The root-mean-square deviation or root-mean-square error is a frequently used measure of the
differences between 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.
Formula 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 error.
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 The normalized root-mean-square deviation
or error is the RMSD divided by the range of observed values of a variable being predicted,
or:
The value is often expressed as a percentage, where lower values indicate less residual
variance.
CV 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:
It is the same concept as the coefficient of variation except that RMSD replaces the
standard deviation.
Applications In meteorology, to see how effectively a mathematical
model predicts the behavior of the atmosphere In bioinformatics, the RMSD is the measure
of the average distance between the atoms of superimposed proteins.
In structure based drug design, the RMSD is a measure of the difference between a crystal
conformation of the ligand conformation and a docking prediction.
In economics, the RMSD is used to determine whether an economic model fits economic indicators.
Some experts have argued that RMSD is less reliable than Relative Absolute Error.
In experimental psychology, the RMSD is used to assess how well mathematical or computational
models of behavior explain the empirically observed behavior.
In GIS, the RMSD is one measure used to assess the accuracy of spatial analysis and remote
sensing.
In hydrogeology, RMSD and NRMSD are used to evaluate the calibration of a groundwater
model.
In imaging science, the RMSD is part of the peak signal-to-noise ratio, a measure used
to assess how well a method to reconstruct an image performs relative to the original
image.
In computational neuroscience, the RMSD is used to assess how well a system learns a
given model.
In Protein nuclear magnetic resonance spectroscopy, the RMSD is used as a measure to estimate
the quality of the obtained bundle of structures.
Submissions for the Netflix Prize were judged using the RMSD from the test dataset's undisclosed
"true" values.
In simulation of energy consumption of buildings, the RMSE and CV(RMSE) are used to calibrate
models to measured building performance.
See also Root mean square
Squared deviations Errors and residuals in statistics
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"Error Measures For Generalizing About Forecasting Methods: Empirical Comparisons".
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