Root mean square deviation


Root mean square deviation

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. RMSD is a good measure of accuracy. 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{ heta} with respect to the estimated parameter heta is defined as the square root of the mean squared error: :operatorname{RMSD}(hat{ heta}) = sqrt{operatorname{MSE}(hat{ heta})} = sqrt{operatorname{E}((hat{ heta}- heta)^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{ heta}_1 = egin{bmatrix} x_{1,1} \ x_{1,2} \ vdots \ x_{1,n}end{bmatrix}qquad mathrm{and} qquadmathbf{ heta}_2 = egin{bmatrix} x_{2,1} \ x_{2,2} \ vdots \ x_{2,n}end{bmatrix}.The formula becomes::operatorname{RMSD}(mathbf{ heta}_1, mathbf{ heta}_2) = sqrt{operatorname{MSE}(mathbf{ heta}_1, mathbf{ heta}_2)} = sqrt{operatorname{E}((mathbf{ heta}_1 - mathbf{ heta}_2)^2)} = sqrt{frac{sum_{i=1}^n (x_{1,i} - x_{2,i})^2}{n.

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.

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

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{ar x}.

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 Cheminformatics, the RMSD is a measure of the distance between a crystal structure conformation and a 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 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. [cite book |title=Applied Groundwater Modeling: Simulation of Flow and Advective Transport |publisher=Academic Press |year=1992 |last=Anderson |first=M.P. |coauthors=Woessner, W.W. |pages=381 |edition=2nd Edition ]
*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 [ [http://www.ocgy.ubc.ca/projects/clim.pred/NN/3.1/model.html Ensemble Neural Network Model ] ] .
*Submissions for the Netflix Prize are 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.

ee also

*Mean squared error
*Quadratic mean
*Squared deviations

References


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