The root mean square deviation (RMSD) or root mean square error (RMSE) is either one of two closely related and frequently used measures of the differences between true or predicted values on the one hand and observed values or an estimator on the other.
The deviation is typically simply a differences of scalars; it can also be generalized to the vector lengths of a displacement, as in the bioinformatics concept of root mean square deviation of atomic positions.
The RMSD of a sample is the quadratic mean of the differences between the observed values and predicted ones.
These deviations are called residuals when the calculations are performed over the data sample that was used for estimation (and are therefore always in reference to an estimate) and are called errors (or prediction errors) when computed out-of-sample (aka on the full set, referencing a true value rather than an estimate).
[1] RMSD is always non-negative, and a value of 0 (almost never achieved in practice) would indicate a perfect fit to the data.
However, comparisons across different types of data would be invalid because the measure is dependent on the scale of the numbers used.
If X1, ..., Xn is a sample of a population with true mean value
with variables observed over T times, is computed for T different predictions as the square root of the mean of the squares of the deviations: (For regressions on cross-sectional data, the subscript t is replaced by i and T is replaced by n.) In some disciplines, the RMSD is used to compare differences between two things that may vary, neither of which is accepted as the "standard".
For example, when measuring the average difference between two time series
, the formula becomes Normalizing the RMSD facilitates the comparison between datasets or models with different scales.
Though there is no consistent means of normalization in the literature, common choices are the mean or the range (defined as the maximum value minus the minimum value) of the measured data:[4] This value is commonly referred to as the normalized root mean square deviation or error (NRMSD or NRMSE), and often expressed as a percentage, where lower values indicate less residual variance.
This is also called Coefficient of Variation or Percent RMS.
[5] This is analogous to the coefficient of variation with the RMSD taking the place of the standard deviation.
MAE possesses advantages in interpretability over RMSD.