Transcript Document

FUR XII 2006 at LUISS in
Roma
A METHODOLOGY
FOR QUALITATIVE
MODELLING OF
ECONOMIC,
FINANCIAL SYSTEMS
WITH EMPHASIS TO
CHAOS THEORY
Tomas Vicha, Mirko Dohnal
Faculty of Business and Management,
Brno University of Technology,
Czech Republic
[email protected]
Introduction to Common Sense
•
•
•
•
The classical quantitative (i.e. analytical or statistical)
analysis is not appropriate for some ill-known,
difficult to measure, very complex and vague
engineering problems
Low information intensity of the qualitative models
offers the highest chance of success in development
of a realistic i.e. applicable model.
Very often we need information such as variables will
be rise or not – trend analysis.
Common-sense models (naive physics, qualitative
model) offer a very flexible formal tool to deal with
realistic engineering tasks.
Qualitative Modelling of Chaos
• Identification of all possible chaotic behaviours
which are qualitatively different can be done by brute
force.
• It means that all possible combinations of numerical
values of all constants used in a model must be
studied.
• Qualitative models can identify not only all
qualitative behaviours, but all possible transitions
among them.
• Complete sets of qualitative behaviours can, to
certain level, characterize traditional quantitative
phase portraits (more precisely state/scenario graph).
Qualitative Algebra – Intro 1.
• A Qualitative value is + … increasing ,positive (k+)
0 … constant, zero (k0)
- … decreasing, negative (k-)
• A qualitative solution is specified if all its n qualitative
variables: x1, x2, ..., xn are described in qualitative triplets:
(X1, DX1, DDX1), (X2, DX2, DDX2), … (Xn, DXn, DDXn),
where Xi is the i-th variable and DXi and DDXi are the first
qualitative and second qualitative time derivations
• A qualitative model has m qualitative solutions. The j-th
qualitative state is the n-triplet:
• (X1, DX1, DDX1), (X2, DX2, DDX2),… (Xn, DXn, DDXn)j,
where j = 1, 2, …, m.
Qualitative Algebra – Intro 2.
Qualitative addiction
+
k+
Xi ko
k-
Xj
k+ ko
k+ k+
k+ ko
? k-
Qualitative multiplication
*
k?
kk-
Xi + Xj = Xs
DXi + DXj = DXs
DDXi + DDXj = DDXs
Xi
k+
k+ k+
ko ko
k- k-
Xj
ko
ko
ko
ko
kkko
k+
Xi * Xj = Xs
Xi * DXi + Xj * DXj = DXs
Methodology
1. Selection of a real-word model
2. Transformation of quantitative equations to
qualitative equations
3. Assignment of follow up conditions
4. Transformation of qualitative equations to
instruction set of a mathematical program –
e.g. Q-SENECA
5. Definition of triplets of a qualitative variable
6. Make output questions
7. Computation of scenarios and transitions
8. Interpretation of result
M. 1) Selection of model
• Damped Oscillation
d 2x
F  kx  bv  m a  m 2
dt
d 2 x b dx k

 x0
2
dt
m dt m
• Undamped Oscillation
d 2x k
 x0
2
m
dt
m … mass on a spring
k … spring constant
-bv … damping force
x … position
v … velocity (dx/dt)
a … acceleration (dv/dt)
M. 2) Transform. to Qualitative Eq.
• Ignore multiplicative quantitative constants
• Additive constants can be solved by adding a
new qualitative variable
Damped Oscillation
Undamped Oscillation
DDX + DX + X = 0
DDX + X = 0
M. 3) Follow up Conditions
• A relation between variables of a model is
known. Thereafter complexity of the model can
be reduced by assignment a follow up condition
between variables.
• It is not important for these models. If necessary,
we could describe a relation of variables by the
instruction set of Q-SENECA.
• For example if two variables have linear
relationship than we can use instruction:
1 23 VAR1 VAR2 0
M. 4) Transform. to Instruction Set
Damped Oscillation
Undamped Oscillation
1 d/t PX DPX 0
2 d/t DPX DDPX 0
3 ADD DDX DX P1
4 ADD P1 X ZERO
1 d/t X DX 0
2 d/t DX DDX 0
3 ADD DDX X ZERO
• Note; 0 is a quantitative constant and that is why we have
to replace it with an auxiliary variable ZERO.
• Operations addition (ADD) and multiplication (MUL)
need two parameters. That’s why we need the auxiliary
variable P1 for the addition of DDX + DX + X.
M. 5) Definition of Triplets
• We can restrict values of triplet (X, DX, DDX) or forbid
computation of values of triplet (X, DX, DDX). The value
of triplet is presented by the symbols from the set {+, -,
0, X (=count), * (=don’t count)}.
DOTAZ
ZERO 000
X XXX
• The program will count with value of variable X and its
first and second derivation … whole triplet of X
M. 6) Make Output Questions
• We should set which variable(s) is(are)
necessary for output.
