THE THEORY OF COMPLEX PROBABILITY AND THE FIRST ORDER RELIABILITY METHOD

The Kolmogorov’s system of axioms can be extended t o 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 ev aluate 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 variab le in R is, the corresponding probability in the wh ole set C is always one, so the outcome of the random exper iment in C can be predicted totally. The result ind icates that chance and luck in R is replaced now by total determinism in C. This new complex probability mode l will be applied to the concepts of degradation and the Remaining Useful Lifetime (RUL), thus to the fi ld of prognostic based on reliability. Therefore, an exam ple of Young modulus will be applied and the First Order Reliability Method (FORM) analysis will be us ed for this purpose.

In engineering systems, the remaining useful lifetime prediction is related deeply to many factors that generally have a chaotic behavior which decreases the degree of our knowledge of the system.
As the Degree of Our Knowledge (DOK for short) in the real universe R is unfortunately incomplete, the extension to the complex universe C includes the contributions of both the real universe R and the imaginary universe M. Consequently, this will result in a complete and perfect degree of knowledge in C = R+M (Pc = 1). In fact, 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 (Pc 2 = DOK-Chf = 1). Thus, the study in the complex universe results in replacing the phenomena that used to be random in R by deterministic and totally predictable ones in C.
This hypothesis is verified in a previous study and paper by the mean of many examples encompassing both discrete and continuous distributions.
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) then 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. As a matter of fact, prognostic analysis consists in the prediction of the remaining useful lifetime of a system at any instant t 0 and during the system functioning. Let us consider a degradation trajectory D(t) of a system where a specific instant t 0 is studied. The instant t 0 means here the time or age that can be measured also by the cycle number N.
Referring to the figure above (Fig. 2), the previous statement means that at the system age t 0 , the prognostic study must give the prediction of the failure instant t N . Therefore, the RUL predicted here at instant t 0 is the following interval: RUL(t 0 ) = t N -t 0 .
In fact, at the beginning (t 0 = 0) (point J), the failure probability P r = 0 and the chaotic factor in our prediction is zero (Chf = 0). Therefore, RUL(t 0 = 0) = t N -t 0 = t N .
If t 0 = t N (point L) then the RUL (t N ) = t N -t N = 0 and the failure probability is one (P r = 1).
If not (i.e., 0<t 0 <t N ) (point K), the probability of the occurrence of this instant and the prediction probability of RUL are both less than one (not certain) due to nonzero chaotic factors. The degree of our knowledge (DOK for short) is consequently less than 1. Thus, by applying here the EKA method, we can determine the system RUL with certainty in C = R + M where Pc = 1 always.
Furthermore, we need in our current study the absolute value of the chaotic factor that will give us the magnitude of the chaotic and random effects on the Science Publications JMSS studied system. This new term will be denoted accordingly MChf or Magnitude of the Chaotic Factor. Hence, we can deduce the following: Moreover, we can define two complementary events E and E with their respective probabilities: rob rob P (E) p and P (E) q 1 p = = = − Then P rob (E) in terms of the instant t 0 is given by: P rob (E) = P r = P rob (t≤t 0 ) = F(t 0 ) where F is the cumulative probability distribution function of the random variable t. Since Let us define the two particular instants: t 0 = 0 assumed as the initial time of functioning (raw state) corresponding to D = D 0 = 0 and t N = the failure instant (wear out state) corresponding to the degradation D = 1.
The boundary conditions are. For t 0 = 0 then D = D 0 (initial damage that may be zero or not) and: 0 rob For t 0 = t N then D = 1 and F(t 0 ) = F(t N ) = P rob (t≤t N ) = 1. Also F(t 0 ) is a non-decreasing function that varies between 0 and 1. In fact, F(t 0 ) is a cumulative function (Fig. 3). In addition, since RUL(t 0 ) = t N -t 0 and 0≤t 0 ≤t N then RUL(t 0 ) is a non-increasing remaining useful lifetime function (Fig. 4).
The square of the norm of Z(t 0 ) is: The Chaotic Factor and the Magnitude of the Chaotic Factor are: Chf(t 0 ) = -2P r (t 0 )[1-P r (t 0 )] = -2P r (t 0 )+2P r 2 (t 0 ) is null when P r (t 0 ) = P r (0) = 0 (point J) or when P r (t 0 ) = P r (t N ) = 1 (point L) and MChf(t 0 ) = |Chf(t 0 )| = 2P r (t 0 )[1-P r (t 0 )] = 2P r (t 0 ) -2P r 2 (t 0 ) is null when P r (t 0 ) = P r (0) = 0 (point J) or when P r (t 0 ) = P r (t N ) = 1 (point L) At any instant t 0 (point K), the probability expressed in the complex set C is: Pc(t 0 ) = P r (t 0 ) + P m (t 0 )/i = P r (t 0 ) + [1-P r (t 0 )] = 1 always Hence, the prediction of RUL(t 0 ) of the system degradation in C is permanently certain. The reliability is the probabilistic evaluation of a limit state of performance on a domain of basic variables. In other words, it is obtained by the computation of the failure probability toward a criterion or a limit state.

