RMSE

Root mean squared error (RMSE) is a measure of the differences between predicted values by a model and the observed values. It is widely used in regression analysis to assess the accuracy of predictive models.

Given n samples with observed values y_i and predicted values ŷ_i, RMSE is defined as sqrt( (1/n) ∑_{i=1}^n (y_i - ŷ_i)^2 ).

The RMSE is in the same units as the observed data, making it easy to interpret. A lower RMSE indicates better fit. RMSE is always nonnegative, and it decreases as the model's predictions improve, though it cannot be negative even if predictions are perfect.

Its squaring of errors makes RMSE more sensitive to large errors (outliers) than MAE. It is differentiable, which facilitates optimization in many learning algorithms. RMSE can be computed on a training or test set; when used on a hold-out or cross-validated set, it provides an estimate of predictive accuracy on unseen data.

RMSE is the square root of MSE; all three are used for regression evaluation. Compared to MAE, RMSE punishes large residuals more heavily.

Variants include weighted RMSE, which accounts for heteroscedasticity or uneven importance of samples, and normalized or relative RMSE, such as RMSE divided by the range or mean of the observed values to enable comparison across datasets. Cross-validation RMSE is RMSE computed on held-out folds, often averaged to estimate generalization error.

Because RMSE depends on the scale of the data, it cannot be used to compare models trained on datasets with different scales without standardization or normalization.