Reliability Estimation in Series Systems: Maximum Likelihood Techniques for Right-Censored and Masked Failure Data

Alex Towell Email: lex@metafunctor.com GitHub: github.com/queelius

---

Abstract

This paper investigates maximum likelihood techniques to estimate component reliability from masked ailure data in series systems. A likelihood model accounts for right-censoring and candidate sets indicative of masked failure causes. Extensive simulation studies assess the accuracy and precision of maximum likelihood estimates under varying sample size, masking probability, and right-censoring time for components with Weibull lifetimes. The studies specifically examine the accuracy and precision of estimates, along with the coverage probability and width of BCa confidence intervals. Despite significant masking and censoring, the maximum likelihood estimator demonstrates good overall performance. The bootstrap yields correctly specified confidence intervals even for small sample sizes. Together, the modeling framework and simulation studies provide rigorous validation of statistical learning from masked reliability data.

See: https://github.com/queelius/reliability-estimation-in-series-systems/blob/bookdown/pdfbook/AlexTowellPaper.pdf

---

Context & Motivation

Reliability in Series Systems is like a chain's strength -- determined by its weakest link.

• Essential for system design and maintenance.

Main Goal: Estimate individual component reliability from failure data.

Challenges:

• Masked component-level failure data.
• Right-censoring system-level failure data.

Our Response:

• Derive techniques to interpret such ambiguous data.
• Aim for precise and accurate reliability estimates for individual components using maximum likelihood estimation (MLE)
• Quantify uncertainty in estimates with bootstrap confidence intervals (CIs).

---

Core Contributions

Likelihood Model for Series Systems.

• Accounts for right-censoring and masked component failure.

Specifications of Conditions:

• Assumptions about the masking of component failures.
• Simplifies and makes the model more tractable.

Simulation Studies:

• Components with Weibull lifetimes.
• Evaluate MLE and confidence intervals under different scenarios.

R Library: Methods available on GitHub.

• See: www.github.com/queelius/wei.series.md.c1.c2.c3

---

What Is A Series System?

Critical Components: Complex systems often comprise critical components. If any component fails, the entire system fails.

• We call such systems series systems.
• Example: A car's engine and brakes.

System Lifetime is dictated by its shortest-lived component:

$$ T_i = \min(T_{i 1}, \ldots, T_{i 5}) $$

where:

• $T_i$ is the lifetime of $i^{\text{th}}$ system.
• $T_{i j}$ is the $j^{\text{th}}$ component of $i^{\text{th}}$ system.

---

Reliability Function

Reliability Function represents the probability that a system or component functions beyond a specified time.

• Essential for understanding longevity and dependability.

Series System Reliability: Product of the reliability of its components:

$$ R_{T_i}(t;\theta) = \prod_{j=1}^m R_j(t;\theta_j). $$

• If any component has low reliability, it can impact the whole system.

• Here, $R_{T_i}(t;\theta)$ and $R_j(t;\theta_j)$ are the reliability functions for the system $i$ and component $j$, respectively.

---

Hazard Function: Understanding Risks

Hazard Function: Measures the immediate risk of failure at a given time, assuming survival up to that moment.

• Reveals how the risk of failure evolves over time.
• Guides maintenance schedules and interventions.

Series System Hazard Function: Sum of the component hazard functions:

$$ h_{T_i}(t;\theta) = \sum_{j=1}^m h_j(t;\theta_j). $$

• Components' risks are additive.

---

Joint Distribution of Component Failure and System Lifetime

Our likelihood model depends on the joint distribution of the system lifetime and the component that caused the failure.

• Formula: Product of the failing component's hazard function and the system reliability function:

$$ f_{K_i,T_i}(j,t;\theta) = h_j(t;\theta_j) R_{T_i}(t;\theta). $$

• Here, $K_i$ denotes component cause of $i^{\text{th}}$ system's failure.

---

Component Cause of Failure

We can use the joint distribution to calculate the probability of component cause of failure.

• Helps predict the cause of failure.
• Derivation: Marginalize the joint distribution over the system lifetime:

$$ \Pr{K_i = j} = E_{\theta} \biggl[ \frac{h_j(T_i;\theta_j)} {h_{T_i}(T_i;\theta_l)} \biggr]. $$

• Well-Designed Series System: Components exhibit comparable chances of causing system failures.
• Relevance: Our simulation study employs a (reasonably) well-designed series system.

