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https://github.com/mihakralj/QuanTAlib.git
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Miha Kralj
180 lines
6.6 KiB
Markdown
180 lines
6.6 KiB
Markdown
# ME: Mean Error (Mean Bias Error)
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> *Sometimes you need to know not just how wrong you are, but which direction you're wrong in.*
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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 (ME) |
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| **Output range** | Any (positive or negative) |
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| **Warmup** | `period` bars |
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| **PineScript** | [me.pine](me.pine) |
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- Mean Error (ME), also known as Mean Bias Error, measures the average error between actual and predicted values while preserving the sign.
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- **Similar:** [MAE](../mae/Mae.md), [MPE](../mpe/Mpe.md) | **Trading note:** Mean Error (bias); positive = systematic overprediction, negative = underprediction.
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- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
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Mean Error (ME), also known as Mean Bias Error, measures the average error between actual and predicted values while preserving the sign. Unlike MAE, ME reveals systematic bias in predictions: whether a model consistently over-predicts or under-predicts.
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## Historical Context
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ME is one of the fundamental error metrics in statistics and forecasting. While MAE and MSE focus on error magnitude, ME fills the critical role of detecting directional bias. A model could have low MAE but significant ME, indicating consistent over or under-prediction that cancels out when measuring magnitude alone.
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## Architecture & Physics
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ME preserves the sign of errors, allowing positive and negative errors to cancel each other. This makes it ideal for detecting systematic bias but unsuitable for measuring prediction accuracy alone.
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### Properties
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* **Can be negative**: ME can be positive, negative, or zero
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* **Positive ME**: Model under-predicts (actual > predicted on average)
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* **Negative ME**: Model over-predicts (actual < predicted on average)
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* **Zero ME**: No systematic bias (but not necessarily accurate)
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* **Same units**: ME is in the same units as the original data
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* **Cancellation**: Errors can cancel out, hiding large individual errors
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## Mathematical Foundation
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### 1. Error Calculation
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For each observation, calculate the signed 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 errors over the period:
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$$ME = \frac{1}{n} \sum_{i=1}^{n} (y_i - \hat{y}_i)$$
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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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$$ME = \frac{S_{new}}{n}$$
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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 me = new Me(period: 20);
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var result = me.Update(actualValue, predictedValue);
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// Batch mode - calculate for entire series
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var results = Me.Calculate(actualSeries, predictedSeries, period: 20);
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// Span mode - zero-allocation for high performance
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Me.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 ME value |
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| **IsHot** | bool | True when buffer is full |
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| **Name** | string | Indicator name (e.g., "Me(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** | ~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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| ME Value | Interpretation |
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| :--- | :--- |
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| **ME > 0** | Systematic under-prediction (actual > predicted) |
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| **ME = 0** | No systematic bias |
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| **ME < 0** | Systematic over-prediction (actual < predicted) |
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## Comparison with Other Metrics
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| Metric | Shows Bias | Units | Use Case |
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| :--- | :--- | :--- | :--- |
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| **ME** | Yes | Same as data | Detect systematic bias |
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| **MAE** | No | Same as data | Average error magnitude |
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| **MSE** | No | Squared units | Penalize large errors |
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| **MPE** | Yes | Percentage | Relative bias |
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## Common Use Cases
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1. **Bias Detection**: Identify if a model consistently over or under-predicts
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2. **Model Calibration**: Use ME to adjust model outputs
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3. **Forecast Evaluation**: Distinguish between random errors and systematic bias
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4. **Trading Signals**: Detect directional bias in price predictions
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## Warning: Cancellation Problem
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ME can be misleading when errors cancel out:
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```csharp
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var me = new Me(4);
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me.Update(110, 100); // Error: +10
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me.Update(90, 100); // Error: -10
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me.Update(110, 100); // Error: +10
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me.Update(90, 100); // Error: -10
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// ME = 0, but individual errors are large!
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```
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Always use ME alongside MAE or MSE to get a complete picture.
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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 signed error
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* **Balanced Errors**: Can return 0 even with large individual errors
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## Related Indicators
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* [MAE](../mae/Mae.md) - Mean Absolute Error (magnitude only)
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* [MSE](../mse/Mse.md) - Mean Squared Error
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* [MPE](../mpe/Mpe.md) - Mean Percentage Error (relative bias) |