![]() ![]() By dividing by n, we keep this measure of error consistent as we move from a small collection of observations to a larger collection (it just becomes more accurate as we increase the number of observations). We should also now have an explanation for the division by n under the square root in RMSE: it allows us to estimate the standard deviation σ of the error for a typical single observation rather than some kind of “total error”. ![]() But then RMSE is a good estimator for the standard deviation σ of the distribution of our errors! This tells us that Σᵢ (ŷᵢ - yᵢ)² / n is a good estimator for E = σ². In fact a sharper form of the central limit theorem tell us its variance should converge to 0 asymptotically like 1/n. The central limit theorem tells us that as n gets larger, the variance of the quantity Σᵢ (ŷᵢ - yᵢ)² / n = Σᵢ (εᵢ)² / n should converge to zero. Notice the left hand side looks familiar! If we removed the expectation E from inside the square root, it is exactly our formula for RMSE form before. ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |