Papers › The Many-to-Many Mapping Between the Concordance Correlation Coefficient and the Mean...
The Many-to-Many Mapping Between the Concordance Correlation Coefficient and the Mean Square Error
Vedhas Pandit, Björn Schuller
We derive the mapping between two of the most pervasive utility functions, the mean square error (MSE) and the concordance correlation coefficient (CCC, ρ_c). Despite its drawbacks, MSE is one of the most popular performance metrics (and a loss function); along with lately ρ_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 Lₚ norm of the errors or of its positive powers (e.g., MSE) is aimed at ρ_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 `MSE₁<MSE₂' does not imply `ρ_(c₁)>ρ_(c₂)', but also provides the precise range for the ρ_c metric for a given MSE. We discover the conditions for ρ_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 Lₚ norm, for an even p. We present newly discovered, albeit apparent, mathematical paradoxes. The study inspires and anticipates a growing use of ρ_c-inspired loss functions e.g., |MSE/(σ_(XY))|, replacing the traditional Lₚ-norm loss functions in multivariate regressions.
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