Errormeasure

Errormeasure is a quantitative metric used to describe the discrepancy between a measured, estimated, or predicted value and a reference value, typically the true value. In measurement theory, statistics, and data analysis, errormeasures are used to assess accuracy, precision, and the quality of models, sensors, or experiments. They can be defined for individual observations or aggregated over datasets, time series, or spatial domains.

Common types of error include absolute error and relative error. Absolute error is the magnitude of the difference between the estimate ŷ and the true value y, computed as e_abs = |ŷ − y|. Relative error scales this difference by the magnitude of the true value, e_rel = |ŷ − y| / |y| when y ≠ 0, providing a unitless measure. Squared error e_sq = (ŷ − y)^2 is also used to emphasize larger discrepancies. In addition, signed error e = ŷ − y keeps track of direction.

Aggregated error measures are widely used to evaluate models and predictions. Mean Absolute Error (MAE) is the average of absolute errors: MAE = (1/n) ∑ |ŷ_i − y_i|. Mean Squared Error (MSE) and Root Mean Squared Error (RMSE) are the average of squared errors and its square root, respectively. Relative error measures include Mean Absolute Percentage Error (MAPE) = (100/n) ∑ |(ŷ_i − y_i)/y_i|, which expresses errors as percentages, though it can be undefined when y_i = 0. Symmetric MAPE (SMAPE) provides a scale-invariant alternative: SMAPE = (100/n) ∑ [2|ŷ_i − y_i| / (|ŷ_i| + |y_i|)].

Practitioners select errormeasures based on the context, balancing interpretability, scale sensitivity, and robustness to outliers. In machine learning, many error measures also function as loss functions guiding model optimization.