4. Multidimensional risk model of complex systems

Consider the task of ensuring the system sustainable functioning. Here you first need to concretize the "sustainability" concept. Typically the stability of the system functioning is interpreted in terms of its safety. Security issues are resolved with the help of risk analysis [25]. Some authors note that growth rates of damage considerably exceed growth rates of the economy [26, 27]. This can be explained with a constant increase of risk in the conditions of a scientific and technical revolution and the forced development of a technosphere [28]. Therefore, we will assume that the functioning stability of the territorial system is intimately connected with risk; the lower the risk level, the more stable the system state. Thus, the diagnosis of system sustainability can be made on the basis of monitoring its risk. This requires adequate risk models.

Let S be some multidimensional stochastic system. Let us consider an adequate representation of this system as a random vector X = (X1, X2, ..., Xl) with a certain probability density pX(x). The development of a complex system and an increase in the efficiency of its functioning are an inevitable cause of increasing risks. Therefore, it is necessary to assess the risks of such systems. Consider the risk model of multidimensional stochastic systems proposed in [29].

Instead of the generally conventional selection of concrete dangerous situations, we will define the geometric area D of adverse outcomes. Formally this area can look arbitrarily depending on a specific objective.

The concept of dangerous states as larger and improbable deviations of a conception of dangerous conditions as large and improbable deviations of random variables from some best provision Θ is mostly distributed. In this case, D represents an external area of an m-axis ellipsoid.

Setting the function of consequences from dangerous situations (risk function) in the form of g(x), we will receive a model for the quantitative assessment of risk [30]:

$$r(X) = \int\int\_{\mathbb{R}^m} \dots \int\_{\mathbb{R}} g(\boldsymbol{\varkappa}) p\_X(\boldsymbol{\varkappa}) d\boldsymbol{\varkappa}.\tag{9}$$

If in Eq. (9) to accept g xð Þ¼ 1 ∀x∈ D and g xð Þ¼ 0 ∀x ∉ D, that r Xð Þ¼ P Xð Þ ∈ D , that is, the risk is estimated as a probability of an unfavorable outcome.

If at an early stage of system analysis is difficult to describe enough precisely the g(x) function, then Eq. (2) becomes an assessment of P(D) and is a convenient initial approximation of risk model.

To define a function g(x) requires a quantitative assessment of consequences for the studied system depending on values of risk factors. It demands to carry out separate research. Let us note that values of the function g(x) are given in the nominal units. But they are usually quite simply interpreted in the respective subject area. The essence of the function g(x) is as follows. It accepts the least nonnegative (e.g., zero) value in a point of Θ or in its neighborhood of U(Θ). Further in each direction during removal from U(Θ) the g(x) function has to increase monotonously. For scaling on each risk factor, we will set some limit values, at which consequences become dangerous (or irreversible). Let us set values g(x) at each

Seventeen coal mining enterprises were investigated [23]. On the basis of two generalized factors (Y1 is the factor characterizing the organization of safe production; Y2 is the factor reflecting the professionalism of the staff), all enterprises were divided into two groups: (1) enterprises with a low level of injury; (2) enterprises with a high level of injury. For the first and second groups of mines,

<sup>¼</sup> ð Þ 2, 42; �0, 31 , H Yð Þ<sup>1</sup>

<sup>¼</sup> ð Þ 3, 74; �0, 70 , H Yð Þ<sup>2</sup>

In this case, the direction of the entropy change vector will differ from Example 1: at the enterprises with a high level of injury, the randomness entropy needs to be

For example, this can be accomplished as follows: to bring the state of the second group of mines to the state of the first group, it is necessary to reduce the dispersion of the factor characterizing the organization of safe production and reduce interrelation with the factor reflecting the professionalism of the staff. This means a more specific and accurate organization of production. The organization of safe production should not depend on the degree of professionalism and competence of staff

Example 3. We investigate the possibilities of entropy modeling on the example of the population analysis in terms of prevention of chronic noninfectious diseases (CNID) by biological risk factors [23]. For carrying out the analysis of change of population entropy depending on the health status, two equal age groups were formed: 18–26 years and 27–35 years. Four risk factors were identified: "total cholesterol," "systolic blood pressure," "body mass index," and "glucose level." The

As the health of the population deteriorates, the total population entropy and the randomness entropy increase. This can be explained by the fact that the additional damaging influence of CNID, in general, is added to the pathological influence of

> Self-organization entropy H(Y)<sup>R</sup>

7.131 �0.578 6.553

8.376 �0.542 7.834

Total entropy H(Y)

Conversely, the self-organization entropy as the deterioration of the health status of the population decreases which corresponds to the strengthening of the narrowness of the interrelations between subsystems. This can be explained by the fact that the development of diseases in the organism happens in many respects and it is interdependent. On the other hand, at the development of diseases, some

reduced, and the self-organization entropy needs to be increased.

risk factors on a human body separately and on all population.

