number. The probability of the outcome Ej at the n th
trial is:
(n)
P{X,= 1} =P,
FREE Il
so that the initial distribution is given by pi(®, If we
make Xn - 1= i and Xn = j, then the system has made
a transition Ei Ej at the n th trial, We now need to
know the probabilities of the various transitions which
may occur in the system, If the trials are independent
we can use equation (1); but if the trials are not
independent we must write:
P {X = X%-1 SS Ma" hr een Ba a} „.++(2)
Thus, a Markov chain is a sequence of consequent trials
such that:
P{X.= i Ba Zn X =a}=p[X =] Kae } (3)
We are often given an initial distribution and the
transition probabilities, and wish to determine the
probable distribution for each random variable Xn. The
Transition probabilities may be handled in matrices, for
which we write; P = Pi ; that is,
Pe {Fir Pız)
Pay Piz Page
Pay Paz“ SPogier
Pag Paz AP3g
A 0)
This is calleda transition matrix, and
all columns sum to unity. It is possible to calculate the
transition probabilities from existing data through
compiling a series of column vectors from which the
distribution at the first stage and the probability of
transitions can be derived, A Markov chain may
pass through a series of states which sometimes lead to a
steady condition which may be absorbing or ergodic,
The above description is very brief, and the subject is
much more complicated, but it does outline the
approach sufficiently for us to move on to a brief
discussion of our own model, We begin by suggesting
that, for each of our 500 metre cells, k, there are n
states of newly cqstructed office space in the t th
interval. Thus, k ij is defined by;
Ki = pl = ilk = i}
AS)
Thus each cell k has an associated N x N matrix of
transition probabilities {xPi;.}
We could compile similar matrices for change of use and
for demolition oß office space. The transition
probabilities, k ij, depend only upon the initial state
and so our model has some of the properties of a Markov
process, As our model stands it is not a full Markov
process because we have not followed our system through
to a stable state. However, it should be possible, by
means of an absorbing process, to follow the flow of
floor space through states which are defined in terms of
age and type of use, to a final state of demolition. It
might provide a way, even if we were only able to
guess at some of the missing parameters, to test some
interesting hypotheses, For example, to the extent that
planning controls and policies are effective, transition
probabilities ought to differ among groups of areas
defined by public policies. Thus the probabilities
limiting office uses and demolition should be higher in
an area in which the authority generally permits new
office construction than in areas where this is not
permitted, if and only if there is a real demand for
office space where the restrictive policy is in effect.
Unfortunately our resources did not permit us to move on
to testing such ideas, but the act of constructing the
model itself lead us to believe that this kind of
approach avoids some of the pitfalls of more
behavioural type modelling and that it has considerable
potential as a powerful tool in urban analysis.
However, as it stands our quasi-Markov approach does
not offer a very full explanation of the phenomena
which it considers. Explanation enters into the testing
of hypotheses through Markovian analysis and to some
extent in assigning transition probabilities, but we felt
that an alternative was needed,
We adopted a more "traditional" approach to our
second model, the Monte Carlo simulation. At the
current stage of urban model building, simulation
continues to offer a number of advantages. As in all
model building it imposes rigorous description upon the
modeller, and it possesses some flexibility for later
modification and addition. Simulation does not provide
the kind of precision which is available in more formal
mathematical treatment but it has made considerable
contributions to urban studies in the past and there were
good reasons for us to adopt such an approach in our
own work,
As with the Markov type model our problem was to
allocate office space to cells over a period of time. We
were in possession of complete data concerning the
amount of office space in each cell in 1957 and in 1962
Thus, given the 1957 situation we hoped to generate the
position for 1962, and compare our simulated pattern
with reality. There were seven stages in the simulation
as follows: First, given the total number of planning
permissions as inputs of office space in terms of new
buildings, change of use and extensions in period t,
estimate the values for period ++ 1%. Second, derive
lists of actual units of office space, from empirically
derived size distributions of the three types of inputs.
Third, allocate each of these individual units of office
space in turn to some cell in the region and up-date the
cell’s stock. Fourth, repeat stages one to three for time
t+2., Fifth, repeat this process until the estimate for
1962 is generated, Sixth, recycle the process until a
distribution of outcomes for 1962 has been generated,
Seventh, compare the most likely outcome with the
actual 1962 position.
The simulation itself rested upon the calculation of an
ARCH + 1(1968) H.4