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Interval Estimation of System Dynamics Model Parameters

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

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Interval Estimation of System Dynamics Model Parameters

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  1. Interval Estimation of SystemDynamics 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

  2. 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 systemlevel General view of the model with two variables (2) 2 parameters to be determined

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

  4. The problem of parameter estimation is overdetermined, because the number of observation exceeds the number of parameters characteristically flawed, becauseinitial data is approximate specific methodsare required 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. 701-709 Advanteges the possibilityofdetermination the set of the model parameters of a given type, providing a satisfactory quality the possibilityof choice from many models of the best according to accepted quality criteria the possibilityof full use of available information - - 4 -

  5. to verify that the calculated and experimental data agree in the deviation, consider the values (5) the condition that the model describes the observed values, leads to a system of inequalities (6) – ith 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 deviationof the calculated data from the experimental - (7) - 5

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

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

  8. Initial data for the system dynamics model of Russian Federation population 8

  9. (8) hypothesis as a model: • Requirements • 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 exceed10% • 3) should provide a reasonable change in the forecasting valueofN: 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. 9

  10. (9) N exp. D exp. I exp. D calc. I calc. N calc. 10

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

  12. The calculation results for the equation Additional conditions: 299552,047 530000,0 137542,5 12

  13. The calculation results for the equation Additional conditions: 393,836 10,0 1524,5 12

  14. The calculation results for the equation Additional conditions: 0,063 100,0 100,23 12

  15. Thank you 1

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