Ability that it would change to face g was found, over prolonged testing, to
Be:
g
Fig. 9/6/1
It will be seen that the populations tend, through dying oscilla-
Tions, to a state of equilibrium, at (44.9, 42.9, 12.2), at which the
System will remain indefinitely. Here “the system” means, of
Course, these three variables.
It is worth noticing that when the system has settled down, and
Is practically at its terminal populations, there will be a sharp con-
Trast between the populations, which are unchanging, and the
Insects, which are moving incessantly. The same pond can thus
Provide two very different meanings to the one word “system”.
(“Equilibrium” here corresponds to what the physicist calls a
“steady state”.)
168
1
2
3
4
5
6
↓
f
1
0.1
0.1
0.5
0.1
0.1
0.1
2
0.1
0.1
0.5
0.1
0.1
0.1
3
0.1
0.1
0.5
0.1
0.1
0.1
4
0.1
0.1
0.5
0.1
0.1
0.1
5
0.1
0.1
0.5
0.1
0.1
0.1
6
0.1
0.1
0.5
0.1
0.1
0.1
Which is x? (Hint: Beware!)
Ex. 4: A compound AB is dissolved in water. In each small interval of time each
molecule has a 1% chance of dissociating, and each dissociated A has an
0.1% chance of becoming combined again. What is the matrix of transition
Probabilities of a molecule, the two states being “dissociated” and “not dis-
Sociated”? (Hint: Can the number of B’s dissociated be ignored ?)
Ex. 5: (Continued.) What is the equilibrial value of the percentage dissociated?
Ex. 6: Write out the transformations of (i) the individual insect’s transitions and
Ii) the population’s transitions. How are they related ?
Ex. 7: How many states appear in the insect’s transitions? How many in the sys-
Tem of populations ?
169
A N I N T R O D UC T I O N T O C Y B E R NE T I C S
I N CESSA N T TR AN SMI SSIO N
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*Ex. 8: If D is the column vector of the populations in the various states, D' the
Vector one step later, and M the matrix of transition probabilities, show that,
In ordinary matrix algebra,
D'=MD, D"=M2D, and D(n) = MnD.
This simple and natural relation is lost if the matrix is written in transposed
Form. Compare Ex. 2/16/3 and 12/8/4.)
These dependencies are characteristic in language, which con-
Tains many of them. They range from the simple linkages of the
Type just mentioned to the long range linkages that make the end-
ing “… of Kantian transcendentalism” more probable in a book
that starts “The university of the eighteenth century…” than in
one that starts “The modern racehorse …”.
Ex.: How are the four transitions C → C, C → D, D → C, and D → D affected
In frequency of occurrence by the state that immediately preceded each oper-
And, in the protocol:
DDCCDCCDDCCDCCDDCCDCCDDCCDDDDDDDDC
C D D D C C D C C D C?
Hint: Classify the observed transitions.)
Dependence on earlier values. The definition of a Markov
Chain, given in S.9/4, omitted an important qualification: the
Probabilities of transition must not depend on states earlier than
The operand. Thus if the insect behaves as a Markov chain it will
be found that when on the bank it will go to the water in 75% of
The cases, whether before being on the bank it was at bank, water,
Or pebbles. One would test the fact experimentally by collecting
The three corresponding percentages and then seeing if they were
all equal at 75%.
Here is a protocol in which the independence does not hold:
AABBABBAABBABBABBABBAABBABBABABA
The transitions, on a direct count, are
↓
A
B
A 3 10
B 10 8
In particular we notice that B is followed by A and B about
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Equally. If we now re-classify these 18 transitions from B accord-
Ing to what letter preceded the B we get:
… AB was followed by A: 2 times
B: 8 ,,
… BB ,,,,,, A: 8 ,,
B: 0 ,,
So what state follows B depends markedly on what state came
Before the B. Thus this sequence is not a Markov chain. Some-
Times the fact can be described in metaphor by saying that the sys-
Tem’s “memory” extends back for more than one state (compare
S.6/21).
This dependence of the probability on what came earlier is a
Marked characteristic of the sequences of letters given by a lan-
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