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Statistical Methods for Fuzzy Data by Reinhard Viertl

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Preface

Statistics is concerned with the analysis of data and estimation of probability distribution and stochastic models. Therefore the quantitative description of data is essential for statistics.

In standard statistics data are assumed to be numbers, vectors or classical functions. But in applications real data are frequently not precise numbers or vectors, but often more or less imprecise, also called fuzzy. It is important to note that this kind of uncertainty is different from errors; it is the imprecision of individual observations or measurements.

Whereas counting data can be precise, possibly biased by errors, measurement data of continuous quantities like length, time, volume, concentrations of poisons, amounts of chemicals released to the environment and others, are always not precise real numbers but connected with imprecision.

In measurement analysis usually statistical models are used to describe data uncertainty. But statistical models are describing variability and not the imprecision of individual measurement results. Therefore other models are necessary to quantify the imprecision of measurement results.

For a special kind of data, e.g. data from digital instruments, interval arithmetic can be used to describe the propagation of data imprecision in statistical inference. But there are data of a more general form than intervals, e.g. data obtained from analog instruments or data from oscillographs, or graphical data like color intensity pictures. Therefore it ...

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