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Miha Kralj
86fe32a682
Co-authored-by: Claude Opus 4.5 <noreply@anthropic.com> Co-authored-by: aider (openrouter/anthropic/claude-sonnet-4) <aider@aider.chat> Co-authored-by: Warp <agent@warp.dev>
126 lines
4.1 KiB
Markdown
126 lines
4.1 KiB
Markdown
# MAE: Mean Absolute Error
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> "When you need to know how wrong you are on average, without the drama of squared errors."
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Mean Absolute Error (MAE) measures the average magnitude of errors in a set of predictions, without considering their direction. It represents the average of the absolute differences between actual and predicted values.
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## Historical Context
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MAE is one of the oldest and most intuitive error metrics in statistics. Its simplicity and interpretability have made it a staple in regression analysis, forecasting, and model evaluation since the early days of statistical analysis.
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## Architecture & Physics
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MAE treats all errors equally, making it more robust to outliers compared to squared-error metrics like MSE. The absolute value operation removes directionality, focusing purely on error magnitude.
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### Properties
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* **Non-negative**: MAE ≥ 0, with 0 indicating perfect prediction
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* **Same units**: Unlike MSE, MAE is in the same units as the original data
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* **Linear sensitivity**: Each unit of error contributes equally to the final metric
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* **Robust**: Less sensitive to outliers than squared-error metrics
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## Mathematical Foundation
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### 1. Absolute Error
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For each observation, calculate the absolute difference between actual and predicted values:
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$$e_i = |y_i - \hat{y}_i|$$
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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 absolute errors over the period:
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$$MAE = \frac{1}{n} \sum_{i=1}^{n} |y_i - \hat{y}_i|$$
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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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$$MAE = \frac{S_{new}}{n}$$
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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 mae = new Mae(period: 20);
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var result = mae.Update(actualValue, predictedValue);
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// Batch mode - calculate for entire series
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var results = Mae.Calculate(actualSeries, predictedSeries, period: 20);
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// Span mode - zero-allocation for high performance
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Mae.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 MAE value |
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| **IsHot** | bool | True when buffer is full |
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| **Name** | string | Indicator name (e.g., "Mae(20)") |
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| **WarmupPeriod** | int | Number of periods before valid output |
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## Performance Profile
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| Metric | Score | Notes |
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| :--- | :--- | :--- |
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| **Throughput** | ~10 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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| MAE Range | Interpretation |
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| :--- | :--- |
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| **0** | Perfect prediction |
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| **Low** | Predictions are close to actual values |
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| **High** | Large average prediction error |
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## Comparison with Other Metrics
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| Metric | Outlier Sensitivity | Units | Interpretation |
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| :--- | :--- | :--- | :--- |
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| **MAE** | Low | Same as data | Average absolute error |
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| **MSE** | High | Squared units | Penalizes large errors more |
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| **RMSE** | High | Same as data | MSE in original units |
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| **MAPE** | Varies | Percentage | Relative error |
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## Common Use Cases
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1. **Forecast Evaluation**: Measure prediction accuracy over time
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2. **Model Comparison**: Compare different prediction models
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3. **Trading Strategy**: Track signal accuracy
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4. **Risk Assessment**: Monitor prediction reliability
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## Edge Cases
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* **Identical Values**: Returns 0 when actual equals predicted
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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 absolute error
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## Related Indicators
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* [MSE](../mse/Mse.md) - Mean Squared Error
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* [RMSE](../rmse/Rmse.md) - Root Mean Squared Error
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* [MAPE](../mape/Mape.md) - Mean Absolute Percentage Error
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