on the application of fuzzy sets theory to the optimal flood control problem arising in water resources systems abstract we first survey the application of fuzzy sets theory to various problems occurring in water resources systems the problem of optimal flood control planning by an appropriate integration of structural and non structural measures with the objective of optimizing the flood damage reduction due to recurrent floods is modeled via fuzzy sets methodologies a two level optimization model appropriate for planning decisions first on a regional or local level and then on a national level is proposed a third phase involving coordination is appended in particular a fuzzy optimization model involving an adroit combination of fuzzy dynamic programming type reasoning and a branch and bound type search procedure is offered as a more utilitarian approach than its crisp equivalent the performance of the algorithm is illustrated with an example

recent developments in nonparametric density estimation advances in computation and the fast and cheap computational facilities now available to statisticians have had a significant impact upon statistical research and especially the development of nonparametric data analysis procedures in particular theoretical and applied research on nonparametric density estimation has had a noticeable influence on related topics such as nonparametric regression nonparametric discrimination and nonparametric pattern recognition this article reviews recent developments in nonparametric density estimation and includes topics that have been omitted from review articles and books on the subject the early density estimation methods such as the histogram kernel estimators and orthogonal series estimators are still very popular and recent research on them is described different types of restricted maximum likelihood density estimators including order restricted estimators maximum penalized likelihood estimators and sieve estimators are discussed where restrictions are imposed upon the class of densities or on the form of the likelihood function nonparametric density estimators that are data adaptive and lead to locally smoothed estimators are also discussed these include variable partition histograms estimators based on statistically equivalent blocks nearest neighbor estimators variable kernel estimators and adaptive kernel estimators for the multivariate case extensions of methods of univariate density estimation are usually straightforward but can be computationally expensive a method of multivariate density estimation that did not spring from a univariate generalization is described namely projection pursuit density estimation in which both dimensionality reduction and density estimation can be pursued at the same time finally some areas of related research are mentioned such as nonparametric estimation of functionals of a density robust parametric estimation semiparametric models and density estimation for censored and incomplete data directional and spherical data and density estimation for dependent sequences of observations

on the maximum deviation between the histogram and the underlying density the limiting joint distribution of the location and size of the maximum deviation between the historgram and the underlying density is derived for smooth densities the location and size of the maximum are asymptotically independent the size has a limiting double exponential distribution and the location has a limiting normal distribution

data compression and histograms in this paper the relationship between code length and the selection of the number of bins for a histogram density is considered for a sequence of iid observations on 0 1 first we use a shortest code length criterion to select the number of bins for a histogram a uniform almost sure asymptotic expansion for the code length is given and it is used to prove the asymptotic optimality of the selection rule in addition the selection rule is consistent if the true density is uniform 0 1 secondly we deal with the problem what is the best achievable average code length with underlying density functionf minimax lower bounds are derived for the average code length over certain smooth classes of underlying densitiesf for the smooth class with bounded first derivatives the rate in the lower bound is shown to be achieved by a code based on a sequence of histograms whose number of bins is changed predictively moreover this best code can be modified to ensure that the almost sure version of the code length has asymptotically the same behavior as its expected value i e the average code length

a hybrid classification algorithm by subspace partitioning through semi supervised decision tree among data mining techniques the decision tree is one of the more widely used methods for building classification models in the real world because of its simplicity and ease of interpretation however the method has some drawbacks including instability the nonsmooth nature of the decision boundary and the possibility of overfitting to overcome these problems several works have utilized the relative advantages of other classifiers such as logistic regression support vector machine and neural networks in combination with a decision tree in hybrid models which avoid the drawbacks of other models some hybrid models have used decision trees to quickly and efficiently partition the input space and many studies have proved the effectiveness of the hybrid methods however there is room for further improvement by considering the topological properties of a dataset because typical decision trees split nodes based only on the target variable the proposed semi supervised decision tree splits internal nodes by utilizing both labels and the structural characteristics of data for subspace partitioning to improve the accuracy of classifiers applied to terminal nodes in the hybrid models experimental results confirm the superiority of the proposed algorithm and demonstrate the detailed characteristics of the algorithm highlightspropose the semi supervised split criterion for decision trees combine the semi supervised decision tree as subspace partitioning with other classifiers experiments on several datasets showed that the proposed method outperforms the existing ones

