Survival Analysis

The hazard function represents the instantaneous risk of failure at time t, given that the asset has survived up to that moment. Unlike simple failure probability, it captures the dynamic nature of risk. In manufacturing, a machine's hazard rate often increases over time due to wear-out failures, but can also spike during early-life infant mortality periods. Key characteristics:

• Conditional probability: Risk is always calculated for assets still in operation
• Non-parametric estimation: Can be modeled without assuming a specific distribution
• Bath-tub curve mapping: Directly visualizes early-life, useful-life, and wear-out phases

Right-censoring is the defining feature that separates survival analysis from standard regression. It occurs when a machine's failure time is unknown because the study ends or the asset is removed from service before failure. Survival models handle three types:

• Type I (Time-censoring): Observation stops at a fixed calendar date while machines are still running
• Type II (Failure-censoring): Study ends after a pre-specified number of failures occur
• Random censoring: Assets are removed for unrelated reasons, such as decommissioning or process changes Ignoring censored data biases predictions toward shorter-than-actual lifetimes.

The Kaplan-Meier estimator is the foundational non-parametric method for computing the survival probability function from censored operational data. It calculates the probability that an asset survives beyond time t by multiplying conditional survival probabilities at each observed failure time. Critical properties:

• Step function: Survival curve drops only at exact failure times, not between events
• Handles staggered entry: Machines commissioned at different dates can be included
• No distribution assumption: Works without knowing if failures follow Weibull, exponential, or other patterns
• Confidence intervals: Greenwood's formula provides variance estimates for the survival curve

The Cox Proportional Hazards (CPH) model is a semi-parametric regression technique that relates sensor covariates to the hazard rate without specifying the baseline hazard's shape. It assumes covariates multiplicatively shift the hazard function. For predictive maintenance:

• Feature coefficients: A positive coefficient for vibration amplitude means higher vibration increases failure risk
• Hazard ratio: Exponentiated coefficients quantify risk multipliers, e.g., a ratio of 2.0 means doubled instantaneous failure risk
• Time-varying covariates: Extended Cox models incorporate sensor readings that change over an asset's lifecycle
• Proportionality assumption: Requires that the effect of a covariate is constant over time, verifiable via Schoenfeld residuals

Accelerated Failure Time (AFT) models provide an alternative to Cox regression by directly modeling the logarithm of failure time as a linear function of covariates. Unlike CPH, which models the hazard rate, AFT models describe how covariates accelerate or decelerate the time scale of degradation. Common distributions:

• Weibull AFT: Handles monotonic hazard rates, widely used for mechanical wear
• Log-normal AFT: Appropriate when failure times follow a log-normal distribution, common in fatigue life analysis
• Log-logistic AFT: Models non-monotonic hazards where risk initially increases then decreases AFT models produce directly interpretable time ratios—a coefficient of 0.5 means the covariate halves expected lifetime.

The competing risks extension of survival analysis handles assets that can fail from multiple mutually exclusive causes, where the occurrence of one failure mode precludes observing others. In manufacturing, a bearing might fail from inner race spalling, outer race fracture, or cage degradation. Key components:

• Cause-specific hazard: The instantaneous risk of failure from a specific mode, treating other modes as censored
• Cumulative incidence function (CIF): The probability of failing from a specific cause by time t, accounting for competing events
• Fine-Gray subdistribution hazard: Models the effect of covariates on a specific failure type without assuming independence between competing risks This framework prevents overestimating individual failure mode probabilities.

The defining characteristic of survival analysis. Right-censoring occurs when a machine is still operational at the end of the study period—its true failure time is unknown, only that it exceeds the observation window. Left-censoring happens when a failure occurred before monitoring began. Interval-censoring indicates failure between two inspection points. Ignoring censored data by treating it as non-failure or discarding it introduces severe bias, underestimating asset reliability. Survival models like the Kaplan-Meier estimator and Cox Proportional Hazards are explicitly designed to incorporate this partial information.

The non-parametric gold standard for estimating the survival function S(t)—the probability that an asset survives beyond time t. It calculates survival probabilities at each distinct failure time by multiplying the conditional probabilities of surviving each interval. Key features:

• Handles right-censored data natively
• Produces a step-function curve that drops only at observed failure events
• Allows visual comparison of survival curves between different asset cohorts using the log-rank test
• Assumes censoring is non-informative (failure and censoring mechanisms are independent)

A semi-parametric regression model that quantifies the effect of covariates on the hazard rate without specifying the baseline hazard function. The model expresses the hazard as h(t|X) = h₀(t) × exp(β₁X₁ + β₂X₂ + ...). The hazard ratio exp(β) indicates how a one-unit increase in a covariate multiplies the risk of failure. Critical assumptions:

• Proportional hazards: The hazard ratio between two assets is constant over time
• Log-linearity: Continuous covariates have a linear effect on the log-hazard
• Used extensively to identify which sensor readings (vibration, temperature) most strongly predict imminent failure

The instantaneous failure rate at time t, conditional on survival up to that moment. Mathematically, h(t) = f(t) / S(t), where f(t) is the probability density of failure and S(t) is the survival function. In manufacturing contexts:

• A constant hazard indicates random, memoryless failures (exponential distribution)
• An increasing hazard models wear-out degradation (Weibull distribution with shape > 1)
• A bathtub curve hazard combines decreasing infant mortality, constant useful life, and increasing wear-out phases
• The cumulative hazard H(t) = -ln[S(t)] provides diagnostic plots for model selection

The most widely used parametric model in reliability engineering due to its flexibility in modeling different failure regimes through its shape parameter β:

• β < 1: Decreasing failure rate (infant mortality, burn-in period)
• β = 1: Constant failure rate (reduces to exponential distribution, random failures)
• β > 1: Increasing failure rate (wear-out, fatigue, corrosion)
• β = 3.4: Approximates a normal distribution The scale parameter η represents the characteristic life—the time at which 63.2% of assets have failed. Weibull analysis is standard for analyzing run-to-failure data from bearing life tests and accelerated life testing.

A framework for scenarios where an asset can experience multiple mutually exclusive failure modes, and the occurrence of one precludes observing others. Examples in manufacturing:

• A motor can fail from bearing seizure or winding insulation breakdown
• A pump can fail from cavitation or seal leakage The cause-specific hazard quantifies the instantaneous risk from each failure type. The cumulative incidence function (CIF) estimates the probability of failing from a specific cause by time t, accounting for competing events. Ignoring competing risks by treating other failures as censoring inflates cause-specific failure probabilities.