The computation algorithm for determination of the socioeconomic development integral indicator based on the methods of factor analysis and expert evaluations has been described in the paper. By taking into account the knowledge and experience of experts, the factor model for evaluation of the level of socio-economic development has been improved. Based on the joint use of the methods of factor analysis and expert evaluation, the algorithm of automated computation for integral indicators has been developed. The approach has increased the reliability of the results of calculations and made it possible to analyze the correlations between indicators in terms of their in uence on the overall socio-economic situation. The developed computation algorithm is used in the educational process within the framework of teaching the discipline "Prognostics of socio-economic processes".
The main problem that arises when using methods of factor analysis in socioeconomic studies is reliable conclusions. In statistical calculations, the importance of speci c indicators for the socio-economic system is not taken into account. In this case, only the weighted average of such indicators is considered. This problem is solved by expert evaluation. Knowledge and experience of experts make it possible to rank the indicators in terms of their importance for ensuring the e ective functioning of the socio-economic system. At the same time, however, expert evaluation fails to establish the correlation between socioeconomic indicators. This task is successfully managed by the factor analysis.
Thus, in order to increase the reliability of the procedures for evaluation of the level of socio-economic development, there was a need to improve the mechanism for determination of the integral indicators on the basis of factor analysis by taking into account knowledge and experience of experts in the calculation procedures. It is an expert statistical option that is most suitable, since taking into account knowledge and experience of experts in calculations signi cantly increases the reliability of the conclusions obtained in the study. At the same time, it gives a chance to perform an extensive socio-economic analysis by establishing correlations between indicators and to determine the in uence of the change of a particular indicator (indicators) on the state of the system. Accordingly, within the framework of this study, the algorithm of automated evaluation of the level of socio-economic development has been developed on the basis of the joint use of the methods of factor analysis and expert evaluation. 2 2.1
Mathematical model for determination of the socio-economic
development integral indicator
In the framework of the presented studies, it is proposed to use a two-stage
approach in order to calculate the socio-economic development integral indicator.
At the rst stage, the dimension of the initial feature space is reduced. Reducing
the dimension of the feature space is based on the use of factor analysis [
The next stage is to obtain the one integral indicator based on the reduced
set of independent factors, which would combine all these factors in the best
way [
Let us directly consider the model in which the factor is the estimated value, in other words, it represents a certain new characteristic of the studied set of objects. The description of the factor in terms of its connection with the set of initial indicators is in the form of an n m matrix of factors A, where n is the number of features, m is the number of factors. The basis for constructing the matrix of factors A is the n m matrix of pairwise correlations R. It re ects the degree of correlation between each pair of initial indicators, while the factor matrix characterizes the correlation between each of the n indicators and the m factors determined during the progress of analysis. In this case, the number of factors m should be signi cantly less than n, and the level of loss of informativeness is negligible.
We assume that there is a set G(i = 1; 2; :::) of observations of a particular studied socio-economic phenomenon. In this case, the phenomenon is described by a set of n(j = 1; 2; :::) features. That is, the information presented in the socio-economic study can be described as a G n matrix : = 11 ::: 1j ::: 1n ::: ::: ::: ::: ::: i1 ::: ij ::: in ; ::: ::: ::: ::: :::
G1 ::: Gj ::: Gn
As a rule, the features selected to describe the socio-economic phenomenon
have di erent dimensions, and therefore di erent scalability. In order to ensure
the possibility of comparing the features of an object and avoiding the in uence
of their dimension, the initial data matrix is usually transformed (normalized),
introducing a single scale for all features. The most common ways of obtaining
a normalized data matrix Zij are standardization [
As a result of the standardization of the indicators, we obtain a G n matrix of the normalized values of the observations. Thus, a normalized matrix is obtained. Now it consists of vectors whose coordinates are indicators of socioeconomic development.
