a geometrical method for removing edge effects from kernel type nonparametric regression estimators abstract we introduce a simple geometric method for removing edge effects from kernel type nonparametric regression estimators it involves reflecting the data set in two estimated points thereby generating a new data set with three times the range of the original data the usual kernel type estimator may be applied to the new enlarged data set without any danger of edge effects this technique is applicable generally to both regularly spaced and randomly spaced designs and admits a natural analog of leave one out cross validation the new cross validation algorithm may be extended to the very ends of the design interval unlike its more conventional counterpart which must be downweighted at the ends of the interval to avoid edge effects

nonparametric density estimation and detection in impulsive interference channels i estimators for pt i see ibid vol 42 no 2 4 p 1684 1697 1994 nonlinear processing significantly enhances detector performance in nongaussian noise relative to that of linear detectors several nonparametric detection schemes for impulsive noise channels are formulated using the nonparametric probability density estimators developed in part i the likelihood ratio test and the small signal locally optimum nonlinearity provide the basis for the formulation of these nonparametric detection schemes several modifications to these basic strategies are used to compensate for inaccuracies in the density estimates in particular for the problem of detecting a known signal in impulsive noise two modifications to the standard likelihood ratio test are considered the first is adapted from robust statistics whereas the second the l sub 1 error based detector is specifically formulated for use with density estimates both schemes are found to perform close to the optimum likelihood ratio detector for a wide variety of impulsive noise densities from the merits of these two tests a new detection scheme that approximates the locally optimum nonlinearity is then developed this detector which uses the nonparametric density estimators developed in part i is shown to perform very well for the wide variety of impulsive and heavy tailed densities considered in the study this nonparametric density estimate based detector is also shown to outperform more conventional nonparametric detectors in impulsive noise

selecting the individual numerical scale and prioritization method in the analytic hierarchy process a 2 tuple fuzzy linguistic approach the validity of the priority vector used in the analytic hierarchy process ahp relies on two factors the selection of a numerical scale and the selection of a prioritization method the traditional ahp selects only one numerical scale e g the saaty scale and one prioritization method e g the eigenvector method for each particular problem for this traditional selection approach there is disagreement on which numerical scale and prioritization method is better in deriving a priority vector in fact the best numerical scale and the best prioritization method both rely on the content of the pairwise comparison data provided by the ahp decision makers by defining a set of concepts regarding the scale function and the linguistic pairwise comparison matrices lpcms of the priority vector and by using lpcms to unify the format of the input and output of ahp this paper extends the ahp prioritization process under the 2 tuple fuzzy linguistic model based on the extended ahp prioritization process we present two performance measure criteria to evaluate the effect of the numerical scales and prioritization methods we also use the performance measure criteria to develop a 2 tuple fuzzy linguistic multicriteria approach to select the best numerical scales and the best prioritization methods for different lpcms in this paper we call this type of selection the individual selection of the numerical scale and prioritization method we also compare this individual selection with traditional selection by using both random and real data and show better results with individual selection

the owa based consensus operator under linguistic representation models using position indexes when using linguistic approaches to solve decision problems we need linguistic representation models the symbolic model the 2 tuple fuzzy linguistic representation model and the continuous linguistic model are three existing linguistic representation models based on position indexes together with these three linguistic models the corresponding ordered weighted averaging operators such as the linguistic ordered weighted averaging operator the 2 tuple ordered weighted averaging operator and the extended ordered weighted averaging operator have been developed respectively in this paper we analyze the internal relationship among these operators and propose a consensus operator under the continuous linguistic model or the 2 tuple fuzzy linguistic representation model the proposed consensus operator is based on the use of the ordered weighted averaging operator and the deviation measures some desired properties of the consensus operator are also presented in particular the consensus operator provides an alternative consensus model for group decision making this consensus model preserves the original preference information given by the decision makers as much as possible and supports consensus process automatically without moderator

on exponential bounds on the bayes risk of the kernel classification rule the exponential distribution free bounds for the kernel classification rule are derived the equivalence of all modes of the global convergence of the rule is established under optimal assumptions on the smoothing sequence also derived is the optimal global rate of convergence of the kernel regression estimate within the class of lipschitz distributions the rate is optimal for the nonparametric regression but not for classifications it is shown using the martingale device that weak strong and complete l sub 1 bayes risk consistencies are equivalent consequently the conditions on the smoothing sequence h sub n to 0 and nh sub n to infinity are necessary and sufficient for bayes risk consistency of the kernel classification rule the rate of convergence of the kernel classification rule is also given

