# The Neutrosophic Lognormal Model in Lifetime Data Analysis: Properties and Applications

Authors: Sultan Salem; Zahid Khan; Hamdi Ayed; Ameni Brahmia; Adnan Amin
Journal: Journal of Function Spaces
 (2021)

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## Abstract

The lognormal distribution is more extensively used in the domain of reliability analysis for modeling the life-failure patterns of numerous devices. In this paper, a generic form of the lognormal distribution is presented that can be applied to model many engineering problems involving indeterminacies in reliability studies. The suggested distribution is especially effective for modeling data that are roughly symmetric or skewed to the right. In this paper, the key mathematical properties of the proposed neutrosophic lognormal distribution (NLD) have been derived. Throughout the study, detailed examples from life-test data are used to confirm the mathematical development of the proposed neutrosophic model. The core ideas of the reliability terms, including the neutrosophic mean time failure, neutrosophic hazard rate, neutrosophic cumulative failure rate, and neutrosophic reliability function, are addressed with examples. In addition, the estimation of two typical parameters of the NLD by mean of maximum likelihood (ML) approach under the neutrosophic environment is described. A simulation experiment is run to determine the performance of the estimated parameters. Simulated findings suggest that ML estimators effectively estimate the unknown parameters with a large sample size. Finally, a real dataset on ball bearings failure times has been considered an application of the proposed model.


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## Body

## 1. Introduction

In anticipating the long-term reliability of electronic components and devices, identifying failure distribution and assessing the failure mechanism is always critical in reliability analysis. To characterize the behavior of the products, many lifespan models have been proposed [1]. These lifetime models have been categorized according to increasing, decreasing, and constant failure patterns [2, 3]. The lognormal distribution has also been called the most often utilized life distribution model in reliability domains [4]. This distribution effectively fits data that is roughly symmetric or skewed to the right [5]. The lognormal model is equally well at fitting a specific set of lifetime data [6]. The lognormal model is commonly employed to model the failure time of components that fail due to stress or fatigue, such as failure caused by chemical reactions or deterioration, for example, diffusion, corrosion, or migration [7]. The lognormal model also provides several advantages over previous models in software reliability [8, 9]. From a statistical viewpoint, a lognormal model is a particular form of the Gaussian family; most traditional texts on reliability and statistical distributions include a complete description of its features in terms of assumed precise data and characteristics parameters [10–13]. In reality, we encounter circumstances when the lifespan data acquired from studies do not fit any commonly used lifetime models. As a result, there is always the possibility of considering alternative statistical distributions to model the failure mechanism of different products.A novel extension of the lognormal distribution is proposed in this paper to broaden its utility in applied statistical research. This extension is sparked by Smarandache’s work on the concept of neutrosophy [14]. A neutrosophic logic is used to assist the investigation of assertions that are either false or true but are also indeterminate, neutral, inconsistent, or anything in between [15–19]. On the mathematical side, every field has a neutrosophic component which is called indeterminacy. Smarandache pioneered the applications of the neutrosophic methods in statistics, precalculus, and calculus to account for inaccuracy in studied variables [20]. In a view of neutrosophic statistics (NS), indeterminacy in statistical modeling has become a study area of interest for many researchers. The neutrosophic concept of statistical modeling has only come to be described in some recent publications [21–23]. Describing data using neutrosophic descriptive methods and neutrosophic probability are discussed in [24, 25]. Applications of neutrosophy in decision-making in quality control are appeared to be quite efficient [26]. The neutrosophic algebraic structures of probability models were first initiated by Salama et al. [27]. The efforts on NS have always mainly concentrated on the applications side of neutrosophy, rarely addressing the algebraic structures of probability distributions in detail.With the primary objective of integrating ambiguous knowledge about the study variables, the concept of the NLD has been explained in this work. Study parameters that are ambiguous cannot be ignored for practical analysis and must be incorporated into the model used to represent a data-generating process. The neutrosophic structure of the lognormal model has never been addressed in earlier studies to the best of our knowledge.The remaining part of the work is presented as follows: in Section2, the neutrosophic extension of the lognormal distribution is developed. The notion of NLD has been demonstrated with some examples in Section 3. The mathematical treatment for the unknown parameters of the NLD is explained in Section 4. In this section, a simulation analysis for indicating the performance of the neutrosophic parameters is also conducted. An expression for the quantile function under the neutrosophic environment is established in Section 5. An application of the proposed model is described in Section 6. Finally, Section 7 summarizes the main research findings.


