Correlations and How to Interpret Them

Abstract

Correlations between observed data are at the heart of all empirical research that strives for establishing lawful regularities. However, there are numerous ways to assess these correlations, and there are numerous ways to make sense of them. This essay presents a bird's eye perspective on different interpretive schemes to understand correlations. It is designed as a comparative survey of the basic concepts. Many important details to back it up can be found in the relevant technical literature. Correlations can (1) extend over time (diachronic correlations) or they can (2) relate data in an atemporal way (synchronic correlations). Within class (1), the standard interpretive accounts are based on causal models or on predictive models that are not necessarily causal. Examples within class (2) are (mainly unsupervised) data mining approaches, relations between domains (multiscale systems), nonlocal quantum correlations, and eventually correlations between the mental and the physical.Here, µ X and µ Y are expectation values of X and Y, respectively, and σ X and σ Y are their standard deviations; E is the expected value operator; cov stands for covariance; and corr for correlation.But more often than not random variables X and Y are not linearly dependent. In this case, ρ X,Y may be greater than 0, but does only insufficiently characterize the dependence between X and Y. Also, different datasets with the same ρ X,Y can be of entirely different origin if they reflect variables that are nonlinearly related (a classic but still impressive example is the "Anscombe quartet" [4]). Even the case ρ X,Y = 0 does typically not reflect vanishing dependence for nonlinear relations. More sophisticated correlation measures have to be used in nonlinear time series analysis [5] to address nonlinar dependence relations properly.If two random (or stochastic) processes {X t } and {Y t } are statistically dependent, then they are correlated. The relevant quantitative measure (for wide-sense stationary processes with constant mean and well-defined variance) is the cross-correlation function τ → R X,Y (τ),where τ = t 1 − t 2 is the time lag between X t and Y t (for complex-valued processes, Y t+τ has to be replaced by its complex conjugate). For temporal correlations within a single process {X t }, this simplifies to the autocorrelation function R X,X (τ) = E[X t X t+τ ] .The Fourier-transform of R X,X (τ) yields the power spectrum of the process {X t } as a function of frequency f :

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