||A mathematical-statistical model describing the sequence of outcomes from a series of trials in probability terms. Some authorities distinguish between a probabilistic model, in which the outcome of individual trials is predicted, and a stochastic model, in which the development of a series of outcomes is modelled (cf. prediction); a stochastic model may therefore include situations in which the outcome of a specific trial is in some way dependent upon the outcome of previous trials. The same process can produce an infinite (or at least very large) number of realizations â€” a sequence of 50 throws of a die, for example, can produce very many different sequences of the numbers 1 to 6 â€” so that a stochastic process is very different from a deterministic one, which can realize only a single outcome from a given set of inputs.
Geographers have applied the concept of a stochastic process to both temporal (such as crop yields and commodity prices) and spatial sequences, and in some cases have combined the two (as in studies of epidemics; cf. diffusion). The underlying time scale may be either discrete or continuous, and the stochastic element is included because the process involved is believed to include either a purely random element or a large number of minor causal factors whose net effect is a quasi-random disturbance of the outcomes.
One geographical usage of these ideas involves the proposal that a point pattern (of settlements or factories, for example) has resulted from a quasi-random process in which each geographical unit of a given size (such as a quadrat) has an equal probability of being the chosen location for such a point. The process can be modelled using a Poisson distribution (cf. frequency distribution), which assumes that the location decisions are independent of each other, but if the positioning of one point in an area changes the probability of a further point being placed there (cf. agglomeration), then the process is stochastic and should be modelled accordingly using an alternative frequency distribution (such as the negative binomial).
Analysis of time series within this approach has focused on three main types of stochastic process: in autoregressive models the value at time t is highly correlated with the values at t-1, t-2 etc., but with a random component; in moving average models, the value at t is determined as a weighted average of preceding and subsequent values; and in Markov processes there is no random element â€” each value is a deterministic function of previous values. If the process proceeds slowly (as, for example, in the evolution of a settlement pattern), two problems may arise: it may be necessary to infer the process from the pattern at just a single or, at best, a small number of cross-sections on the time scale; and most models assume stationarity â€” that the process operates constantly over either or both of time (temporal stationarity) or space (spatial stationarity). Where these assumptions regarding stationarity are valid, it may be possible to filter the data (i.e. to remove temporal or spatial trends); where they are not, sophisticated analytical procedures are required (as in the expansion method: Jones and Casetti, 1991).Â (AMH)
Reference Jones, J.P. and Casetti, E., eds, 1991: Applications of the expansion method. London: Routledge.
Suggested Reading Bennett, R.J. 1979: Spatial time series. London: Pion.Â Hoel, P.G., Port, S.C. and Stone, C.J. 1972: Introduction to stochastic processes. Boston: Houghton Mifflin.