Chaos by Smith Leonard;

Author:Smith, Leonard;
Language: eng
Format: epub
Publisher: Oxford University Press, UK
Published: 2007-09-14T16:00:00+00:00

The decay of certainty: information without correlation

When it comes to predicting what a system will do next, data on the recent state of the system often provide more information than data on some long past state of the system. In the 1920s, Yule wanted to quantify the extent to which data on this year’s Sun spots provide more information about the number of spots that will appear next year than ten-year-old data do. Such a statistic would also allow him to quantitatively compare properties of the original data with those of time series generated by models. He invented what is now called the auto-correlation function (or ACF), which measures the linear correlation between states k iterations apart. When k is zero the ACF is one, since any number is perfectly correlated with itself. If the time series reflects a periodic cycle, the ACF decreases from one as k increases, and then returns to equal one whenever k is an exact multiple of the period. Given data from a linear stochastic system the ACF is of great value, but as we will soon see, it is of less use when faced with observations from a nonlinear system. Nevertheless, some statisticians went so far as to define determinism as linear correlation; many are still reeling from this misstep. It is well known that correlation does not imply causation; the study of chaos has made it clear that causation does not imply (linear) correlation either. The correlation between consecutive states of the Full Logistic Map is zero despite the fact that the next state is completely determined by the current state. In fact, its ACF is zero for every separation in time. How then are we to detect relationships in nonlinear systems, much less quantify predictability, if a mainstay of a century of statistical analysis is blind to such visible relationships? To answer this question, we first introduce base two.

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