The invariance of the peak point(s) in a non-symmetrical graph via CETD matrix under varying α-levels

Abstract

Events or attributes occur at different ages or times but, in some circumstances, for effective planning and policy formulation, the peak point, where the events or attributes has its peak value, is of interest. Usually, the graphs depicting peak values are not symmetrical. In determining the peak point(s) of events that occur over time, a set of $ \alpha_s $ of $ \alpha $-levels, chosen from an antisymmetric interval $ (0, 1] $, was used on an ATD matrix. This was done to obtain an RTD matrix which was then aggregated to obtain a CETD matrix. Most authors chose $ \alpha $ without any condition. The problem associated with this was that two different sets of $ \alpha $ may not necessarily produce the same peak point for the same data set. In this study, the condition to guarantee that the row which had the highest sum (the peak value) in a CETD matrix was invariant, regardless of the set of $ \alpha $-levels, was established. To establish the authenticity of this method, there were experiments conducted and numerical examples were given in this paper.

Keywords

• peak value
• average time dependent matrix
• revised time dependent matrix
• combined effect time dependent matrix
• fuzzy matrix