periodic and partially periodic representations of su n q the gelfand zetlin basis is adapted tosu n q forq a root of unit extra parameters are incorporated in the matrix elements of the generators to obtain all the invariants corresponding to the augmented center a crucial identity is derived and proved which guarantees the periodicity of the action of the generators full periodicity is relaxed by stages some raising and lowering operators remaining injective while others become nilpotent with corresponding changes in the dimension of the representation in the extreme case of highest weight representations all the raising and lowering operators are nilpotent as an alternative approach an auxiliary algebra giving all the periodic representations is presented an explicit solution of this system forn 3 while fully equivalent to the g z basis turns out to be much simpler

finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple lie algebra let g be a complex simple lie algebra we show that whent is not a root of 1 all finite dimensional representations of the quantum analogu t g are completely reducible and we classify the irreducible ones in terms of highest weights in particular they can be seen as deformations of the representations of the classical u g

chiral potts model as a descendant of the six vertex model we observe that then state integrable chiral potts model can be considered as a part of some new algebraic structure related to the six vertex model as a result we obtain a functional equation which is supposed to determine all the eigenvalues of the chiral potts model transfer matrix

on the stability of j homomorphisms abstract the main purpose of this paper is to prove the generalized hyers ulam rassias stability of j homomorphisms between j algebras

efficient bootstrap simulation on donne deux idees pour renforcer une simulation la premiere est d equilibrer les echantillons simules la seconde est de faire un usage explicite des approximations qui ne demandent pas de simulation

model evaluation based on the sampling distribution of estimated absolute prediction error the construction of a reliable practically useful prediction rule for future responses is heavily dependent on the adequacy of the fitted regression model in this article we consider the absolute prediction error the expected value of the absolute difference between the future and predicted responses as the model evaluation criterion this prediction error is easier to interpret than the average squared error and is equivalent to the misclassification error for a binary outcome we show that the prediction error can be consistently estimated via the resubstitution and crossvalidation methods even when the fitted model is not correctly specified furthermore we show that the resulting estimators are asymptotically normal when the prediction rule is nonsmooth the variance of the above normal distribution can be estimated well with a perturbation resampling method with two real examples and an extensive simulation study we demonstrate that the interval estimates obtained from the above normal approximation for the prediction errors provide much more information about model adequacy than their point estimate counterparts

approximate homomorphisms and derivations between c ternary algebras in 1940 ulam proposed the famous ulam stability problem in this paper we introduce a general cauchy jensen functional equation and prove the generalized ulam stability of c ternary homomorphisms and c ternary derivations in c ternary algebras for the general cauchy jensen equation

representations of uq sl n at roots of unity the gelfand zetlin basis for representations of uq sl n is improved to better fit the case when q is a root of unity the usual q deformed representations as well as the nilpotent periodic cyclic semi periodic semi cyclic and also some atypical representations are now described with the same formalism

the problem of s m ulam for approximately multiplicative mappings in the present paper we study the problem of s m ulam for approximately multiplicative mappings and we consider new open problems

estimating a time dependent concordance index for survival prediction models with covariate dependent censoring given a predictive marker and a time to event response variable the proportion of concordant pairs in a data set is called concordance index a specifically useful marker is the risk predicted by a survival regression model this article extends the existing methodology for applications where the length of the follow up period depends on the predictor variables a class of inverse probability of censoring weighted estimators is discussed in which the estimates rely on a working model for the conditional censoring distribution the estimators are consistent for a truncated concordance index if the working model is correctly specified and if the probability of being uncensored at the truncation time is positive in this framework all kinds of prediction models can be assessed and time trends in the discrimination ability of a model can be captured by varying the truncation time point for illustration we re analyze a study on risk prediction for prostate cancer patients the effects of misspecification of the censoring model are studied in simulated data

two group classification via a biobjective margin maximization model abstract in this paper we propose a biobjective model for two group classification via margin maximization in which the margins in both classes are simultaneously maximized the set of pareto optimal solutions is described yielding a set of parallel hyperplanes one of which is just the solution of the classical svm approach in order to take into account different misclassification costs or a priori probabilities the roc curve can be used to select one out of such hyperplanes by expressing the adequate tradeoff for sensitivity and specificity our result gives a theoretical motivation for using the roc approach in case misclassification costs in the two groups are not necessarily equal