According to the factor model [
Zj = f (F1; F2; ::: ; Fm; Uj ) ;
The application of factor analysis to the matrix of pairwise correlations
between the initial indicators, on the basis of which the statistical weight of the
factor is determined, makes it possible to represent the initial indicators by
factors using the principal component analysis [
Zj = n X ajpFp; p=1
Z = AF;
The coe cients ajm are called factor loadings and characterize the signi cance of each of the factors for describing the j-th feature. Factor loadings are correlation coe cients between the initial indicators and factors. Let us write the expression (4) in vector form:
where F = (F1; F2; ::: ; Fn)T is a centered random column-vector of uncorrelated principal components; Z = (Z1; Z2; ::: ; Zn)T is a centered random columnvector of initial features; A = (aij ) is a nonrandom matrix of factor loadings of random values Zi on the components Fj (i = 1; 2; : : : ; n; j = 1; 2; : : : ; n).
We consider that = M(ZZT) is a covariant matrix of vector Z. Being symmetric and positive de nite, it has n positive eigenvalues 1; 2; ::: ; n. Let us assume that 1 > 2 > ::: > n. We denote: det j
jIj = 0; xj = j xj ;
If xj = (x1j ; x2j ; :::; xnj )T are normalized eigenvectors-columns of matrix , which correspond to their eigenvalues j (j = 1; 2; :::; n), then for all j = 1; 2; :::; n the following equalities are valid: where I is an n-th order unit matrix. From this follows: (6) (7) (8) (9) (10) (11) (12) (13) (14)
DFj0 = j (j = 1; 2; :::; n); xpT xj = n X xipxij = pj = i=1 1; p = j ; 0; p 6= j xjT xp = j xjT xp = j ; p = j ; 0 p 6= j
Let us introduce the matrix X = (x1; x2; :::; xn). Since, taking into account (8) and (9) then Let us assume that and since
XT
X =
;
F0 = XT Z;
MF0 = M(XT Z) = XT MZ; then F0 is a centered vector, and since that is, F0 is a vector of the principal components F, which is calculated in accordance with (14) as follows:
Fj = n X xij Zi (j = 1; 2; :::; n); i=1
Let us nd the matrix of factor loadings A. Using the orthogonality of the matrix X and the equation (12) we obtain:
XF = XXT Z = XX 1Z = Z; taking into account (5) the expression (17) can be written as
A = X; in other words, the factor loadings aij are the components of the eigenvectors xij of the matrix of pairwise correlation indicators.
In fact, for analysis, n0 < n rst principal components are used, which exhaust
at least 60 70% of the initial random variables [
i (i = 1; 2; ::: ; n);
That is, for statistical determination of weighting coe cients it is possible to use calculated earlier eigenvalues of the correlation matrix of initial indicators. The larger the di erence in the values of objects by factor, the greater the statistical weight of this factor.
The main disadvantage of statistical methods is the reliability of the conclusions, in particular, in this statistical mechanism of determining the integral indicators, the weight of the factor is determined by the dispersion of initial indicators, which is not always reliable in socio-economic studies, since in this case the importance of indicators for the socio-economic system is not taken into account. Therefore, in the framework of this study, in order to increase the reliability of the algorithm for evaluating the level of socio-economic development based on factor analysis, the implementation of an expert evaluation procedure in the mechanism of determining the weight of factors is proposed. The conclusions that change from method to method depend on the subjectivity of choosing the method of processing expert evaluations. In connection with this circumstance, it seems expedient in the automation procedures of expert evaluation of weighting coe cients of factors to make a simultaneous use of the method of median Mi and the method of scoring of indicators yi. An important feature of this mechanism of ranking socio-economic indicators by various experts is the possibility of minimizing the factor of subjectivity of expert evaluations by virtue (16) (17) (18) (19) (20) of the following procedures: 1) nding the density of the correlation between an arbitrary number of ranked features; 2) nding the density between the results of ranking of the two experts; 3) evaluating the consistency of expert conclusions in a group of more than two experts.
In order to solve the rst problem, as a rule, the Spearman's rank correlation
coe cient is used [
Sj = n X yij ; i=1 yij = yij /Sj ; KK = 12PC (L2m m2
1 ); i=1 j=1
L (m + 1) 2 12 A ; where L is the number of experts, m is the number of evaluated parameters, rij is the rank of the i-th element, assigned by the j-th expert.
The evaluation of competence is carried out with the help of a control examination, on the assumption that the correct answers to the questions are not known in advance. The mechanism is based on the processing of normalized scoring. The essence of the calculation is as follows:
1. The number of experts Lj is determined, who take part in the examination and must rank the indicators by means of their evaluation yij , for example, using a 10-point scale.
