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how does kriging arrive at areal level prediction?

annfeng1

I have searched the ESRI kriging-related documentation but cannot seem to find relevant information on this. It's rather straightforward with the prediction feature from the geostatistical analysis extention to predict an areal-level value from a continuous surface interplated from kriging (e.g. ordinary kriging). But what are the maths behind this process? Is it some sort of moving average based on continous values for each areal unit? How is the average calculated? I only find the basic formula for point-location predicted values, but not for areal level values. Any help or pointers in this regard is greatly appreciated.

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annfeng1

Thanks Eric for clarifying this. I doubt if i can apply GGS though since the input layer must be from Simple kriging which is not the case here. But i have a better understanding of block kriging. Thanks for your help.

EricKrause

For block predictions, it's just an areal average. Block standard errors are more complicated than that, and I think that's what your colleague is getting at.

GA Layer to Grid performs block predictions from a kriging model you've already made. It does not calculate block standard errors, which is the big difference from true "block kriging". True block kriging tries to reduce calculations by incorporating the areal units into the kriging step, and it handles the standard errors for the polygons differently (the calculation of block standard errors is not a simple areal average of the point standard errors). Sorry for the terminology issues; I probably should have consistently said "block prediction" rather than "block kriging."

Again, we decided not to implement full block kriging because geostatistical simulations do the same job a lot better.

annfeng1

Thanks Eric but i want to properly interpret the difference between block prediction and block kriging, if there is any. A fellow researcher of mine is pretty adamant block kriging is NOT post hoc block prediction. I don't think block kriging is included in the kriging methods available in GA though. You seem to suggest block kriging is synonymous with block prediction. please entertain my curiosity and thrash this out if you could. Thanks so much!

EricKrause

Good to hear you found the tool. I highly recommend using it to aggregate predictions to polygons.

When researchers talk about block kriging, they're usually talking about things that are a lot more complicated than just averaging to polygons. They usually are talking about analyzing how the point-to-point semivariogram compares to the polygon-to-polygon semivariogram after doing areal averaging of the discretized surface. But if you just want to aggregate kriging predictions to polygons, you don't really need to worry about all that.

annfeng1

Sorry i misspoke about the GA toolbox. I was looking at the tool bar. Yes i do have GGS in the toolbox. very remiss of me not to check your image closer.

annfeng1

No mine only has explore data, geostatistical wizard, and create subsets. Here is what i was told 'In ArcGIS 10, this tool moved into an additional tool box (Simulation toolbox) which was not present in ArcGIS 931'. So mystery solved. The reason i ask about block kriging is that i think i have it confused with block prediction. The latter seems to be what i was doing, while block kriging seems much more complicated as it involves "averging values of the covariances over some grid of point locations within each block". So average covariance instead of individual covariance structure was adopted in block kriging, but I haven't done anything of this sort. I don't think you accomplish block kriging in the prediction phase, or is it just another term for block prediction?

EricKrause

Does your Geostatistical Analyst toolbox look like the picture I've attached?

And yes, block kriging is just taking averages of predictions within the areal regions.

gatoolbox.png

annfeng1

I am pretty sure i don't have this simulation tool in the GA toolset. Guess it's a lower version than 9.3 although my Arcview is 9.3. There seems to be no way to tell the version of my GA extension from the built-in GA help menu! I hope you are not exasperated by me asking this: the zonal statistics or block estimates are simple (arithmatic?) averages of cell values per block? is there local regression invovled? or kernel interpolation at the centroid of each block? Many thanks!

EricKrause

Thank you Eric for helping me understand the areal prediction details. To my knowledage, when interpolating a surface from say ordinary kriging, the default is filled contour display; but when further predicting to arbitrary areal unit, you mean the gridded cell values are averaged per area/block? So even when the surface looks continous, the underlying predicted values are still cell-discretized (for a cell size of my choice i understand)?

The filled contour display is just for visual representation. The surface actually is continuous, but in order to make areal predictions, the surface is discretized. This is true for both block kriging and geostatistical simulations.

And a separate question, when the areal units are variable in size, do larger areal units have higher prediction uncertainty (or higher estimated variance), or lower? Thanks again for your insight! I will have to make do with GA for now since i don't have a GGS license...

GGS is a tool in the Geostatistical Analyst toolbox (in the Simulation toolset), so if you have a GA license, you have access to the tool. The tool was added in ArcGIS 9.3.

In general, larger areal units will have smaller standard errors from both block kriging and geostatistical simulations.

annfeng1

Thank you Eric for helping me understand the areal prediction details. To my knowledage, when interpolating a surface from say ordinary kriging, the default is filled contour display; but when further predicting to arbitrary areal unit, you mean the gridded cell values are averaged per area/block? So even when the surface looks continous, the underlying predicted values are still cell-discretized (for a cell size of my choice i understand)?

And a separate question, when the areal units are variable in size, do larger areal units have higher prediction uncertainty (or higher estimated variance), or lower? Thanks again for your insight! I will have to make do with GA for now since i don't have a GGS license...

EricKrause

GA Layer to Grid employs block kriging where the raster/grid cells are the "areal units." We don't have block kriging to arbitrary areal units because GGS does the same job better. If your goal is to predict to areal units, we highly recommend Geostatistical simulations over block kriging or zonal statistics.

All GA Layer to Grid is doing is making predictions on a lattice within each raster cell, then taking the average of the predictions and assigning it to the raster cell. By default, it only predicts a single point at the center of the raster cell, but you can specify the dimensions of the lattice (3 by 2, for example).

We recommend "Cressie, N (1993) Statistics for spatial data, Wiley, New York" for mathematical analysis of block kriging. However, this book can be intimidating for people without a substantial background in statistics.

For a more example-driven text that is tailored to Geostatistical Analyst and GIS users:
http://esripress.esri.com/display/index.cfm?fuseaction=display&websiteID=194

Note: Neither GGS nor block kriging are "Bayesian" methods. GGS can be called a Monte Carlo method (though that phrase is rarely used), and block kriging is simply areal averaging of the surface.

annfeng1

Thanks for explaining to me the steps Eric. I am not sure if I understand it correctly but I think I should have phrased the question in terms of block kriging, or at least that's what I blieve is behind the areal-level prediction. Because I am interested in estimating areas or blocks, instead of mapping a continuous surface, I used the prediction (in the krigging output menu) feature to give an averaged value for each areal unit. What I want to know is how is this areal-level average calculated behind the scene. There is little help in the way of documentation. Are you saying the Bayesian approach is used in areal-level prediction? Could you point me to some papers, book chapters that explain the approach? Thanks a lot!

EricKrause

The best way to do this in Geostatistical Analyst is to use the Gaussian Geostatistical Simulations tool. You first create a Simple kriging layer in the Geostatistical Wizard, then you use this layer as input to the GGS tool. Make sure to condition on the features/field that you used to interpolate, then specify the statistical polygons where you would like to make areal predictions.

This process is conditionally simulating many point values in each of the polygons, then it's calculating statistics on these simulated points within each polygon.

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