Soft Computing https://doi.org/10.1007/s00500-018-3067-3
FOCUS
Beyond precision: accelerated life testing for fuzzy life time data Muhammad Shafiq1 · Muhammad Atif2 · Reinhard Viertl3
© Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract Reliability analysis comprises statistical analysis techniques that make inferences based on life time data. Swift progress has been observed in life time data analyses during the last few decades. Accelerated life testing models are regarded as the most popular techniques for engineering life time data analysis. Their main aim is to model life times under different stress levels that are more severe than the usual stress level. The existing techniques consider life times as precise measurements and do not contemplate the imprecision of observations. In fact, life time measurements are not precise quantities but more or less fuzzy. Therefore, in addition to standard statistical tools, fuzzy model approaches are also essential. The current study generalizes some parametric and nonparametric classical estimation procedures for accelerated life testing in order to accommodate both fuzziness and random variation. The proposed estimators cover both uncertainties, which make them more applicable and practicable for life time analysis. The results of fuzzy life times are considered under various stress conditions, and comparisons with precise life time analysis are further presented in examples. Keywords Accelerated life testing · Characterizing function · Fuzzy number · Non-precise data · Real measurements
1 Introduction Statistics is the science of learning from data. Data are usually presented in the form of numbers or vectors and are obtained from measurements of some phenomena. Similar to any research-oriented activity, the obtained measurements are usually recorded as precise numbers for further evaluation and inference. Countless techniques are available in classical statistics to model and draw inferences from the collected precise Communicated by A. Genovese and G. Bruno.
B
Muhammad Shafiq
[email protected] Muhammad Atif
[email protected] Reinhard Viertl
[email protected]
1
Institute of Numerical Sciences, Kohat University of Science and Technology, Kohat, Pakistan
2
Department of Statistics, University of Peshawar, Peshawar, Pakistan
3
Institute of Statistics and Mathematical Methods in Economics, Vienna University of Technology, Vienna, Austria
measurements. The main concern of classical statistical techniques is to model variation among the precise observations in the best possible way. However, in practical situations, there are two kinds of uncertainty in data: variation among the observations and the imprecision of single observations (i.e., fuzziness) (Viertl 2011). Unequivocally, modern measurement science argues that in practical applications, the exact measurement of a continuous real variable is not possible (Barbato et al. 2013). Hence, real measurements are not precise but are fuzzy. The common variables used in daily life include the life time of an object, the amount of rainfall, the amount of carbon emissions, the color intensity of light, height, age. These types of variables are continuous but are always measured as precise numbers. One should keep in mind that variation among observations is different from fuzziness. Classical statistics (stochastic models) consider only the variation among precise observations and ignore the fuzziness. By doing so, information may be lost and misleading results obtained (Viertl 2011). Therefore, the fuzziness of every single observation should be considered and modeled using a modern fuzzy numbers approach. To consider the imprecision of single observations, the idea of fuzzy sets was introduced by Zadeh (1965). According to him, in the physical world, one may classify some
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animals or plants such as cows, horses, and birds, but some have unclear statuses such as bacteria and starfish. Additionally, if we consider some examples of linguistic descriptions such as a class of good students, a class of good-looking women, and high blood pressure, in these situations, one cannot characterize them in classical mathematical set notations. In the same way, there are many situations for which we cannot define precise criteria for the membership of a set (Zadeh 1965). A binary valued characteristic function, called an indicator function, has significant importance in classical set theory. It is usually denoted by 1ϒ (·) and is used to represent whether an element υ is in a subset ϒ of a universal set U , as shown in Eq. (1): 1 if υ ∈ ϒ ∀ υ ∈ U. 1ϒ (υ) = 0 if υ ∈ /ϒ
(1)
In fuzzy set theory, the idea of a classical set was generalized, i.e., two-valued logic was progressed to multi-valued logic. Therefore, the indicator function from classical set notations was extended to the membership function μϒ ∗ (·) of a fuzzy subset ϒ ∗ of U , as shown in Eq. (2): ⎧ ⎫ if υ is for sure in ϒ ∗ ⎨1 ⎬ μϒ ∗ (υ) = δ if υ belongs to ϒ ∗ to some degree δ ∈ (0 , 1) ⎩ ⎭ 0 if υ does not belong to ϒ ∗ ∀ υ ∈ U.
(2)
For more details, see Gil and Hryniewicz (2009).