2. The amount of scores is calculated, determined by each expert on all indicators:
3. A table of normalized scores for each expert is calculated, by dividing the points of each indicator by the expert's score: 4. The weighted sums of relative scores for each expert are calculated: Sj = n X yij i=1 n X yi i=1 , ! n ; 5. The sum of the obtained weighted evaluations is calculated: (21) (22) (23) (24) (25) 7. The average group competence of experts is calculated:
Kj = Sj S;
L Kavg = X Kj ,
L; j=1
Experts, whose signi cance of their competencies is closest to the average group competence, are considered to be the most competent, and then the evaluations of only the most competent experts are taken into account. Thus, expert scoring evaluation of the i-th indicator on the basis of the joint use of the medians clustering and the method of scoring evaluations will be de ned as:
Then the weight of the i-th indicator evaluated by N most competent experts will be equal to:
S =
L X Sj ; j=1 6. The coe cients of expert competence are determined by dividing the weighted sum of the relative points of the expert into the total sum of the weighted evaluations:
Yi = (Mi + yi)/2; ci = Yi , L
X Yi; i=1 qi = m X cij ; j=1 and the weight of the i-th factor, according to expert evaluations, will be determined as the sum of the weight of each indicator included in this factor: where m is the number of indicators included in the i-th factor. As a result, we obtain a set of dimensionless coe cients qi, i = 1; 2; : : : ; n (n is a number of factors).
It should also be noted that the proposed mechanism of expert evaluation
makes it possible to implement program realization of the expert-statistical
procedure for determining the weighting coe cients of factors. The generalized
weight of factors that takes into account both the weight of the factor,
determined on the basis of expert evaluations, and the weight of the factor determined
statistically, can be obtained as the weighted average of these two evaluations [
X (qi + i); i=1 (26) (27) (28) (29) (30) (31) (32) n n where qi = qi P qi, i = i P i are expert and statistical (factor analyi i sis) weighted coe cients of the factor, respectively. Thus, the integral indicator is calculated as the sum of factors with the corresponding weighted average weighting coe cients wi:
Ij = n X wiFij (j = 1; 2; :::; n); i=1 (33) where n is a number of factors; Fij is the value of the i-th factor for the j-th object. The best is an object with a larger value of the integral indicator. 2.2
Computation algorithm of the socio-economic development level One of the main aspects of developed and existing models is ensuring the possibility of the processing automatization of socio-economic information on the basis of modern computer facilities. The presented model of determination of socioeconomic development integral indicators formalizes the settlement procedures and makes it possible to develop an algorithm for automated data processing of socio-economic research, based on joint use of methods of expert evaluation and factor analysis. Figure 1 illustrates a general scheme of the developed algorithm for determining the socio-economic development integral indicator.
The initial stage of the algorithm is characterized by the entering of values of indicators of socio-economic development and expert evaluations. Thus, the initial data base is formed in the form of the matrix of indicators (1) and the matrix of expert evaluations of indicators. Data from statistical directories can be used as indicators. Further actions within the framework of the presented algorithm are related to realization of the principal component analysis and mechanisms of expert evaluation (Fig. 1). According to the principal component analysis, a matrix of indicators of socio-economic development is initially formed, followed by its reduction to a single scale of measurements. Then, bringing the indicators to the normal distribution law and calculating the matrix of pairwise correlations are carried out. For this matrix, its eigenvalues and eigenvectors are calculated. The following actions are associated with the multiplication of the normalized matrix of indicators and the matrix of eigenvectors, which results in a matrix of factors. Factors are normalized, the dispersion is determined for them. Further it can be used in the analysis of integral indicators.