the equivalence of weak strong and complete convergence in l_1 for kernel density estimates theorem 1 let k be a nonnegative borel measurable function on rd with f k x dx 1 then the following conditions are equivalent i j 0 in probability as n oo some f ii jn o 0 in probability as n s oo all f iii jn s 0 almost surely as n oo all f iv jn 0 exponentially as n oc i e for all e 0 there exist r no 0 such that p jne no all f v lim0h 0 and limnnhd oo also v implies iv when k is merely absolutely integrable and f k x dx 1 0

a finite difference method for the one dimensional time dependent schrodinger equation on unbounded domain a finite difference scheme is proposed for the one dimensional time dependent schrodinger equation we introduce an artificial boundary condition to reduce the original problem into an initial boundary value problem in a finite computational domain and then construct a finite difference scheme by the method of reduction of order to solve this reduced problem this scheme has been proved to be uniquely solvable unconditionally stable and convergent some numerical examples are given to show the effectiveness of the scheme

decision making with the ahp why is the principal eigenvector necessary in this paper it is shown that the principal eigenvector is a necessary representation of the priorities derived from a positive reciprocal pairwise comparison judgment matrix a aij when a is a small perturbation of a consistent matrix when providing numerical judgments an individual attempts to estimate sequentially an underlying ratio scale and its equivalent consistent matrix of ratios near consistent matrices are essential because when dealing with intangibles human judgment is of necessity inconsistent and if with new information one is able to improve inconsistency to near consistency then that could improve the validity of the priorities of a decision in addition judgment is much more sensitive and responsive to large rather than to small perturbations and hence once near consistency is attained it becomes uncertain which coefficients should be perturbed by small amounts to transform a near consistent matrix to a consistent one if such perturbations were forced they could be arbitrary and thus distort the validity of the derived priority vector in representing the underlying decision

a note on the numerical solution of the semilinear schrodinger equation abstract the unique solvability of local and nonlocal boundary value problems for the semilinear schrodinger equation in a hilbert space is investigated the convergence estimates for the solution of difference schemes are established some numerical examples illustrating the methods described in our work are given applicability of these methods to nonlinear schrodinger equation is discussed

bayes error estimation using parzen and k nn procedures the use of k nearest neighbor k nn and parzen density estimates to obtain estimates of the bayes error is investigated under limited design set conditions by drawing analogies between the k nn and parzen procedures new procedures are suggested and experimental results are given which indicate that these procedures yield a significant improvement over the conventional k nn and parzen procedures we show that by varying the decision threshold many of the biases associated with the k nn or parzen density estimates may be compensated and successful error estimation may be performed in spite of these biases experimental results are given which demonstrate the effect of kernel size and shape parzen the size of k k nn and the number of samples in the design set

eigenvector and logarithmic least squares abstract the eigenvector method deals with two questions simultaneously closeness and order together they belong to the field topology of order the metric idea of closeness is inadequate to judge what is a good approximation to data involving order relations there is usually a condition which relates order in the derived scale to order in the data this is precisely the kind of condition captured by the eigenvector but not by the logarithmic least squares estimate

transitive calibration of the ahp verbal scale abstract one of the strengths of the analytic hierarchy process ahp is that it allows decision makers to specify their preferences using a verbal scale yet as is well known a strict reliance on the corresponding saaty 1 9 numeric scale can induce some inconsistency hence we argue that in certain situations it may be appropriate to calibrate the verbal scale the result of this calibration is a geometric scale based on a single parameter we present some limited evidence that this geometric scale marginally outperforms the saaty 1 9 scale moreover we suggest that this calibration can be used to do a simple sensitivity analysis in cases where judgements are uncertain

kernels for nonparametric curve estimation summary the choice of kernels for the nonparametric estimation of regression functions and of their derivatives is investigated explicit expressions are obtained for kernels minimizing the asymptotic variance or the asymptotic integrated mean square error imse the present proof of the optimality of the latter kernels is restricted up to order k 5 these kernels are also of interest for the nonparametric estimation of probability densities and spectral densities a finite sample study indicates that higher order kernels asymptotically improving the rate of convergence may become attractive for realistic finite sample size suitably modified kernels are considered for estimating at the extremities of the data in a way which allows to retain the order of the bias found for interior points

