CC BY
6
(Al r\A2)<u(A3 r\A4),
So
P{AX u A4)> P((A, nA2)u(A3n A4)). Thus
R(t)>R(t).
The inequality can be proved for any n by the induction. It assures that the reliability of the parallel system with the independent components is greater than (or equal) to the reliability of that system with dependent components.
Computing the reliability of the real systems we often assume that the components life lengths are independent even though the random variables describing the life lengths are dependent. That example shows that such assumption leads towards careless conclusions. The real parallel system may have significantly lower reliability. Moreover, we come to the similar conclusions if we take under consideration more general assumption about the association of the random variables Tu T2. ...,T„ [1].
Example 2.
Assume as previously that a non-negative random variable £/, , describing time to failure of the component caused by the source z, has a Weibull distribution with parameters
a,,A,,, i = l,2,...,n +1. Let n = 3. Then for u, > 0
G1(u1) = P(U1 >u1) = e-x""a' , / = 1,2,3,4. As previously (Xj =1.2 ,A,j = 0.1, a, = 2, =0.2,
a3=2.2, A,3=0.1, a4=3, A,4=0.2.
Using (16) we obtain the reliability function of the parallel system with dependent components. For t > 0 it satisfies
R(t)
Figure 2. The graph of the reliability function of the parallel system with dependent components
R(0 = Gi (0 G2 (0 + G2 (0 G3 (0 + G3 (0 G4 (0
_ g-(0.1fL2+0.2f2) _|_^-(0.2f2+0.1f2-2) + g-(ai'2'2+0-2'3) _g-oii^+o^+o.»2-2) _ g-(0.2r+0.1i2-2+0.2i3)
Figure 2. Presents its graph. 4. Conclusion
The reliability of a series system with dependent (in the considered sense) life lengths of components is greater than (or equal) to the reliability of that system with independent life lengths of components. Assuming the independence of the life lengths of the components even though the random variables describing the life lengths are dependent, we make a mistake but that error is „safe" because the real series system has a greater reliability. The estimation of the reliability function is very conservative. The reliability of the parallel system with the independent components is greater than (or equal) to the reliability of that system with dependent components.
Computing the reliability of the real systems we often assume that the components life lengths are independent even though the random variables describing the life lengths are dependent. The examples presented here show that such assumption leads towards careless conclusions. The real parallel system may have significantly lower reliability.
References
[1] Barlow, R.E & Proshan, F. (1975). Statistical theory of reliability and life testing. Holt, Reinhart and Winston Inc. New York.
[2] Limnios, N. Dependability analysis of semi-Markov system. Reliab. Eng. and Syst. Sa/,55:203-207.
[3] Navarro, J.M., Ruiz, C.J. & Sandoval. (2005). A note on comparisons among coherent systems with dependent components using signatures, Statist. Probab. Lett. 72 179-185.
[4] Rausand, M. A. & Hoyland. (2004). System Reliability Theory; Models, Statistical Methods and Applications", 2nd edition, Wiley, New York.
Guo Renkuan
University of Cape Town, Cape Town, South Africa
An univariate DEMR modelling on repair effects
Keywords
grey theory, DEMR, repair effect, credibility measure theory, random fuzzy variable Abstract
Repairable system analysis is in nature an evaluation of repair effects. Recent tendency in reliability engineering literature was estimating system repair effects or linking repair to certain covariate to extract repair impacts by imposing repair regimes during system reliability analysis. In this paper, we develop a differential equation motivated regression (abbreviated as DEMR) model with a random fuzzy error term based on the axiomatic framework of self-dual fuzzy credibility measure theory proposed by Liu [5] and grey differential equation models. The fuzzy variable indexes the random fuzzy error term will be used to facilitate the evaluation of repair effects. We further propose a parameter estimation approach for the fuzzy variable (repair effect) under the maximum entropy principle.
