Yule’s Nonsense Correlations

Differential equations describing a fluid continuum are, in a sense, non-physical in that they seek to describe the behaviour of physical quantities over an infinite range of scales. A granular description of the natural world is more realistic and more closely related to observation. A description based on time series and discrete time spectral analysis  is both granular and confined to a finite range of scales. The variance density spectral estimate is defined as the discrete Fourier Transform of the variance-covariance function of the sample. It follows that the periodogram of an unselfcorrelated, N-long, sample time series is a consistent but biased estimator of the population spectrum. Models fitted to time series can be examined by testing residuals for self-correlation using well established methods such a Ljung-Box. Autoregressive Moving Average (ARMA) models are particularly suited to this process and allow peaks due to sinusoidal components to be tested via the F-distribution.

Time series having power law spectra with negative indices, i.e. with “red” spectra, are common in nature due to integration effects. These often appear to be deterministic functions of elapsed time or to be correlated with the time series of similar red spectra quantities. This is known as spurious regression or spurious correlation and was first noted by Yule (1926) as “Nonsense Correlations”. Such spurious effects may be avoided by fitting an ARMA model and testing its drift coefficient for significance.

Reid, J. (2017) “There is no significant trend in global average temperature”. Energy & Environment, 28, 3, pp 302-317 doi: 10.1177/0958305X16686447

Reid, J. (2019) The Fluid Catastrophe Cambridge Scholars Publishing, Newcastle upon Tyne, U.K.