nonparametric density estimation for classes of positive random variables a kernel based density estimator for positive random variables is proposed and analyzed in particular a nonparametric estimator is developed which takes advantage of the fact that positive random variables can be represented as the norms of random vectors by appropriately choosing the dimension of the assumed vector space the estimator can be structured to exploit a priori knowledge about the density to be estimated the asymptotic properties e g pointwise and l sub 1 consistency of this density estimator are investigated and found to be similar to the desirable features of the standard kernel estimator an upper bound on the expected value of the l sub 1 error is also derived which provides insight into the behavior of the estimator upon using this upper bound the optimal form for the estimator i e the kernel function the smoothing factor etc is selected via a minimax strategy in addition this upper bound is used to compare the asymptotic performance of the proposed estimator to that of the standard kernel estimator and to boundary corrected kernel estimators numerical examples illustrate that the proposed scheme outperforms the standard and boundary corrected estimators for a variety of density types

integrating three representation models in fuzzy multipurpose decision making based on fuzzy preference relations abstract the purpose of this paper is to study a fuzzy multipurpose decision making problem where the information about the alternatives provided by the experts can be of a diverse nature the information can be represented by means of preference orderings utility functions and fuzzy preference relations and our objective is to establish a general model which cover all possible representations firstly we must make the information uniform using fuzzy preference relations as uniform preference context secondly we present some selection processes for multiple preference relations based on the concept of fuzzy majority fuzzy majority is represented by a fuzzy quantifier and applied in the aggregation by means of an owa operator whose weights are calculated by the fuzzy quantifier we use two quantifier guided choice degrees of alternatives a dominance degree used to quantify the dominance that one alternative has over all the others in a fuzzy majority sense and a non dominance degree that generalises orlovski s non dominated alternative concept the application of the two above choice degrees can be carried out according to two different selection processes a sequential selection process and a conjunction selection process

a comparative study of the numerical scales and the prioritization methods in ahp abstract in the analytic hierarchy process ahp a decision maker first gives linguistic pairwise comparisons then obtains numerical pairwise comparisons by selecting certain numerical scale to quantify them and finally derives a priority vector from the numerical pairwise comparisons in particular the validity of this decision making tool relies on the choice of numerical scale and the design of prioritization method by introducing a set of concepts regarding the linguistic variables and linguistic pairwise comparison matrices lpcms and by defining the deviation measures of lpcms we present two performance measure algorithms to evaluate the numerical scales and the prioritization methods using these performance measure algorithms we compare the most common numerical scales the saaty scale the geometrical scale the ma zheng scale and the salo hamalainen scale and the prioritization methods the eigenvalue method and the logarithmic least squares method in addition we also discuss the parameter of the geometrical scale develop a new prioritization method and construct an optimization model to select the appropriate numerical scales for the ahp decision makers the findings in this paper can help the ahp decision makers select suitable numerical scales and prioritization methods

an approach to computing with words based on canonical characteristic values of linguistic labels herrera and martinez initiated a 2 tuple fuzzy linguistic representation model for computing with words cw which offers a computationally feasible method for aggregating linguistic information that are represented by linguistic variables with equidistant labels through counting indexes of the corresponding linguistic labels lawry introduced an alternative approach to cw based on mass assignment theory that takes into account the underlying definitions of the words recently we provided a new proportional 2 tuple fuzzy linguistic representation model for cw that is an extension of herrera and martinez s model and takes into account the underlying definitions of linguistic labels of linguistic variables in the process of aggregating linguistic information by assigning canonical characteristic values ccvs of the corresponding linguistic labels in this paper we study further into cw based on ccvs of linguistic labels to provide a unifying link between lawry s framework and herrera and martinez s 2 tuple framework as well as allowing for computationally feasible cw our approach is based on a formal definition of ccv functions of a linguistic variable which is introduced under the context of its proportional 2 tuple linguistic representation model as continuation of our earlier works and on a group voting model that is for the probabilistic interpretation of the whole semantics of the ordered linguistic terms set of an arbitrary linguistic variable after the general framework developed in the former part of this paper we focus on a particular linguistic variable probability we show that for a linguistic probability description the expectation of its posterior conditional probability is a canonical characteristic value of the linguistic probability description then a calculus for reasoning with linguistic syllogisms and inference from linguistic information is introduced it is investigated under the context that linguistic quantifiers in such linguistic syllogisms can be arbitrary linguistic probability description and that the related linguistic information can be linguistic facts overfacts or underfacts intrinsically our approach to this calculus is with computing with words based on canonical characteristic values of linguistic labels

direct approach processes in group decision making using linguistic owa operators abstract in a linguistic framework several group decision making processes by direct approach are presented these processes are designed using the linguistic ordered weighted averaging lowa operator to do so first a study is made of the properties and the axiomatic of lowa operator showing the rationality of its aggregation way and secondly we present the use of lowa operator to solve group decision making problems from individuals linguistic preference relations