2 Accelerated life testing (ALT) A life time can be simply defined as the time to the occurrence of a specified event. Life time data analysis began in the 20th century, and now, it is considered one of the most important fields of statistics. Based on the developments of the last fifty years, numerous standard statistical tools are available for the analysis and modeling of precise life time observations (Lee and Wang 2013). Increasing the life time of mechanical components has always been a prime interest for industry. To estimate the reliability of a component, various life tests were performed based on available environmental conditions. Accelerated life testing (ALT) contains popular standard statistical tools to assess life time and make inferences about various mechanical products/systems without waiting longer than the mean life time of the product. For the reliability tests of mechanical components, the tests are usually performed under different stress levels that are more severe than the usual stress level (Nelson 2009).
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The ALT performance of the product under higher stress levels is evaluated over a time period, and the failure times are recorded. Therefore, the failure times cannot be obtained in the form of accurate numbers since the possibility of an occurrence at the next inspection interval is anticipated. Therefore, great ambiguity is found in the failure time if the interval is too long. Since the ALT observations are transformed to normal circumstances, the ambiguity/uncertainty would also shift and impact the accuracy and credibility of the estimated outcomes. Furthermore, the exact failure time is impossible to attain since the judgment criteria of various inspectors might differ. Moreover, the conclusion of an exact failure time is not accurate (Xu et al. 2015). In Viertl (2004), it was stated that the data obtained in accelerated life testing are more or less fuzzy. Therefore, in dealing with life time observations, in addition to standard statistical tools, fuzzy number approaches are necessary, more suitable, and realistic.
3 Fuzzy models and fuzzy numbers Some preliminary concepts of fuzzy set theory are explained according to (Viertl 2011) below.
3.1 Fuzzy numbers Let t ∗ be a fuzzy number characterized by a characterizing function ξ(·), which is a real-valued function of one real variable obeying the following conditions: 1. ξ : R → [0, 1]. 2. For all δ ∈ (0, 1], δ-cut is defined as Cδ (t ∗ ) := {t ∈ R : ξ(t) ≥ δ}, which is a finite union of compact intervals [a j,δ , b j,δ ],
δ i.e.,Cδ (t ∗ ) = kj=1 [a j,δ , b j,δ ] = ∅. 3. ξ(·) has a bounded support such that supp[ξ(·)] := [t ∈ R : ξ (t) > 0 ] ⊆ [a0 , a1 ] with − ∞ < a0 < a1 < ∞. For a fuzzy number, if all δ-cuts are non-empty closed bounded intervals, they are called fuzzy intervals.
3.2 Remark The family (Cδ (t ∗ ); δ ∈ (0 , 1]) is nested. Therefore, for δ1 < δ2 , we have Cδ1 (t ∗ ) ⊇ Cδ2 (t ∗ ).
Beyond precision: accelerated life testing for fuzzy life time data
3.3 Fuzzy vectors A p-dimensional fuzzy vector t ∗ is determined by a realvalued function of p real variables t1 , t2 , . . . , t p by a vectorcharacterizing function ζ (., . . . , .) that satisfies the following three conditions: 1. ζ : R p → [0 , 1]. 2. For all δ ∈ (0 , 1], the δ-cut Cδ (t ∗ ) := t ∈ R p : ζ (t) ≥ δ is a non-empty, finite union of simply connected and closed sets. 3. The support of ζ (., . . . , .) defined by supp[ζ (., . . . , .)] := t ∈ R p : ζ (t) > 0 is a bounded set. If all δ-cuts of a p-dimensional fuzzy vector are simply connected compact sets, then the corresponding pdimensional fuzzy vector is called the p-dimensional fuzzy interval. From a stochastic quantity T , if describing life times, consider a sample of size n such that t1 , t2 , . . . , tn . Then, each ti is an element of the observation space MT ⊆ [0 , ∞), and (t1 , t2 , . . . , tn ) is an element of the sample space. The sample space is the Cartesian product of n copies of MT such that MT × MT × · · · × MT = MTn . When dealing with fuzzy observations, the situation is not the same as for precise numbers. If there is a sample of n fuzzy observations ti∗ , i = 1(1)n, each with the characterizing function ξi (·), that are fuzzy elements of MT , then (t1∗ , t2∗ , . . . , tn∗ ) is not a fuzzy element of MTn . Therefore, to obtain a fuzzy element (fuzzy vector) of MTn without losing the fuzziness, usually the so-called minimum t-norm is used. For the vector-characterizing function ζ (., . . . , .) of the combined fuzzy sample t ∗ , by applying the minimum tnorm, ζ (t1 , t2 , . . . , tn ) = min { ξ1 (t1 ), ξ2 (t2 ), . . . , ξn (tn )} ∀ (t1 , t2 , . . . , tn ) ∈ Rn , a fuzzy element of MTn ⊆ Rn is obtained.
