Nonlinear Stochastic Modeling with Heterogeneous Covariates for Degradation Analysis Applied to Wax Lubrication Layer
Abstract
:1. Introduction
1.1. Background
1.2. Motivating Example
1.3. Literature Review
1.4. Contribution and Outline
- (1)
- By addressing the challenge of modeling the unknown and nonlinear behavior of wax creep under the influence of temperature, we provide a comprehensive analysis of this complex phenomenon.
- (2)
- By utilizing a new degradation model based on the Wiener process, we describe the inherent relationships between creep and time without assuming a specific parametric form.
- (3)
- By considering that the working environment is different even if the covariates preset are the same, we view covariates as random variables to capture the heterogeneity among degradation data.
- (4)
- We propose a nonparametric estimator to estimate the nonlinear function concerning time and derive the corresponding asymptotic results.
2. Methodology
2.1. Nonlinear Wiener Process Models
2.2. Semiparametric Estimation via ML
2.3. The EM Algorithm for the MLE
3. Numerical Experiments
4. Case Study
4.1. Data Overview
4.2. Estimation Results
5. Conclusions
Supplementary Materials
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Estimate | 0.1080 | 0.4957 | 0.4601 | 0.0998 | 0.4984 | 0.5043 |
SD | 0.0236 | 0.0322 | 0.0656 | 0.0028 | 0.0076 | 0.0263 |
Estimate | 0.1010 | 0.4886 | 0.4840 | 0.0999 | 0.4960 | 0.4963 |
SD | 0.0145 | 0.0274 | 0.0501 | 0.0032 | 0.0096 | 0.0230 |
Estimate | 0.1235 | 0.4914 | 0.4751 | 0.1155 | 0.4981 | 0.4861 |
SD | 0.0326 | 0.0288 | 0.0667 | 0.0247 | 0.0078 | 0.0427 |
Estimate | 0.1099 | 0.4907 | 0.4720 | 0.0994 | 0.4974 | 0.4871 |
SD | 0.0257 | 0.0286 | 0.0582 | 0.0113 | 0.0103 | 0.0388 |
Estimate | 0.4886 | 0.4769 | 0.4923 | 0.5025 | 0.4974 | 0.4933 |
SD | 0.0429 | 0.0350 | 0.1141 | 0.0125 | 0.0097 | 0.0313 |
Estimate | 0.4885 | 0.4663 | 0.5202 | 0.4977 | 0.4989 | 0.4906 |
SD | 0.0378 | 0.0419 | 0.1478 | 0.0129 | 0.0126 | 0.0285 |
Estimate | 0.4851 | 0.4836 | 0.4751 | 0.5009 | 0.4975 | 0.4936 |
SD | 0.0299 | 0.1217 | 0.0667 | 0.0128 | 0.0086 | 0.0285 |
Estimate | 0.4789 | 0.5099 | 0.4720 | 0.4991 | 0.4979 | 0.4986 |
SD | 0.0342 | 0.1360 | 0.0582 | 0.0119 | 0.0097 | 0.0284 |
Estimate | 0.0209 | 0.0142 | 0.0101 | 0.0068 | 0.0494 | 0.0347 | 0.0242 | 0.0148 |
SD | 0.0158 | 0.0103 | 0.0061 | 0.0037 | 0.0167 | 0.0118 | 0.0081 | 0.0050 |
Estimate | 0.0054 | 0.0040 | 0.0027 | 0.0017 | 0.0175 | 0.0110 | 0.0077 | 0.0044 |
SD | 0.0028 | 0.002 | 0.0014 | 0.0006 | 0.0082 | 0.0042 | 0.0028 | 0.0016 |
Estimate | 0.0247 | 0.0193 | 0.0147 | 0.0097 | 0.0671 | 0.0403 | 0.0295 | 0.0172 |
SD | 0.0183 | 0.0159 | 0.0134 | 0.0082 | 0.0390 | 0.0166 | 0.0113 | 0.0086 |
Estimate | 0.0107 | 0.0068 | 0.0050 | 0.0032 | 0.0321 | 0.0195 | 0.0136 | 0.0085 |
SD | 0.0055 | 0.0038 | 0.0034 | 0.0020 | 0.0134 | 0.0065 | 0.0033 | 0.0024 |
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Li, S.; Tian, Y.; Wang, D. Nonlinear Stochastic Modeling with Heterogeneous Covariates for Degradation Analysis Applied to Wax Lubrication Layer. Mathematics 2025, 13, 872. https://doi.org/10.3390/math13050872
Li S, Tian Y, Wang D. Nonlinear Stochastic Modeling with Heterogeneous Covariates for Degradation Analysis Applied to Wax Lubrication Layer. Mathematics. 2025; 13(5):872. https://doi.org/10.3390/math13050872
Chicago/Turabian StyleLi, Shixiang, Yubin Tian, and Dianpeng Wang. 2025. "Nonlinear Stochastic Modeling with Heterogeneous Covariates for Degradation Analysis Applied to Wax Lubrication Layer" Mathematics 13, no. 5: 872. https://doi.org/10.3390/math13050872
APA StyleLi, S., Tian, Y., & Wang, D. (2025). Nonlinear Stochastic Modeling with Heterogeneous Covariates for Degradation Analysis Applied to Wax Lubrication Layer. Mathematics, 13(5), 872. https://doi.org/10.3390/math13050872