improving quality of sample entropy estimation for continuous distribution probability functions abstract entropy is a one of the key parameters characterizing state of system in statistical physics although the entropy is defined for systems described by discrete and continuous probability distribution function pdf in numerous applications the sample entropy is estimated by a histogram which in fact denotes that the continuous pdf is represented by a set of probabilities such a procedure may lead to ambiguities and even misinterpretation of the results within this paper two possible general algorithms based on continuous pdf estimation are discussed in the application to the shannon and tsallis entropies it is shown that the proposed algorithms may improve entropy estimation particularly in the case of small data sets

oversmoothed nonparametric density estimates abstract the optimal histogram for a sample of size n from a density defined on the entire real line requires at least 2n 1 3 bins under mild smoothness conditions similar bounds exist for the frequency polygon and kernel estimators values near these bounds give nearly optimal results for a variety of smooth densities providing good quick density estimates

statistical inference for dea estimators of directional distances in productivity and efficiency analysis the technical efficiency of a production unit is measured through its distance to the efficient frontier of the production set the most familiar non parametric methods use farrell debreu shephard or hyperbolic radial measures these approaches require that inputs and outputs be non negative which can be problematic when using financial data recently chambers et al 1998 have introduced directional distance functions which can be viewed as additive rather than multiplicative measures efficiency directional distance functions are not restricted to non negative input and output quantities in addition the traditional input and output oriented measures are nested as special cases of directional distance functions consequently directional distances provide greater flexibility however until now only free disposal hull fdh estimators of directional distances and their conditional and robust extensions have known statistical properties simar and vanhems 2012 this paper develops the statistical properties of directional d estimators which are especially useful when the production set is assumed convex we first establish that the directional data envelopment analysis dea estimators share the known properties of the traditional radial dea estimators we then use these properties to develop consistent bootstrap procedures for statistical inference about directional distance estimation of confidence intervals and bias correction the methods are illustrated in some empirical examples

on the relationship between stability of extreme order statistics and convergence of the maximum likelihood kernel density estimate by michel broniatowski paul deheuvels and luc devroyeiuniversite pierre et marie curie universite pierre et marie curie andmcgill universitylet f be a density on the real line and let f be the kernel estimate of fin which the smoothing factor is obtained by maximizing the cross validatedlikelihood product according to the method of duin and habbema hermansand vandenbroek under mild regularity conditions on the kernel and f weshow among other things that f jf f 0 almost surely if and only if thesample extremes of f are strongly stable

the essential histogram the histogram is widely used as a simple exploratory display of data but it is usually not clear how to choose the number and size of bins we construct a confidence set of distribution functions that optimally address the two main tasks of the histogram estimating probabilities and detecting features such as increases and modes in the distribution we define the essential histogram as the histogram in the confidence set with the fewest bins thus the essential histogram is the simplest visualization of the data that optimally achieves the main tasks of the histogram the only assumption we make is that the data are independent and identically distributed we provide a fast algorithm for the essential histogram and illustrate our methodology with examples an r package is available on cran

research in statistical graphics abstract three active areas of research in statistical graphics are methodology computing and graphical perception the five articles in this special section provide a good representation of these areas dynamic graphical methods which involve a tight weaving of methodological and computing issues are a particularly exciting topic of research dynamic graphics will soon have a major impact on how data are analyzed evaluation of graphical inventions is difficult certain visualization issues can be evaluated by studies in graphical perception at the moment however the performance of most graphical methods can be studied only by rigorous field testing in which the methods are used to analyze a large number of data sets a bibliography is provided to allow readers to begin a study of any of the major areas of statistical graphics

asymptotically optimal cells for a historgram on examine les proprietes de l histogramme lorsque les cellules sont arbitraires on obtient une expression asymptotique pour l infimum de l erreur quadratique moyenne integree mise sur toutes les partitions

