Synonyms
Best linear unbiased prediction; BLUP
Definition
Prediction of ex-sample spatially dependent dependent variables not only uses ex-sample independent variables in conjunction with sample estimates of the associated parameters, but also uses the sample residuals and the spatial relations between the sample observations and the ex-sample observations to produce Best Linear Unbiased Predictions (BLUP). For spatial econometric models that specify space via weight matrices, BLUP has a simple and computationally feasible form.
Historical Background
In a GIS context, there are many motivations for conducting spatial prediction. First, a need often exists for accurate prediction, as in the case of real estate where accurate valuation requires use of data from neighboring properties. Second, predictions are smoother than the underlying data, and facilitate construction of easily understood maps. Third, diagnostic maps of predictions and associated residuals can be used to assess the...
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Notes
- 1.
Pace and LeSage (2004) combined spatial autoregressive models and GWR. Using a large bandwidth (m) for GWR resulted in smooth predictions, but led to spatially dependent residuals, whereas spatial autoregressive local estimation allowed independent selection of m for smoothness without creation of obvious spatial dependence.
- 2.
For details on derivation of these estimates see LeSage and Pace (2004).
- 3.
- 4.
Note, β or λ do not change across the n separate predictions. These could change as well, but the focus is on spatial prediction instead of estimation, and so assume a known, common β and λ exist.
Recommended Reading
Anselin, L.: Spatial econometrics: methods and models. Kluwer, Dordrecht (1988)
Brunsdon, C., Fotheringham, A.S., Charlton, M.E.: Geographically Weighted Regression: A Method for Exploring Spatial Non-stationarity, Geographical Analysis, vol 28, pp. 281–298 (1996)
Cressie, N.: Statistics for Spatial Data. Revised edition. John Wiley, New York (1993)
Goldberger, A.: Best Linear Unbiased Prediction in the Linear Model, Journal of the American Statistical Association, vol. 57, pp. 369–375 (1962)
Harville, D.: Matrix Algebra from a Statistician's Perspective. Springer-Verlag, New York (1997)
Kelejian, H., Prucha, I.: Prediction Efficiencies in Spatial Models with Spatial Lags, University of Maryland (2004)
LeSage, J., Pace, R.K.: A Matrix Exponential Spatial Specification, Journal of Econometrics (2006)
LeSage, J., Pace, R.K.: Conditioning upon All the Data: Improved Prediction via Imputation, Journal of Real Estate Finance and Economics, vol. 29, pp. 233–254 (2004)
LeSage, J., Pace, R.K.: Introduction. In: LeSage, J.P. and Pace, R.K. (eds.) Advances in Econometrics: vol. 18: Spatial and Spatiotemporal Econometrics, pp. 1–30. Elsevier Ltd, Oxford (2004)
Pace, R.K., Barry, R.: Quick Computation of Spatial Autoregressive Estimators, Geographical Analysis, vol. 29, pp. 232–246 (1997)
Pace, R.K., LeSage, J.: Spatial Autoregressive Local Estimation. In: Mur, J., Zoller, H., Getis A. (eds.) Recent Advances in Spatial Econometrics, pp. 31–51. Palgrave Publishers (2004)
Robinson, G.K.: That BLUP is a Good Thing: The Estimation of Random Effects, Statistical Science, vol. 6, pp. 15–32 (1991)
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Pace, R., LeSage, J. (2008). Spatial Econometric Models, Prediction. In: Shekhar, S., Xiong, H. (eds) Encyclopedia of GIS. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-35973-1_1266
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