# A Gamma Process with Three Sources of Variability

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## Abstract

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## 1. Introduction

## 2. The Simple Gamma Process

## 3. The Gamma Process with Three Sources of Variability

## 4. The Proposed Method

- Fitting the degradation data to the gamma process with random scale;
- Obtaining the PDF $f\left({\zeta}_{i}\right)$ via convolution as defined in (4);
- Characterizing the CDF ${F}_{G3}\left({t}_{\zeta}\right)$.

- For the first step, consider that a degradation test has been conducted and that measurements about a performance characteristic of interest $\left({Z}_{i}\left({t}_{j}\right)\right)$ have been observed at times ${t}_{j}$ for $i=1,2,...,n$ devices under test, and $j=1,2,...,m$ measurements. Then, the degradation increments $\Delta {Z}_{i}\left({t}_{j}\right)={Z}_{i}\left({t}_{j}\right)-{Z}_{i}\left({t}_{j-1}\right)$ follow a gamma distribution as ${f}_{\Delta {Z}_{i}\left({t}_{j}\right)}\left(\Delta {z}_{ij}|v\Delta {t}_{j},{u}_{i}\right)$, where ${u}_{i}$ is considered to be a random parameter described by a gamma distribution as $f\left({u}_{i}|{a}_{u},{b}_{u}\right)$. The PDF of $\Delta {Z}_{i}\left({t}_{j}\right)$ with random ${u}_{i}$ is obtained by solving the integral in (2), which has a closed form as,$${f}_{\Delta {Z}_{i}\left({t}_{j}\right)}\left(\Delta {z}_{ij}|v,{a}_{u},{b}_{u}\right)=\frac{\Delta {z}_{ij}^{v\left(\Delta {t}_{j}\right)-1}{b}_{u}^{{a}_{u}}\Gamma \left(v\left(\Delta {t}_{j}\right)+{a}_{u}\right)}{\Gamma \left(v\left(\Delta {t}_{j}\right)\right)\Gamma \left({a}_{u}\right){\left(\Delta {z}_{ij}+{b}_{u}\right)}^{v\left(\Delta {t}_{j}\right)+{a}_{u}}}.$$The estimation of the parameters ${\theta}_{1}=\left(v,{a}_{u},{b}_{u}\right)$ from (6), can be performed by considering the MLE method. For this, the log-likelihood function is defined as,$${l}_{1}\left({\theta}_{1}\right)=\sum _{i=1}^{n}\sum _{j=1}^{m}ln\left({f}_{\Delta {Z}_{i}\left({t}_{j}\right)}\left(\Delta {z}_{ij}|v,{a}_{u},{b}_{u}\right)\right).$$Thus, $\widehat{{\theta}_{1}}$ is obtained by maximizing ${l}_{1}\left({\theta}_{1}\right)$ with respect $\left(v,{a}_{u},{b}_{u}\right)$. This process can be implemented in the R software by considering the optim routine.It can be noted, that for the characterization of the first-passage time CDF in (5), the estimated parameters $\left(\widehat{v},\widehat{{a}_{u}},\widehat{{b}_{u}}\right)$ are required. Furthermore, the PDF $f\left({\zeta}_{i}\right)$ is also needed.
- For the second step, the convolution operation in (4) is performed by considering that both ${\omega}_{i}$ and ${z}_{i}^{0}$ are random variables that are described by different PDFs. For this, consider that both characteristics are measured for every device under test. Thus, if n measurements from the devices under test for ${\omega}_{i}$ and ${z}_{i}^{0},i=1,2,\dots ,n$ are observed, then several PDFs can be considered to fit the datasets in the aims of finding the best fitting distributions $f\left({\omega}_{i}|{a}_{\omega},{b}_{\omega}\right)$ and $f\left({z}_{i}^{0}|{a}_{{z}^{0}},{b}_{{z}^{0}}\right)$. These distributions are considered to perform the convolution operation in in (4) to obtain $f\left({\zeta}_{i}\right)$.Fortunately, the integral in (4) can be solved by using the R software by considering the next code, Note that the functions in f.O and f.Z can be defined by any distribution. The d"densityname" should be completed as dnorm, dlnorm, dweibull, dgamma if the normal, lognormal, Weibull, or gamma distributions are considered. If other distributions are of interest, the corresponding name should be completed according to the R environment. In f.X, the integral is performed according to (4).
- For the third step, numerical integration implemented in R is considered to characterize the first-passage time CDF. Note from (5) that the estimated parameters $\left(\widehat{v},\widehat{{a}_{u}},\widehat{{b}_{u}}\right)$ obtained according to step 1 are required. Furthermore, as $f\left({\zeta}_{i}\right)$ has been obtained via the convolution operation described in step 2. Thus, the characterization of the CDF can be obtained by solving the corresponding integral. The code developed in R is presented next, From this code, note that the integrand denotes the multiplication of (3) and $f\left({\zeta}_{i}\right)$. Specifically, in $f1$, the CDF from (3) is presented, the estimated parameters via (7) are required. While, in $f2$ the PDF $f\left({\zeta}_{i}\right)$ obtained via convolution is presented, and x represents ${\zeta}_{i}$.The CDF ${F}_{G3}\left({t}_{\zeta}\right)$ is then obtained in cdf, where the integral is defined for the previously presented integrand. The sapply function is considered to obtain the corresponding probabilities for a given sequence of ${t}_{\zeta}$. Please note that, from ${F}_{G3}\left({t}_{\zeta}\right)$, the reliability function can be defined as, ${R}_{G3}\left({t}_{\zeta}\right)=1-{F}_{G3}\left({t}_{\zeta}\right)$, and the mean time to failure (MTTF) can be obtained as,$$MTTF={\int}_{0}^{\infty}{R}_{G3}\left({t}_{\zeta}\right)d{t}_{\zeta}.$$

