Grégoire Dubois
Institute of Mineralogy & Petrography, University of Lausanne, Switzerland
gregoire.dubois@usa.net

Figure 1.  Sampling locations of the 467 measurements made in Switzerland

Figure 2.  Subset of 167 samples from Figure 1.

2. Nearest Neighbours Index (NNI)
The nearest neighbours index (NNI) (Clark and Evans, 1954) compares the distances between nearest samples to distances that would be expected by chance. The NNI is defined as the ratio of the mean of the Nearest Neighbours distances (NN_dist), that is

where N is the number of sampling points and, to the mean of the Nearest Neighbours distances for a uniform distribution of the points . This mean random distance (MRD) is defined as

where S_total is the total surface of the investigated area. The NNI is thus equal to

The NNI is close to 1 when the sampling points have an uniform spatial distribution. When NNI < 1, the samples are more clustered than expected compared to a random distribution. In the contrary, an NNI > 1 indicates a dispersion of the samples. The statistics of the distances to the nearest neighbours as well as the NNI are given in Table 1. The surface used as total surface is  the area of Switzerland (41 293 km²).

One will note two limitations to the method. If the whole data set can be described by the NNI, it does not indicate the clusters. Secondly, the geometry of the analysed surface can be complex and have an impact on the spatial distribution of the sampling points.

3. Fractal dimension of sampling networks
    Mandelbrot (1982) introduced the term of "fractal" which is used to describe continuous spatial phenomenon that present spatial correlation at different scales. For a linear fractal function, the Hausdorff-Besicovitch (D) dimension can vary between 1 (it can be completely derivated) and 2, the function is so irregular that it covers the whole space of 2 dimensions. Lovejoy et al. (1986) have applied fractals to describe de heterogeneity of measuring networks. When the heterogeneity of the network appears at different scales of a space of dimension E, it can be characterised by a fractal dimension D_m. The authors have also shown that each time D_m< E, phenomena of fractal dimension D_p < E - D_m cannot be detected by the network even if the density of the monitoring stations is infinite. D_m is defined by the variation of the average number n(R) of observations found within a circle with a varying radius R centered on each point of the monitoring network. A set of measures has a fractal dimension D_m if it satisfies the following condition

n(R)R ^Dm

Figure 3. Estimation of the fractal dimension of the 467 points data set.

Table 2. Fractal dimension of the sampling networks for both the 467 and 167 points data set. n (0-80 km) Drs (0-300 km) Drs 467 1.71 0.75 167 1.63 0.57

The second one is due to border effects. The circle used to define the average number of points falling within R will find only a part of those found when the circle is somewhere in the middle of the data (Tessier et al., 1994 ; Doswell and Lasher-Trapp, 1997).

4. Morisita Index
Another method is Morisita's Index (Morisita, 1959; Cressie, 1993, p. 578, 590-591). The analysed surface is divided into rectangular cells of equal size d and the index is defined as

where N is the number of points in the sampling network, n_i is the number of samples found within the i^th cell, and Q is the total number of cells. By displaying the values of the index against the size of the cells, one can investigate the degree of contagion of the sampling network, that is the probability that two points from the network fall within the same cell.

For a regular distribution of the samples in space, I_d  increases with the size of the cells to reach the value 1.

If the spatial distribution of the samples is random but homogeneous, I_d is independent of the size of cells and fluctuates around a mean value of 1.

When clusters are present, I_d becomes greater than 1.

Figure 4. Morisita's diagram for the 467 points data set

Figure 5. Morisita's diagram for the 167 points data set

The maximum of the index, that is the typical size of a cluster, is reached for a cell size of 19 km for 467 points and 22 km for 167 points.

5. Thiessen/Voronoi polygons
Thiessen polygons, also known as Voronoi polygons or Dirichlet cells  (Thiessen, 1911; Okabe et al., 1992) have the property to contain only one measurement and to have a geometry that will include all the data points that are closer to the measurement than to any other measurement. Isolated observations will therefore have polygons that will be larger than those associated to clustered data. These polygons are also frequently used to define weights that are used to decluster the data when an attempt is made to obtain statistics that are representative of the studied phenomenon over the whole surface of interest (Isaaks and Srivastava, 1989). Histograms of the areas of these polygons  might help to describe quantitatively the homogeneity of the sampling network: by knowing the average cell size, which represents the cell size one would have in case of an homogeneous network, one can evaluate the impact of extreme values on the sampling network. The Thiessen polygons of the 467 points data set are shown in Figure 6.