mutual information identifies spurious hurst phenomena in resting state eeg and fmri data long range memory in time series is often quantified by the hurst exponent h a measure of the signal s variance across several time scales we analyze neurophysiological time series from electroencephalography eeg and functional magnetic resonance imaging fmri resting state experiments with two standard hurst exponent estimators and with the time lagged mutual information function applied to discretized versions of the signals a confidence interval for the mutual information function is obtained from surrogate markov processes with equilibrium distribution and transition matrix identical to the underlying signal for eeg signals we construct an additional mutual information confidence interval from a short range correlated tenth order autoregressive model we reproduce the previously described hurst phenomenon h 0 5 in the analytical amplitude of alpha frequency band oscillations in eeg microstate sequences and in fmri signals but we show that the hurst phenomenon occurs without long range memory in the information theoretical sense we find that the mutual information function of neurophysiological data behaves differently from fractional gaussian noise fgn for which the hurst phenomenon is a sufficient condition to prove long range memory two other well characterized short range correlated stochastic processes ornstein uhlenbeck cox ingersoll ross also yield h 0 5 whereas their mutual information functions lie within the markovian confidence intervals similar to neural signals in these processes which do not have long range memory by construction a spurious hurst phenomenon occurs due to slow relaxation times and heteroscedasticity time varying conditional variance in summary we find that mutual information correctly distinguishes long range from short range dependence in the theoretical and experimental cases discussed our results also suggest that the stationary fgn process is not sufficient to describe neural data which seem to belong to a more general class of stochastic processes in which multiscale variance effects produce hurst phenomena without long range dependence in our experimental data the hurst phenomenon and long range memory appear as different system properties that should be estimated and interpreted independently

review papers recent developments in nonparametric density estimation abstract advances in computation and the fast and cheap computational facilities now available to statisticians have had a significant impact upon statistical research and especially the development of nonparametric data analysis procedures in particular theoretical and applied research on nonparametric density estimation has had a noticeable influence on related topics such as nonparametric regression nonparametric discrimination and nonparametric pattern recognition this article reviews recent developments in nonparametric density estimation and includes topics that have been omitted from review articles and books on the subject the early density estimation methods such as the histogram kernel estimators and orthogonal series estimators are still very popular and recent research on them is described different types of restricted maximum likelihood density estimators including order restricted estimators maximum penalized likelihood estimators and sieve estimators are discussed where re

estimating the false discovery rate using nonparametric deconvolution summary given a set of microarray data the problem is to detect differentially expressed genes using a false discovery rate fdr criterion as opposed to common procedures in the literature we do not base the selection criterion on statistical significance only but also on the effect size therefore we select only those genes that are significantly more differentially expressed than some f fold e g f 2 this corresponds to use of an interval null domain for the effect size based on a simple error model we discuss a naive estimator for the fdr interpreted as the probability that the parameter of interest lies in the null domain e g log2 2 1 given that the test statistic exceeds a threshold we improve the naive estimator by using deconvolution that is the density of the parameter of interest is recovered from the data we study performance of the methods using simulations and real data

exploratory projection pursuit abstract a new projection pursuit algorithm for exploring multivariate data is presented that has both statistical and computational advantages over previous methods a number of practical issues concerning its application are addressed a connection to multivariate density estimation is established and its properties are investigated through simulation studies and application to real data the goal of exploratory projection pursuit is to use the data to find low one two or three dimensional projections that provide the most revealing views of the full dimensional data with these views the human gift for pattern recognition can be applied to help discover effects that may not have been anticipated in advance since linear effects are directly captured by the covariance structure of the variable pairs which are straightforward to estimate the emphasis here is on the discovery of nonlinear effects such as clustering or other general nonlinear associations among the variables although arbitrary