Methodology
• Identify the limit states that govern the lifetime of the structure • Identify the basic parameters intervening in the limit state • Deduce their probability density functions • Compute the failure probability that expresses the risk when the limit states are not satisfied The approximate method FORM is an iterative procedure that allows to calculate an index of reliability (denoted β).
The index β is the distance between the origin and the limit state function G(t) in a standard space. Once we have calculated β we can deduce the failure probability P r = Φ(-β).
In FORM approximation the real (usually nonlinear) limit state is replaced by its tangent plane at a specific point called the Most Probable Failure Point (MPFP). This point is the closest point on G(t) to the origin.
The limit state G(t) divides the space into two regions: • First region where G(t) > 0 called safe region • And the second region where G(t) ≤ 0 called failure region

Work Plan
We choose, in a general case, N random variables, correlated and of any density functions, as well as a nonlinear limit state function. This method is based on the following iterative algorithm: • Transforming of basic random variables into standard normal variables N(0,1) • Transforming the limit state from the original space to the standard normal space • Search of the MPFP point by replacing the limit state surface by its tangent hyper-plane at the same point • Calculate the index β and the probability of failure P f

Description of the Algorithm
The transformation from the basic state to the normalized state is implicit in the algorithm. The steps are the following (Fig. 6): Let the limit state equation be: g (z) where, z = z1,z2,z3,…..zn is the random vector of the limit state, therefore.

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The gradient at MPFP is assumed to be: The equivalent normal standard deviation and mean value of non-normal variables: 3) Calculate the intermediate parameters:

Application of FORM to Prognostic
In this part, we study the extended Kolmogorov axioms in the context of reliability by defining a limit state G that describes the lifetime margin of the system. For each value of an instant t 0 we determine its corresponding probability of survival or of the Remaining Useful Lifetime (RUL).
We have: Where: G(t 0 ) = The limit state of lifetime. t N = The fixed lifetime of the system which follows a normal distribution N(0.0006 N t ; 1) t 0 = An arbitrary instant that varies from 0 to t N and which follows a normal distribution When RUL (t 0 ) is zero or negative then we have a case of t 0 ≥ t N that means that we have a system failure that cannot live until the instant t 0 . In the other case where t 0 < t N , the system can live above the instant t 0 and we have a case of success.
The probability: is computed by the FORM (First Order Reliability Method) procedure that uses a reliability index β. β = -Φ -1 [P r (t 0 )] where P r (t 0 ) is the cumulative probability and Φ is the normal cumulative distribution function. Hence, Φ -1 is the inverse of Φ and P r (t 0 ) = Φ (-β).
In the extended Kolmogorov's axioms, the real part of probability is taken here P r (t 0 ). As we make the instant t 0 vary between 0 and t N , then P r (t 0 ) varies between 0 and 1 (Fig. 7).
Knowing that we take t 0 and t N as two normal random variables where the value of t N corresponds to nearly 5798 number of cycles (critical value: N c ). After a reliability calculation using a Matlab program, we deduce a value of P r (t 0 ) for each value of instant t 0 . For this set of P r (t 0 ) we have computed and ploted the extended Kolmogorov's parameters and components Chf(t 0 ), MChf(t 0 ), DOK(t 0 ), Pc(t 0 ), P m (t 0 )/i. Therefore, we get the following figures ( Fig. 8 and 9).

