Sustainable Development Indicators in Benchmarks of Russia’s Regional Policy

We have considered the problem concerning the substantiation of sustainable development indicators and the improvement of the standard of living of the population in Russian regions. To quantify the standard of living of the population, we have used a common indicator such as the Human Development Index. The authors present an approach to managing the region in the form of optimization problems aimed at improving the standard of living of the region’s population by increasing the probability of its classification as a region with a higher standard of living. The optimization problem is solved based on the identified relationship between the standard of living of the region’s population and its socioeconomic indicators.


Introduction
Determining practical measurable indicators of the sustainable development concept is of essence for its implementation. There is a wide range of approaches and indicators offered by leading international organizations (UN, World Bank, OECD, European Commission, etc.). These indicators are used in the program-based approach to territory management as markers determining the system sustainability.
Russian regions are developing their systems of sustainable development indicators using the current database of the official Russian statistics which, according to S.N. Bobylev [1], makes it much easier to obtain necessary information and provides an adequate assessment of the regional progress in sustainable development. Since 2012, the general Russian regional development strategy has been determined by the so called ''May Decrees'' that contain clear benchmarks in such areas as education, science, healthcare, economy, demography and utilities. The goal of this research is to understand whether the target indicators of the regional policy are well justified, and to what extent their accomplishment will promote the region's sustainable development and improve the standard of living of its population.

Building a logistic regression by the criterion of maximum likelihood
where i x is the vector of values of the i-th object = ( ; m is the number of features of each object; n is the number of observations. The classification is performed using the function taking values in the range of (0; 1). The threshold value is (2) sets a separating linear border described with the hyperplane equation Let us introduce the function Let us set the D1 area of possible x values for the first class as , and for the second class as 2 = { : ( ) > 0}. Then ∀ ∈ 1 ℎ( ) < 0.5 and ∀ ∈ 2 ℎ( ) > 0.5. If x belongs to the hyperplane (3), then . Consequently, for the arbitrary observation * x , the probability of its placement into the first class is equal to ( * ∈ 1 ) = 1 − ℎ( * ), and to the second class - . It is established [2] that the maximization of the likehood logarithm is equivalent to the minimization To assess the coefficient vector b, iteration descent algorithms are used to solve extremum problems, e.g. the Newton-Raphson iteration algorithm is offered [2][3]. It involves the following. As a zero approximation, the classification problem can be accomplished using the multivariable linear regression Then the iteration process starts, on the k-th step of which the coefficient vector bis specified (k) : , where Q′(b (k) ) is a vector of first derivatives (gradient) of the Q(b) functional in the b (k) point, Q′′(b (k) ) is a matrix of second derivatives (hessian) of the Q(b) functional in the b (k) point,andhk is a step value that can be equivalent to 1, although its selection at each step can increase the convergence rate. The Newton-Raphson method is described in a number of textbooks on optimization methods [4]. However, the minimization problem (5) is not generally accomplished as, with a correct classification of all precedents, it has the lower bound at infinity equal to zero [5]. The quality of the built separating the border with a formal descent will not be related to the value of the target function Q(b) any more. The growth in the component of the b vector results in the unconstrained growth of values -yib T xi that can take very large values, which causes inherent computational errors or even stoppage of the algorithm due to memory overflow [6]. For this reason, the descent strategy needs to be adjusted. Gradient methods are intended for a descent to a local minimum. As the target function has a lower bound at infinity and there can be points, in which the gradient is close to zero, we deem it reasonable to proceed to zero-order methods. In [5], the authors offer an iteration descent mechanism. It comprises a sequence of zero-order minimization problems based on random searching. At every k-th step, the length of the coefficient vector b (k) is fixed. It is gradually increased until the target function reaches the required value.

