Monitoring of a condition of Monitoring of a condition of economic systems on the economic systems on the

basis of the analysis of basis of the analysis of dynamics of entropydynamics of entropy

A.N. Tyrsin1, O.V. Vorfolomeeva2

1 – Science and Engineering Center «Reliability and Resource of Large Systems and Machines», Ural Branch,

Russian Academy of Sciences, Ekaterinburg2 – Chelyabinsk Stat University, Chelyabinsk

2

Entropy role in difficult, open systems

Prigogine I.: Bifurcation – turning point in system development: choice from several new conditions; bifurcations are provoked by change of the operating parameter of system, entropy increase. Overcoming of a point of bifurcation is accompanied by decrease in entropy, self-organization.

Klimontovich U.L.: There is "a norm of a randomness" (entropy level) for normal functioning of system; deviations from norm mean "illness". If "treatment" approaches a condition of open system to norm, self-organization process takes place.

-------------------------------------------------------------------- Entropy characterizes system functioning.

3

S -multidimensional random variable

If the random vector Y has a

multidimensional normal distribution then

)..,,,( 21 mYYYY

RY ln21)()(

1

m

iiYHH

4

Entropy - probabilistic model

Y1

Y2

Y3

Y4Y5

growth points

31YYr

52YYr

53YYr

41YYr

42YYr

51YYr

21YYr

32YYr

43YYr

45YYr

H(Y2)

H(Y3)H(Y1)

H(Y5) H(Y4)

The Entropy - probabilistic model allows to allocate elements of difficult system and communication between them as separate variables

System and its components:

5

Statement 1. Let X1, X2 - two continuous random variables with the finite variances, defined on the all numerical axis and described by one-type distributions. Then

12, 2

2, 1, 2 – variances and parameters of scale of random variables X1 и X2.

1

2

1

212 lnln)()(

XHXH

6

Example 1. 1) For normally distributed random variables X1 and X2 with variances 1

2 and

22 the difference of entropies is equal

2) For exponential distributed random variables X1 and X2 with scale

parameters 1, 2 the difference of entropies is equal

3) Let's consider random variables X1 and X2, distributed under the lognormalny law with parameters of scale , and form parameter s. Entropy for the lognormalny law with scale parameters and form s is equal Taking into account that dispersion is

equal, receive

1

21212 ln)2ln()2ln()()(

eeXHXH

1

2121212 ln)ln()ln()ln()ln()()(

eeeeXHXH

11

e 22

e

)2ln()( esXH 22 22 )1( ss ee

1

2

1

21212 lnln)2ln()2ln()()(

esesXHXH

7

Statement 2. Let we have two systems of continuous random variables

and Then the difference of joint entropies of

system of random variables is equal

),...,,( )1()1(2

)1(1

)1(mYYYY ),...,,( )2(2(

2)2(

1)2(

mYYYY

m

k YYY

YYYm

k Y

Y

kk

kk

k

k

R

RHHH

22

.../

2.../

1

)1()2(

)1(1

)1(1

)1(

)2(1

)2(1

)2(

)1(

)2(

1

1ln

21ln)()()(

YYY

8

where

the coefficients of

determination of the corresponding dependences

2.../.../ )(

1)(

1)()()(

1)(

1)( 1 j

kjj

kj

kj

kjj

k YYYYYYYR

2.../ )(

1)(

1)( j

kjj

k YYYR

mk ...,,3,2 2,1j

9

Denoting

we will present the system as

where ,  it is the

increments of entropy at the expense of changes of dispersions and correlations of random variables .

m

k Y

Y

k

kH1 )1(

)2(

ln)(

ΣY

m

k YYY

YYY

kk

kk

R

RH

22

.../

2.../

)1(1

)1(1

)1(

)2(1

)2(1

)2(

1

1ln

21)( RY

RΣ YYY )()()( HHH

ΣY)(H RY)(H

10

The analysis of entropy-dynamic The analysis of entropy-dynamic model in economicsmodel in economics

 There are following ideas in economics that are underlay in the practical application of entropy-dynamic model: 

Hypothesis : The behavior of system can be considered as stochastic 

Formation of system of signs by means of the factorial analysis

Monitoring of a condition of system in dynamics (analysis of change of entropy)

11

Example 2. Let's consider the list of macroeconomic indicators from the section "Main Socio-economic Indexes of the Russian Federation" annually published by the State committee on statistics of the Russian Federation of collections "Russia in Figures" from 2000 to 2011.

It was established on the basis of the factorial analysis that the initial system can be represented in the form of three factors (main components)

which are explained by 93,2% of all variation of initial signs. Factor: Y1 – national wealth, factor Y2 – deficiency (surplus) of

the budget taking into account a rate of national currency and unemployment rate in the country, factor Y3 – price index of producers of the industry.

),,( 321 YYYY

12

Then, let's carry out the comparative analysis of behavior of a macro system in two periods (before 2005 and after) on the basis of the analysis if entropy of a random vector.

Then we will receive

This result can testify to deterioration of macroeconomic indicators in the second period integrally, caused by the economic crisis in comparison with the fact that the first period was characterized by the growth of economic development of the country.

256,2084,1172,1)()()( RΣ YYY HHH

13

The analysis of change of each of a component shows that the growth of entropy of a randomness was affected mostly by the second element of the system (Y2) and on the increase in entropy of self-organization - weakening of the interaction between components Y1 and Y2.

14

Entropy modeling of dynamics of stochastic system is offered. In it's basis there is a representation of the system in the form of the random vector. Each of vector's components presents a continuous random variable. This approach allows solving problems of monitoring of the condition of stochastic systems in economics.

Entropy - dynamic model doesn't show quantitative change of studied parameters, but gives deeper assessment of influence of this change. For example, if it is known that any average value of a quantitative index went down, then with the help entropy – dynamic the dynamic model could answer whether this decrease was uniform and organized or not.

 Entropy-dynamic model investigates the system fully. Results can be received both on separate elements of the system, and on the whole system that is almost impossible to analyze at a quantitative assessment of indicators of system.

ConclusionsConclusions ::