By Hans Bandemer
"Mathematics of Uncertainty" offers the fundamental rules and foundations of uncertainty, protecting the fields of arithmetic during which uncertainty, variability, imprecision and fuzziness of knowledge are of value. This introductory e-book describes the elemental rules of the mathematical fields of uncertainty from basic interpolation to wavelets, from blunders propagation to fuzzy units and neural networks. The booklet offers the therapy of difficulties of interpolation and approximation, in addition to statement fuzziness which may basically effect the preciseness and reliability of statements on sensible relationships. The notions of randomness and chance are tested as a version for the range of commentary and dimension effects. along with those uncomplicated rules the ebook additionally offers equipment of qualitative info research resembling cluster research and category, and of overview of sensible relationships reminiscent of regression research and quantitative fuzzy information research.
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Extra resources for Handling Uncertainty by Mathematics
Between neighbouring data there are neither maxima nor minima. As the aim of the task an approximate prediction of the functional values between neighbouring data is requested. Moreover, in this subsection it is assumed that the distances between the data points are small, and, for the next consideration, that the function is diﬀerentiable. First, the problem of interpolation is considered. Because of the preceding assumptions for a function in question, which maps the (one-dimensional) x-axis into the (one-dimensional) y-axis, one can use the well-known Taylor-formula f (x) = f (x0 ) + (x − x0 )f (ξ); ξ ∈ [x0 , x] .
The problem of impreciseness, however, is of great importance for the choice of a meaningful numerical preciseness for the procedure of the method of ﬁnite elements. Mostly things are overdone here and a precision is demanded, which suggests a real precision of the solution for the problem at hand proving totally virtual, if the impreciseness of the model and of the initial and boundary values is taken into account. A well known method for an approximate representation of a graphically given periodical function is its expansion in a Fourier series .
On the one hand the intervals for the input values should be speciﬁed as precisely as possible, which can be eﬀected in practical cases only with some arbitrariness. Hence application of interval mathematics is recommended, as a rule, if the speciﬁed intervals are small when compared with the variability of the quantities considered, hoping that in such cases minor misspeciﬁcations remain without perceptible consequences. On the other hand interval mathematics demands for a practically inﬁnitely precise performance of the chosen procedure, in order to include the changings of the intervals always towards the “safe” side.