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Miha Kralj
183 lines
6.8 KiB
Markdown
183 lines
6.8 KiB
Markdown
# SMAPE: Symmetric Mean Absolute Percentage Error
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> *MAPE punishes based on who's right; SMAPE punishes based on how different they are.*
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| Property | Value |
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| ---------------- | -------------------------------- |
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| **Category** | Error Metric |
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| **Inputs** | Actual vs Predicted (dual input) |
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| **Parameters** | `period` |
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| **Outputs** | Single series (SMAPE) |
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| **Output range** | $\geq 0$ |
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| **Warmup** | `period` bars |
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| **PineScript** | [smape.pine](smape.pine) |
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- Symmetric Mean Absolute Percentage Error addresses a fundamental asymmetry in MAPE: the fact that over-predictions and under-predictions of the sam...
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- **Similar:** [MAPE](../mape/Mape.md), [MAAPE](../maape/Maape.md) | **Trading note:** Symmetric MAPE; bounded 0–200%, handles zeros better than MAPE. Common in forecasting competitions.
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- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
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Symmetric Mean Absolute Percentage Error addresses a fundamental asymmetry in MAPE: the fact that over-predictions and under-predictions of the same magnitude receive different penalties. SMAPE uses the average of actual and predicted values in the denominator, creating a metric that treats both directions equally.
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## Architecture & Physics
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SMAPE computes the symmetric percentage error for each observation:
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$$\text{SMAPE} = \frac{200}{n} \sum_{i=1}^{n} \frac{|\text{actual}_i - \text{predicted}_i|}{|\text{actual}_i| + |\text{predicted}_i|}$$
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The factor of 200 (rather than 100) scales the result to match traditional percentage ranges.
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### Symmetry Explained
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Consider predicting a value of 80 when actual is 100, versus predicting 100 when actual is 80:
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**MAPE calculations:**
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* Case 1: $100 \times |100-80|/100 = 20\%$
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* Case 2: $100 \times |80-100|/80 = 25\%$
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**SMAPE calculations:**
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* Case 1: $200 \times |100-80|/(100+80) = 22.2\%$
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* Case 2: $200 \times |80-100|/(80+100) = 22.2\%$
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SMAPE assigns identical penalties regardless of which value is larger.
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## Mathematical Foundation
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### 1. Point-wise Symmetric Error
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For each observation:
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$$e_i = 200 \times \frac{|\text{actual}_i - \text{predicted}_i|}{|\text{actual}_i| + |\text{predicted}_i|}$$
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### 2. Rolling Average
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Over a period $n$:
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$$\text{SMAPE}_t = \frac{1}{n} \sum_{i=t-n+1}^{t} e_i$$
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### 3. Bounds
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SMAPE is bounded between 0% and 200%:
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* **0%**: Perfect prediction (actual = predicted)
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* **200%**: Maximum error (one value is 0, other is non-zero)
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* **100%**: Occurs when |actual - predicted| = (|actual| + |predicted|)/2
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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** | 18 ns/bar | O(1) via running sum |
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| **Allocations** | 0 | Zero-allocation hot path |
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| **Complexity** | O(1) | Constant per update |
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| **Symmetry** | 10/10 | Primary advantage |
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| **Zero Handling** | 8/10 | Better than MAPE |
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| **Scale Independence** | 9/10 | Percentage-based |
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| **Interpretability** | 7/10 | 200% scale less intuitive |
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## Usage
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```csharp
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// Streaming mode - symmetric error measurement
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var smape = new Smape(20);
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// These two scenarios give identical SMAPE
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smape.Update(actual: 100.0, predicted: 80.0); // Under-prediction
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smape.Update(actual: 80.0, predicted: 100.0); // Over-prediction
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double symmetricError = smape.Last.Value;
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// Batch mode - historical analysis
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var actual = new TSeries { 100, 105, 98, 102, 101 };
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var predicted = new TSeries { 95, 100, 95, 100, 100 };
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var results = Smape.Calculate(actual, predicted, period: 3);
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// Span mode - zero-allocation bulk processing
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Span<double> output = stackalloc double[1000];
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Smape.Batch(actualSpan, predictedSpan, output, period: 20);
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```
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## Interpretation Guide
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| SMAPE Value | Interpretation | Model Quality |
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| :--- | :--- | :--- |
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| **0-10%** | Excellent accuracy | Production-ready |
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| **10-25%** | Good accuracy | Suitable for most applications |
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| **25-50%** | Moderate accuracy | May need improvement |
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| **50-100%** | Poor accuracy | Significant errors |
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| **100-200%** | Very poor accuracy | Model needs redesign |
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## Comparison with MAPE
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| Scenario | MAPE | SMAPE | Winner |
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| :--- | :--- | :--- | :--- |
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| Actual=100, Pred=80 | 20% | 22.2% | Similar |
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| Actual=80, Pred=100 | 25% | 22.2% | SMAPE (symmetric) |
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| Actual=0, Pred=100 | Undefined | 200% | SMAPE (defined) |
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| Actual=100, Pred=0 | 100% | 200% | Context-dependent |
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| Interpretation | Familiar | Less intuitive | MAPE |
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## Common Pitfalls
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### 1. The 200% Scale
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SMAPE ranges from 0% to 200%, not 0% to 100%. This can cause confusion when comparing with MAPE:
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```csharp
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// SMAPE = 50% is roughly equivalent to MAPE ≈ 33-40%
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// The relationship is non-linear
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```
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### 2. Both Values Near Zero
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When both actual and predicted approach zero, SMAPE approaches 0% (perfect):
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```csharp
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// actual = 0.001, predicted = 0.002
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// |diff| = 0.001, sum = 0.003
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// SMAPE = 200 * 0.001 / 0.003 = 66.7%
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// This may not reflect actual model quality
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```
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### 3. Sign Insensitivity
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Like MAPE, SMAPE doesn't indicate bias direction. A model consistently over-predicting by 10% looks identical to one consistently under-predicting by 10%.
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**Solution**: Pair SMAPE with MPE for complete analysis.
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## Variant: Armstrong's SMAPE
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Some implementations use the mean (divide by 2) in the denominator:
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$$\text{SMAPE}_{\text{Armstrong}} = \frac{100}{n} \sum \frac{|\text{actual} - \text{predicted}|}{(|\text{actual}| + |\text{predicted}|)/2}$$
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This scales to 0-100% but is mathematically equivalent to the 0-200% version. QuanTAlib uses the 0-200% convention to match the original formulation.
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## See Also
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* [MAPE](../mape/Mape.md) - Asymmetric percentage error
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* [MPE](../mpe/Mpe.md) - Signed percentage error for bias
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* [MAE](../mae/Mae.md) - Absolute error without scaling |