Full text: ARCH+ : Studienhefte für architekturbezogene Umweltforschung und -planung (1968, Jg. 1, H. 1-4)

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
	        

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