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I am trying to measure uncertainty associated with a mean value for spatial data. I have:

  • A polygon representing a spatial district.
  • A raster dataset that was produced via a Random Forests model, with associated RMSE.

I want to calculate the average value of the raster data within the district. My understanding is that I should simply calculate the mean, followed by the standard error of the data to represent how far the sample mean is from the true mean.

My issue is that I don't know whether to treat the raster data as a population or a sample. Including all raster pixels accounts for all values within the district, which drives the standard error close to 0. However, taking only a sample of the raster values from within the district seems incorrect as it is an arbitrary decision to omit available data.

I am currently propagating uncertainty by taking the root sum of squares of both the standard error as well as the model RMSE. I think this is correct, but the standard error contributes almost no uncertainty given that there are a very large number of raster values (>10,000).

Can anyone provide clarification on how to think about this problem? I have not been able to find much material that describes how to think about summarizing raster data from a traditional sampling approach. References or additional reading would greatly be appreciated.

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    $\begingroup$ The raster data are a sample. Your most fundamental problem is that the SEs of the pixels are unlikely to be enough information: you also need to covariances between pixel estimates. There is a principled way to handle this in the geostatistical theory of kriging. Even though you're not using a kriging estimator, these geostatistical methods are conceptually helpful and often can be directly applied to estimating the SE of any linear functional of the estimates (of which a spatial mean is one example). $\endgroup$ – whuber Aug 13 at 16:09
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    $\begingroup$ How did you estimate the variances? After all, a covariance is a variance. For instance, if you can estimate the variance of the sum of two pixels and the variances of the pixels separately, then you have an estimate of their covariance. $\endgroup$ – whuber Aug 13 at 17:02
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    $\begingroup$ There's no way, then, to obtain a valid standard error for the sum over a region. Because any raster when subdivided into sufficiently small pixels exhibits strong positive spatial correlation, at least over short distances, it is likely the SE formula you propose (which is valid for uncorrelated pixels) will grossly underestimate the true SE. Conceivably you could come up with some kind of pseudo-SE (which might nevertheless be useful) by analyzing the spatial correlation of those pixels. $\endgroup$ – whuber Aug 13 at 18:20
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    $\begingroup$ Interesting. I think this hits the nail on the head, because the SE appears to be heavily underestimated, which is what I've been struggling with. I find this concerning because for many spatial model products (which drive environmental policy and decision-making), it would appear there's no way of obtaining uncertainty measures associated with summary statistics (e.g., how much forest carbon is in a particular jurisdiction). Anyways, thanks very much for your help! $\endgroup$ – jbukoski Aug 13 at 18:34
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    $\begingroup$ Your impression is correct: most raster products come with woefully inadequate (and often terribly misleading) representations of sources of error and uncertainty. I have written about some of these issues on Geographic Information Systems, but I don't recall addressing your particular question there. Others might have, though, so consider searching there. $\endgroup$ – whuber Aug 13 at 18:37

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