## 2. Proposed Neutrosophic Model with Some Useful Reliability Characteristics

IfX~=lnT~ follows a neutrosophic normal distribution, a random variable T~>0 is said to follow the NLD with the density function:
(1)ωnt~=12πσn~tˇexp−lnt~−μn22σ2nI0,∞t~;μn,σn>0,whereμn=μl,μu is the neutrosophic location, σn=σl,σu is the neutrosophic shape parameters on the log scale, and T~ denotes the failure time of a system or a component in a reliability context. For the selected values of μn and σn, the neutrosophic density PDF~ is sketched in Figure 1.Figure 1
The density function of NLD with neutrosophic parameters (a)σn=0.3,0.6 and (b) μn=0,0.3.
(a)(b)Figure1 shows the general statistical pattern of thePDF~ when it is assumed that indeterminacy, respectively, exists in the scale and shape parameters of the distribution. The shaded part in Figure 1 indicates the neutrosophic region due to imprecision in the defined parameters of the distribution. Figure 1 area under the neutrosophic curve, say from t1 to t2, provides the failure probability of a system that is assumed to follow the statistical pattern of the NLD.Likewise, the other important function of the NLD is the neutrosophic cumulative functionCDF~. This CDF~ is used to estimate the probability that how many operational objects have failure time less than or equal to the prescribed time, say t1. The CDF~ of the NLD is given by
(2)Ψt~=Flnt~−μnσn.Note that NLD convert to the existing lognormal distribution whenμl=μu=μ and σl=σu=σ, where the F. represents the distribution function of the neutrosophic standardized normal model. The derivation of (2) under the neutrosophic environment will be discussed later in this section. However, the curve of CDF~ for imprecise values of μn and σn is given in Figure 2.Figure 2
TheCDF~ curve for the vague parameter values (a) σn=0.3,0.6 and (b) μn=0,0.3.
(a)(b)In Figure2(a), σn is crisp value but μnis given in neutrosophic interval, whereas in Figure 2(b), μn is crisp value while σn is given in the uncertain form.All essential features of the proposed NLD, such as moments, shape coefficients, and the moment generating function, are based on the algebraic framework of the neutrosophic numbers [28].If we assume the two bounded real intervalsT1=p1,q1 and T2=p2,q2 and ∆ denotes the basic arithmetic operations, then
(3)p1,q1Δp2,q2=γ1,γ2,where
(4)γ1,γ2=pΔq∣p1≤p≤q1,p2≤q≤q2.We further assume that zero is not included in the intervalp2,q2 and if ∆ denotes the division operation, then (4) further can be expressed as
(5)T1+T2=p1+p2,q1+q2,T1−T2=p1−q2,q1−p2,T1T2=p1q2,q1p2,T1∗T2=ϑ1,ϑ2,where ϑ1=minp1p2,p1q2,q1p2,q1q2 and ϑ2=maxp1p2,p1q2,q1p2,q1q2.Numerical results of the proposed NLD have been obtained by using interval arithmetic.Definition 1.
Neutrosophic data extends the classic data that contain some imprecise, vague, or indeterminacy in some or all values. In general terms, it can be represented as(6)x=constant+I,where I∈u,l; for example, 7+I where I∈3,3.5.In addition to specific patterns of the components or system reliability that are best described by the NLD in form ofPDF~ and CDF~ curves, a practitioner may take interest to know some other beneficial distributional properties of the NLD, which can be established in the theorems.Theorem 2.
