A rule-of-thumb for the variable bandwidth selection in kernel hazard rate estimation

In nonparametric curve estimation the decision about the type of smoothing parameter is critical for the practical performance. The nearest neighbor bandwidth as introduced by Gefeller and Dette 1992 for censored data in survival analysis is specified by one parameter, namely the number of nearest neighbors. Bandwidth selection in this setting is rarely investigated although not linked closely to the frequently studied fixed bandwidth. We introduce a selection algorithm in the hazard rate estimation context. The approach uses a newly developed link to the fixed bandwidth which identifies the variable bandwidth as additional smoothing step. The procedure gains further data-adaption after fixed bandwidth smoothing. Assessment by a Monte Carlo simulation and a clinical example demonstrate the practical relevance of the findings. --

Bias in nearest-neighbor hazard estimation

In nonparametric curve estimation, the smoothing parameter is critical for performance. In order to estimate the hazard rate, we compare nearest neighbor selectors that minimize the quadratic, the Kullback-Leibler, and the uniform loss. These measures result in a rule of thumb, a cross-validation, and a plug-in selector. A Monte Carlo simulation within the three-parameter exponentiated Weibull distribution indicates that a counter-factual normal distribution, as an input to the selector, does provide a good rule of thumb. If bias is the main concern, minimizing the uniform loss yields the best results, but at the cost of very high variability. Cross-validation has a similar bias to the rule of thumb, but also with high variability. --hazard rate,kernel smoothing,bandwidth selection,nearest neighbor bandwidth,rule of thumb,plug-in,cross-validation,credit risk

A Rule-of-Thumb for the Variable Bandwidth Selection in Kernel Hazard Rate Estimation

In nonparametric curve estimation the decision about the type of smoothing parameter is critical for the practical performance. The nearest neighbor bandwidth as introduced by Gefeller and Dette 1992 for censored data in survival analysis is specified by one parameter, namely the number of nearest neighbors. Bandwidth selection in this setting is rarely investigated although not linked closely to the frequently studied fixed bandwidth. We introduce a selection algorithm in the hazard rate estimation context. The approach uses a newly developed link to the fixed bandwidth which identifies the variable bandwidth as additional smoothing step. The procedure gains further data-adaption after fixed bandwidth smoothing. Assessment by a Monte Carlo simulation and a clinical example demonstrate the practical relevance of the findings

A General Kernel Functional Estimator with Generalized Bandwidth - Strong Consistency and Applications

We consider the problem of uniform asymptotics in kernel functional estimation where the bandwidth can depend on the data. In a unified approach we investigate kernel estimates of the density and the hazard rate for uncensored and right-censored observations. The model allows for the fixed bandwidth as well as for various variable bandwidths, e.g. the nearest neighbor bandwidth. An elementary proof for the strong consistency of the generalized estimator is given that builds on the local convergence of the empirical process against the cumulative distribution function and the Nelson-Aalen estimator against the cumulative hazard rate, respectively

Image Segmentation Using Active Contours Driven by the Bhattacharyya Gradient Flow

©2007 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or distribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.DOI: 10.1109/TIP.2007.908073This paper addresses the problem of image segmentation by means of active contours, whose evolution is driven by the gradient flow derived froman energy functional that is based on the Bhattacharyya distance. In particular, given the values of a photometric variable (or of a set thereof), which is to be used for classifying the image pixels, the active contours are designed to converge to the shape that results in maximal discrepancy between the empirical distributions of the photometric variable inside and outside of the contours. The above discrepancy is measured by means of the Bhattacharyya distance that proves to be an extremely useful tool for solving the problem at hand. The proposed methodology can be viewed as a generalization of the segmentation methods, in which active contours maximize the difference between a finite number of empirical moments of the "inside" and "outside" distributions. Furthermore, it is shown that the proposed methodology is very versatile and flexible in the sense that it allows one to easily accommodate a diversity of the image features based on which the segmentation should be performed. As an additional contribution, a method for automatically adjusting the smoothness properties of the empirical distributions is proposed. Such a procedure is crucial in situations when the number of data samples (supporting a certain segmentation class) varies considerably in the course of the evolution of the active contour. In this case, the smoothness properties of the empirical distributions have to be properly adjusted to avoid either over- or underestimation artifacts. Finally, a number of relevant segmentation results are demonstrated and some further research directions are discussed

