This article is devoted to the use of Bayesian networks to analyze the possibility of economic growth in Ukraine. It was found that at the maximum level of external investment, direct investment in Ukraine will increase and this creates the conditions for increasing the country's economic growth. It has also been shown that Noisy-max nodes, compared to General nodes, provide a relatively high initial accuracy. General nodes require retesting. However, Noisy-max nodes entail an increase in time and computational cost.
Economic growth can be considered a major factor in the well-being and prosperity of the country. Industrialization, technology development, and innovation activity are widening the gap between developed countries and developing countries. The innovative development of enterprises is one of the basic needs of the national economy.
The activities of the enterprise reveal innovations by transforming and reforming production through the use of inventions or various opportunities for the release of new goods, the opening of new sources of raw materials, markets, modernization of production, etc., ie the implementation of new combinations of factors of production. Innovative activity is a factor that gives a dynamic character to the economy and has a two-sided influence: on the one hand, it opens new opportunities for economic expansion, on the other hand, it requires a change of traditional directions for further development [1].
Today, there are 42,564 industrial enterprises in Ukraine, representing 12.4% of their total. In the countries of the European Union (EU), the share of enterprises engaged in innovation activity is about 53%. The largest number of innovative enterprises among EU countries is in Germany (79.3% of the total number of enterprises), the smallest in Bulgaria (27.1% of the total number of enterprises) [2]. Theories and models of economic growth highlight the ways in which current economic activity can influence future economic events. Therefore, it will be advisable to determine the informative economic indicators that have the greatest impact on the dynamics of the economic growth of Ukraine.
This will create the necessary prerequisites for the growth of production and expanded reproduction of GDP in order to increase the welfare of the country's population. This paper presents the results of studies on the development of probability-determined models based on Bayesian networks to assess the degree of economic development of Ukraine. Analysis of the country's economic growth is associated with the level of external investment resource, the level of internal investment potential, financial development and the level of manufacturability (innovation) of industrial enterprises.
The aim of the work is to develop static Bayesian network models based on noisy-MAX nodes to analyze the country's economic growth trends. 2
A mathematical model that, when analyzing the financial, investment, and economic indicators of an enterprise would help assess the level of economic growth of a country, requires the availability of input data. Bayesian network methods, with a certain degree of probability, make it possible to achieve the goal.
Having input indicators, such as level of manufacturability (innovativeness), level of financial security (financial development), UAH, an indicator of external investment, UAH, internal investment potential, UAH, which interact with each other as shown in Figure 1, it is necessary to design a static Bayesian network for assessing the country's economic growth opportunities.
Considering that one of the problems in the development of Bayesian networks is the exponential increase in the number of parameters in conditional probability tables (CPT), this study proposes a technique for using noisy-MAX nodes to model economic processes.
The Noisy-MAX node, which in the case of noisy variable reduces to the noisyOR, consists of a child node,Y, taking on nY possible values that can be labeled from 0 tо nY 1 , and N parents, Pa Y X1,, X N , which usually represent the causes of Y. Each Xi has a certain zero value, so that Xi = 0 represents the absence of Xi. Two basic axioms define the Noisy-MAX [3].
P Y 0 X i 0i 1,
The degree reached by Y, is the maximum of the degrees produced by the X, if they were acting independently:
PY y x PY y X i xi , X j 0 j , ji, (2) where x represents a certain configuration of the parents of Y, x = (x1,…, xN). The parameters for link Xi →Y – are the probabilities that the effect assumes a certain value y, when Xi takes on the value xi, and all the other causes of Y are absent: cyxi PY y X i xi , X j 0 j , ji, (1) (3)
If Xi has nXi values, the number of parameters required for the link Xi →Y is (nXi 1) (nY 1) – because of Equation 1. Since all the variables involved in a noisy OR are binary, this model only requires one parameter per link. Alternatively, it is possible to define new parameters:
Cyxi PY y X i xi , X j 0 j , ji cyxi , y y so that Equation 2 can be rewritten as:
PY y x1,, xn Cyxi ,
i
PY 0 x Py x PY y x PY y 1x if y 0 if y 0 (4) (5) (6) 3
An analysis of the current state, tendency of innovation activity of industrial enterprises of Ukraine and generalization of theoretical approaches, directions, and measures of increasing the innovative activity of the country is covered in [1].
Аuthors [4,5] concluded that the actions of large financial institutions have significant implications for the stability of the entire economic system and may be a threat to economic risk.
Network models have become attractive for modeling dependencies in real-world phenomena because of their ease of representation and the ability to provide an intuitive way visualization and interpretation of complex economic relations [6].
To understand vulnerabilities in the financial system, the idea of what-if network analysis has proven to be a promising tool that can help monitor the interconnectedness of financial institutions and markets. This has led to a significant increase in research on the statistical properties of network indicators to analyze systemic risk and the possibility of economic development of the country [7].
Directional acyclic graphical models and graphical Gaussian models were dealt with by the authors [8-10] and achieved results in their research.
However, when analyzing and modeling economic processes, risks, and to predict future results, Bayesian network models have proven themselves best [11, 12]. They are currently widely used for working with discrete data [13].
As experimental data for assessing the economic growth of Ukraine, that take into account the statistical capabilities and the modification of existing methods for studying the financial activity of an enterprise (Table 1) macroeconomic indicators were used.
