comparing classifiers when the misallocation costs are uncertain receiver operating characteristic roc curves are popular ways of summarising the performance of two class classification rules in fact however they are extremely inconvenient if the relative severity of the two different kinds of misclassification is known then an awkward projection operation is required to deduce the overall loss at the other extreme when the relative severity is unknown the area under an roc curve is often used as an index of performance however this essentially assumes that nothing whatsoever is known about the relative severity a situation which is very rare in real problems we present an alternative plot which is more revealing than an roc plot and we describe a comparative index which allows one to take advantage of anything that may be known about the relative severity of the two kinds of misclassification

truncation error bounds for limit periodic continued fractions many special functions of mathematical physics have representations in the form of limit k periodic continued fractions f k 1 b n where limn b n b or lim n b kn i i where k is a positive integer and 0 i k 1 for computation of such functions by continued fractions one needs sharp bounds for the truncation error left f f n right resulting when the value f is replaced by the nth approximate f n in this paper we present bounds for limit 4 periodic continued fractions k 1 bn some of these bounds are shown to be best relative to limited given information

a shamanskii like self adaptive levenberg marquardt method for nonlinear equations abstract in this paper we are concerned with the shamanskii like self adaptive levenberg marquardt methods for nonlinear equations we consider two choices of levenberg marquardt parameter one of them is the standard self adaptive levenberg marquardt parameter the other is nonmonotone self adaptive levenberg marquardt parameter by using the nonmonotone technique of grippo lampariello and lucidi under the error bound condition which is weaker than nonsingularity we show that the shamanskii like self adaptive levenberg marquardt methods converge with q order m 1 numerical experiments show the new algorithms are efficient and promising

random products of contractions in banach spaces we show that the random product of a finite number of w contractions converges weakly in all smooth reflexive banach spaces if one of the contractions is compact then the convergence is uniform let x 1 be a real banach space and t x x a linear operator recall that t is called a contraction if i tvi i v i for all vectors v in x we say that a contraction t satisfies condition w if whenever v is bounded and ivni itvni 0 it follows that the weak limn o vn tvn 0 the algebraic semigroup s generated by a possibly infinite set of contractions will be said to satisfy condition w if for any bounded sequence of vectors vn c x and a sequence of words wn from s such that i v n wnvni 0 the weak limn oo vn wnvn 0 now let t1 t2 be a sequence of contractions that satisfy condition w and let r be a self mapping of the set of natural numbers a random product of these contractions is the sequence sn n 1 2 defined by sn tr n tr n 1 tro 1 our purpose in this paper is to study the convergence properties of such products such a study is not only of intrinsic interest but is also motivated by applications to the numerical solution of partial differential equations 2 14 and linear inequalities 19 approximation theory 16 9 and computer tomography 18 11 we begin by noting several properties of contractions which satisfy condition w we shall call such contractions w contractions for short we then consider the algebraic semigroup generated by a finite number of w contractions and establish a weak convergence theorem for random products of these contractions in all smooth reflexive banach spaces theorem 1 this seems to be the first such result outside hilbert space it includes the hilbert space theorem due to amemiya and ando 1 we continue with a weak convergence result for semigroups generated by an infinite sequence of contractions theorem 2 finally we obtain a uniform convergence theorem for random received by the editors october 24 1988 and in revised form april 4 1989 1980 mathematics subject classification 1985 revision primary 47a05 secondary 47b05 65j10

truncation error bounds for the composition of limit periodic linear fractional transformations abstract baltus c truncation error bounds for the composition of limit periodic linear fractional transformations journal of computational and applied mathematics 46 1993 395 404 suppose i tn w is a sequence of linear fractional transformations tn w t w aw b w d ad b or a b 0 d where fixed points un u vn v with d v d u

the denjoy wolff theorem for condensing holomorphic mappings if b is the open unit ball of a strictly convex banach space x and f b b is holomorphic condensing with respect to and fixed point free then there exists b such that the sequence lcub fn of the iterates of f converges in the compact open topology to the constant map taking the value

goodness of fit and related inference processes for quantile regression abstract we introduce a goodness of fit process for quantile regression analogous to the conventional r2 statistic of least squares regression several related inference processes designed to test composite hypotheses about the combined effect of several covariates over an entire range of conditional quantile functions are also formulated the asymptotic behavior of the inference processes is shown to be closely related to earlier p sample goodness of fit theory involving bessel processes the approach is illustrated with some hypothetical examples an application to recent empirical models of international economic growth and some monte carlo evidence

the denjoy wolff theorem in the open unit ball of a strictly convex banach space letxbe a complex strictly convex banach space with an open unit ballb for each compact holomorphic and fixed point free mappingf b bthere exists bsuch that the sequence fn of iterates offconverges locally uniformly onbto the constant map taking the value

comparing images using the hausdorff distance the hausdorff distance measures the extent to which each point of a model set lies near some point of an image set and vice versa thus this distance can be used to determine the degree of resemblance between two objects that are superimposed on one another efficient algorithms for computing the hausdorff distance between all possible relative positions of a binary image and a model are presented the focus is primarily on the case in which the model is only allowed to translate with respect to the image the techniques are extended to rigid motion the hausdorff distance computation differs from many other shape comparison methods in that no correspondence between the model and the image is derived the method is quite tolerant of small position errors such as those that occur with edge detectors and other feature extraction methods it is shown that the method extends naturally to the problem of comparing a portion of a model against an image

