Rejection Rule Research Articles

Wavelet Thresholding with Bayesian False Discovery Rate Control

The false discovery rate (FDR) procedure has become a popular method for handling multiplicity in high-dimensional data. The definition of FDR has a natural Bayesian interpretation; it is the expected proportion of null hypotheses mistakenly rejected given a measure of evidence for their truth. In this article, we propose controlling the positive FDR using a Bayesian approach where the rejection rule is based on the posterior probabilities of the null hypotheses. Correspondence between Bayesian and frequentist measures of evidence in hypothesis testing has been studied in several contexts. Here we extend the comparison to multiple testing with control of the FDR and illustrate the procedure with an application to wavelet thresholding. The problem consists of recovering signal from noisy measurements. This involves extracting wavelet coefficients that result from true signal and can be formulated as a multiple hypotheses-testing problem. We use simulated examples to compare the performance of our approach to the Benjamini and Hochberg (1995, Journal of the Royal Statistical Society, Series B57, 289-300) procedure. We also illustrate the method with nuclear magnetic resonance spectral data from human brain.

On optimal reject rules and ROC curves

In this paper we make the connection between two approaches for supervised classification with a rejection option. The first approach is due to Tortorella and is based on ROC curves and the second is a generalisation of Chow’s optimal rule.

A ROC-based reject rule for dichotomizers

Many complex classification tasks involve a discrimination between two classes. Since in such cases a classification error could frequently have serious consequences, the classifiers employed should ensure a very high reliability to avoid erroneous decisions. Unfortunately this is difficult to obtain in real situations where the classifier can meet samples very different from those examined in the training phase. Moreover, the cost for a wrong classification can be so high that it is convenient to reject the sample which gives raise to an unreliable result. However, despite its relevance, a reject option specifically devised for dichotomizers (i.e. two-class classifiers) has not been yet proposed. This paper presents a novel reject rule for dichotomizers, based on the Receiver Operating Characteristic curve. The rule minimizes the expected classification cost, defined on the basis of classification and error costs peculiar for the application at hand. Experiments performed with different classifier architectures on several data sets publicly available confirmed the effectiveness of the proposed reject rule.

Reducing the classification cost of support vector classifiers through an ROC-based reject rule

This paper presents a novel reject rule for support vector classifiers, based on the receiver operating characteristic (ROC) curve. The rule minimises the expected classification cost, defined on the basis of classification and the error costs for the particular application at hand. The rationale of the proposed approach is that the ROC curve of the SVM contains all of the necessary information to find the optimal threshold values that minimise the expected classification cost. To evaluate the effectiveness of the proposed reject rule, a large number of tests has been performed on several data sets, and with different kernels. A comparison technique, based on the Wilcoxon rank sum test, has been defined and employed to provide the results at an adequate significance level. The experiments have definitely confirmed the effectiveness of the proposed reject rule.

Adaptively truncated maximum likelihood regression with asymmetric errors

We consider robust estimators for the linear regression model with asymmetric (or symmetric) error distribution. We assume that the error model belongs to a location-scale family of distributions. Since in the asymmetric case the mean response is very often the parameter of interest and scale is a main component of mean, we do not assume that scale is a nuisance parameter. First, we show how to convert an ordinary robust estimate for the usual model with symmetric errors to an estimate for the more general model with asymmetric errors. Then, in order to improve efficiency, we introduce the truncated maximum likelihood or TML-estimator. A TML-estimate is computed in three steps: first, an initial high breakdown point estimate is computed; then, observations that are unlikely under the estimated model are rejected; finally, the maximum likelihood estimate is computed with the retained observations. The rejection rule used in the second step is based on a cut-off parameter that can be tuned to attain the desired efficiency while maintaining the breakdown point of the initial estimator (e.g., 50%). Optionally, one can use a new adaptive cut-off that, asymptotically, does not reject any observation when the data are generated according to the model. Under the model, the influence function of this adaptive TML-estimator (or ATML-estimator) coincides with the influence function of the maximum likelihood estimator. The ATML-estimator is, therefore, fully efficient at the model; nevertheless, its breakdown point is not smaller than the breakdown point of the initial estimator. We evaluate the TML- and ATML-estimators for finite sample sizes with the help of simulations and discuss an example with real data.

Heuristic Rejection in Interval Global Optimization

Based on the investigation carried out in Ref. 1, this paper incorporates new studies about the properties of inclusion functions on subintervals while a branch-and-bound algorithm is solving global optimization problems. It is found that the relative place of the global minimum value within the inclusion function value of the objective function at the current interval indicates mostly whether the given interval is close to a minimizer point. This information is used in a heuristic interval rejection rule that can save a considerable amount of computation. Illustrative examples are discussed and an extended numerical study shows the advantages of the new approach.

