SMAPE - Symmetric Mean Absolute Percentage Error

In time series forecasting, the Symmetric Mean Absolute Percentage Error (SMAPE) is notorious for having multiple conflicting definitions across literature. To completely eliminate ambiguity, permetrics explicitly implements four distinct variants categorized into two mathematical paradigms:

1. The Original Paradigm (Armstrong, 1985)

In the original formulation [8], the absolute error is divided by the mean of the absolute actual and predicted values: \((|y_i| + |\hat{y}_i|) / 2\). This factor of \(1/2\) in the denominator flips up to become a multiplier of \(2\) in the numerator.

SMAPE (Original Percentage)

The classic percentage variant utilized in the M3-Competition.

\[\text{SMAPE}(y, \hat{y}) = \frac{100\%}{N} \sum_{i=1}^{N} \frac{2 \cdot |y_i - \hat{y}_i|}{|y_i| + |\hat{y}_i|}\]

Range: [0, 200%]
Best possible score: 0.0
Function alias: evaluator.SMAPE() or evaluator.symmetric_mean_absolute_percentage_error()

SMAPE_NP (Original No-Percentage)

The raw ratio of the original Armstrong formula (omitting the 100% multiplier).

\[\text{SMAPE\_NP}(y, \hat{y}) = \frac{1}{N} \sum_{i=1}^{N} \frac{2 \cdot |y_i - \hat{y}_i|}{|y_i| + |\hat{y}_i|}\]

Range: [0, 2]
Best possible score: 0.0
Function alias: evaluator.SMAPE_NP()

2. The Simplified Paradigm (Makridakis, 1993)

Proposed by Spyros Makridakis [9] as an alternative to bound the upper limit naturally to 1.0 (or 100%). It divides the absolute error directly by the sum of the absolutes.

SMAPE_S (Simplified No-Percentage)

The raw decimal variant of the simplified formula.

\[\text{SMAPE\_S}(y, \hat{y}) = \frac{1}{N} \sum_{i=1}^{N} \frac{|y_i - \hat{y}_i|}{|y_i| + |\hat{y}_i|}\]

Range: [0, 1]
Best possible score: 0.0
Function alias: evaluator.SMAPE_S()

SMAPE_S_P (Simplified Percentage)

The 100-scaled percentage variant of the simplified formula.

\[\text{SMAPE\_S\_P}(y, \hat{y}) = \frac{100\%}{N} \sum_{i=1}^{N} \frac{|y_i - \hat{y}_i|}{|y_i| + |\hat{y}_i|}\]

Range: [0, 100%]
Best possible score: 0.0
Function alias: evaluator.SMAPE_S_P()

Mathematical Disambiguation & Rules 

1. Zero-Division Protection: In all four implementations, if both the actual value and the predicted value at timestamp \(i\) are strictly zero (\(y_i = 0\) and \(\hat{y}_i = 0\)), the element-wise error is explicitly evaluated as 0.0 to prevent NaN propagation.

2. Asymmetry Warning: Despite the name “Symmetric”, all SMAPE variants inherently penalize over-forecasting (\(\hat{y}_i > y_i\)) more heavily than under-forecasting (\(\hat{y}_i < y_i\)).

Example to use SMAPE metrics:

from numpy import array
from permetrics.regression import RegressionMetric

## For 1-D array
y_true = array([3, -0.5, 2, 7])
y_pred = array([2.5, 0.0, 2, 8])

evaluator = RegressionMetric(y_true, y_pred)
print(evaluator.symmetric_mean_absolute_percentage_error())
print(evaluator.SMAPE_NP())
print(evaluator.SMAPE_S())
print(evaluator.SMAPE_S_P())

## For > 1-D array
y_true = array([[0.5, 1], [-1, 1], [7, -6]])
y_pred = array([[0, 2], [-1, 2], [8, -5]])

evaluator = RegressionMetric(y_true, y_pred)
print(evaluator.SMAPE(multi_output="raw_values"))