ONE-DIMENSIONAL ANALYTICAL GEOMETRY OF MULTIDIMENSIONAL SPACES
Journal Title: Наукові праці. Серія "Комп’ютерні технології" - Year 2017, Vol 307, Issue 295
Abstract
Analytical geometry allows to visualize complex functions of the original estimation of the decision-making system F(Z), where Z = (z1, z2, ..., ze ..., zE) is a vector of input variables. The value of F(Z) can be determined by the algorithm or its function F(Z) can be expanded in a series of elementary nonlinear or linear functions of the reduced estimates e e Å å Å å å å F Z f w m z 1 1 ( ) , where e = 1, 2, ..., E are the sequence numbers of the input variables ze and functions fе = wетеzе; we = 0 ... 1 - the weighted coefficients of influence of the value of ze on the output function F(Z) calculated or determined by the expert under the condition E e e w 1 1 (weights wе are visually displayed in the multidimensional input space by the width along the abscissa with and their numerical value along the ordinate axis); mе = 0 ... 1 is the coefficient of the expert’s accounting for deficiencies (penalties), which reduce the received value of ze. Then each variable ze and each function fе acquires the same metric with the output function F(Z), and the multidimensional minimized estimate of the intellectual system becomes «one-dimensional». With the help of calculations and expert estimates, the variables ze can be determined, refined and geometrically visualized on the input axes, and the output function F(Z) can be reflected on one output coordinate axis – in one-dimensional space with the output metric F(Z). The developed theory is a separate direction of analysis in Parallel Coordinates, which was first applied by F. M. d’Ocagn. Due to the one-dimensionality of the inputs and the output of the analytic geometry under consideration, only the simplest geometric objects can be visualized on the coordinate axes: a point; positive, negative and positive-negative segments; the sum of the obtained segments. Although it is impossible to construct complex geometric shapes of n-dimensional space on one initial axis, at the same time, the described onedimensional analytical geometry covers modeling, solving and visualization of a large volume of multidimensional problems, the analysis of which requires practice.
Authors and Affiliations
V. Yа. Kutkovetskyi
Keywords
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