Robert Hable

Consistency of support vector machines using additive kernels for additive models

Support vector machines (SVMs) are special kernel based methods and have been among the most successful learning methods for more than a decade. SVMs can informally be described as kinds of regularized M-estimators for functions and have demonstrated their usefulness in many complicated real-life problems. During the last few years a great part of the statistical research on SVMs has concentrated on the question of how to design SVMs such that they are universally consistent and statistically robust for nonparametric classification or nonparametric regression purposes. In many applications, some qualitative prior knowledge of the distribution P or of the unknown function f to be estimated is present or a prediction function with good interpretability is desired, such that a semiparametric model or an additive model is of interest. The question of how to design SVMs by choosing the reproducing kernel Hilbert space (RKHS) or its corresponding kernel to obtain consistent and statistically robust estimators in additive models is addressed. An explicit construction of such RKHSs and their kernels, which will be called additive kernels, is given. SVMs based on additive kernels will be called additive support vector machines. The use of such additive kernels leads, in combination with a Lipschitz continuous loss function, to SVMs with the desired properties for additive models. Examples include quantile regression based on the pinball loss function, regression based on the @e-insensitive loss function, and classification based on the hinge loss function.

Habitat selection by a large herbivore at multiple spatial and temporal scales is primarily governed by food resources

Habitat selection can be considered as a hierarchical process in which animals satisfy their habitat requirements at different ecological scales. Theory predicts that spatial and temporal scales should co-vary in most ecological processes and that the most limiting factors should drive habitat selection at coarse ecological scales, but be less influential at finer scales. Using detailed location data on roe deer (Capreolus capreolus) inhabiting the Bavarian Forest National Park, Germany, we investigated habitat selection at several spatial and temporal scales. We tested (i) whether time-varying patterns were governed by factors reported as having the largest effects on fitness, (ii) whether the trade-off between forage and predation risks differed among spatial and temporal scales and (iii) if spatial and temporal scales are positively associated. We analysed the variation in habitat selection within the landscape and within home ranges at monthly intervals, with respect to land-cover type and proxys of food and cover over seasonal and diurnal temporal scales. The fine-scale temporal variation follows a nycthemeral cycle linked to diurnal variation in human disturbance. The large-scale variation matches seasonal plant phenology, suggesting food resources being a greater limiting factor than lynx predation risk. The trade-off between selection for food and cover was similar on seasonal and diurnal scale. Habitat selection at the different scales may be the consequence of the temporal variation and predictability of the limiting factors as much as its association with fitness. The landscape of fear might have less importance at the studied scale of habitat selection than generally accepted because of the predator hunting strategy. Finally, seasonal variation in habitat selection was similar at the large and small spatial scales, which may arise because of the marked philopatry of roe deer. The difference is supposed to be greater for wider ranging herbivores. This article is protected by copyright. All rights reserved.

Data-based decisions under imprecise probability and least favorable models

Data-based decision theory under imprecise probability has to deal with optimization problems where direct solutions are often computationally intractable. Using the @C-minimax optimality criterion, the computational effort may significantly be reduced in the presence of a least favorable model. Buja [A. Buja, Simultaneously least favorable experiments. I. Upper standard functionals and sufficiency, Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete 65 (1984) 367-384] derived a necessary and sufficient condition for the existence of a least favorable model in a special case. The present article proves that essentially the same result is valid in case of general coherent upper previsions. This is done mainly by topological arguments in combination with some of Le Cams decision theoretic concepts. It is shown how least favorable models could be used to deal with situations where the distribution of the data as well as the prior is allowed to be imprecise.

Finite approximations of data-based decision problems under imprecise probabilities

In decision theory under imprecise probabilities, discretizations are a crucial topic because many applications involve infinite sets whereas most procedures in the theory of imprecise probabilities can only be calculated for finite sets so far. The present paper develops a method for discretizing sample spaces in data-based decision theory under imprecise probabilities. The proposed method turns an original decision problem into a discretized decision problem. It is shown that any solution of the discretized decision problem approximately solves the original problem. In doing so, it is pointed out that the commonly used method of natural extension can be most instable. A way to avoid this instability is presented which is sufficient for the purpose of the paper.

Estimation of scale functions to model heteroscedasticity by regularised kernel-based quantile methods

A main goal of regression is to derive statistical conclusions on the conditional distribution of the output variable Y given the input values x. Two of the most important characteristics of a single distribution are location and scale. Regularised kernel methods (RKMs) – also called support vector machines in a wide sense – are well established to estimate location functions like the conditional median or the conditional mean. We investigate the estimation of scale functions by RKMs when the conditional median is unknown, too. Estimation of scale functions is important, e.g. to estimate the volatility in finance. We consider the median absolute deviation (MAD) and the interquantile range as measures of scale. Our main result shows the consistency of MAD-type RKMs.

Robustness Versus Consistency in Ill-Posed Classification and Regression Problems

It is well-known from parametric statistics that there can be a goal conflict between efficiency and robustness. However, in so-called ill-posed problems, there is even a goal conflict between consistency and robustness. This particularly applies to certain nonparametric statistical problems such as nonparametric classification and regression problems which are often ill-posed. As an example in statistical machine learning, support vector machines are considered.

On the Consistency of the Bootstrap Approach for Support Vector Machines and Related Kernel-Based Methods

It is shown that bootstrap approximations of support vector machines (SVMs) based on a general convex and smooth loss function and on a general kernel are consistent. This result is useful for approximating the unknown finite sample distribution of SVMs by the bootstrap approach.