Geostat_Bimodal data

[EifLeffie]

Hi,

I am using geostat analysis to particularly kriging to interpolate my data but I cant seem to get a better result. I am new to kriging and geostat and am doing some readings and research about it but its taking so much time that I cant finish the interpolation as early as planned.

I would highly appreciate if anyone could give me an advise on this. The data that I am interpolating is bimodal and I wonder if there is an appropriate procedure for this. Thank u so much for any advise you can give.

Eif

[JeffreyEvans]

Have you checked "stationarity" assumptions? It sounds like you may have some serious nonstationarity in your data. A polynomial trend removal will not account for second-order effects. Violation of even the most relaxed model of stationarity can have a negative effect on Kriging estimates. ArcGIS has the LISA model available, in the Spatial Statistics Toolbox, for testing stationarity.

[EricKrause]

The Local Moran's I test in spatial statistics has the potential to detect nonstationarity, but it's really looking for local outliers.  If the data is stationary, you won't find local outliers; however, the lack of local outliers does not imply stationarity, so be careful.

For investigating stationarity, I suggest using the Voronoi Map ESDA tool with Type set to Entropy or StDev.  One advantage of the Voronoi Map is that it works with quantiles, so it's nonparametric.  Local Moran's I comes with distributional assumptions.

[EifLeffie]

Have you checked "stationarity" assumptions? It sounds like you may have some serious nonstationarity in your data. A polynomial trend removal will not account for second-order effects. Violation of even the most relaxed model of stationarity can have a negative effect on Kriging estimates. ArcGIS has the LISA model available, in the Spatial Statistics Toolbox, for testing stationarity.

Thanks for the reply Jevans. I have not checked yet for the "stationarity" of the dataset. The dataset is actually annual growth of trees. Pardon me but I am not yet that well verse in GIS processing. What is LISA model?

Thanks,
Eif

[EifLeffie]

The Local Moran's I test in spatial statistics has the potential to detect nonstationarity, but it's really looking for local outliers.  If the data is stationary, you won't find local outliers; however, the lack of local outliers does not imply stationarity, so be careful.

For investigating stationarity, I suggest using the Voronoi Map ESDA tool with Type set to Entropy or StDev.  One advantage of the Voronoi Map is that it works with quantiles, so it's nonparametric.  Local Moran's I comes with distributional assumptions.

Thanks Eric. I am not familiar with how the Voronoi map works and this is sort of abstract to me. Is there a paper on how this works? I have attached the image of the voronoi for both entropy and stDev and I am confused on how to interpret it.

Any help is highly appreciated.

Thanks,
Eif

[EricKrause]

LISA = Local Indicators of Spatial Association

He is referring to this geoprocessing tool:
http://help.arcgis.com/en/arcgisdesktop/10.0/help/index.html#//005p0000000z000000.htm

[EricKrause]

Thanks Eric. I am not familiar with how the Voronoi map works and this is sort of abstract to me. Is there a paper on how this works?

If your data is stationary, you expect to see randomness in the colors of the Voronoi polygons when they're symbolized by entropy or standard deviation.  The idea is that the local variation should be roughly constant across the surface; you should not have areas with much more erratic data than others.

Looking at your two Voronoi maps, it looks like you have some nonstationarity, but it doesn't look very drastic.  The StDev symbolization seems more clustered, but the Entropy symbolization doesn't look too bad, and I prefer to use Entropy when looking for stationarity.

If you want, you can send your data to ekrause@esri.com, and I'll see if I can fit a good kriging model.

[EifLeffie]

If your data is stationary, you expect to see randomness in the colors of the Voronoi polygons when they're symbolized by entropy or standard deviation.  The idea is that the local variation should be roughly constant across the surface; you should not have areas with much more erratic data than others.

Looking at your two Voronoi maps, it looks like you have some nonstationarity, but it doesn't look very drastic.  The StDev symbolization seems more clustered, but the Entropy symbolization doesn't look too bad, and I prefer to use Entropy when looking for stationarity.

Thanks much Eric.

If you want, you can send your data to ekrause@esri.com, and I'll see if I can fit a good kriging model.

I have sent you the file via email. I highly appreciate the help.

Best regards,

Eif

[JeffreyEvans]

Eric,
I am curious to why you prefer entropy as a measure of nonstationarity? Since the the empirical semivariogram is based on binned variance/2 and the standard model of nonstationarity is based on mean and variance, would not the standard deviation better represent how this assumption is violated?

Without looking at the data this is pure conjecture, but I am wondering if the data is limited to two distinct modes is there a possibility that it is assuming a binomial form? There is a possibility that indicator Kriging may be more appropriate. Since indicator Kriging is nonparametric, how sensitive is it to nonstationarity and distributional assumptions, compared to the linear form?

[EricKrause]

I prefer Entropy because it works with classified bin values rather than the raw data.  You're right that using Standard Deviation more directly checks the assumption of stationarity, but I've found that it is often too sensitive to deviations.  A couple relatively non-extreme outliers will often completely throw off the Voronoi map, giving the impression that the dataset is highly nonstationary when the deviation from stationarity is not actually very extreme. 

I took a look at the data, and it isn't binomial data.  I couldn't fit a good kriging model to the data in the graphics, but I found a seemingly good model using Kernel Smoothing.