sMAPE: Symmetric Mean Absolute Percentage Error

• sMAPE is a scale-independent metric that normalizes absolute errors by the average of predictions and actual values, ensuring robust comparisons across datasets.
• Its symmetric design equally penalizes over- and under-predictions, mitigating instability often seen with near-zero target values.
• Widely applied in forecasting, scientific modeling, and neural network interpretability, sMAPE provides fair model evaluation and actionable performance insights.

The Symmetric Mean Absolute Percentage Error (sMAPE) is a scale-independent, symmetric percentage error metric widely applied in regression, time series forecasting, imputation, neural network interpretability, and scientific modeling. Its fundamental property is the normalization of error by the average magnitude of prediction and target, enabling robust comparison across diverse datasets and preventing instability near zero values. sMAPE is favored over traditional metrics such as Mean Absolute Percentage Error (MAPE) due to its balanced treatment of over- and under-prediction and its suitability for scientific, medical, and engineering domains where both target and predicted values may vary widely or approach zero.

1. Formal Definition and Comparison with Related Error Metrics

sMAPE quantifies the relative error between forecasts or predicted values ($\hat{y}_i$) and observed or ground truth values ($y_i$) using the formula

$$\text{sMAPE} = \frac{100\%}{n} \sum_{i=1}^{n} \frac{|y_i - \hat{y}_i|}{\frac{1}{2}(|y_i| + |\hat{y}_i|)}$$

Variants exist depending on application context—for example, some domains apply $2|y_i - \hat{y}_i| / (|y_i| + |\hat{y}_i|)$ or use a denominator without the factor of two. The distinguishing trait of sMAPE, relative to MAPE ($|y_i - \hat{y}_i| / |y_i|$), is its symmetric treatment of both prediction and ground truth, preventing division-by-zero instability and exaggerated errors for near-zero actuals (Lovell et al., 2019, Shukla et al., 7 Aug 2025, Asadi et al., 22 Aug 2025).

In tabular comparison:

Metric	Formula	Stability Near Zero
MAE	$\frac{1}{n} \sum_i |y_i-\hat{y}_i|$	Stable, but not scale-normalized
MAPE	$$\frac{100\%}{n} \sum_i \frac{|y_i-\hat{y}_i|}{|y_i|}$$	Unstable for $|y_i|\rightarrow0$
sMAPE	see above	Robust to $|y_i| \approx 0, |\hat{y}_i| \approx 0$

MAPE's susceptibility to large or infinite errors for $|y_i| \to 0$ is widely documented (Myttenaere et al., 2015, Shukla et al., 7 Aug 2025). sMAPE's normalization by the average of $|y_i|$ and $|\hat{y}_i|$ reduces this distortion, making it suitable for domains with low or variable target magnitudes.

2. Symmetry, Scale-Invariance, and Robustness

sMAPE's symmetry ensures that $|y_i-\hat{y}_i|$ is penalized equally whether the error results from over-prediction or under-prediction. This feature is crucial for fair model assessment in multi-scale or heterogeneous datasets (Mohottige et al., 2021, Asadi et al., 22 Aug 2025). Its scale-invariance—error reported in percent relative to mean magnitude—supports direct comparison across models trained on domains with different scales (e.g., ATM withdrawals, axial strain in rocks, classroom occupancy, or Cobb angles in medical imaging) (Vangala et al., 2020, Shukla et al., 7 Aug 2025).

sMAPE also avoids undefined behavior when either target or prediction is zero, unlike MAPE which diverges. For example, in star formation history reconstruction, many bins have near-zero SFRs; sMAPE remains well-behaved and bounded (Lovell et al., 2019).

3. Application Domains and Model Evaluation

sMAPE features prominently as an accuracy metric across:

• Forecasting: Used with neural and statistical models for ATM withdrawals (Vangala et al., 2020), salt-rock deformation (Shukla et al., 7 Aug 2025), missing data imputation (Saad et al., 8 Jun 2024), and occupancy estimation (Mohottige et al., 2021). Practitioners rely on sMAPE for percentage-based comparison across ATMs, time series, or rooms, regardless of magnitude.
• Medical Imaging: Used to assess automated and ML-based Cobb angle extraction for scoliosis diagnosis (Dubost et al., 2019, Chen et al., 2022). sMAPE enables relative error assessment across scans with widely varying curvature magnitudes.
• Scientific Modeling: In quantum computing, sMAPE is used to assess ground-state energy prediction accuracy in VQE optimizers (Zhang et al., 2 May 2025), supplementing metrics such as mean relative error (MRE) to provide noise-robust low-error evaluation.
• Neural Interpretability: sMAPE is deployed in layer-wise relevance propagation for evaluating the discrepancy between predicted and backpropagated neuron contributions, supporting scale-free, interpretable visualization (Bhati et al., 7 Dec 2024).
• Temporal Reasoning and QA: sMAPE quantifies numeric prediction errors in LLM temporal QA tasks, offering a continuous metric where exact match yields only binary feedback (Abbood et al., 20 Sep 2025).