The reliability function of the NLD is1−Flnt~–μl/σl,1−Flnt~–μu/σu.Proof.
The reliability function of NLD is defined as(7)ξt~=PT~>t~=1−PT~<t~=1−∫0t~ωNt~dt~=1−∫0t~ωlt~,ωut~dt~=1−∫0t~ωlt~dt~,1−∫0t~ωut~dt~=ξl,ξu,where ξl=1−∫0tωlt~dt~=1−Flnt~−μl/σl and ξu=1−∫0tωut~dt~=1−Flnt~−μu/σu, hence proved.The functionξt~ is particularly helpful in reliability studies, which connects a unit’s age to its chance of living to that age. In the presence of indeterminacy in parameters, the function ξt~ of the NLD would be displayed as shown in Figure 3. This function may be used to determine the failure rate function, the conditional function, and the PDF~.Corollary 3.
The hazard functionRt~ of the NLD is Flnt~−μn/σn/t~σn1−Flnt~−μn/σn.Proof.
The ratio of theωt~ to ξt~results in the desired Rt~.Corollary 4.
The distribution functionCDF~ of the NLD isFlnt~−μn/σn.Proof.
TheCDF~ can be yielded by solving the following expression:
(8)Ψt~=∫0t~ωt~dt~.
By assumingT~=expZσn+μn, we can write
(9)=PΖ≤lnt~−μnσn,Ψt~=Flnt~−μnσn.Figure 3
Reliability function of the NLD with neutrosophic parameters (a)σn=0.3,0.6 and (b) μn=0,0.3.
(a)(b)Consequently, the hazard rate of components with a specific aget~ is described by an interval rate of death in neutrosophy philosophical terms.Theorem 5.
The median of the NLD isexpμl,expμu.Proof.
The medianM~ of NLD can be found as
(10)∫0M~Ψt~dt~=12,12,(11)∫0M~Ψlt~dt~,∫0M~Ψut~dt~=12,12,where Ψlt~=Flnt~−μl/σl and Ψut~=Flnt~−μu/σu.
Analytical simplification of (11) implies
(12)lnM~−μlσl=0,lnM~−μuσu=0.
Implying thereby(13)M~=expμl,expμu.Corollary 6.
First quantileQ1~ and the third quantile Q3~ of the NLD are expμl+σlF−11/4,expμu+σuF−11/4 and expμl+σlF−13/4,expμu+σuF−13/4, respectively.Proof.
By definitionQ1~ and Q3~ correspond to solutions of the following expressions,
(14)∫0Q1~Ψt~dt~=14,14,∫0Q3~Ψt~dt~=34,34.
Following theorem5 implies
(15)Q1~=expμl+σlF−114,expμu+σuF−114,Q3~=expμl+σlF−134,expμu+σuF−134,where F−1. is the quantile point of the standard normal variate.Theorem 7.
The neutrosophic average time to failure of the NLD isexpμn+σn2/2(16)=∫0∞t~ωut~,t~ωlt~dt~seeFigure2,=∫0∞tˇ12πσltˇexp−lnt~−μl22σ2ldt~,∫0∞tˇ12πσutˇexp−lnt~−μu22σ2udt~.
The transformationZ~=lnT~ yields
(17)=expσl2+μl2−μl22σl2∫−∞∞1σl√2πexp−z~−σl2+μl22σl2dz~,expσu2+μu2−μu22σu2∫−∞∞1σu√2πexp−z~−σu2+μu22σu2,=expμl+σl22,expμu+σu22,=expμn+σn22.Theorem 8.
The variance of the NLD iseσn2−1e2μn+σn2.Proof.
By definition, variance is(18)ϑt~=Et~2−Ω2,where ϑt~ stands for neutrosophic variance
Now,(19)Et~2=∫0∞t~2ωnt~dt~.
It follows(20)Et~2=∫0∞tˇ12πσlexp−lnt~−μl22σ2ldt~,∫0∞tˇ12πσuexp−lnt~−μu22σ2udt~.
Consider the transformationZ~=lnT~−μn/σn.
Thus, (20) yields
(21)=exp2σl2+2μl∫−∞∞12πexp−z~−2σl22dz~,exp2σu2+2μu∫−∞∞12πexp−z~−2σu22dz~.
Simplifying further provides(22)=exp2σl2+2μl,exp2σu2+2μu,=exp2σn2+2μn.
Thus, from (18), we can write the variance as
(23)ϑt~=eσn2−1e2μn+σn2,where ϑt~∈eσl2−1e2μl+σl2,eσu2−1e2μu+σu2.In a neutrosophy context, it is also possible to establish the other characteristics of the NLD. To further comprehend the underlying notion of the NLD, several examples of the proposed model are provided.


