Re: [ai-geostats] Sum of predicted values

[Pete Gething]

Dear List   Following up from my question regarding the variance associated with the sum of a set of values predicted via Kriging..   Many thanks to Pierre Goovaerts, Donald Myers, Isobel Clark, Yetta Jager and Christopher Taylor for their responses. I have compiled these below, along with my original query.   Best wishes,   Pete   ################################### Original Query:   Dear list,   I have Kriged predictions of a continuous variable at a set of 1700 points. I want to sum these values and obtain an estimate of the overall prediction variance based on the kriging variances of the individual points (i.e., taking into account the spatial correlation between points). The data are approximately Gaussian.   I would expect there to be a standard solution to this problem, but I'm having difficulty finding examples - can anyone help me out, or point me to a reference?   Thanks in advance,   Pete     ####################################  Pierre Goovaerts:   Hi Pete,   This is a classical example where stochastic simulation would allow an easy quantification of the uncertainty attached to the aggregated value. Just generate a series of realizations of your process over these 1700 points, sum each set of simulated values, and use the empirical distribution of simulated block values as a model of uncertainty. You can find an example in Goovaerts, P. 2001. Geostatistical modelling of uncertainty in soil science. Geoderma, 103: 3-26. <http://www.terraseer.com/training/geostats/geoder01.pdf>   that you can download from my webpage.   Cheers, Pierre   ####################################  Donald Myers:   Two observations 1.  It would be possible to directly krige the sum, then you will get only one kriging variance, this is almost the same as "block" kriging.  Look at the derivation of the "block" kriging equations. 2. There are no distribution assumptions in the derivation of the kriging equations. It is true that as a weighted average the kriging estimator is sensitive to the presence of data values that are unusually large or small (relative to the majority of the data values), to that extent skewed distributions do not give very good results. In addition it is generally harder to estimate and model the variogram in the case of a skewed distribution because the sample variogram is a weighted average (of squared differences, which makes it more sensitive). Finally is it really the sum of the estimated values that you want?  Or do you want an estimate of an average value over a region? Donald E. Myers ####################################   Isobel Clark:   Pete   There is a standard solution but you won't like it ;-)   You can't do it from the kriging variances for a point grid. You have to go back to the kriging.   You need to record for each grid point, the weight allocated to each of your samples. Sum these for each sample over the 1700 points and divide each total by 1700. You now have the effective estimator for the global mean using all of your samples.   The standard estimation variance consists of three terms (1)-(2)-(3):   (1) Twice the weighted average of the semi-variogram between each sample and the area being estimated. If your area is large (compared to the size of the total sample set and the range of influence) you can assume the semi-variogram between each sample and the global area is equal to the sill. That would make first term 2xtotal sill.   (2) the cross product term: take each pair of samples (i,j) and calculate weight(i) x weight(j) x semi-variogram(i,j). Remember you have to take i,j and j,i (or multiply by 2).   (3) the within-area variance. This is the average of all the semi-variogram values between all of your 1700 points on the estimated grid. If your area is large (greater than, say, 5 ranges of influence) this will be within 2-3% of the total sill. There is a shortcut approach, but this is only valid for estimates of large sub-areas within a much larger area. If your data set is not large, it is easier to do a direct kriging of the overall area and get the appropriate kriging variance. Our public domain kriging game will do this for up to 29 samples. Download from http://www.kriging.com and follow links for "kriging game", data sets and tutorials.   Hope this helps Isobel   ####################################   Yetta Jager:   I would use conditional simulation, summing over maps in each realization.  The variance among replicate totals is the answer.  The attached does this for variances on a cdf. Yetta   ####################################  Christopher Taylor :   Pete, I recently published a paper using this same technique, but the number of points were much higher (20,000+), and so including the correlation between points was too complicated.  In our case, the variogram suggested a very short range and so we fealt comfortable estimating the global variance by simply summing the krige variances.  There are techniques described in texts: Rivoirard et al. 2000. Geostatistics for Estimating Fish Abundance.  Blackwell Science Notation generally comes from Ripley 1981. Spatial Statistics. I have seen recent examples in the fisheries literature: Kern, JW; Coyle, KO. 2000. Global block kriging to estimate biomass from acoustic surveys for zooplankton in the western Aleutian Islands. Canadian journal of fisheries and aquatic sciences Vol. 57, no. 10, pp. 2112-2121. 2000. -Chris ########################################################################

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