3.4 Remark The corresponding δ-cuts of the combined fuzzy sample are obtained through the Cartesian product of the δ-cuts of n fuzzy numbers as n Cδ [ξi (·)] ∀ δ ∈ (0 , 1]. Cδ [ζ (., . . . , .)] = ×i=1
From the δ-cuts, we can reconstruct the characterizing functions using the following lemma.
3.5 Lemma Denote by 1 A (·) the indicator function of the set A ⊆ R for any characterizing function ξ(·) of a fuzzy number t ∗ ; then, the following is valid:
ξ (t) = max δ·1Cδ (t ∗ ) (t) : δ ∈ [0 , 1] ∀ t ∈ R. For the proof, see Viertl (2011).
3.6 Remark
One should note that not all families Aδ ; δ ∈ (0, 1] of nested finite unions of compact intervals are the δ-cuts of a fuzzy number. Nonetheless, the following construction lemma holds:
3.7 Construction lemma
δ a j,δ , b j,δ be a Let Aδ ; δ ∈ (0, 1] with Aδ = kj=1 nested family of non-empty subsets of R. Then, the characterizing function ξ(·) of the generated fuzzy number is given by ξ (t) = sup δ·1 Aδ (t) : δ ∈ [0, 1] ∀ t ∈ R. See Viertl and Hareter (2006).
3.8 Extension principle This is the generalized form of an arbitrary function G: ℵ → for a fuzzy argument ω∗ in ℵ with the corresponding membership function ψ : ℵ → [0 , 1]. Then, the generated fuzzy value y ∗ = G(ω∗ ) is defined to be the corresponding fuzzy element in for which the membership function (·) is obtained by (y) sup {ψ(ω) : ω ∈ ℵ, G(ω) = y} if ∃ ω : G(ω) = y := 0 if ω : G(ω) = y ∀ y ∈ . For more details, see Klir and Yuan (1995).
3.9 Theorem For a continuous function F: Rn → R and fuzzy ndimensional interval t ∗ , the following holds: Cδ F(t ∗ ) =
min F(t) , max∗ F(t)
t∈Cδ (t ∗ )
t∈Cδ (t )
∀ δ ∈ (0 , 1].
For the proof, see Viertl (2011). Realizing the importance of fuzziness, some work has been performed related to fuzzy life time analysis, such as fuzzy Bayesian inference (Frühwirth-Schnatter 1993), reliability estimation based on fuzzy life time data (Viertl and Gurker 1995), estimation of the Weibull distribution using
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the fuzzy least squares method (Hung and Liu 2004), fuzzy Bayesian estimation of life time data (Wu 2004), Bayesian reliability analysis of fuzzy life time data (Huang et al. 2006), reliability estimation with the Rayleigh distribution based on fuzzy life time data (Pak et al. 2013), modeling of future life times as a fuzzy random variable (Shapiro 2013), generalized Kaplan Meier estimation for fuzzy survival times (Shafiq and Viertl 2015b), and use of empirical reliability functions based on fuzzy life time data (Shafiq and Viertl 2015a). However, in most of the publications, fuzziness is still ignored. It was already discussed in Viertl (2004) that the life time observations recorded in accelerated life testing are not precise numbers but fuzzy. Since then, only a few references [Gonzalez-Gonzalez et al. (2014), Liu et al. (2015), Xu et al. (2015), Shafiq and Atif (2015) and Shafiq et al. (2016)] can be found that deal with fuzzy information in accelerated life testing approaches. Therefore, the classical analysis techniques related to ALT need to be generalized for fuzzy life times.
To obtain θ i,δ and θ i,δ , the proposed estimators are explained in Sect. 5.1.
4.2 Power rule model The power rule model for precise data in Viertl (1988) is defined as θ (S) = SCA , where S is the stress level and A and C are unknown constants. For the estimation of A and C, the modified power rule model is given as Cˆ θˆi = θˆ (Si ) = ˆ , A
k k where S = i=1 Sri / i=1 ri . In the given equation, ri denotes the number of failures under stress level Si for i = 1(1)k. If θi , i = (1)k are considered independent of each other, then the maximum likelihood estimators for A and C can be obtained using the following equations:
4 Estimation based on ALT
k
Some parametric and nonparametric estimation procedures for accelerated life testing are given below.
