Graphs show different error functions (Relative Squared Error (RSE), lnRSE and RSE + lnRSE ) plotted against λ and k over large and small ranges of the values.
For all models, AshCalc displays a plot of thickness against Area, the total estimated tephra deposit, the estimates for the model parameters, the mean relative squared error in the fit of the model to the data and the equation for T x).
For example, the Fitch-Margoliash tree [see Figure S1 in Additional file 1] gives 0.199 relative squared errror, that is, 0.199 = ∑ i, j (T i, j) - D i, j))/ D i, j), where T i, j) is the pairwise distance given by the tree, and gives 1.51%averagee percent standard deviation (APSD) [ 51, 52], a measure of the typical error of T i, j) relative to D i, j).
Another way of expressing the prediction error is through the Relative Squared Error (RS E– Equation 6), Root Relative Squared Error (RRS E– Equation 7), and Relative Absolute Error (RAE – Equation 8).
Otherwise, a good way of expressing the prediction performance is through relative errors such as the Mean Relative Error (MRE – Equation 4) and Mean Relative Squared Error (MRS E– Equation 5).
Several metrics to compare our models with baselines are provided in Tables 1 and 2. In particular, we report the Mean Absolute Error (MAE), the Mean Squared Error (MSE), the Root Mean Squared Error (RMSE), the Relative Squared Error (RSE), the Relative Absolute Error RAEE), and (R^{2}).
The model efficiency was estimated by RMSE (Root mean square error) and by RSE (Relative square error).
When combined with the anhydrous ammonia leak test, the overall mean relative bias is 0.135 and mean relative square error is 0.068, which is within the valid range for dispersion simulation.
Based on the experience that it is not easy to choose an un-biased stopping rule, we selected the image that had the smallest relative square error (RSE) within 1000 iterations as the deblurred image of the tested method.
In DIYABC, the latter include the relative square root of the mean square error, the relative square root of the mean integrated square error, the relative mean absolute deviation, the 95%and50%0% coverages and the factor 2. See Annex 2 in Supplementary Material for more details.
For evaluating the quality of point estimation, we consider the square root of the frequentist relative mean squared error.