Health status Randomness entropy

H(Y)<sup>V</sup>

18–26 Healthy 5.500 �0.514 4.986

27–35 Healthy 5.731 �0.299 5.432

Patient 7.847 �0.696 7.151

Patient 8.720 �0.781 7.939

¼ 2, 11,

¼ 3, 04:

respectively, we have.

strongly.

Age (years)

Table 1.

28

hð Þ<sup>1</sup> <sup>V</sup> ;hð Þ<sup>1</sup> R 

Sustainability Assessment at the 21st Century

hð Þ<sup>2</sup> <sup>V</sup> ;hð Þ<sup>2</sup> R 

results of the analysis are given in Table 1.

Apparently healthy

Apparently healthy

Entropy levels in different groups of people.

such point equal to some value, for example 1. For convenience, it is desirable to impose a number of restrictions on the function g(x): convexity, continuity, etc.

In [30] the variant of the task of the g(x) function in the form of a paraboloid is given. By way of illustration in Figure 4, the example of the risk function for a case m ¼ 2 is shown. The ellipse describing the area D of admissible values of risk factors and lying on the Ox1x2 (r = 0) plane is shown by black color. The paraboloid above the plane represents possible values of risk r(X). White points on the plane are values of risk factors; to them there correspond points on paraboloid surface which set risk values; the image of the border of an ellipse D is shown in the form of the black line. All corresponding couples of points (values of risk factors and risk values) are connected among themselves by vertical dashed lines.

In the problems of risk monitoring, along with risk assessment, r Xð Þ on all risk factors of X1, X<sup>2</sup> ..., Xm of the multidimensional system is expedient to estimate the contribution of each factor to total risk. We introduce a random vector X� <sup>k</sup> ¼ X1, …, Xk�1,Xkþ<sup>1</sup> ð Þ , …, Xm . Then the absolute change of risk of the multidimensional system due to the addition of factor Xk is equal:

$$
\Delta r(X\_k) = r(X) - r\left(X\_k^{-}\right). \tag{10}
$$

pXð Þ¼ x

DOI: http://dx.doi.org/10.5772/intechopen.89287

by setting the g(x) function accordingly.

is a coefficient of correlation between X1 and X2.

we will accept A<sup>1</sup> ¼ A<sup>2</sup> ¼ … ¼ Am ¼ A, Aj ¼ bj=σ <sup>j</sup>.

sionality factor and narrowness of correlations.

levels are given in Table 2.

Realization of a standard normal random vector.

Figure 5.

31

for monitoring of risk of Sverdlovsk region in 1999–2017.

ance matrix, <sup>σ</sup>ii <sup>¼</sup> <sup>σ</sup><sup>2</sup>

1 � j j RX

1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ <sup>2</sup><sup>π</sup> <sup>m</sup>j j <sup>Σ</sup> <sup>p</sup> exp � <sup>1</sup>

where <sup>a</sup> <sup>¼</sup> ð Þ <sup>a</sup>1, <sup>a</sup>2, ::, am <sup>T</sup>—a vector of expectations, <sup>Σ</sup> <sup>¼</sup> <sup>σ</sup>ij � �

Probabilistic-Entropic Concept of Sustainable Development of the Example of Territories

<sup>i</sup> —dispersion of factor Xi.