i=1
ri · θˆi ·
θˆi =
ri
j=1 ti,( j)
k
+ (n i − ri )ti,(ri ) ri
,
(3)
where ti,( j) , j = 1(1)ri denotes the observed ri ordered times of failure under the stress level Si . As discussed earlier, life time observations are more or less fuzzy and parameter estimators are functions of life time observations. Therefore, these estimators need to be general∗ , j = ized for fuzzy life time observations, i.e., θˆi∗ = f (ti,( j) ∗ 1(1)ri ), where ti,( j) denotes the fuzzy life times of ri failed units under the stress level Si (Viertl 1988). ˆ∗ a fuzzy value, Since the parameter estimator
θi becomes ∗ its δ-cuts are denoted as Cδ θˆ = θ i,δ , θ i,δ ∀ δ ∈ i
(0 , 1].
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S Aˆ i
S
· ln
S i
S
=0
(5)
and
4.1 Constant stress level Consider an exponential distribution, which is one of the most popular distributions for addressing life time data. Its density is defined as f (t|θ ) = θ1 exp − ( θt ) for t > 0, θ > 0 with a failure rate of λ = θ1 . To estimate the parameter under different stress levels S1 , S2 , . . . , Sk , we obtain the data set {Si , n i , ri } , i = 1(1)k, where n i is the number of units at stress Si and ri the number of failures at stress level Si . The following estimator is used:
(4)
Si S
Cˆ =
Aˆ · θˆi · Si S . k r i=1 i
i=1 ri
(6)
This is done by solving the first equation for Aˆ and then ˆ the second one for C. However, as discussed earlier, life time observations are not precise numbers but are fuzzy. Therefore, the estimators for θi obtained using fuzzy life time data are written as θˆi∗ , i = 1(1)k. The estimates of A and C will also be fuzzy because they are functions of θˆi∗ . They are defined by Cˆ ∗ Cˆ ∗ = f 1 (θˆi∗ ), Aˆ ∗ = f 2 (θˆi∗ ) and θˆ ∗ (Si ) = ˆ ∗ . A Si S
Therefore, their δ-cuts are denoted by Cδ ( Aˆ ∗ ) = Aδ , Aδ ∀ δ ∈ (0 , 1], ∀ δ ∈ (0 , 1], Cδ (Cˆ ∗ ) = C δ , C δ and Cδ (θˆ ∗ (Si )) = θ δ (Si ) , θ δ (Si ) ∀ δ ∈ (0 , 1].
Beyond precision: accelerated life testing for fuzzy life time data
For the estimation of lower and upper ends of the δ-cuts of A∗ , C ∗ and θˆ ∗ (Si ), the proposed estimators are explained in Sect. 5.2
4.3 Step-stress model According to Balakrishnan et al. (2007), step-stress life testing is a particular accelerated life testing approach that observes the units under various stress levels for some prespecified time. The number of failures is recorded under the initial stress level, which is then further elevated to another level. At this level, the number of failures is recorded again, and the process keeps repeating. Consider a simple two-step-stress test in which the life times follow exponential distributions. Ex(θ1 ) and Ex(θ2 ) are at stress level S1 for a specified time τ , and then, the stress level is change to S2 . The cumulative distribution function of the time to failure of a unit is denoted by G(·): G(t) =
G 1 (t) = F1 (t | θ1 ) G 2 (t) = F2 (t − τ + u | θ2 )
for 0 < t < τ for τ ≤ t < ∞,
Cδ (g ∗ (t)) = g δ (t) , g δ (t) ∀ δ ∈ (0 , 1]. Let n units be placed under stress level S1 for a specified time τ . Then, the stress is changed to S2 until r total failures occur, with r < n. The experiment is terminated on the occurrence of the r th failure/death. This type of censoring is called type II censoring. Let n 1 be the number of failed units up to time τ and the total failures be fixed to r , showing type II censored data. Then, under conditions 1 ≤ n 1 ≤ r − 1, the likelihood function is denoted as L(θ1 , θ2 ; t1,(1) , t1,(2) , . . . , t1,(n 1 ) , t2,(n 1 +1) , . . . , t2,(r ) ). The corresponding maximum likelihood estimators can be written as n 1 t1,( j) + (n − n 1 )τ j=1 θˆ1 = (7) n1 and r θˆ2 =
where t for t ≥ 0 ; θk ≥ 0, Fk (t | θk ) = 1 − exp − θk θ2 u = τ, θ1 and ⎧
⎨G 1 (t) = 1 − exp − t
θ1 G(t) = ⎩G 2 (t) = 1 − exp − τ + θ1
τ −t θ2
for 0 < t < τ for τ ≤ t < ∞,
with corresponding density
g(t) =
and
⎧ ⎨g1 (t) = ⎩g2 (t) =
1 θ1 1 θ2
exp −
exp −
t θ1 t−τ θ2
−
τ θ1
for 0 < t < τ for τ ≤ t < ∞.
