Chapter 1 Introduction
3.4 Dynamic Behaviors in Stochastic Systems
In the next three sections, engineering systems will be modeled as systems of random variables, and their interactions can be quantified by mutual information. Therefore, their modularity can be measured in the framework established in the previous section. This section discusses the modularity of dynamic behaviors in stochastic systems.
Usually, engineering products can be modeled as stochastic systems since they have stochastic behaviors, stochastic input (including random noises) from random working environments. From dynamic views, design products can be viewed as mappings, which map inputs from environments and initial states to outputs. For any subsystem of a product, the mapping can also be viewed as one which maps inputs from environments and other subsystems and its initial states to outputs to environments and other subsystems. Let us begin with two simple dynamic subsystems, shown in Figure 3.8. VectorUi, i= 1,2 represent the inputs,Vi, i= 1,2 represent the outputs of subsystems to the outside environment, andSi, i= 1,2,represent the state status of subsystemi. Wij represents the outputs from subsystemito subsystemj.
The simple way to quantify the couplings between dynamic behaviors of two subsystems is degree of freedom (DOF). Given the states of one subsystem, the number of independent states of the other subsystem (DOF) can briefly tell the strength of couplings between the two subsystems.
The shortcomings of DOF are 1) they only have integer values, that is, they are 0,1 functions and therefore can not quantify the situations between totally independent and fully dependent; 2)usually they are limited to rigid body systems. More general measures can be defined on the basis of real physical interaction strength such as power and forces between different modules. However, it is not a good way by the following four arguments. First, the same value probably has a different significance for different physical properties. Second, physical properties have units and different properties have different units. For example, force can use Newton, while distance uses meter. Even one physical property can use different scale units, such as meter and millimeter for distance. It is difficult to
U
*V
*S
+U
,V
,S
-f
.W
*,f
/W
,*Figure 3.8: A model of two-way clustered stochastic systems
compare different physical values with different units. Third, in some systems, especially dynamic system, very small real values of a physical interaction can dramatically affect the behaviors of the systems [109]. That is, the real value of a physical interaction is not necessary a good indicator of interaction between two subsystems from the view of modularity. The last point is that sometimes the physical types of interactions are not very clear. For example, it is difficult to tell what the physical interaction is between the two particles in the example shown in Figure 3.8 in this section.
The dynamical states of one system can be viewed as random variables. Based on the discussion in the previous section, interactions can be viewed as information flow, which can be quantified by mutual information. Mutual information has no physical units, is applicable to any system which has uncertainty (randomness) and has nice physical semantic meaning. From the view of information flow, if there are strong couplings between two systems, one system’s states can be very much inferred from the other system’s states, and therefore there is large mutual information between them.
Systems of random variables are relatively simple since the interactions between different units are symmetric and non-directional, and there are no inputs from the outside environment or outputs to the environments, i.e., they are closed systems. The physical stochastic systems are more com- plex and could be open, that is, there are possible inputs from outside environments and outputs to the environments. Observers in stochastic systems care how the states of one subsystem are affected by the states of the other subsystem and the inputs to the other subsystem from the outside environments. Considering the interactions between systems and environments, the interaction of
A
B
l
L
L
(x A
, y A
) (x
B , y
B )
A
B
Figure 3.9: Two particles on a lattice.
subsystemi= 1,2 on subsystem j= 1,2 can be defined as
Cij =I(Si:Sj, Uj) =H(Si)−H(Si|(Sj, Uj)), i, j= 1,2, i6=j. (3.29)
According to this definition, interactions in physical stochastic systems are asymmetric and direc- tional. That is,Cij is not necessarily equal toCji.
By replacing I(S1 :S2) and I(S2 :S1) byI(S1 :S2, U2) and I(S2:S1, U1), respectively, in the framework to calculate the modularity a system of random variables, the same framework can also be used to calculate the modularity of dynamic behaviors of stochastic systems.
Let us consider the following simple example, shown in Figure 3.9. Suppose there are two particles AandBrandomly moving around in a two-dimensional lattice, and their locations are limited inside the square: 0≤xi≤100,0≤yi≤100. There is a soft link of lengthl connecting the two particles.
Suppose only locations of the two particles, denoted as (XA, YB) and (XB, YB), are concerned. The question is how strong the interaction between the two particles is. One extreme case is when l is larger than the diagonal length of the square. In this case, there is no interaction since the position of particleAdoes not affect that of particleBand vice versa. The other extreme case is whenl= 0.
In this case, the position of particleB can immediately give that of particleAand vice versa. That is, the two particles have the strongest interaction. How then strong is the interaction under the situation between the two extreme cases?
Assume the prior probability distribution of (XA, YA, XB, YB) is uniform over (1,· · · , N)4. The posterior probability distribution of (XA, YA, XB, YB) can then be calculated according to the fol-
0 20 40 60
80 100
0 20 40 60 80 100
8 8.5 9 9.5 10 10.5
x 10−5
Y X
Probability
Figure 3.10: Posterior probability of (XA, YA).
lowing constraint:
P(xA, yA, xB, yB) = 0 ifp
(xA−xB)2+ (yA−yB)2> l.
The result of the case wherel= 100 is shown in Figure 3.10. Then,
H(S1) =
N
X
i=1 N
X
j=1
P(XA=i, YA=j) log 1
P(XA=i, YA=j)= 13.287,
and in this simple example, there are no inputs from outside the system, i.e.,Ui doesn’t appear in the calculation of the couplingCij. We have
H(S1|S2) =
N
X
i=1 N
X
j=1
P(XB =i, YB=j)H(S1|XB=i, YB =j) = 13.252.
By symmetry, the couplings
C21=C12=H(S1)−H(S1|S2) = 0.0343
With the length of link varying from 0 to 500, the change of coupling is shown in Figure 3.11.
It is shown in the figure that the coupling decreases exponentially as the length of the link increase, which is consist with our intuition.
0 30 60 90 120 150 0
2 4 6 8 10 12 14
Link length (l) I(S1:S2)
Figure 3.11: Mutual information vs. the length of link connecting the two particles.