Root Mean Squared Error (RMSE)

Formula

\[ \mathrm{RMSE} = \sqrt{\frac{1}{n}\sum_{i=1}^n (y_i-\hat y_i)^2} \]

Plot

fn: abs(x)
xmin: -3
xmax: 3
ymin: 0
ymax: 3.2
height: 280
title: Root squared error for a single residual (|r|)

Parameters

• \(y_i\): true value
• \(\hat y_i\): prediction
• \(n\): number of samples

What it means

Typical prediction error in the same units as the target.

What it's used for

• Interpretable regression error metric.
• Comparing models on the same target scale.

Key properties

• \(\mathrm{RMSE} = \sqrt{\mathrm{MSE}}\).
• More sensitive to large errors than MAE.

Common gotchas

• Still sensitive to outliers.
• Not scale-free; avoid comparing across targets.

Example

If \(\mathrm{MSE}=1.667\), then \(\mathrm{RMSE}=1.291\).

How to Compute (Pseudocode)

Input: true values y[1..n], predictions y_hat[1..n]
Output: rmse

sum_sq <- 0
for i from 1 to n:
  residual <- y[i] - y_hat[i]
  sum_sq <- sum_sq + residual^2

mse <- sum_sq / n
rmse <- sqrt(mse)
return rmse

Complexity

• Time: \(O(n)\)
• Space: \(O(1)\) additional space
• Assumptions: \(n\) is the number of paired predictions/targets