a random forest approach for competing risks based on pseudo values random forest is a supervised learning method that combines many classification or regression trees for prediction here we describe an extension of the random forest method for building event risk prediction models in survival analysis with competing risks in case of right censored data the event status at the prediction horizon is unknown for some subjects we propose to replace the censored event status by a jackknife pseudo value and then to apply an implementation of random forests for uncensored data because the pseudo responses take on values on a continuous scale the node variance is chosen as split criterion for growing regression trees in a simulation study the pseudo split criterion is compared with the gini split criterion when the latter is applied to the uncensored event status to investigate the resulting pseudo random forest method for building risk prediction models we analyze it in a simulation study of predictive performance where we compare it to cox regression and random survival forest the method is further illustrated in two real data sets copyright 2013 john wiley sons ltd

a fixed point method for perturbation of bimultipliers and jordan bimultipliers in c ternary algebras let a be a c ternary algebra a c bilinear t a a a is called a c ternary algebra bi multiplier if it satisfies t abc d t a b cd t a bcd abt c d for all a b c d a also the mapping t a a a is a called c ternary algebra jordan bimultiplier if it satisfies t aaa a t a a aa t a aaa aat a a for all a a using the fixed point method we investigate the generalized hyers ulam rassias stability of bimultipliers and jordan bimultipliers in c ternary algebras the concept of hyers ulam rassias stability originated from the th m rassias stability theorem that appeared in his paper th m rassias proc am math soc 72 297 1978

bandwidth choice for nonparametric classification it is shown that for kernel based classification with univariate distributions and two populations optimal bandwidth choice has a dichotomous character if the two densities cross at just one point where their curvatures have the same signs then minimum bayes risk is achieved using bandwidths which are an order of magnitude larger than those which minimize pointwise estimation error on the other hand if the curvature signs are different or if there are multiple crossing points then bandwidths of conventional size are generally appropriate the range of different modes of behavior is narrower in multivariate settings there the optimal size of bandwidth is generally the same as that which is appropriate for pointwise density estimation these properties motivate empirical rules for bandwidth choice

the estimation of prediction error having constructed a data based estimation rule perhaps a logistic regression or a classification tree the statistician would like to know its performance as a predictor of future cases there are two main theories concerning prediction error 1 penalty methods such as cp akaike s information criterion and stein s unbiased risk estimate that depend on the covariance between data points and their corresponding predictions and 2 cross validation and related nonparametric bootstrap techniques this article concerns the connection between the two theories a rao blackwell type of relation is derived in which nonparametric methods such as cross validation are seen to be randomized versions of their covariance penalty counterparts the model based penalty methods offer substantially better accuracy assuming that the model is believable

on the stability of functional equations in banach spaces the paper is devoted to some results on the problem of s m ulam for the stability of functional equations in banach spaces the problem was posed by ulam 60 years ago

improvements on cross validation the 632 bootstrap method abstract a training set of data has been used to construct a rule for predicting future responses what is the error rate of this rule this is an important question both for comparing models and for assessing a final selected model the traditional answer to this question is given by cross validation the cross validation estimate of prediction error is nearly unbiased but can be highly variable here we discuss bootstrap estimates of prediction error which can be thought of as smoothed versions of cross validation we show that a particular bootstrap method the 632 rule substantially outperforms cross validation in a catalog of 24 simulation experiments besides providing point estimates we also consider estimating the variability of an error rate estimate all of the results here are nonparametric and apply to any possible prediction rule however we study only classification problems with 0 1 loss in detail our simulations include smooth prediction rules like fisher s linear discriminant fun

periodic and flat irreducible representations ofsu 3 q we construct all the periodic irreducible representations ofu su 3 q forq am root of unity their dimensions arek 2m 2 fork 1 m onlyk 1 m 2 for evenm their interest is that they could be a tool to generalize the chiral potts model by truncation of these representations we construct flat representations ofu su 3 q in which all the multiplicities of the weights are set to 1