• For damped oscillation we need to
compute outputs for X and DX.
# X DX
M. 7) Computation of Scenarios
Damped Oscillation
State No. X
1 + + 2 + + 3 + + 4 + 0 5 + - +
6 + - +
7 + - +
8 + - 0
9 + - 10 0 + 11 0 + 12 0 + 13 0 0 0
14 0 - +
15 0 - +
16 0 - +
17 - + +
18 - + 0
19 - + 20 - + 21 - + 22 - 0 +
23 - - +
24 - - +
25 - - +
DX
+ - +
+ - 0
+ - 0 - +
- + +
- + 0
- + - 0 +
- - +
+ - +
+ - 0
+ - 0 0 0
- + +
- + 0
- + + + + 0 + - +
+ - 0
+ - 0 + - + +
- + 0
- + -
Undamped Oscillation
State No. X
v = DX
1 + + + - 2 + 0 0 - 0
3 + - - - +
4 0 + 0
+ 0 5 0 0 0
0 0 0
6 0 - 0
- 0 +
7 - + +
+ + 8 - 0 +
0 + 0
9 - - +
- + +
a = DDX
- - +
- 0 +
- + +
0 - 0
0 0 0
0 + 0
+ - + 0 + + -
Damped Oscillation
DDX + DX + X = 0
Undamped Oscillation
DDX + X = 0
M. 7) Computation of Transitions
• Information as:
from: scenario no. to: scenario no.
Damped
Undamped
M. 8) Interpretation of Result
• The interpretation depends on the purpose of our
modelling.
• For example undamped oscillation has 9 scenarios. 8 are
accessible and 1 is only for an extreme situation (no
input energy – no oscillation, isolated scenario). All of 8
scenarios can be starting or terminating state/scenario.
For each scenario is only one successor.
• In this point it is good to compare qualitative results with
quantitative system, see picture on the next slide.
• Note: qualitative values of variable X / position (or DX /
velocity) are described above (or under) time axis.
M. 8) Interpretation of Result
Predator Prey – Lotka Volterra
• Predator prey – Lotka Voltera model (PPLV) is one of the
oldest known chaotic models and was first used in
theoretical biology by Lotka (1925) and Volterra (1931)
• The basic non-linear differential equations concerning
this mathematical approach are
x  x  xy
y   y  xy
• where x is the prey, y the predator, α, β, γ and δ are all
positive constants and the prime (') refers to x and y time
derivatives.
PPLV Phase Space Picture
1) model A, B
parameters:
α=1, β=1,
γ=1, δ=1
1)
Time space portrait
Phase space portrait
2)
2) model C, D
parameters:
α=1, β=0.2,
γ=1, δ=0.5
PPLV – Economic Using
d (K )
 K   ( K )(GDP)
dt
d (GDP)
  (GDP)   ( K )(GDP)
dt
• Where K is the prey or productive capacity or fixed
capital stock and GDP the predator or gross domestic
product (GDP) or simply production (by Kumimura ).
• An approach to the economical equilibrium theory has
also been applied by Goodwin to another different
context, that is, the employment capacity being the prey
and employees salary being the predator.
PPLV Qualitative Analysis
• The equations can be qualitatively rewritten as:
DX + X*Y = X
DY + Y = X*Y
• PPLV qualitative computation is made for two settings:
general setting (XXX) and positive settings of variables
X, Y (+XX, see methodology chapter). The results are
shown in the table. Model II. is illustrated by the graph
of transitions, see next page.
Model
I. PPLV general
II. PPLV positive
Values of X,Y No. Scenarios Theoretical No. Sc. Transitions
XXX
75
729
291
+XX
41
81
168
PPLV State/scenario graph
Source Code for Q-Seneca
Qualitative equations of
PPLV:
DX + X*Y = X
DY + Y = X*Y
1 d/t X DX 0
2 d/t Y DY 0
3 MUL X Y P1
4 ADD DX P1 X
5 ADD DY Y P1
DOTAZ
X +XX
Y +XX
#XY
Conclusion
• A result analysis is set of scenarios showing us general
rules about a model. Moreover, set of qualitative scenarios
and consequently qualitative transitions is a super set of
existing solutions, because of potential spurious solutions.
• The qualitative forecasting itself is very flexible.
• Possible to perform any union or intersection of different
models. Useful feature for verification of models and their
simplification. This aspect is important for development of
classical quantitative prediction models.
• The classical statistical approach to model development is
always based on a (semi) subjective choice of
approximation curves.
• The total number of qualitative scenarios is usually high.
Multidimensional problems can have more than 10 000
scenarios.
Thank you for your attention
A METHODOLOGY
FOR QUALITATIVE
MODELLING OF
ECONOMIC,
FINANCIAL SYSTEMS
WITH EMPHASIS TO
CHAOS THEORY
Tomas Vicha, Mirko Dohnal
Faculty of Business and Management,
Brno University of Technology,
Czech Republic
[email protected]