If we consider the stress levels as precise numbers and life time observations as fuzzy, then the upper and lower δ-level curves of the fuzzy cumulative distribution function of the time to event of a unit and the fuzzy probability density function defined through the extension principle are denoted by G ∗ (·) and g ∗ (·), respectively. For the corresponding δ-level curves, we obtain for all t ≥0 Cδ (G ∗ (t)) = G δ (t) , G δ (t) ∀ δ ∈ (0 , 1],
j=n 1 +1 (t2,( j)
− τ ) + (n − r )(t2,(r ) − τ ) r − n1
.
(8)
However, as discussed earlier, life time observations are fuzzy. Therefore, the likelihood function and parameter estimators must be generalized for fuzzy life time observations. The likelihood function for the fuzzy data can be generalized by the extension principle. It is denoted as
∗ ∗ ∗ ∗ ∗ . , t1,(2) , . . . , t1,(n , t , . . . , t L ∗ θ1 , θ2 ; t1,(1) 2,(n 2,(r ) ) +1) 1 1 For fuzzy life time observations, the generalized parameter estimators are denoted as θˆ1∗ and θˆ2∗ . Since these estimators will be functions of fuzzy life time observations, their δ-cuts are denoted by
Cδ θˆ1∗ = θ 1,δ , θ 1,δ ∀ δ ∈ (0 , 1] and
Cδ θˆ2∗ = θ 2,δ , θ 2,δ ∀ δ ∈ (0 , 1]. The estimation of the lower and upper ends of the δ-cuts is explained in Sect. 5.3.
4.4 Nonparametric estimation According to Shaked et al. (1979), let S1 , S2 , . . . , Sk denote the stress levels and ti,l , i = 1(1)k, l = 1(1)n i be the
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corresponding life times under accelerated stress levels. Let F1 , F2 , . . . , Fk be the corresponding Cdfs of the life times. the number of units tested under stress level Si , Let n i be k n i is the total number of units tested. where N = i=1 n i ti,l . The data will be transformed into T i = n1i l=1 The scale factor between Fi and F j is denoted by αi j , where
αi j = Si /S j
γ
ξi (t) 1 0.8 0.6 0.4 0.2 0
for i = j,
(9)
20
30
t [time]
40
Fig. 1 Fuzzy sample under stress S1 = 1 ξi (t)
and
lnSi − lnS j . γ = ln αi j
(10)
αˆ i j = T i /T j for i = j,
1 0.8 0.6
The estimator of αi j is given by
0.4
(11)
0.2 0
and the estimate of γ can be obtained as γˆi j = ln(αˆ i j ) lnSi − lnS j for i = j,
ln T i Tj γˆi j = S for i = j. ln Sij
k
S ln Sij ln T i Tj .
2 k k Sj i=1 j=i+1 ln Si
i=1
(12) (13)
20
30
t [time]
40
5.1 Constant stress levels and fuzzy life times Under constant stress levels Si , i = 1(1)k, the generating family of the fuzzy estimator θˆi∗ from Sect. 4.1 for the parameters is defined by
Aδ θˆi∗ =
ri
j=1 t i,( j),δ
+ (n i − ri )t i,(ri ),δ ri
ri
k
j=i+1
(14)
The estimator γˆ is based on precise life time observations and needs to be generalized for fuzzy life time observations. The generalized fuzzy estimator of γ is denoted by γˆ ∗ and has δ-cuts Cδ (γˆ ∗ ) = γ δ , γ δ ∀ δ ∈ (0 , 1]. The estimators through which the values of γ δ and γ δ can be obtained are explained in Sect. 5.4.
5 Proposed algorithms To obtain the approximate δ-cuts for different fuzzy estimators, the algorithms are given below. In the proposed algorithms, precise stress levels are considered.
