Abstract
The Kolmogorov’s system of axioms can be extended to encompass the imaginary set of numbers and this b y adding to the original five axioms an additional th ree 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 distrib ution of the input random variable in R is, the correspondin g 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 l uck in R is replaced now by total determinism in C. This new complex probability model will be applied to the c oncepts of degradation and the Remaining Useful Lifetime (RUL), thus to the field of prognostic.
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 Knowledge to lead to a probability in C equal to one (Pc2 = Degree of Our Knowledge (DOK)-Chf = 1)
The Degree of Our Knowledge DOK: DOK is the measure of our certain knowledge (100% probability) about the expected event, it does not include any uncertain knowledge:
I established a tight link between the new theory and degradation or the remaining useful lifetime
Summary
Abou Jaoude et al (2010); Abou Jaoude (2013; 2005; 2007); Bell (1992); Benton (1996); Boursin (1986); Chen et al (1997); Cheney and Kincaid (2004); Dacunha-Castelle (1999); Dalmédico 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); Greene (2000; 2004) 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. Prob (E) in terms of the instant t0 is given by: Prob (E) = Pr = Prob (t≤t0) = F(t0) where F is the cumulative probability distribution function of the random variable t.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