Statistical methods of automatic spectral classification and their application to the Hamburg/ESO Survey

We employ classical statistical methods of multivariate classification for the exploitation of the stellar content of the Hamburg/ESO objective prism survey (HES). In a simulation study we investigate the precision of a three-dimensional classification (Te ,l ogg ,( Fe/H)) achievable in the HES for stars in the eective temperature range 5200 K 10 (typically corresponding to BJ < 16:5). The accuracies in logg and (Fe/H) are better than 0:68 dex in the same S=N range. These precisions allow for a very ecient selection of metal-poor stars in the HES. We present a minimum cost rule for compilation of complete samples of objects of a given class, and a rejection rule for identification of corrupted or peculiar spectra. The algorithms we present are being used for the identification of other interesting objects in the HES data base as well, and they are applicable to other existing and future large data sets, such as those to be compiled by the DIVA and GAIA missions.

Registration of Multiple Acoustic Range Views for Underwater Scene Reconstruction

This paper proposes a technique for the three-dimensional reconstruction of an underwater environment from multiple acoustic range views acquired by a remotely operated vehicle. The problem is made challenging by the very noisy nature of the data, the low resolution, and the narrow field of view. Our main contribution is a new global registration technique to distribute registration errors evenly across all views. Our approach does not use data points after the first pairwise registration, for it works only on the transformations. Therefore, it is fast and occupies only a small amount of memory. Experimental results suggest the global registration technique is effective in equalizing the error. Moreover, we introduce a statistically sound thresholding (the X84 rejection rule) to improve ICP robustness against noise and nonoverlapping data.

Controversies in Clostridium difficile testing

Recent reports of two nosocomial outbreaks of Clostridium difficile-associated disease caused by toxin A-deficient strains emphasize that these strains can cause disease. Laboratories using an assay that detects only toxin A as their primary diagnostic test risk misdiagnosis of cases or outbreaks in the institutions they serve. Repeat testing can account for a significant portion of a laboratory's C. difficile testing workload. Published data are available to support laboratory rules for rejection of repeat stool specimens within 7 days of an initial specimen. There are also substantial published data to support laboratory rejection of formed stools sent to the laboratory for C. difficile testing.

A CLASSIFICATION RELIABILITY DRIVEN REJECT RULE FOR MULTI-EXPERT SYSTEMS

In this paper we propose a reject rule applicable to a Multi-Expert System (MES). The rule is adaptive to the given domain and allows the achievement of the best trade-off between reject and error rates as a function of the costs attributed to errors and rejects in the considered application. The results of the method are particularly effective since the method does not rely on particular statistical assumptions, as other reject rules. An experimental analysis carried out on publicly available databases is reported together with a comparison with other methods present in the literature.

Graphical conditional moment tests

Summary Tests of misspecification or other defects in microeconometric models are usually numerical and based on an average over the sample. In this paper, it is shown that certain types of misspecification can be concealed by averaging, and a framework is developed which allows the specification to be checked in the near neighbourhood of each sample point, rather than for the sample as a whole. The framework involves the application of a scatter-plot smoothing technique to a pair of auxiliary variables derived from the estimated model. Using the sampling distribution of the location of the smooth under correct specification, a rejection rule based on the location of the realized smooth is established. Thus the smooth itself is used as a test statistic. A natural application of the procedure is to tests of equi-dispersion in count data models, and we find that it is particularly useful in the case in which a sample contains elements of both under- and over-dispersion, since it is in this case that numerical tests based on averaging are most likely to lack power. Two specific applications, one on takeover bids and the other on worker absenteeism, are used to illustrate the procedure.

A heuristic rejection criterion in interval global optimization algorithms

This paper investigates the properties of the inclusion functions on subintervals while a Branch-and-Bound algorithm is solving global optimization problems. It has been found that the relative place of the global minimum value within the inclusion interval of the inclusion function of the objective function at the actual interval mostly indicates whether the given interval is close to minimizer point. This information is used in a heuristic interval rejection rule that can save a big amount of computation. Illustrative examples are discussed and a numerical study completes the investigation.

Improving Feature Tracking with Robust Statistics

This paper addresses robust feature tracking. The aim is to track point features in a sequence of images and to identify unreliable features resulting from occlusions, perspective distortions and strong intensity changes. We extend the well-known Shi–Tomasi–Kanade tracker by introducing an automatic scheme for rejecting spurious features. We employ a simple and efficient outliers rejection rule, called X84, and prove that its theoretical assumptions are satisfied in the feature tracking scenario. Experiments with real and synthetic images confirm that our algorithm consistently discards unreliable features; we show a quantitative example of the benefits introduced by the algorithm for the case of fundamental matrix estimation. The complete code of the robust tracker is available via ftp.