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10
Fig. 2 Fuzzy sample under stress S2 = 2.5
An overall estimator of γ can be written as the weighted average of the γˆi j s:
γˆ =
10
j=1 t i,( j),δ +(n i
ri
− ri )t i,(ri ),δ
,
∀ δ ∈ (0 , 1], (15)
where t i,( j),δ , t i,( j),δ ∀ δ ∈ (0 , 1] are the δ-cuts of ∗ . the fuzzy life times ti,( j) The characterizing function of θˆi∗ is obtained by the Construction Lemma for the characterizing functions. Example 1 For the construction of the characterizing functions of the fuzzy estimates, as an example, we have S1 = 1, n 1 = 5, r1 = 4 and S2 = 2.5, n 2 = 5, r2 = 4. The characterizing functions of the observed fuzzy life times are given in Fig. 1. Based on the fuzzy samples mentioned in Figs. 1 and 2, the characterizing functions of the fuzzy parameter estimates obtained through Eq. (15) are depicted in Fig. 3. The above fuzzy parameter estimates θˆ1∗ and θˆ2∗ based on the fuzzy life times are the mean life times under stress 1 and stress 2, respectively. These estimates are more suitable for
Beyond precision: accelerated life testing for fuzzy life time data
ξ(θ)
ξ(A)
1
1
θˆ2∗
0.8
0.8
0.6
0.6
0.4
0.4
θˆ1∗
0.2 0
10
20
0.2
30
θ
40
0
the realistic life times since they cover both the stochastic variation and the fuzziness of the life times. Assuming r1 = 4 precise observed life times (6, 10, 15, 18) under stress S1 = 1 and r2 = 4 life times (1, 3, 9, 11) under stress level S2 = 2.5, the corresponding standard parameter estimates are θˆ1 = 28.33 and θˆ2 = 8.75. These estimates are based on precise measurements that neglect fuzziness. This means that one of the uncertainties is completely ignored, which makes these estimates unrealistic.
C δ = max k
Aδ = min
ri · θ i,δ ·
Si
A
Si
ri · θ i,δ
i=1
Aδ = max
ri · θ i,δ ·
i=1
ri · θ i,δ ·
Si S
Si
A · ln
S
A
· ln
· θ i,δ k
k
· θ i,δ · k i=1 ri
i=1 ri
,
Aδ Si S
,
Aδ Si S
,
Aδ ⎫ ⎪ ⎪ ⎬ · Si S
i=1 ri
⎪ ⎪ ⎭
∀ δ ∈ (0, 1].
(19)
∀ δ ∈ (0, 1],
Aδ θˆ ∗ (S1 ) =
Si S
Si S
Cδ Cδ , ( S1/S) Aδ ( S1/S) Aδ
∀ δ ∈ (0, 1]
(20)
∀ δ ∈ (0, 1].
(21)
and = 0,
Aδ θˆ ∗ (S2 ) =
=0
∀ δ ∈ (0, 1], (17)
Aδ
Aδ k · θ i,δ · Si ri · θ i,δ · Si i=1 S S , , k k i=1 ri i=1 ri
Aδ ⎫ k ⎪ Si ⎪ ⎬ i=1 ri · θ i,δ · S ∀ δ ∈ (0, 1], (18) k ⎪ ⎪ i=1 ri ⎭ i=1 ri
· θ i,δ · k i=1 ri
Si S
= 0,
⎧
Aδ k ⎪ ⎨ i=1 ri · θ i,δ · Si S C δ = min , k ⎪ r ⎩ i=1 i k
⎪ ⎪ ⎩
Aδ
(16) k i=1
k
A
8
The characterizing functions for Cˆ ∗ and Aˆ ∗ are obtained by the above-mentioned Construction Lemma. The fuzzy estimate for the parameters θ (Si ) is obtained by the following generating families:
i=1 k
· θ i,δ · k i=1 ri
i=1 ri
i=1 ri
· ln S S A Si Si = 0 · · ln S S
⎧ ⎪ ⎪ ⎨ k
i=1 ri
k
The generating families ofintervals of the fuzzy estimators Aˆ ∗ and Cˆ ∗ are denoted as Aδ , Aδ and C δ , C δ , respectively. To obtain them, we have the following equations:
6
and
5.2 Power rule model and fuzzy life times
k
4
Fig. 4 Characterizing function of Aˆ ∗ based on the data from Example 1
Fig. 3 Characterizing functions of θˆ1∗ and θˆ2∗
2
Cδ
Cδ , S A ( S2/S) Aδ ( 2/S) δ
From these generating families of intervals, the characterizing functions are obtained by the Construction Lemma mentioned in Sect. 3.7. Based on the fuzzy parameter estimates θˆi∗ , i = 1, 2 depicted in Fig. 3 and the stress levels mentioned in Example 1, the characterizing function of the fuzzy estimate Aˆ ∗ obtained through Eqs. (16) and (17) is depicted in Fig. 4. Based on the fuzzy parameter estimates θˆi∗ , i = 1, 2 and Aˆ ∗ and the stress levels mentioned in Example 1, the characterizing function of the fuzzy estimates Cˆ ∗ obtained through Eq. (18) and (19) is depicted in Fig. 5. On the basis of the fuzzy estimates Aˆ ∗ and Cˆ ∗ and the stress levels mentioned in Example 1, the characterizing
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ξi (t)
ξ(C)
1 0.8 0.6 0.4 0.2
1 0.8 0.6 0.4
0
0.2 0
10
20
30
40
C
Fig. 5 Characterizing function of Cˆ ∗ based on the data from Example 1
