Root-mean-square (RMS) error, also known as RMS deviation, is a frequently used measure of the differences between values predicted by a model or an estimator and the values actually observed. These individual differences are called residuals when the calculations are performed over the data sample that was used for estimation, and are called prediction errors when computed out-of-sample. The differences between values occur because of randomness or because the estimator doesn't account for information that could produce a more accurate estimate.
Root-mean-square error serves to aggregate the magnitudes of the errors in predictions for various times into a single measure of predictive power. It is also a good measure of accuracy, but only to compare forecasting errors of different models for a particular variable and not between variables, as it is scale-dependent.
RMS error is the square root of mean squared error (MSE), which is a risk function corresponding to the expected value of the squared error loss or quadratic loss. MSE measures the average of the squares of the "errors. " The MSE is the second moment (about the origin) of the error, and thus incorporates both the variance of the estimator and its bias. For an unbiased estimator, the MSE is the variance of the estimator. Like the variance, MSE has the same units of measurement as the square of the quantity being estimated.
Computing MSE and RMSE
If Y^\hat { Y } is a vector of
This is a known, computed quantity given a particular sample (and hence is sample-dependent). RMS error is simply the square root of the resulting MSE quantity.
RMS Error for the Regression Line
In terms of a regression line, the error for the differing values is simply the distance of a point above or below the line. We can find the general size of these errors by taking the RMS size for them:
This calculation results in the RMS error of the regression line, which tells us how far above or below the line points typically are. In general, about 68% of points on a scatter diagram are within one RMS error of the regression line, and about 95% are within two. This is known as the 68%-95% rule.