statistical methods for eliciting probability distributions elicitation is a key task for subjectivist bayesians although skeptics hold that elicitation cannot or perhaps should not be done in practice it brings statisticians closer to their clients and subject matter expert colleagues this article reviews the state of the art reflecting the experience of statisticians informed by the fruits of a long line of psychological research into how people represent uncertain information cognitively and how they respond to questions about that information in a discussion of the elicitation process the first issue to address is what it means for an elicitation to be successful that is what criteria should be used our answer is that a successful elicitation faithfully represents the opinion of the person being elicited it is not necessarily true in some objectivistic sense and cannot be judged in that way we see that elicitation as simply part of the process of statistical modeling indeed in a hierarchical model at which point the likelihood ends and the pr

rates of convergence for minimum contrast estimators we shall present here a general study of minimum contrast estimators in a nonparametric setting although our results are also valid in the classical parametric case for independent observations these estimators include many of the most popular estimators in various situations such as maximum likelihood estimators least squares and other estimators of the regression function estimators for mixture models or deconvolution the main theorem relates the rate of convergence of those estimators to the entropy structure of the space of parameters optimal rates depending on entropy conditions are already known at least for some of the models involved and they agree with what we get for minimum contrast estimators as long as the entropy counts are not too large but under some circumstances large entropies or changes in the entropy structure due to local perturbations the resulting the rates are only suboptimal counterexamples are constructed which show that the phenomenon is real for non parametric maximum likelihood or regression this proves that under purely metric assumptions our theorem is optimal and that minimum contrast estimators happen to be suboptimal

modeling tangential vector fields on a sphere abstractphysical processes that manifest as tangential vector fields on a sphere are common in geophysical and environmental sciences these naturally occurring vector fields are often subject to physical constraints such as being curl free or divergence free we start with constructing parametric models for curl free and divergence free vector fields that are tangential to the unit sphere through applying the surface gradient or the surface curl operator to a scalar random potential field on the unit sphere using the helmholtz hodge decomposition we then construct a class of simple but flexible parametric models for general tangential vector fields which are represented as a sum of a curl free and a divergence free components we propose a likelihood based parameter estimation procedure and show that fast computation is possible even for large data sets when the observations are on a regular latitude longitude grid characteristics and practical utility of the proposed methodology are illustrated th

bayesian calibration of inexact computer models bayesian calibration is used to study computer models in the presence of both a calibration parameter and model bias the parameter in the predominant methodology is left undefined this results in an issue where the posterior of the parameter is suboptimally broad there has been no generally accepted alternatives to date this article proposes using bayesian calibration where the prior distribution on the bias is orthogonal to the gradient of the computer model problems associated with bayesian calibration are shown to be mitigated through analytic results in addition to examples supplementary materials for this article are available online

bayesian measures of model complexity and fit summary we consider the problem of comparing complex hierarchical models in which the number of parameters is not clearly defined using an information theoretic argument we derive a measure pd for the effective number of parameters in a model as the difference between the posterior mean of the deviance and the deviance at the posterior means of the parameters of interest in general pd approximately corresponds to the trace of the product of fisher s information and the posterior covariance which in normal models is the trace of the hat matrix projecting observations onto fitted values its properties in exponential families are explored the posterior mean deviance is suggested as a bayesian measure of fit or adequacy and the contributions of individual observations to the fit and complexity can give rise to a diagnostic plot of deviance residuals against leverages adding pd to the posterior mean deviance gives a deviance information criterion for comparing models which is related to other information criteria and has an approximate decision theoretic justification the procedure is illustrated in some examples and comparisons are drawn with alternative bayesian and classical proposals throughout it is emphasized that the quantities required are trivial to compute in a markov chain monte carlo analysis

reflections on value regions limit regions and truncation errors for continued fractions for continued fractionsk a n 1 the concept of limit region is discussed and its use for obtaining modified truncation error estimates is illustrated on examples a certain strategy for numerical computation of limit regions is presented and illustrated on examples

general notions of statistical depth function statistical depth functions are being formulated ad hoc with increasing popularity in nonparametric inference for multivariate data here we introduce several general structures for depth functions classify many existing examples as special cases and establish results on the possession or lack thereof of four key properties desirable for depth functions in general roughly speaking these properties may be described as affine invariance maximality at center monotonicity relative to deepest point and vanishing at infinity this provides a more systematic basis for selection of a depth function in particular from these and other considerations it is found that the halfspace depth behaves very well overall in comparison with various competitors

complex dynamics of the limit periodic system f n z f n 1 f n z f n f abstract the sequence f n z is one kind of generalization of limit periodic continued fractions the convergence divergence of f n z relative to that of the iterative sequence f n z is determined when n uniformly on a domain s when maps the closure of s into s f n z for all z s if n rapidly on s the sequence f n uniformly shadows n f n z n z and the two sequences show similar complex dynamics a priori uniform tail end bounds are provided n 1 o o n m z g n 1 o o g n m z n m where for example the first expansion may represent a modified continued fraction and the second another limit periodic structure with g n

