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By Kurt Marti, Yuri Ermoliev, Marek Makowski

Support for addressing the on-going international adjustments wishes options for brand spanking new clinical difficulties which in flip require new strategies and instruments. A key factor issues an enormous number of irreducible uncertainties, together with severe occasions of excessive multidimensional results, e.g., the weather swap. The quandary is worried with huge, immense bills as opposed to tremendous uncertainties of maximum affects. conventional medical methods depend upon actual observations and experiments. but no adequate observations exist for brand spanking new difficulties, and "pure" experiments, and studying by way of doing should be pricey, harmful, or very unlikely. additionally, the to be had historic observations are usually infected through prior activities, and guidelines. therefore, instruments are provided for the specific remedy of uncertainties utilizing "synthetic" details composed of accessible "hard" information from old observations, the result of attainable experiments, and medical evidence, in addition to "soft" facts from specialists' critiques, and scenarios.

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21) From these equations we can also see that NSD D 1 if the whole support of xQ k is located to the left of yk . In a more general case, when the emission in the base year is given as a fuzzy number (due to the fuzziness of the emission factors), the commitment may be verified by the calculation of the NSD index for the relation yQk xQ k . 20), but the value of ˛? is now calculated from the following expression: ˛? ˛? D yk;L : ˛? 22) In the most general case we may want to know in advance, after the evaluations of emissions in years numbered from 0 (base year) to n have been observed, if the commitment in the year k is likely to be fulfilled.

2 On Joint Modelling of Random Uncertainty and Fuzzy Imprecision 17 It has to be stressed, however, that once the probability distributions describing uncertain phenomena have been defined, the mechanisms of the theory of probability are sufficient for further analysis of uncertain phenomena. For this reason probabilistic methods, such as Bayesian methods, are generally used in practice. Basic principles of the application of the Bayesian methods are described in classical textbooks by Raiffa and Schleifer [69] and De Groot [11].

For example, statistical methods for testing fuzzy hypotheses have been considered in the papers by Saade and Schwarzlander [73], Saade [72], Watanabe and Imaizumi [90], Arnold [1, 2] Taheri and Behboodian [80], and Grzegorzewski and Hryniewicz [39]. When the data are also fuzzy, interesting solutions have been proposed in the papers by Arnold [1], Casals et al. [6], Kruse and Meyer [53], Saade [72], Saade and Schwarzlander [73], Son et al. [79], Watanabe and Imaizumi [90], R¨omer and Kandel [70], and Montenegro et al.

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