# MdAPE: Median Absolute Percentage Error

> "When you need relative errors but can't trust the outliers."

Median Absolute Percentage Error (MdAPE) combines the scale-independence of percentage errors with the robustness of median statistics. It provides a measure of typical relative prediction accuracy that remains stable even when some predictions are dramatically wrong.

## Historical Context

MdAPE arose as a natural combination of two statistical improvements: using percentages for scale-independence (like MAPE) and using medians for robustness (like MdAE). This hybrid approach addresses both the scale problem of MAE and the outlier sensitivity of MAPE.

## Architecture & Physics

MdAPE first normalizes each error as a percentage of the actual value, then finds the median of these percentages. This two-stage approach provides both relative context and outlier resistance.

### Properties

* **Scale-independent**: Comparable across different data magnitudes
* **Outlier-robust**: Extreme errors don't skew results
* **Percentage-based**: Results are interpretable as "typical % error"
* **Non-negative**: MdAPE ≥ 0, with 0 indicating perfect prediction

## Mathematical Foundation

### 1. Absolute Percentage Error

For each observation, calculate the percentage error:

$$e_i = \frac{|y_i - \hat{y}_i|}{|y_i|} \times 100$$

Where:
* $y_i$ = actual value
* $\hat{y}_i$ = predicted value

### 2. Median Calculation

Find the middle value of the sorted percentage errors:

$$MdAPE = \text{median}(e_1, e_2, ..., e_n)$$

### 3. Running Update (O(1))

QuanTAlib uses a sorted ring buffer for efficient median retrieval:

$$MdAPE = \begin{cases}
e_{(n+1)/2} & \text{if } n \text{ is odd} \\
\frac{e_{n/2} + e_{n/2+1}}{2} & \text{if } n \text{ is even}
\end{cases}$$

## Implementation Details

### Usage Patterns

```csharp
// Streaming mode - update with each new observation
var mdape = new Mdape(period: 20);
var result = mdape.Update(actualValue, predictedValue);

// Batch mode - calculate for entire series
var results = Mdape.Calculate(actualSeries, predictedSeries, period: 20);

// Span mode - zero-allocation for high performance
Mdape.Batch(actualSpan, predictedSpan, outputSpan, period: 20);
```

### Parameters

| Parameter | Type | Description |
| :--- | :--- | :--- |
| **period** | int | Lookback window for median calculation (must be > 0) |

### Properties

| Property | Type | Description |
| :--- | :--- | :--- |
| **Last** | TValue | Most recent MdAPE value (in percentage) |
| **IsHot** | bool | True when buffer is full |
| **Name** | string | Indicator name (e.g., "Mdape(20)") |
| **WarmupPeriod** | int | Number of periods before valid output |

## Performance Profile

| Metric | Score | Notes |
| :--- | :--- | :--- |
| **Throughput** | ~25 ns/bar | O(1) with sorted buffer |
| **Allocations** | 0 | Uses pre-allocated buffers |
| **Complexity** | O(1) | Constant time per update |
| **Accuracy** | 10/10 | Exact calculation |
| **Timeliness** | 9/10 | No lag beyond the period |
| **Robustness** | 10/10 | Immune to outliers |

## Interpretation

| MdAPE Range | Interpretation |
| :--- | :--- |
| **0%** | Perfect prediction |
| **0-5%** | Excellent accuracy |
| **5-10%** | Good accuracy |
| **10-20%** | Acceptable accuracy |
| **> 20%** | Poor accuracy |

## Comparison with MAPE

| Scenario | MAPE | MdAPE |
| :--- | :--- | :--- |
| **Normal distribution** | Similar values | Similar values |
| **Single 1000% error** | Heavily inflated | Unchanged |
| **Asymmetric errors** | Biased | Representative |
| **Zero actual values** | Undefined | Undefined (uses substitution) |

## Common Use Cases

1. **Retail Forecasting**: Track typical accuracy across SKUs with varying prices
2. **Financial Analysis**: Evaluate prediction quality ignoring market crashes
3. **Model Selection**: Choose models based on typical rather than average performance
4. **Operations Research**: Measure forecast reliability for planning

## Edge Cases

* **Zero Actual Values**: Substitutes with small epsilon to avoid division by zero
* **NaN Handling**: Uses last valid value substitution
* **Single Input**: Not supported (requires two series)
* **Period = 1**: Returns current absolute percentage error
* **All Perfect**: Returns 0%

## Related Indicators

* [MAPE](../mape/Mape.md) - Mean Absolute Percentage Error (uses mean)
* [MdAE](../mdae/Mdae.md) - Median Absolute Error (non-percentage)
* [SMAPE](../smape/Smape.md) - Symmetric MAPE (different normalization)