truncation error bounds for modified continued fractions with applications to special functions much recent work has been done to investigate convergence of modified continued fractions mcf s following the proof by thron and waadeland 35 in 1980 that a limit periodic mcfk a n 1 x 1 with mathop lim limits n to infty a n a in mathbb c infty tfrac 1 4 andnth approximant g n s n x 1 frac a 1 1 frac a 2 1 cdots frac a n 1 1 frac a n 1 x 1 converges more rapidly to its limitf than the ordinary reference continued fractionk a n 1 withnth approximantf n s n 0 herex 1 denotes the smaller in modulus of the two fixed points oft w a 1 w the present paper gives truncation error bounds for bothf n andg n that exploit the limit periodic property lima n a certain a posteriori bounds given forg n are shown to be best possible relative to the given limited information available this is the first instance in which truncation error bounds for this problem have been shown to be best possible also included in this paper are results on speed of convergence a practical method for constructing the bounds and applications to a number of special functions the given numerical examples indicate that the error bounds are indeed sharp for the function arctanz we give graphical contour maps of the number of significant digits in the approximationsf n z g n z andp n z thenth partial sum of the maclaurin series forz in a key region of the complex plane these maps help us compare the various approximations with each other and add to our understanding of their convergence behavior

multivariate type g matern stochastic partial differential equation random fields for many applications with multivariate data random field models capturing departures from gaussianity within realizations are appropriate for this reason we formulate a new class of multivariate non gaussian models based on systems of stochastic partial differential equations with additive type g noise whose marginal covariance functions are of matern type we consider four increasingly flexible constructions of the noise where the first two are similar to existing copula based models in contrast with these the last two constructions can model non gaussian spatial data without replicates computationally efficient methods for likelihood based parameter estimation and probabilistic prediction are proposed and the flexibility of the models suggested is illustrated by numerical examples and two statistical applications less

regularized discriminant analysis abstract linear and quadratic discriminant analysis are considered in the small sample high dimensional setting alternatives to the usual maximum likelihood plug in estimates for the covariance matrices are proposed these alternatives are characterized by two parameters the values of which are customized to individual situations by jointly minimizing a sample based estimate of future misclassification risk computationally fast implementations are presented and the efficacy of the approach is examined through simulation studies and application to data these studies indicate that in many circumstances dramatic gains in classification accuracy can be achieved

a general method for comparing probability assessors a probability assessor or forecaster is a person who assigns subjective probabilities to events which will eventually occur or not occur in this paper a general method for comparing forecasters after a finite number of trials is introduced the general method is proven to include calculating all proper scoring rules as special cases it also includes comparison of forecasters in all simple two decision problems as special cases the relationship between the general method and calibration is also explored the general method is also translated into a method for deciding who will give better forecasts in the future

information divergence and risk for binary experiments we unify f divergences bregman divergences surrogate regret bounds proper scoring rules cost curves roc curves and statistical information we do this by systematically studying integral and variational representations of these objects and in so doing identify their representation primitives which all are related to cost sensitive binary classification as well as developing relationships between generative and discriminative views of learning the new machinery leads to tight and more general surrogate regret bounds and generalised pinsker inequalities relating f divergences to variational divergence the new viewpoint also illuminates existing algorithms it provides a new derivation of support vector machines in terms of divergences and relates maximum mean discrepancy to fisher linear discriminants

computing bayes factors by combining simulation and asymptotic approximations abstract the bayes factor is a ratio of two posterior normalizing constants which may be difficult to compute we compare several methods of estimating bayes factors when it is possible to simulate observations from the posterior distributions via markov chain monte carlo or other techniques the methods that we study are all easily applied without consideration of special features of the problem provided that each posterior distribution is well behaved in the sense of having a single dominant mode we consider a simulated version of laplace s method a simulated version of bartlett correction importance sampling and a reciprocal importance sampling technique we also introduce local volume corrections for each of these in addition we apply the bridge sampling method of meng and wong we find that a simulated version of laplace s method with local volume correction furnishes an accurate approximation that is especially useful when likelihood function evaluations are costly a simple bridge sampli

weather forecasting for weather derivatives we take a simple time series approach to modeling and forecasting daily average temperature in u s cities and we inquire systematically as to whether it may prove useful from the vantage point of participants in the weather derivatives market the answer is perhaps surprisingly yes time series modeling reveals conditional mean dynamics and crucially strong conditional variance dynamics in daily average temperature and it reveals sharp differences between the distribution of temperature and the distribution of temperature surprises as we argue it also holds promise for producing the long horizon predictive densities crucial for pricing weather derivatives so that additional inquiry into time series weather forecasting methods will likely prove useful in weather derivatives contexts

confidence intervals around bayes cost in multi state diagnostic settings to estimate optimal performance a critical feature of diagnostic testing is correctly classifying subjects based upon specified thresholds of some measure the commonly employed youden index determines a test s optimal thresholds by maximizing the correct classification rates for a diagnostic scenario an alternative to the youden index is the cost function bayes cost bc bc determines a test s optimal setting by minimizing the sum of all misclassification rates from the test unlike the youden index bc can consider a priori costs of all the diagnostic outcomes including class specific misclassifications regardless of the number of classes delta method approximate confidence intervals around bc are derived under the assumption of normally distributed classes as a means for quantifying a test s performance and comparing classifiers at their optimal settings in a multi state diagnostic framework a simulation study is conducted to demonstrate the performance of the derived confidence intervals that are found to perform well especially for sample sizes of 50 or larger in each diagnostic class finally the proposed methods are applied to a four class breast tissue classification problem where four possible discriminatory features are compared under varying decision cost structures using the confidence intervals around bc the best feature for classification is selected and the optimal thresholds and their 95 confidence intervals are determined

