Mathematics of Uncertainty: Ideas, Methods, Application by Hans-Walter Bandemer, H. Bandemer,

By Hans-Walter Bandemer, H. Bandemer,

"Mathematics of Uncertainty" presents the elemental principles and foundations of uncertainty, protecting the fields of arithmetic within which uncertainty, variability, imprecision and fuzziness of information are of value. This introductory booklet describes the elemental principles of the mathematical fields of uncertainty from basic interpolation to wavelets, from blunders propagation to fuzzy units and neural networks. The publication offers the therapy of difficulties of interpolation and approximation, in addition to commentary fuzziness that can primarily impact the preciseness and reliability of statements on useful relationships. The notions of randomness and chance are tested as a version for the range of remark and size effects. along with those simple rules the ebook additionally provides tools of qualitative info research akin to cluster research and class, and of overview of practical relationships reminiscent of regression research and quantitative fuzzy facts research.

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This approach was improved extensively and in different directions. The function need no longer be periodical (the period is “infinitely long”). For this case (orthogonal) systems of functions are developed, in which each single function is different from zero only on a finite interval, which decreases in length with increasing index of the function. This approach was generalized in different respects. For a comprehension of this direction of development the consideration of a suggestion by Haas (1910) will suffice, called already classical today.

The corresponding values of the first and second derivatives of the neighbouring curves should be equal, yields the desired unique solution of the system. This spline technique can be extended to arbitrary many intervals, for which procedure the continuity of the crossing at all the single data points is demanded (if using cubical parabolae up to the second derivative). This technique reminds of pliable curve templates, by which smooth curves can be drawn through a few single points. This task is now taken over by an appropriate software, which presents the result on the screen.

1 Simple Error Propagation An approach, which can be already called classical, for inclusion of impreciseness caused by measurement or observation is the consideration of error propagation. Here the starting point is the information situation that the imprecision of certain argument variables xj can be specified by intervals, in which the “true” values of the corresponding variables are situated with certainty. The generalization to multidimensional closed (convex) domains for several argument variables simultaneously makes sense theoretically, but will not be treated here.

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