QuanTAlib/lib/errors/smape/Smape.md

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

// 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:

// 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):

// 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 - Asymmetric percentage error
• MPE - Signed percentage error for bias
• MAE - Absolute error without scaling