a convergence question inspired by stieltjes and by value sets in continued fraction theory abstract let v be a subset of the complex plane c let fn n 1 be a sequence of self mappings of v i e fn v v the question is then under what conditions will the sequence fn w f1 f2 fn w n 1 2 3 of composite maps converge to a constant function in v in this paper we give a survey of some of the answers and open problems connected with this question such answers have applications in dynamical systems schur analysis continued fractions and other similar structures like infinite exponentials infinite radicals

general convergence of continued fractions on introduit la notion de convergence generale des fractions continues qu on compare a celles de convergence ordinaire et de convergence forte on etablit quelques unes de ses proprietes

a family of nonparametric statistics for comparing diagnostic markers with paired or unpaired data summary in this paper we study a broad class of nonparametric statistics for comparing two diagnostic markers one can compare the sensitivities of these diagnostic markers over restricted ranges of specificity by selecting an appropriate statistic from this class as special cases one can compare the entire area under the receiver operator curve hanley mcneil 1982 or one can compare the sensitivities at a fixed common specificity usually we would recommend a comparison based on an average of sensitivities over a restricted high level of specificities test procedures and confidence intervals are based on asymptotic normality these procedures are applicable for paired data in which both diagnostic markers are performed on each subject and for unpaired data the procedures may be used to compare two real functions of multiple diagnostic markers as well as to compare individual markers

predictive model selection we consider the problem of selecting one model from a large class of plausible models a predictive bayesian viewpoint is advocated to avoid the specification of prior probabilities for the candidate models and the detailed interpretation of the parameters in each model using criteria derived from a certain predictive density and a prior specification that emphasizes the observables we implement the proposed methodology for three common problems arising in normal linear models variable subset selection selection of a transformation of predictor variables and estimation of a parametric variance function interpretation of the relative magnitudes of the criterion values for various models is facilitated by a calibration of the criteria relationships between the proposed criteria and other well known criteria are examined

composite binary losses we study losses for binary classification and class probability estimation and extend the understanding of them from margin losses to general composite losses which are the composition of a proper loss with a link function we characterise when margin losses can be proper composite losses explicitly show how to determine a symmetric loss in full from half of one of its partial losses introduce an intrinsic parametrisation of composite binary losses and give a complete characterisation of the relationship between proper losses and classification calibrated losses we also consider the question of the best surrogate binary loss we introduce a precise notion of best and show there exist situations where two convex surrogate losses are incommensurable we provide a complete explicit characterisation of the convexity of composite binary losses in terms of the link function and the weight function associated with the proper loss which make up the composite loss this characterisation suggests new ways of surrogate tuning as well as providing an explicit characterisation of when bregman divergences on the unit interval are convex in their second argument finally in an appendix we present some new algorithm independent results on the relationship between properness convexity and robustness to misclassification noise for binary losses and show that all convex proper losses are non robust to misclassification noise

line pattern retrieval using relational histograms this paper presents a new compact shape representation for retrieving line patterns from large databases the basic idea is to exploit both geometric attributes and structural information to construct a shape histogram we realize this goal by computing the n nearest neighbor graph for the lines segments for each pattern the edges of the neighborhood graphs are used to gate contributions to a two dimensional pairwise geometric histogram shapes are indexed by searching for the line pattern that maximizes the cross correlation of the normalized histogram bin contents we evaluate the new method on a database containing over 2 500 line patterns each composed of hundreds of lines

minimax regret classifier for imprecise class distributions the design of a minimum risk classifier based on data usually stems from the stationarity assumption that the conditions during training and test are the same the misclassification costs assumed during training must be in agreement with real costs and the same statistical process must have generated both training and test data unfortunately in real world applications these assumptions may not hold this paper deals with the problem of training a classifier when prior probabilities cannot be reliably induced from training data some strategies based on optimizing the worst possible case conventional minimax have been proposed previously in the literature but they may achieve a robust classification at the expense of a severe performance degradation in this paper we propose a minimax regret minimax deviation approach that seeks to minimize the maximum deviation from the performance of the optimal risk classifier a neural based minimax regret classifier for general multi class decision problems is presented experimental results show its robustness and the advantages in relation to other approaches

on a notion of data depth based on random simplices for a distribution f on mathbb r p and a point x in mathbb r p the simplical depth d x is introduced which is the probability that the point x is contained inside a random simplex whose vertices are p 1 independent observations from f mathematically and heuristically it is argued that d x indeed can be viewed as a measure of depth of the point x with respect to f an empirical version of d cdot gives rise to a natural ordering of the data points from the center outward the ordering thus obtained leads to the introduction of multivariate generalizations of the univariate sample median and l statistics this generalized sample median and l statistics are affine equivariant

continued fractions with applications i introductory examples ii more basics iii convergence criteria iv continued fractions and three term recurrence relations v correspondence of continued fractions vi hypergeometric functions vii moments and orthogonality viii pade approximants ix some applications in number theory x zero free regions xi digital filters and continued fractions xii applications to some differential equations appendix some continued fraction expansions references index