1. Introduction
Repairable system analysis is in nature an evaluation of repair effects. Recent tendency in reliability engineering literature was estimating system repair effects or linking repair to certain covariate to extract repair impacts by imposing repair regimes to the system. Guo [3], [4] proposed an approach to isolate repair effects in terms of grey differential equation modelling, particularly, the one-variable first order differential equation model, abbreviated as GM (1,1) model, initiated by Deng [2]. The efforts of modelling of system repair effects in terms of grey differential equation models has attracted attention from because it is easy to calculated, for example, in Microsoft Excel. However, there were two fundamental problems necessary to be addressed. The first issue is the nature of the GM(1,1) model. In The second fundamental problem is GM(1,1) model is a deterministic approach and is just a delicate approximation approach and in nature ignores the regression error structure, which may be very reasonable if the sample size is too small, however, in general, Deng's approach results in information loss, particularly he used the adjective word "grey", implying grey uncertainty involved, but there was not uncertainty structure build up to describe "grey uncertainty". In other words, the existing GM(1,1)
model has a good idea without a convincingly rigorous mathematical foundation yet. In this paper, we will review the coupling principle materialization in GM(1,1) model in section 2. In section 3, will propose a families of first order differential equation motivated regression models under unequal-gaped data, which is suitable for the usages in system functioning time analysis. In section 4, we argue that the differential equation motivated regression model is a coupling regression model with random fuzzy error terms in nature. In section 5, review Liu's [5] fuzzy credibility measure theory and then discuss the random fuzzy variable theory in order to establish the differential equation motivated regression models as a coupling regression with random fuzzy error terms. In section 6, we will discuss the parameter estimation for the fuzzy variable repair effect indexing the random fuzzy error terms of the differential equation motivated regression modelling on system functioning time sequence under maximum entropy principle. Section 7 concludes the paper.
2. An univariate DEMR model
The success of GM(1,1) model lies on the following two aspects: data accumulative generation operator (abbreviated as AGO), which is the partial sum operation in algebra, and a simple regression model
coupled with a first-order linear constant coefficient differential equation model, which Deng [2] called is as whitening differential equation or the shadow differential equation. Let JS*0) =(x(0) (1), x(0) (2),..., x(0) («)) be a data sequence, and the partial sum with respect to
'(1)(*) = 5>CO)(0, k = 2,3,4,
J).
(1)
7=1
and the mean of two consecutive partial sums, which is used as an approximation to the primitive function
of x(1)(i)
zm{k) = ^[xm{k) + xm{k-1)].
(2)
Definition 1. Given a (strictly positive) discrete real-vaiued data sequence X"'> =(i(0)(l), x(0)(2),..., x "(n)). the equation
x(0)(£)=a + p(-z(1) (Â:)) + i
(3)
k = 2,3,4,- ,n,
"coupled" with the first-order constant coefficient linear ordinary differential equation.
(4)
x(0)(k) = a + p(- z(1)(*))+6t,k = 2,3,4,• ,;
is called a univariate DEMR model with respect to the data sequence = (x(0)(l), x(0) (2), ..., x(0)(n)) • Parameter |3 is called the developing coefficient, parameters is the grey input, term x(u) is called a grey derivative and term xl \k) is called the ktb 1-AGO of value (partial sum in fact). Furthermore, the differential equation dxm/dt + |3xl J = a in Eq. (4) is called the whitening differential equation or the shadow equation of the grey differential equation Eq. (3) by Deng [2]. The unknown parameter values (a,|3) can be estimated in terms of a standard regression. Note that Eq. (3) can be re-written as in a simple regression form,
yk =a + pxt +efr, k = 2,3,4,* (5)
where
yt=x(0\k), xk=-zm{k\ k = 2,3,4,. ,/;. (6)
The estimate for regression parameter pair (a, p), denoted as (a.b), can be calculated by,
(a,b)T =(xtXYXtY
where
(7)
"l "x(0)(2)~
x = 1 • • , Y = x(0)(3) •
1 -zm(n)
(8)
The grey filtering-prediction equation is thus
£C0)(k) = i?1-1 (k) - j^1-1 (k -\), where
(9)
&r> {k +1) =
a
m-b
-bk
a + —. b
(10)
Note that Eq. (10) is the discrete version of the solution to the differential equation (Eq. (4))
x(1) =
a
x (0) - —
P
a
+ —.