20
40
60
80
100
120
t [time]
Fig. 7 Fuzzy sample 3 ξi (t) 1 0.8
ξ(θ) 1
0.6 0.4
θˆ∗ (S2 )
0.8
0.2
0.6
0
0.4
0
10
20
30
40
50
θ
Fig. 6 Characterizing functions for θˆ ∗ (Si ) based on the data from Example 1
functions of the fuzzy parameter estimates θˆ ∗ (Si ), i = 1, 2, are depicted in Fig. 6. On the other hand, based on the above artificial precise data, the classical estimates for A and C are Aˆ = 1.28 and Cˆ = 14.48. In these estimates, fuzziness is ignored, which leads to non-suitable results for the realistic life times. Similarly, for the precise observation given above, the standard parameter estimates are θˆ (S1 ) = 28.26 and θˆ (S2 ) = 8.76. These estimates are based on incomplete information, which makes them unrealistic. Figures 4, 5 and 6 contain the characterizing functions of the fuzzy estimates obtained through the generalized proposed estimators. These estimates are based on both the stochastic variation and the fuzziness of the life times, which make them more realistic.
5.3 Step-stress model and fuzzy life times For the step-stress model explained in Sect. 4.3, the fuzzy estimates of the corresponding parameters can be obtained through the following equations:
Aδ θˆ1∗ =
n 1
j=1 t 1,( j),δ
n 1
n1
j=1 t 1,( j),δ
+ (n − n 1 )τ
n1
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+ (n − n 1 )τ
10
15
20
t [time]
25
Fig. 8 Fuzzy sample before time τ = 21
θˆ∗ (S1 )
0.2
5
,
∀ δ ∈ (0, 1]
(22)
ξi (t) 1 0.8 0.6 0.4 0.2 0
20
40
60
80
100
120
t [time]
Fig. 9 Fuzzy sample after time τ = 21
and
r j=n 1 +1 (t 2,( j),δ −τ )+(n − r )(t 2,(r ),δ − τ ) Aδ θˆ2∗ = , r − n1 r j=n 1 +1 (t 2,( j),δ − τ ) + (n − r )(t 2,(r ),δ − τ ) r − n1 ∀ δ ∈ (0, 1]. (23) From these generating families of intervals, using the mentioned Construction Lemma, the characterizing functions can be obtained. Example 2 For the construction of the characterizing functions of fuzzy parameter estimates, as an example, n = 20, r = 16, n 1 = 4, τ = 21 and S1 = 1, S2 = 2.5 are considered. Figure 7 shows the characterizing functions of the overall fuzzy sample before a specified time τ and after that time. Figures 8 and 9 show the separate samples before and after the specified time τ = 21, respectively. In Fig. 10, the characterizing function of the fuzzy estimate based on the fuzzy life times from Figs. 8 and 9 is
Beyond precision: accelerated life testing for fuzzy life time data
ξ(θ)
ξi (t)
1
1
θˆ2∗
0.8
0.8 0.6
0.6 0.4
0.4
θˆ1∗
0.2 0
20
40
60
80
100
0.2 0
θ
Fig. 10 Characterizing functions of θˆ1∗ and θˆ2∗ based on the algorithm from Sect. 5.3
2
4
6
5.4 Nonparametric estimation for fuzzy life times If we consider precise stress levels S1 , S2 , . . . , Sk and fuzzy ∗ , l = 1(1)n and i = 1(1)k having life time observations ti,l i ∗ δ-cuts Cδ (ti,l ) = t i,l,δ , t i,l,δ ∀δ ∈ (0 , 1], then the δcuts of the mean of these life time observations can be written as ni ni
∗ 1 1 t i,l,δ , t i,l,δ Cδ T i = ∀δ ∈ (0 , 1]. ni ni l=1
l=1
Defining T i,δ =
ni 1 t i,l,δ ni l=1
and
T i,δ =
ni 1 t i,l,δ , ni l=1
the
lower and upper ends of the generating family of intervals [γ δ , γ δ ]; ∀ δ ∈ (0 , 1] for the fuzzy estimator of γ can be calculated as
t [time]
10
Fig. 11 Fuzzy sample under S1 = 30
⎡
depicted. The result is obtained through the generalized proposed estimators. This estimate covers both uncertainties, i.e., the stochastic variation and the fuzziness of the life times, which makes it more realistic in daily life applications. Assuming n = 20 precise life time data points, r = 16, n 1 = 4, τ = 21, S1 = 1, and S2 = 2.5, i.e., the censored sample (3, 8, 12, 14) before τ = 21 under stress S1 and r − n 1 data (26, 31, 37, 42, 49, 53, 60, 71, 75, 83, 97, 98) under stress S2 = 2.5, the corresponding standard parameter estimates are θˆ1 = 93.75 and θˆ2 = 64.83. These estimates do not consider the fuzziness of the life time data, which makes these estimates non-realistic.
