Key shapes

These shapes are commonly found in measures traced over time.  When significantly different states are found 'before' and 'after', they reflect the development of the transition between them.

Integral of an 'S' curve
A.. A simple change of direction.   Periods reflecting different steady states (Black lines) when nature behaves as if following different rules.   A shape connecting them (Red line) shows progression of change during the period that connects before and after.   The mathematical uniqueness of this shape is that an infinite number of derivatives may exist and all have the same sign.   In a perfect continuum, for change to begin or end there must be finite periods during which an infinite number of derivatives exist and have the same sign, a 'law of continuity'.

A step transition, 'S' curve

B. A simple change of state.   This shape is also the rates of change (first derivative) of A and the accumulated effect (the integral) of a simple whole event as in C.

Derivative of an 'S' curve
C.  A simple whole event.   A passing 'bump' on a curve going from nothing to nothing, from before beginning to after the end.

Actual data may contain influences from many different things at once, displaying overlapping effects.  The periods of constant behavior may follow a 'constant' rule that is quite complex.  There might be too little data or the processes of organizational transition might be expressed in different ways in the measure being used at different times.   It is also quite common that the 'states' of behavior being monitored are in fact not highly organized and neither constant nor transitional states can be found.

ed. 3/20/99