---

Likelihood Model: Data Generating Process

The data generating process (DGP) is the underlying process that generates the data. Green elements are observed, red elements are latent:

• Right-Censored lifetime: $S_i = \min(T_i, \tau_i)$
• Event Indicator: $\delta_i = 1_{{T_i < \tau_i}}$
• Candidate Set: $\mathcal{C}_i$ related to components

---

Likelihood Function

Likelihood Function measures how well model explains the data:

• Right-Censored data ($\delta_i = 0$).
• Candidate Sets or Masked Failure data ($\delta_i = 1$)

Masked Data

Sys	Right-Censored Lifetime ($S_i$)	Event ($\delta_i$)	Candidate ($\mathcal{C}_i$)
1	$1.1$	1	${1,2}$
2	$5$	0	$\emptyset$

---

Likelihood Function: Total Likelihood

Each system contributes to total likelihood via its likelihood contribution:

$$ L(\theta|\text{data}) = \prod_{i=1}^n L_i(\theta|\text{data}_i) $$

where data$_i$ is for $i^{\text{th}}$ system and $L_i$ is its contribution.

---

Likelihood Contribution: Right-Censoring

Right-Censoring: For the $i^{\text{th}}$ system, if right-censored ($\delta_i = 0$) at duration $\tau$, its likelihood contribution is proportional to the system reliability function evaluated at $\tau$:

$$ L_i(\theta) \propto R_{T_i}(\tau;\theta). $$

• We only know that a failure occurred after the right-censoring time.
• This is captured by the system reliability function.

Key Assumptions:

• Censoring time ($\tau$) independent of parameters.
• Event indicator ($\delta_i$) is observed.
• Reasonable in many cases, e.g., right-censoring time $\tau$ predetermined by length of study.

---

Likelihood Contribution: Candidate Sets

Masking Component Failure: If the $i^{\text{th}}$ system fails ($\delta_i = 1$), it is masked by a candidate set $\mathcal{C}_i$. Its likelihood contribution is complex and we use simplifying assumptions to make it tractable.

• Condition 1: The candidate set includes the failed component: $\Pr\{K_i \in \mathcal{C}_i\} = 1$.

---

Likelihood Contribution: Candidate Sets (Cont'd)

• Condition 2: The condition probability of a candidate set given a cause of failure and a system lifetime is constant across conditioning on different failure causes within the candidate set:

$$ \Pr\{C_i = c_i | T_i = t_i, K_i = j\} = \Pr\{C_i = c_i | T_i = t_i, K_i = j'\} \text{ for $j,j' \in c_i$.} $$

• Condition 3: The masking probabilities when conditioned on the system lifetime and the failed component aren't functions of the system parameter.

---

Likelihood Contribution: Derivation for Candidate Sets

Take the joint distribution of $T_i$, $K_i$, and $\mathcal{C}_i$ and marginalize over $K_i$:

$$ f_{T_i,C_i}(t_i,c_i;\theta) = \sum_{j=1}^m f_{T_i,K_i}(t_i,j;\theta)\Pr{}_{!\theta}\{C_i = c_i | T_i = t_i, K_i = j\}. $$

Apply Condition 1 to get a sum over candidate set:

$$ f_{T_i,C_i}(t_i,c_i;\theta) = \sum_{j \in c_i} f_{T_i,K_i}(t_i,j;\theta)\Pr{}_{\theta}\{C_i = c_i | T_i = t_i, K_i = j\}. $$

---

Likelihood Contribution: Derivation for Candidate Sets (Cont'd)

Apply Condition 2 to move probability outside the sum:

$$ f_{T_i,C_i}(t_i,c_i;\theta) = \Pr_{\theta} \{C_i = c_i | T_i = t_i, K_i = j'\} \sum_{j \in c_i} f_{T_i,K_i}(t_i,j;\theta). $$

Apply Condition 3 to remove the masking probability's dependence on $\theta$:

$$ f_{T_i,C_i}(t_i,c_i;\theta) = \beta_i \sum_{j \in c_i} f_{T_i,K_i}(t_i,j;\theta). $$

Result: $L_i(\theta) \propto \sum_{j \in c_i} f_{T_i,K_i}(t_i,j;\theta) = R_{T_i}(t_i;\theta) \sum_{j \in c_i} h_j(t_i;\theta_j)$.

---

Bootstrap Confidence Intervals (CIs)

Confidence Intervals (CI) help capture the uncertainty in our estimate.

• Normal assumption for constructing CIs may not be accurate.
  • Masking and censoring.
• Bootstrapped CIs: Resample data and obtain MLE for each.
  • Use percentiles of bootstrapped MLEs for CIs.
• Coverage Probability: Probability the interval covers the true parameter value.
  • Challenge: Actual coverage may deviate to bias and skew in MLEs.
• BCa adjusts the CIs to counteract bias and skew in the MLEs.

---

Challenges with Masked Data

Like any model, ours has its challenges:

• Convergence Issues: Nearly flat likelihood regions can occur.

