18 Modular Analysis of Complex Systems with Numerically Described Multidimensional Probability Distributions 173 Fig. 18.1 Objectives, components and tools of Darmstadt Risk Analysis Method DRAM [6] 18.4 Using Probability Values The advantages of using probability distributions to describe the values of parameters has already been addressed in the introduction. In the context of this paper it seems not necessary to deepen this argumentation. The concept is broadened by considering certain dependencies, what eventually leads to multidimensional probability distributions. 18.5 Numerical Described Multidimensional Probability Distributions (NDMPD) Numerical Described Multidimensional Probability Distributions (NDMPD) describe such multidimensional probability distributions by numerical values. In principle they are a systematic collection of probability values. An one-dimensional distribution can be described by a scale (names for nominal and ordinal scales; numbers as class limits for cardinal scales), which divides the value range into classes, and probability values for each class. If such a distribution describes a variable, the probability values indicate for each class the probability, that the value of the variable will lay in this class. Depending on the scale type, additional classes are added for very high, very low and undefined values. Multidimensional distributions have an additional dimension for each dependency, that shall be considered. A twodimensional NDMPD is effectively a set of one-dimensional distributions, one for each class of the dependency scale. Then, the probability values are not to be interpreted absolutely, but Bayes’ probabilities in relation to the dependency (pj dependency). This is continued for each dependency (Fig. 18.2). Thus, a five-dimensional NDMPD then consists of 5 scale descriptions and–with about 10 classes per scale/dimension–of 105 D100,000 (related) probability values. That sounds much, but is no problem for actual computers. To the contrary: the systematic structure of NDMPD allows the treatment of complex NDMPD by machine, even if many NDMPD are connected by dependencies. The method is designed in that way, that the algorithm recognizes, if two NDMPD are referenced to the same scale and considers the entity with this scale as mutual dependency of the two variables described by the NDMPD, and considers it in the calculation process.
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