Contributions in Merlin

Präferenzen für Umweltgüter hängen immer vom Kontext ab, in dem der einzelne seine Entscheidung trifft. Da viele Umweltbelastungen nicht unmittelbar erfahrbar sind, müssen ökologische Handlungsalternativen in der konkreten Situation erst einmal "in den Sinn" kommen. Wahrnehmungshilfen wie Meßwerte, Okobilanzen oder Umweltlabels können dabei ökologische Präferenzen wachrufen.

This article addresses some of the valuation problems, in the Black and Scholes setting of a geometric Brownian motion for the underlying asset dynamics, for options whose pay-off is related to the terminal price of the stock and an arithmetic average of fixing and/or involves stopping times related to excursions. In all cases, we are able to provide at least the Laplace transform in time of the option price under a form whose complexity varies with the number of exotic features. We emphasize that we do not give closed form formulas for the general case, but we aim to develop a methodology which may be used in many cases.

Computers work in a very precise, tireless, and fast manner. Nevertheless mechanisms by which computers could learn themselves and perform very simple everyday tasks arc virtually unknown. Such mechanisms that allow machines to learn from individual experiences and relate them adequately to other similar situations involve the key issue of how to generalize from past experiences to deal with future situations. Hence, the goal of this thesis is to investigate the laws governing reliability in learning. This document describes a constructive approach to building learning machines for real-world applications, which try to solve many well-studied issues and problems originating from mathematics and real-world computing. The concept presented is that there arc learning machines, which have generally valid structural constraints incorporated in their architecture in order to achieve satisfactory performance for mastering challenging real-world tasks. Such structural constraints arc given and theoretically evaluated by the constructive and consistent generalization theory developed by Vapnik and Chervonenkis (Vapnik 1995b). Based on this theory, the factors governing generalization arc further utilized to classify and compare learning machines that exploit invariants. These include, among the various, discussed learning machines, learning from hints (Abu-Mostafa 1994), estimating the tangent of invariances by “Tangent Prop” (Simard and LeCun 1994), the group-invariance theorem (Minsky 1961), and the hyper-plane point invariance theorem (Nillson 1965). This comparison is conducted from the perspective of including generally valid geometric restrictions, i.c., geometrical structures learned by the learning machine arc adjusted such that they comply with the sensed data. Then two now neural network paradigms Me described: 1. Structurally Constrained Resource-Allocating Neural Networks (SCRAN), and 2. Topological Growing Structures (TGS). Both methods allocate incrementally new neurons and regulate their structure to comply with the sensed or measured data sets. Both rely on geometric constructs founded on sample vectors. The first kind of network belongs to the class of supervised architectures, whereas the second belongs to the class of unsupervised ones. The first algorithm is based on the assumption of continuity and monotony. The assumption applies to samples of one category belonging to the same partition. For such a partition, the monotony continuity constraint is defined such that all points between the center of gravity of a partition and its samples belong to the same partition. This geometric construct is generally concave. It is the base construct of the learning algorithm, which is adjusted by an empirical risk minimization scheme. Furthermore, the complexity of this algorithm is studied by the above-mentioned generalization theory. This and other benchmarking problems corroborate the superior generalization power of SCRANs. The second algorithm that implements learning by constraints is established by constructing a topological projection of the data onto its inherent minimized subspace by a self-organizing unsupervised learning scheme. The projection, motivated by Kohonen’s self-organizing feature maps, is constructed by adapting the number of neurons, their weights spawning up the subspace as well as their interconnection topology. Thus, the tessellation is constructed such that the neuronal weights arc only placed in parts of the state space that rue effectively visited by the problem studied. Another important property of the mapping is that it is distribution preserving. These methods are applied to tasks such as credit worthiness analysis, differentiation between mines and rocks, empirical effective VC-dimension estimation, and phoneme analysis. To conclude, our two network paradigms, namely SCRANs and TGSs, arc compared to the others mentioned above by applying the presented generalization scheme and the principle of topological learning by structural constraints. Then the various issues and problems arc reevaluated from the perspective of the learning machine, thereby adapting its structural constraints and architecture to the contemplated problem domain. This important principle is called Topological Induction, and it is the key principle for learning machines in the real world.

null

null

null