Logistic regression as a mathematical model of relationship between the human development index and the socioeconomic indicators of the region
To characterize the standard of living of the region's population, we use the Human Development Index (HDI). Let us consider the values of the informative socioeconomic indicators of thei-th region of Russia as the vector xi. We have formed two classes of regions: the first class comprises regions with a low HDI (HDI < D), and the second class includes regions with a high HDI (HDI  D), where D is a threshold value. For the first D1 class, let the value of the binary variable membership be y = -1, and for the second class D2, let it be y = 1. Let us consider a region with a set of actual values of its socioeconomic indicators as the vector x*. According to (3), we are assessing W(x*).
, the region will fall into the first class with the probability , the region will fall into the second class with , the case is uncertain: the region can be placed both into the first class and into the second class with an equal probability of 0.5. Thus, a probabilistic assessment of the standard of living of the region's population can be made. The region can be classified among regions with a high standard of living with the probability h(x*), and among regions with a low standard of living with the probability can be used as the value of the target function: the higher its value, the more likely it is that the region's population will be deemed as having a high standard of living.
The features of the object in question are determined by the location of the point x* relative to the hyperplane (3). They can be numerically described, for example, using the distance between the point x* and the hyperplane (3) is always orthogonal to the hyperplane (3), its direction is set by the vector b, and its length -by the value q(x*). As the coefficient vector sets the hyperplane (3) equation with the accuracy up to the multiplier, it is expedient to normalize it for definiteness purposes when solving Let us assume that By equating the differential and multiplying both parts of the latter formula by 100%, , the coefficient bk with the variable Xk shows the percentage, by which the ratio of the probability of the region's classification as one with a high standard of living (D2) to the probability of classifying the region among regions with a low standard of living (D1) will change, if Xk increases by one unit, with the fixed values of other variables. The derivative of the function Z(x) for the variable Xk is directly proportionate to the coefficient bk. The growth (reduction) of the function Z(x) is equivalent to the growth (reduction) of h(x). Accordingly, to increase the value h(x), with positive values bk the increments k x  must also be positive, or negative, if the values are negative.
One of the main issues related to the use of logistic regression is the ability to establish a statistically reliable HDI-based classification of regions. Let us consider the classification of regions in two classes [7]: with a low HDI (y=-1) and with a high HDI (y=1). Based on the statistics for 2013, two classes of precedents (training samples) were formed. The first class of regions with a low HDI (maximum 0.84) included 28 subjects of the Russian Federation. The second class of regions with the HDI of over 0.85 included 31 subjects of the Russian Federation. To improve the reliability of results, we removed from the total array of statistical data 24 constituent entities of the Russian Federation with very low and very high HDIs, as well as those whose HDI was close to the average value, and those with data gaps. As a result, we obtained a system of statistically significant indicators which enabled us to accurately classify the observations for the training samples. The coefficients of the separating hyperplane (3) equation built on the basis of the training sample are given in Table 1 . This is quite reasonable and consistent with Keynesian theory. One of the main requirements for the model is its adequacy to the described system or value. The coefficients of the hyperplane (2) equation were built using the data for 2013 (Table 1). We examined the adequacy of the diagnostic model (1) based on the statistical data released by Rosstat for the next year of 2014, using the case of the subjects of the Ural Federal District. Table 2 presents the actual HDI values resulting from the assessment of the h(x) probabilities over 2013-2014. It should be noted that in 2013 the h(x) probabilities were based on training samples; and in 2014 -on test statistics available under the formula (2). When establishing the quantitative values of attributes X4, X5, X12, and X13 for 2014, we recalculated the monetary indicators taking into account the inflation estimated through the consumer price index. The results presented in Table 2 indicate the adequacy of the model. The increase in the HDI values for the Kurgan, Sverdlovsk and Chelyabinsk Regions correlates with the increase in the h(x) probabilities for all the three regions. The Tyumen Region has the maximum value of its HDI in both cases, which corresponds to the maximum probability h(x) that is practically equal to 1.

