On dealing with spatially correlated residuals in remote sensing and GIS

Nicholas A. S. Hamm 1, Peter M. Atkinson 2 and Edward J. Milton 3
School of Geography
University of Southampton
Southampton SO17 3AT
United Kingdom
Tel.: +44 7887 578 442; Fax: +44 23 8059 3295
nick@hamm.org1
pma@soton.ac.uk2
ejm@soton.ac.uk3

Abstract
Key assumptions in standard regression models are that the residuals are independent and identically distributed. These assumptions are often not met in practice. In particular, the issue of spatial correlation amongst the residuals has had limited attention in the remote sensing and GIS literature. This paper discusses approaches that use familiar authorized models to specify the covariance amongst the residuals. The model parameters were estimated using a maximum likelihood approach (ML or REML). The accuracy of the approach was investigated using simulated data and found to give accurate estimates of the error variance (σ 2 ), although estimates of and range (a) and nugget component (s) were less accurate. The approach was found to be robust to choice of covariance function (exponential or spherical). The approach was then extended to deal with heteroskedastic residuals by incorporating a weighting component analogous to that used in weighted  least squares. This was also shown to yield accurate results for  σ 2 , a and s. Finally, possibilities for extending these approaches to prediction are considered.

Keywords: correlated residuals, regression, GIS, remote sensing, variogram

In: Caetano, M. and Painho, M. (eds). Proceedings of the 7th International Symposium on Spatial Accuracy Assessment in Natural Resources and Environmental Sciences, 5 – 7 July 2006, Lisboa, Instituto Geográfico Português

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