In these applications, sMAPE facilitates robust reporting (median errors of 2–11% for high-performing models) and interprets results across scenarios with over- or under-prediction, model generalization, and noise injection.

4. Theoretical Considerations and Consistency

Theoretical analysis of sMAPE (and its predecessor, MAPE) explores its suitability for empirical risk minimization and universal consistency within regression contexts. For MAPE, it has been rigorously proven that minimizing empirical risk is equivalent to weighted MAE regression where weights are $1/|y|$, and universal consistency is achievable provided the response variable is strictly bounded away from zero, the function class is uniformly bounded, and model complexity grows slowly with sample size (Myttenaere et al., 2015).

A plausible implication is that the strong theoretical foundations of MAPE consistency, under boundedness assumptions and covering number/Vapnik–Chervonenkis dimension analyses, carry over to sMAPE. This is particularly likely if the denominator term $((|p|+|y|)/2)$ in sMAPE is guaranteed to remain above some threshold. Ensuring this boundedness is necessary for uniform convergence of empirical risk minimization and for extending statistical learning theory arguments to sMAPE.

5. Practical Implications, Limitations, and Model Selection

sMAPE's practical benefits are evidenced by superior reliability in imputation (Saad et al., 8 Jun 2024), deep learning tasks (Lovell et al., 2019, Shukla et al., 7 Aug 2025), and model selection procedures—particularly when comparing heterogeneous or multimodal targets (as observed in RF circuit metric prediction (Asadi et al., 22 Aug 2025)). sMAPE equips practitioners to select models (e.g., RF vs. LSTM) based on relative rather than absolute errors, accounting for differences in intermittent demand or rare events (Vangala et al., 2020). Its boundedness and symmetry produce interpretable model rankings, even when underlying value distributions are skewed or contain near-zero values.

Limitations arise when the denominator $(|y_i| + |\hat{y}_i|)$ is very small, potentially amplifying noise. In practice, values are clipped or error definitions are adjusted (e.g., error defined as zero when both prediction and ground truth equal zero (Abbood et al., 20 Sep 2025)). Model selection based on sMAPE may also lead to increased interquartile range (IQR), reflecting error variability across samples—users should supplement sMAPE with distributional assessments and secondary metrics (MAE, RMSE, or MASE).

6. sMAPE in Benchmarking, Challenge Ranking, and Robustness Assessment

In competitive and benchmarking settings, sMAPE is often adopted as the official ranking metric due to its neutrality and quantitative interpretability (Dubost et al., 2019, Shukla et al., 7 Aug 2025). For example, medical imaging challenges employ sMAPE to compare automated ML systems to manual assessments, facilitating unbiased ranking despite wide-ranging case difficulty. In time series imputation and forecasting, sMAPE underpins both algorithmic comparisons and statistical significance testing of model improvements (Saad et al., 8 Jun 2024).

Its role extends to quantitative robustness assessment in scenario testing—e.g., evaluating generalization across disrupted input distributions in CNN-based galaxy spectral inference (Lovell et al., 2019), and quantifying degradation under added observational noise. In temporal reasoning and QA, sMAPE identifies whether model prediction errors are close misses or large failures, offering more actionable insight than binary exact match (Abbood et al., 20 Sep 2025).

7. Specialized Variants, Parameterization, and Metric Selection

Some studies adopt customized sMAPE variants to address specific normalization needs (e.g., multiplying by 200 instead of 100 when averaging over multiple targets or using alternate denominator averaging (Dubost et al., 2019)). Metric selection is governed by domain-specific requirements; sMAPE is often chosen where scale-independence, symmetry, and bounded error reporting are required.

Modelers must explicitly report the sMAPE variant and parameterization employed, reflecting on the metric’s interpretative domain—whether for single-target time series, multidimensional regression, clinical angle estimation, or quantum chemical energy prediction. Supplementing sMAPE with secondary metrics (MSE, RMSE, MASE) facilitates comprehensive performance evaluation and mitigates metric-specific artifacts.

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In sum, sMAPE functions as a principled, symmetric, and scale-independent percentage error metric, serving as a foundation for robust evaluation, model selection, benchmarking, and scientific reporting across regression, forecasting, imputation, and interpretability domains. Its adoption is substantiated by theory, empirical reliability, and practical utility in scenarios where conventional metrics are inadequate or unstable.