maximization of auc and buffered auc in binary classification in binary classification performance metrics that are defined as the probability that some error exceeds a threshold are numerically difficult to optimize directly and also hide potentially important information about the magnitude of errors larger than the threshold defining similar metrics instead using buffered probability of exceedance bpoe generates counterpart metrics that resolve both of these issues we apply this approach to the case of auc the area under the roc curve and define buffered auc bauc we show that bauc can provide insights into classifier performance not revealed by auc while being closely related as the tightest concave lower bound and representable as the area under a modified roc curve additionally while auc is numerically difficult to optimize directly we show that bauc optimization often reduces to convex or linear programming extending these results we show that auc and bauc are special cases of generalized bauc and that popular support vector machine svm formulations for approximately maximizing auc are equivalent to direct maximization of generalized bauc we also prove bauc generalization bounds for these svm s as a central component to these results we provide an important novel formula for calculating bpoe the inverse of conditional value at risk using this formula we show that particular bpoe minimization problems reduce to convex and linear programming

the skill plot a graphical technique for evaluating continuous diagnostic tests we introduce the skill plot a method that it is directly relevant to a decision maker who must use a diagnostic test in contrast to roc curves the skill curve allows easy graphical inspection of the optimal cutoff or decision rule for a diagnostic test the skill curve and test also determine whether diagnoses based on this cutoff improve upon a naive forecast of always present or of always absent the skill measure makes it easy to directly compare the predictive utility of two different classifiers in an analogy to the area under the curve statistic related to roc analysis finally this article shows that the skill based cutoff inferred from the plot is equivalent to the cutoff indicated by optimizing the posterior odds in accordance with bayesian decision theory a method for constructing a confidence interval for this optimal point is presented and briefly discussed

mixtures of g priors for bayesian model averaging with economic application this paper examines the issue of variable selection in linear regression modeling where there is a potentially large amount of possible covariates and economic theory offers insufficient guidance on how to select the appropriate subset in this context bayesian model averaging presents a formal bayesian solution to dealing with model uncertainty the main interest here is the effect of the prior on the results such as posterior inclusion probabilities of regressors and predictive performance the authors combine a binomial beta prior on model size with a g prior on the coefficients of each model in addition they assign a hyperprior to g as the choice of g has been found to have a large impact on the results for the prior on g they examine the zellner siow prior and a class of beta shrinkage priors which covers most choices in the recent literature the authors propose a benchmark beta prior inspired by earlier findings with fixed g and show it leads to consistent model selection inference is conducted through a markov chain monte carlo sampler over model space and g the authors examine the performance of the various priors in the context of simulated and real data for the latter they consider two important applications in economics namely cross country growth regression and returns to schooling recommendations for applied users are provided

another look at results of wolff and julia type forj algebras abstract domains of invariance for certain classes of holomorphic mappings on the open unit ball of a j algebra are given

compositions of analytic functions of the form f n z f n 1 f n z f n z f z abstract frequently in applications a function is iterated in order to determine its fixed point which represents the solution of some problem in the variation of iteration presented in this paper fixed points serve a different purpose the sequence f n z is studied where f 1 z f 1 z and f n z f n 1 f n z with f n f many infinite arithmetic expansions exhibit this form and the fixed point of f may be used as a modifying factor z to influence the convergence behaviour of these expansions thus one employs rather than seeks the fixed point of the function f

strict definiteness of integrals via complete monotonicity of derivatives let k be a nonnegative integer and let 0 r be a c function with k k completely monotone and not constant if 6 0 is a signed measure on any euclidean space rd with vanishing moments up to order k 1 then the integral rd rd x y 2 d x d y is strictly positive whenever it exists for general d no larger class of continuous functions seems to admit the same conclusion examples and applications are indicated a section on bilinear integrability might be of independent interest

of quantiles and expectiles consistent scoring functions choquet representations and forecast rankings summary in the practice of point prediction it is desirable that forecasters receive a directive in the form of a statistical functional for example forecasters might be asked to report the mean or a quantile of their predictive distributions when evaluating and comparing competing forecasts it is then critical that the scoring function used for these purposes be consistent for the functional at hand in the sense that the expected score is minimized when following the directive we show that any scoring function that is consistent for a quantile or an expectile functional can be represented as a mixture of elementary or extremal scoring functions that form a linearly parameterized family scoring functions for the mean value and probability forecasts of binary events constitute important examples the extremal scoring functions admit appealing economic interpretations of quantiles and expectiles in the context of betting and investment problems the choquet type mixture representations give rise to simple checks of whether a forecast dominates another in the sense that it is preferable under any consistent scoring function in empirical settings it suffices to compare the average scores for only a finite number of extremal elements plots of the average scores with respect to the extremal scoring functions which we call murphy diagrams permit detailed comparisons of the relative merits of competing forecasts

a predictive approach to the bayesian design problem with application to normal regression models a predictive decision theoretic approach is developed for the bayesian design problem the loss functions used are fair bayes or proper scoring rules and are quadratic measures of distance between probability measures optimal bayesian designs are those which minimise the preposterior risk for the decision problem such designs typically depend on both the prior distribution and the loss function the results are applied to certain normal regression models where explicit optimal designs are constructed

noisy logo recognition using line segment hausdorff distance logo recognition is of great interest in the document and shape analysis domain in order to develop a recognition method that is robust to employ under adverse conditions such as different scale orientation broken curves added noise and occlusion a modified line segment hausdorff distance is proposed in this paper the new approach has the advantage to incorporate structural and spatial information to compute dissimilarity between two sets of line segments rather than two sets of points the proposed technique has been applied on line segments generated from logos with encouraging results clear cut distinction between the correct and incorrect matches has been observed this suggests a strong potential for logo and shape recognition system

