> "When outliers scream but you need to hear the whisper of typical performance."
Median Absolute Error (MdAE) measures the middle value of all absolute errors. Unlike MAE which averages errors, MdAE finds the median, providing exceptional robustness against outliers and extreme values.
## Historical Context
MdAE emerged from robust statistics, where the median has long been preferred over the mean for its resistance to outliers. In forecasting and machine learning, MdAE provides a more stable measure of typical prediction accuracy when data contains anomalies or heavy-tailed distributions.
## Architecture & Physics
MdAE maintains a sorted view of errors through a specialized ring buffer. When new errors arrive, they replace the oldest while maintaining sort order, enabling O(1) median retrieval. This makes MdAE both robust and efficient.
### Properties
* **Outlier-robust**: Unaffected by extreme values
* **Non-negative**: MdAE ≥ 0, with 0 indicating perfect prediction
* **Same units**: Results are in the same units as the original data
* **Stable**: Small changes in data produce small changes in output
## Mathematical Foundation
### 1. Absolute Error
For each observation, calculate the absolute difference:
$$e_i = |y_i - \hat{y}_i|$$
Where:
* $y_i$ = actual value
* $\hat{y}_i$ = predicted value
### 2. Median Calculation
Find the middle value of the sorted errors:
$$MdAE = \text{median}(e_1, e_2, ..., e_n)$$
For odd n: middle element
For even n: average of two middle elements
### 3. Running Update (O(1))
QuanTAlib uses a sorted ring buffer for efficient median retrieval:
$$MdAE = \begin{cases}
e_{(n+1)/2} & \text{if } n \text{ is odd} \\
\frac{e_{n/2} + e_{n/2+1}}{2} & \text{if } n \text{ is even}
\end{cases}$$
## Implementation Details
### Usage Patterns
```csharp
// Streaming mode - update with each new observation
