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Miha Kralj
35a6702b06
Deep review of all indicator categories verified .md headers against .cs WarmupPeriod, parameters, inputs, and outputs. Fixes include warmup corrections, parameter documentation, output type accuracy, and Pine Script alignment.
187 lines
6.7 KiB
Markdown
187 lines
6.7 KiB
Markdown
# SMAPE: Symmetric Mean Absolute Percentage Error
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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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### TL;DR
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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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- Parameterized by `period`.
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- Output range: $\geq 0$.
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- Requires `period` bars of warmup before first valid output (IsHot = true).
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- Validated against TA-Lib, Skender, and Tulip reference implementations where available.
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> "MAPE punishes based on who's right; SMAPE punishes based on how different they are."
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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
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