
spacetime 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 observationunit 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, ejt1 and ejt2 (the moving average coefficients b1 and b2):
{img src=show_image.php?name=bkhumgeofm30.gif }
This is the wellknown timeseries autoregressivemoving average (or ARMA) model. It is a \'blackbox\' model in that it does not explain the population changes causally, but models and extrapolates them statistically. Such models can have quite good shortterm forecasting ability.
Spacetime forecasting models extend the singleregion 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 spacetime ARMA model (STARMA) can be written (using only onelag 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 blackbox 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 blackbox structure (the same criticism is made of such blackbox 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. 





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