Multiclassification: reject criteria for the Bayesian combiner

In the present paper we propose a method for determining the best trade-off between error rate and reject rate for a multi-expert system (MES) using the Bayesian combining rule. The method is based on the estimation of the reliability of each classification act and on the evaluation of the convenience of rejecting the input sample when the reliability is under a threshold, evaluated on the basis of the requirements of the application domain. The adaptability to the given domain represents an original feature since, till now, the problem of defining a reject rule for an MES has not been systematically introduced, and the few existing proposals seldom take into account the requirements of the domain. The method has been widely tested with reference to the recognition of handwritten characters coming from a standard database. The results are also compared with those provided by employing the well-known Chow’s rule.

CLASS-SELECTIVE REJECTION RULE TO MINIMIZE THE MAXIMUM DISTANCE BETWEEN SELECTED CLASSES

Class-selective rejection scheme assigns to an input pattern a subset of classes that are most likely to identify the pattern, and not simply reject it. Conventional class-selective rejection rules can minimize the error rate for a given average number of classes. In this paper, an example of unnatural classification is shown by using conventional rules and a new optimum criterion for establishing a class-selective rejection rule is proposed. Its optimality and upper-bound of error rate are proven. Finally, two examples are provided to illustrate various aspects of this optimum decision rule.

Optimum tradeoff between class-selective rejection error and average number of classes

The paper first reviews the recently proposed optimum class-selective rejection rule. This rule provides an optimum tradeoff between the error rate and the average number of (selected) classes. Then, a new general relation between the error rate and the average number of classes is presented. The error rate can be directly computed from the class-selective reject function, which in turn can be estimated from unlabelled patterns, by simply counting the rejected classes. Theoretical as well as practical implications are discussed, and some future research directions are proposed.

Resistant and Test-Based Outlier Rejection: Effects on Gaussian One- and Two-Sample Inference

Resistant and sequential test-based procedures for the detection of multiple outliers are compared in Gaussian one- and two-sample problems. Subsequent to statistical rejection of putative outliers, operating characteristics of hypothesis tests applied to samples that are outlier-free may differ from their nominal values. Outlier rejection rules may be calibrated, however, to control the rate of erroneous outlier labeling in pure Gaussian samples. The small-sample behavior of resistant location and scale estimators used in outlier detection is examined through simulation, yielding new formulas for calibration of boxplot-based and shorth-based outlier-rejection rules. Effects of different outlier criteria and calibration rubrics on postrejection hypothesis tests are compared in a variety of contamination settings and real-world examples. Tests conducted subsequent to calibrated rejection have approximately nominal operating characteristics whether or not outliers are present. The selection of an outlier cr...

Resistant and Test-Based Outlier Rejection: Effects on Gaussian One- and Two-Sample inference

Resistant and sequential test-based procedures for the detection of multiple outliers are compared in Gaussian one- and two-sample problems. Subsequent to statistical rejection of putative outliers, operating characteristics of hypothesis tests applied to samples that are outlier-free may differ from their nominal values. Outlier rejection rules may be calibrated, however, to control the rate of erroneous outlier labeling in pure Gaussian samples. The small-sample behavior of resistant location and scale estimators used in outlier detection is examined through simulation, yielding new formulas for calibration of boxplot-based and shorth-based outlier-rejection rules. Effects of different outlier criteria and calibration rubrics on postrejection hypothesis tests are compared in a variety of contamination settings and real-world examples. Tests conducted subsequent to calibrated rejection have approximately nominal operating characteristics whether or not outliers are present. The selection of an outlier criterion can have an impact on inference and thus becomes a potential component of model uncertainty. Properly calibrated tools should be widely available so that this component of uncertainty becomes better understood; we describe public-domain software implementing a variety of detection techniques.

The optimum class-selective rejection rule

Class-selective rejection is an extension of simple rejection. That is, when an input pattern cannot be reliably assigned to one of the N classes in an N-class problem, it is assigned to a subset of classes that are most likely to issue the pattern, instead of simply being rejected. By selecting more classes, the risk of making an error can be reduced, at the price of subsequently having a larger remaining number of classes. The optimality of class-selective rejection is therefore defined as the best trade-off between error rate and average number of selected classes. Formally, the trade-off study is embedded in the framework of decision theory. The average expected loss is expressed as a linear combination of error rate and average number of classes. The minimization of the average expected loss, therefore, provides the best trade-off. The complexity of the resulting optimum rule is reduced, via a discrete convex minimization, to be linear in the number of classes. Upper-bounds on error rate and average number of classes are derived. An example is provided to illustrate various aspects of the optimum decision rule. Finally, the implications of the new decision rule are discussed.

Rejection rules in theM/G/1 queue

We consider aM/G/1 queue modified such that an arriving customer may be totally or partially rejected depending on a r.v. (the barricade) describing his impatience and on the state of the system. Three main variants of this scheme are studied. The steady-state distribution is expressed in terms of Volterra equations and the relation to storage processes, dams and queues with state-dependent Poisson arrival rate is discussed. For exponential service times, we further find the busy period Laplace transform in the case of a deterministic barricade, whereas for exponential barricade it is shown by a coupling argument that the busy period can be identified with a first passage time in an associated birth-death process.