AE - Absolute Error

Absolute Error (AE) measures the magnitude of difference between an actual value and a predicted value for each individual data point, without considering the direction of the error.

Unlike aggregated metrics (such as MSE or RMSE), AE computes the element-wise error between a pair of lists, tuples, or NumPy arrays.

\[\text{AE}_i = | y_i - \hat{y}_i |\]

Description

Advantages:

  • Perfect interpretability: Expressed in the exact same units as the response variable. It directly answers the question: “How far off was this specific prediction?”

  • Outlier robustness: Because it scales linearly (not squared), AE does not disproportionately inflate the error for extreme outliers.

Disadvantages:

  • Non-aggregated: AE is an element-wise metric. To evaluate overall model performance across an entire dataset, it must be mathematically aggregated into metrics like Mean Absolute Error (MAE) or Sum of Absolute Errors (SAE).

  • Mathematical optimization: The absolute value function is non-differentiable at exactly zero (\(y_i = \hat{y}_i\)), which can make it less ideal as a direct loss function for gradient-based algorithms.

Properties

  • Best possible score: 0.0 (Smaller value is better).

  • Range: [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 Absolute Error for each element
print("AE: ", evaluator.AE())

## 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)
# Calculate Absolute Error for multi-dimensional array
print("AE (Multi-output): ", evaluator.AE())