the knowledge gradient algorithm for a general class of online learning problems we derive a one period look ahead policy for finite and infinite horizon online optimal learning problems with gaussian rewards our approach is able to handle the case where our prior beliefs about the rewards are correlated which is not handled by traditional multiarmed bandit methods experiments show that our kg policy performs competitively against the best known approximation to the optimal policy in the classic bandit problem and it outperforms many learning policies in the correlated case

exact sample conditioned mse performance of the bayesian mmse estimator for classification error part i representation in recent years biomedicine has been faced with difficult high throughput small sample classification problems in such settings classifier error estimation becomes a critical issue because training and testing must be done on the same data a recently proposed error estimator places the problem in a signal estimation framework in the presence of uncertainty permitting a rigorous solution optimal in a minimum mean square error sense the uncertainty in this model is relative to the parameters of the feature label distributions resulting in a bayesian approach to error estimation closed form solutions are available for two important problems discrete classification with dirichlet priors and linear classification of gaussian distributions with normal inverse wishart priors in this work part i of a two part study we introduce the theoretical mean square error mse conditioned on the observed sample of any estimate of the classifier error including the bayesian error estimator for both bayesian models thus bayesian error estimation has a unique advantage in that its mathematical framework naturally gives rise to a practical expected measure of performance given an observed sample in part ii of the study we examine consistency of the error estimator demonstrate various mse properties and apply the conditional mse to censored sampling

bayesian minimum mean square error estimation for classification error part i definition and the bayesian mmse error estimator for discrete classification with the advent of high throughput genomic and proteomic technologies in conjunction with the difficulty in obtaining even moderately sized samples small sample classifier design has become a major issue in the biological and medical communities with small samples training data error estimation becomes mandatory yet none of the popular error estimation techniques have been rigorously designed based on statistical inference and optimization in this investigation we place classifier error estimation into the framework of optimal mean square error mse signal estimation in the presence of uncertainty which results in a bayesian approach to error estimation based on a parameterized family of feature label distributions with the prior distribution of the parameters governing the choice of feature label distribution these bayesian error estimators are optimal when averaged over a given family of distributions unbiased when averaged over a given family and all samples and analytically address a trade off between robustness modeling assumptions and accuracy minimum mean square error in this paper part i of a two part study we define the minimum mean square error mmse error estimator discuss its basic properties provide closed form analytic estimator representation for discrete classifiers with both non informative and informative prior distributions and examine the performance and robustness of the mmse error estimator via simulations in part ii of this paper in this same issue of ieee transactions on signal processing we address all of these issues in particular closed form representation for linear classification in the gaussian model with known and unknown covariance matrices for both the discrete and gaussian cases the mmse error estimator has especially good performance for distributions having moderate true errors

on delay differential equations with impulses abstract sufficient conditions are obtained respectively for the asymptotic stability of the trivial solution of x t ax t j 1 bjx tj t tj t tj and for the existence of a nonoscillatory solution conditions are also obtained for all solutions to be oscillatory the asymptotic behaviour of an impulsively perturbed delay logistic equation is investigated as an extension to a nonlinear equation

the domino problem of the hyperbolic plane is undecidable in this paper we prove that the general tiling problem of the hyperbolic plane is undecidable by proving a slightly stronger version using only a regular polygon as the basic shape of the tiles the problem was raised by a paper of raphael robinson in 1971 in his famous simplified proof that the general tiling problem is undecidable for the euclidean plane initially proved by robert berger in 1966

discrete optimal bayesian classification with error conditioned sequential sampling when in possession of prior knowledge concerning the feature label distribution in particular when it is known that the feature label distribution belongs to an uncertainty class of distributions governed by a prior distribution this prior knowledge can be used in conjunction with the training data to construct the optimal bayesian classifier obc whose performance is on average optimal among all classifiers relative to the posterior distribution derived from the prior distribution and the data typically in classification theory it is assumed that sampling is performed randomly in accordance with the prior probabilities on the classes and this has heretofore been true in the case of obc in the present paper we propose to forego random sampling and utilize the prior knowledge and previously collected data to determine which class to sample from at each step of the sampling specifically we choose to sample from the class that leads to the smallest expected classification error with the addition of the new sample point we demonstrate the superiority of the resulting nonrandom sampling procedure to random sampling on both synthetic data and data generated from known biological pathways highlightsa sampling algorithm for training the optimal bayesian classifier is introduced the algorithm works based on minimization of the expected error on the uncertainty class of prior knowledge using a zipf model we show that our sampling algorithm leads to a less true error on average than random sampling our algorithm shows robustness even in case when prior knowledge drifts away from true distributions an example on data from p53 network shows that our method works well on from real pathway data as well

analytical study of performance of linear discriminant analysis in stochastic settings this paper provides exact analytical expressions for the first and second moments of the true error for linear discriminant analysis lda when the data are univariate and taken from two stochastic gaussian processes the key point is that we assume a general setting in which the sample data from each class do not need to be identically distributed or independent within or between classes we compare the true errors of designed classifiers under the typical i i d model and when the data are correlated providing exact expressions and demonstrating that depending on the covariance structure correlated data can result in classifiers with either greater error or less error than when training with uncorrelated data the general theory is applied to autoregressive and moving average models of the first order and it is demonstrated using real genomic data

