ME - Max Error

The Max Error (ME) (also known as Maximum Residual Error) is a regression metric that captures the absolute worst-case error between the predicted values and the true values.

Unlike average-based metrics (like MAE or RMSE) that aggregate errors across the entire dataset, Max Error strictly isolates and reports the single largest deviation.

\[\text{ME}(y, \hat{y}) = \max_{i=1}^{N} (| y_i - \hat{y}_i |)\]

Description

Advantages:

  • Safety-Critical Benchmarking: This is the ultimate evaluation metric for high-stakes environments (e.g., industrial sensor calibration, clinical dosage prediction, autonomous driving) where bounding the absolute maximum failure is far more critical than average performance. If a single severe error can break a machine or harm a patient, ME is the metric you must monitor.

  • Worst-Case Guarantee: It establishes a strict upper bound on the model’s error. If your ME is 2.0, you can mathematically guarantee that no prediction in your test set was off by more than 2.0.

Disadvantages:

  • Extreme Outlier Vulnerability: By definition, ME focuses entirely on a single data point. A single corrupted ground-truth label, a broken sensor reading, or an unpredictable anomaly will cause the ME to explode. This can falsely make a model look terrible, hiding the fact that it might be perfectly fitted to the other 99.9% of the data.

  • Not Differentiable: Because it relies on the strict maximum operator, it is generally not smooth and cannot be used directly as a loss function for training neural networks via gradient descent.

Properties

  • Best possible score: 0.0 (Indicates a perfectly fitted model with zero error across all data points).

  • Range: [0.0, +inf)

Example Usage

from numpy import array
from permetrics.regression import RegressionMetric

## 1. For 1-D array (Single-output)
y_true = array([3, -0.5, 2, 7])
y_pred = array([2.5, 0.0, 2, 8])

evaluator = RegressionMetric(y_true, y_pred)
# Calculate Max Error
print("ME: ", evaluator.max_error())

## 2. For > 1-D array (Multi-output)
y_true = array([[0.5, 1], [-1, 1], [7, -6]])
y_pred = array([[0, 2], [-1, 2], [8, -5]])

evaluator = RegressionMetric(y_true, y_pred)
# Return an array of scores for each column
print("ME (Multi-output): ", evaluator.ME(multi_output="raw_values"))