Figure 6. Thiessen polygons of the 467 sampling points.

Figure 7. Frequency histogram of the surfaces of the Thiessen polygons shown in Figure 6.

    The frequency histogram of the surfaces of these polygons (Figure 7) shows a skewed distribution: while the ideal cell size, that is the total surface divided by the number of samples, has a surface of 88.13 km², one can observe few polygons having extremely large areas and many small polygons highlighting the presence of clustered data.
    Two drawbacks appear when applying such a method. The first one is due to the fact that the borders of the polygons are frequently defined   by a convex hull which is constructed on the basis of the outern points. If such an approach is acceptable if one is working only within the geographical limits defined by the sampling network, a bias may be introduced when the shape of the analysed region does not correspond to the convex hull. Geographic Information Systems (GIS) facilitate the use of additional information that could better define the boundaries of the region of interest. For the here presented case study, the sampling network is expected to provide information for the whole surface of Switzerland. Reconstructing the Thiessen polygons with the help of the country borders will let appear a new map which is shown in Figure 8.

 

Figure 8. Thiessen/Voronoi polygons  of the 467 sampling points and country borders.

Figure 9. Comparison of the surfaces of the Thiessen/Voronoi polygons generated with and without country borders.

    The differences between the surfaces of the Thiessen polygons of the two maps (Figures 6 and  8) are shown in Figure 9. The use of country borders has clearly reduced the impact of the very large polygons found in the South of the country .
A second problem in using such an approach is that points can be clustered and still have relatively large Thiessen polygons as shown in Figure 10.

6. Coefficient of Representativity (CR)
One the basis of the preceding observations, a new measure that would combine both the surfaces of Thiessen polygons and the distance of each point to its nearest neighbour is proposed. This measure, which we will call Coefficient of Representativity (CR), is a product of two terms:

• a term called A which will take into account the surface of the Thiessen polygon. It is equal to the ration of the surface of the Thiessen polygon (S_Th) to the ideal surface it should have to obtain a homogeneous monitoring network. This surface is simply defined as the mean surface (S_m), that is the total area of the investigated area S_Total, divided by the number of sampling points N. A is therefore defined by

• the second term, B, is equal to the ratio of the squared distance between a point and its nearest neighbour (NN_dist) to the mean surface of the Thiessen polygons.

The choice of the squared distance is based on desire to compare the structure of a monitoring network to a reference, which is here a regular grid where points are distributed in the middle of each cell of the grid. For easier calculation, the regular square grid has been kept as reference and in such case, NN_dist² = S_Th and B = 1. If the point is closer to a neighbouring point, then B < 1 and in case the point is more distant than the theoretical distance, then B > 1.

The properties of the CR are summarised in Table 3.

Because a CR can be attributed to any sampling point, one can create maps of CR that should help to identify clustered data as well as regions where few data are available. For mapping purposes, one could use the logarithm of CR,

instead of the CR so that the CR < 1 become negative. Figure 11 displays the values taken by log₁₀(CR) for the 467 (top) and 167 (bottom) sampling points. Isolated points as well as those that are clustered are put in evidence, respectively by log₁₀ (CR) > 0 and a log₁₀ (CR) < 0.

Figure 11. Levels of log₁₀(CR) for 467 (top) and 167 (bottom) sampling points.

The mean, the standard deviation and the coefficient of variation (CV) of the values taken by the CR also allow the description of the structure of the sampling networks (Table 4).

