Hans Bandemer

Fuzzy data analysis

1. Introduction. 2. Basic Notions. 3. Basic Notions of Data Analysis. 4. Fuzzy Data. 5. Qualitative Analysis. 6. Quantitative Analysis. 7. Evaluation of Methods. Bibliography. Subject Index. List of Symbols.

Evaluating explicit functional relationships from fuzzy observations

Starting from ideas due to Zadeh and Sugeno we suggest two methods to transfer information given by fuzzy observations to fuzzy sets on the parameter region of a given explicit functional relationship: expected cardinality and fuzzy expectation. Moreover we mention their use in soft modelling when several functional relationships are to be compared for the same fuzzy observations. The procedures are illustrated by an example.

Bayesian fuzzy kriging

The paper contains a combination of two approaches generalising the usual kriging technique for prediction in fields: the Bayesian approach incorporating prior knowledge on the field and the fuzzy set approach reflecting uncertainty w.r.t. observation impreciseness and specification vagueness. The presentation includes a numerical example.

Once more: optimal experimental design for regression models (with discussion)

The paper reconsiders some of the recent developments in experimental design for linear regression models. Atfirst, some attention is paid to a discussion of the advantages and limitations of a decision–theoretically oriented approach to the compound problem including the choice of set–up, estimator and experimental design, with special emphasis on the use of prior knowledge and on robustness inverstigations. In the main part of the paraper we review and discuss our results obtained since 1979 concerning BAYESian experimental design and experimental design in case of correalted observations. As a main feaurer we can point out an analogy between continuous experimental designing and linear regression estimation which opens the possibility to use experimental designs methods for the ocnstructions of optimal lineal estimators. This is demosntrated, in praticular, for the fields of minimaz linear estiamtion with a restircted parameter space and best linear unbiased estimation of the trend functions of random ...

On a fuzzy-theory-based computer-aided particle shape description

Abstract A central problem of particle shape analysis consists in the characterization of the shape of any natural of produced particle by a small number of parameters, which are to be evaluated from a two-dimensional picture of the particle, e.g. from its projection. We suggest using the grey-tone picture obtained by a camera to compute a sympathy function with respect to a suitable family of planar shapes. The computation is performed by means of the image processing equipment ROBOTRON A 6471. The sympathy function defines a fuzzy set on the parameter space. The procedure is applied to a quartz particle.

A fuzzy approach to stability of fuzzy controllers

This paper deals with stability analysis of fuzzy controllers. A new definition embedding the notion of stability into fuzzy set theory is suggested. In this context the introduction of a gradual stability is discussed. Investigating stability needs a model of the process. One of the essential assumptions for the model used here is that the process has an approximatively linear structure within a sufficiently small time interval Δt. Possible deviations between the model and the real system are taken into account by specifying fuzzy parameters. For the sake of simplicity, however, only a simple version of the model is considered in the present paper. Moreover, some facts about the influence of the defuzzifier and the inference engine on the stability statement are presented. By using this model and the suggested definition of stability some criteria ensuring stability are derived.

Fuzzy projection pursuits

Abstract A well-known method in data analysis to recognize patterns in multivariate point-shaped data consists in a series of projections into, usually two-dimensional, linear subspaces; this procedure is called projection pursuit. Possibly, it can be controlled by some statistically based functional of interestingness. The present paper gives a fuzzy version of this procedure, assuming fuzzy data to be handled and projections performed to rules from fuzzy theory, perhaps controlled by some measure of fuzziness.

Evaluating a functional relationship from picture data of a chemical diffusion process—a case study

Abstract Experiments on the behaviour of solid materials with respect to the formation of new phases caused by chemical diffusion result in pictures showing the (random) penetration in a certain moment. Since the (surely rather complicated) law of probability of this process is unknown, the usually applied elementary methods from mathematical statistics become pure heuristics. Interpreting the phase (penetration) depth as a fuzzy variable allows a more realistic representation of the experimental results and the evaluation of a fuzzy functional relationship between phase depth and time, from which, e.g., useful conclusions are possible with respect to reasons and time laws of phase formation. The paper contains a real-world example.

Optimum Experimental Design for a Bayes Estimator in Linear Regression

We consider the problem of optimum experimental design for a Bayes estimator of the response surface in a linear regression model. We give a characterization of optimal designs which minimize the Bayes risk of this estimator. On the basis of this characterization we obtain a sufficient condition on the information matrix of an optimal design and a condition for the existence of optimal one-point designs. These conditions are then used to determine optimal designs in a simple linear regression model.

A special measure of uncertainty

Abstract A new measure of uncertainty is presented, which evaluates the ‘hardness’ of decision if there are two or more competing decisions. The case of ‘dead heat’ is starting point for the construction of this measure, which is then generalized to the case of fuzzy decisions and arbitrary universe.