Ref: DR97900
1. The simplest is a straight line (S.L.) fitted to
historical data. The S.L. is a particular case of a polynomial. Other
polynomials can be fitted to historical data. A S.L. assumes a constant
quantity increase or decrease per period. This is usually only realistic for a
few periods. When this model is used for time value calculations, the value of
a series of increasing or decreasing payments can be estimated using the
expressions for a gradient series.
2. An assumption of a constant fractional increase
of the current population from period to period gives exponential growth.
This can be estimated using the compound interest model for future value.
Exponential growth is only realistic as long as there appears to be no
limits to growth. Many systems appear to grow in this fashion for the
initial periods until some capacity constraint begins to take place.
3. A model that takes into account capacity constraints
is the GOMPERTZ:
Y
= kabx
Y
<- k x a * b * x (APL
notation)
Depending on the
values of k, a, b or their logs the curve may take on a variety of shapes.
When log a < 0,
and 0 < b
When log a < 0,
and b > 1 the curve starts at the upper limit k & decreases with a lower
asymptote.
When log a > 0,
and 0 < b < 1 the curve give a negative exponential decay from the
maximum to a lower limit k.
Log a > 0, and
b > 1 produces exponential growth beginning at k.
The x = 0 position
of all these curves is about the 1/3 or 2/3 points.
End to date 060214, ams