Interval Estimation of System Dynamics Model Parameters Spivak S.I. – prof., Bashkir State University Kantor O.G. – senior staff scientist, Institute of Social and Economic Research, Ufa Scientific Centre of RAS Salahov I.R. – post graduate student, Bashkir State University 1

System dynamics – method for the study of complex systems with nonlinear feedback Founder –Jay Forrester (professor of the Massachusetts Institute of Technology ) (1) and - positive and negative growth rate of the system level General view of the model with two variables 2 (2) parameters to be determined

parameter estimates Expansion of equations (2) in a Maclaurin series 3 Stage 1 Stage 2 Expansion of equations (2) in a Taylor series centered at point and interval estimates of the parameters (3) (4)

4 The problem of parameter estimation is overdetermined, because the number of observation exceeds the number of parameters characteristically flawed, because initial data is approximate interval estimation of model parameters (founder Kantorovich L.V.) Kantorovich L.V. On some new approaches to computational methods and the processing of observations / / Siberian Mathematical Journal, 1962, vol.3, №5, p Advanteges the possibility of determination the set of the model parameters of a given type, providing a satisfactory quality the possibility of choice from many models of the best according to accepted quality criteria the possibility of full use of available information specific methods are required

5 to verify that the calculated and experimental data agree in the deviation, consider the values the condition that the model describes the observed values, leads to a system of inequalities – i th measurement error problem of determining the parameters of the system dynamics models can be reduced to solving a series of linear programming problems Results : point estimates of the system dynamics models parameters optimal deviation of the calculated data from the experimental - - (5)(5) (6)(6) (7)

6 In general, the point estimates obtained do not guarantee satisfactory results in the numerical integration of (2) It is important to determine the range of the model parameters variation for each model parameter two linear programming are solved : Result : interval estimates of the model parameters the possibility of organizing a numerical experiment to “customize" the model (2)

I I N D N D – system rates – system levels * * – unaccounted factors The system dynamics model of Russian Federation population N – population of RF, pers. D - per capita income, rub./pers. per year I - consumer price index, share units S – auxiliary variable that shows the real cash income, which has the country's population for the year in response to changing prices 7 construction system dynamics models of acceptable precision and calculation of forecasting estimates Purpose:

Initial data for the system dynamics model of Russian Federation population 8 Year Population of Russian Federation, pers. (N) Per capita income, rub./pers. per year (D) Consumer price index, share units (I) ,41, ,81, ,21, ,01, ,41, ,81, ,61, ,81, ,01, ,41, ,21, ,81,088

9 hypothesis as a model: Elements software package 1. The direct problem solution by numerical integration of system (8) with the aid of the Runge-Kutta method. 2. The initial approximation of model parameters chosen through the translation of the differential equations system (8) to integral equations by Simpson’s rule. 3. Determination of variation ranges of the coefficient in which the conditions are adequately described. 4. Defining the parameters that provide the best value optimization criteria. (8) Requirements 1)the unknown parameters of the system dynamics model must provide a given deviation of calculated and experimental data: 2) in all three equations mean error of approximation does not exceed 10% 3) should provide a reasonable change in the forecasting value of N:

10 Population of Russian Federation, people January г. January г. The average annual - according to the Federal State Statistics Service , , ,3 - according to the model (9)142042, , ,4 Error919,6 (0,64%)244,1 (0,17%)581,9 (0,41%) (9) N exp. D exp. I exp. N calc. D calc. I calc.

11 advisable to determine the final form of system dynamics models based on analysis of a database of information relevance of the proposed method for determining the ranges of model parameters variation on the basis of the approach of L.V.Kantorovich General view of the model: - parameters to be determined

530000, , The calculation results for the equation Point estimationsminmax a 1 -a 2 22, ,7423,4 α1α1 5,000,005,00 β1β1 1,02 5,00 γ1γ1 -5,005,00 α2α2 1,410,111,412 β2β2 0,00 1,03 γ2γ2 4,061,474, ,5 Additional conditions:

10,0 393, The calculation results for the equation Point estimationsminmax a 3 -a , ,3 α3α3 0,130,000,13 β3β3 0,320,320,320,33 γ3γ3 -1,18-1,21-1,14 α4α4 2,000,002,00 β4β4 0,00 2,00 γ4γ4 1,99-2,002, ,5 Additional conditions:

100,0 0, The calculation results for the equation Point estimationsminmax a 5 -a , ,6-686,59 α5α5 0,00 2,99 β5β5 0,320,320,320,33 γ5γ5 1,70-1,993,00 α6α6 0,003,00 β6β6 0,01 3,00 γ6γ6 1,70-2,003,00 100,23 Additional conditions:

Thank you 1