COMPLEX PROBABILITY THEORY AND PROGNOSTIC

The Kolmogorov’s system of axioms can be extended t o 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 execu ted 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 predic ted 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 on epts of degradation and the Remaining Useful Lifetime (R UL), thus to the field of prognostic.

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 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 below (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 quantity: 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 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 studied system. This new term will be denoted accordingly MChf or Magnitude of the Chaotic Factor. Hence, we can deduce the following:

JMSS
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.
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 F (t 0 ) = P rob (t≤0) = 0.
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).
Hence, the prediction of RUL(t 0 ) of the system degradation in C is permanently certain.
Let us consider thereafter many probability distributions to model the function F(t 0 ). With a probability density function:

APPLICATION TO DIFFERENT
and a cumulative distribution function: With the two boundaries a = 0 and b = t N then: We have taken the domain for the uniform variable t 0 = [0, t N = 1000] and dt 0 = 0.1 then: if 0 t 1000 1000 = ≤ ≤ 2.1.1. The Real Probability P r : We note that P r (t 0 ) is a non-decreasing function.

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We note that P m (t 0 )/i is a non-increasing function.

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 (with probability less than 100%): Which is a parabola concave upward having a vertex (a minimum) at: Which is a parabola concave upward having a vertex (a minimum) at: 3 N 0 t t 500 0.5 10 , 0.5 2 Therefore, we can infer the magnitude of the chaotic factor MChf: Which is a parabola concave downward having a vertex (a maximum) at: 3 N 0 t t 500 0.5 10 , 0.5 2

Pc: The Probability in the Complex Set C:
Thus we deduce that in the set C, we have a complete knowledge of the random variable since Pc = 1.

The EKA Parameters Analysis in the Prognostic of Degradation:
We note from the figure below that the DOK is maximum (DOK = 1) when MChf is minimum (MChf = 0) (points J & L) and that means when the magnitude of the chaotic factor (MChf) decreases our certain knowledge increases.
At the beginning P r (t 0 )= t 0 /t N = 0/t N = 0, the system is intact (zero damage: D = 0) and has zero chaotic factor before any usage, at this instant DOK(0) = 1 and RUL(0) = t N -0 = t N with Pc(0) = 1. Afterward, 0<t 0 <t N , RUL(t 0 ) = t N -t 0 with P r (t 0 ) = t 0 /t N ≠ 0 and Pc(t 0 ) = 1 and 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 = t N /2 = 500 (point K) where RUL(t N /2) = t N /2 = 500. Since the real probability P r is a uniform distribution MChf will intersect with DOK at the point (t N /2 = 500, 0.5) (point K).
With the increase of the time of functioning, MChf returns to zero and the DOK returns to 1 where we reach total damage (D = 1) and hence the total failure of the system (point L). At this last point the failure here is certain, P r (t N ) = t N /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 (Fig. 10).  We note that the same logic and analysis apply concerning the degradation and the remaining useful lifetime for the all the six probability distributions.
and a cumulative distribution function: We have taken the domain for the logarithmic variable t 0 = [0, t N = 1000] where dt 0 = 0.1.

The Real Probability P r :
We note that P r is a non-decreasing function.

The Complementary Probability P m /i:
if 0 t t 1000 We note that P m /i is a non-increasing function.

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 (with probability less than 100%):

The Chaotic Factor Chf and MChf:
( which is a curve concave upward having a minimum at:

JMSS
Therefore, we can infer the magnitude of the chaotic factor MChf: Which is a curve concave downward having a maximum at: Thus we deduce that in the set C, we have a complete knowledge of the random variable since Pc = 1.

The Intersection
So P r (t 0 ) and P m (t 0 )/i intersect at (377.54, 0.5). Moreover, the minimum of DOK and the maximum of MChf occur at (377.54, 0.5).

The EKA Parameters Analysis in the Prognostic of Degradation:
In this case, we note from the figure below that the DOK is maximum (DOK = 1) when MChf is minimum (MChf = 0) (points J & L). Afterward, MChf starts to increase with the decrease of DOK until it reaches 0.5 at t 0 = 377.54 (point K). Since the real probability P r is a logarithmic distribution (convex curve) it will intersect with DOK at the point (377.54, 0.5) (point K). With the increase of t 0 , MChf returns 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). We note that the point K is no more at the middle of DOK since the distribution is not anymore uniform and symmetric.
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 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 (Fig. 11).

The Power Probability Distribution:
With a probability density function: and a cumulative distribution function: We have taken the domain for the power variable t 0 = [0,t N = 1000] and dt 0 = 0.1.

The Real Probability P r :
We note that P r is a non-decreasing function.

The Complementary Probability P m /i:
We note that P m /i is a non-increasing function.
2. 1000 1000 1000 1000 Thus we deduce that in the set C, we have a complete knowledge of the random variable since Pc = 1. So we conclude that P r (t 0 ), P m (t 0 )/i, DOK and MChf all intersect at (951.7, 0.5) (Fig. 14-16).

The EKA Parameters Analysis in the Prognostic of Degradation:
In this case, we note from the figure below that the DOK is maximum (DOK=1) when MChf is minimum (MChf = 0) (points J & L). Afterward, MChf starts to increase with the decrease of DOK until it reaches 0.5 at t 0 = 951.7 (point K). Since the real probability P r is a power distribution it will intersect with DOK at the point (951.7, 0.5) (point K). With the increase of t 0 , MChf returns 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 L the failure here is certain, P r (t N ) = t N /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 no more at the middle of DOK since the distribution is neither uniform nor symmetric (Fig. 14).