game theory maximum entropy minimum discrepancy and robust bayesian decision theory we describe and develop a close relationship between two problems that have customarily been regarded as distinct that of maximizing entropy and that of minimizing worst case expected loss using a formulation grounded in the equilibrium theory of zero sum games between decision maker and nature these two problems are shown to be dual to each other the solution to each providing that to the other although topsoe described this connection for the shannon entropy over 20 years ago it does not appear to be widely known even in that important special case we here generalize this theory to apply to arbitrary decision problems and loss functions we indicate how an appropriate generalized definition of entropy can be associated with such a problem and we show that subject to certain regularity conditions the above mentioned duality continues to apply in this extended context this simultaneously provides a possible rationale for maximizing entropy and a tool for finding robust bayes acts we also describe the essential identity between the problem of maximizing entropy and that of minimizing a related discrepancy or divergence between distributions this leads to an extension to arbitrary discrepancies of a well known minimax theorem for the case of kullback leibler divergence the redundancy capacity theorem of information theory for the important case of families of distributions having certain mean values specified we develop simple sufficient conditions and methods for identifying the desired solutions we use this theory to introduce a new concept of generalized exponential family linked to the specific decision problem under consideration and we demonstrate that this shares many of the properties of standard exponential families finally we show that the existence of an equilibrium in our game can be rephrased in terms of a pythagorean property of the related divergence thus generalizing previously announced results for kullback leibler and bregman divergences

effective scoring rules for probabilistic forecasts this paper studies the use of a scoring rule for the elicitation of forecasts in the form of probability distributions and for the subsequent evaluation of such forecasts given a metric distance function on a space of probability distributions a scoring rule is said to be effective if the forecaster s expected score is a strictly decreasing function of the distance between the elicited and true distributions two simple well known rules the spherical and the quadratic are shown to be effective with respect to suitable metrics examples and a practical application in foreign exchange rate forecasting are also provided

model choice a minimum posterior predictive loss approach summary model choice is a fundamental and much discussed activity in the analysis of datasets nonnested hierarchical models introducing random effects may not be handled by classical methods bayesian approaches using predictive distributions can be used though the formal solution which includes bayes factors as a special case can be criticised we propose a predictive criterion where the goal is good prediction of a replicate of the observed data but tempered by fidelity to the observed values we obtain this criterion by minimising posterior loss for a given model and then for models under consideration selecting the one which minimises this criterion for a broad range of losses the criterion emerges as a form partitioned into a goodness of fit term and a penalty term we illustrate its performance with an application to a large dataset involving residential property transactions

evaluating probabilities asymmetric scoring rules proper scoring rules are over evaluation measures that reward accurate probabilities specific rules encountered in the literature and used in practice are invariably symmetric in the sense that the expected score for a perfectly calibrated probability assessor or model generating probabilities is minimized at a probability of one half a family of asymmetric scoring rules that provide better measures of the degree of skill inherent in the probabilities and render scores that are more comparable in different situations is developed here one member of this family a quadratic asymmetric rule is applied to evaluate an extensive set of precipitation probability forecasts from the u s national weather service connections to previous characterizations of proper scoring rules are investigated and some relevant issues pertaining to the design of specific asymmetric rules for particular inferential and decision making problems are discussed briefly

bipartite ranking a risk theoretic perspective we present a systematic study of the bipartite ranking problem with the aim of explicating its connections to the class probability estimation problem our study focuses on the properties of the statistical risk for bipartite ranking with general losses which is closely related to a generalised notion of the area under the roc curve we establish alternate representations of this risk relate the bayes optimal risk to a class of probability divergences and characterise the set of bayes optimal scorers for the risk we further study properties of a generalised class of bipartite risks based on the p norm push of rudin 2009 our analysis is based on the rich framework of proper losses which are the central tool in the study of class probability estimation we show how this analytic tool makes transparent the generalisations of several existing results such as the equivalence of the minimisers for four seemingly disparate risks from bipartite ranking and class probability estimation a novel practical implication of our analysis is the design of new families of losses for scenarios where accuracy at the head of ranked list is paramount with comparable empirical performance to the p norm push

should scoring rules be effective a scoring rule is a reward function for eliciting or evaluating forecasts expressed as discrete or continuous probability distributions a rule is strictly proper if it encourages the forecaster to state his true subjective probabilities and effective if it is associated with a metric on the set of probability distributions recently the property of effectiveness which is stronger than strict properness has been proposed as a desideratum for scoring rules for continuous forecasts for reasons of monotonicity in keeping the forecaster close to his true probabilities since in practice the forecast must be chosen from a low dimensional set of admissible distributions it is shown in this paper that what effectiveness implies beyond strict properness is not a monotonicity property but a transitivity property which is difficult to justify behaviorally the logarithmic scoring rule is shown to violate the transitivity property and hence is not effective the l1 and l metrics are shown to allow no effective scoring rules some potential difficulties in interpreting admissible forecasts are also discussed

