions with application to image compression this paper is focused on the special properties of functions monotonicity lipschitz continuity that are invariant with fuzzy transform on the basis of the monotonicity invariance we propose an efficient algorithm for improved image compression and reconstruction based on fuzzy transform

łukasiewicz transform and its application to compression and reconstruction of digital images we define the lukasiewicz transform as a residuated map and a homomorphism between semimodules over the semiring reducts of an mv algebra then we describe the lukasiewicz transform based ltb algorithm for image processing demonstrating its applicability

a primal dual bicriteria distributed algorithm for capacitated vertex cover in this paper we consider the capacitated vertex cover problem which is the variant of vertex cover where each node is allowed to cover a limited number of edges we present an efficient deterministic distributed approximation algorithm for the problem our algorithm computes a 2 epsilon approximate solution which violates the capacity constraints by a factor of 4 epsilon in a polylogarithmic number of communication rounds on the other hand we also show that every efficient distributed approximation algorithm for this problem must violate the capacity constraints our result is achieved in two steps we first develop a 2 approximate sequential primal dual algorithm that violates the capacity constraints by a factor of 2 subsequently we present a distributed version of this algorithm we demonstrate that the sequential algorithm has an inherent need for synchronization which forces any naive distributed implementation to use a linear number of communication rounds the challenge in this step is therefore to achieve a reduction of the communication complexity to a polylogarithmic number of rounds without worsening the approximation guarantee

from a gaussian mixture model to nonadditive fuzzy systems this work explores how a kind of probabilistic system namely the gaussian mixture model gmm can be translated to an additive fuzzy system we will prove the mathematical equivalence between the conditional mean of a gmm and the defuzzified output of a generalized fuzzy model gfm the relationship between a gmm and a gfm and the conditions for gmm to gfm translation will be made explicit in the form of theorems the work will then extend to special cases of the gfm specifically the mamdani larsen and takagi sugeno fuzzy models the possibility of reverse translation that is from a gfm to a gmm will also be discussed finally we will consider the generality of a gmm specifically how it can approximate other distribution functions

a new fuzzy approximation method to cauchy problems by fuzzy transform we present new numeric methods based on the first and second degree f transform for solving the cauchy problem we show that they outperform the second order rungekutta method especially when a right hand function is oscillating and or a solution is requested on a long interval moreover a new numeric method to solve fuzzy initial value problem using the f transform is presented the suitability of the presented new methods is theoretically justified and illustrated on various examples

the logical content of triangular bases of fuzzy sets in łukasiewicz infinite valued logic abstract continuing to pursue a research direction that we already explored in connection with godel dummett logic and ruspini partitions we show here that ukasiewicz logic is able to express the notion of pseudo triangular basis of fuzzy sets a mild weakening of the standard notion of triangular basis en route to our main result we obtain an elementary logic independent characterisation of triangular bases of fuzzy sets

forecasting seasonal time series based on fuzzy techniques abstract this paper is devoted to a method for the forecasting of seasonal time series the core of our approach is based on the fuzzy transform and fuzzy natural logic fnl techniques under the assumption that a time series can be additively decomposed into a trend cycle a seasonal component and an irregular fluctuation the forecasting is a combination of individual forecasting of each of these constituents more precisely the trend cycle and the seasonal component are predicted with the help of fuzzy transform pattern recognition and fuzzy natural logic techniques to model the irregular component we apply the box jenkins approach in the paper we compare the suggested method with two other well known methods namely stl see 14 and arima ones

approximation properties of fuzzy transforms we enlarge the class of fuzzy transforms f transforms by considering different types of fuzzy partitions new types of f transforms are constructed based on b splines shepard kernels bernstein basis polynomials and favard szasz mirakjan type operators we study approximation properties of f transforms particular cases of the classical korovkin s theorems are presented together with new error estimates for different types of fuzzy transforms it turns out that f transforms can be studied in the framework of classical approximation theory

fuzzy solutions to partial differential equations adaptive approach a new technique using an adaptive fuzzy algorithm to obtain the solutions to a class of partial differential equations pdes is presented the design objective is to find a fuzzy solution to satisfy precisely the pdes with boundary conditions according to the adaptive scheme of fuzzy logic systems a fuzzy solution with adjustable parameters for the pde is first described then a set of adaptive laws for tuning the free parameters in the consequent part is derived from minimizing an appropriate error function in addition an elegant approximated error bound between the exact solution and the proposed fuzzy solution with respect to the number of fuzzy rules and solution errors has also been derived furthermore the convergence of error equations in mesh points is also discussed from the energy perspective in this paper we show that the proposed method can solve a variety of pdes encountered in engineering comparisons are also made with solutions obtained by the finite element method

a simple parallel algorithm for the maximal independent set problem two basic design strategies are used to develop a very simple and fast parallel algorithms for the maximal independent set mis problem the first strategy consists of assigning identical copies o