The next stage of the developed algorithm for automated determination of socio-economic development integral indicators is the procedure for determining the number of N factors included in the integral indicator (Fig. 1) on the basis of a series of eigenvalues of the matrix of pairwise correlations of socio-economic indicators and the given boundary value L due to dispersion factors of normalized parameters. The contribution of factors in the description of the total dispersion of the entire set of n socio-economic indicators is compared with the given limit value L of the dispersion of the normalized parameters, with achievement of which the factorization is stopped by the determination of N factors, or in other words - a sampling of the minimum number of factors with maximal eigenvalues i is made, the sum values of which are not less than nL:
N X i=1 i nL (34)
It is also worth noting that for analysis, one should use such number of
factors that exhaust at least 60 70% of the dispersion of the initial random
variables [
The parallel branch of the algorithm (see Fig. 1) is associated with the
program implementation of the expert evaluation mechanisms. For said purpose,
initially, using the above procedure, the determination of the competence of
experts is carried out, and then, by calculating Kendall's coe cient of concordance,
one gets an evaluation of the consistency of their conclusions. Thus, as a result of
the calculations, only the agreed conclusions of the competent experts remain.
On the basis of these data, a group of experts is determined, the conclusions
of which will participate in the evaluation. Using experts' performance of the
ranking of indicators on the basis of the medians clustering and the method of
arithmetic mean value, taking into account the competence of experts, the
scoring evaluation of the indicators is calculated. Then, the weight of the indicators
is determined according to expert evaluations [
The nal stage of the algorithm is the determination of weighting coe cients,
the calculation of integral indicators and the visualization of the results of data
processing. Weighting coe cients for each factor are calculated by a combination
of expert and statistical weight. The statistical weighting factors of the factors
included in the integral indicator are determined by the formula (20), based on
the eigenvalues of the pairwise correlations matrix of normalized socio-economic
indicators. The expert weight of the factors included in the integral indicator is
calculated by the formula (31). The generalized weight of the factors wi, which
takes into account both the weight of the factor, determined on the basis of
expert estimates, and the weight of the factor determined statistically, we obtain
by the formula (32). In order to directly determine the integral indicators, it
is necessary to combine the calculated factors into a single indicator. Since all
factors are independent, the combination is carried out using a simple linear
convolution [
Modeling the process of regional socio-economic development assessing Consider the process of assessing the level of regional socio-economic development in accordance with the developed computation algorithm for integral indicators of socio-economic development (Fig. 1) on the example of the Vinnytsia region districts.
According to the National State Statistics Service of Ukraine [
The matrix of factors F is obtained by multiplying the normalized matrix of socio-economic indicators (Table 1) into the matrix of the eigenvektors of the pairwise correlations matrix. The obtained factors are normalized by the formula (2). The normalized factor matrix is used to calculate the matrix of correlations between factors and indicators of socio-economic development, that is required for the interpretation of factors.
On the basis of the calculated eigenvalues of the pairwise correlations matrix (Table 2) and the given threshold L of the dispersion for normalized socioeconomic indicators (Table 1), the formula (34) determines the number of N factors in the integral indicator. In this case, the number of main components (factors) must be used, which exhaust at least 60-70% of the variance of the initial random variables. For example, at a given threshold of 0.6, from Table 2 it is necessary to select N factors with maximal eigenvalues, the sum of values of which is not less than 0; 6 11 = 6; 6. The sum of the rst three eigenvalues is 7.49, that is, the integral index consists of the rst three factors (N = 3) that explain approximately 68% (see formula 35) of the variance of the initial data (Table 2). The calculate matrix of correlations V between the normalized socio-economic indicators and the factors shows, which indicators are included in the given three factors (with the value of the variance of the indicators should not be less than the given limit value of 0.6).
Table 3 shows the structure of factors: the coe cient of correlation between the indicators and factors in which they are included, statistical (20) and expert (31) weights coe cients and weighted average weight coe cient of factors (32). The rst factor included the rst four socio-economic indicators - 1) the number of cars per 1000 people; 2) services rendered per unit of population; 3) natural increase (reduction) of the population; 4) the level of registered unemployment. The second factor included the eleventh indicator - emissions of pollutants. The third factor entered the seventh indicator - the ratio m2 of the built housing to the population.
