An important step for any modeling study is to compare the modeled estimates with observed/reliable data. The original chord index (also known as the original Willmott index) was often used to measure how the estimates generated by the model simulate the observed data. However, in its original version, such an index can cause the user to mistakenly select a predictive model. Therefore, this study compared the sensitivity of the original match index with its two more recent versions (modified and refined) and provided an easy-to-use R code capable of calculating these three indices. First, the sensitivity of the indices was assessed by Monte Carlo experiments. These controlled simulations took into account different types of errors (systematic, random and systematic + random) and error sizes. Using the R code, we also conducted a case study in which the clues are supposed to indicate that the empirical Thornthwaite model provides poor estimates of baseline daily evapotranspiration compared to the standard Penman-Monteith method (FAO56). Our results suggest that the original match index may indeed mistakenly select a poorly functioning predictive model. Our results also suggest that new versions of this index overcome this problem and lead to stricter ratings.
Although the refined Willmott index represents the widest possible range of values, it does not inform the user when a predictive model overestimates or underestimates the simulated data, which does not result in any additional information on the data already provided by the modified version. None of the indexes represents the error as a linear function of its size relative to the observed process. Unlike previous studies, Ji & Gallo9 explicitly designed an index that would meet the symmetry criteria. This index, which is proposed for the comparison of remote sensing images, is defined as follows: where P and O are respectively, the predicted and observed values and the 95% confidence interval of each index value were estimated by the Bootstrap approach, as recommended by Willmott et al. (1985) Willmott, C. J., Ackleson, S. G., Davis, R. E., Feddema, J. J., Klink, K.M., Legates, D. R., O`donnell, J. and Rowe, C.M. (1985).
Statistics to evaluate the performance of the model. Zeitschrift für geophysikalische Forschung, 90, 8995-9005. dx.doi.org/10.1029/JC090iC05p08995.dx.doi.org/10.1029/JC090iC05p089. . In this case, the denominator μ is defined by adding the differences of all points of X and Y with respect to , the mean of X. The original version was based on squared deviations, but was later modified15 using absolute deviations, arguing that MAD (or in this case MAE, since they refer to errors between predictions and observations instead of deviation) is a more natural measure of mean error and is less ambiguous than RMSD (or RMSE)12. A refinement of index16 was intended to remove predictions from the denominator, but as others have argued14, this amounts to resizing the expression of the efficiency coefficient while losing the interesting reference point of. Again, these indexes do not meet the symmetry requirement.
The seasonal variability of the three indices calculated by the R code is consistent with the theoretical context mentioned above. In addition, the results presented in Figure 5 are also consistent with those of the Monte Carlo simulations, which suggested that dref and dmod are stricter than dorig and should therefore be preferred to their original version. This statement is particularly true for the summer season, when the results in Figure 5, taking into account the 95% confidence interval, showed that Dorig can reach values of up to 0.80 when mean absolute errors greater than 0.80 mm∙day-1 are present. Since dorig=1 indicates a perfect model, the original version of the Willmott index may cause the user to mistakenly accept that the TW model is (at least) suitable for estimating the daily amounts of ETo in Campinas during the summer season. On the other hand, the upper limits (95% confidence interval) of all dmod values remained below 0.60 in all seasons. Of course, the dmod values are consistent with the theoretical context mentioned above, suggesting that the TW model cannot be applied to estimate the daily amounts of ETo in Campinas-SP regardless of the season. Overall, it has been defined as an index based on it, the meaning of which would be independent of n. For this reason, A`c is still used by OxCal as a threshold for Aoverall if errors are not correlated.
If the errors are correlated (as with shaky combinations and matches), An is used instead. An indexed contract tends to contain fewer options than a fixed-price contract: the difference between the contract price and the market price is generally smaller and less volatile. This reduces arbitrage trading opportunities in the market and therefore the value of the option. The scope of the index remains a pragmatic extension of r and is therefore used in a context where a linear functional agreement is desired. It is not intended as a tool to investigate new functional associations in the data (e.B. the maximum information coefficient32). However, its use could go beyond symmetrical comparison of dataset matching and complement the collection of existing methods2 to characterize the performance of the model against a reference. The index has also been demonstrated here with case studies of spatiotemporal raster data, but it should also be usable for any pair of vectors of any data type, just like r. Given the level of error used in the Monte Carlo simulations, as well as in the case of the study, our results suggest that the original version of the willmott index may cause the user to mistakenly select a predictive model that generates poor estimates. This statement is consistent with previous studies.
Our results also suggest that the two new versions of this index (modified and refined) overcome such a problem, resulting in stricter evaluations of predictive models. Therefore, they should be preferred to the original version. The 100% deviations have the same meaning as for individual agreements. .
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