P
(11)
The typical goodness-of-fit measure of GM(1,1) model is the (absolute) relative error described by Deng [2], i.e.
e(k) =
x(0)(£)-£<0)(£)
x(0\k)
k = 2,3,4,- ,n, (12)
and the model efficiency is defined as 1
E =
n -
7 !>(/).
1 I=T
(13)
The nature of the univariate DEMR model can be identified as that the model couples a differential equation model and a simple regression model together organically. The form of the motivated differential equation (i.e., Deng's whitening differential equation) in Eq. (4) determines the form of the coupling regression (i.e., CREG) in Eq. (3). The data assimilated parameter pair («./>) in CREG determines the system parameter pair (a, p). The coupling translation rule is listed in Table 1.
Table 1. Coupling Rule in Univariate DEMT Model
Term | Motivated DE | Coupling REG
Translation between MDE and CREG
Intrinsic feature Continuous Discrete
Independent Variable t k
1st-order Derivative dx (1)(t )/ dt x (0)(k )
2st-order Derivative d(2) x(1) (t) / dt2 x (-1)(k )
Primitive function x (1)(t) z (1)(k )
Model Formation dt +ß x<1)(i ) = a x( 0)( k ) + bz(l){ k ) = a
Parameter Coupling
Parameter (a, ß) (a, b)
Dynamics (Solution) x(1)W = [ x'» -f] e-^ f®(k+1) = [x(M) - - ] e-bk + - L b û b
Filtering (Prediction) x(0)(t) = [a-ßx(0)(1) ] e-ßt €0)(t) = x€I>(k+1) - 0\k)
In DEMR modelling, the motivated differential equation and the coupling regression model are not separable but are organic integration. The DEMR models are differential equation motivated but defined by system data. A DEMR model starts with a motivated differential equation, then the coupling regression model is specified in the form "translated" from the form of the motivated differential equation, in return, in terms of coupling regression model, the parameters specifying the motivated differential equation are estimated under Z2-optimality, and finally, the solution to the motivated differential equation (or the discredited solution) equipped with data-assimilated parameters is used for system analysis or prediction. In nature a DEMR model is a coupling of a motivated differential equation and a regression formed by the discredited version of the motivated differential equation. We call the "translation" rule in grey differential equation modelling as a coupling principle.
3. Unequal-gapped differential equation motivated regression model with term of product of exponential and sine function
The basic form of the first order linear differential equation with constant term in right side is
dx „ dt ■ î \ --+ fx = ae sin(wt +v)
dt
(14)
Note here, the proposal of the motivated differential equation in Eq. (14) is featured by the term aeSt sin wt to replacing the constant term a in Eq. (4) with an intention that the fluctuating pattern of e5t sin (w t + to) will help the model goodness-of-fit. Then the solution to Eq. (14) is
x = xh + xp
where
xh = C0e
(15)
(16)
is the solution to the homogeneous equation
dx
--+ax = 0
dt
(17)
while a particular solution to the motivated differential equation Eq. (14) takes a form
xp = eu(4, sin(wt +v) + B0cos(wt +v)). (18)
Note that xp satisfies Eq. (14), thus substitute the particular solution into Eq. (14), we obtain
dxp -t+bxp
= A0ôe5t sin (wt +v) + B0ôe5t cos(wt +v) + A0weSt cos(wt +v) + B0weSt sin (wt +v) + A0 ße5t sin (wt +v) + B0 ße5t cos(wt +v) (wt +v), (19)
= ae5t sin
which leads to an equation system by comparing the coefficients of term e5t sin (wt + v) and term
e5t (wt + to) respectively,
A0(S +ß) - B0w =a A0w + B0(d +ß ) = 0
(20)
Solving the linear equation Eq. (19), we obtain the coefficients A and B0 respectively as follows
A =
0 ~ „ 2
(S + p )a CO 2 + (P +5
(21)
Bo=-~
aco
co + (P +Ô)-
In theory, the expressions of Ail and B0 will determine the particular solution xp
x = Af,eSt sin(co/ +tn )
+ B()eSt cos(co í+tü)
(22)
which will result in the general solution to Eq. (14) as
x = Cj<rp' + Ane5t sin(coi +tn)
+ BneSt cos(coí+tü)
(23)
Note that for the unequal-gapped data sequence, X(0) = (x(0) (tx),x(0) (t2),- ,x(0) (t„)), the coupling (or translation) rule is slightly different from the equal-gapped data sequence.