discrimination slope and integrated discrimination improvement properties relationships and impact of calibration discrimination slope defined as the slope of a linear regression of predicted probabilities of event derived from a prognostic model on the binary event status has recently gained popularity as a measure of model performance it is as a building block for the integrated discrimination improvement that equals the difference in discrimination slopes between the two models being compared several authors have pointed out that it does not make sense to apply the integrated discrimination improvement and discrimination slope when working with mis calibrated models whereas others have raised concerns about the ability of improving discrimination slope without adding new information in this paper we show that under certain assumptions the discrimination slope is asymptotically related to two other r squared measures one of which is a rescaled version of the brier score known to be proper furthermore we illustrate how a simple recalibration makes the slope equal to the rescaled brier r squared metric we also show that the discrimination slope can be interpreted as a measure of reduction in expected regret for the gini brier regret function using theoretical and practical examples we illustrate how all of these metrics are affected by different levels of model mis calibration in particular we demonstrate that simple recalibration ascertaining calibration in the large and calibration slope equal to 1 are not sufficient to correct for some forms of mis calibration we conclude that r squared metrics including the discrimination slope offer an attractive choice for quantifying model performance as long as one accounts for their sensitivity to model calibration copyright 2016 john wiley sons ltd

compositions of contractions we prove that if d is a simply connected open domain in the complex plane c e is a closed subset of d and fn n 1 are functions analytic in d such that fn d e for all n then the compositions fn z f1of2o ofn z for n 1 2 3 converge uniformly in d to a constant function some generalizations are also presented

random iterations fixed points and invariant crf horospheres in complex banach spaces abstract for holomorphic noncontractive maps on not necessarily bounded domains in complex banach spaces we establish the conditions guaranteeing locally uniform convergence of random iterations and study the existence of fixed points and boundary behaviour of iterations in particular we show that the problem concerning the existence of the horospheres determined by caratheodory reiffen finsler pseudometrics defined on unbounded domains has the solution and we prove new results of type of julia s lemma and wolff s theorem

frontiers in complex dynamics rational maps on the riemann sphere occupy a distinguished niche in the general theory of smooth dynamical systems first rational maps are complexanalytic so a broad spectrum of techniques can contribute to their study quasiconformal mappings potential theory algebraic geometry etc the rational maps of a given degree form a finite dimensional manifold so exploration of this parameter space is especially tractable finally some of the conjectures once proposed for smooth dynamical systems and now known to be false seem to have a definite chance of holding in the arena of rational maps in this article we survey a small constellation of such conjectures centering around the density of hyperbolic rational maps those which are dynamically the best behaved we discuss some of the evidence and logic underlying these conjectures and sketch recent progress towards their resolution

a graphical analysis of cost sensitive regression problems several efforts have been done to bring roc analysis beyond binary classification especially in regression however the mapping and possibilities of these proposals do not correspond to what we expect from the analysis of operating conditions dominance hybrid methods etc in this paper we present a new representation of regression models in the so called regression roc rroc space the basic idea is to represent over estimation on the x axis and under estimation on the y axis the curves are just drawn by adjusting a shift a constant that is added or subtracted to the predictions and plays a similar role as a threshold in classification from here we develop the notions of optimal operating condition convexity dominance and explore several evaluation metrics that can be shown graphically such as the area over the rroc curve aoc in particular we show a novel and significant result the aoc is equal to the error variance multiplied by a factor which does not depend on the model the derivation of rroc curves with non constant shifts and soft regression models and the relation with cost plots is also discussed

roc curves for regression receiver operating characteristic roc analysis is one of the most popular tools for the visual assessment and understanding of classifier performance in this paper we present a new representation of regression models in the so called regression roc rroc space the basic idea is to represent over estimation against under estimation the curves are just drawn by adjusting a shift a constant that is added or subtracted to the predictions and plays a similar role as a threshold in classification from here we develop the notions of optimal operating condition convexity dominance and explore several evaluation metrics that can be shown graphically such as the area over the rroc curve aoc in particular we show a novel and significant result the aoc is equivalent to the error variance we illustrate the application of rroc curves to resource estimation namely the estimation of software project effort

a unified view of performance metrics translating threshold choice into expected classification loss many performance metrics have been introduced in the literature for the evaluation of classification performance each of them with different origins and areas of application these metrics include accuracy unweighted accuracy the area under the roc curve or the roc convex hull the mean absolute error and the brier score or mean squared error with its decomposition into refinement and calibration one way of understanding the relations among these metrics is by means of variable operating conditions in the form of misclassification costs and or class distributions thus a metric may correspond to some expected loss over different operating conditions one dimension for the analysis has been the distribution for this range of operating conditions leading to some important connections in the area of proper scoring rules we demonstrate in this paper that there is an equally important dimension which has so far received much less attention in the analysis of performance metrics this dimension is given by the decision rule which is typically implemented as a threshold choice method when using scoring models in this paper we explore many old and new threshold choice methods fixed score uniform score driven rate driven and optimal among others by calculating the expected loss obtained with these threshold choice methods for a uniform range of operating conditions we give clear interpretations of the 0 1 loss the absolute error the brier score the auc and the refinement loss respectively our analysis provides a comprehensive view of performance metrics as well as a systematic approach to loss minimisation which can be summarised as follows given a model apply the threshold choice methods that correspond with the available information about the operating condition and compare their expected losses in order to assist in this procedure we also derive several connections between the aforementioned performance metrics and we highlight the role of calibration in choosing the threshold choice method

efficiently locating objects using the hausdorff distance the hausdorff distance is a measure defined between two point sets here representing a model and an image the hausdorff distance is reliable even when the image contains multiple objects noise spurious features and occlusions in the past it has been used to search images for instances of a model that has been translated or translated and scaled by finding transformations that bring a large number of model features close to image features and vice versa in this paper we apply it to the task of locating an affine transformation of a model in an image this corresponds to determining the pose of a planar object that has undergone weak perspective projection we develop a rasterised approach to the search and a number of techniques that allow us to locate quickly all transformations of the model that satisfy two quality criteria we can also efficiently locate only the best transformation we discuss an implementation of this approach and present some examples of its use