## 3. Illustrative Examples

The concept of NLD has been illustrated in this section using some examples from the field of reliability research.Example 9.
Failure time (in months) of a bearing used in the washing machine is adequately followed by a neutrosophic lognormal random variable with parametersμn=2.1,2.9 and σn=0.85,0.99. What is the probability that the lifespan will exceed 14 months?
Solution From theCDF~(24)PT~>t~=1−Ψt~,=1−Ψ.
Consequently,(25)=1−Fln14−2.1,2.90.85,0.99,=1−F−0.307,0.544,=0.293,0.620.Thus, the failure probability that bearing lifetime exceeds 14 months with given neutrosophic parameters is approximately29,62%.Example 10.
The time to failure of a semiconductor laser (in hours) is modeled as an NLD, with [9000, 15000] hours and [20000, 23000] hours being the neutrosophic mean and standard deviation, respectively. Find the parametersμn and σn2 of the NLD.
Solution given that(26)expμn+σn22=9000,12000,(27)eσn2−1e2μn+σn2=200002,230002.
Using the transformation,(28)xn=expμn,yn=expσn2.
Equations (26) and (27) can be written as
(29)xnyn=9000,12000,(30)xn2ynyn−1=200002,230002.
Squaring (29) and substituting into (30) provides the solution for yn as
(31)yn=4.67,5.94.
Thus,(32)expσn2=4.67,5.94.Further simplification of (32) provides
(33)σn2=1.54,1.78.Using (30) into (29) provides the following solution:
(34)xn=4164.70,4923.66,expμn=4164.70,4923.66.Thus, from (32), we can write
(35)μn=8.33,8.52.


## 4. Estimation Procedure

In this section, the neutrosophic maximum likelihood estimateML~ technique has been developed for estimating the distributional parameters of NLD. Assume a sample of values Tj,i=1,2,..n taken from the NLD. Which neutrosophic parameter values for an observed sample should be used? To answer this, we have to determine these unknown values by the likelihood function of the proposed distribution. Therefore, ML~ the function of the NLD is characterized by
(36)Ynμn,σ2n∣t~=∏j=1nωnt~j.For the seriest~jj=1,2,,⋯n, the log-likelihood of (36) is given by
(37)Ynμn,σ2n∣t~=∑j=1nlnt~j−nln2πσ2n2−nμn22σ2n+∑j=1nlnt~jσ2n−∑j=1nlnt~j22σ2n.To maximizeYn., the gradient concerning unknown quantities μn and σ2n is given by
(38)∂Ynμn,σ2n∣t~∂μn=∑j=1nlnt~jσ2n−2nμn2σ2n,(39)∂Ynμn,σ2n∣t~∂σ2n=−n2σ2n+∑j=1nlnt~j−μn22σ2n2.Setting the gradian (38) and (39) equal to zero provides the simultaneous solution as
(40)μ^n=∑j=1nlnt~jn,σ∧2n=∑j=1nlnt~j−∑j=1nlnt~j/n2n,where μ^n=μ^l,μ^u and σ∧2n=σ∧2l,σ∧2u are the required estimators of the parameters μn and σ2n, respectively.Note that theML~ method has been developed here to estimate the parameters of the NLD; however, other estimation approaches may also be implemented similarly. The performance of ML~ estimators is evaluated in terms of the neutrosophic average biased AB~ and neutrosophic root mean square error RMS~ as defined below [5]:
(41)AB~=∑j=1Nθ^j−θNN,RMS~=∑j=1Nθ∧j−θN2N,whereθ^j represents the estimators μ^n or σ^n in the repeated runs.Using theR programming language, a Monte Carlo simulation is carried out with varied sample sizes and fixed values of the parameters.μn=1.5,2 and σn=1,1. An uncertain dataset is originated from the NLD with the parameters as mentioned earlier, and simulation analysis is replicated for a total of N=105 times with sample sizes of n=5,10,20, 50, 150, and 300, respectively. The performance measures of ML~ estimators are then computed and given in Table 1Table 1
Performance ofML~ estimators for simulated data.
Sample sizeμ^nσ^n2AB~RMS~AB~RMS~5[0.072, 0.119][4.352, 7.176][2.848, 4.696][7.285, 12.011]10[0.019, 0.032][3.035, 5.003][2.017, 3.326][6.096, 10.051]20[0.009, 0.0164][2.177, 3.590][1.398, 2.305][5.347, 8.816]50[0.009, 0.015][1.371, 2.259][0.774, 1.276][4.073, 6.715]150[0.006, 0.011][0.787, 1.299][0.415, 0.685][2.807, 4.628]300[0.004, 0.007][0.556, 0.917][0.238, 0.393][2.176, 3.589]We can see from the results given in Table1 that as the sample size n grows, the AB~ and RMS~ decrease. This demonstrates that neutrosophic estimators provide better reliability and efficiency in estimating moderate or more extensive sample sizes.


