This article focuses on the use of Bayesian networks for analyzing the growth relationship of Ukraine's gross domestic product (GDP) from the volume of investment in the transport industry and offers a comparative description of the use of different structural training algorithms. It is shown that Noisy-max nodes as compared to General nodes provide relatively high initial accuracy. General nodes require a repeated validation procedure. When using the Hirerical sampling method, the accuracy of the network result with General nodes remains unchanged, and with Noisy-max nodes, it increases (in our case, by 12.32%). However, Noisy-max nodes entail an increase in time and computational costs.
Successful implementation of the investment policy will contribute to the implementation of one of the main country's economy tasks to increasing the number of main domestic investment resources sources. This will create the necessary prerequisites for the production growth and expanded reproduction of GDP in order to increase the population's well-being.
Therefore, it would be advisable to determine the informative investment indicators that have the greatest impact on the dynamics of Ukraine’s GDP. It is necessary to develop a model of the relationship between capital investment and GDP.
In [
It is believed that investment is the material basis for the economy's structuring.
Solving the problem of investment will mean the beginning of not only the economy's
restructuring but also its stabilization and subsequent growth [
The term "investment" has several meanings. First, it means the purchase of shares, bonds with the expectation of certain financial results. Secondly, real assets, for example, machinery, equipment necessary for the production and sale of any goods. In the broadest sense, investments provide the mechanism necessary to finance the growth and development of the country's economy, region, industry or enterprise.
Successful implementation of the investment policy will contribute to the implementation of one of the main tasks of the country's economy to increasing the number of main sources of domestic investment resources. This will create the necessary prerequisites for the production growth and expanded reproduction of GDP to increase the population well-being.
An important problem is the identification of causal relationships between causes or factors affecting GDP. Therefore, it would be advisable to determine informative investment indicators that have the greatest impact on the dynamics of Ukraine's GDP. The results presented in the article concern research on the development of probabilistic deterministic models using Bayesian networks to identify the factors affecting investment in the transport industry of Ukraine’s GDP.
Considering that one of the difficulties in the development of Bayesian networks is an exponential increase in the parameters number in their conditional probability tables (CPT), this study proposes a technique for using noisy-MAX nodes to simulate economic processes.
The purpose of the study is constructing a Bayesian network model using noisyMAX nodes for analyzing the dependence of Ukraine's GDP growth on the volume of investments in the transport industry. 2
For a set of events X i ,i 1, , N that are related, and a set of learning data D d1 , ,dn ,di xi1 xi2
xiN , is given. Here the subscript is the observation amount, and the upper one is the variable amount, n –is the amount of observations, each observation consists of N N 2 variables, and each j -th variable j 1, , N has A j 0,1, , j 1 j 2 conditions. Based on a given training sample, you need to build an acyclic graph connecting the event sets Xi , i=1,…,N . In addition, each BN structure g G is represented by a set N of predecessors P1 , , PN , that is, for each vertex j 1, , N , P j it is a variety of parent vertices, such that P j X 1 , , X N\X j . In this study, the Bayesian network of modeling investment processes in transport and their impact on GDP are built using noisy-MAX nodes. To do this, we have events X i ,i 1, , N that are affected by the uncertainties of a different nature. And also we have data describing these events. 3
There is a wide variety of Bayesian networks applications, including design [
The use of Bayesian networks in socio-economic research is widely considered in
[
The advantage of using Bayesian networks is their resistance to incomplete,
inaccurate and noisy information. In these cases, the result will reflect the most likely
outcome of events [
One of the main problems of Bayesian networks is the rapid increase in parameters
while increasing the number of parents. To solve this problem, the most widely used
are Noisy-MAX [
A pair <G, В> called a Bayesian network (BN), when the first part of G is a acyclic directed graph corresponding to random variables. When each variable is autonomous of its parents in G, so a graph is written as a composition of autonomous conditions. The second part of the pair, B, is the composition of parameters defining the network. It composed of parameters Qxi | pa(Xi ) P(xi | pa(Xi )) for each possible xi value from Xi and pa( X i ) from Pa( X i ) , where Pa( X i ) means the variable Xi parents set in G . Each variable Xi in graph G is suggested as a vertex. If we consider more than only one graph, then we use the notation to identify the parents PaG ( X i ) in graph G.
The
BN’s cumulative probability
B is determined by the equation PB ( X 1,..., XN ) iN1 PB (Xi | Pa(Xi )) .
The BN represents a model for getting probabilistic dependencies, as well as the absence of these dependencies. At the same time, the A→B relationship can be causal, when event A causes B to occur. So that is, at the time, when there is a mechanism by which value is accepted by A affects the value adopted by B. When all BN’s connections are causal, so BN is called causal.
From the existing discretization methods (hierarchical discretization, discretization on the same width of classes, discretization on the same number of points inside the classes) for the existing data set a hierarchical discretization method was chosen [18,19.].
Structural methods of BN training are algorithms, such as: the Bayesian Search, the Essential Graph Search, the TAN. In our study, the Greedy Thick Thinning algorithm is used.
The essence of the Greedy algorithm for constructing the BN structure is as
follows. The structure learning algorithm, called Greedy Thick Thinning (GTT), is found
on the approach of Bayesian search. GTT was proposed in [
Arcs are added to maximize the marginal likelihood P (D | S). This process is repeated until the addition of the arc leads to a positive increase. This phase is called “thickening”.
