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# space-time forecasting models

Statistical models that attempt to forecast the evolution of variables over both time and space (sets of regions). These models are of the general regression form, and they forecast the value of a variable and an observation-unit in terms of (a) its own past values, (b) lagged spatial diffusion effects, and (c) lagged exogenous or explanatory variables. The simplest form is (a), where a variable (such as population) in region j at time t is predicted by regression on its own earlier values (the \'autoregressive\', or memory, effect) and also perhaps by its delayed response to the impact of random shocks ejt, ejt-1 and ejt-2 (the moving average coefficients b1 and b2):

{img src=show_image.php?name=bkhumgeofm30.gif }

This is the well-known time-series autoregressive-moving average (or ARMA) model. It is a \'black-box\' model in that it does not explain the population changes causally, but models and extrapolates them statistically. Such models can have quite good short-term forecasting ability.

Space-time forecasting models extend the single-region ARMA model to include (b), spatial diffusion between regions, so that pjt is also dependent on population changes in nearby regions and hence trends in population change diffuse across the map. Defining the average (or weighted average) of population for regions adjacent to region j at time t as Lpjt, the space-time ARMA model (STARMA) can be written (using only one-lag terms) as:

{img src=show_image.php?name=bkhumgeofm31.gif }

The final term allows random shocks to spill over between regions also. distance decay can also be built into the definition of the weights L. The STARMA model is still a black-box model, and the introduction of (c), lagged exogenous variables (such as employment opportunities, EMP), is essential for a causal model:

{img src=show_image.php?name=bkhumgeofm32.gif }

The exogenous variable EMP then has to be itself extrapolated (which is possible by STARMA) before population can be forecast more than one period ahead. However, STARMA models have not proved as useful as first hoped, mainly because of their black-box structure (the same criticism is made of such black-box models in forecasting for macroeconomic policy), and modellers have found form (c) the most relevant, allowing conditional forecasts to be made based on different assumptions about the exogenous variables.Â (LWH)

Suggested Reading Bennett, R.J. 1979: Spatial time series. London: Pion.

##### space, human geography and

Other Terms : sport, geography of | mercantilist model | pluralism
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