A10 - A10 Index
===============

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The **A10 Index** is a strictly threshold-based regression metric widely used in engineering and empirical models. It quantifies the proportion of predictions that deviate by no more than **±10%** from the actual ground truth values.

.. math::

    \text{A10}(y, \hat{y}) = \frac{1}{N} \sum_{i=1}^{N} \begin{cases} 1, & \text{if } \frac{|y_i - \hat{y}_i|}{|y_i|} \leq 0.1 \\ 0, & \text{otherwise} \end{cases}

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Description
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**Advantages:**
	* **Exceptional interpretability:** Directly translates into business, engineering, or clinical success criteria (e.g., "85% of our predictions fall within the strict 10% margin of error").
	* **Robust to extreme outliers:** Unlike RMSE or MSE, massive prediction errors do not disproportionately skew the final score. An outlier is simply counted as a failure (score = 0).

**Disadvantages:**
	* **Rigid threshold (Cliff effect):** A prediction with a 10.1% error is penalized exactly the same as a prediction with a 500% error. It completely ignores "near-misses".
	* **Undefined for zero targets:** Because it divides by the actual value (:math:`y_i`), the metric calculation will crash or become undefined if the ground truth data contains absolute zeros.

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Properties
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* **Best possible score:** ``1.0`` (Higher is better, meaning 100% of samples are within the ±10% tolerance zone).
* **Range:** ``[0.0, 1.0]``
* **Mathematical Reference:** `MDPI Applied Sciences <https://www.mdpi.com/2076-3417/9/18/3715/htm>`_

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Example Usage
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.. 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 A10 Index
    print("A10 Index: ", evaluator.A10())

    ## 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("A10 Index (Multi-output): ", evaluator.A10(multi_output="raw_values"))