if lul � r then
4.2.4 Estimation of Ratios
In many applications, one has to estimate quantities of the fonn:
(4.1)
e.g. percentages of trees with some characteristics, or mean timber volume per tree, etc. Note that in most circumstances, it would be wrong to consider quantities like:
th fu . z1 (x) . all cldi . as e nction --- 1s gener y not even a tive.
Z2(x)
The technique we propose to estimate r(V0 ) , and particularly its mean square error, differs from the standard procedure as presented in (Journel and Huijbregts, 1978). Instead, it rests upon a geostatistical refonnulation of Fieller's theorem (Cox, 1967), which is more in line with the design-based approach.
For each p e R1 we define the following random variable:
(4.2)
Note that from the definitions, one has :
(4.3)
Let 8 (p) be the double kriging estimate of S(p); for each given p , one needs to determine the * variograms of the "predictions" z1 (x) -pz2 (x) and of the "residuals" E1 (x) -pe2(x). This can be done either by direct model fitting of the empirical variograms of these two new processes, or by using pre-existing models of variograms and cross-variograms together with the relation
'Yu-9v (h) = Yu (h) + p2yv (h) - 2pyu_v(h) for any two processes
U,V.
Obviously, the first technique is more time conswning, but also more robust and instructive, since it models the relevant variogram directly.* *
Toe estimate r = r (� ) of the ratio r(V0) is defined, because of (4.3), as the solution of the equation
Toe procedure is iterative, namely:
* * *
8 (p) = O, i.e. 8 (r ) = 0
(4.4)
(4.5)
* *
where
z,
( � ) ,z
2 (� ) are the double kriging estimates of the true values z 1 (V0) . Z2 (V0) . The scheme (4.5) is based on the following argument:* *
**
Set ,;.+1 = rn + En in (4.2) to obtain 0(rn+1) = 0(,;. ) - Enz2( � ) . Because of (4.3), this suggests immediately to set:
which is precisely the iteration scheme (4.5).
* * We assume convergence and define Jim rn = r
Let 0 (p) = 8(p) + e(p), where e(p) is the estimation error satisfying E e(p) = O and
2
* * * * *
E e (p) = MSE(S (p)) . Hence, one can write 0 = 8 (r ) = 0(r )+ e(r ) and therefore
so that r ( � ) is asymptotically unbiased, with the mean square error approximately given by: *
(4.6)
in perfect analogy with the design-based approach (Mandallaz, 1991). In practice it appears that 0, 1 , 2 iterations suffice. It is worth noting that the above procedure also reduces the bias of the first iteration estimate.
4.2.5 Illustration
We now briefly illustrate double kriging and compare it with design-based regression techniques. The data originates from an intensive inventory performed on a Swiss Plateau forest enterprise of 218 ha.
Systematic cluster sampling was used to sample the auxiliary information; this information was provided by a stand map based on aerial photographs. The nominal cluster size was 5 (note that the effective cluster size, i.e. the number of points of a cluster falling into the forest area, is also a random variable,with a mean value of 4) ; the first phase sample resulted in 298 clusters and 1203 points. The ter-restrial inventory rests upon 300 m2 plots with the same cluster structure on a 1 :4 subgrid (73 cluster, 298 plots). A small area of 17 ha has been fully inventoried with deter-mination of the tree co
ordinates (4784 trees) to allow for validation. The quantities of interest are stem and basal area densities, as well as the percentage of non-healthy trees. The prediction model is a pure main effects ANOV A whose factors are:
(1) development stages (4 levels) (2) degree of mixture (2 levels) (3) crown closure index (2 levels)
The model is fitted with the actual data (the coefficients of determination were 0.5 for stem and 0.2 for basal area, Mandallaz, 1991, 1993). External models based on a yield table give essentially the same results. To estimate the density of non-healthy trees, the prediction of the total stem density was multiplied by a prediction of the percentage of non-healthy trees as obtained by a logistic model with the same factors as the above ANOV A model. The calculations required were perfonned with the software BLUEPACK (Ecole des Mines de Paris, 1990) on a VAX 9000-420. The results are summarized in tables I and 2 below.
Table 1: Entire Domain, 218 ha
The superiority of double kriging, with respect to bias and error, is striking for the small area estimation problem; a finding which is further substantiated if one splits the small area in squares of either 1 ha or 0.25 ha. The true values lie in the 95% confidence intervals for slightly more than 95% of
the squares. For such small areas the design-based estimates are either not even defined or usually useless.
The above results and other analysis of this data set provide statistical evidence that:
(1) The larger the domain, the smaller the difference between the kriging and design-based point and error estimates.
(2) The smaller the domain, the more pronouced the superiority of double kriging in terms of bias and error.
There are further empirical and theoretical arguments to support the view that these findings should hold for a wide range of forest inventories (Matheron, 1989; Mandallaz, 1991, 1993);
4.2.6 Conclusions
Double kriging is a simple geostatistical method particularly well suited to combined forest inventories based on double sampling schemes. It appears to be more efficient than the design-based regression estimate, especially for small area estimation problems.
4.2. 7 References
BLUEPACK, 1990: Manual, ENSM, Paris.
Cox, D.R., 1967: Fieller's theorem and a generalization. Biometrika 54, 567-572.
CRESSIE, N., 1991: Statistics for Spatial Data. New York, John Wiley & Sons.
JOURNEL, A.G.; HUIJBREGTS, C., 1978: Mining Geostatistics. London, Academic Press.
MANDALLAZ, D., 1991: A unified approach to sampling theory for forest inventory based on infinite population and super-population models. PhD thesis no 9378, ETH Ztlrich, Chair of foest inventory and planning, ETII Ztlrich.
MANDALLAZ, D., 1993: Geostatistical methods for double sampling schemes: application to combined forest inventories, Chair of forest inventory and planning, ETII Zurich.
MATHERON, G., 1970: La theorie des variables regionalisees et ses applications, Cahiers du centre de morphologie mathematique, no 5, Fontainebleau, France.
MATHERON, G., 1989: Estimating and choosing. Berlin, Springer.
4.3 Estimating the Probability and .Amount of Decayed Wood in Standing