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Eigen-based Approach for Leverage Parameter
Identification in System Dynamics
Abstract—Eigen-based approaches are useful tools in analyzing
dynamic systems. They can be used to identify the leverage
points that drive the observed behaviour. The structure under
investigation is system parameter. This work distracts from the
traditional focus on the eigenvalue to the behaviour mode weight.
It is known that not only the behaviour mode (e λt ), but also the
weight influences the state behaviour. Further investigation finds
the weight can be decomposed into eigenvectors and system initial
condition. An analytical methodology to compute the weight
elasticity over the parameter is proposed here. An experiment
on the labour-inventory model is performed to both validate the
methodology and render the implications for policy design.
Index Terms—leverage points identification, parameter analysis,
numerical weight analysis
I. Introduction
System dynamics (SD) is a computer-aided approach to
policy analysis and design. The heart of SD lies at exploring
the rule of “structure drives behaviour”. SD approaches have
a series benefits:
1) grasp counterintuitive behaviors;
2) tell a “system story” about policy outcomes, and highlight
the potential leverage points;
3) simplify system archetypes.
Eigen-based approaches including eigenvalues and eigenvector
analysis are widely used in the SD research (see [1]). Eigenvector
analysis has started to attract more attention recently. [2]
first proposed its application in identifying dominant feedback
loops. A numerical weight analysis was carried out by [3] to
show the leverage structure points.
II. Methodologyofweightanalysis
We propose an analytical methodology of weight analysis
with respect to the system parameters. It is carried out in the
following steps.
1) For an n-order system, the behavior of the state variable
xi is expressed by Eq. (1).
xi(t)=e tλ1 r1iℓ H
1 x(0)
+etλ2 r2iℓ H
2 x(0)
Jinjing Huang, Enda Howley and Jim Duggan
+...+ etλn rniℓ H
n x(0)
(1)
where t is time,λis eigenvalue, r andℓare right and
left eigenvectors, and x(0) is system initial conditions.
The term under brace is weight.
2) Computing the weight elasticity (ε) with respect to a
parameter p in Eq. (2).
j X0
ε w ∂w ji/w ji
ji =
∂p/p =∂ r jiℓH p
∂p w ji
⎛
= ⎜⎝ ∂r ji
∂p ℓHj
+ r ∂ℓ
ji
H⎞
j
⎟⎠ X0∗
∂p
p
w ji
(2)
153
3) The weight elasticity is decomposed into the calculation
of eigenvector sensitivity over the parameter:
∂r ji/∂p,∂ℓ H
j /∂p, which is solved by [4].
III. Experiment Results
The labour-inventory model is shown in the diagram below.
Part of the analysis results are presented in Table I. It confirms
Fig. 1. Stock and flow diagram of the labor-inventory model
outcomes from both methods match each other.
Parameter
Inventory Labor
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IAT -8.7623 -7.661 -8.6363 -7.6334
WIPAT 6.2983 6.0812 6.0653 5.7435
MCT -23.697 -18.305 -22.034 -19.699
SWW 15.13 15.131 14.13 12.857
PRO 15.13 15.131 14.13 12.857
VAT 2.9925 3.2873 1.2854 1.312
LAT 2.0097 1.7556 2.248 2.0165
ATTFV -4.5658 -5.4077 -5.5783 -5.8229
TABLE I
Partialresultsofanalytical (1stcol.)andnumerical (2ndcol.)weight
elasticitytoparametersassociatedwith 1stmode
References
[1] M. Saleh, R. Oliva, C. E.Kampmann, and P. I.Davidsen, “A comprehensive
eigenvalue analysis of system dynamics models.” The 23rd
International Conference of the System Dynamics Society, 2005.
[2] P. Goncalves, “Eigenvalue and eigenvector analysis of linear dynamic
systems.” The 24th International Conference of the System Dynamics
Society, 2006.
[3] M. Saleh, R. Oliva, C. E.Kampmann, and P. I.Davidsen, “Eigenvalue
analysis of system dynamics models: Another perspective.” The 24rd
International Conference of the System Dynamics Society, 2006.
[4] J. Huang, E. Howley, and J. Duggan, “An eigenvector approach for
analysing linear feedback systems.” The 2010 International Conference
of the System Dynamics Society, 2010.
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