finite difference method for multipoint nonlocal elliptic parabolic problems a finite difference method for solving the multipoint elliptic parabolic partial differential equation with a nonlocal boundary condition is considered stable difference schemes accurate to first and second orders for this problem are presented stability almost coercive stability and coercive stability for the solution of these difference schemes are obtained the theoretical statements for the solution of these difference schemes are supported by numerical examples

a simulation based evaluation of the approximate and the exact eigenvector methods employed in ahp abstract the analytic hierarchy process ahp is a popular multi criteria decision making method it provides ratio scale measurements of the priorities of elements in various levels of a hierarchy these priorities are obtained through pairwise comparisons of elements in one level with reference to each element in the immediate higher level saaty has suggested two methods viz the approximate eigenvector method aev and the exact eigenvector method eev for calculating the priorities research on ahp has led to conflicting views on the use of these two methods this paper presents a simulation analysis to evaluate them the simulation analysis uses the concept of approximating a continuous pairwise comparison cpc matrix by its closest discretized pairwise comparison dpc matrix this exercise supports saaty s theoretical statements and empirically concludes that the eev method is to be preferred over the aev method for the calculation of priority vectors

nonlocal boundary value problems for the schrodinger equation abstract in the present paper the nonlocal boundary value problem i d u d t a u f t 0 t t u 0 m 1 p m u m 0 1 2 p t for the schrodinger equation in a hilbert space h with the self adjoint operator a is considered stability estimates for the solution of this problem are established two nonlocal boundary value problems are investigated the first and second order of accuracy difference schemes for the approximate solutions of this nonlocal boundary value problem are presented the stability of these difference schemes is established in practice stability inequalities for the solutions of difference schemes for the schrodinger equation are obtained a numerical method is proposed for solving a one dimensional schrodinger equation with nonlocal boundary condition a procedure involving the modified gauss elimination method is used for solving these difference schemes the method is illustrated by giving numerical examples

multiperson decision making based on multiplicative preference relations abstract a multiperson decision making problem where the information about the alternatives provided by the experts can be presented by means of different preference representation structures preference orderings utility functions and multiplicative preference relations is studied assuming the multiplicative preference relation as the uniform element of the preference representation a multiplicative decision model based on fuzzy majority is presented to choose the best alternatives in this decision model several transformation functions are obtained to relate preference orderings and utility functions with multiplicative preference relations the decision model uses the ordered weighted geometric operator to aggregate information and two choice degrees to rank the alternatives quantifier guided dominance degree and quantifier guided non dominance degree the consistency of the model is analysed to prove that it acts coherently

on arbitrarily slow rates of global convergence in density estimation let a density f on rd be estimated by fn x x1 xn where x rd fn is a borel measurable function of its arguments and x1 xn are independent random vectors with common density f let p 1 be a constant one of the main results of this note is that for every sequence fn and for every positive number sequence an satisfying lim an 0 there exists an f such that e left smallint f n left x right f left x right p dx right a n infinitely often

estimating criterion weights using eigenvectors a comparative study abstract in addition to saaty s eigenvector there are infinitely many eigen weight vectors which can be constructed for any given data of estimated weight ratios as the judgment of the ratios are dependent on personal experience learning situations and state of mind inconsistencies and degree of easiness or judgment on these individual ratios can be different we study the properties of the different eigen weight vectors including that of saaty and that recently proposed by cogger and yu a general framework for the construction of eigen weight vectors incorporating that confidence in obtaining the individual ratios can be different will be proposed and discussed

fuzzy linguistic pert a model for program evaluation and review technique pert under fuzzy linguistic contexts is introduced in this fuzzy linguistic pert network model each activity duration is represented by a fuzzy linguistic description aggregation and comparison of the estimated linguistic expectations of activity durations are manipulated by the techniques of computing with words cw to provide suitable contexts for this purpose we first introduce several variations of basic linguistic labels of a linguistic variable such as weighted linguistic labels generalized linguistic labels and weighted generalized linguistic labels etc and then based on the notion of canonical characteristic value ccv function of a linguistic variable we develop some related cw techniques for aggregation and comparison of these linguistic labels afterward using a computing technique of linguistic probability introduced by zadeh and based on the new developed cw techniques for weighted generalized linguistic labels we investigate the associated linguistic expectation pert network of a fuzzy linguistic pert network also throughout the paper several examples are used to illustrate related notions and applications