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Pearson (1902) seemed convinced of the idea that correlated error could be genuine, and attributed this to the individuals who performed observations. In the context of semi-automated upper air observations and pendulum measurements, Mahalanobis (1947) proposed to take a closer look at the genuine nature of correlated error. In a broad address, Kruskal (1987) seems to suggest that a closer look had not yet been performed. However, because a modern definition of dependence may place little constraint on error correlation (Edelmann et al. 2021), perhaps the nature of correlated error may benefit from another look, even after so many years. If correlated error has both genuine physical and statistical aspects, then we ask to what extent an overlap may exist. A quantitative approach is emphasised, and motivation is given for initially completing a measurement model, or regression model, by accommodating partially overlapping notions of equation error and representation error, where both are taken as signal terms (i.e., as genuine). Implications are that error cross-correlation has the interpretation of nonlinear association, error that is uncorrelated is a lack of association, and all three (linear, nonlinear, and lack of association) are needed to describe the signal of interest. An extended sampling model is shown to offer multiple solutions, with two being comparable to ordinary and reverse linear regression, but offering better bounds on systematic error or bias. A few practical applications are highlighted and a quantitative framework for exploring model solutions under controlled settings is given. Edelmann, D., Móri, T.F., Székely, G.J., 2021. On relationships between the Pearson and the distance correlation coefficients. Stat. Prob. Lett. 169, 1–6, doi:10.1016/j.spl.2020.108960. Kruskal, W., 1988: Miracles and statistics: The casual assumption of independence. J. Amer. Statist. Assoc., 83, 929-940. Mahalanobis, P. C., 1947: Summary of a lecture on the combination of data from tests conducted at different laboratories (reported by J. Tucker Jr.). Amer. Soc. Test. Materials Bulletin, 144, 64-66. Pearson, K., 1902: On the mathematical theory of errors of judgement, with special reference to the personal equation. Phil. Trans. Roy. Soc. London, 198, 235–299.
Topic : Theme 1: Oceans and Hydrology.
Reference : T1-B14
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