Information Technology Reference
In-Depth Information
1
)
j
=
1
U
j
−
U
U
j
+
k
−
U
N
−
k
/(
−
N
k
r
k
≡
;
N
k
,
(4.3)
j
=
1
U
j
−
U
2
N
(
/
)
1
N
where the sums of the sequences in (
4.3
) are calculated using the average value
N
1
N
U
=
U
j
(4.4)
j
=
1
for a sequence of
N
computer-generated coin tosses. The autocorrelation coefficient
determines whether the value of the dynamical process at one instant of time influ-
ences its value at subsequent times. The autocorrelation coefficient is normalized to one
such that when
k
=
0, the absence of time lag, there is complete self-determination of
the data. The magnitude of this coefficient as a function of
k
determines how much the
present value is determined by history and how much the present will determine the
future. The autocorrelation function is graphed versus the lag time in Figure
4.2
for
the Bernoulli sequence. The correlation is seen to plummet to zero in one step and to
fluctuate around this no-influence value of zero for the remainder of the time.
There are many phenomena in the physical, social and biological sciences for which
we use random-walk models to describe their evolution over time. For example, just
as Bachelier had foreseen [
4
], the stock market with its responses to inflation and the
level of employment, its discount of taxes, its reaction to wars in various parts of the
world, all of which are uncontrollable, does in fact buffet the price of stock in a manner
not unlike Igen Hauz's charcoal or Brown's pollen motes. This simple reasoning has
also been applied to a bacterium's search for food. The bacterium moves in a straight
line for a while and then tumbles to reorient itself for another straight-line push in a
random direction. In each of these applications of the random walk the rationale is
that the best model is the simplest one that can explain all the available experimental
data with the fewest assumptions. Alternative models are those that make predictions
1
0.5
0
-0.5
-1
2.5
5
7.5 10
Lag time
12.5
15
17.5
20
The discrete autocorrelation function is calculated for a Bernoulli sequence. The coefficient has
an initial value of one and drops to zero within one unit of delay time, the time it takes to carry
out one step. The correlation subsequently fluctuates around zero for arbitrarily long delay times.
Figure 4.2.