roots of unity representations of quantum groups representations of quantum groupsu e g n g n any semi simple lie algebra of rankn are constructed from arbitrary representations of rankn 1 quantum groups for e a root of unity representations which have the maximal dimension and number of free parameters for irreducible representations arise as special cases

multiobjective optimization of classifiers by means of 3d convex hull based evolutionary algorithms the receiver operating characteristic roc and detection error tradeoff det curves are frequently used in the machine learning community to analyze the performance of binary classifiers recently the convex hull based multiobjective genetic programming algorithm was proposed and successfully applied to maximize the convex hull area for binary classification problems by minimizing false positive rate and maximizing true positive rate at the same time using indicator based evolutionary algorithms the area under the roc curve was used for the performance assessment and to guide the search here we extend this research and propose two major advancements firstly we formulate the algorithm in detection error tradeoff space minimizing false positives and false negatives with the advantage that misclassification cost tradeoff can be assessed directly secondly we add complexity as an objective function which gives rise to a 3d objective space as opposed to a 2d previous roc space a domain specific performance indicator for 3d pareto front approximations the volume above det surface is introduced and used to guide the indicator based evolutionary algorithm to find optimal approximation sets we assess the performance of the new algorithm on designed theoretical problems with different geometries of pareto fronts and det surfaces and two application oriented benchmarks 1 designing spam filters with low numbers of false rejects false accepts and low computational cost using rule ensembles and 2 finding sparse neural networks for binary classification of test data from the uci machine learning benchmark the results show a high performance of the new algorithm as compared to conventional methods for multicriteria optimization

hypersymmetry a z3 graded generalization of supersymmetry we propose a generalization of non commutative geometry and gauge theories based on ternary z3 graded structures in the new algebraic structures we define all products of two entities are left free the only constraining relations being imposed on ternary products these relations reflect the action of the z3 group which may be either trivial i e abc bca cab generalizing the usual commutativity or non trivial i e abc jbca with j e 2 i 3 the usual z2 graded structures such as grassmann lie and clifford algebras are generalized to the z3 graded case certain suggestions concerning the eventual use of these new structures in physics of elementary particles and fields are exposed

evaluating prediction rules for t year survivors with censored regression models suppose that we are interested in establishing simple but reliable rules for predicting future t year survivors through censored regression models in this article we present inference procedures for evaluating such binary classification rules based on various prediction precision measures quantified by the overall misclassification rate sensitivity and specificity and positive and negative predictive values specifically under various working models we derive consistent estimators for the above measures through substitution and cross validation estimation procedures furthermore we provide large sample approximations to the distributions of these nonsmooth estimators without assuming that the working model is correctly specified confidence intervals for example for the difference of the precision measures between two competing rules can then be constructed all of the proposals are illustrated with real examples and their finite sample properties are evaluated through a simulation study

model selection in estimating equations summary model selection is a necessary step in many practical regression analyses but for methods based on estimating equations such as the quasi likelihood and generalized estimating equation gee approaches there seem to be few well studied model selection techniques in this article we propose a new model selection criterion that minimizes the expected predictive bias epb of estimating equations a bootstrap smoothed cross validation bcv estimate of epb is presented and its performance is assessed via simulation for overdispersed generalized linear models for illustration the method is applied to a real data set taken from a study of the development of ewe embryos

the covariance inflation criterion for adaptive model selection we propose a new criterion for model selection in prediction problems the covariance inflation criterion adjusts the training error by the average covariance of the predictions and responses when the prediction rule is applied to permuted versions of the data set this criterion can be applied to general prediction problems e g regression or classification and to general prediction rules e g stepwise regression tree based models and neural nets as a by product we obtain a measure of the effective number of parameters used by an adaptive procedure we relate the covariance inflation criterion to other model selection procedures and illustrate its use in some regression and classification problems we also revisit the conditional bootstrap approach to model selection

