10.10. How the network can serve as a tool for transformation of an
input space dimension?
(Translation by Rafał Opiał, rafal.opial@op.pl)
After that short trip to blue seas and humming palms, lets now come back to next scientific problem.
An unique characteristic of Kohonen Network is that within its responses there exists a topological
representation of an input signal space. Of course it applies for neighborhood and its consequences.
In figures that you saw above, and on those you will see during your own usage of the described
program, there are blue dots representing neurons, connected to each other with red lines. These
lines, as you know, connect neurons considered as adjacent. At the beginning these lines – similarly
as dots itself – are distributed randomly. During teaching you will notice thou that a network during
learning will develop a regular net consisted of red lines. What, in fact, the net represents? Well, it is
an expression of rule consisting in that adjacent neurons will tend to signal and detect adjacent
points from an input signal space! This way the order coming out of the fact that some points
representing input signals are close together will be transposed to a network in which adjacent
points will be signaled only by adjacent neurons.
In all the figures seen so far, two notions: closeness (similarity) of input signals and adjacency
(neighborhood) of neurons in a network could be easily associated because input signal space (and of
course weights space) was two dimensional (as in Fig. 10.13 or 10.16) and during that time also a
network topology was two dimensional (as in Fig. 10.9 or 10.11). So, there existed natural
correspondence between such notions as “an input signal lying higher than the previous signal” (in
this sense it had a higher value of a second component) and “an input neuron lying higher than the
previous neuron” (in this sense that it lies, according to agreed numbering – in the previous row (Fig.
10.42).
Fig. 10.42. Example of same dimension of network topology and data space
Such situation is not the only one possible. One can easily imagine a one dimensional network which
will learn to recognize two dimensional signals (Fig. 10.43).
Fig. 10.43. Example of different dimension of network topology and data space
You may study how behaves a network that makes such a conversion of two dimensional input signal
space to one dimensional structure of a network, because my program allows you to create one
dimensional networks (chains of neurons).
Such untypical structure of a network you may obtain by giving one dimension of a network (the best
the first one) equal 1. The second dimension is then worth to define rather big, for example 100,
because then the network behaves in an interesting way. It learns though pretty fast, so there is no
need to limit yourself (in this case).
Fig. 10.44. Mapping of two-dimensional input space into one dimensional neural topology
Fig. 10.45. More complicated example of mapping of two-dimensional input space into one
dimensional neural topology
Fig. 10.46. Yet another example of mapping of two-dimensional input space into one dimensional
neural topology
Take a look at figures 10.44, 10.45 and 10.46. It looks from them that one dimensional network
pretty reasonably fits in a highlighted area of an input signal space – this guarantees two pretty
important advantages.

Firstly, a chain of neurons being a one dimensional network fills (better or worse) the whole
selected area of an input signal space. It means that for every point of two dimensional input
signal space there is a representative in a one dimensional neural network that will indicate
its occurrence. There are no points or areas in the multidimensional input space that are
„orphaned”.

