The logic behind Mielke7`s original design of the index, based on all sorts of permutations between the elements in the two data sets, intuitively suggests how its denominator is actually the maximum possible value that the average sum of squares can reach. Because of the mathematical properties of these square deviations, it is possible to rewrite this index in an expression based on variances rather than permutations, which greatly simplifies the calculation. Unfortunately, we have not been able to generalize the (easily predictable) index structure to use with other deviation metrics, such as average absolute deviations. However, the demonstration of how the inssystem can be disentangled from the systematic contribution to the agreement through self-destruction could be applied to any other type of metric. Boundaries between a lower limit (z.B. 0) that does not correspond to a chord and a ceiling (z.B. 1) corresponding to a perfect agreement. One of the consequences is that higher values should always show greater agreement. To summarize the result of this analysis, it is possible to see that all metrics have at least one gap: in one way or another, the small values of the index represent a greater counterintuitive concordance. For all of them, it is also unclear how they can be related to the correlation coefficient. In addition, Ji & Gallos AC has a highly undesirable behavior in the presence (but also in the absence) of bias. While the Mielke index is computational intensive, the index, with its simplified expression, seems to be a suitable candidate for comparing data sets if the correlation is zero or positive. However, the mathematical formulation proposed by the author does not explicitly indicate how the correlation coefficient is related.
We think this last point is worth considering, as the index user usually has a clear understanding of what means a correlation value, but is not familiar with the values taken by the chord index itself. Willmott, C. J., Robeson, S.M. and Matsuura, K. (2011). A refined index of the model`s performance. Int. J. Climatol.
DOI: 10.1002/joc.2419, the last term being proportional to the covariance between X and Y. One way to do this is to create an index that explicitly contains this covariance term in the denominator and limit it so that it is always positive: you will find a description of all the symbols in the equations above in Table 3. Willmott et al. (2011) proposed a new index, dr, and compared the dr to the mean absolute error (EED) of the records, which logically varies with MAE. This must however be compared to the absolute mean relative error, as MAE can vary with different models/data sets, while the value of the mean absolute relative error can be the same (i.e. no change in the relative model). In this study, the dr index does not follow the logical trend within a given data set, as in Table 2 (combined analysis); and also ambiguous between different sentences (first year and combined data) – with the PMARE value. Similar inconsistencies are also observed for random records (Table 4, 1 and 3rd data series – with PMARE). Dimensionless. It is therefore independent of the unit of measurement.
It makes it easy to compare the correspondence between different pairs of data sets (if each pair z.B. different units) or in a different parameter space (for example. .