Abstract
The Kolmogorovs system of axioms can be extended to encompass the imaginary set of numbers and this by adding to the original five axioms an additional three axioms. Hence, any experiment can thus be executed in what is now the complex set C (Real set R with real probability + Imaginary set M with imaginary probability). The objective here is to evaluate the complex probabilities by considering supplementary new imaginary dimensions to the event occurring in the real laboratory. Whatever the probability distribution of the input random variable in R is, the corresponding probability in the whole set C is always one, so the outcome of the random experiment in C can be predicted totally. The result indicates that chance and luck in R is replaced now by total determinism in C. This new complex probability model will be applied to the concepts of degradation and the Remaining Useful Lifetime (RUL), thus to the field of prognostic based on reliability. Therefore, an example of Young modulus will be applied and the First Order Reliability Method (FORM) analysis will be used for this purpose.
Highlights
In order to have a certain prediction of any event it is necessary to work in the complex universe C in which the chaotic factor is quantified and subtracted from the Degree of Our Knowledge to lead to a probability in C equal to one (Pc2 = decreases our certain knowledge (DOK)-Chf = 1)
I used for this purpose the very well known First Order Reliability Method or FORM analysis for short
I established a tight link between the new theory and degradation or the remaining useful lifetime and reliability
Summary
Abou Jaoude et al (2010); Abou Jaoude (2013a; 2013b; 2005; 2007; 2012); Bell (1992); Benton (1996); Boursin (1986); Chan Man Fong et al (1997); Cheney and Kincaid (2004); Dacunha-Castelle (1996); Dalmedico Dahan et al (1992); Dalmedico Dahan and Peiffer (1986); Ekeland (1991); Feller (1968); Finney et al (2004); Gentle (2003); Gerald and Wheatley (1999); Gleick (1997) and Greene (2000) firstly, the Extended Kolmogorov’s Axioms (EKA for short) paradigm can be illustrated by the following figure (Fig. 1). From the Extended Kolmogorov’s Axioms (EKA), we can deduce that if we add to an event probability in the real set R the imaginary part M (like the lifetime variables) we can predict the exact probability of the remaining lifetime with certainty in C (Pc = 1). We can apply this idea to prognostic analysis through the degradation evolution of a system. It is obtained by the computation of the failure probability toward a criterion or a limit state
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