fisher information and criticality in the kuramoto model of nonidentical oscillators we use the fisher information to provide a lens on the transition to synchronization of the kuramoto model of nonidentical frequencies on a variety of undirected graphs we numerically solve the equations of motion for a n 400 complete graph and n 1000 small world scale free uniform random and random regular graphs for large but finite graphs of small average diameter the fisher information f as a function of coupling shows a peak closely coinciding with the critical point as determined by kuramoto s order parameter or synchronization measure r however for graphs of larger average diameter the position of the peak in f differs from the critical point determined by estimates of r on the one hand this is a finite size effect even at n 1000 however we show across a range of topologies that the fisher information peak points to a transition for smaller graphs that indicates structural changes in the numbers of locally phase synchronized clusters often directly from metastable to stable frequency synchronization solving explicitly for a two cluster ansatz subject to gaussian noise shows that the fisher infomation peaks at such a transition we discuss the implications for fisher information as an indicator for edge of chaos phenomena in finite coupled oscillator systems

multinomial approximations for nonparametric experiments which minimize the maximal loss of fisher information certain nonparametric product experimentsp n n can asymptotically be approximated by multinomial experiments obtained by a finite interval partition of the sample space the real line for specific familiesp n defined in terms of bounded fisher information and monotone likelihood ratios with bounded derivatives we study the problem to calculate a partition which is optimal in the sense that it minimizes the maximal loss of fisher information caused by the discretization this leads to a minimax problem by considering partitions of the sample space intok intervals which have equal probability under a densityh and then lettingk we obtain an expansion for the quantity loss of fisher information which is of orderk 2 under regularity conditions the corresponding minimax problem for the first order term of this expansion is shown to be the unique solution of a free boundary problem

on stochastic complexity and nonparametric density estimation summary we use the concepts of stochastic complexity description length and model selection to develop data based methods for choosing smoothing parameters in nonparametric density estimation in the case of histogram estimators we derive a simple exact formula for stochastic complexity when the prior distribution of cell probabilities is uniform over the class of all possible choices the formula depends only on the data and the smoothing parameter which is readily chosen according to the criterion of minimum stochastic complexity approaches based on stochastic complexity and description length are shown to be asymptotically equivalent in certain circumstances they produce a degree of smoothing which is almost optimal from the viewpoint of minimizing l or supremum distance but which smooths a little more than is optimal in the sense of minimizing l distance for any finite value of r

optimal bandwidth selection for re substitution entropy estimation abstract a new fusion approach of selecting an optimal bandwidth for re substitution entropy estimator re is presented in this study when approximating the continuous entropy with density estimation two types of errors will be generated entropy estimation error type i error and density estimation error type ii error these two errors are all strongly dependent on the undetermined bandwidths firstly an experimental conclusion based on 24 typical probability distributions is demonstrated that there is some inconsistency between the optimal bandwidths associated with these two errors secondly two different error measures for type i and type ii errors are derived a trade off between type i and type ii errors is a fundamental and potential property of our proposed method called re i ii thus the fusion of these two errors is conducted and an optimal bandwidth for re i ii is solved finally the experimental comparisons are carried out to verify the estimation performance of our proposed strategy the discretization method is deemed to be the necessary preprocessing technology for the calculation of continuous entropy traditionally so the nine mostly used unsupervised discretization methods are introduced to give comparison of their computational performances with that of re i ii and five most popular estimators for entropy approximation are also plugged into our comparisons splitting data estimator sde cross validation estimator cve m spacing estimator mse m n spacing estimator mnse and nearest neighbor distance estimator nnde the simulation studies on 24 different typical density distributions show that re i ii can obtain the better estimation performance among the involved methods meanwhile the estimation behaviors of different entropy estimation methods are also revealed based on the comparative results the empirical analysis demonstrates that re i ii is more insensitive to data and a better generalizable way for the estimation of continuous entropy re i ii makes it possible for a handy optimal bandwidth to be derived from a given dataset

survey of utilisation of fuzzy technology in medicine and healthcare the complexity of biological systems unlike physical science applications makes the development of computerised systems for medicine not a straightforward algorithmic solution because of the inherent uncertainty which arises as a natural occurrence in these types of applications human minds work from approximate data extract meaningful information from massive data and produce crisp solutions fuzzy logic provides a suitable basis for the ability to summarise and extract from masses of data impinging upon the human brain those facts that are related to the performance of the task at hand in practice a precise model may not exist for biological systems or it may be too difficult to model in these cases fuzzy logic is considered as an appropriate tool for modelling and control since our knowledge and experience are directly contained and presented in control strategies without explicit mathematical models this paper surveys the utilisation of fuzzy logic in medical sciences with an analysis of its possible future penetration