a robust alternative to the schemper henderson estimator of prediction error in clinical applications the prediction error of survival models has to be taken into consideration to assess the practical suitability of conclusions drawn from these models different approaches to evaluate the predictive performance of survival models have been suggested in the literature in this article we analyze the properties of the estimator of prediction error developed by schemper and henderson 2000 biometrics 56 249 255 which quantifies the absolute distance between predicted and observed survival functions we provide a formal proof that the estimator proposed by schemper and henderson is not robust against misspecification of the survival model that is the estimator will only be meaningful if the model family used for deriving predictions has been specified correctly to remedy this problem we construct a new estimator of the absolute distance between predicted and observed survival functions we show that this modified schemper henderson estimator is robust against model misspecification allowing its practical application to a wide class of survival models the properties of the schemper henderson estimator and its new modification are illustrated by means of a simulation study and the analysis of two clinical data sets

on foundation of the generalized nambu mechanics we outline basic principles of a canonical formalism for the nambu mechanics a generalization of hamiltonian mechanics proposed by yoichiro nambu in 1973 it is based on the notion of a nambu bracket which generalizes the poisson bracket a binary operation on classical observables on the phase space to the multiple operation of higher ordern 3 nambu dynamics is described by the phase flow given by nambu hamilton equations of motion a system of ode s which involvesn 1 hamiltonians we introduce the fundamental identity for the nambu bracket a generalization of the jacobi identity as a consistency condition for the dynamics we show that nambu bracket structure defines a hierarchy of infinite families of subordinated structures of lower order including poisson bracket structure which satisfy certain matching conditions the notion of nambu bracket enables us to define nambu poisson manifolds phase spaces for the nambu mechanics which turn out to be more rigid than poisson manifolds phase spaces for the hamiltonian mechanics we introduce the analog of the action form and the action principle for the nambu mechanics in its formulation dynamics of loops n 2 dimensional chains for the generaln ary case naturally appears we discuss several approaches to the quantization of nambu mechanics based on the deformation theory path integral formulation and on nambu heisenberg commutation relations in the latter formalism we present an explicit representation of the nambu heisenberg relation in then 3 case we emphasize the role ternary and higher order algebraic operations and mathematical structures related to them play in passing from hamilton s to nambu s dynamical picture

stability of n lie homomorphisms and jordan n lie homomorphisms on n lie algebras the motivation of this paper is to apply the hypers ulam stability problems of some kinds of functional equations to the classes of n lie homomorphisms and n lie algebras by using the structures of n lie homomorphisms and n lie algebras in this paper the generalized hyers ulam rassias stability of n lie homomorphisms and jordan n lie homomorphisms on n lie algebras associated to the generalized cauchy jensen rassias additive functional equation are investigated using the fixed point methods

quantum group sigma models field theoretic models for fields taking values in quantum groups are investigated first the authors consider the suq 2 sigma model q real expressed in terms of basic notions of noncommutative differential geometry they discuss the case in which the sigma model fields are represented as products of conventional sigma fields and of the coordinate independent algebra an explicit example is provided by the uq 2 sigma model with qn 1 in which case quantum matrices uq 2 are realized as 2n 2n unitary matrices open problems are pointed out

on the stability of a pexiderized gol ab schinzel equation in this paper we establish the stability of the pexiderized email protected b schinzel functional equation f x yf x g x h y under the condition that lim t 0h tx exists and is non zero

some comments on cp i study the typical configuration of a cp plot when the number of variables in the regression problem is large and there are many weak effects i show that a particular configuration that is very commonly seen can arise in a simple way i give a formula by means of which the risk incurred by the minimum cp rule can be estimated

how biased is the apparent error rate of a prediction rule abstract a regression model is fitted to an observed set of data how accurate is the model for predicting future observations the apparent error rate tends to underestimate the true error rate because the data have been used twice both to fit the model and to check its accuracy we provide simple estimates for the downward bias of the apparent error rate the theory applies to general exponential family linear models and general measures of prediction error special attention is given to the case of logistic regression on binary data with error rates measured by the proportion of misclassified cases several connected ideas are compared mallows s cp cross validation generalized cross validation the bootstrap and akaike s information criterion

induced module construction for highest weight representations of uq gl n at roots of unity a method is investigated for inducing highest weight representations for the quantum group uq gl n from the canonical subalgebra uq gl n 1 when q is a root of unity we classify the irreps into two types typical and atypical where the former is a generalization of the class of irreps with maximal dimensionality the structures of both the typical and atypical irreps are studied in particular a sufficiency condition is given for an irrep to be typical as examples we consider flat representations induced from a one dimensional representation of the canonical subalgebra and representations induced from vector representation