minimax classifiers based on neural networks the problem of designing a classifier when prior probabilities are not known or are not representative of the underlying data distribution is discussed in this paper traditional learning approaches based on the assumption that class priors are stationary lead to sub optimal solutions if there is a mismatch between training and future real priors to protect against this uncertainty a minimax approach may be desirable we address the problem of designing a neural based minimax classifier and propose two different algorithms a learning rate scaling algorithm and a gradient based algorithm experimental results show that both succeed in finding the minimax solution and it is also pointed out the differences between common approaches to cope with this uncertainty in priors and the minimax classifier

strictly proper scoring rules prediction and estimation scoring rules assess the quality of probabilistic forecasts by assigning a numerical score based on the predictive distribution and on the event or value that materializes a scoring rule is proper if the forecaster maximizes the expected score for an observation drawn from the distributionf if he or she issues the probabilistic forecast f rather than g f it is strictly proper if the maximum is unique in prediction problems proper scoring rules encourage the forecaster to make careful assessments and to be honest in estimation problems strictly proper scoring rules provide attractive loss and utility functions that can be tailored to the problem at hand this article reviews and develops the theory of proper scoring rules on general probability spaces and proposes and discusses examples thereof proper scoring rules derive from convex functions and relate to information measures entropy functions and bregman divergences in the case of categorical variables we prove a rigorous version of the

trademark shapes description by string matching techniques abstract this work considers the possibility of measuring the shape similarity distances of two dimensional objects the method proposed is articulated in two stages chain coding the objects contours and applying string distances to the coded contours the distance values obtained by such a procedure are then associated to the general geometrical characteristics of the object comparison of the distance values of two objects gives an indication of their similarity the implementation of a shape similarity distance based on this concept is considered in depth its theoretical and practical implications are examined the effectiveness of the method is tested with trademarks a class of images of considerable practical interest the results are encouraging and point to the feasibility of the task undertaken by using the proposed approach

random iterations of holomorphic contractions in locally convex spaces and of weaker contractions in uniform spaces conditions which guarantee the uniform convergence of random iterations of holomorphic contractions on unbounded domains in complex hausdorff locally convex and sequentially complete topological vector spaces are established also conditions concerning the convergence of random iterations of weaker contractions in hausdorff uniform spaces are given

coherent dispersion criteria for optimal experimental design we characterize those coherent design criteria which depend only on the dispersion matrix assumed proper and nonsingular of the state of nature which may be a parameter vector or a set of future observables and describe the associated decision problems connections are established with the classical approach to optimal design theory for the normal linear model based on concave functions of the information matrix implications of the theory for more general models are also considered

dynamic two strip algorithm in curve fitting abstract in this paper a new dynamic strip algorithm for fitting a curve with lines is presented the algorithm first finds the best fitted left hand side and right hand side strips at each point on the curve a figure of merit is computed based on the fitting results then a local maximum detection process is applied to pick out points of high curvature the approximated curve is one with the high curvature points connected by straight lines the global structure of a shape which is insensitive to a scale and orientation changes is obtained by merging the results from several runs of the algorithm with different minimum strip widths

probabilistic forecasts calibration and sharpness summary probabilistic forecasts of continuous variables take the form of predictive densities or predictive cumulative distribution functions we propose a diagnostic approach to the evaluation of predictive performance that is based on the paradigm of maximizing the sharpness of the predictive distributions subject to calibration calibration refers to the statistical consistency between the distributional forecasts and the observations and is a joint property of the predictions and the events that materialize sharpness refers to the concentration of the predictive distributions and is a property of the forecasts only a simple theoretical framework allows us to distinguish between probabilistic calibration exceedance calibration and marginal calibration we propose and study tools for checking calibration and sharpness among them the probability integral transform histogram marginal calibration plots the sharpness diagram and proper scoring rules the diagnostic approach is illustrated by an assessment and ranking of probabilistic forecasts of wind speed at the stateline wind energy centre in the us pacific northwest in combination with cross validation or in the time series context our proposal provides very general nonparametric alternatives to the use of information criteria for model diagnostics and model selection

a two dimensional accuracy based measure for classification performance abstract accuracy has been used traditionally to evaluate the performance of classifiers however it is well known that accuracy is not able to capture all the different factors that characterize the performance of a multiclass classifier in this manuscript accuracy is studied and analyzed as a weighted average of the classification rate of each class this perspective allows us to propose the dispersion of the classification rate of each class as its complementary measure in this sense a graphical performance metric which is defined in a two dimensional space composed by accuracy and dispersion is proposed to evaluate the performance of classifiers we show that the combined values of accuracy and dispersion must fall within a clearly bounded two dimensional region different for each problem the nature of this region depends only on the a priori probability of each class and not on the classifier used thus the performance of multiclassifiers is represented in a two dimensional space where the models can be compared in a more fair manner providing greater awareness of the strategies that are more accurate when trying to improve the performance of a classifier furthermore we experimentally analyze the behavior of seven different performance metrics based on the computation of the confusion matrix values in several scenarios identifying clusters and relationships between measures as shown in the experimentation the graphical metric proposed is specially suitable in challenging highly imbalanced and with a high number of classes datasets the approach proposed is a novel point of view to address the evaluation of multiclassifiers and it is an alternative to other evaluation measures used in machine learning