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

on the histogram as a density estimator l 2 theory let f be a probability density on an interval i finite or infinite i includes its finite endpoints if any and f vanishes outside of i let x1 x k be independent random variables with common density f the empirical histogram for the x s is often used to estimate f to define this object choose a reference point xosi and a cell width h let nj be the number of x s falling in the j th class interval

difference distance and similarity as a basis for fuzzy decision support based on prototypical decision classes the use of prototypical decision classes as a basis for fuzzy decision support is outlined it is shown that given certain assumptions about the structure of both numeric and non numeric linguistic variables trapezoidal fuzzy sets can be used to model linguistic terms in particular the notion of a fuzzy index is introduced to model sets of linguistic terms for which there is no formal measurement scale the similarity between prototypes and input cases is measured as a function of the difference or distance between fuzzy sets the difference measure is itself a fuzzy set to reflect the underlying uncertainties in the original linguistic model similarity profiles for different decision classes are output as trapezoidal fuzzy sets the approach can be applied to the rapid development of decision aids using standard business software

multifractal diffusion entropy analysis optimal bin width of probability histograms abstract in the framework of multifractal diffusion entropy analysis we propose a method for choosing an optimal bin width in histograms generated from underlying probability distributions of interest the method presented uses techniques of renyi s entropy and the mean squared error analysis to discuss the conditions under which the error in the multifractal spectrum estimation is minimal we illustrate the utility of our approach by focusing on a scaling behavior of financial time series in particular we analyze the s p500 stock index as sampled at a daily rate in the time period 1950 2013 in order to demonstrate a strength of the method proposed we compare the multifractal spectrum for various bin widths and show the robustness of the method especially for large values of q for such values other methods in use e g those based on moment estimation tend to fail for heavy tailed data or data with long correlations connection between the spectrum and renyi s q parameter is also discussed and elucidated on a simple example of multiscale time series

on the distribution of spacings between zeros of the zeta function etude numerique de la distribution des espacements des zeros de la fonction zeta de riemann observation de certains phenomenes inatendus

on polynomial based projection indices for exploratory projection pursuit we develop asymptotic theory for two polynomial based methods of estimating orientation in projection pursuit density approximation one of the techniques uses legendre polynomials and has been proposed and implemented by friedman 1 the other employs hermite functions issues of smoothing parameter choice and robustness are addressed it is shown that each method can be used to construct sqrt n consistent estimates of the projection which maximizes distance from normality although the former can only be employed in that manner when the underlying distribution has extremely light tails the former can be used very generally to measure low frequency departure from normality

medical diagnosis fuzzy sets and degrees of belief abstract a model simulating the medical diagnostic process is presented diagnostic groups are considered as fuzzy sets partial knowledge is quantified by belief functions the process is considered as the evaluation by the clinician of his degree of belief concerning the belonging of his patient to a fuzzy set given fuzzy and partial informations

frequency polygons theory and application abstract in this article i investigate the theoretical properties and applications of the frequency polygon which is constructed by connecting with straight lines the mid bin values of a histogram for estimating an unknown probability density function using a random sample the frequency polygon is shown to dominate the histogram with respect to the criterion of integrated mean squared error achieving the same rate of convergence to zero of the integrated mean squared error as non negative kernel estimators data based algorithms for constructing frequency polygons are discussed and illustrated one is based on a histogram with bin width equal to 2 15sn 1 5 where s is an estimate of the standard deviation from a sample of size n another approach is based on the method of generalized crossvalidation the bivariate frequency polygon is also investigated comparisons are made between the frequency polygon and other density estimators examples with data in one and two dimensions are presented

nonparametric estimation of a probability density on a riemannian manifold using fourier expansions supposing a given collection y 1 cdots y n of i i d random points on a riemannian manifold we discuss how to estimate the underlying distribution from a differential geometric viewpoint the main hypothesis is that the manifold is closed and that the distribution is sufficiently smooth under such a hypothesis a convergence arbitrarily close to the n 1 2 rate is possible both in the l 2 and the l infty senses

