{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/on-many-to-many-mapping-between-concordance","title":"The Many-to-Many Mapping Between the Concordance Correlation Coefficient and the Mean Square Error","arxiv_id":"1902.05180","date":"2019-02-14","proceeding":null,"authors":["Vedhas Pandit","Björn Schuller"],"abstract":"We derive the mapping between two of the most pervasive utility functions, the mean square error ($MSE$) and the concordance correlation coefficient (CCC, $\\rho_c$). Despite its drawbacks, $MSE$ is one of the most popular performance metrics (and a loss function); along with lately $\\rho_c$ 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 $\\rho_c$ 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 `$\\rho_{c_1}>\\rho_{c_2}$', but also provides the precise range for the $\\rho_c$ metric for a given $MSE$. We discover the conditions for $\\rho_c$ 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 $\\rho_c$-inspired loss functions e.g., $\\left|\\frac{MSE}{\\sigma_{XY}}\\right|$, replacing the traditional $L_p$-norm loss functions in multivariate regressions.","url_abs":"https://arxiv.org/abs/1902.05180v6","url_pdf":"https://arxiv.org/pdf/1902.05180v6.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"on-many-to-many-mapping-between-concordance","repo_url":"https://github.com/vedhasua/mse_ccc_corollary","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":{"status":"ok"}}],"tasks":[{"task_slug":"sentiment-analysis","task_name":"Sentiment Analysis"},{"task_slug":"time-series","task_name":"Time Series Analysis"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/1902.05180","atlas_url":"https://app.syntology.ai/?focus=1902.05180","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}