Table 2. Coupling Principle in unequal gapped GM( 1,1) Model.
Term Motivated DE Coupling REG
Model Formation
Intrinsic Continuous Discrete
feature
Independent Variable t tk
Response *<°>(0
lst-order Derivative dxm(t)/dt
2ntl-order Derivative d2xm(t)/dt2 fk - fk-1
Primitive function ¿l\r) >PHtk)
Data Assimilation in Model
Parameter M) (a, b)
Dynamic law —+ ßx = cte51 sin(co/+ tu) dt x<°< (t, ) = at'- sin(mi, + tu) + p(-;(,) (t, ))
Dynamics (Solution) x(1)(í ) = Cle-f" -h^e0' sin (toî + to) +B0e°' cos((Df + to) .r m(tl)=v-t"> +AQeét¡c sin(coífc + ra) +B0e5t¡c cos^í^. + ra)
Filtering
(Prediction) +(yi0co+ C0S(C0/+ îïï) + (yi0CO+ COS^/j. + îïï)
+(yíflS - B0 co) e01 sin(co/ + îïï) +(^8- i?0co) es sin(co/i + îïï)
The coupling regression is
x(0\tk) = aestt sin(co^ +tn)
+ ß(-z(1)(^)) + s„ k = 2,3,4,. ,//,(24)
where
(h ) = - (^-i ) + - (0) (h )(h - h-i )
k = 2,3,4,* ,n.
(25)
The parameter pair (a,p) is obtained by least-square estimation (a,bf = [xTx) 1XTY , where
X =
e5'2 sin(co+tü) ~zm{tx) e5ti sin(coí2 +tn) -zm{t2)
e°'"sm(cQt„+w) -zm{tn)
Y =
- (O
*(0)C2)
*l0)(tn)
(26)
since 5 and co are given (in a manner by trials and errors).
Formally, we have a DEMR model as
dx n 8t • / \ --b ßx = ae sin(co t+xs)
x(0) {tk ) = aest sin(coi +Ttj ) + ß (- zm {tk ))
(27)
+st.
4. Fuzzy repair effect structure
In standard regression modelling exercises, it is often to assume that the error terms s;, / = 1.2.« ,n are random with zero mean and constant variance, i.e., E[e,.] = 0 and VAR[b,] = c2, 7 = 1,2,' ,n. It is typically assuming a normal distribution with zero mean and constant variance, i.e., n(o,o2 ).
Furthermore, as we pointed out that a grey differential equation model is a motivated differential equation motivated regression, which takes the form translated from the motivated differential equation, as shown in Table 1 for GM(1,1) case. However, we should be fully aware that translation back and forward between the motivated differential equation and the coupling
regression will bring in new error which is different from the random sampling error7V^0,c2). The errors
brought in come from the steps of the usage of difference x(0) (A-) = x(1) (k)-x(1) (Ar-l)to replace the derivative (clx/clt ); k and the usage of the average accumulated partial sum z"' (^Jto replace the primitive function xMl{tk) during the translation
between the motivated differential equation and the coupling regression.
Our simulation studies have shown that the coupling-introduced error is dependent upon the grids size A, or equivalent to the total number of approximation N. The simulation evidences have shown that the larger the number of approximating grid, or equivalently, the smaller the approximating grid, the coupling translation error is smaller. However, the coupling translation error and the approximating grid do not hold a deterministic functional relation. What we can see is the functional relation has a certain degree of belongingness. In other words, the coupling translation process induces a fuzzy error term, denoted as q with a membership function. We perform a simulation study of the error occurrence frequencies of approximating cos (71/2) by
(sin (n /2) - sin (n /2 + Ax))jAr.
error's frequency Chart
Ms) = <
5 + 0
o
o - s
o
if-o<s<0 if 0<s<o
otherwise
(28)
which has a fuzzy mean zero.