        CV's greater than 1 indicate the presence of high erratic values that underline the irregularity of the sampling network. The CV for the 467 points data set is lower than for 167 sampling points and shows therefore an improvement in the spatial distribution of the sampling points even if clusters are still present (the mean value of the CR is < 1). The proposed method presents, however, a limitation. Points for which the CR = 1 (or log₁₀CR = 0) can not be automatically classified within the class of those points that fall in the middle of Thiessen polygons that have a surface that is equal to the expected average surface. A measurement located within a large polygon but close to another point can indeed have a CR = 1.
        If the CR can not be used directly to weight the samples prior to a declustering because of the contagion problem, that is that two points very close to each other will have a low CR, one can still make use of the CR's obtained for each point for  a declustering problem. The point that has the lowest CR can be averaged to its nearest neighbour and a new sampling network is obtained.  By proceeding in such a way and after several iterations, the CR of the network will tend to 1. A drawback is that such a process can be time consuming since the construction of the polygons has to be repeated after each iteration. Moreover, the use of boundaries that have a complex geometrical shape might require from the user to check to which polygons corresponds a single point since a single polygon can be split in several pieces when "clipping" it with the boundaries. The attempt to convert an irregular sampling network in a more regular one is similar to triangulations techniques that fill the holes with points until all triangles have the same sizes (Kanevsky and Savelieva, 1995). The main difference comes from the averaging of the data that will reduce systematically the amount of information while preserving the most important one while the triangulation method requires the sampling of new locations.

7. Conclusions
    "All samples are equal but some are more equal than others...". Isolated points require more attention than clustered data even if these last ones are essential for the analysis of microscale variations of the studied phenomenon. The Coefficient of Representativity should help decision makers to improve sampling campaigns by identifying undersampled areas as well as for the declustering of the data in order to obtain statistics that are representative of the analysed phenomenon. One can argue that the use of a single value to describe the complexity of sampling network is somehow limitating. It does, however, allow the comparison of different sampling strategies within a defined region. The CR benefits from the information provided not only by both the NNI and the surfaces of the Thiessen polygons, but also by the boundaries of the region of interest which clearly influences the spatial structure of a monitoring network. Used in combination with semivariograms, it should also provide an interesting information on the impact of each sample during an estimation problem.

References
Burgess, T. M.,  R. Webster and  A. B. McBratney (1981). Optimal interpolation and isarithmic mapping of soil properties. IV. Sampling strategy. Journal of Soil Science, 32: 643-659.
Burrough, P. A. (1991). Sampling Designs for Quantifying Map Unit Composition. In Spatial Variabilities of Soils and Landforms, Soil Science Society of America, SSSA Special Publication no. 28, pp. 89-125.
Clark, P. J. and  F. C. Evans (1954). Distance to nearest neighbor as a measure of spatial relationships in populations. Ecology, 35: 445-453.
Cressie, N. (1993). Statistics for spatial data. John Wiley & Sons Inc., Revised Edition, 900 p.
Doswell III C. A. and  S. Lasher-Trapp (1997). On measuring the degree of irregularity in an observing network. Journal of Atmospheric and Oceanic Technology, 14: 120-132.
Isaaks E. H. and  Srivastava R. M. (1989). An introduction to applied geostatistics. Oxford University Press.
Kanevsky M. and  Savelieva E. (1995). Environmental monitoring networks and quantitative description of clustering. In: Proceedings of the annual conference of the International Association for Mathematical Geology, IAMG, Osaka, Japan. pp. 31-32.
Lovejoy S., D. Schertzer and  P. Ladoy (1986). Fractal characterization of inhomogeneous geophysical measuring networks. Nature, 319: 43-44.
Mandelbrot, B. B. (1982). The fractal geometry of nature. W. H. Freeman.
Matheron G. (1963). Principles of geostatistics. Economical Geology, 58: 1246-1266
McBratney, A. B., R. Webster and T. M. Burgess (1981). The design of optimal sampling schemes for local estimation and mapping of ragionalized variables. I. Theory and method.  Computer & Geosciences 7(4): 331-334.
Morisita, M. (1959). Measuring of the dispersion and analysis of distribution patterns. Memoires of the Faculty of Science, Kyushu University, Series E. Biology. 2: 215-235.
Okabe A., B. Boots & K. Sugihara (1992). Spatial Tessellations. Concept and Applications of Voronoi Diagrams. Wiley and Sons.
Oliver M. A. and  Webster R. (1986). Combining nested and linear sampling for determining the scale and form of spatial variation of regionalized variables. Geographical Analysis, 18(3): 227-242.
Tessier Y., S. Lovejoy and D. Schertzer (1994). Multifractal analysis and simulation of the global meteorological network. Journal of Applied Meteorology, 33: 1572-1586.
Thiessen, A. H. (1911). Precipitation average for large areas. Monthly Weather Review, 39: 1082-1084.