2 ð Þ x � a

The use of a Gaussian random vector is based on the central limit theorem [31]. As approbation on a number of examples has shown, such idealization is not so critical, and if there are any bases to consider that density of probabilities is a component of the vector of X having more extended tails, then this can be corrected

Example 4. Let us consider a two-dimensional Gaussian random vector which components have a zero average and single dispersion. In Figure 5 the example of realization of such accidental vector is shown for: (a) ρ ¼ 0, 4; (b) ρ ¼ 0, 9, where ρ

From Figure 5 we see that the probability of large deviations of the random vector from the origin increases with the increase in the closeness of the correlation. Example 5. Let us estimate the probability of P(D) for a Gaussian random vector

<sup>1</sup>=<sup>m</sup> [32], where j j RX is a continuant of a complete correlation matrix (Deð Þ¼ X 0 is independence of components, and Deð Þ¼ X 1 is a rigorous linear relation). Let us consider the following cases: Deð Þ¼ X 0, Deð Þ¼ X 0, 5, Deð Þ¼ X 1. The results of the calculation of P(D) are given in Figure 3. For descriptive reasons

The increase in the probability of an unfavorable outcome is influenced by both the increase in the dimension of m and an increase in the narrowness of correlation communication between the components of a random vector of X. Let us note that even the average narrowness of correlation (Deð Þ¼ X 0, 5) leads to a significant increase in the probability of an unfavorable outcome. The effect increases with the increase in values Aj that correspond to less probable, but more dangerous, adverse outcomes. Therefore, risk modeling should take into account both a multidimen-

Example 6. Approbation of risk model of a multidimensional stochastic system

Let us execute monitoring of Sverdlovsk region on the dynamics of macroeconomic risk factors, taken as an interval of 9 years. Risk factors and their threshold

of X, with the different narrowness of correlation communication Deð Þ¼ X

The analysis of schedules in Figure 6 indicates the following.

<sup>T</sup>Σ�<sup>1</sup>

� �,

ð Þ x � a

<sup>m</sup>�m—a covari-

Dividing Δr Xð Þ<sup>k</sup> of the risk r X� k , we will receive the relative change of risk of the multidimensional system by the addition of factor Xk:

$$
\delta r(X\_k) = \Delta r(X\_k) / r(X\_k^-). \tag{11}
$$

Let us note that along with a contribution to the common risk of one factor, Eqs. (10) and (11) allow us to estimate influence and groups of factors.

Monitoring risk on the basis of the model in Eqs. (9)–(11) consists of serial estimation in time of the actual values of r(X), Δr Xð Þ<sup>k</sup> , δr Xð Þ<sup>k</sup> ,j ¼ 1, 2, …, m, and also dynamics of their change.

Let us consider the most common case when X has a joint normal distribution with a probability density:

Figure 4. An example of a two-dimensional risk functions.

Probabilistic-Entropic Concept of Sustainable Development of the Example of Territories DOI: http://dx.doi.org/10.5772/intechopen.89287

$$p\_X(\mathbf{x}) = \frac{1}{\sqrt{\left(2\pi\right)^m |\Sigma|}} \exp\left\{-\frac{1}{2} (\mathbf{x} - a)^T \Sigma^{-1} (\mathbf{x} - a)\right\},$$

where <sup>a</sup> <sup>¼</sup> ð Þ <sup>a</sup>1, <sup>a</sup>2, ::, am <sup>T</sup>—a vector of expectations, <sup>Σ</sup> <sup>¼</sup> <sup>σ</sup>ij � � <sup>m</sup>�m—a covariance matrix, <sup>σ</sup>ii <sup>¼</sup> <sup>σ</sup><sup>2</sup> <sup>i</sup> —dispersion of factor Xi.

The use of a Gaussian random vector is based on the central limit theorem [31]. As approbation on a number of examples has shown, such idealization is not so critical, and if there are any bases to consider that density of probabilities is a component of the vector of X having more extended tails, then this can be corrected by setting the g(x) function accordingly.

Example 4. Let us consider a two-dimensional Gaussian random vector which components have a zero average and single dispersion. In Figure 5 the example of realization of such accidental vector is shown for: (a) ρ ¼ 0, 4; (b) ρ ¼ 0, 9, where ρ is a coefficient of correlation between X1 and X2.

From Figure 5 we see that the probability of large deviations of the random vector from the origin increases with the increase in the closeness of the correlation.

