Root-mean-square deviation

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The root-mean-square deviation (RMSD) or root-mean-square error (RMSE) is a frequently used measure of the differences between value (Sample and population values) predicted by a model or an estimator and the values actually observed. Basically, the RMSD represents the sample standard deviation of the differences between predicted values and observed values. 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 RMSD serves to aggregate the magnitudes of the errors in predictions for various times into a single measure of predictive power. RMSD is 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.[1]


The RMSD of an estimator \hat{\theta} with respect to an estimated parameter \theta is defined as the square root of the mean square error:

\operatorname{RMSD}(\hat{\theta}) = \sqrt{\operatorname{MSE}(\hat{\theta})} = \sqrt{\operatorname{E}((\hat{\theta}-\theta)^2)}.

For an unbiased estimator, the RMSD is the square root of the variance, known as the standard error.

The RMSD of predicted values \hat y_t for times t of a regression's dependent variable y is computed for n different predictions as the square root of the mean of the squares of the deviations:

\operatorname{RMSD}=\sqrt{\frac{\sum_{t=1}^n (\hat y_t - y)^2}{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 x_{1,t} and x_{2,t}, the formula becomes

\operatorname{RMSD}= \sqrt{\frac{\sum_{t=1}^n (x_{1,t} - x_{2,t})^2}{n}}.

Normalized root-mean-square deviation[edit]

The normalized root-mean-square deviation or error (NRMSD or NRMSE) is the RMSD divided by the range of observed values of a variable being predicted,[citation needed] or:

\mathrm{NRMSD} = \frac{\mathrm{RMSD}}{y_\max -y_\min}

The value is often expressed as a percentage, where lower values indicate less residual variance.

CV (RMSD)[edit]

The coefficient of variation of the RMSD, CV(RMSD), or more commonly CV(RMSE), is defined as the RMSD normalized to the mean of the observed values:[2]

 \mathrm{CV(RMSD)} = \frac {\mathrm{RMSD}}{\bar y}.

It is the same concept as the coefficient of variation except that RMSD replaces the standard deviation.


See also[edit]


  1. ^ Hyndman, Rob J. Koehler, Anne B.; Koehler (2006). "Another look at measures of forecast accuracy". International Journal of Forecasting 22 (4): 679–688. doi:10.1016/j.ijforecast.2006.03.001. 
  2. ^ Retrieved 8 August 2014.  |first1= missing |last1= in Authors list (help); Missing or empty |title= (help)
  3. ^ J. Scott Armstrong and Fred Collopy (1992). "Error Measures For Generalizing About Forecasting Methods: Empirical Comparisons". International Journal of Forecasting 8 (1): 69–80. doi:10.1016/0169-2070(92)90008-w. 
  4. ^ Anderson, M.P.; Woessner, W.W. (1992). Applied Groundwater Modeling: Simulation of Flow and Advective Transport (2nd ed.). Academic Press. 
  5. ^ Ensemble Neural Network Model