Julian Belz

Interpolation and extrapolation: Comparison of definitions and survey of algorithms for convex and concave hulls

Any data based method is vulnerable to the problem of extrapolation, nonetheless there exists no unified theory on handling it. The main topic of this publication is to point out the differences in definitions of extrapolation and related methods. There are many different interpretations of extrapolation and a multitude of methods and algorithms, which address the problem of extrapolation detection in different fields of study. We examine popular definitions of extrapolation, compare them to each other and list related literature and methods. It becomes apparent, that the opinions what extrapolation is and how to handle it, differ greatly from each other. We categorize existing literature and give guidelines to choose an appropriate definition of extrapolation for a present problem. We also present hull algorithms, from classic approaches to recent advances. The presented guidelines and categorized literature enables the reader to categorize a present problem, inspect relevant literature and apply suitable methods and algorithms to solve a problem, which is affected by extrapolation.

Extended Deterministic Local Search Algorithm for Maximin Latin Hypercube Designs

This paper introduces the Extended Deterministic Local Search (EDLS) algorithm for Latin Hypercube (LH) designs. The main goal of the algorithm is to improve an existing algorithm towards a better uniformity of the data distribution, while maintaining a good computational performance. After presenting background information about LH designs and how to assess their quality (choice of loss function), the EDLS algorithm is explained and compared to two other algorithms for LH designs.

Input Selection Using Local Model Network Trees

Abstract This paper presents an input selection wrapper approach using local model network trees. This model class allows the distinction in two input spaces - the rule premises input space and the rule consequents input space. Therefore the input selection can take place in both or just in one of these input spaces. As we will show, this leads to an improved model accuracy and an improved understanding of the dependencies between the inputs and the output. The introduced input selection algorithm is applied to one artificial data set and to the auto miles per gallon data set, see Frank and Asuncion [2010], to show the algorithms abilities.

Order of experimentation for metamodeling tasks.

The order in which measurements are carried out, determines the accuracy of models in early stages of the measurement process, i.e. while measurements are still in progress. Reliable models in early stages of the data acquisition phase allow for model-based investigations like optimization runs or an earlier switching to an active learning phase. This paper compares different methods to determine the order of experimentation for regression problems in metamodeling tasks. The data distribution and the data density in the input space are varied for several randomly generated synthetic functions in order to find the most promising determination strategy for the order of experimentation. As an application example, all strategies are also applied to a computational fluid dynamics (CFD) metamodel. The order of experimentation based on the intelligent k-means clustering algorithm turns out to be the best overall order-determination strategy.

Automated order determination strategies for nonlinear dynamic models

A crucial part during the generation of nonlinear dynamic models is the determination of an appropriate model order. Five automated order determination strategies are compared. One model-based and four model-free approaches are investigated. We evaluated the performances of all methods with four artificial test processes and two noise levels. In an external dynamics approach, local model networks are trained with the determined (lagged) inputs and outputs that are found through the automated order determination strategies. An independent noise-free data set reveals the simulation quality of the estimated models. Most of the filter methods are unreliable since their performance varies strongly. Most robust is the wrapper method, which achieves good results in general. We show that in some cases even the model yielded through the incorporation of prior-knowledge is outperformed by some of the models resulting from the presented order determination methods.

Proposal for a function generator and extrapolation analysis

We propose a polynomial-based function generator to support decision-making in the context of experimental modeling (identification). The function generator tries to imitate regression problems in engineering applications. Stochastic elements ensure high variability between generated functions, while the user is able to choose a general complexity level defined by the strength of the nonlinearity and the order of interactions. An extension to overcome unfavorable properties of the polynomial-based structure is made. The ability to generate an arbitrary amount of test functions offers the possibility to statistically secure decisions in the development of algorithms or for the modeling task at hand. To demonstrate the abilities of our proposed function generator, it is utilized to pick a strategy for the design of experiments that should be used for the metamodeling of a centrifugal fan. We show, that for the application at hand the inclusion of all corners in the experimental design is destructive for the meta models generalization performance.

Order of experimentation and advisability of corner measurements

Abstract Basically two questions are examined regarding the design of experiments. First, for what settings is the placement of design points in the corners of the input space advisable in order to achieve a good generalization performance. And second, what order of experimentation leads to the best model quality in early stages of the measurement process. The effect of the explicit addition of corner measurements to three space-filling experimental designs is investigated. It turns out that for most cases the incorporation of corner points harms the model quality. Two new methods to determine the order of experimentation are compared to one active learning strategy and a simple randomization of the measurement sequence. Both order determination methods prove to outperform the random sequences significantly and are only slightly inferior to the active learning strategy.