Invariants and Homomorphisms Implicit in, and the Invalidity of the Mean-Variance Framework and Other Causality Approaches: Some Structural Effects

Many aspects of modern statistical analysis, Data Science and optimization are based almost entirely on the Mean–Variance (M-V) Framework and its elements—Variance, Semivariance, Correlation and Covariance. This chapter explains why these measures are very inaccurate and don’t reflect reality and also introduces Invariants for analysis of rates-of-change and Pattern Formation. That is, some of the illustrated limitations of the M-V Framework are Invariants that present new opportunities in computing and computational methods in various fields including Optimization, Pattern Formation, Chaos and Evolutionary Computation, given the discussions in Sandfeld and Zaiser ( Modelling and Simulation in Materials Science and Engineering, 23 (6), 065005, 2015), Kriener et al. ( Frontiers of Computational Neuroscience, 7 , 187–191, 2014), Fenn et al. ( Physics Review E, 84 , 61–65, 2011), Preis et al. ( Scientific Reports, 2 , Article number: 752, 2012), Kenett et al. ( International Journal of Bifurcation & Chaos, 22 , 1250181, 2012), Pearson ( Philosophical Transactions of the Royal Society of London Series A, 186 , 343–414, 1895), Fuwape and Ogunjo ( CBN Journal of Applied Statistics, 4 (2), 129–134, 2013), Menna et al. ( International Journal of Modern Physics C, 13 (1), 31–39, 2002), Egozcue ( Cogent Mathematics, 2 (1), 991082, 2015), and Andrade et al. ( Physica D: Nonlinear Phenomena, 223 (2), 139–145, 2006), all of which omitted the limitations. One of the biggest problems inherent in the M-V Framework is that its main components (Variance, Covariance, Correlation and Semivariance) measure the results but not the causes of variation and covariation.