We define an activity dependent branching ratio that allows comparison of
different time series Xt​. The branching ratio bx​ is defined as bx​=E[ξx​/x]. The random variable ξx​ is the value of the next signal given
that the previous one is equal to x, so ξx​={Xt+1​∣Xt​=x}. If
bx​>1, the process is on average supercritical when the signal is equal to
x, while if bx​<1, it is subcritical. For stock prices we find bx​=1
within statistical uncertainty, for all x, consistent with an ``efficient
market hypothesis''. For stock volumes, solar X-ray flux intensities, and the
Bak-Tang-Wiesenfeld (BTW) sandpile model, bx​ is supercritical for small
values of activity and subcritical for the largest ones, indicating a tendency
to return to a typical value. For stock volumes this tendency has an
approximate power law behavior. For solar X-ray flux and the BTW model, there
is a broad regime of activity where bx​≃1, which we interpret as an
indicator of critical behavior. This is true despite different underlying
probability distributions for Xt​, and for ξx​. For the BTW model the
distribution of ξx​ is Gaussian, for x sufficiently larger than one, and
its variance grows linearly with x. Hence, the activity in the BTW model
obeys a central limit theorem when sampling over past histories. The broad
region of activity where bx​ is close to one disappears once bulk dissipation
is introduced in the BTW model -- supporting our hypothesis that it is an
indicator of criticality.Comment: 7 pages, 11 figure