exact performance of error estimators for discrete classifiers discrete classification problems abound in pattern recognition and data mining applications one of the most common discrete rules is the discrete histogram rule this paper presents exact formulas for the computation of bias variance and rms of the resubstitution and leave one out error estimators for the discrete histogram rule we also describe an algorithm to compute the exact probability distribution of resubstitution and leave one out as well as their deviations from the true error rate using a parametric zipf model we compute the exact performance of resubstitution and leave one out for varying expected true error number of samples and classifier complexity number of bins we compare this to approximate performance measures computed by monte carlo sampling of 10 repeated 4 fold cross validation and the 0 632 bootstrap error estimator our results show that resubstitution is low biased but much less variable than leave one out and is effectively the superior error estimator between the two provided classifier complexity is low in addition our results indicate that the overall performance of resubstitution as measured by the rms can be substantially better than the 10 repeated 4 fold cross validation estimator and even comparable to the 0 632 bootstrap estimator provided that classifier complexity is low and the expected error rates are moderate in addition to the results discussed in the paper we provide an extensive set of plots that can be accessed on a companion website at the url http ee tamu edu edward exact discrete

regular production systems and triangle tilings we discuss regular production systems as a tool for analyzing tilings in general as an application we give necessary and sufficient conditions for a generic triangle to admit a tiling of and show that almost every triangle that admits a tiling is weakly aperiodic we pause for informal discussion of a variety of other applications such as non quasi isometric maps between regular tilings non periodic archimedean tilings growth and decidability most generally regular production systems provide a model for the organized growth of surfaces along a front subject to local rules

bayesian minimum mean square error estimation for classification error part ii linear classification of gaussian models in this paper part ii of a two part study we derive a closed form analytic representation of the bayesian minimum mean square error mmse error estimator for linear classification assuming gaussian models this is presented in a general framework permitting a structure on the covariance matrices and a very flexible class of prior parameter distributions with four free parameters closed form solutions are provided for known scaled identity and arbitrary covariance matrices we examine performance in small sample settings via simulations on both synthetic and real genomic data and demonstrate the robustness of these error estimators to false gaussian modeling assumptions by applying them to johnson distributions

exact sample conditioned mse performance of the bayesian mmse estimator for classification error part ii consistency and performance analysis in part i of a two part study on the mse performance of bayesian error estimation we have derived analytical expressions for mse conditioned on the sample for bayesian error estimators and arbitrary error estimators in two bayesian models discrete classification with dirichlet priors and linear classification of gaussian distributions with normal inverse wishart priors here in part ii we examine the consistency of bayesian error estimation and provide several simulation studies that illustrate the concept of conditional mse and how it may be used in practice a salient application is censored sampling where sample points are collected one at a time until the conditional mse reaches a stopping criterion

a hierarchical strongly aperiodic set of tiles in the hyperbolic plane we give a new construction of strongly aperiodic set of tiles in h 2 exhibiting a kind of hierarchical structure simplifying the central framework of margenstern s proof that the domino problem is undecidable in the hyperbolic plane margenstern 2008 16

the persistence of nonoscillatory solutions of delay differential equations under impulsive perturbations abstract we obtain a sufficient condition for the persistence of nonoscillatory solutions of the delay differential equation under impulsive perturbations

moments and root mean square error of the bayesian mmse estimator of classification error in the gaussian model the most important aspect of any classifier is its error rate because this quantifies its predictive capacity thus the accuracy of error estimation is critical error estimation is problematic in small sample classifier design because the error must be estimated using the same data from which the classifier has been designed use of prior knowledge in the form of a prior distribution on an uncertainty class of feature label distributions to which the true but unknown feature distribution belongs can facilitate accurate error estimation in the mean square sense in circumstances where accurate completely model free error estimation is impossible this paper provides analytic asymptotically exact finite sample approximations for various performance metrics of the resulting bayesian minimum mean square error mmse error estimator in the case of linear discriminant analysis lda in the multivariate gaussian model these performance metrics include the first second and cross moments of the bayesian mmse error estimator with the true error of lda and therefore the root mean square rms error of the estimator we lay down the theoretical groundwork for kolmogorov double asymptotics in a bayesian setting which enables us to derive asymptotic expressions of the desired performance metrics from these we produce analytic finite sample approximations and demonstrate their accuracy via numerical examples various examples illustrate the behavior of these approximations and their use in determining the necessary sample size to achieve a desired rms the supplementary material contains derivations for some equations and added figures highlightsthe bayesian minimum mean square error bmmse error estimator of lda is studied the first second and cross moments of the bmmse estimator with true error are considered both conditional and unconditional performance metrics are analyzed for the first time the kolmogorov double asymptotic is used in a bayesian setting asymptotically exact finite sample approximations of performance metrics are derived

oscillation criteria for impulsive delay differential equations abstract some new results on oscillation and nonoscillation for delay differential equations with impulses are established a counterexample is given which shows that a known oscillation criterion is incorrect

