Geoscience Reference
In-Depth Information
as being a data array
x
= (
x
1
, …,
x
n
) ∈ ℜ
n
(
n
-dimensional Euclidean space), and the output from
the network as being another data array
y
= (
y
1
, …,
y
m
) ∈ ℜ
m
(
m
-dimensional Euclidean space with
m
<
n
). In this notation, ℜ
n
is the
n
-dimensional input space of real numbers with
x
being an ele-
ment of ℜ
n
and ℜ
m
is the
m
-dimensional output space of real numbers with
y
being an element of
ℜ
m
. The CNN, when viewed in such a manner, can therefore be thought of as being just one simple
function Φ: ℜ
n
→ℜ
m
.
The PEs contain internal
transfer functions
, and it is the implementation of these functions, in
association with the weighted connections, which will in combination generate Φ: the so-called
network function.
CNNs can be differentiated according to the following criteria:
•
Their node characteristics, that is, properties of the PEs
•
Their network topologies, that is, pattern of connections between the PEs (also termed
network architecture)
•
The method that is used to determine their connection weights (called learning rules,
learning algorithms, machine learning or network training)
13.4 CHARACTERISTICS OF THE PROCESSING ELEMENTS
Most notation in the field of neural networks is focused on the PEs, their chosen method of
arrangement into multiple layers and the weighted connections that exist between them. Figure 13.2
shows the internal workings of a generic PE. This is the basic PE that is associated with a CNN
and elements of this nature would be found occupying general (non-dedicated) positions within the
overall structure, that is, this PE (1) accepts inputs from other PEs and (2) sends its output signal
(activation) to other PEs. Those PEs that are not dedicated input or output units will maintain this
general form and function and thereby provide the fundamental non-linear computing capabilities
that exist within each CNN.
To make matters simple, and where there is no confusion, we shall use the notation
u
i
to refer to
both the PE and the numerical activation (output) of that unit. Each element
u
i
computes a single
(numerical) unit output or activation value. The input and output signal from the PEs can be in the
form of discrete numbers, usually taking on values {0, 1} or {-1, 0, 1}, or it can be in the form of
continuous values that will in most cases range between 0 and 1 or −1 and +1.
Figure 13.2 shows that the PE,
u
i
, gets
k
input signals,
u
= {
u
1
, …,
u
k
} which all arrive via
the incoming connections that impinge on element
u
i
. Note that the connected elements are
w
i
0
=
θ
i
u
0
=1
u
1
w
il
v
i
f
i
(.)
i
(.)
u
i
=
i
(
f
i
(
u
0
,...,
u
k
))
w
ik
u
k
Input vector
Integrator
function
Activation
function
Output signal
FIGURE 13.2
Generic PE
u
i
. (From Fischer, M.M.,
Environ. Plann. A
, 30(10), 1873, 1998.)
