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 values (sample and population values) predicted by a model or an estimator and the values actually observed. 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]

Formula[edit]

The RMSD of an estimator with respect to an estimated parameter is defined as the square root of the mean square error:

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

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

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 and , the formula becomes

Normalized root-mean-square deviation[edit]

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:[2]

or .

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. In many cases, especially for smaller samples, the sample range is likely to be affected by the size of sample which would hamper comparisons.

When normalising by the mean value of the measurements, the term coefficient of variation of the RMSD, CV(RMSD) may be used to avoid ambiguity.[3] This is analogous to the coefficient of variation with the RMSD taking the place of the standard deviation.

Applications[edit]

See also[edit]

References[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. ^ "Coastal Inlets Research Program (CIRP) Wiki - Statistics". Retrieved 4 February 2015. 
  3. ^ "FAQ: What is the coefficient of variation?". Retrieved 4 February 2015. 
  4. ^ J. Scott Armstrong & Fred Collopy (1992). "Error Measures For Generalizing About Forecasting Methods: Empirical Comparisons" (PDF). International Journal of Forecasting 8 (1): 69–80. doi:10.1016/0169-2070(92)90008-w. 
  5. ^ Anderson, M.P.; Woessner, W.W. (1992). Applied Groundwater Modeling: Simulation of Flow and Advective Transport (2nd ed.). Academic Press. 
  6. ^ Ensemble Neural Network Model
  7. ^ ANSI/BPI-2400-S-2012: Standard Practice for Standardized Qualification of Whole-House Energy Savings Predictions by Calibration to Energy Use History