A STUDY OF SENSITIVE TEST
Professor Ching T. Yang
Shanghai Jiao Tong and Fudan University
Ansheng Cao
Management School of Fudan University
ABSTRACT. It is necessary to test a newly built model to make
sure that it does simulate the actual system. The test may include
many aspects, such as sensitive test, comparison.test with history
data, etc. This paper studies sensitive test and presents some re-
sult of the findings.
In studying the structure of a given system, the endogenous varia-
bles and exogenous variables are given explicitly. If we can not
tell the difference between them correctly,it may confuse the real
meaning of the sensitive test.
In the present analysis we find that the conditions for non-sensi-
tivity of variables to a model are: 1) the change is made for one
variable only and 2) the change is small. The conditions for sen-
sitivity of variables to a model are: 1) two or more variables
change at the same time and 2) the change is great. A-model may
also become sensitive for a small change of the exogenous variable.
The above results have been thoroughly investigated and the me-
thods of sensitive test are presented,
The system always changes from one stable condition to another
stable condition in which it is non-sensitive to. variable changes,
put if the system is in a state between two stable conditions, it
is completely possible to become sensitive to. variable changes, So
we may ~oint out that the model builder and analyst are reauired
to find out the correct boundary of stable conditions,
Finally, the simple economic long wave model is tested to confirm
the above results,
Recent evidence (System Dynamics Review vol.4,No.1-2,1988) points
chaotic modes do exist in the range, of models commonly explored
by most students and practitioners of System Dynamics. Experiments
also demonstrated that chaos can in fact be produced by de i
making processes of real people. Chaos thus appears to be
mode of behavior not cnly in physical systems, but also in social
and economic systems. However, chaos is a steady-stzte pheononena
over very long time frame and many policy-oriented models are con-
cerned with transient dynamics with time horizon much shorter than
those used in chaotic dynamics, Over such extended time horizons
the parameters of the system can't be considered static, but will
themselves evolve with learning and evolutionary processes.
With such a viewpoint in mind we have explored a study of sensitive
test in the following way.
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I. The basic criterion of sensitive test
Which svstem is called stable or unstah’e svstem ?-A stable svstem
is that its -ronertw (behavior) is re ular and varies with the re-
gularity. An unstable svstem is that its property can't be found
regular. In fig.1, we suppose curve 1 stands for the behavior of
system 1, curve 2 for system 2. We can see the system 1 is stable
because the variation of curve 1 is regular (periodic oscilation).
System 2 is unstable.
Curve 2
Fig 1: curve 1 for system 1, curve 2 for system 2
We have learned that a model to be effective must be non-sensi-
tive to its endogenous variables without changes of exogenous va-
riables. This principle must be looked for in the characteristics
‘of a real system. In so doing we consider the system to be dealt
with must be non-linear either in a stable or unstable state.That
means the system is either in a steady-state pheonomena, or vary-
ing with transient dynamics tending to be stable on its time ho-
rizon. We can't imagine that a system is in a position of unstable
state for ever with no changes of exogenous variables. So in the
following analysis we shall discuss on the premise that a system
itself has always a tendency to become stable without changes of
exogenous variables.
By endogenous variables we mean those variables being determined by
the inner structure of a system and‘by exogenous variables we mean
those variables not included in the inner system structure. They both
have effects on the behavior of the system but of different nature.
The former is non-sensitive to the system, which the latter is tending
to change the system from stable state to unstable state.
That a system has a tendency to be stable doesn't mean the system
doesn't change. In the real system, exogenous variables always
changes.and these changes will cause the behavior of the system
changes slightly or violently. So the system in the real world is
always changing in process of being stable-unstable-stable... and
so on, Now we may conclude that the only way for a stable system
to become unstable is due to the variation of exogenous variables
and a system is non-sensitive to its endogenous variables.
In the following analysis, let us call endogenous variable a vari-
able and exogenous variable a parameter for simplicity in writing.
302
Il, Three principles of sensitive test
We present here three principles for sensitive test giving clear
indications for a stable system.
Principle 1: A tiny variation of a variable can't cause the system
behavior to change greatly, A system is always non-
sensitive to its variable.
The charicteristic of a system having a tendency to be stable as-
sures that principle 1 is tenable. If a model reflects a system
correctly but is sensitive to a variable, we can be sure that the
system is in a position of unstable state.
Principle 2: If two or more variables make tiny variations at the
same time, it may cause a.system behavior to change
greatly.
When several variables make tiny variations at the same time and
if these variables' effects offset each other, it may have little
effect on the behavior of a system which is non-sensitive. But if
these variables' effects amplify each other, the behavior of the
system which is sensitive may have ‘great changes. So we can't
guarantee a system being Mon-sensitive to tiny variations of seve-
ral variables at the same time.
Principle 3: It may cause the behavior of a system to change great-
ly if a variable has a great change.
A system is non-sensitive to a tiny variation of a variable, but
if the variation becomes great enough, it may certainly make the
system to change greatly.
III, Sensitive test and policy analysis
Sensitive test and policy analysis both examine the behavior of a
system by changes of variable and parameter, But the former use en-
dogenous variables and the latter use exogenous variables for the
test and analysis, respectively.
A parameter's effect on a system is witnessed by variables, so we
can discuss policy analysis more easily with the help of sensitive
est.
If change of a parameter only makes a variable tiny change, we can
see the system must be non-sensitive to the parameter according to
principle 1; If change of a parameter makes a variable great change
or makes several variables change together, the system may be sen-
sitive according to principle 2 and 3.
