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https://github.com/mihakralj/QuanTAlib.git
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Miha Kralj
162 lines
6.1 KiB
Markdown
162 lines
6.1 KiB
Markdown
# MRAE: Mean Relative Absolute Error
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> *When you need to understand your error in the context of what you're predicting.*
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| Property | Value |
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| ---------------- | -------------------------------- |
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| **Category** | Error Metric |
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| **Inputs** | Actual, Predicted (dual series) |
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| **Parameters** | `period` |
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| **Outputs** | Single series (MRAE) |
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| **Output range** | $\geq 0$ |
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| **Warmup** | `period` bars |
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| **PineScript** | [mrae.pine](mrae.pine) |
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- Mean Relative Absolute Error (MRAE) measures the average magnitude of errors relative to the actual values.
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- **Similar:** [RAE](../rae/Rae.md), [MASE](../mase/Mase.md) | **Trading note:** Mean Relative Absolute Error; ratio of errors to benchmark errors. Scale-free comparison metric.
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- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
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Mean Relative Absolute Error (MRAE) measures the average magnitude of errors relative to the actual values. This normalization makes the metric scale-independent and easier to interpret across different datasets.
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## Historical Context
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MRAE emerged as an alternative to MAPE for situations where relative error measurement is important but where the issues with percentage-based metrics (like undefined values when actuals are zero) need to be handled differently. It provides a bounded, interpretable measure of prediction accuracy.
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## Architecture & Physics
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MRAE divides each absolute error by the actual value, providing context for the error magnitude. The error of 5 means something different when predicting 10 versus predicting 1000, and MRAE captures this distinction.
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### Properties
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* **Scale-independent**: Comparable across different data magnitudes
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* **Non-negative**: MRAE ≥ 0, with 0 indicating perfect prediction
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* **Interpretable**: A value of 0.1 means 10% average relative error
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* **Denominator sensitivity**: Undefined when actual values are zero (handled via substitution)
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## Mathematical Foundation
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### 1. Relative Absolute Error
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For each observation, calculate the relative error:
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$$e_i = \frac{|y_i - \hat{y}_i|}{|y_i|}$$
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Where:
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* $y_i$ = actual value
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* $\hat{y}_i$ = predicted value
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### 2. Mean Calculation
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Average the relative errors over the period:
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$$MRAE = \frac{1}{n} \sum_{i=1}^{n} \frac{|y_i - \hat{y}_i|}{|y_i|}$$
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### 3. Running Update (O(1))
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QuanTAlib uses a ring buffer with running sum for O(1) updates:
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$$S_{new} = S_{old} - e_{oldest} + e_{newest}$$
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$$MRAE = \frac{S_{new}}{n}$$
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## Implementation Details
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### Usage Patterns
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```csharp
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// Streaming mode - update with each new observation
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var mrae = new Mrae(period: 20);
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var result = mrae.Update(actualValue, predictedValue);
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// Batch mode - calculate for entire series
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var results = Mrae.Calculate(actualSeries, predictedSeries, period: 20);
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// Span mode - zero-allocation for high performance
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Mrae.Batch(actualSpan, predictedSpan, outputSpan, period: 20);
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```
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### Parameters
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| Parameter | Type | Description |
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| :--- | :--- | :--- |
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| **period** | int | Lookback window for averaging (must be > 0) |
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### Properties
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| Property | Type | Description |
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| :--- | :--- | :--- |
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| **Last** | TValue | Most recent MRAE value |
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| **IsHot** | bool | True when buffer is full |
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| **Name** | string | Indicator name (e.g., "Mrae(20)") |
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| **WarmupPeriod** | int | Number of periods before valid output |
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## Performance Profile
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### Operation Count (Streaming Mode)
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O(1) per bar. Single-pass scalar transformation of (actual, forecast) pair; no lookback window required.
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| Operation | Count | Cost (cycles) | Subtotal |
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| :--- | :---: | :---: | :---: |
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| Error computation (subtract, abs/square/log) | 1-3 | ~3-8 cy | ~5-15 cy |
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| Running accumulator update (EMA or sum) | 1 | ~4 cy | ~4 cy |
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| **Total** | **2-4** | — | **~9-19 cycles** |
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Streaming update requires only the current actual/forecast pair and running state. ~10-15 cycles/bar typical.
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### Batch Mode (SIMD Analysis)
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| Operation | Vectorizable? | Notes |
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| :--- | :---: | :--- |
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| Element-wise error computation | Yes | Independent per bar; fully vectorizable with `Vector<double>` |
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| Reduction (sum/mean) | Yes | Parallel reduction; AVX2 gives 4x speedup |
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| Log/exp components | Partial | Transcendental ops; polynomial approx for SIMD |
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Batch SIMD: 4x-8x speedup for large windows. ~3-5 cy/bar amortized in vectorized batch mode.
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| Metric | Score | Notes |
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| :--- | :--- | :--- |
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| **Throughput** | ~15 ns/bar | O(1) update complexity |
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| **Allocations** | 0 | Uses pre-allocated ring buffer |
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| **Complexity** | O(1) | Constant time per update |
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| **Accuracy** | 10/10 | Exact calculation |
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| **Timeliness** | 9/10 | No lag beyond the period |
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| **Smoothness** | 7/10 | Moderate smoothing |
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## Interpretation
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| MRAE Range | Interpretation |
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| :--- | :--- |
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| **0** | Perfect prediction |
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| **0 - 0.1** | Excellent (< 10% average relative error) |
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| **0.1 - 0.3** | Good (10-30% average relative error) |
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| **> 0.3** | Poor (> 30% average relative error) |
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## Comparison with Other Metrics
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| Metric | Scale-Independent | Zero-Safe | Symmetry |
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| :--- | :--- | :--- | :--- |
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| **MRAE** | Yes | No (uses substitution) | No |
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| **MAPE** | Yes | No | No |
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| **MAE** | No | Yes | Yes |
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| **SMAPE** | Yes | Partially | Yes |
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## Common Use Cases
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1. **Financial Forecasting**: Compare prediction accuracy across different asset prices
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2. **Demand Forecasting**: Normalize errors across products with varying sales volumes
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3. **Model Comparison**: Compare models on datasets with different scales
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4. **Time Series Analysis**: Track relative prediction quality over time
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## Edge Cases
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* **Zero Actual Values**: Substitutes with small epsilon (1e-10) to avoid division by zero
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* **NaN Handling**: Uses last valid value substitution
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* **Single Input**: Not supported (requires two series)
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* **Period = 1**: Returns current relative absolute error
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## Related Indicators
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* [MAE](../mae/Mae.md) - Mean Absolute Error (non-relative)
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* [MAPE](../mape/Mape.md) - Mean Absolute Percentage Error
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* [SMAPE](../smape/Smape.md) - Symmetric Mean Absolute Percentage Error |