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I am using Spatial Statistics tool (GWR regression)of the ArcGIS 10.3 for my PhD research, the smallest geographical area in the study is called a ward. There are 27 wards in the study area, these wards formed the polygons while industries (SMEs) and their coordinates formed the points. After I have aggregated the point data and the polygons and ran the GWR regression, I always get results from the analysis. However, in all the results ArcGIS suggested feature class be not less than 30. Meanwhile, the number of my polygons (feature class) is 27.

What do I do?

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Sorry to spoil the party but, there are several issues with the modeling approach that you are describing.

1) It is unlikely that you have enough observations to adequately represent global or localized spatial variation. In general statistical terms, I would also imagine that you may have a power issue in the model with very little in the way of effect size. Have you quantified the spatial autocorrelation using a global statistic such as Moran's-I or a Joins-count (in the absence of a marked process) or a local statistic (eg., LISA, Gi*)?

2) You are misrepresenting the spatial process by using points to represent a conditional spatial process. In other words, you cannot use a point location to represent an area (polygon). Any observed autocorrelation structure would be an artifact of the aggregation units.

3) By aggregating industries (points) to the "wards" polygons you are introducing a change of support (MAUP) problem. This could most certainly introduce a spurious relationship and as it stands any statistical model results would be quite unsupported. Without seeing the data one can only speculate but, the MAUP issue in the data may be adequately addressed via a Monte Carlo simulation that was designed to "prove" that the modifiable units (wards) represent a statistically stable set of aggregate units. You can read on the MAUP issue starting with Openshaw (1983) and Cressie (1996) then dozens of subsequent papers from various spatially related disciplines.

4) A GWR model is indicated when nonstationarity is present in the data. Given how you have structured the data it is very unlikely that you have any 2nd order spatial variation in your data. Any 1st order autocorrelation would likely be an artifact of how you structured your data. If a robust spatial exploratory data analysis (ESDA) were to be performed on the data at hand it is unlikely that any spatial model would be well supported. It would seem that, given the conditional nature of the wards, a mixed effects model with the random effect being the wards, would be in order.

In applying GWR you are accepting a trade off between missing 1st order variation (global trend) and representing 2nd order trend (localized variation). Because of this is not a generalized spatial model that is applied to just any data with an observed spatial structure. If the autocorrelation is representing a general 1st order spatial process, a GWR is not supported. There are also several some serious issues (Paez et. al., 2011; Wheeler & Tiefelsdorf 2005) with the GWR approach, indicating considerable instability in resulting coefficients, to the point of potential random results. These issues need to be discussed and weighed before applying the model. As much as I dislike this method I will occasionally apply it in exploratory analysis. The GWR approach would be so much more satisfying if a first-order trend were accounted for, without this I always feel that the model suffers from over-fit.

At this juncture, I would highly recommend talking with a spatial statistician, or anybody well versed in spatial modeling, to seek advice on the best way to structure your data and a supported modeling approach. Aside from exploratory analysis, I would also look somewhere other than ArcGIS to perform your statistical modeling.

References

Cressie, Noel A (1996). Change of support and the modifiable areal unit problem. Geographical Systems. 3(2–3):159–180.

Openshaw, S. (1983) The Modifiable Areal Unit Problem. CATMOG 38.

Openshaw, S. (1984) Ecological fallacies and the analysis of areal census data. Environment and Planning A 16, 17—31.

Páez,A., S. Farber, & D. Wheeler (2011). A simulation based study of geographically weighted regression as a method for investigating spatially varying relationships. Environment and Planning 43(12):2992–3010.

Wheeler, D. & M. Tiefelsdorf, (2005). Multicollinearity and correlation among local regression coefficients in geographically weighted regression. Journal of Geographical Systems, 7:161–187.

  • I never thought it that way though, really appreciate your suggestions. Thanks. – Salisu Abubakar Dec 17 '17 at 11:52

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