Taxonomy of bi-logistic curves

Wavelets often overlap in time, though this is not a necessary condition. Depending on the order and magnitude of the overlap, the aggregate curve can take on a wide range of appearances. Figure 3 shows a taxonomy of bi-logistic processes, with the Fisher-Pry transform of the two component logistics on the right.

Figure 3: A Bi-logistic taxonomy. Fisher-Pry decompositions are shown in the right column.

Panel A is an example of a ``sequential'' bi-logistic; the second pulse does not start growing until the first pulse has nearly reached its saturation level $\kappa_1$. This shape bi-logistic characterizes a system which pauses between growth phases.

Panel B is an example of a ``superposed'' bi-logistic, where the second pulse begins growing when the first pulse has reached about 50% of saturation. This bi-logistic growth model characterizes systems that contain two processes of a similar nature growing concurrently except for a displacement in the midpoints of the curves.

Panel C shows a ``converging'' bi-logistic, where a first wavelet is joined by a second faster, steeper wavelet; the two pulses culminate at about the same time. Often a late adopter of a technology, having learned from the experiences of an early adopter, will advance faster, resulting in a smaller $\Delta t$.

Panel D shows a ``diverging'' bi-logistic, where two logistic growth processes begin at the same time but grow with different rates and carrying capacities defined from the start.

Panels C and D show the merits of Loglet analysis; their curves are S-shaped but asymmetric, so they do not appear to be logistic, yet indeed they are made up of logistic components.