The explanation for this phenomenon is shown in fig.5: all three time-series are generated by the same set of equations, the well-known Lorenz equations [21].
Fig.5:
(x-z) projection of the Lorenz Attractor
Thus we can see that a geometrical analysis might prove much more powerful in cases like this than a quantitative statistical analysis. Thus we have to make sure that we capture all the relevant variables in a global model, even when there is no statistical correlation among some of them. This, of course raises the question of nonlinear indicators for relevancy or redundancy a question that is the topic of a very active area of current research. The conclusion from this example should not be that ``everything is connected with everything'' in a näive way. From the theory of synergetics [12,13] we know that there are slaved variables and order parameters. For low-dimensional models we need to find out what these order parameters are and we need nonlinear methods to do that.