Interval Estimation of System Dynamics Model Parameters Spivak S.I. – prof., Bashkir State University Kantor O.G. – senior staff scientist, Institute of.

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

x xx

General view of the model with two variables

2

xxdt

dx

4433

2211

yxayxadt

dy

yxayxadt

dx

43

21

(2)

4,1i,,,a iii parameters to be determined

parameter estimates Expansion of equations (2) in a Maclaurin series

3

43

21

aadt

dy

aadt

dx

Stage 1

Stage 2

Expansion of equations (2) in a Taylor series centered at

4,1i,0,0,a ii0i

4044

044

043

033

033

03

2022

022

021

011

011

01

ylnaxlnaaaylnaxlnaaadt

dy

ylnaxlnaaaylnaxlnaaadt

dx

4,1i,a0i

point and interval estimates of the parameters 4,1i,,,a iii

(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

4,1i,,,a iii

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

Advantegesthe 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 criteriathe 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

,экспрасч

j

jjdt

dx

dt

dx

m,1j

the condition that the model describes the observed values, leads to a system of inequalities

– ith measurement error ,jj m,1j

problem of determining the parameters of the system dynamics models can be reduced to solving a series of linear programming problems

iii ,,amin

,j m,1j

0,iii iii

Results:point estimates of the system dynamics models parameters

optimal deviation of the calculated data from the experimental

-

- *

(5)

(6)

(7)

i

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 :

i,,a iii

min

,*j m,1j

,iii iii

m,1j

,iii iii

i,,a iii

max

,*j

Result: interval estimates of the model parameters

0i

0ii ,

0i

0ii ,

the possibility of organizing a numerical experiment to “customize" the model (2)

0i

0ii a,aa

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 yearI - 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

I

DNS

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

YearPopulation of Russian

Federation, pers. (N)

Per capita income , rub./pers. per year

(D)

Consumer price index, share units

(I)

1998 147802133 12122,4 1,844

1999 147539426 19906,8 1,365

2000 146890128 27373,2 1,202

2001 146303611 36744,0 1,186

2002 145649334 47366,4 1,151

2003 144963650 62044,8 1,120

2004 144168205 76923,6 1,117

2005 143474219 97342,8 1,109

2006 142753551 122352,0 1,090

2007 142220968 151232,4 1,119

2008 142008800 179287,2 1,133

2009 141904000 202282,8 1,088

9

hypothesis as a model:

4321 kk2

kk1 SNaSNa

dt

dN

65 k4

k3 IaDa

dt

dD

87 k6

k5 SaIa

dt

dI

Elements software package1. 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)

61 a,,a

61 a,,a

61 a,,a

Requirements1) 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:

1expcalc tNtN 2

expcalc tDtD 3expcalc tItI T,1t

TNTNtTN expcalc

10

3,003,0205,022 SN1,64SN10139,8dt

dN

I9900D560dt

dD 35,0

092,04,0 S0072,0I131,0dt

dI

%74,6AAA2I

2D

2N

Population of Russian Federation, people

January 12010 г.

January 12011 г.

The average annual

- according to the Federal State Statistics Service 142962,4 142914,1 142938,3

- according to the model (9) 142042,8 142670,0 142356,4

Error 919,6 (0,64%) 244,1 (0,17%) 581,9 (0,41%)

(9)

%13,0AN %86,5AI %33,3AD

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

222111 IDNaIDNadt

dN21

444333 IDNaIDNadt

dD43

666555 IDNaIDNadt

dI65

General view of the model:

- parameters to be determined 6,1i,,,,a iiii

530000,0

299552,047

12

*N 0

2a01a

The calculation results for the equation

Point estimations min max

a1-a2 22,03 -14151439,74 23,4

α1 5,00 0,00 5,00

β1 1,02 1,02 5,00

γ1 5,00 -5,00 5,00

α2 1,41 0,11 1,412

β2 0,00 0,00 1,03

γ2 4,06 1,47 4,06

137542,5

222111 IDNaIDNadt

dN21

Additional conditions: 50 1 50 1 50 2 50 2

55 1 55 2

10,0

393,836

12

04a0

3a

The calculation results for the equation

Point estimations min max

a3-a4 -7173,5 -7173,5 -3459,3

α3 0,13 0,00 0,13

β3 0,32 0,32 0,33

γ3 -1,18 -1,21 -1,14

α4 2,00 0,00 2,00

β4 0,00 0,00 2,00

γ4 1,99 -2,00 2,00

1524,5

Additional conditions:

444333 IDNaIDNadt

dD43

*D

20 3 20 3 20 4 20 4

22 3 22 4

100,0

0,063

12

06a0

5a

The calculation results for the equation

Point estimations min max

a5-a6 -686,59 -1559,6 -686,59

α5 0,00 0,00 2,99

β5 0,32 0,32 0,33

γ5 1,70 -1,99 3,00

α6 3,00 0,00 3,00

β6 0,01 0,01 3,00

γ6 1,70 -2,00 3,00

100,23

Additional conditions: 30 5 30 5 30 6 30 6

32 5 32 6

666555 IDNaIDNadt

dI65

*I

Thank you  

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