To calculate the integral indicators, it is necessary to implement the mechanism for determining weight factor factors using expert evaluation. The basis for calculations is a table with score points of socio-economic indicators, put forward by experts. As experts participating in the evaluation, employees of the Regional Economic Development Department of Vinnytsia Region State Admin4,566 1,334 1,193 0,931 0,765 0,587 0,495 0,364 0,244 0,098 0,022 %(Fi) 41,51% 12,12% 10,85% 8,46% 6,95% 5,34% 4,50% 3,31% 2,22% 0,89% 0,20% P % 41,5% 53,6% 64,4% 72,9% 79,9% 85,2% 89,7% 93,1% 95,2% 96,2% 96,4% istration were used in this study. In order to minimize the subjectivity of expert assessments, an evaluation of the consensus of the experts' conclusions in the group and the de nition of experts competencies is initially carried out. To do this, the Kendal Concordance Coe cient (KK) is used. Calculation is carried out by formulas (21) and (22). The result of calculating the coe cient Kendel shows that it is close to the unit KK = 0,815. That is, it can be concluded that expert assessments are consistent and the composition of the expert group need not be changed. To assess the competencies of experts, an approach based on the processing of normalized ball scores is used. First, on the basis of scores of experts, the amount of points calculated by a particular expert is calculated (23). Then, by dividing each ball into the sum of all the scores arranged by this esperty (24), we obtain a normalized ball scores of each expert. The average normalized value is calculated for each indicator. On the basis of the obtained results, by formula (25), we calculated the weighted sum of the relative points of each expert. In Table 4 the results of the weighted sum of expert assessments calculations are presented. Also, the sum of the weighted evaluations obtained by experts is (2.3463). Subsequently, by dividing the weighted sum of expert points by the sum of the weighted assessments of all experts, the competencies of each expert (27) and the average group competence of experts (28) are calculated. Analyzing the competencies of experts (Table 4), it is evident that 5 experts have competences that are closest to the group's core competencies. That is, the most competent experts are 1, 4, 10, 11 and 20 experts. It is the ballroom evaluation of the indicators by these experts (Table 4) will be used in the future for calculations. Experts estimate the medians and the average values of ball scores for indicators. Based on the median and the mean values of the ball scores, the formula (29) determines the group expert scores for each indicator Yi (Table 5). The expert weight of each indicator Ci is determined by the formula (30), their values are given in Table 5. The weight coe cient qi of the factor is estimated by the experts (31), as the sum of the expert weight of each indicator included in this factor (see Table 3). The statistical weight coe cient i of factor is determined on the basis of the eigenvalues of the pairwise correlations matrix of normalized socio-economic indicators (Table 2) by the formula (20). The generalized weight of the factor wi by the formula (32).
To calculate the integral index Ij , it is necessary to combine the calculated factors. Calculations made show (Table 3) that weighted average weight coefcients of factors are: 0,3697, 0,1881 and 0,4422 for 1st, 2nd and 3rd factors, respectively. By multiplying the obtained factors by the corresponding weighted average weight coe cients of factors, by formula (33) we obtain the values of integral indicators that allow to rank the regions in terms of their socio-economic status. In table 6 results of calculations of factors and integral indicators of social and economic development of districts of Vinnytsia region are presented. 3
In the presented algorithm of evaluation of the socio-economic development level, the increase of its reliability is not due to a growing number of sources of initial data on the basis of which the factors are determined, but due to the implementation of expert evaluation procedure in the mechanism of determining weighting coe cients of factors. Thus, taking into account the knowledge and experience of experts in determining the weighting coe cients of factors, we introduce the importance of speci c indicators in the factor model of the evaluation of the level of socio-economic development, or, in other words, the intensity of their in uence on the state of the socio-economic system. The obtained correlation dependencies can be used, for example, to detect correlations between the indicators and features that determine the socio-economic development (regression) of individual regions, etc. The main advantages of the developed algorithm for determining the integral indicators are: the use of the whole set of initial data, which excludes the possibility of distorting the content of the socio-economic model; ensuring the possibility of operative work with large socio-economic data bulk; taking into account knowledge and experience of experts in building a single socio-economic development integral indicator. The proposed algorithm for the determination of integral indicators makes it possible to implement a uni ed approach to data analysis and to ensure the e ciency of constructing integral indicators. It should also be noted that in the context of processing automatization of socio-economic data, the expert-statistical algorithm proposed provides the possibility of program implementation of the procedure for determining the socio-economic development integral indicators. The developed computation algorithm is used in the teaching of the discipline "Prognostics of socio-economic processes" in conducting a laboratory workshop on the topic "Determining the level of socio-economic development on the basis of expert-statistical method" and "Modeling the in uence of the socio-economic indicators values on the general level of regional socio-economic development".