Root-mean-square deviation

Root-mean-square deviation (RMSD) or 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 observed. The RMSD represents the square root of the second sample moment of the differences between predicted values and observed values or the quadratic mean of these differences. These deviations are called residuals when the calculations are performed over the data sample that was used for estimation and are called errors (or prediction errors) when calculated 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 measure of accuracy, to compare forecasting errors of different models for a particular dataset and not between datasets, as it is scale-dependent.

Definition

The RMSD of an estimator \\( \hat{\theta} \\) with respect to an unknown parameter \\( \theta \\) is defined as the square root of the mean square error: \\[ \text{RMSD}(\hat{\theta}) = \sqrt{\text{MSE}(\hat{\theta})} = \sqrt{\mathbb{E}[(\hat{\theta} - \theta)^2]} \\] For a predicted value \\( \hat{y}_i \\) and an observed value \\( y_i \\), the MSE is calculated as: \\[ \text{MSE} = \frac{1}{n} \sum_{i=1}^{n} (\hat{y}_i - y_i)^2 \\] where \\( n \\) is the number of observations.

Applications

RMSD is commonly used in climatology, forecasting, and regression analysis to verify experimental results. It is especially relevant in the fields of Machine Learning, Data Science, and Statistics, where it helps in validating models and algorithms. In Chemistry and Physics, RMSD is used to compare the difference between the positions of atoms in molecular structures or simulation results.

Advantages and Disadvantages

The main advantage of RMSD is its interpretability in terms of measurement units. It gives an estimation of the standard deviation of the unexplained variance, and it can be used to compare the forecasting errors of different models. However, RMSD has its disadvantages; it is sensitive to outliers, and its value can increase dramatically with the presence of a few large errors among the data points. This sensitivity makes it less robust as a measure of average error, especially in datasets with significant outliers.

Comparison with Other Measures

RMSD is often compared with other measures of fit, such as the mean absolute error (MAE) or the mean absolute percentage error (MAPE). While RMSD gives a relatively high weight to large errors (due to squaring), MAE provides a linear scale of errors, which some researchers prefer for its interpretability. The choice between RMSD, MAE, and MAPE depends on the context of the analysis and the specific requirements of the application.

See Also

• Mean squared error
• Mean absolute error
• Mean absolute percentage error
• Standard deviation
• Variance