PREDICTIONS – 10 Years Later - Santa Fe Institute

APPENDIX A
one—except during the saturating phase; that is, while in transition between
the end of the phasing in and the beginning of the phasing out.
The generalized equations for the market shares of n competitors
are:
1
f i (t) = ————— ,
1 + e –(at+b)
i ≠ j
and
n
f j (t) = 1 – ∑ f i (t)
i=1
i ≠ j
Fitting an S-curve on a Set of Data Points
There are many ways to fit a mathematical function onto a sequence of
data points. The method used extensively in this book, and the one
likely to produce the best results, involves iterations through a computer
program which tries to minimize the following sum
(F i – D i ) 2
———— ∑
i W i
where F i is the value of the function, D i is the data value and W i is the
weight we may want to assign, all at time t i . The function is originally
evaluated unintelligently by assigning arbitrary starting values to the
parameters of F, but the program performs a trial-and-error search
through many iterations to determine those values for which the sum
becomes as small as possible.
For S-curve fitting, the function F takes the form of Equation (2),
while for one-to-one substitutions it takes the form (3) or simply the
straight-line expression: at + b.
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