Root Mean Squared Error

machine-learning statistics regression

Definition

Root Mean Squared Error

The root mean squared error (RMSE) is the square root of the mean squared error. For targets $y_1,\dots ,y_N$ and predictions ${\hat{y}}_1,\dots ,{\hat{y}}_N$ with residuals $r_i=y_i-{\hat{y}}_i$,

$$\mathrm{RMSE}=\sqrt{\frac{1}{N}\sum _{i=1}^N(y_i-{\hat{y}}_i{)}^2}=\sqrt{\frac{1}{N}\sum _{i=1}^N{r}_i^2}.$$

Because MSE squares the target’s units, RMSE takes the square root to restore them: predicting a price in euros, MSE is in euros squared, while RMSE is in euros. RMSE therefore reads on the same scale as the target, which is why it is the more interpretable evaluation metric for regression. The square root preserves MSE’s disproportionate penalty on large residuals — an error of size $2c$ still contributes $\sqrt{4c^2}=2c$ to the sum, twice the contribution of an error of size $c$ — so RMSE is dominated by large errors just as MSE is.

Relation to MSE

RMSE is a monotone increasing function of MSE:

$$\mathrm{RMSE}=\sqrt{\mathrm{MSE}}.$$

The two metrics rank models identically (the squaring is preserved through the root), so the choice between them is one of units and interpretability, not of optimisation.