## 5. Quantile Function

The quantile functionQF~ of NLD may be obtained by solving the equation for Y shown below:
(42)Yj=F−1ϑj,whereϑj is distributed uniformly with parameters 0 and 1, i.e.,ϑj~U0,1.In the case of NLD,(43)Yj=lnT~j−μnσn,j=1,2,⋯.Statistical speaking, theQF~ is employed to generate and extend quantile analogs of conventional moment-based descriptive metrics. This function QF~ may be used to generate random data that corresponds to the density specified in (1). Analytical properties of NLD can be validated by utilizing the simulated data from (24).InR software, the NLD can be easily simulated to view the validity of theory-based derived results. We set μn=1.5,2 and σn=1,1 in the NLD and produce 10000 samples at random from U0,1. The neutrosophic random samples are then generated for NLD with the aid of (42). The exact findings for distributional properties of NLD and the simulated outcomes at the above-specified values of parameters are shown in Table 2Table 2
Comparisons of the simulated results with analytical results of the NLD.
Distributional propertiesExact resultsSimulated resultsMeanΩ[7.38, 12.17][7.38, 12.18]Standard deviationϑt~[9.72, 16.02][9.68, 15.98]First quartileQIN[2.28, 3.75][2.20, 3.81]MedianM~[4.47, 7.37][4.48, 7.38]Third quartileQ3N[8.78, 14.48][8.80, 14.46]Thus, simulated results in Table2 are in great agreement with those obtained from the analytical properties of the NLD.


## 6. Illustrative Application

Manufactured items that are vulnerable to wear and tear are typically evaluated for durability with a view to determining their useful failure times. This lifetime data is essential in various fields, including engineering, biomedical, and social sciences. For the real application of NLD, data used for analysis are taken from Lawless [29]. This data represents the number of million spins for each of the 23 bearing balls before failure had resulted. The lognormal and Weibull models have been applied to this life test data. However, the lognormal model is slightly appropriate than the Weibull model with fitting estimates μ^=4.16 and σ^=0.533. The histogram and essential quantile plot of the original data are depicted in Figures 4 and 5, respectively.Figure 4
Histogram of the bearing balls data.Figure 5
The quantile plot of the bearing balls data.Clearly, Figure4 shows that data are skewed to the right, whereas Figure 5 indicates how well the lognormal model fitted the failure time measurements. Figure 6 also supports the best fitting of the lognormal distribution.Figure 6
Theoretical and empirical CDF plot.At first, data are the precise measurements; but, for illustrative purposes, we interpret them as indeterminate sample values for certain bearing balls, as shown in Table3.Table 3
Imprecise data on failure times of 23 bearing balls.
Failure time data17.88, 28.92, [33.00, 40.6], 41.52, 42.12, 45.60,[48, 52.2], 51.84, 51.96, 54.12, 55.56, 67.80, 68.64,68.64, 68.88, 84.12, 93.12, 98.64, 104.12,[105.4,112], 127.92, 128.04, 173.40Table3 indicates that failure times on certain bearing balls such as [33.00, 40.6], [48, 52], and [105.40, 112] are not accurately recorded to precise values but are given in intervals. Indeed, vague or incomplete information in the sample leads to the inappropriateness of the existing lognormal model. On the other hand, the proposed distribution can easily be employed to analyze neutrosophic set of measurements. The descriptive measures of the proposed NLD are given in Table 4.Table 4
Neutrosophic summary of the bearing balls dataset.
Descriptive measuresMeanΩ[4.141, 4.180]Standard deviationϑt~[0.513, 0.521]First quantileQ1~[44.610, 45.550]MedianM~[63.423, 64.390]Third quantileQ3~[90.172, 91.0048]From Table4, it can be viewed that the essential numerical characteristics of ball bearings data are in intervals on account of certain indeterminacies in the sample. Thus, the proposed model can be applied to analyze the data, which follows the NLD.


## 7. Conclusions

The NLD as a new generic version of the lognormal distribution has been suggested in this study. The structural features of the proposed model have been elaborately discussed. The analytical results for the neutrosophic descriptive measures and the other associated properties are obtained. The NLD precisely reflects the failure trends of many inservice components. The estimation technique has been developed and described with examples under vague information in the observed data. Furthermore, the notion of the neutrosophic quantile function is introduced that can be used to validate the analytical results of the proposed NLD. To verify the performance of the calculated neutrosophic parameters, a simulation analysis has been carried out. The results of the simulations show that imprecisely defined sample data with a considerable size can be employed in order to accurately predict an unknown parameter. The illustrative application also validates the practicality of the NLD in applied statistical problems.We believe that by broadening the scope of NLD in reliability research, this neutrosophic generalization of the classical model can be extended for other statistical models.



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