The Exponential Probability Distribution:
With a probability density function: and a cumulative distribution function: We have taken the domain for the logarithmic variable t 0 = [0,t N = 1000] and dt 0 = 0.1.

The Real Probability P r :
We note that P r is a non-decreasing function.

The Complementary Probability P m /i:
We note that P m /i is a non-increasing function.

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 (with probability less than 100%): Which is a curve concave upward having a minimum at: Which is a curve concave upward having a minimum at: ( ) 0 t 861.37, 0.5 = − Therefore, we can infer the magnitude of the chaotic factor MChf: Which is a curve concave downward having a maximum at: Pc 1 Thus we deduce that in the set C, we have a complete knowledge of the random variable since Pc = 1.  So P r (t 0 ) and P m (t 0 )/i intersect at (861.37, 0.5). Moreover, the minimum of DOK and the maximum of MChf occur at (861.37, 0.5).

The EKA Parameters Analysis in the Prognostic of Degradation:
In this case, we note from the figure above that the DOK is maximum (DOK=1) when MChf is minimum (MChf=0) (points J & L). Afterward, the magnitude of the chaotic factor MChf starts to increase with the decrease of DOK until it reaches 0.5 at t 0 =861.37 (point K). Since the real probability P r is an exponential distribution it will intersect with DOK at the point (861.37, 0.5) (point K). With the increase of t 0 , 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 L the failure here is certain, P r (t N ) = t N /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 no more at the middle of DOK since the exponential distribution is not symmetric (Fig. 17).

The Normal Probability Distribution:
With a probability density function: and a cumulative distribution function: We have taken the domain for the normal variable t 0 = [0, t N = 1000], dt 0 = 0.1, t 500 = (Mean) and σ t = 150 (Standard deviation).
Note that:

The Real Probability P r :
We note that P r (t 0 ) is a non-decreasing function.

The Complementary Probability P m /i:
We note that P m (t 0 )/i is a non-increasing function.

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 (with probability less than 100%).

The Chaotic Factor Chf and
Which is a curve concave upward having a minimum at: Since the normal distribution is symmetric about the mean which is t 500 Therefore, we can infer the magnitude of the chaotic factor MChf: exp .dt 2 σ 2πσ Which is a curve concave downward having a maximum at: At t t 500, 0.5 Since the normal distribution is symmetric about the mean which is t 500 Science Publications JMSS 2.5.5. Pc: The Probability in the Complex Set C: Thus we deduce that in the set C, we have a complete knowledge of the random variable since Pc = 1. exp .dt 1 2 σ 2πσ 1 1 t t 1 exp .dt, 0.5 2 σ 2 2πσ t t 500

The Intersection Point:
and P So P r (t 0 ) and P m (t 0 )/i intersect at (500, 0.5). Moreover, the minimum of DOK and the maximum of MChf occur at (500, 0.5).

The EKA Parameters Analysis in the Prognostic of Degradation:
In this case, we note from the figure that the DOK is maximum (DOK = 1) when MChf is minimum (MChf = 0) (points J & L). Afterward, the magnitude of the chaotic factor MChf starts to increase with the decrease of DOK until it reaches 0.5 at t 0 = 500 (point K). Since the real probability P r is a normal distribution it will intersect with DOK at the point (500, 0.5) (point K). With the increase of t 0 , 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 L the failure here is certain, P r (t N ) = t N /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 at the middle of DOK since the normal distribution is symmetric (Fig. 20).

The Lognormal Probability Distribution:
With a probability density function: and a cumulative distribution function: We have taken the domain for the lognormal variable t 0 = [0,t N = 1000], dt 0 = 0.1, t 5.3 = (mean) and σ t = 0.7 (standard deviation).
Note that: 6.1. The Real Probability P r : We note that P r (t 0 ) is a non-decreasing function.
We note that P m (t 0 )/i is a non-increasing function.

The EKA Parameters Analysis in the Prognostic of Degradation:
In this case, we note from the figure below that the DOK is maximum (DOK=1) when MChf is minimum (MChf = 0) (points J & L). Afterward, the magnitude of the chaotic factor MChf starts to increase with the decrease of DOK until it reaches 0.5 at t 0 = 200.34 (point K). Since the real probability P r is a lognormal distribution it will intersect with DOK at the point (200.34, 0.5) (point K). With the increase of t 0 , 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). We note that the point K is no more at the middle of DOK since the lognormal distribution is not symmetric.
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 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 (Fig. 23).

CONCLUSION
In this study I applied the theory of Extended Kolmogorov Axioms to different probability distributions: the uniform, the logarithmic, the power, the exponential, the normal and the lognormal cumulative probability distributions. In addition, I established a tight link between the new theory and degradation or the remaining useful lifetime. Hence, I developed the theory of "Complex Probability" beyond the scope of the previous first and second paper 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 , I have determined its corresponding probability of survival or of the remaining useful lifetime RUL(t 0 ) = t N -t 0 . In other words, at each instant t 0 , RUL(t 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 instant t 0 = 0 where D(t 0 ) = 0 until the failure instant t N where D(t 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). 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".