Piecewise Linear RUL Degradation Model

Learning Objectives

By the end of this section, you will:

1. Understand the linear RUL problem—why raw RUL labels are problematic
2. Derive the piecewise linear model that caps RUL at a maximum value
3. Justify the RUL maximum of 125 cycles based on physical and practical considerations
4. Apply the mathematical formulation to transform raw labels
5. Understand evaluation implications of capped RUL during testing

Why This Matters: The choice of RUL target formulation significantly impacts model performance. The piecewise linear model is now standard practice in RUL prediction research—understanding why enables you to make informed decisions for other prognostics problems.

The Linear RUL Problem

The raw RUL label is simply cycles remaining until failure:

This creates a linear decrease from $T_{\text{max}} - 1$ at cycle 1 to 0 at failure. However, this formulation has fundamental problems.

Problem 1: No Visible Degradation Early

In the early phases of engine life, sensor readings are essentially identical:

Problem 2: High Variance Labels

Linear RUL creates high variance in early-life labels:

Training on high-variance labels leads to high-variance predictions.

Problem 3: Asymmetric Importance

In practice, predicting "RUL = 200" vs "RUL = 300" matters much less than "RUL = 20" vs "RUL = 30":

• High RUL: No maintenance action needed either way
• Low RUL: Critical decision point for scheduling maintenance

Equal weighting of all RUL errors ignores this practical asymmetry.

Piecewise Linear Degradation Model

The piecewise linear model addresses these problems by capping RUL at a maximum value:

Visual Representation

The piecewise linear RUL function has two phases:

1. Constant phase: RUL = $\text{RUL}_{\text{max}}$ for early cycles (no visible degradation)
2. Linear decay phase: RUL decreases linearly as degradation becomes apparent

1RUL
2 ^
3 |
4 |  ___________________    <- Constant at RUL_max
5 |                     \
6 |                      \
7 |                       \   <- Linear decay
8 |                        \
9 |                         \
10 +-------------------------->  Cycle
11 0                         T_max

Physical Interpretation

The piecewise linear model reflects a physical reality:

Early in life, degradation is imperceptible. The engine operates within normal parameters, and sensor readings cannot distinguish a "young" engine from a "slightly older" engine. Only after sufficient wear accumulates do degradation signatures become detectable.

Setting RUL = $\text{RUL}_{\text{max}}$ for early cycles says: "We know the engine is healthy, but we cannot predict exactly how long it will last."

Choosing RUL Maximum

The choice of $\text{RUL}_{\text{max}}$ is important. Common values in the literature include 125, 130, and 150 cycles.

Why 125 Cycles?

We use $\text{RUL}_{\text{max}} = 125$ based on:

1. Literature standard: Most published work uses 125, enabling fair comparison
2. Statistical analysis: Degradation signatures become detectable around 100-130 cycles before failure
3. Practical maintenance: 125 cycles (~125 flights) provides sufficient planning window

Analysis Supporting RUL Max

Effect on Label Distribution

Capping at 125 dramatically changes the label distribution:

Mathematical Formulation

Let's formalize the piecewise linear RUL computation.

Definition

For an engine with total lifetime $T$ cycles, the piecewise linear RUL at cycle $t$ is:

Or equivalently using the min function:

Transition Point

The transition from constant to linear occurs at:

Implementation

1# Piecewise linear RUL computation
2def compute_piecewise_rul(cycle, total_cycles, rul_max=125):
3    """
4    Compute piecewise linear RUL.
5
6    Args:
7        cycle: Current cycle number (1-indexed)
8        total_cycles: Total cycles until failure
9        rul_max: Maximum RUL cap (default 125)
10
11    Returns:
12        Capped RUL value
13    """
14    linear_rul = total_cycles - cycle
15    return min(rul_max, linear_rul)

Training and Evaluation Implications

The piecewise linear model affects both training and evaluation.

Training Implications

• Reduced label variance: Model sees consistent RUL = 125 for healthy engines
• Focused learning: Model concentrates on the degrading phase where prediction matters
• Gradient stability: Loss gradients are more stable without extreme RUL targets

Evaluation Implications

During evaluation, we compare predicted RUL to ground truth. For test samples, the ground truth may be below or above 125:

Scoring Functions

Standard metrics (RMSE, MAE) are computed on capped RUL:

Where $y_i^{\text{(clipped)}} = \min(125, y_i^{\text{(true)}})$.

Summary

In this section, we introduced the piecewise linear RUL model:

1. Linear RUL problems: Indistinguishable early-life readings, high label variance, asymmetric importance
2. Piecewise solution: Cap RUL at maximum value (125 cycles)
3. Physical motivation: Degradation is imperceptible until ~100-130 cycles before failure
4. Formula: $\text{RUL}_{\text{PL}} = \min(125, T - t)$
5. Benefits: Reduced variance, focused learning, stable training
6. Evaluation: Apply same clipping to test labels for fair comparison

Looking Ahead: The piecewise linear model gives us continuous RUL targets. But for multi-task learning, we also need discrete targets. In the next section, we will discretize RUL into health states—categorical labels that enable our classification task and provide regularization for the regression.

With the piecewise RUL model defined, we are ready to create the health state classification targets.