ML Models for PdM

Types of PdM Models

Different ML approaches address different PdM questions:

Deep Learning for PdM

Deep learning has shown promise for PdM, particularly for directly modeling raw sensor time series:

• 1D-CNNs: Convolutional networks applied to sensor time series can automatically learn relevant temporal patterns without manual feature engineering.
• LSTMs/GRUs: Recurrent networks capture long-range dependencies across multiple process runs (e.g., slow drift over hundreds of runs).
• Transformer-based models: Attention mechanisms can identify which time steps and which sensors are most predictive of impending failure.
• Autoencoders: Learn a compressed representation of "normal" equipment behavior. Large reconstruction error = abnormal behavior.

In practice, gradient-boosted trees (XGBoost, LightGBM) on engineered features often outperform deep learning in this domain due to limited training data and the effectiveness of domain-informed features.

Survival Analysis and RUL Estimation

Survival analysis is the statistical backbone of PdM. The central object is the survival function:

S(t) = P(T > t)

i.e. the probability that a component is still alive at time t. The complement is the cumulative failure probability F(t) = 1 − S(t), and the instantaneous failure rate (hazard) is h(t) = f(t) / S(t).

1. The Weibull model — the workhorse

Fab equipment lifetimes are routinely fit with the two-parameter Weibull distribution:

S(t) = exp(−(t/η)^β)    h(t) = (β/η)(t/η)^β−1

2. Cox proportional hazards — using covariates

Adds an exponential effect of covariates x on the baseline hazard:

h(t | x) = h₀(t) · exp(β·x)

This lets you say, e.g., "a 10% higher RF reflected power doubles the instantaneous failure rate," without committing to a specific h₀ shape.

3. RUL from a Weibull HI model

Once you have an estimated S(t) and the component has already survived to time t_now, the Remaining Useful Life is the expectation:

RUL(t_now) = E[T − t_now | T > t_now] = ∫_tnow^∞ [S(u)/S(t_now)] du

import numpy as np
from scipy.special import gamma

def weibull_rul(t_now: float, eta: float, beta: float) -> float:
    """Remaining useful life under a Weibull lifetime distribution.

    Mean lifetime is eta * Gamma(1 + 1/beta); conditional mean
    given survival to t_now uses numeric integration of S(u)/S(t_now).
    """
    if t_now < 0:
        raise ValueError("t_now must be non-negative")
    # Integrate from t_now to a horizon ~5x mean
    horizon = 5 * eta * gamma(1 + 1 / beta)
    u = np.linspace(t_now, horizon, 4000)
    S = np.exp(-(u / eta) ** beta)
    S_now = np.exp(-(t_now / eta) ** beta)
    return np.trapezoid(S / S_now, u)

# Example: focus ring with eta=1500 RF-hours, beta=2.5 (wear-out)
print(f"RUL at 800 RF-hr: {weibull_rul(800, 1500, 2.5):.0f} hours")
print(f"RUL at 1400 RF-hr: {weibull_rul(1400, 1500, 2.5):.0f} hours")