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Markov process (or Markov chain)

A type of stochastic process: if the probability of being in a state at time t is wholly dependent upon the state(s) at some preceding time(s), it is said to be a Markov process. Where only the immediately preceding state is considered it is said to be a first order Markov process, but higher order processes (i.e. dependence on earlier states) can be modelled. The process is often represented by a transition probability matrix in which the rows and columns represent states and the cells represent the probabilities of movement between states. For example, land-use changes between the categories \'residential\', \'commercial\', and \'industrial\' could be modelled by the matrix:

|||The matrix may be interpreted as follows: between time t and time t + 1, 90 per cent of the residential land remains in the same use, 6 per cent is transferred to commercial use, and 4 per cent becomes industrial; the second and third rows may be interpreted similarly. Note that each of the rows sums to unity (no land is lost or gained in the conversion process). The repeated operation of such a matrix results in a stable distribution between states, which is independent (in general) of the initial distribution. But for such a repeated operation to be realistic it is necessary to demonstrate (or assume) that the transition matrix is constant over a number of time intervals (referred to as stationary). The Markov model has been used to study the growth of firms (movement between size categories) and the migration of firms and households (movement between geographical locations).Â (AMH)

Suggested Reading Collins, L., Drewett, R. and Ferguson, R. 1974: Markov models in geography. Statistician 23: 179-210.
State at time t+1:|
Res.|Comm.|Ind.|
State|Res.|0.90|0.06|0.04|
at time|Comm.|0.02|0.85|0.13|
t|Ind.|0.00|0.10|0.90|

Marxian economics

Other Terms : underconsumption | gross domestic product (GDP) | modifiable areal unit problem
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