asymptotic maximum order statistic for sir in kappa mu shadowed fading using tools from extreme value theory evt it is proved that when the user signal and the interferer signals undergo independent and non identically distributed i n i d kappa mu shadowed fading the limiting distribution of the maximum of l independent and identically distributed i i d signal to interference ratio sir random variables rvs is a frechet distribution it is observed that this limiting distribution is close to the true distribution of maximum for maximum sir evaluated over moderate l furthermore moments of the maximum rv is shown to converge to the moments of the frechet rv in addition the rate of convergence of the actual distribution of the maximum to the frechet distribution is derived and analyzed for different kappa and mu parameters finally results from the stochastic ordering are used to analyze the variation in the limiting distribution with respect to the variation in source fading parameters these results are then used to derive upper bound for the rate in full array selection fas schemes for the antenna selection and the asymptotic outage probability and the ergodic rate in maximum sum capacity msc scheduling systems

akaike s information criterion and kullback leibler loss for histogram density estimation we take a detailed look at akaike s information criterion aic and kullback leibler cross validation klcv in the problem of histogram density estimation two different definitions of number of unknown parameters inaic are considered a careful description is given of the influence of density tail properties on performance of both types ofaic and onklcv a number of practical conclusions emerge in particular we find thataic will often give problems when used with heavy tailed unbounded densities but can perform quite well with compactly supported densities in the latter case both types ofaic produce similar results and those results will sometimes be asymptotically equivalent to the ones obtained fromklcv however depending on the shape of the true density theklcv method can fail to balance bias and variance components of loss with the result thatklcv andaic may produce very different results

the design of stable sparse wavefield extrapolators using projections onto convex sets abstractwe present the results of poststack explicit depth migration of the well known 2d seg eage salt model zero offset seismic data using sparse wavefield extrapolators the extrapolators are designed to be sparse by forcing some of the very small complex valued coefficients magnitude values to be zero the proposed extrapolators design method combines the previously reported modified projections onto convex sets mpocs for designing explicit depth frequency space f x wavefield extrapolation operators with hard thresholding of the small extrapolators coefficients magnitude the real and imaginary parts of the mpocs operators with small magnitudes are replaced by zeros during the mpocs algorithm iterations the migrated result of the seg eage salt model data using such sparse designed operators shows comparable migrated results using the nonsparse version of the mpocs extrapolation operators as well as the image obtained using the well known phase shift plus interpolation pspi migration techn

measurement and valuation of a fuzzy mathematical model for medical diagnosis classical mathematical models for medical diagnosis which have been computerized are known to perform very poorly when compared to diagnoses made by the physician factors which contribute to their poor performance relate to the omission by these models of important information on the patient such as symptoms of past undiagnosed diseases which can only be vaguely recalled by the patient other deficiencies include failure to model the stage of development of the disease and certain intrinsically fuzzy aspects of the pertinent information nets that are needed to develop a medical hypothesis models which attempted to remedy these shortcomings were developed and presented by the authors elsewhere in this effort we describe a study in which our fuzzy diagnosis models were computerized validated and compared with a mock physician hypothesis as well as existing mathematical models the example involved a medical hypothesis concerning a medical condition of valvular heart disease the fuzzy sets together with their membership functions were measured using a distance criterion measure similar to the one employed by kochen to quantify fuzzy adjectives in psychology the results show that while there were discrepancies between the fuzzy model s and the physician s hypotheses the model s performance was vastly superior to that of existing mathematical models while the fuzzy models did well in naming the diseases they performed less satisfactorily in the specification of the disease stage development a factor completely ignored by classical models improvement in measurement aspects and the choice of diseases possessing discriminant attributes will greatly increase the accuracy of these models it is however shown that a fuzzy systems approach is not only practical but results in models of greater validity than those based on classical set theoretic approaches

estimating model discrepancy a quantity 2 is defined as the discrepancy between an actual and an assumed statistical model an estimator is developed for the case in which data are grouped the estimator is based on pearson