Example 5. Let us estimate the probability of P(D) for a Gaussian random vector of X, with the different narrowness of correlation communication Deð Þ¼ X 1 � j j RX <sup>1</sup>=<sup>m</sup> [32], where j j RX is a continuant of a complete correlation matrix (Deð Þ¼ X 0 is independence of components, and Deð Þ¼ X 1 is a rigorous linear relation). Let us consider the following cases: Deð Þ¼ X 0, Deð Þ¼ X 0, 5, Deð Þ¼ X 1. The results of the calculation of P(D) are given in Figure 3. For descriptive reasons we will accept A<sup>1</sup> ¼ A<sup>2</sup> ¼ … ¼ Am ¼ A, Aj ¼ bj=σ <sup>j</sup>.

The analysis of schedules in Figure 6 indicates the following.

The increase in the probability of an unfavorable outcome is influenced by both the increase in the dimension of m and an increase in the narrowness of correlation communication between the components of a random vector of X. Let us note that even the average narrowness of correlation (Deð Þ¼ X 0, 5) leads to a significant increase in the probability of an unfavorable outcome. The effect increases with the increase in values Aj that correspond to less probable, but more dangerous, adverse outcomes. Therefore, risk modeling should take into account both a multidimensionality factor and narrowness of correlations.

Example 6. Approbation of risk model of a multidimensional stochastic system for monitoring of risk of Sverdlovsk region in 1999–2017.

Let us execute monitoring of Sverdlovsk region on the dynamics of macroeconomic risk factors, taken as an interval of 9 years. Risk factors and their threshold levels are given in Table 2.

Figure 5. Realization of a standard normal random vector.

such point equal to some value, for example 1. For convenience, it is desirable to impose a number of restrictions on the function g(x): convexity, continuity, etc. In [30] the variant of the task of the g(x) function in the form of a paraboloid is given. By way of illustration in Figure 4, the example of the risk function for a case m ¼ 2 is shown. The ellipse describing the area D of admissible values of risk factors and lying on the Ox1x2 (r = 0) plane is shown by black color. The paraboloid above the plane represents possible values of risk r(X). White points on the plane are values of risk factors; to them there correspond points on paraboloid surface which set risk values; the image of the border of an ellipse D is shown in the form of the black line. All corresponding couples of points (values of risk factors and risk

In the problems of risk monitoring, along with risk assessment, r Xð Þ on all risk factors of X1, X<sup>2</sup> ..., Xm of the multidimensional system is expedient to estimate the contribution of each factor to total risk. We introduce a random vector

Δr Xð Þ¼ <sup>k</sup> r Xð Þ� r X�

δr Xð Þ¼ <sup>k</sup> Δr Xð Þ<sup>k</sup> =r X�

Let us note that along with a contribution to the common risk of one factor,

Monitoring risk on the basis of the model in Eqs. (9)–(11) consists of serial estimation in time of the actual values of r(X), Δr Xð Þ<sup>k</sup> , δr Xð Þ<sup>k</sup> ,j ¼ 1, 2, …, m, and

Let us consider the most common case when X has a joint normal distribution

k

k

, we will receive the relative change of risk of

: (10)

: (11)

values) are connected among themselves by vertical dashed lines.

multidimensional system due to the addition of factor Xk is equal:

k

Eqs. (10) and (11) allow us to estimate influence and groups of factors.

the multidimensional system by the addition of factor Xk:

Dividing Δr Xð Þ<sup>k</sup> of the risk r X�

Sustainability Assessment at the 21st Century

also dynamics of their change.

with a probability density:

Figure 4.

30

An example of a two-dimensional risk functions.

<sup>k</sup> ¼ X1, …, Xk�1,Xkþ<sup>1</sup> ð Þ , …, Xm . Then the absolute change of risk of the

X�

Figure 6.

Dependences of lgP(D) on threshold level A: (a) De(X) = 0; (b) De(X) = 0.5; (c) De(X) = 1. Designations: Row 1 (m = 1), row 2 (m = 2), row 3 (m = 3), row 4 (m = 4), row 5 (m = 5).


#### Table 2.

Macroeconomic risk factors of the region.

We consider that random vector X has a joint normal distribution.

In Figures 7 and 8, results of the calculation of the probability of unfavorable outcome P(D) and risk r(X) for the threshold levels of risk factors K are shown.

Analysis of the results of monitoring of multidimensional risk in the Sverdlovsk region showed the following:

5. Formation of the sustainable development concept

as sustainable development of the complex system.

Figure 7.

Figure 8.

