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Mean Absolute Error#

Mean Absolute Error is a risk metric corresponding to the expected value of the absolute error loss or \(l1\)-norm loss.

Definition (Mean Absolute Error)#

Given a dataset of \(n\) samples indexed by the tuple pair \((x_i, y_i)\), the mean absolute error (MAE) is defined as:

\[ \textbf{MAE} = \dfrac{\sum_{i=1}^n |\hat{y}_i - y_i|}{n} \]

Theorem (Optimality)#

In simple words, this is the property of the median minimizes the sum of absolute error (\({\ell}_{1}\) loss). This is important that one realises this when doing linear regression, and in complement to the mean minimizes the root mean squared error.

For some proofs:

Implementation of MAE#

import numpy as np

def mean_absolute_error_(y_true: np.ndarray, y_pred: np.ndarray) -> float:
    """Mean absolute error regression loss.

    Args:
        y_true (np.ndarray): Ground truth (correct) target values.
        y_pred (np.ndarray): Estimated target values.

    Shape:
        y_true: (n_samples, )
        y_pred: (n_samples, )

    Returns:
        loss (float): The mean absolute error.

    Examples:
        >>> y_true = [3, -0.5, 2, 7]
        >>> y_pred = [2.5, 0.0, 2, 8]
        >>> mean_absolute_error_(y_true, y_pred)
        0.5
    """
    y_true = np.asarray(y_true).flatten()
    y_pred = np.asarray(y_pred).flatten()
    loss = np.mean(np.abs(y_true - y_pred))
    return loss

>>> y_true = [3, -0.5, 2, 7]
>>> y_pred = [2.5, 0.0, 2, 8]
>>> mean_absolute_error_(y_true, y_pred)

0.5