We have know that a system always changes from a stable state to an
unstable state, there must exist a boundary for a variable within
which the system, will be stable no matter how the variable changes.
A good model builder will not only examine the sensitivity of the
system, but also find the stable boundary of variables. If he (or
she) can make it, certainly it's helpful for a decision maker to get
more correct decisions.
IX. Sensitive test for a simplified economic long wave model
Fig.2 shows a flow diagram of the said model and Fig 3 shows a
simulation of it ( equations are in appendix). Where ALC, DPG, DDC,
303
COR are parameters and the others are variables. Through the test we
find the model is non-sensitive to variables.
Fig.2 S.D. flow diagram of said model
1PC 2 UOC
3} 60.000
3} 30.000
3} 18.000
3} 0.0 N 4
a , ? r : ; r 5
00 75.000 ~—*¥50.000. ~~ 225.000 300.000
Time
Fig.3 Simulation result for said model
ALC=20, DDC=3, COR=6
304
Fig.4 shows the result of test with variable PO amplified 1.1 times
as Fig.3 (Now-PO=(PC/COR)*1.1, compared with PO equation in appendix).
Compared with Fig.3, we see that the original behavior of the model
doesn't change (periodic oscilation), If we let CAR be amplified 1.1
times as Fig.2 and DR 0.9 times respectively, the results are similar
to Fig.3. So the model is non-sensitive to the variables.
1 PC 2 voc
3} 60.000 4
4) 48.000
4} 30.000
4} 18.000 Sr A
4} 00 ? , ? 7 ; T #: ‘
00 75.000 150.000 225,000 300.000
Time
Fig.4: amplifying PO 1.1 times
Now let the two variables CAR,DR ‘change at the same time, with CAR
amplified 1,1 time as Fig.3 and DR 0.9 times. The result is shown in
Fig.5. The behavior of the model is changed from a periodic oscila-
tion to an approximate steady line. It means that the model is sen-
sitive to the two variables changing at the same time. This is in
accordance with principle 2.
1 Pe : 2 vO
3} 60,000
3} 45.000 4
4} 30.000
4} 15.000 4
\ ' A ‘ 1
1 a B # — #
00 Sener 160000 295.000 300,000
Tine
Fig.5: amplifying CAR 1.1 times and DR 0.9 times
Fig.S: amplifying CAR 1.1 times and DR 0.9 times
305
From Fig.4, we see that the model is non-sensitive to the tiny varia-
tion of PO, but if PO changes greatly, say amplified i.5 times origi-
nal equation, the result is similar to Fig.5, The model exhibits sen-
sitivity to the variable PO great change. This is in accordance with
principle 3.
Now suppose our aim is to make the behavior exhibit an approximate
line. This aim can easily be reached by taking PO 1,5 times the ori-
ginal equation. In eq.PO.K=PC.K/COR, the aim can also be reached by
Changing parameter COR from 6 yr to 4 yr. That means if capital-out-
put ratio could be made 4 yr, the economic long wave may disappear.
If we lower the variable DR(depreciation rate), the péak of the pe-
riodic oscilation will become small What does it mean ? From eq.
DR.KL=PC.K/ALC, we can see that lowering DR means enlarging ALC (
average lifetime of capital). That is to say, if ALC is enlarged,
the economic wave oscilation could be make smaller, But in the pre-
sent real world the eliminating period of capital is, in fact, becom-
ing shorter and shorter. It will certainly cause more periodic osci-
lations of economy in the world.
From above analysis, it is quite clear that the study of sensitive
test as outlined in this paper is very useful and helpful to those
who are involved in sensitive test, policy analysis and decision
making. The authors hope that after being familiarised with the new-
ly published issues of chaos, the present study may be further. ela-
borated at a later date.
Reference
1, “INTRODUCTION TO SYSTEM DYNAMICS MODELING WITH DYNAMO"
George P.Richardson, Alexander L.Pugh III
2, "INDUSTRIAL DYNAMICS" Jay W. Forrester
3. "BIFURCATION AND CHAOTIC BEHAVIOR IN A SIMPLE MODEL OF THE
ECONOMIC LONG WAVE"
Steen Rasmussen, Erik Mosekilde, and John D, Sterman
4. "SYSTEM DYNAMICS REVIEW" vol.4, No.1-2,1988.
306
Apendix: the equation of a simplified economic long wave model
L
z
Par aawrazer
> a> be > ww
PC.K=PC.J+DT* (CAR, JK-DR, JK) production capital
PC=50
UOC .K=U0C.J+DT* (ORC.JK-CAR.JK) unfilled orders for capital
uoc=10
ALC=20 average lifetime of capital
CAR. KL=PR.K-DPG capital acquisition rate
COR=6 capital / output ratio
DDC=3 desired delivery delay
DP. K=UOC.K/DDC+DPG desired production
DPG=1 desired production of goods
DPO.K=DP.K/PO.K desired production/potential
output
DR.KL=PC.K/ALC depreciation rate
ORC.KL=(PC.K/ALC) *MDP.K order rate for capital
PO.K=PC.K/COR potential output
PR.K=PO*CUF.K production rate
CUF.K=TABLE(TCUF,DPO.K,0,0.2,2) capital utilization factor
TCUF=0/0.2/0.4/0.6/0.8/1.0/1.1/1.15/1.18/1.19/1.2
MDP.K=TABLE(TMDP,DPO.K,0,0.2,2) multiplier from desired produc-
tion
TMDP=0/0.1/0.2/0.3/0.5/1.0/2/3/3.5/3.9/4