33

Estimates of P(D) in the Sverdlovsk region.

DOI: http://dx.doi.org/10.5772/intechopen.89287

Estimates of r(X) in the Sverdlovsk region.

sustainable development, they need to be taken into account together.

indicators Yj, characterizing the functioning of the system, that is,

Consideration of examples shows that "development" and "sustainability" characterize various aspects of the complex systems operation. And for ensuring

Probabilistic-Entropic Concept of Sustainable Development of the Example of Territories

Hypothesis 2. We will understand dynamics consisting of available trends of the balanced change of a vector entropy while maintaining an acceptable level of risks

For this purpose, we combine the vector entropy model and the risk model of a multidimensional stochastic system. Moreover, it is necessary to consider elements of the system (components of the random vector Z), both as risk factors Xi and as

Z ¼ X∪Y ¼ Z1 ð Þ , Z2, …, Z<sup>n</sup> , max ð Þ l, m ≤n≤ l þ m:


Probabilistic-Entropic Concept of Sustainable Development of the Example of Territories DOI: http://dx.doi.org/10.5772/intechopen.89287

#### Figure 7. Estimates of P(D) in the Sverdlovsk region.

Figure 8. Estimates of r(X) in the Sverdlovsk region.

## 5. Formation of the sustainable development concept

Consideration of examples shows that "development" and "sustainability" characterize various aspects of the complex systems operation. And for ensuring sustainable development, they need to be taken into account together.

Hypothesis 2. We will understand dynamics consisting of available trends of the balanced change of a vector entropy while maintaining an acceptable level of risks as sustainable development of the complex system.

For this purpose, we combine the vector entropy model and the risk model of a multidimensional stochastic system. Moreover, it is necessary to consider elements of the system (components of the random vector Z), both as risk factors Xi and as indicators Yj, characterizing the functioning of the system, that is,

$$\mathbf{Z} = \mathbf{X} \cup \mathbf{Y} = (Z\_1, Z\_2, \dots, Z\_n), \text{ max } (l, m) \le n \le l + m.$$

We consider that random vector X has a joint normal distribution.

dangerous situations has been growing slightly since 2014.

3. In the Sverdlovsk region until 2010, the main contribution to regional instability was made by factor X6, then X<sup>2</sup> became such a factor, and since

2015, the main contribution to instability was made by factor X7.

region showed the following:

Macroeconomic risk factors of the region.

Figure 6.

population

rubles

Table 2.

32

region have increased (decreased risks).

In Figures 7 and 8, results of the calculation of the probability of unfavorable outcome P(D) and risk r(X) for the threshold levels of risk factors K are shown. Analysis of the results of monitoring of multidimensional risk in the Sverdlovsk

Dependences of lgP(D) on threshold level A: (a) De(X) = 0; (b) De(X) = 0.5; (c) De(X) = 1. Designations: Row

Risk factors Threshold levels

X1—real income movement, in % to the previous year 79.93 X2—the ratio of the average size of pension to subsistence minimum of pensioners 0.66 X3—morbidity on 1000 people of the population 960

X5—wear of fixed assets on the end of the year, % 71.33

X7—quantum index of gross regional product, % to the previous year 88.4 X8—unemployment rate, in % 18

X4—mortality from external causes, number of the dead on 100,000 people of the

X6—the volume of budget revenues per capita, in the prices of 2017, thousand

Kj

322.1

21.75

1 (m = 1), row 2 (m = 2), row 3 (m = 3), row 4 (m = 4), row 5 (m = 5).

Sustainability Assessment at the 21st Century

1.During the initial period, the greatest socioeconomic instability (the highest risk values) was observed. Then gradually the dynamics of sustainability in the

2.After the sanctions were imposed, the risk began to increase. The lower rate of growth of r(X) than P(D) indicates that the probability of occurrence of very

The case n <l þ m appears, when X∩Y 6¼ ∅.

Within the framework of the proposed concept, along with the tasks of monitoring complex systems discussed above, it is possible to solve management problems (development of control recommendations).

6. Discussion of results

DOI: http://dx.doi.org/10.5772/intechopen.89287

stability is provided due to an acceptable risk level.

able development of Sverdlovsk region.

multidimensional risk models.

application to territorial systems.

real data.

7. Conclusion

of risk.