filtering out high frequencies in time series using f transform abstract in this paper we focus on application of fuzzy transform f transform to analysis of time series under the assumption that the latter can be additively decomposed into trend cycle seasonal component and noise we prove that when setting properly width of the basic functions the inverse f transform of the time series closely approximates its trend cycle this means that the f transform almost completely removes the seasonal component and noise the obtained theoretical results are demonstrated on two artificial time series whose trend cycle is precisely known and on three real time series at the same time comparison with three classical methods namely stl method ssa method and low pass butterworth filter is also provided

nonstationary langevin equation statistical properties and application to explain effects observed in cardiological time series using the langevin equation we develop the model of a stochastic process subject to a given time dependent regulatory mechanism the effects of this nonstationarity on the statistical properties of the time series i e on global and conditional probability densities and on the moments of the distribution are derived application of these results on simple model trends allows one to approximate cardiological data and thus to explain effects recently observed in the reconstruction of the deterministic part of the langevin equation for time series of heart rate

fuzzy relation equations for coding decoding processes of images and videos we adopt fuzzy relation equations with continuous triangular norms for compression decompression processes of grey images colour images in the rgb space and frames of videos by comparing the results of the reconstructed images with standard methods like jpeg and mpeg 4 any image is subdivided in blocks and each block is coded decoded using arbitrary fuzzy sets as coders in the fuzzy equations we evaluate the peak signal to noise ratio psnr on the decompressed images for several values of the compression rates as measure of the quality of images and frames reconstructed the original frames of a video are classified in intra frames and predictive frames by using a similarity measure based on the well known lukasiewicz t norm

properties of fuzzy transform obtained from l p minimization and a connection with zadeh s extension principle abstract in 2006 perfilieva proved that the components fk of the fuzzy transform are actually minimizers of certain weighted l2 type distances obtained from a fuzzy partition inspired by this property we consider and analyze the properties of a fuzzy transform when the components fk are obtained as minimizers with respect to weighted lp type distances where p 1 in this way we can generate new approximation operators based on the fuzzy transform method for these operators we obtain good rates of uniform convergence and what is more they satisfy all the properties required for an effective approximation operator such as additivity and homogeneity if we consider the extended fuzzy transform and inverse fuzzy transform respectively in this setting of lp type distances then we get some nice supplementary properties like monotonicity of the operator and preservation of monotone functions finally we consider fuzzy valued versions of the l1 and l2 type fuzzy transforms proposed recently as quantile and expectile smoothing tools and we prove that the l1 type fuzzy valued transform of a continuous monotonic function f coincides with the zadeh s fuzzy extension fep evaluated at suitable fuzzy numbers obtained from the basic functions of the used fuzzy partition

equality relations as a basis for fuzzy control abstract the aim of this paper is to introduce a fuzzy control model with well founded semantics in order to explain the concepts applied in fuzzy control assuming that the domains of the input and output variables for the process are endowed with equality relations that reflect the indistinguishability of values lying closely together the use of triangular and trapezoidal membership functions can be justified and max inference where is a t norm turns out to be a consequence of our model distinguishing between a functional and a relational view of the control rules it is possible to explain when defuzzification strategies like mom or coa are appropriate or lead to undesired results

fuzzy transform as an additive normal form in this paper we recall a class of approximating formulas since they arose as a generalization of classical normal forms we denote them by the same term moreover we will show that the fuzzy transform f transform introduced in perfilieva i fuzzy approach to solution of differential equations with imprecise data application to reef growth problem in r v demicco g j klir eds fuzzy logic in geology academic press amsterdam 2003 pp 275 300 chapter 9 perfilieva i fuzzy transforms in j f peters a skowron eds transactions on rough sets ii rough sets and fuzzy sets lecture notes in computer science vol 3135 2004 pp 63 81 as an approximation method for continuous functions can be viewed as a particular case of normal form therefore the results valid for normal forms may be applied to f transforms as well

fuzzy transforms of higher order approximate derivatives a theorem in many practical applications it is useful to represent a function f x by its fuzzy transform i e by the average values f i f x a i x dx a i x dxover different elements of a fuzzy partitiona 1 x a n x for which a i x 0 and i 1 na i x 1 it is known that when we increase the number n of the partition elements a i x the resulting approximation gets closer and closer to the original function for each value x 0 the values f i corresponding to the function a i x for which a i x 0 1 tend to f x 0 in some applications if we approximate the function f x on each element a i x not by a constant but by a polynomial i e use a fuzzy transform of a higher order we get an even better approximation to f x in this paper we show that such fuzzy transforms of higher order and even sometimes the original fuzzy transforms not only approximate the function f x itself they also approximate its derivative s for example we have f i x 0 f x 0

fuzzy transforms theory and applications the technique of direct and inverse fuzzy f transforms of three different types is introduced and approximating properties of the inverse f transforms are described all three types of the direct f transform are transformations from a function space to a finite dimensional vector space the first ordinary f transform is constructed on the basis of the ordinary algebra of reals while the other two types of the f transform are constructed on the basis of residuated lattice the core idea of the technique of f transforms is a fuzzy partition of a universe into fuzzy subsets factors clusters granules etc we claim that for a sufficient representation of a function defined on this universe we may consider its average values over fuzzy subsets from the partition thus a function can be associated with a mapping from a set of fuzzy subsets to the set of its thus obtained average values a number of theorems establishing best approximation properties of the inverse f transforms are proved in fact three types of the inverse f transform are the best approximations in average from below and from above respectively as one of many possible applications we present a method of image compression and reconstruction on the basis of the f transform

