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Universal Kriging with External Drift

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12-23-2012 05:59 PM
Tae-Jung_JonathanKwon
Deactivated User
Hi

I am looking for a way to use UK with external drift and the external drift. Or in others words, is there any way to perform a regression kriging on ArcGIS? (i.e., kriging with underlying trend..).
Thanks for your input in advance.
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27 Replies
Tae-Jung_JonathanKwon
Deactivated User
Please clarify exactly which step(s) you are confused with.

Hello

I would like to know how exactly we can do the external drift kriging using ArcGIS. I couldnot figure it out(got confused) from the above discussion.
Also can anyone tell whether it is possible to do it for a time series of data. I have the hourly precipitation data for some staions. I would like to interpolate it for the entire area.

Thanks in advance! 🙂
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Tae-Jung_JonathanKwon
Deactivated User
Hi Eric,

So I carried out the analysis using a variable called mean surface temperature from 40 stations.
Following the steps that I outlined, I linearly regressed using some auxiliary information (e.g., easting, northing, altitude, relative topography, and distance to water). And I obtained the good results (R^2 = 89%).
I calculated the residuals and tried to krig using simple Kriging available in ArcGIS 10.1 (since I know that sum of all residuals is equal to zero)... but the result of krigging was poor.. (please see the attached..)
How would you interpret this phenomenon? You thoughts on this would be appreciated.
Thanks,
[ATTACH=CONFIG]21416[/ATTACH][ATTACH=CONFIG]21417[/ATTACH]


Sorry I didn't know it was that simple.. I should have looked into more closely.. I thought such feature would be available when running the kriging..

Thank you!
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EricKrause
Esri Regular Contributor
It could be a couple of things.  First, regression kriging is a fairly complicated method for only 40 data points.  You may just not have enough data to apply such a complicated model. 

Second, the residuals of your OLS model don't appear to display spatial autocorrelation.  Looking at the crossvalidation statistics and the empirical semivariances (the blue crosses in the semivariogram screen), your residuals seem to be independent and normally distributed.  Sampling more data may reveal more spatial patterns that can be used in kriging, but I don't see anything in those diagnostics that indicates the residuals of your OLS model need to be kriged at all.
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Tae-Jung_JonathanKwon
Deactivated User
[ATTACH=CONFIG]21426[/ATTACH][ATTACH=CONFIG]21427[/ATTACH]

Okay. I understand.  The example that I showed you does not really require any further analysis or need to be krigged at all since the residuals are independent and normally distributed (or since the regression result is very good).

You mentioned about the spatial autocorrelation. I have attached another example. For this, I took the standard deviation of surface temperature as a means to describe the variability of surface temperature (VST). And the variable that I regressed earlier was mean surface temperature (MST). The regression model using the same auxiliary variable revealed that the VST were highly correlated with northing, altitude, and distance to water (which makes intuitive sense..)
Then I ran the simple krigging on the VST residuals (see the attached files), and the result seemed to be okay. There was a spatial dependence between these 40 locations to a certain extent and the residuals were not normally distributed.. I also expected this to be "not good" since the location information (i.e., northing) had already been taken into account when calibrating the model using linear regression.. Since the models for both VST and MST were calibrated using the similar locational attributes/auxiliary variables, I am having difficulties understanding as to why simple krigging would give such different results on the residuals for VST and MST.. Any thoughts or suggestions on this? My speculation would be the difference in the overall quality of the regression models: MST (R^2=89%) and VST (R^2=63%).

Thanks!


It could be a couple of things.  First, regression kriging is a fairly complicated method for only 40 data points.  You may just not have enough data to apply such a complicated model. 

Second, the residuals of your OLS model don't appear to display spatial autocorrelation.  Looking at the crossvalidation statistics and the empirical semivariances (the blue crosses in the semivariogram screen), your residuals seem to be independent and normally distributed.  Sampling more data may reveal more spatial patterns that can be used in kriging, but I don't see anything in those diagnostics that indicates the residuals of your OLS model need to be kriged at all.
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EricKrause
Esri Regular Contributor
The mean and variance of the temperature are two different variables, and one may display spatially autocorrelated residuals while the other doesn't.  There may be a physical reason for this, but I don't know what it could be.
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Tae-Jung_JonathanKwon
Deactivated User
Hi Eric,

I have a question regarding the difference of the underlying theories between simple kriging (SK) and ordinary kriging (OK). Literature well describes the nature of their difference; SK assumes that the mean is known and constant over the entire domain whereas OK assumes that the mean is unknown.

We also know that SK requires knowledge of the mean to solve the problem of finding weights that minimize the variance of the estimation error, but OK elegantly discards this requirement by filtering out the mean by performing a constrained optimization such that sum of OK weights equals 1.

My confusion comes from one literature stating that SK not only assumes the expectation of the random field to be known (i.e., known mean), but also relies on a covariance function to be known beforehand. [ATTACH=CONFIG]29456[/ATTACH]
My understanding is that regardless of which method to use (either SK or OK), covariance must be determined from semivariogram analysis when interpolating the values (kriging estimate and error variance). However this contradicts with the statement provided above (i.e., covariance function needs to be known beforehand).

Could you explain if there is any part that I misunderstood or if the statement is not true? Your answer will be greatly appreciated.
Thank you.
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EricKrause
Esri Regular Contributor
Everything you said is correct.  This has always been a known problem for simple and ordinary kriging; you have to estimate the semivariogram from the data, but then you have to assume that your estimation is completely correct.  In practice, this means that standard errors will almost always be underestimated.

This is one of several problems that we addressed with Empirical Bayesian Kriging in ArcGIS 10.1.  Instead of estimating a single semivariogram and assuming it is correct, EBK simulates many semivariograms, so you end up with an entire spectrum that we weight by likelihood.  By accounting for some uncertainty in the estimation of the semivariogram, this weighted spectrum does a much better job of estimating the covariance structure than relying on a single semivariogram.  If you work with some data using ordinary, simple, and empirical Bayesian kriging, you'll notice that EBK usually gives larger standard errors than the others.  This might seem like a disadvantage, but the larger standard errors will usually be more accurate.  For example, if you use simple or ordinary kriging and make 90% confidence intervals, it will probably only capture 75% of the data, whereas 90% confidence intervals from EBK should capture closer to 90% of the data.
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Tae-Jung_JonathanKwon
Deactivated User
Hi Eric,

I am now trying to use semivariogram to analyse/characterize the spatial patterns of friction measurements collected at an equal interval (every minute) by a mobile unit. The measurements are GPS tagged so they can be visualized on the map as attached below. (the lines represent a road network)
[ATTACH=CONFIG]32629[/ATTACH]

I am trying to build a semivarogram model for each run (collected at different days) and compare their similarities/differences by examining the model parameters (i.e., sill, nugget, range..)

Here I have attached the diagram of datasets collected at two different days (Excel file is also attached)
[ATTACH=CONFIG]32630[/ATTACH]

As you can see, the friction measurements collected on these two days are similar to each other and thus I would expect that their spatial patterns are also similar... But their semivariograms show that they are quite different to each other..

For Day1
[ATTACH=CONFIG]32631[/ATTACH]

For Day2
[ATTACH=CONFIG]32632[/ATTACH]

Would you be able to reason why this is the case? I would expect that, for instance, their ranges should be similar but they aren't. I understand that the range (and other parameters) could vary subject to how you fit the empirical semivarainces by using different models (gaussian, exponential...etc), but there still exists a large difference in their spatial characteristics.

Any comments/suggestions would be appreciated...
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