Root mean square deviation

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This article is about the general meaning of RMSD. For the application of RMSD to Bioinformatics, see Root mean square deviation (bioinformatics) .

The root mean square deviation (RMSD) (also root mean square error (RMSE)) is a frequently-used measure of the differences between values predicted by a model or an estimator and the values actually observed from the thing being modeled or estimated. These individual differences are also called residuals, and the RMSD serves to aggregate them into a single measure of predictive power.

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

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

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

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 distance between two oblong objects, expressed as random vectors

\mathbf{\theta}_1 = \begin{bmatrix} x_{1,1} \\ x_{1,2} \\ \vdots \\ x_{1,n} \end{bmatrix} \qquad \mathrm{and} \qquad \mathbf{\theta}_2 = \begin{bmatrix} x_{2,1} \\ x_{2,2} \\ \vdots \\ x_{2,n} \end{bmatrix} ,

The formula becomes:

\operatorname{RMSD}(\mathbf{\theta}_1, \mathbf{\theta}_2) = \sqrt{\operatorname{MSE}(\mathbf{\theta}_1, \mathbf{\theta}_2)} = \sqrt{\operatorname{E}((\mathbf{\theta}_1 - \mathbf{\theta}_2)^2)} = \sqrt{\frac{\sum_{i=1}^n (x_{1,i} - x_{2,i})^2}{n}}

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[edit] Nondimensional forms of the root mean squared deviation

Nondimensional forms of the RMSD are useful because often one wants to compare RMSDs with different units, which is cumbersome. There are two approaches: normalize the RMSD to the range of the observed data, or normalize to the mean of the observed data.

[edit] Normalized root mean squared deviation

The normalized root mean squared deviation or error (NRMSD or NRMSE) is the RMSD divided by the range of observed values, or:

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

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

[edit] CV(RMSD)

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

c_{v,\mathrm {RMSD}} = \frac {\mathrm{RMSD}}{\bar x}

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

[edit] Applications

[edit] References

  1. ^ Anderson, M.P.; Woessner, W.W. (1992). Applied Groundwater Modeling: Simulation of Flow and Advective Transport, 2nd Edition, Academic Press, 381. 
  2. ^ Ensemble Neural Network Model

[edit] See also