a genetic algorithm for generating fuzzy classification rules abstract fuzzy classification systems based on fuzzy logic are capable of dealing with cognitive uncertainties such as vagueness and ambiguity involved in classification problems to build a fuzzy classification system the key is to find an optimal set of fuzzy rules machine learning methods such as fuzzy neural networks and fuzzy decision tree induction have been applied to learn fuzzy rules but they may be trapped into local optimal based on the principle of natural evolution and global searching a genetic algorithm is promising in obtaining better results but it is usually less efficient in this paper a fuzzy genetic algorithm fga is developed to generate fuzzy classification rules several techniques such as multi value logic coding composite fitness function viability check and rule extraction are used to improve the efficiency and the effectiveness of the algorithm two experiments are conducted to demonstrate the performance of fga and to compare fga with other machine learning algorithms

on the universalr matrix of u_q widehat sl _2 at roots of unityat roots of unity we show that the action of the universalr matrix of the affine u q widehat sl 2 quantum algebra whenq is a root of unity can be renormalized by some scalar factor to give a well defined nonsingular expression satisfying the yang baxter equation it can be reduced to intertwining operators of representations corresponding to chiral potts if the parameters of these representations lie on the well known algebraic curve

estimating the error rate of a prediction rule improvement on cross validation abstract we construct a prediction rule on the basis of some data and then wish to estimate the error rate of this rule in classifying future observations cross validation provides a nearly unbiased estimate using only the original data cross validation turns out to be related closely to the bootstrap estimate of the error rate this article has two purposes to understand better the theoretical basis of the prediction problem and to investigate some related estimators which seem to offer considerably improved estimation in small samples

estimation and accuracy after model selection classical statistical theory ignores model selection in assessing estimation accuracy here we consider bootstrap methods for computing standard errors and confidence intervals that take model selection into account the methodology involves bagging also known as bootstrap smoothing to tame the erratic discontinuities of selection based estimators a useful new formula for the accuracy of bagging then provides standard errors for the smoothed estimators two examples nonparametric and parametric are carried through in detail a regression model where the choice of degree linear quadratic cubic is determined by the cp criterion and a lasso based estimation problem

properties of bagged nearest neighbour classifiers summary it is shown that bagging a computationally intensive method asymptotically improves the performance of nearest neighbour classifiers provided that the resample size is less than 69 of the actual sample size in the case of with replacement bagging or less than 50 of the sample size for without replacement bagging however for larger sampling fractions there is no asymptotic difference between the risk of the regular nearest neighbour classifier and its bagged version in particular neither achieves the large sample performance of the bayes classifier in contrast when the sampling fractions converge to 0 but the resample sizes diverge to 1 the bagged classifier converges to the optimal bayes rule and its risk converges to the risk of the latter these results are most readily seen when the two populations have well defined densities but they may also be derived in other cases where densities exist in only a relative sense cross validation can be used effectively to choose the sampling fraction numerical calculation is used to illustrate these theoretical properties

special representations of at the roots of unity we show how to adapt the gelfand zetlin basis for describing the special representations of when q is a root of unity the explicit construction of special representations is presented in detail for n 3

a center sliding bayesian binary classifier adopting orthogonal polynomials a center sliding bayesian design adopting orthogonal polynomials for binary pattern classification is studied in this paper essentially a bayesian weight solution is coupled with a center sliding scheme in feature space which provides an easy tuning capability for binary classification the proposed method is compared with several state of the art binary classifiers in terms of their solution forms decision thresholds and decision boundaries based on the center sliding bayesian framework a novel orthogonal polynomial classifier is subsequently developed the orthogonal polynomial classifier is evaluated using two representative orthogonal polynomials for feature mapping our experimental results show promising potential of the orthogonal polynomial classifier since it achieves both desired accuracy and computational efficiency highlightsproposed a center sliding bayesian classifier with deterministic solution proposed to adopt orthogonal polynomials for efficient feature expansion developed an easy tuning mechanism for the classification framework provided an extensive evaluation of the proposed classifier

