Information Technology Reference
In-Depth Information
The probability density for the decision-making process is then given by the inverse
Fourier transform of the characteristic function
∞
1
2
)
=
FT
−
1
e
−
ikq
p
(
q
,
t
[
φ(
k
,
t
)
;
q
]≡
φ(
k
,
t
)
dk
,
(3.154)
π
−∞
so that inserting (
3.153
)into(
3.154
) and carrying out the indicated integration yields
Dt
exp
2
1
−
(
q
−
Ct
)
p
(
q
,
t
)
=
√
4
.
(3.155)
2
Dt
π
It is clear from this form of the probability density that the average value of the
dynamical variable is given by
Q
(
t
)
=
Ct
,
(3.156)
so the mode of the distribution moves linearly in the direction of the drift. The variance
of the dynamic variable is given by
[
Q
]
2
2
σ
(
)
≡
(
)
−
Q
(
)
=
,
t
t
t
2
Dt
(3.157)
just as in classical diffusion.
Bogacz
et al
.[
11
] model the interrogation paradigm by asking whether, at the inter-
rogation time
T
, the value of the dynamic variable lies above or below zero. If the
hypothesis H
+
is appropriate, a correct decision occurs if
Q
(
T
)>
0 and an incorrect
(
)<
one if
Q
0. The average error rate (ER) can therefore be determined from the
probability that the solution
Q
T
lies below zero, so that, using the probability density
(
3.155
), the average error rate is given by the expression
(
T
)
−
Ct
/
√
Dt
1
√
2
e
−
u
2
/
2
du
ER
=
(3.158)
π
−∞
and the integral on the rhs is the error function
x
1
√
2
e
−
u
2
/
2
du
erf
(
x
)
≡
.
(3.159)
π
−∞
In the DDM the decision is made when
Q
(
t
)
reaches one of two fixed thresholds,
usually selected symmetrically at the levels
z
. Figure
3.5
depicts some sample
paths of the solution to the Langevin equation with a constant drift; these are separate
realizations of the trajectory
Q
+
z
and
−
. From Figure
3.5
it is evident that the short-time aver-
age of the trajectory does not necessarily move in the direction determined by the sign
of the constant drift
C
. The fluctuations produce erratic motion, sometimes resulting
in the trajectory crossing the wrong threshold, indicating how uncertainty can lead to
the wrong decision. The mathematics required to formally determine the probability of
crossing one of the two thresholds indicated by the dark horizontal lines for the first
time comes under the heading of first-passage-time problems. Consequently, in order to
determine the first-passage-time probability density we must develop the calculus for
probabilities, which we do later. For the time being we examine how to model decision
making with increasing levels of complexity.
(
t
)