Таble 1. Matrix of economic indicators
X1 Х11 Х12 Х13 Х2 Х21 Х22 Х3 Х31 Х32 Х4 Х41 Х42 Y
The share of enterprises engaged in innovation The share of the proceeds of innovation enterprises Profitability of operating activities of industrial enterprises, %
National currency loans for a term of 5 years to residents (excluding deposittaking corporations), average value, UAH million Foreign currency loans to residents (excluding deposit-taking corporations) for a term of 5 years, average value, UAH million
Foreign direct investment in Ukraine Interest rates on term deposits attracted, %
Average propensity to save Average annual dollar rate, UAH
The level of economic growth (nominal GDP at actual prices)
In our study, we are dealing with a set of statistical data that are interconnected (Fig. 1) consisting of 14 indicators for the period 2005-2018. The matrix of indicators is divided into four blocks that most fully characterize the financial-economic and business activity of enterprises, as well as the course of economic processes in the country (Table 1). The resulting indicator Y is an integral indicator of the level of economic growth of Ukraine. 4.2
A Bayesian network (BN) is a pair <G, В>, in which the first component G is a directed acyclic graph corresponding to random variables [14, 15]. Each variable is independent of its parents in G. So, the graph is written as a set of independence conditions. The set of parameters defining the network is the second component B. It contains parameters Qxi | pa( X i ) P(xi | pa( X i )) for each possible xi value from Xi and pa( X i ) from pa( X i ) , where pa( X i ) denotes the set of parents of the variable Xi in G . Each variable Xi is represented as a vertex. We use the notation to identify the parents paG ( X i ) if we consider more than one graph.The total joint probability of BN is calculated by the formula PB ( X 1,..., X N ) iN1 PB ( X i | pa( X i )) .
BN is a probabilistic model for representing probabilistic dependencies, as well as the absence of these dependencies. At the same time, the A→B relationship is causal, when event A causes B to occur, that is, when there is a mechanism whereby the value accepted by A affects the value adopted by B.
Validation was proposed for the first time in 1977 in [16]. Validation of the network that we design was carried out according to the algorithm for maximizing expectations. The algorithm finds local optimal estimates of the maximum likelihood of arguments. The concept of the algorithm is that if we knew the values of all nodes, then training would be simple at some step M. Therefore, at stage E, estimations of the expected likelihood value are made, including latent variables, as if we were able to observe them. In step M, the maximum likelihood values of the parameters are estimated using the maximization of the expected likelihood values obtained in step E. Then, the algorithm performs step E using the parameters obtained in step M again and so on.
The goal of parametric learning is to find the most likely θ variables that explain the data. Let D={D1,D2,…,DN} be a composition of the learning data, where D1={x1[l],x2[l],…,xn[l]} consists of instances of Bayesian network nodes. So the learning parameter is quantified by a log-likelihood function, denoted as LD(θ) [17]. 5
When developing the BN, the GeNIe 2.4 Academic software environment was used. The original Bayesian network model is built on General nodes. The block diagram of the BN of the country's economic growth is presented in Fig. 2.
When developing the BN, the GeNIe 2.4 Academic software environment was used. As can be seen from Fig. 1, the network contains 5 key nodes:
X1 - the level of manufacturability (innovation), X2 - the level of financial security (financial development), UAH X3 - external investment, UAH X4 - domestic investment potential, UAH,
Y – the level of economic growth.
It should be noted that due to the specifics of the Bayesian networks, all the conclusions of this model regarding the information sought are probabilistic in nature and are presented in the form of a ranked list (according to the values of the probability of fidelity of a particular conclusion).
Data were taken from 2005 to 2018. The dynamics of changes in the initial indicators for the observed period are presented in Figure 3. All nodes have five states: s1, s2, s3, s4, s5.
For example, for the node X1, the intervals of state discretization will be as follows: s1_below_85879; s2_85879_114478; s3_114478_150998; s4_150998_218982; s5_218982_up
We carry out, parameterization, and network validation on the nodes of General. The initial overall accuracy of the network was 48.8%, the accuracy of the result was 42%. At the next stage of the study, we changed the type of all nodes to Noisy-max with five states s1-s5, and the resulting node Y.
The network remains the same, the data file also does not change. We carry out training in parameters, primary validation. The overall accuracy of the network increased by 6.6% (and amounted to 55.4%), the accuracy of the result increased by 4% (and amounted to 46%).
Next, we analyze the sensitivity [18] of the network using influence charts. Repeated training in parameters and repeated validation led to an increase in overall accuracy by 2% (from 55.4% to 57.4%), and the accuracy of the result increased by 14% (from 46% to 60%). The results are shown in Table 2: Таble 2. Comparison of the initial accuracy of the model with accuracy after sensitivity analysis on nodes Noisy - MAX
6
Overall network accuracy,% 55,4
Accuracy of the result ,% 46,0
Overall network accuracy,% 57,4
Accuracy of the result ,% 60,0 If the level of innovation is increased to the maximum, the share of enterprises' bargaining will increase by 7% (from 34% to 41%), and this, in turn, will lead to an increase in the country's economic growth by 12% (from 24% to 36%), as shown in Figure 4.
At the maximum level of external investment, the level of innovation will increase by 4% (from 14% to 23%), direct investment in Ukraine will increase by 28% (from 19% to 47%), the tendency of the population to save will increase by 14% (from 28% to 42%). All this together will create conditions for increasing the country's economic growth, which will increase by 42% (from 24% to 66%), as shown in Figure 5. This article has conducted a comparative study of the behavior of Noise Max nodes and common nodes when designing a Bayesian network. Noisy-max nodes have been shown to provide relatively high initial accuracy compared to conventional nodes. Shared nodes require retesting. However, Noisy-max nodes entail an increase in time and computational cost.
Along with this, experiments were conducted to assess the general trends of the country's economic growth potential. It was found that at the maximum level of external investment, direct investment in Ukraine will increase by 28% (from 19% to 47%) and this creates the conditions for increasing the country's economic growth, which will increase by 42% (from 24% to 66%).
In our future research, we will try to trace the country's economic growth over time using the dynamic Bayesian network tool.