7,294 research outputs found
Sharp Oracle Inequalities for Aggregation of Affine Estimators
We consider the problem of combining a (possibly uncountably infinite) set of
affine estimators in non-parametric regression model with heteroscedastic
Gaussian noise. Focusing on the exponentially weighted aggregate, we prove a
PAC-Bayesian type inequality that leads to sharp oracle inequalities in
discrete but also in continuous settings. The framework is general enough to
cover the combinations of various procedures such as least square regression,
kernel ridge regression, shrinking estimators and many other estimators used in
the literature on statistical inverse problems. As a consequence, we show that
the proposed aggregate provides an adaptive estimator in the exact minimax
sense without neither discretizing the range of tuning parameters nor splitting
the set of observations. We also illustrate numerically the good performance
achieved by the exponentially weighted aggregate
Spatial aggregation of local likelihood estimates with applications to classification
This paper presents a new method for spatially adaptive local (constant)
likelihood estimation which applies to a broad class of nonparametric models,
including the Gaussian, Poisson and binary response models. The main idea of
the method is, given a sequence of local likelihood estimates (``weak''
estimates), to construct a new aggregated estimate whose pointwise risk is of
order of the smallest risk among all ``weak'' estimates. We also propose a new
approach toward selecting the parameters of the procedure by providing the
prescribed behavior of the resulting estimate in the simple parametric
situation. We establish a number of important theoretical results concerning
the optimality of the aggregated estimate. In particular, our ``oracle'' result
claims that its risk is, up to some logarithmic multiplier, equal to the
smallest risk for the given family of estimates. The performance of the
procedure is illustrated by application to the classification problem. A
numerical study demonstrates its reasonable performance in simulated and
real-life examples.Comment: Published in at http://dx.doi.org/10.1214/009053607000000271 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Weighted Constraints in Fuzzy Optimization
Many practical optimization problems are characterized by someflexibility in the problem constraints, where this flexibility canbe exploited for additional trade-off between improving theobjective function and satisfying the constraints. Especially indecision making, this type of flexibility could lead to workablesolutions, where the goals and the constraints specified bydifferent parties involved in the decision making are traded offagainst one another and satisfied to various degrees. Fuzzy setshave proven to be a suitable representation for modeling this typeof soft constraints. Conventionally, the fuzzy optimizationproblem in such a setting is defined as the simultaneoussatisfaction of the constraints and the goals. No additionaldistinction is assumed to exist amongst the constraints and thegoals. This report proposes an extension of this model forsatisfying the problem constraints and the goals, where preferencefor different constraints and goals can be specified by thedecision-maker. The difference in the preference for theconstraints is represented by a set of associated weight factors,which influence the nature of trade-off between improving theoptimization objectives and satisfying various constraints.Simultaneous weighted satisfaction of various criteria is modeledby using the recently proposed weighted extensions of(Archimedean) fuzzy t-norms. The weighted satisfaction of theproblem constraints and goals are demonstrated by using a simplefuzzy linear programming problem. The framework, however, is moregeneral, and it can also be applied to fuzzy mathematicalprogramming problems and multi-objective fuzzy optimization.wiskundige programmering;fuzzy sets;optimalisatie
Stability and aggregation of ranked gene lists
Ranked gene lists are highly instable in the sense that similar measures of differential gene expression may yield very different rankings, and that a small change of the data set usually affects the obtained gene list considerably. Stability issues have long been under-considered in the literature, but they have grown to a hot topic in the last few years, perhaps as a consequence of the increasing skepticism on the reproducibility and clinical applicability of molecular research findings. In this article, we review existing approaches for the assessment of stability of ranked gene lists and the related problem of aggregation, give some practical recommendations, and warn against potential misuse of these methods. This overview is illustrated through an application to a recent leukemia data set using the freely available Bioconductor package GeneSelector
- …