compression and decompression of images with discrete fuzzy transforms in foregoing papers we have used the compression decompression method of images based on the concept of discrete fuzzy transform and its inverse of a function f defined on a real interval with respect to the fuzzy sets a 1 a n forming a fuzzy partition of such interval here we make a detailed experimental comparison with the similar method based on the fuzzy transforms f 6 and f 7 of f defined via a continuous triangular norm and its corresponding residuum respectively we consider some images of sizes 256x256 pixels extracted from the well known database corel galery arizona directory by using the same compression rate in both methods we have that the psnr peak signal to noise ratio obtained with the discrete fuzzy transform and its inverse of f is more higher than the psnr determined with the operators f 6 and f 7 defined via the usual lukasiewicz product and minimum triangular norms moreover we compare our results with the classical jpeg method for values of compression rate approximately equal to those used in the previous methods

arithmetic fuzzy models it is well known that a fuzzy rule base can be interpreted in different ways from a logical point of view the conjunctive interpretation is preferred while from a practical point of view the disjunctive interpretation has been dominantly present each of these interpretations results in a specific fuzzy relation that models the fuzzy rule base basic interpolation requirements naturally suggest a corresponding inference mechanism the direct image for the conjunctive interpretation and the subdirect image for the disjunctive interpretation interpolation then corresponds to solvability of some system of fuzzy relational equations in this paper we show that other types of fuzzy relations which are closely related to takagi sugeno t s models are of major interest as well these fuzzy relations are based on addition and multiplication only from which we get the name arithmetic fuzzy models under some mild requirements these fuzzy relations turn out to be solutions of the same systems of fuzzy relational equations the impact of these results is both theoretical and practical there exist simple solutions to systems of fuzzy relational equations other than the extremal solutions that have received all the attention so far which are moreover easy to implement

approximation of extensional fuzzy relations over a residuated lattice we investigate a relationship between extensionality of fuzzy relations and their lipschitz continuity on generalized metric spaces the duality of these notions is shown and moreover two particular applications of the extensionality property in the field of approximation are given

a measure of market volatility based on f transform abstract volatility is a tool of fundamental importance for capturing price movements in financial markets in general terms volatility is the tendency of prices to vary along a period of time measuring volatility is a quantitative proxy of uncertainty in prices there are different definitions of volatility measure one of the most common is based on the standard deviation of daily returns in this paper we propose an alternative definition based on f transform and its inverse the method is compatible with the theory of risk originally proposed by duncan luce as experimental setting we considered the nifty 50 stock market index within the period september 2000 and february 2017 results show that the proposed measure is smoother and better correlated between stocks

approximation of fuzzy numbers by max product bernstein operators in this paper firstly we extend to an arbitrary compact interval the definition of the nonlinear bernstein operators of max product kind by proving that their order of uniform approximation is the same as in the particular case of the unit interval then similarly to the particular case of the unit interval we will prove that these operators preserve the quasi concavity too since these properties will help us to generate in a simple way fuzzy numbers of the same support it turns out that these results are very suitable in the approximation of fuzzy numbers moreover these operators approximate the non degenerate segment core with a good order of approximation in addition in the case when the fuzzy numbers are given in parametric form the bernstein max product operator generates a sequence of fuzzy numbers whose widths expected intervals ambiguities and values approximate with a convergent rate the width the expected interval the ambiguity and the value of the approximated fuzzy number finally we obtain a quantitative approximation with respect to a well known integral type metric for some subclasses of functions leading to applications to the approximation of some subclasses of fuzzy numbers

fuzzy transforms method in prediction data analysis we present a prediction method based on fuzzy transforms for forecasting problems and we compare it with the well known wang mendel s one another comparison is made on the local linear wavelet neural network method via forecasting time series we apply these concepts to a mackey glass chaotic time series and to the monthly nao climatic index time series

an interval valued inversion of the non additive interval valued f transform as shown in a recent paper the fuzzy transform can be reinterpreted as a sampling process with an ill known sampling kernel or point spread function this reinterpretation leads to an interval valued direct fuzzy transform the non additive fuzzy transform it provides the convex set of all sampled values that should have been obtained by sampling a continuous signal with a convex set of sampling kernels its dual transform the non additive inverse fuzzy transform allows computation of an interval valued reconstruction of a sampled signal which is the set of all values that would have been obtained by interpolating or approximating this sampled signal when using a convex set of reconstruction kernels in this paper we consider using a new interval valued inversion process within this new framework to upsample a signal i e reconstruct a high resolution signal with a low resolution signal in a semi blind context as illustrated in the experimental part of the paper the main advantage of using this method compared to previous techniques is its inherent robustness with respect to modeling the sampling process