In the previous chapter, it was said that the use of artificial neural networks can be attributed to intelligent methods. As mentioned above, artificial intelligence is widely used in all areas, including in the financial sector. Consider what artificial neural networks are.

The expression «neural networks» was formed in the 40s of the 20th century in the field of researchers who studied the principles of organization and functioning of biological neural networks [9].

It should be noted that artificial neural networks are an attempt to simulate the information processing capabilities of the nervous systems. In order to understand this concept, one should consider the basic properties of biological neural networks [10–11].

Note that artificial neural networks differ in the type of neurons from which they are made, as well as in the way they are interconnected.

There are two main classes of neural networks: the first class is artificial neural network with direct connection and the second is recurrent artificial neural network. Direct artificial neural networks are organized into cascading layers of neurons. Here, each layer contains neurons, which receives input from neurons in the previous layer and transmitting outputs to neurons in the next layer, where the first network layer is called the input and the last is called the output [12].

Nowadays, under the McCulloch-Pitts neuron, a multi-input non-linear converter with weighted input signals is received, which is shown in figure 2.1.

Figure 2.1 – Neuron McCullough-Pitts

The inputs of the jth neuron receive n signals x₁, x₂, ..., x_n, which are weighted by amplifiers that realize synaptic weights, after which the weighted values w_j1x₁, w_j2x₂,..., w_jnx_n  together with the threshold value θ_j , also called the bias signal are fed to adder ∑, as a result of which an internal signal u_j is generated.

The soma of a biological neuron is modeled using some nonlinear function ψ (u_j), which is called in the artificial neural network theory either the activation or transfer function of the formal neuron [13].

The McCulloch-Pitts model can be written as

(1.1)

or

(1.2)

where

As for the functional characteristics of individual neurons and networks in general, they are determined by the type of activation functions used. Currently, many other transformations are used, some of which are shown in figure 2.2.

a) relay function; b) threshold function of the relay; c) linear threshold function; d) modular function; e) sigmoid function

Figure 2.2 – Activation functions of neurons

с) d)

e)

Figure 2.2 (continuation)

We all know that the main property of a biological brain is its ability to learn. This means that the concept of «learning» is the key in the theory of artificial neural networks [14–17]. The learning process itself is considered as adaptation of parameters to solve the problem by optimizing the accepted quality criterion. To date, the most popular and obvious by far is the «with a teacher» learning paradigm, schematically presented in figure 2.3 [13].

In this scheme, the «teacher» knows information about the external environment, given in the form of a sequence or packet of input vectors x, as well as the «correct response» to these signals, presented in the form of a training signal d. Of course, the reaction of the untrained network y differs from the «correct» reaction of the teacher, resulting in the error e = d − y.

In the learning process, it is necessary to adjust the artificial neural network parameters so that some scalar function of the error E (e) reaches its minimum value. A network is considered trained, which in some, as a rule, statistical sense repeats the teacher’s reaction.

Figure 2.3 – Teacher training schedule

Since information about the external environment is usually unsteady in nature, the learning process goes on continuously, for which one or another recurrent procedure is used. There is also «without a teacher» training, or it is also called self-training, when the correct reaction to environmental signals is unknown [13].

Figure 2.4 shows the most applicable types of artificial neural networks related to statistical and dynamic classes [18].

Neural networks mimic some aspects of the human brain and demonstrate its capabilities such as the ability to non-formal learning, generalization and clustering of unclassified information, the ability to independently make predictions based on already presented time series.

Neural networks have the following advantages [3, 19, 20]:

- universality; simplicity;

- the absence of the «curse of dimension» problem.

At the same time, neural networks have a number of serious drawbacks:

- the complexity of building a network architecture for a specific task;

- difficulty in interpreting learning outcomes.

It should be noted that the use of neural network technology is advisable in cases where the formalization of the decision process is difficult or even impossible. artificial neural networks are a very powerful modeling tool, since they are non-linear in nature.

Figure 2.4 – Classification of Neural Networks

Today we have many linear modeling methods. It is worth noting that in the problems of risk analysis, linear modeling methods in most cases are not applicable.