# SMAPE: Symmetric Mean Absolute Percentage Error

> "MAPE punishes based on who's right; SMAPE punishes based on how different they are."

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.

## Architecture & Physics

SMAPE computes the symmetric percentage error for each observation:

$$\text{SMAPE} = \frac{200}{n} \sum_{i=1}^{n} \frac{|\text{actual}_i - \text{predicted}_i|}{|\text{actual}_i| + |\text{predicted}_i|}$$

The factor of 200 (rather than 100) scales the result to match traditional percentage ranges.

### Symmetry Explained

Consider predicting a value of 80 when actual is 100, versus predicting 100 when actual is 80:

**MAPE calculations:**

* Case 1: $100 \times |100-80|/100 = 20\%$
* Case 2: $100 \times |80-100|/80 = 25\%$

**SMAPE calculations:**

* Case 1: $200 \times |100-80|/(100+80) = 22.2\%$
* Case 2: $200 \times |80-100|/(80+100) = 22.2\%$

SMAPE assigns identical penalties regardless of which value is larger.

## Mathematical Foundation

### 1. Point-wise Symmetric Error

For each observation:

$$e_i = 200 \times \frac{|\text{actual}_i - \text{predicted}_i|}{|\text{actual}_i| + |\text{predicted}_i|}$$

### 2. Rolling Average

Over a period $n$:

$$\text{SMAPE}_t = \frac{1}{n} \sum_{i=t-n+1}^{t} e_i$$

### 3. Bounds

SMAPE is bounded between 0% and 200%:

* **0%**: Perfect prediction (actual = predicted)
* **200%**: Maximum error (one value is 0, other is non-zero)
* **100%**: Occurs when |actual - predicted| = (|actual| + |predicted|)/2

## Performance Profile

| Metric | Score | Notes |
| :--- | :--- | :--- |
| **Throughput** | 18 ns/bar | O(1) via running sum |
| **Allocations** | 0 | Zero-allocation hot path |
| **Complexity** | O(1) | Constant per update |
| **Symmetry** | 10/10 | Primary advantage |
| **Zero Handling** | 8/10 | Better than MAPE |
| **Scale Independence** | 9/10 | Percentage-based |
| **Interpretability** | 7/10 | 200% scale less intuitive |

## Usage

```csharp
// Streaming mode - symmetric error measurement
var smape = new Smape(20);

// These two scenarios give identical SMAPE
smape.Update(actual: 100.0, predicted: 80.0);  // Under-prediction
smape.Update(actual: 80.0, predicted: 100.0); // Over-prediction

double symmetricError = smape.Last.Value;

// Batch mode - historical analysis
var actual = new TSeries { 100, 105, 98, 102, 101 };
var predicted = new TSeries { 95, 100, 95, 100, 100 };
var results = Smape.Calculate(actual, predicted, period: 3);

// Span mode - zero-allocation bulk processing
Span<double> output = stackalloc double[1000];
Smape.Batch(actualSpan, predictedSpan, output, period: 20);
```

## Interpretation Guide

| SMAPE Value | Interpretation | Model Quality |
| :--- | :--- | :--- |
| **0-10%** | Excellent accuracy | Production-ready |
| **10-25%** | Good accuracy | Suitable for most applications |
| **25-50%** | Moderate accuracy | May need improvement |
| **50-100%** | Poor accuracy | Significant errors |
| **100-200%** | Very poor accuracy | Model needs redesign |

## Comparison with MAPE

| Scenario | MAPE | SMAPE | Winner |
| :--- | :--- | :--- | :--- |
| Actual=100, Pred=80 | 20% | 22.2% | Similar |
| Actual=80, Pred=100 | 25% | 22.2% | SMAPE (symmetric) |
| Actual=0, Pred=100 | Undefined | 200% | SMAPE (defined) |
| Actual=100, Pred=0 | 100% | 200% | Context-dependent |
| Interpretation | Familiar | Less intuitive | MAPE |

## Common Pitfalls

### 1. The 200% Scale

SMAPE ranges from 0% to 200%, not 0% to 100%. This can cause confusion when comparing with MAPE:

```csharp
// SMAPE = 50% is roughly equivalent to MAPE ≈ 33-40%
// The relationship is non-linear
```

### 2. Both Values Near Zero

When both actual and predicted approach zero, SMAPE approaches 0% (perfect):

```csharp
// actual = 0.001, predicted = 0.002
// |diff| = 0.001, sum = 0.003
// SMAPE = 200 * 0.001 / 0.003 = 66.7%
// This may not reflect actual model quality
```

### 3. Sign Insensitivity

Like MAPE, SMAPE doesn't indicate bias direction. A model consistently over-predicting by 10% looks identical to one consistently under-predicting by 10%.

**Solution**: Pair SMAPE with MPE for complete analysis.

## Variant: Armstrong's SMAPE

Some implementations use the mean (divide by 2) in the denominator:

$$\text{SMAPE}_{\text{Armstrong}} = \frac{100}{n} \sum \frac{|\text{actual} - \text{predicted}|}{(|\text{actual}| + |\text{predicted}|)/2}$$

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.

## See Also

* [MAPE](../mape/Mape.md) - Asymmetric percentage error
* [MPE](../mpe/Mpe.md) - Signed percentage error for bias
* [MAE](../mae/Mae.md) - Absolute error without scaling