## 5. Simulation Study

## 6. Implementation in a Case Study

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Degradation trajectories with different sources of variation. (

**a**) With three sources of variation. (

**b**) With no sources of variation.

**Figure 2.**Illustration for the considered distributions for ${\omega}_{i}$ and ${z}_{i}^{0}$ and the convoluted distribution for $f\left(\zeta \right)$. (

**a**) Illustration of the obtained distribution via convolution for scenario 1. (

**b**) Illustration of the obtained distribution via convolution for scenario 2. (

**c**) Illustration of the obtained distribution via convolution for scenario 3. (

**d**) Illustration of the obtained distribution via convolution for scenario 4. (

**e**) Illustration of the obtained distribution via convolution for scenario 5.

**Figure 3.**Comparison of estimated CDFs for the five scenarios considering the four gamma process models with different sources of variability. (

**a**) Estimated CDFs for the first scenario. (

**b**) Estimated CDFs for the second scenario. (

**c**) Estimated CDFs for the third scenario. (

**d**) Estimated CDFs for the fourth scenario. (

**e**) Estimated CDFs for the fifth scenario.

**Figure 4.**Comparison of estimated reliabilities for the five scenarios considering the four gamma process models with different sources of variability. (

**a**) Estimated reliabilities for the first scenario. (

**b**) Estimated reliabilities for the second scenario. (

**c**) Estimated reliabilities for the third scenario. (

**d**) Estimated reliabilities for the fourth scenario. (

**e**) Estimated reliabilities for the fifth scenario.

**Figure 5.**Densities for the initial level of degradation, the threshold, and the convolution of both for the case study.

**Figure 6.**Estimated reliabilities for the gamma process under different scenarios of variability for the case study.

Scenario | ${\mathit{\omega}}_{\mathit{i}}$ | ${\mathit{z}}_{\mathit{i}}^{0}$ |
---|---|---|

1 | Lognormal | Normal |

${f}_{L}\left({\omega}_{i}|{a}_{\omega}=0.7,{b}_{\omega}=0.05\right)$ | ${f}_{N}\left({z}_{i}^{0}|{a}_{{z}^{0}}=0.5,{b}_{{z}^{0}}=0.09\right)$ | |

2 | Weibull | Weibull |

${f}_{W}\left({\omega}_{i}|{a}_{\omega}=20,{b}_{\omega}=2.3\right)$ | ${f}_{W}\left({z}_{i}^{0}|{a}_{{z}^{0}}=5.6,{b}_{{z}^{0}}=0.44\right)$ | |

3 | Gamma | Weibull |

${f}_{G}\left({\omega}_{i}|{a}_{\omega}=396.6,{b}_{\omega}=0.0056\right)$ | ${f}_{W}\left({z}_{i}^{0}|{a}_{{z}^{0}}=5.5,{b}_{{z}^{0}}=0.5\right)$ | |

4 | Normal | Gamma |

${f}_{N}\left({\omega}_{i}|{a}_{\omega}=2.2,{b}_{\omega}=0.12\right)$ | ${f}_{G}\left({z}_{i}^{0}|{a}_{{z}^{0}}=23,{b}_{{z}^{0}}=0.05\right)$ | |

5 | Lognormal | Gamma |

${f}_{L}\left({\omega}_{i}|{a}_{\omega}=0.7,{b}_{\omega}=0.05\right)$ | ${f}_{G}\left({z}_{i}^{0}|{a}_{{z}^{0}}=23,{b}_{{z}^{0}}=0.05\right)$ |

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**MDPI and ACS Style**

Rodríguez-Picón, L.A.; Méndez-González, L.C.; Pérez-Olguín, I.J.C.; Hernández-Hernández, J.I. A Gamma Process with Three Sources of Variability. *Symmetry* **2023**, *15*, 162.
https://doi.org/10.3390/sym15010162

**AMA Style**

Rodríguez-Picón LA, Méndez-González LC, Pérez-Olguín IJC, Hernández-Hernández JI. A Gamma Process with Three Sources of Variability. *Symmetry*. 2023; 15(1):162.
https://doi.org/10.3390/sym15010162

**Chicago/Turabian Style**

Rodríguez-Picón, Luis Alberto, Luis Carlos Méndez-González, Iván Juan Carlos Pérez-Olguín, and Jesús Israel Hernández-Hernández. 2023. "A Gamma Process with Three Sources of Variability" *Symmetry* 15, no. 1: 162.
https://doi.org/10.3390/sym15010162