Secondly, for the objects in an input signal space that lie close to each other (that is they are
in some way similar one to another) – correspond adjacent neurons in the one dimensional
chain of neurons. Unfortunately, it is likely so, but it is not always, so you should expect
errors (fig. 10.47), but still in most cases, the fact that some two states of an input signal
space are being represented by two adjacent neurons implies that they are similar states.
Points
representing
similar input
signals
Distant neurons recognizing
similar input signals
Neighboring neurons
recognizing similar
input signals
Points
representing
similar input
signals
Fig. 10.47. Example of proper and non proper representation of similarity of input signal
in neural network topology
Let's together consider what meaning may it have. Now in a great number of tasks related to
informatics we encounter a similar and a very difficult problem. For getting the knowledge related to
some phenomenon and some processes there are being gathered enormous quantities of data. E. g.
to figure out the current state of a nuclear reactor in a big power plant it is necessary to measure and
evaluate hundreds different parameters. The same refers to a blast furnace in a steelworks, a multiengine aircraft during takeoff and also to a big company which we would like to effectively manage.
An actual state of each of these big systems is described with hundreds and thousands of numerical
parameters, we may imagine it then as a point in a space of great number of dimensions. This
multidimensional space is necessary, because a result of each measurement, an effect of each
observation, a state of each signaling device – should be put on a separate axis. As certain processes
evolve, values of every parameter change and a point corresponding to a state of the considered
system changes its position. Therefore for full estimation of a plane safety, stability of a reactor,
effectiveness of a blast furnace or a company profitability – all the time it is needed to evaluate
position of a point in a multidimensional space. Specifically – the thing is about a position of such
point that represents an actual state relative to areas for which we may assign a particular practical
meaning: stable activity of a reactor – or a symptoms of overheating, good quality of produced pig
iron – or its defective composition, favorable growth of a company – or a specter of bankruptcy.
In order to control and inspect a considered process we must have a current and certain information
about its state, what is a trend of change and where it heads for. When trying to solve this kind of
problems, in contemporary engineering, generally we use solutions where all original data is
gathered and passed to a person which makes decisions. As an effect of that we can often find giant
control rooms with many indicators and blinking lights, as the ones from nuclear power plants, that
you may know from movies, or cabins of supersonic jets where every square centimeter is filled with
some devices, meters and indicators, as well as kilometers long printouts containing many numbers
and charts of economic rates on which businessmen rack their brains.
Such solutions though are fatally ineffective in practice. This mainly comes out of the fact that no
man is capable of effective inspect, control and analyze of thousands of input data. In addition an
operator of nuclear power plant, pilot of an aircraft or a chief executive of a company does not need
such detailed data. He needs a synthetic global information: everything goes right or – there is
happening something wrong. Such an information may be produced by a Kohonen Network.
Imagine that you have built a network where to its every neuron come few hundreds or even a few
thousands of signals related to every collected measurement data. Such network is not necessarily
more difficult to program than a network with two inputs, but it will require more space in
computer's memory and more time during simulation of its working. Imagine that such network
predicts two dimensional proximity of neurons, which you use so that an input signal of every neuron
you will display in some (predetermined) point on a screen, and signals of neighbors you will display
in neighboring rows and columns in order to visualize their relation. After teaching the network with
using a Kohonen method you will obtain a tool making a specific “mapping” of multidimensional,
difficult to evaluate, data, to a plane of one screen, which can be easily looked over, evaluated and
interpreted. Image on a screen will provide data that can be interpreted as following:

Every practically occurring combination of input signals will be represented by exactly one
neuron (“winner”), which will detect and signal occurrence of this exact situation. If you will
depict on a screen only an input signal of this particular neuron, you will obtain an image of a
luminous point moving on a screen. If, based on previous studies, you will remember, which
areas of screen correspond to good states of a supervised process and which to dangerous
states – then by observing where actually the luminous point lies and where is it going – you
can obtain a synthetic evaluation of situation from a perspective of the supervised process.

On a screen you can also display output signals of neurons without making a discrimination
following from usage of a principle of competition. You can make an agreement so that
different values of output signals will be displayed in different colors and then a change in a
supervised process will be displayed as a colorful mosaic. It carries much more information
about a state of a supervised process and after obtaining some practice lets deep and
accurate reasoning about observed trends, however at first sight is pretty illegible.
Ways for depicting results of Kohonen network working can be more. But yet, note on one common
feature of them: they are always two dimensional images, convenient of being looked over with a
glance and relatively easy to interpret. These images do not carry information about detailed values
of particular data (they can be easily reached out when needed), but they show synthetic, overall
image which is so very necessary to person that makes decisions and evaluates results. These images
lower a detail level of the data being analyzed, but thanks to averaging and generalization described
above, this way may be obtained very good results.