fusion rules of the lowest weight representations of rm osp _q 1 2 at roots of unity polynomial realization the degeneracy of the lowest weight representations of the quantum superalgebra and their tensor products at exceptional values of q is studied the main features of the structures of the finite dimensional lowest weight representations and their fusion rules are illustrated using realization of group generators as finite difference operators acting in the space of the polynomials the complete fusion rules for the decompositions of the tensor products at roots of unity are presented the appearance of indecomposable representations in the fusions is described using clebsh gordan coefficients derived for general values of q and at roots of unity

normalized uq sl 3 gel fand weyl zetlin basis and new summation formulas for q hypergeometric functions an explicit realization of the normalized gel fand weyl zetlin gwz basis for uq sl 3 in terms of polynomial functions in three variables real or complex is given the construction uses two different realizations of the uq sl 3 unnormalized gwz basis which were given previously and whose transformation properties were not known it turns out that finding these properties enables us to find the gwz dependent proportionality constant between these two realizations the scalar product is also fixed by this in both unnormalized realizations and then by normalization the normalized gwz states are obtained as by products new summation formulas are obtained which seem new also for q 1 the main new formula is a double sum which is given in terms of the proportionality constant mentioned above this double sum can be written as single sum over a q 3f2 hypergeometric function or as a q hypergeometric function of two variables

isomorphisms between c ternary algebras abstract in this paper we prove the hyers ulam rassias stability of homomorphisms in c ternary algebras and of derivations on c ternary algebras for the following cauchy jensen additive mappings 0 1 f x y 2 z f x y 2 z f x 2 f z 0 2 f x y 2 z f x y 2 z f y 0 3 2 f x y 2 z f x f y 2 f z these are applied to investigate isomorphisms between c ternary algebras the concept of hyers ulam rassias stability originated from the th m rassias stability theorem that appeared in his paper th m rassias on the stability of the linear mapping in banach spaces proc amer math soc 72 1978 297 300

multiobjective genetic optimization of diagnostic classifiers with implications for generating receiver operating characteristic curves it is well understood that binary classifiers have two implicit objective functions sensitivity and specificity describing their performance traditional methods of classifier training attempt to combine these two objective functions or two analogous class performance measures into one so that conventional scalar optimization techniques can be utilized this involves incorporating a priori information into the aggregation method so that the resulting performance of the classifier is satisfactory for the task at hand the authors have investigated the use of a niched pareto multiobjective genetic algorithm ga for classifier optimization with niched pareto ga s an objective vector is optimized instead of a scalar function eliminating the need to aggregate classification objective functions the niched pareto ga returns a set of optimal solutions that are equivalent in the absence of any information regarding the preferences of the objectives the a priori knowledge that was used for aggregating the objective functions in conventional classifier training can instead be applied post optimization to select from one of the series of solutions returned from the multiobjective genetic optimization the authors have applied this technique to train a linear classifier and an artificial neural network ann using simulated datasets the performances of the solutions returned from the multiobjective genetic optimization represent a series of optimal sensitivity specificity pairs which can be thought of as operating points on a receiver operating characteristic roc curve all possible roc curves for a given dataset and classifier are less than or equal to the roc curve generated by the niched pareto genetic optimization

model selection via multifold cross validation a natural extension of the simple leave one out cross validation cv method is to allow the deletion of more than one observations in this article several notions of the multifold cross validation mcv method have been discussed in the context of variable selection under a linear regression model we show that the delete d mcv criterion is asymptotically equivalent to the well known fpe criterion two computationally more feasible methods the r fold cross validation and the repeated learning testing criterion are also studied the performance of these criteria are compared with the simple leave one out cross validation method simulation results are obtained to gain some understanding on the small sample properties of these methods

on special classes of n algebras we define n algebras as linear spaces on which the internal composition law involves n elements m v n v it is known that such algebraic structures are interesting for their applications to problems of modern mathematical physics using the notion of a commutant of two subalgebras of an n algebra we distinguish certain classes of n algebras with reasonable properties semisimple abelian nilpotent solvable we also consider a few examples of n algebras of different types and show their properties