Then the arcs are removed step by step until the removal of the arc leads to a positive increase in P (D | S). This phase is called “thinning”. The algorithm is quite effective due to the fact that it is exposed to the trap of local maxima. In GTT, you can use two priorities. The priority of BDeu provides an equal score through the equivalent of Marcov’s classes. The priority of K2 is constant in all variables and, as a rule, is used to maximize P(G|D) when directly searching for a space of graphs.
Validation of the developed network was carried out according to the algorithm of
maximizing the expectation, which was proposed for the first time in 1977 in [
Therefore, at stage E, calculations of the expected likelihood value (expectation of the likelihood) are made, including latent variables, as if we were able to observe them. In step M, the maximum values of parameters’ likelihood are calculated (maximum likelihood estimates) of the parameters using the maximization of the expected likelihood values obtained in step E. Next, the algorithm again performs step E using the parameters obtained in step M and so on.
A whole series of such algorithms was developed, based on the algorithm of
maximizing the expectation [
This algorithm builds networks using penalty probabilistic values, which include values, derived from Bayesian information criteria, the principle of minimum description length, and others.
Noisy-MAX node is made up of a child node Y, accepting
nY values that may be
tagged from 0 tо nY 1 , and N parents, Pa Y X1,, X N , representing causes of
Y. If Xi =0 , it means the absence of X i . If all the reasons are missing, the result is
also missing, then the Noisy-MAX are determined according to the formula [
P Y 0 | Xi 0i 1 P Y y | x P Y y | Xi xi , X j 0j ,ji
i x represents a special combination of the parents of Y, x x1,..., xN . The probabilities that the result will take a certain value of y, in the case when X i equals a certain value of xi , provided that all the other reasons for Y are absent , are the parameters for the link Xi Y : If the degrees’ maximum produced by X acted independently, the degree reached by Y is determined by the formula: cyxi P Y y | Xi xi X j 0j,ji (1) (2) (3) (4) (5) (6) 5
In developing the BN, the Genie 2.3 software environment was used. The initial nodes type is General, each node has 5 states from s1 to s5.
If Xi takes nXi values, so according to formula 1, the amount of parameters required for the link Xi Y will be nXi 1 nY 1 . This model needs a reference to only one parameter, if all the variables included in Noisy are binary. We can determine the new parameters using the following formula: the formula 2 can be represented as:
Cyxi P Y y | Xi xi , X j 0j , ji cyxi , y y P Y y | x1 , , xn Cyxi , i CPT can be obtained given the following conditions:
P Y 0 | x if y 0 P y | x
P Y y | x P Y y 1| x if y 0
The following macroeconomic indicators for the period from 2012 (1st quarter) to 2017 (IV quarter) - 24 points were taken as experimental data for calculating the dependence of Ukraine's GDP growth from the volume of investment in the transport industry: x1 - the volume of investments in land and pipeline transport of actual prices; x2 - the volume of investment in water transport; x3 - the volume of investment in air transport; x4 - the volume of investments in warehousing and auxiliary activities in the field of transport; x5 - the volume of investments in postal and courier activities.
The variety of available data can be divided into two sets: 16 measurements is training sample A, 8 measurements is test sample B (Table 1). 9455,6 56,9 74,6 372,2 340,4 2852,1 5640,6 53,7 285,3 833130 894022 At the first stage of the available in the GeNie2.3 Academic software environment, the methods of structural learning select the appropriate method. Figure 1 presents the results of selection. As a result of the experiment, a Bayesian network consisting of 6 nodes was obtained. After parametric learning, primary validation was carried out. The model has achieved 22.9% level of accuracy during the test. After revalidation, the accuracy of the model was 54.34%.
In the next step, we changed the type of all nodes to Noisy with five states from s1 to s5, the resulting node Y. The network remains the same, the data file also does not change. We conduct parametric learning, primary validation and sensitivity analysis. Comparison of accuracy is presented in the table 2: Next, we apply Hirerical discretization method. We use the Greedy algorithm, we repeat all the steps: structural training, parametric training, validation, sensitivity analysis and re-validation. the accuracy of the result remained unchanged 54,34%. Comparison of accuracy after changing the sampling method is given in table 3.
We apply Hirerical discretization method now to Noisy nodes, then using the Greedy algorithm, we repeat all the steps: structural learning, parametric learning, validation, sensitivity analysis and repeated validation. The accuracy of the entire network decreased slightly - 21.21%, but the accuracy of the result was higher by 12.32% and amounted to 66.66%. During the selection of the structural learning algorithm, it was revealed that the Greedy algorithm turned out to be an adequate method when working with the existing data set.
With Noisy-max nodes, the required resulting accuracy is achieved immediately after the initial validation. This suggests that for a network with this type of nodes there is no need for sensitivity analysis and re-validation (Table 2).
When using the Hirerical discretization method, the accuracy of the result with the General nodes remains unchanged, and with Noisy-max nodes, it increases by 12.32% (Table 3).
When using General nodes, the EM execution time during the validation process was 13 seconds. For Noisy nodes, the EM algorithm spent three times as much computing time (37 seconds). On small data sets with a small BN size, such time costs can be neglected. However, as the network increases, the time costs (and hence the computing power) will be tangible and this will have to be taken into account. 7
Noisy-max nodes, compared to General nodes, provide relatively high initial accuracy. General nodes require a repeated validation procedure. When using the Hirerical discretization method, the accuracy of the network result with General nodes remains unchanged, and with Noisy-max nodes, it increases (in our case, by 12.32%). However, Noisy-max nodes entail an increase in time and computational costs.
In future studies, it is planned to use the dynamic Bayesian network approach in order to trace the levels of key indicators in different time slices.