However, in the modelling of system functioning times, we further note that the repair will reset the system dynamic rule so that the repair impact may be understood as a fuzzy variable having a triangular membership
z-a
b-a c-z
c-b 0
if a < z <b if b < z <c otherwise
(29)
The fuzzy mean of the fuzzy repair effect is thus
1
E(r) = -(a + 2b + c),
(30)
which provides a repair effect structure. Therefore, the "composite" fuzzy "error" term appearing in the differential equation motivated regression for modelling a system function time will be
c, =£,+/;, 7 = 2,3,' ,n, (31)
with a triangular membership function, i.e.,
Figure 1. Error occurrence frequency
Therefore, in general the error terms of a differential equation motivated regression model (i.e., grey differential equation in current grey theory literature) is fuzzy because the vague nature of the error occurrences.
As a standard exercise, the fuzzy error component ei may be assumed as triangular fuzzy variable with a membership function
(w) =
w -a +T3
b -a +T3 c +T3 - w
c +T3 -b 0
if a-w <w <b //b<w<c+tn (32) otherwise
because the sum of two triangular fuzzy variables is still a triangular fuzzy variable. The total error
=(ri+ei) + si, / = 2,3,.
(33)
which is a sequence of random fuzzy variables because the summation nature of a random fuzzy variable and a fuzzy variable according to Liu [5]. Now, we reach a point that the random fuzzy variable concept is involved and therefore it is necessary to
have a quick review on the relevant theoretical foundation.
5. A random fuzzy variable foundation
First we need to review the fuzzy credibility measure theory foundation proposed by Liu [5], then we will establish the normal random fuzzy variable theory for a facilitation of error analysis in the differential equation motivated regression models. Let © be a nonempty set, and 2® the power set on © . Each element, let us say, A a ©, A e 2'~ is called an event. A number denoted as Cr{/i}. 0 < Cr {^4} < 1, is assigned to event A e 2'~, which indicates the credibility grade with which event A e 2" occurs. Cr {^4} satisfies following axioms given by Liu [5]: Axiom 1. Cr{©} = 1.
Axiom 2. Cr{-}is non-decreasing, i.e., whenever
Axiom 3. Cr{-} is self-dual, i.e., for any A^2&,Cr {A}+ Cr{Ac} = \.
Axiom 4. Cr {* ,4 } a 0.5 = sup [Cr {4}] for any {4}
i
with Cr{4} < 0.5 .
Axiom 5. Let set functions Crk{-} : 2@t —» [0,1] satisfy Axioms 1-4, and 0 = 0,x02x' x®p, then:
C/"M2,. ,0J=C^1}aC/-J2(a' aC/-^}(34)
for each ^.e,,« ,0je2®.
Definition 5.1. Liu [5] Any set function Cr: 2q ® [0,1] satisfies Axioms 1-4 is called a (v,a)-credibility measure (or classical credibility measure). The triple (0,2®/>) is called the (v,a)-credibility measure space.
Definition 5.2. Liu [5] A fuzzy variable c, is a mapping from credibility spacc (0,2®, O") to the set of real numbers, i.e., E, : (©, 2®, Cr) —» R.
Definition 5.3. Liu [5] The (induced) membership function of a fuzzy variable E, on (©,2®,Cr) is:
H(x) = (2Cr% = x}) a 1, x g R
(35)
Conversely, for given membership function the credibility measure is determined by the credibility inversion theorem.
Theorem 5.4. Liu [5] LetE, be a fuzzy variable with membership function m. Then forVi? d R,
sup|j,(x)+l-sup|j,(x) (36)
2 V xeB xeBc J
As an example, if the set B is degenerated into a point x, then
Cr{£ =x} = -in(x) + l-supnCy)\ Vxg R(37)
2 v y*x J
Definition 5.5. Liu [5] The credibility distribution <D:R->[0,1] of a fuzzy variable on (©,2®,Cr) is
0(x) = Cr^ G0|^(0)<x}.
