A20 - A20 Index
===============

.. toctree::
   :maxdepth: 3

.. contents:: Table of Contents
   :local:
   :depth: 2

The **A20 Index** :cite:`van2023groundwater` is an empirical evaluation metric that quantifies the proportion of predictions falling within a **±20% deviation** from the experimental (actual) values.

A higher A20 score indicates better predictive accuracy, demonstrating that a larger percentage of the model's predictions are practically acceptable within this 20% tolerance margin.

.. math::

    \text{A20}(y, \hat{y}) = \frac{1}{N} \sum_{i=1}^{N} \begin{cases} 1, & \text{if } \frac{|y_i - \hat{y}_i|}{|y_i|} \leq 0.2 \\ 0, & \text{otherwise} \end{cases}

-------------------------------------------------------------------------------

Description
-----------

**Advantages:**
	* **High interpretability:** Easily understood by non-technical stakeholders (e.g., "85% of predictions are within a 20% error margin").
	* **Outlier resilience:** Extreme deviations do not disproportionately skew the metric; they are simply counted as outside the tolerance threshold (score = 0).

**Disadvantages:**
	* **Rigid threshold (Cliff effect):** A prediction with a 20.1% error is penalized exactly the same as a prediction with a 200% error. It treats "near-misses" and "massive failures" equally.
	* **Zero-target vulnerability:** Because the formula divides by the actual value (:math:`y_i`), the calculation will become undefined (division by zero) if the ground truth data contains absolute zeros.

-------------------------------------------------------------------------------

Properties
----------

* **Best possible score:** ``1.0`` (Higher is better; 100% of samples fall within the ±20% tolerance zone).
* **Range:** ``[0.0, 1.0]``

-------------------------------------------------------------------------------

Example Usage
-------------

.. code-block:: python
    :emphasize-lines: 10, 18

    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 A20 Index
    print("A20 Index: ", evaluator.A20())

    ## 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("A20 Index (Multi-output): ", evaluator.A20(multi_output="raw_values"))