oscillation and stability of linear impulsive delay differential equations the main results of this paper are that the oscillation and the stability of a linear impulsive delay differential equation are equivalent respectively to the oscillation and the stability of a corresponding linear delay differential equation without impulses the results of this paper can be used to improve and derive some of the known results in the literature

optimal classifiers with minimum expected error within a bayesian framework part ii properties and performance analysis in part i of this two part study we introduced a new optimal bayesian classification methodology that utilizes the same modeling framework proposed in bayesian minimum mean square error mmse error estimation optimal bayesian classification thus completes a bayesian theory of classification where both the classifier error and our estimate of the error may be simultaneously optimized and studied probabilistically within the assumed model having developed optimal bayesian classifiers in discrete and gaussian models in part i here we explore properties of optimal bayesian classifiers in particular invariance to invertible transformations convergence to the bayes classifier and a connection to bayesian robust classifiers we also explicitly derive optimal bayesian classifiers with non informative priors and explore relationships to linear and quadratic discriminant analysis lda and qda which may be viewed as plug in rules under gaussian modeling assumptions finally we present several simulations addressing the robustness of optimal bayesian classifiers to false modeling assumptions companion website http gsp tamu edu publications supplementary dalton12a

a strongly aperiodic set of tiles in the hyperbolic plane we construct the first known example of a strongly aperiodic set of tiles in the hyperbolic plane such a set of tiles does admit a tiling but admits no tiling with an infinite cyclic symmetry this can also be regarded as a regular production system 5 that does admit bi infinite orbits but admits no periodic orbits

oscillatory properties of the first order nonlinear advanced and delayed differential inequalities on donne des conditions suffisantes pour que des inegalites differentielles retardees ou avancees aient des solutions oscillatoires

aperiodic tilings positive scalar curvature and amenability of spaces the object of this paper is to begin a geometric study of noncompact spaces whose local structure has bounded complexity manifolds of this sort arise as leaves of foliations of compact manifolds and as their universal covers we shall introduce a coarse homology theory using chains of bounded complexity and study some of its first properties the most interesting result characterizes when h uf x vanishes as an analogue and strengthening of f0lner s amenability criterion for groups in terms of isoperimetric inequalities see 4 one can view this result as producing a successful infinite ponzi scheme on any nonamenable space each point with only finite resources gives to some of its neighbors some of these resources yet receives more from the remaining neighbors as one can imagine this is useful for eliminating obstructions on noncompact spaces this has a number of applications we present two of them the first produces tilings that are unbalanced on any nonamenable polyhedron unbalanced tilings are automatically aperiodic and this gives many examples of sets of tiles that tile only aperiodically unfortunately imbalance is a particularly unsubtle reason for aperiodicity so that the aperiodic tilings of euclidean space penrose tilings are necessarily not accessible to our method on the other hand most other simply connected noncompact symmetric spaces even have unbalanced tilings using our criterion the second application regards characteristic numbers of manifolds whose universal covers have positive scalar curvature we prove a converse to a theorem of roe we show that for any nonamenable group f one can find a spin manifold with fundamental group f with nonzero a genus whose universal cover has a uniformly positive scalar curvature metric of bounded geometry in the natural strict quasi isometry class

tiling spaces are inverse limits let m be an arbitrary riemannian homogeneous space and let be a space of tilings of m with finite local complexity relative to some symmetry group and closed in the natural topology then is the inverse limit of a sequence of compact finite dimensional branched manifolds the branched manifolds are finite unions of cells constructed from the tiles themselves and the group this result extends previous results of anderson and putnam of ormes radin and sadun of bellissard benedetti and gambaudo and of gahler in particular the construction in this paper is a natural generalization of gahler s

asymptotic error rates of the w and z statistics when the training observations are dependent in this paper consideration is given to the properties of the classification statistics w and z which were developed for use in discrimination problems with independent training observations the relative behaviour of these two statistics when the training observations are dependent is investigated for training observations following a stationary autoregressive process of order p the asymptotic expansion of the expected error rates associated with w and z are derived up to and including terms of the second order with respect to the reciprocals of the sample sizes it is shown that neither z nor w is absolutely superior to the other numerical results are given to show that their relative performance is dependent on the extent of correlation among the training observations and the size of the separation between the two populations as measured by the mahalanobis distance

discrimination with autocorrelated observations abstract the effect of serial correlation on the multivariate sample linear discriminant function when the parameters are estimated under the assumption that the observations are independent has been studied by lawoko and mclachlan this paper investigates the performance of the univariate sample linear discriminant function when the parameters are estimated under a model which takes the correlation into consideration this is contrasted with the performance of the sample discriminant function when the parameters are estimated under the assumption that the observations are independent the intraclass correlation model assumed here is the univariate autoregressive process of order p

oscillation of nonlinear delay impulsive differential equations and inequalities the oscillatory and asymptotic behavior of first order nonlinear delay impulsive differential equations and inequalities is studied some new sharp sufficient conditions for oscillation and nonoscillation of solutions of the equations and inequalities are obtained many known results are improved in particular the conditions are necessary and sufficient if the equations and inequalities are autonomous