pareto evolutionary neural networks for the purposes of forecasting or classification tasks neural networks nns are typically trained with respect to euclidean distance minimization this is commonly the case irrespective of any other end user preferences in a number of situations most notably time series forecasting users may have other objectives in addition to euclidean distance minimization recent studies in the nn domain have confronted this problem by propagating a linear sum of errors however this approach implicitly assumes a priori knowledge of the error surface defined by the problem which typically is not the case this study constructs a novel methodology for implementing multiobjective optimization within the evolutionary neural network enn domain this methodology enables the parallel evolution of a population of enn models which exhibit estimated pareto optimality with respect to multiple error measures a new method is derived from this framework the pareto evolutionary neural network pareto enn the pareto enn evolves a population of models that may be heterogeneous in their topologies inputs and degree of connectivity and maintains a set of the pareto optimal enns that it discovers new generalization methods to deal with the unique properties of multiobjective error minimization that are not apparent in the uni objective case are presented and compared on synthetic data with a novel method based on bootstrapping of the training data shown to significantly improve generalization ability finally experimental evidence is presented in this study demonstrating the general application potential of the framework by generating populations of enns for forecasting 37 different international stock indexes

estimating the confidence interval for prediction errors of support vector machine classifiers support vector machine svm is one of the most popular and promising classification algorithms after a classification rule is constructed via the svm it is essential to evaluate its prediction accuracy in this paper we develop procedures for obtaining both point and interval estimators for the prediction error under mild regularity conditions we derive the consistency and asymptotic normality of the prediction error estimators for svm with finite dimensional kernels a perturbation resampling procedure is proposed to obtain interval estimates for the prediction error in practice with numerical studies on simulated data and a benchmark repository we recommend the use of interval estimates centered at the cross validated point estimates for the prediction error further applications of the proposed procedure in model evaluation and feature selection are illustrated with two examples

an introduction to kernel based learning algorithms this paper provides an introduction to support vector machines kernel fisher discriminant analysis and kernel principal component analysis as examples for successful kernel based learning methods we first give a short background about vapnik chervonenkis theory and kernel feature spaces and then proceed to kernel based learning in supervised and unsupervised scenarios including practical and algorithmic considerations we illustrate the usefulness of kernel algorithms by discussing applications such as optical character recognition and dna analysis

double bagging combining classifiers by bootstrap aggregation the combination of classifiers leads to substantial reduction of misclassification error in a wide range of applications and benchmark problems we suggest using an out of bag sample for combining different classifiers in our setup a linear discriminant analysis is performed using the observations in the out of bag sample and the corresponding discriminant variables computed for the observations in the bootstrap sample are used as additional predictors for a classification tree two classifiers are combined and therefore method and variable selection bias is no problem for the corresponding estimate of misclassification error the need of an additional test sample disappears moreover the procedure performs comparable to the best classifiers used in a number of artificial examples and applications

representations of quantumso 8 and related quantum algebras we study irreducible representations of the quantum groupue so 8 when e is a primitivelth root of unity by a theorem of de concini and kac there is a finite number of such representations associated to each point of a complex algebraic variety of dimension 28 and the generic representation has dimensionl12

approximate ternary jordan derivations on banach ternary algebras let a be a banach ternary algebra over a scalar field r or c and x be a ternary banach a module a linear mapping d a a x x is called a ternary jordan derivation if d xxx a d x xx x xd x x x xxd x x for all x a in this paper we investigate ternary jordan derivations on banach ternary algebras associated with the following functional equation f x y z 4 f 3x y 4z 4 f 4x 3z 4 2f x moreover we prove the generalized ulam hyers stability of ternary jordan derivations on banach ternary algebras

on approximation of approximately linear mappings by linear mappings abstract assume a is a normed linear space b is a banach space and f a b is a mapping approximately linear we solve the following ulam problem give conditions in order for a linear mapping near an approximately linear mapping to exist