  • Ambiguity in masked, censored data
  • Complexities of estimating latent parameters.

• Bootstrap Issues: Relies on the empirical sampling distribution.

  • May not represent true variability for small samples.
  • Censoring and masking compound issue by reducing the effective sample size.

---

Challenges with Masked Data (Cont'd)

• Mitigation: In simulation, discard non-convergent samples for MLE on original data but retain all resamples for CIs.
  • More robust assessment at the cost of possible bias towards "well-behaved" data.
  • Convergence Rates reported to provide context.

---

Simulation Study: Series System with Weibull Components

We consider the following series system parameters in our simulation study.

Component	Shape $(k_j)$	Scale ($\lambda_j$)	Failure Probability ($\Pr{K_i}$)
1	1.26	994.37	0.17
2	1.16	908.95	0.21
3	1.13	840.11	0.23
4	1.18	940.13	0.20
5	1.20	923.16	0.20

---

Series System Parameters (Cont'd)

Lifetime of $j^{\text{th}}$ component of $i^{\text{th}}$ system: $T_{i j} \sim \mathrm{Weibull}(k_j,\lambda_j)$.

• Based on (Guo, Niu, and Szidarovszky 2013)
• Extended to include components 4 and 5
  • Shapes greater than 1 indicates wear-outs.
  • Probabilities comparable: reasonably well-designed.
• Focus on Components 1 and 3 (most and least reliable) in study.

---

Synthetic Data and Simulation Values

How is the data generated in our simulation study?

• Component Lifetimes (latent $T_{i 1}, \cdots, T_{i m}$) generated for each system.
  • Observed Data is a function of latent components.
• Right-Censoring amount controlled with simulation value $q$.
  • Quantile $q$ is probability system won't be right-censored.
  • Solve for right-censoring time $\tau$ in $\Pr\{T_i \leq \tau\} = q$.
  • $S_i = \min(T_i, \tau)$ and $\delta_i = 1_{\{T_i \leq \tau\}}$.
• Candidate Sets are generated using the Bernoulli Masking Model.
  • Masking level controlled with simulation value $p$.
  • Failed component (latent $K_i$) placed in candidate set (observed $\mathcal{C}_i$).
  • Each functioning component included with probability $p$.

---

Bernoulli Masking Model: Satisfying Masking Conditions

The Bernoulli Masking Model satisfies the masking conditions:

• Condition 1: The failed component deterministically placed in candidate set.
• Condition 2 and 3: Bernoulli probability $p$ is same for all components and fixed by us.
  • Probability of candidate set is constant conditioned on component failure within set.
  • Probability of candidate set, conditioned on a component failure, only depends on the $p$.

Future Research: Realistically conditions may be violated.

• Explore sensitivity of likelihood model to violations.

---

Performance Metrics

Objective: Evaluate the MLE and BCa confidence intervals' performance across various scenarios.

• Visualize the simulated sampling distribution of MLEs and $95\%$ CIs.
• MLE Evaluation:
  • Accuracy: Bias
  • Precision: Dispersion of MLEs
    • $95\%$ quantile range of MLEs.
• 95\% CI Evaluation:
  • Accuracy: Coverage probability (CP).
    • Correctly Specified CIs: CP near $95\%$ ($>90\%$ acceptable).
  • Precision: Width of median CI.

---

Scenario: Impact of Right-Censoring

Assess the impact of right-censoring on MLE and CIs.

• Right-Censoring: Failure observed with probability $q$: $60\%$ to $100\%$.
  • Right censoring occurs with probability $1-q$: $40\%$ to $0\%$.
• Bernoulli Masking Probability: Each component is a candidate with probability $p$ fixed at $21.5\%$.
  • Estimated from original study (Guo, Niu, and Szidarovszky 2013).
• Sample Size: $n$ fixed at $90$.
  • Small enough to show impact of right-censoring.

---

Scale Parameters

• Dispersion: Less censoring improves MLE precision.
  • Most reliable component more affected by censoring.
• Bias: MLE positively biased; decreases with less censoring.
• Median CIs: Tracks MLE dispersion.

---

Shape Parameters

• Show a similar pattern as scale parameters.

---

Coverage Probability and Convergence Rate

• Coverage (left figure): CIs show good empirical coverage.
  • Scale parameters correctly specified (CP $\approx 95\%$)
  • Shape parameters good enough (CP $> 90\%$).
• Convergence Rate (right figure): Increases with less censoring.
  • Caution: Dips below $95\%$ with more than $30\%$ censoring.