8
⎢ γ δ = min ⎢ ⎣
k k i=1 j=i+1
T S i,δ ln Sij ln T j,δ
2 Sj i=1 j=i+1 ln Si ⎤
k k Sj T i,δ ln ln i=1 j=i+1 Si T j,δ ⎥ ⎥
⎦ 2 k k Sj i=1 j=i+1 ln Si k
k
, (24)
∀δ ∈ (0, 1] and ⎡
⎢ γ δ = max ⎢ ⎣
k k i=1 j=i+1
T S i,δ ln Sij ln T j,δ
2 Sj i=1 j=i+1 ln Si ⎤
k k Sj T i,δ ln ln i=1 j=i+1 Si T j,δ ⎥ ⎥
⎦ 2 k k Sj i=1 j=i+1 ln Si k
k
,
∀δ ∈ (0 , 1]. (25) From these generating families of intervals, the characterizing function of the fuzzy estimator γˆ ∗ can be obtained through the above-mentioned Construction Lemma. Example 3 Below, are the characterizing functions of three fuzzy samples and the fuzzy parameter estimates under S1 , S2 , and S3 are given (Figs. 11, 12, 13, 14). On the other hand, assuming precise observations (0.4, 1.2, 2.5, 4.5, 7), (1.5, 3, 6, 10, 13), and (6, 10, 14, 18, 22) under stress levels S1 = 30, S2 = 20, and S3 = 10, respectively, the corresponding standard estimate is γˆ = 12.74. In this estimate, one uncertainty, i.e., fuzziness, is completely ignored, which means that this result is based on incomplete information. Therefore, it can lead to non-representative results.
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ξi (t) 1 0.8 0.6 0.4 0.2 0
5
10
15
t [time]
20
Fig. 12 Fuzzy sample under S2 = 20 ξi (t) 1 0.8 0.6 0.4
Therefore, in dealing with life time analysis, fuzzy number approaches are more suitable than precise life time models. Therefore, it is worth mentioning that in addition to the variation among the observations, the fuzziness of the observations must be added into the inference. In this study, some parametric and nonparametric classical estimation procedures for accelerated life testing are generalized to address the fuzziness of the observations. The proposed estimators are based on both uncertainties, which make them more applicable and suitable for life time analysis. The results of simulated fuzzy life times obtained under various stress conditions are presented and explained in examples. Since fuzzy set approaches utilize all the available information, i.e., fuzziness and stochastic variation, the results of the proposed estimators are more realistic.
0.2 0
10
20
t [time]
30
Fig. 13 Fuzzy sample under S3 = 10
Compliance with ethical standards Conflict of interest In this paper, the authors have no conflict of interest. Human and animal rights Furthermore, this study does not have any involvement with human-related data and only used computersimulated data.
ξ(γ) 1 0.8 0.6 0.4
References
0.2 0
1
2
3
4
γ
Fig. 14 Characterizing function of the fuzzy parameter estimate γˆ∗
Conclusion Statistics is now used in almost every field of life. Its use in medical and engineering sciences has been drastically developed over the last five decades. In engineering applications, for life time optimization, units are usually tested under various stress levels that are different than usual stress levels. Life time data obtained under different stress levels are modeled through accelerated life testing models. These models are based on precise life time observations and cover only the variation among the observations. However, it has already been shown that life time observations are not precise numbers but are more or less fuzzy. This clearly shows that accelerated life testing models ignore the fuzziness, which is unrealistic and may result in misleading results. The fuzziness of single observations must be integrated into the process. Otherwise, non-realistic results are obtained that hide the uncertainty obtained from fuzziness.
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