Miha Kralj
6.3 KiB
MdAPE: Median Absolute Percentage Error
When you need relative errors but can't trust the outliers.
| Property | Value |
|---|---|
| Category | Error Metric |
| Inputs | Actual, Predicted (dual series) |
| Parameters | period |
| Outputs | Single series (Mdape) |
| Output range | \geq 0 |
| Warmup | period bars |
| PineScript | mdape.pine |
- Median Absolute Percentage Error (MdAPE) combines the scale-independence of percentage errors with the robustness of median statistics.
- Similar: MAPE, MdAE | Trading note: Median Absolute Percentage Error; robust version of MAPE for skewed error distributions.
- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
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
// 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
Operation Count (Streaming Mode)
O(1) per bar. Single-pass scalar transformation of (actual, forecast) pair; no lookback window required.
| Operation | Count | Cost (cycles) | Subtotal |
|---|---|---|---|
| Error computation (subtract, abs/square/log) | 1-3 | ~3-8 cy | ~5-15 cy |
| Running accumulator update (EMA or sum) | 1 | ~4 cy | ~4 cy |
| Total | 2-4 | — | ~9-19 cycles |
Streaming update requires only the current actual/forecast pair and running state. ~10-15 cycles/bar typical.
Batch Mode (SIMD Analysis)
| Operation | Vectorizable? | Notes |
|---|---|---|
| Element-wise error computation | Yes | Independent per bar; fully vectorizable with Vector<double> |
| Reduction (sum/mean) | Yes | Parallel reduction; AVX2 gives 4x speedup |
| Log/exp components | Partial | Transcendental ops; polynomial approx for SIMD |
Batch SIMD: 4x-8x speedup for large windows. ~3-5 cy/bar amortized in vectorized batch mode.
| 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
- Retail Forecasting: Track typical accuracy across SKUs with varying prices
- Financial Analysis: Evaluate prediction quality ignoring market crashes
- Model Selection: Choose models based on typical rather than average performance
- 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%