Locally adaptive density estimation on Riemannian manifolds

In this paper, we consider kernel type estimator with variable bandwidth when the random variables belong in a Riemannian manifolds. We study asymptotic properties such as the consistency and the asymptotic distribution. A simulation study is also considered to evaluate the performance of the proposal. Finally, to illustrate the potential applications of the proposed estimator, we analyse two real examples where two different manifolds are considered

Bias in nearest-neighbor hazard estimation

bandwidth, rule of thumb. In nonparametric curve estimation, the smoothing parameter is critical for performance. In order to estimate the hazard rate, we compare nearest neighbor selectors that minimize the quadratic, the Kullback-Leibler, and the uniform loss. These measures result in a rule of thumb, a cross-validation, and a plug-in selector. A Monte Carlo simulation within the three-parameter exponentiated Weibull distribution indicates that a counter-factual normal distribution, as an input to the selector, does provide a good rule of thumb. If bias is the main concern, minimizing the uniform loss yields the best results, but at the cost of very high variability. Cross-validation has a similar bias to the rule of thumb, but also with high variability

Consistency of the kernel density estimator - a survey

Various consistency proofs for the kernel density estimator have been developed over the last few decades. Important milestones are the pointwise consistency and almost sure uniform convergence with a fixed bandwidth on the one hand and the rate of convergence with a fixed or even a variable bandwidth on the other hand. While considering global properties of the empirical distribution functions is sufficient for strong consistency, proofs of exact convergence rates use deeper information about the underlying empirical processes. A unifying character, however, is that earlier and more recent proofs use bounds on the probability that a sum of random variables deviates from its mean

Representación efectiva de dinámicas fisiológicas mediante fuzzy rough set: una revisión

The latest generation of biomedical systems record at short time intervals the physiological dynamic in large databases. The correct interpretation of the information is difficult to obtain by the expertise of a single physician, so the decision is based only on some selected variables. Effective representation of physiological variables by fuzzy Rough Set type 1 can be applied to characterize and extract relevant information from physiological dynamics, however the disadvantages of these techniques are the complexity of their algorithms and the high computational cost, therefore it is necessary to apply fuzzy rough set type 2 techniques , associated with axiomatic methods through low and high diffuse approximation operators as primitive concepts for generating a dimension reduction system with a tendency to lower computational cost in biomedical engineering applications. This article reviews the state of the art of effective representation of physiological dynamics using fuzzy rough set, in order to determine the ability of these techniques to be included in automatic decision making procedures that support the clinical opinion of a specialist.Los sistemas biomédicos de última generación registran en intervalos cortos de tiempo la dinámica fisiológica mediante grandes bases de datos. La interpretación adecuada de la información difícilmente puede hacerse por la experticia de un sólo médico, por lo tanto la toma de decisiones se basa sólo en algunas variables seleccionadas. La representación efectiva de variables fisiológicas mediante fuzzy rough set tipo 1 puede ser aplicada para caracterizar y extraer la información relevante de la dinámica fisiológica; sin embargo, estas técnicas poseen el problema de la complejidad de sus algoritmos y alto costo computacional; por lo tanto, se requiere aplicar técnicas de fuzzy rough set tipo 2, asociadas a métodos axiomáticos a través de operadores de aproximación difusa baja y alta como conceptos primitivos para generar un sistema de reducción de dimensiones con tendencia a la disminución de costo computacional en aplicaciones de ingeniería biomédica. En este artículo se presenta la revisión del estado del arte sobre representación efectiva de dinámicas fisiológicas mediante fuzzy rough set, con el fin de determinar la capacidad que poseen este tipo de técnicas para ser incluidas en procedimientos automáticos de toma de decisiones que apoyen el concepto clínico de un especialista