---

Key Takeaways: Right-Censoring

Right-censoring has a notable impact on the MLE:

• MLE Precision:
  • Improves notably with reduced right-censoring levels.
  • More reliable components benefit more from reduced right-censoring.
• Bias:
  • MLEs show positive bias, but decreases with reduced right-censoring.
• Convergence Rates:
  • MLE convergence rate improves with reduced right-censoring.
  • Dips: $< 95\%$ at $> 30\%$ right-censoring.

BCa confidence intervals show good empirical coverage.

• CIs offer reliable empirical coverage.
• Scale parameters correctly specified across all right-censoring levels.

---

Scenario: Impact of Failure Masking

Assessing the impact of the failure masking level on MLE and CIs.

• Bernoulli Masking Probability: Vary Bernoulli probability $p$ from $10\%$ to $70\%$.
• Right-Censoring: $q$ fixed at $82.5\%$.
  • Right-censoring occurs with probability $1-q$: $17.5\%$.
  • Censoring less prevalent than masking.
• Sample Size: $n$ fixed at $90$.
  • Small enough to show impact of masking.

---

Shape Parameters

• Dispersion: Precision decreases with masking level ($p$).
• Bias: MLE positively biased and increases with masking level.
  • Applies a right-censoring like effect to the components.
• Median CIs: Tracks MLE dispersion.

---

Scale Parameters

• These graphs resemble the last ones for shape parameters.

---

Coverage Probability and Convergence Rate

• Coverage: Caution advised for severe masking with small samples.
  • Scale parameter CIs show acceptable coverage across all masking levels.
  • Shape parameter CIs dip below $90\%$ when $p > 0.4$.
• Convergence Rate: Increases with less masking.
  • Caution: Dips under $95\%$ when $p > 0.4$ (consistent with CP behavior).

---

Key Takeaways: Masking

The masking level of component failures profoundly affects the MLE:

• MLE Precision:
  • Decreases with more masking.
• MLE Bias:
  • Positive bias is amplified with increased masking.
  • Masking exhibits a right-censoring-like effect.
• Convergence Rate:
  • Commendable for Bernoulli masking levels $p \leq 0.4$.
    • Extreme masking: some masking occurs $90\%$ of the time at $p = 0.4$.

The BCa confidence intervals show good coverage:

• Scale parameters maintain good coverage across all masking levels.
• Shape parameter coverage dip below $90\%$ when $p > 0.4$.
  • Caution advised for severe masking with small samples.

---

Scenario: Impact of Sample Size

Assess the mitigating affects of sample size on MLE and CIs.

• Sample Size: We vary the same size $n$ from 50 to 500..
• Right-Censoring: $q$ fixed at $82.5\%$
  • $17.5\%$ chance of right-censoring.
• Bernoulli Masking Probability: $p$ fixed at $21.5\%$
  • Some masking occurs $62\%$ of the time.

---

Scale Parameters

• Dispersion: Increasing sample size improves MLE precision.
  • Extremely precise for $n \geq 250$.
• Bias: Large positive bias initially, but diminishes to zero.
  • Large samples counteract right-censoring and masking effects.
• Median CIs: Track MLE dispersion. Very tight for $n \geq 250$.

---

Shape Parameters

• These graphs resemble the last ones for scale parameters.

---

Coverage Probability and Convergence Rate

• Coverage: Good empirical coverage.

  • Correctly specified CIs for $n > 250$.

• Convergence Rate: Total convergence for $n \geq 250$.

  • Caution advised for estimates with $n < 100$ in specific setups.

---

Key Takeaways: Sample Size

Sample size has a notable impact on the MLE:

• Precision: Very precise for large samples ($n > 200$).
• Bias: Diminishes to near zero for large samples.
• Coverage: Correctly specified CIs for large samples.
• Convergence Rate: Total convergence for large samples.

Summary

Larger samples lead to more accurate, unbiased, and reliable estimations.

• Mitigates the effects of right-censoring and masking.

---

Conclusion

MLE Performance:

• Right-censoring and masking introduce positive bias for our setup.
  • More reliable components are more affected.
• Shape parameters harder to estimate than scale parameters.
• Large samples can mitigate the affects of masking and right-censoring.

BCa Confidence Interval Performance:

• Width of CIs tracked MLE dispersion.
• Good empirical coverage in most scenarios.

Big Picture

MLE and CIs robust despite masking and right-censoring challenges.

---

Future Work and Discussion

Directions to enhance learning from masked data:

• Relax Masking Conditions: Assess sensitivity to violations and and explore alternative likelihood models.
• System Design Deviations: Assess estimator sensitivity to deviations.
• Homogenous Shape Parameter: Analyze trade-offs with the full model.
• Bootstrap Techniques: Semi-parametric approaches and prediction intervals.
• Regularization: Data augmentation and penalized likelihood methods.
• Additional Likelihood Contributions: Predictors, etc.