Acknowledgements

35

Thus, on the basis of the use of two original models—a vector entropy and multidimensional risk—it was succeeded to formalize the new concept of sustainable development of complex systems. Both models are successfully approved on

Probabilistic-Entropic Concept of Sustainable Development of the Example of Territories

This concept can be implemented by means of monitoring of the studied system. As observed parameters efficiency factors of the functioning of a system and its risk factors are used. The direction of development is given by an entropy vector, and

Management recommendations are formed in the form of the solution to an extreme problem Eq. (12). This problem is solved by methods of penalty functions. Currently, the work is at a stage of practical approbation of monitoring of sustain-

1.The probability-entropy concept of sustainable development of complex stochastic systems is formulated. It is based on vector entropy and

2.According to the formulated concept, the sustainable development of a

3.The proposed concept of sustainable development has been tried out in

The present study was carried out by financial support of the Russian Fund of Fundamental Research grant № 17-01-00315 "Development of models and methods of monitoring, forecasting, and control optimization of multidimensional regional

socioeconomic systems based on entropy and minimax approaches."

complex system will be understood as the dynamics consisting of the tendency of a balanced change in vector entropy while maintaining an acceptable level

The idea of vector entropy control is the transfer of the vector h(Z) from the state h Z<sup>0</sup> � � <sup>¼</sup> <sup>h</sup><sup>0</sup> <sup>V</sup>; <sup>h</sup><sup>0</sup> R � � in the state h Zð Þ¼ \* ð Þ hV\*; hR\* , which corresponds to the effective functioning of the stochastic system.

For a Gaussian system, the vector entropy control consists in directing the entropy from some initial point h<sup>0</sup> <sup>V</sup>; <sup>h</sup><sup>0</sup> R � � <sup>¼</sup> H Z<sup>0</sup> � � <sup>V</sup>; H Z<sup>0</sup> � � R � � with the covariance matrix Σ<sup>0</sup> to the final point ð Þ hV\*; hR\* with a minimal change of the covariance matrixΣ<sup>0</sup> <sup>¼</sup> <sup>σ</sup><sup>0</sup> ij n o and the expectations vector <sup>a</sup><sup>0</sup> and acceptable risk (Figure 9).

The problem of the vector entropy control of the Gaussian system to ensure sustainable development will take the form:

$$\begin{cases} \begin{aligned} \mathbf{G}(\boldsymbol{\Sigma}) &= \sum\_{i=1}^{n} \sum\_{j=1}^{n} \left( \sigma\_{\bar{\mathcal{y}}} - \sigma\_{\bar{\mathcal{y}}}^{0} \right)^{2} + \sum\_{i=1}^{j} \left( a\_{i} - a\_{i}^{0} \right)^{2} \to \min\_{a, \boldsymbol{\Sigma}}, \\ \boldsymbol{H}(\boldsymbol{Y})\_{\boldsymbol{V}} &= \boldsymbol{A}, \\ \boldsymbol{H}(\boldsymbol{Y})\_{\boldsymbol{R}} &= \boldsymbol{B}, \\ \boldsymbol{r}(\boldsymbol{X}) &\leq \boldsymbol{r}, \\ \boldsymbol{\Sigma} &\in \mathcal{G}\_{\boldsymbol{\Sigma}}, \quad a \in \mathcal{H}\_{a}, \\ \sigma\_{\bar{\mathcal{y}}}^{2} &< \sigma\_{\bar{\mathcal{u}}} \sigma\_{\bar{\mathcal{y}}}, \ \sigma\_{\bar{\mathcal{y}}} = \sigma\_{\bar{\mathcal{y}}}, \ \sigma\_{\bar{\mathcal{u}}} > 0 \ \ \ \mathbf{1} \leq i, j \leq n, \\ \boldsymbol{\Sigma} &> 0, \end{aligned} \tag{12}$$

where A ¼ hV\*, B ¼ hR\*, a—the expectations vector of the components Xi, i = 1, 2, ..., l.

The last constraint in Eq. (12) means positive definiteness of the matrix Σ. Note that the performance criterion in Eq. (12) may be different, depending upon the characteristics of a particular system S.

Probabilistic-Entropic Concept of Sustainable Development of the Example of Territories DOI: http://dx.doi.org/10.5772/intechopen.89287
