The Many-to-Many Mapping Between the Concordance Correlation Coefficient and the Mean Square Error

Code

• github.com/vedhasua/mse_ccc_corollary
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Abstract

We derive the mapping between two of the most pervasive utility functions, the mean square error (MSE) and the concordance correlation coefficient (CCC, _c). Despite its drawbacks, MSE is one of the most popular performance metrics (and a loss function); along with lately _c in many of the sequence prediction challenges. Despite the ever-growing simultaneous usage, e.g., inter-rater agreement, assay validation, a mapping between the two metrics is missing, till date. While minimisation of L_p norm of the errors or of its positive powers (e.g., MSE) is aimed at _c maximisation, we reason the often-witnessed ineffectiveness of this popular loss function with graphical illustrations. The discovered formula uncovers not only the counterintuitive revelation that `MSE_1<MSE_2' does not imply `_c_1>_c_2', but also provides the precise range for the _c metric for a given MSE. We discover the conditions for _c optimisation for a given MSE; and as a logical next step, for a given set of errors. We generalise and discover the conditions for any given L_p norm, for an even p. We present newly discovered, albeit apparent, mathematical paradoxes. The study inspires and anticipates a growing use of _c-inspired loss functions e.g., |MSE_XY|, replacing the traditional L_p-norm loss functions in multivariate regressions.

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