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We note from the figure that the DOK is maximum (DOK = 1) when absolute value of Chf which is MChf is minimum (MChf = 0) (points J & L), that means when the magnitude of the chaotic factor (MChf) decreases our certain knowledge (DOK) increases. Afterward, MChf starts to increase during the functioning due to the environment and intrinsic conditions thus leading to a decrease in DOK until they both reach 0.5 at t 0 = 1500 (point K). The real probability P r and the complementary probability P m /i will intersect with DOK also at the point (1500, 0.5) (point K). With the increase of t 0 , the Chf and MChf return to zero and the DOK returns to 1 where we reach total damage (D = 1) and hence the total certain failure (P r = 1) of the system (point L). At this last point the failure here is definite, P r (t N ) = 1 and RUL(t N ) = t N -t N = 0 with Pc(t N ) = 1, so the logical explanation of the value DOK = 1 follows.
We note that the point K is not at the middle of DOK since the probability of failure distribution evaluated by FORM is not symmetric.
Furthermore, at each instant t 0 , the remaining useful lifetime RUL(t 0 ) is certainly predicted in the complex set C with Pc maintained as equal to one through continuous compensation between DOK and Chf. This compensation is from instant t 0 = 0 where D(t 0 ) = 0 until the failure instant t N where D(t N ) = 1.

The Cube of Probability Components
In the following figure, we represent the extended Kolmogorov's probability components P r and P m /i in a three dimensional graph in terms of t and of each other (Fig. 10).
It is important to mention that if we rescale the time axis to an interval [0,1] so the minimal value of DOK is where N = 5798 cycles corresponds to t N. This last important point is clearly shown in this cube and in the following one. From the cube below, we can notice that the probability Pc in the complex set C = R + M is obtained at each instant t 0 as the sum of P r and P m /i and is always equal to one.

The Cube of Probability Parameters
In the following figure, we represent the extended Kolmogorov's probability parameters DOK and Chf in a three dimensional graph in terms of t and of each other (Fig. 11).

Example: Application to Young Modulus
We consider once again the Young modulus example previously treated in the first paper on extended kolmogorov's axioms and "Complex Probability Theory".
Let E be the Young modulus in a material bar domain (Fig. 12) and we assume that it follows a Normal Gaussian distribution.
The limit state considered here for FORM analysis is: When RUL (E 0 ) is zero or negative then we have the case of E 0 ≥ E N that means that we have a system failure that cannot live until E 0 . In the other case where E 0 <E N , the system can live above E 0 and we have a case of success.
The real Probability of failure is given by: The reliability index β = -Φ -1 (P r ).
In the extended Kolmogorov's axioms, the real part of probability is taken here as P r (E 0 ). As we make the E 0 vary between 0 and E N , then P r (E 0 ) varies between 0 and 1 (Fig. 13).
Let E be the mean value of E and is taken to be equal to 29575 Ksi. Let σ E be the standard deviation of E and is equal to 1507 Ksi. Let In the real domain R we have:

∫ ∫
We have also: and the chaotic factor is: Moreover, the Magnitude of the chaotic factor is: Therefore, we say that: and if Chf 0 = 0⇒ |Z 0 | 2 = 1 in other words, if the chaotic factor is zero, then the degree of our knowledge is 1 or 100%.
In addition, we say that: 2 0 Pc = Degree of our knowledge + Magnitude of the Chaotic factor = 1. and if MChf 0 = 0⇒ |Z 0 | 2 = 1, in other words, if the magnitude of the chaotic factor is zero, then the degree of our knowledge is 1 or 100%.
Numerically, we write: Consequently, we can say that. The degree of our knowledge DOK= |Z 0 | 2 = 0.5, the chaotic factor Chf 0 =-0.5 and the magnitude of the chaotic factor MChf 0 = 0.5.
What is interesting here is thus we have quantified both the degree of our knowledge and the chaotic factor of the event as well as the correspondent magnitude of the chaotic factor.
Notice that: If E 0 increases to become=4000 then both |Z 0 | 2 and Chf 0 increase and MChf 0 decreases.
Therefore we can infer that: ( ) for every E 0 in the real set R. We note from the figure below that the DOK is maximum (DOK = 1) when absolute value of Chf which is MChf is minimum (MChf = 0) (points J & L), that means when the magnitude of the chaotic factor (MChf) decreases, our certain knowledge (DOK) increases. Afterward, MChf starts to increase during the functioning due to the environment and intrinsic conditions thus leading to a decrease in DOK until they both reach 0.5 at E 0 = 29575 (point K). The real probability P r and the complementary probability P m /i will intersect with DOK also at the point (29575, 0.5) (point K). With the increase of E 0 , the Chf and MChf return to zero and the DOK returns to 1 where we reach total damage (D = 1) and hence the total certain failure (P r = 1) of the system (point L). At this last point the failure here is definite, P r (E N ) = 1 and RUL(E N ) = E N -E N = 0 with Pc(E N ) = 1, so the logical explanation of the value DOK = 1 follows ( Fig. 14 and 15).
We note that the point K is not at the middle of DOK since the probability of failure distribution evaluated by FORM is not symmetric.

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Furthermore, at each E 0 , the remaining useful lifetime RUL(E 0 ) is certainly predicted in the complex set C with Pc maintained as equal to one through a continuous compensation between DOK and Chf. This compensation is from E 0 = 0 where D(E 0 ) = 0 until failure at E N where D(E N ) = 1.

The Cube of Probability Components
In the figure above, we represent the extended Kolmogorov's probability components P r and P m /i in a three dimensional graph in terms of E and of each other (Fig. 16).
It knowing that E = 114317 corresponds to E N. This last important point is clearly shown in this cube and in the following one. From the cube above, we can notice that the probability Pc in complex set C = R + M is obtained at each value E 0 as the sum of P r and P m /i and is always equal to one.

The Cube of Probability Parameters
In the figure above, we represent the extended Kolmogorov's probability parameters DOK and Chf in a three dimensional graph in terms of E and of each other (Fig. 17).

CONCLUSION
In this study I applied the theory of Extended Kolmogorov Axioms to Prognostic based on Reliability. I used for this purpose the very well known First Order Reliability Method or FORM analysis for short. Consequently, I established a tight link between the new theory and degradation or the remaining useful lifetime and reliability. Hence, I developed the theory of "Complex Probability" beyond the scope of the previous three papers on this topic. As it was proved and illustrated, when the degradation index is 0 or 1 and correspondingly the RUL is t N or 0 then the Degree of Our Knowledge (DOK) is one and the chaotic factor (Chf and MChf) is 0 since the state of the system is totally known. During the process of degradation (0<D<1) we have: 0.5<DOK <1, -0.5<Chf <0 and 0<MChf <0.5. Notice that during the whole process of degradation we have Pc = DOK -Chf = DOK + MChf = 1, that means that the phenomenon which seems to be random and stochastic in R is now deterministic and certain in C = R + M and this after adding to R the contributions of M and hence after subtracting the chaotic factor from the degree of our knowledge. Moreover, for each value of an instant t 0 or E 0 , I have determined their corresponding probability of survival or of the remaining useful lifetime RUL(t 0 ) = t N -t 0 or RUL(E 0 ) = E N -E 0 . In other words, at each instant t 0 or E 0 , RUL(t 0 ) or RUL(E 0 ) are certainly predicted in the complex set C with Pc maintained as equal to one through a continuous compensation between DOK and Chf. This compensation is from instant t 0 = 0 where D(t 0 ) = 0 until the failure instant t N where D(t N )=1 and this compensation is also from E 0 = 0 where D(E 0 ) = 0 until failure at E N where D(E N )=1. Furthermore, using all these graphs illustrated throughout the whole paper, we can visualize and quantify both the system chaos (Chf and MChf) and the system certain knowledge (DOK and Pc). Additionally, an application to Young modulus was successfully done here. This is certainly very interesting and fruitful and shows once again the benefits of extending Kolmogorov's axioms and thus the originality and usefulness of this new field in mathematics that can be called verily: "The Complex Probability and Statistics Paradigm".