Supported Fault Types & Mathematical Background

1. Drift

   Linear change over time:

   \[y_i = x_i + (i - s)\, r, \quad i \in [s, e)\]

   • \(r \in \mathbb{R}\) is the drift rate

   • By default: \(r \sim \mathcal{U}(0, 0.0001)\, d\, \bar{x}\)

   • \(d \in \{-1, 1\}\) controls the drift direction

   • \(\bar{x}\) is the mean of the original signal over the fault interval

2. Offset

   Constant bias added to the values:

   \[y_i = x_i + b, \quad i \in [s, e)\]

   • \(b = r\, \bar{x}\) is the offset magnitude

   • \(r \in \mathbb{R}\) is the offset rate

3. Gaussian Noise

   Random noise with a normal distribution:

   \[y_i = x_i + \epsilon_i, \quad \epsilon_i \sim \mathcal{N}(\mu, \sigma^2), \quad i \in [s, e)\]

   • Defaults: \(\mu = 0\)

   • \(\sigma\) is the standard deviation of the signal over the fault interval

4. Uniform Noise

   Random noise with a uniform distribution:

   \[y_i = x_i + \epsilon_i, \quad \epsilon_i \sim \mathcal{U}(a, b), \quad i \in [s, e)\]

   • Defaults: \(a = -\sigma\), \(b = \sigma\)

   • \(\sigma\) is the standard deviation of the signal over the fault interval

5. NaN

   Replace values with missing data:

   \[\begin{split}y_i = \begin{cases} \mathrm{NaN}, & i \in [s, e) \\ x_i, & \text{otherwise} \end{cases}\end{split}\]

6. Stuck Value

   Replace values with a constant:

   \[\begin{split}y_i = \begin{cases} c, & i \in [s, e) \\ x_i, & \text{otherwise} \end{cases}\end{split}\]

   • \(c\) may be specified directly or generated as \(c = \bar{x}\left(1 + d\, u\right)\)

   • \(u \sim \mathcal{U}(0.01, 0.1)\)

Notes

• Faults are applied over the range [start, stop).

• The class preserves the original values for comparison.

• Useful for testing robustness of sensor data processing algorithms, anomaly detection, and fault-tolerant systems.