Root mean square deviation

The root mean square deviation (RMSD) or root mean square error (RMSE) is a frequently-used measure of the difference between values predicted by a model or an estimator and the values actually observed from the thing being modeled or estimated. These differences are also called residuals.

The RMSD of an estimator $$\hat{\theta}$$ with respect to the estimated parameter $$\theta$$ is defined as the square root of the mean squared error:
 * $$\operatorname{RMSD}(\hat{\theta}) = \sqrt{\operatorname{MSE}(\hat{\theta})} = \sqrt{\operatorname{E}((\hat{\theta}-\theta)^2)}$$

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}}$$

Normalized root mean squared
The normalized root mean squared error is the RMSE divided by the range of observed values, or:
 * $$\mathrm{NRMSE} = \frac{\mathrm{RMSE}}{x_\mathrm{max}-x_\mathrm{min}}$$

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

Applications

 * In Bioinformatics, the RMSD is the measure of the average distance between the atoms of superimposed proteins.
 * In Cheminformatics, the RMSD is a measure of the distance between a crystal structure conformation and a molecular docking result.
 * In Economics, the RMSD is used to determine whether an economic model fits economic indicators.
 * In Experimental Psychology, the RMSD is used to assess how well models of perception explain the abilities of the human senses.
 * In GIS, the RMSE is one measure used to assess the accuracy of spatial analysis and remote sensing.
 * In Hydrogeology, RMSE and NRMSE are used to evaluate the calibration of a groundwater model.
 * In Imaging Science, the RMSD is one measure used to assess how well a method to reconstruct an image performs relative to the original image.
 * In Computational neuroscience, the RMSE is used to assess how well a system learns a given model.
 * Submissions for the Netflix Prize are judged using the RMSE from the test dataset's undisclosed "true" values.