optimal expected distance separating halfspace one recently proposed criterion to separate two data sets in discriminant analysis is to use a hyperplane which minimizes the sum of distances to it from all the misclassified data points here all distances are supposed to be measured by way of some fixed norm while misclassification means lying in the wrong halfspace in this paper we study the problem of determining such an optimal halfspace when points are distributed according to an arbitrary random vector x in rd in the unconstrained case in dimension d we prove that any optimal separating halfspace always balances the misclassified points moreover under polyhedrality assumptions on the support of x there always exists an optimal separating halfspace passing through d affinely independent points these results extend in a natural way when different norms or a fixed gauge are used to measure distances and we allow constraints modeling that certain points are forced to be correctly classified

an approximate analytical approach to resampling averages using a novel reformulation we develop a framework to compute approximate resampling data averages analytically the method avoids multiple retraining of statistical models on the samples our approach uses a combination of the replica trick of statistical physics and the tap approach for approximate bayesian inference we demonstrate our approach on regression with gaussian processes a comparison with averages obtained by monte carlo sampling shows that our method achieves good accuracy

at roots of unity the quantum group at roots of unity is introduced by means of duality pairings with the quantum algebra its irreducible representations corresponding to the type of the quantum algebra are constructed through the universal t matrix the irreducible representations corresponding to the type are also found an invariant integral on this quantum group is given endowed with that some properties like unitarity and orthogonality of the irreducible representations are discussed

a perturbation of ring derivations on banach algebras abstract suppose a is a banach algebra and suppose f a a is an approximate ring derivation in the sense of hyers ulam rassias this stability phenomenon was introduced for the first time in the subject of functional equations by th m rassias th m rassias on the stability of the linear mapping in banach spaces proc amer math soc 72 1978 297 300 if a has an approximate identity or if a is semisimple and commutative then we prove that f is an exact ring derivation

block regularized repeated learning testing for estimating generalization error abstract repeated learning testing rlt is a popular cross validation method that is usually adopted in estimation and comparison of algorithm performance in machine learning however the variance of the estimator of the generalization error in the rlt method is easily affected by random partitions poor data partitioning may cause a large overlap between any two training sets of rlt and enlarge the variance in the rlt estimator of the generalization error thus in this study we constrain numbers of overlapping samples between any two training sets and construct a novel data partitioning schema in which rlt is called the block regularized rlt brlt we theoretically prove that the brlt estimator has a smaller variance than does the rlt estimator specifically the variance of the rlt estimator reaches its minimum when samples in a data set equally occur in all training sets and all the numbers of overlapping samples are equal furthermore we provide easy to use construction algorithms of brlt s partition set for several training set size and partition count settings by adopting two level orthogonal arrays we also illustrate brlt s optimal properties with several simulated and real world examples

flat periodic representations ofu q g we give explicit expression of flat periodic representations when they exist of the quantum analogues of simple lie algebras and their affine extensions for a parameter of deformationq equal to a root of unity by flat periodic we mean that these representations have no highest weight and that all the weights have multiplicity 1

irregular uq sl 3 representations at roots of unity via gel fand weyl zetlin basis the gel fand weyl zetlin gwz description of the uq sl 3 irregular irreps at roots of unity is explicitly given those are irreps fixed by the same parameters as the unitary irreducible representations uirs of su 3 yet having dimensions smaller than their classical counterparts the reason being that to obtain an irregular irrep one has to make factorization of an additional submodule this description is made geometrically transparent by an arrangement of the standard su 3 gwz basis in a hexagonal pyramid which is valid for any q and seems new even for q 1 the pyramid has as a base the standard hexagon which gives the weight space of the uirs of su 3 in the plane of third component of isospin iz and hypercharge y while third dimension of this pyramid is related to the isospin i algebraically this arrangement is related to a one to one correspondence between the abstract gwz states and monomials in the algebra of raising generators uq g however those monomials are not in the standard poi

on the predictive risk in misspecified quantile regression abstract in the present paper we investigate the predictive risk of possibly misspecified quantile regression functions the in sample risk is well known to be an overly optimistic estimate of the predictive risk and we provide two relatively simple asymptotic characterizations of the associated bias also called expected optimism we propose estimates for the expected optimism and the predictive risk and establish their uniform consistency under mild conditions our results hold for models of moderately growing size and allow the quantile function to be incorrectly specified empirical evidence from our estimates is encouraging as it compares favorably with cross validation