(38)
The credibility distribution ®(x) is the accumulated credibility grade that the fuzzy variable E, takes a value less than or equal to a real number x g R. Generally speaking, the credibility distribution <D is neither left-continuous nor right-continuous.
Theorem 5.6. Liu [5] Let be a fuzzy variable on (©,2,(7'] with membership function (j,. Then its credibility distribution,
1
f
O(x) = - sup nOO +1 - sup nCv) L Vx g R (39)
jL \ X x X x J
Definition 5.7. Liu [5] Let <D be the credibility distribution of the fuzzy variable ^ . Then function (|): R —»[0, +co) of a fuzzy variable is called a credibility density function such that,
&(*)= \$(y)<fy, Vx«
R
(40)
Now we are ready to state the normal random fuzzy variable theory for the error analysis in the repairable system modelling.
Liu [5] defines a random fuzzy variable as a mapping from the credibility space (0,2®, Cr) to a set of random variables. We would like to present a definition similar to that of stochastic process in probability theory and expect readers who are familiar
with the basic concept of stochastic processes can understand the comparative definition.
Definition 5.8. A random fuzzy variable, denoted as X = {Xp(0), 9e©} , is a collection of random variables
X b defined on the common probability space
(W,A,Pr) and indexed by a fuzzy variable p(0)
defined on the credibility space (©,2®, Cr ). Similar to the interpretation of a stochastic process, X = {Xt, t e R+), a random fuzzy variable is a
bivariate mapping from (Wx©,A x2®) to the space
(R, B). As to the index, in stochastic process theory,
index used is referred to as time typically, which is a positive (scalar variable), while in the random fuzzy variable theory, the "index" is a fuzzy variable, say, b. Using uncertain parameter as index is not starting in random fuzzy variable definition. In stochastic process theory we already know that the stochastic
process X = { Xt(w), weW) uses stopping time
t (w), w OW, which is an (uncertain) random variable as its index.
In random fuzzy variable theory, we may say that that average chance measure, denoted as ch, plays a similar role similar to a probability measure, denoted as Pr, in probability theory.
Definition 5.9. Liu and Liu [6] Let x be a random fuzzy variable, then the average chance measure denoted by ch {}, of a random fuzzy event {X < x),
is
ch{x < x}= J Cr {q e ©| Pr{x(e)< x)>ajda . (41)
0
Then function Y () is called as average chance distribution if and only if
Y(x) = ch{X < x}.
(42)
Liu [5] stated that if a random variable h has zero mean and a fuzzy variable Z , then the sum of the two, h + Z , results in a random fuzzy variable X . Now, it
is time to find the average chance distribution for a
d
normal random fuzzy variable x: N(z,s2), where Z is
a triangular fuzzy variable and c2 is a given positive real number. Note that fuzzy event {e e© :Pr{X(e) < x)>a)
o{6g© : o[ ^^a} (43)
e©: x >Ç(e)+aF-1(a )}
û
)e e © :z(e)< x-aF-1(a)}
(43)
The fuzzy mean is assumed to have a triangular membership function
mz(w) =
w - a,
z
bç - aç
cv - b 0
az < w < bz
bz < w < az
otherwise
(44)
and
F(w) = Cr{Ç < w} =
w - a
ç
2(bç - aç) w + cç - 2bç
2(cç - bç) 1
w < av
aç < w < bç
bç < w < cç
(45)
w > cç
which gives the credibility distribution for the fuzzy mean, Z .
Then the critical step is to derive the expression of Cr(e ) e® Pr{x (w ,e ) < xj>a). For normal random fuzzy variable with a triangular fuzzy mean,
{ç (e ):Pr{X (w,e ) < x}>a}
o{e e© : ç (e ) < x-aF-1(a)}.
(46)
Then the range for the integration of the integrand Cr {e e © : Z (e ) < x -cF-1 (a )j with respect to a is listed in Table 3.
g(a)
Range for a
Cr{qOQ:z(q)J x- sF- '(a)}
c —w
v
0