Optimization models based on logistic regression
Direct consequences of the set effect on the object are estimated by the value of the target function in the problems (6)- (8). Indirect consequences can be evaluated in each particular case based on the value of the vector of socioeconomic indicators received from the accomplishment of the optimization problem. The main disadvantage of the problems (7) and (8) is the complexity of setting the cost functions vj(j). We can circumvent this restriction by presenting the problem as follows: The essence of the problem (9) is as follows. It is necessary to achieve the transition of the region to the state h(x) = p0 with a minimum mean square relative change in the values of the components of the socioeconomic indicators vector. This problem definition does not require explicit specification of the labor intensity of the changes in the Xj indicators. The weighting coefficients j allow us to take into account the peculiarities of changing the variables, e.g. labor intensity of the change, the importance of the change in each of the variables. If the a priori information about the Xj indicators included in (2) is missing, we take all weighting coefficients j equal to 1. The zero value of the weighting coefficient can be used where the index of the bj coefficient has an opposite direction as compared to the expected one (e.g. in this problem, it is the coefficient b1=-2,381<0 with the aggregate birth rate coefficient), which makes it possible to record the actual value of Xj. It should be noted that this can be attributed to the following dependency: the birth rate is higher in regions with a higher standard of living as compared to regions with a low standard of living. This issue cannot be solved within the framework of the optimization problems offered in this publication, as the coefficient vector is determined based on the training sample and reflects the current socioeconomic environment in the country.
Additional restrictions in (10) allow us to factor in the specifics of the changes in the socioeconomic indicators Xj. Let us consider the possibilities of using optimization problems by the example of (9), assuming ρj = 1, j = 1, 2, …, m. We will solve the problem (9) using the penalty function method [3]. We will test the logistic regression as the management model for three subjects of the Ural Federal District, namely the Kurgan Region, the Sverdlovsk Region and the Chelyabinsk Region. According to their HDI values, each of these regions has a different standard of living (low, high and medium, respectively). The solution of the optimization problem (10) allowed us to calculate the values of socioeconomic indicators, by reaching thereof the optimum is attained, i.e. the best state of the system ensuring the improvement in the standard of living of the population.
The Kurgan Region can be ranked among the regions with a low HDI. The probability of its classification as a region with a high HDI: h(x 0 ) = 0.005. Table 3 provides the results of management aimed at reaching the threshold level (h(x * ) = 0.5) for the Kurgan Region, which were obtained by solving the problem (9). To compare the estimate values of socioeconomic indicators, Table 3   improve the standard of living of its population. Therefore, the estimate (optimal) values of indicators cannot serve as the new proposed benchmarks, but should be taken into consideration when optimizing the costs and specifying the priorities of further development. For the Kurgan Region, such priorities in improving the standard of living of its population should include: (1) Increasing life expectancy (X2) up to 70 years (in general, this level was achieved in Russia as early as in 2012, but for this region, with its initially low indicator, this is an optimal and realistic target); (2) Developing mortgage lending (X3), which has been significantly reduced in recent years as a result of macroeconomic instability; (3) Increasing the investment activity in the region (X4), which would help kick-start the manufacture, attract free labor and improve the paying capacity of the population; (4) Reducing the unemployment rate (X10) to 6.7%.
The Sverdlovsk Region has a relatively high HDI. The solution results of the optimization problem, which provided for the attainment of a higher HDI corresponding to the increase in the h(x) probability of up to 0.95, are presented in Table 4. According to the calculations, an important condition for probable improvements in the standard of living of the population in the Sverdlovsk Region is economic growth, which is impossible without additional investments (X4). Development of mortgage lending (X3) will help raise investments in the construction sector, a driver of the region's economic growth. The Chelyabinsk Region is classified as a region with a high HDI with a probability h(x 0 ) = 0.805, which is average for Russian regions, but lower than in the Sverdlovsk region. When solving the optimization problem for improving the standard of living of its population, we considered three options of increasing the probability h(x). The results of solving the problem for three probabilities h(x 1 ) = 0.805, h(x 2 ) = 0.9, and h(x 3 ) = 0.95 are provided in Table 5.