Statistical inference for random T-tessellations models. Application to agricultural landscape modeling

Abstract

The Gibbsian T-tessellation models allow the representation of a wide range of spatial patterns. This paper proposes an integrated approach for statistical inference. Model parameters are estimated via Monte Carlo maximum likelihood. The simulations needed for likelihood computation are produced using an adapted Metropolis-Hastings-Green dynamics. In order to reduce the computational costs, a pseudolikelihood estimate is derived and then used for the initialization of the likelihood optimization. Model assessment is based on global envelope tests applied to the set of functional statistics of tessellation. Finally, a real data application is presented. This application analyzes three French agricultural landscapes. The Gibbs T-tessellation models simultaneously provide a morphological and statistical characterization of these data.

Appendices

A. Algorithm for approximating a landscape pattern by a T-tessellation

The algorithm first approximates a landscape pattern by a line-based tessellation containing the L-vertices, the I-vertices and the X-vertices (see Figs. 8 and 17). In the second step, these vertices are converted into T-vertices.

The algorithm requires the numerical representation of the landscape composed from the list of the plots coordinates, the list of the holes coordinates (the holes correspond to non-arable areas) and the coordinates of the outer domain.

Fig. 17

Transformation of segment-based tessellation into a T-tessellation: deletion of (a) I-vertex; (b) L-vertex; and (c) X-vertex. The original vertices are marked by dots, while the new T-vertices are marked by triangles. Dotted lines depict the removed edges and dashed lines depict the inserted edges

Grouping polygon sides

The algorithm starts by grouping the sides of polygons that are close and nearly aligned. This is done by hierarchical clustering based on the single linkage method. A measure of dissimilarity between the sides can be adjusted to account for specific landscape features. Hereafter, the following dissimilarity matrix was applied:

$$\begin{aligned} d(c_i,c_j)= \frac{A(C(c_i,c_j))}{l(c_i)l(c_j)}+\frac{2 d_{\min }(c_i,c_j)}{{l(c_i)+l(c_j)}}, \end{aligned}$$

(40)

where $A(C(c_i,c_j))$ denotes the area of the convex envelope of the sides $c_i$ and $c_j$, $d_{\min }(c_i,c_j)$ is the smallest Euclidean distance between $c_i$ and $c_j$, and l(.) denotes the length of the side.

Fig. 18

A cluster of polygon sides replaced by a representative segment (in blue)

Choice of representative segments

The clusters of sides are then replaced by representative segments. For a given cluster, a regression line is fitted to the endpoints of its sides. A representative segment is defined as the minimal segment containing the projections of all endpoints on the regression line (see Fig. 18). The segments representing the side clusters are inserted into a landscape domain. The resulting segment-based tessellation contains three types of vertices: I-vertices with one incident edge, L-vertices with two incident edges, and X-vertices formed by the crossing segments (see Fig. 17).

Transformation into T-tessellation

In order to transform an I vertex into a T-vertex, the incident edge is either removed or lengthened until it meets another edge. The modification that induces the smallest variation in total internal length of tessellation is selected. An L-vertex is transformed into a T-vertex by lengthening one of the incident edges. The choice of the edge is also based on the comparison of length variation.

The algorithm starts by iteratively removing the I-vertices: the vertex whose suppression induces the smallest variation in length is selected at each iteration. The L-vertices are then removed following the same rule. The resulting tessellation contains the T-vertices and the X-vertices. An X-vertex can be suppressed by moving the endpoint of one of its four incident edges along a neighboring edge. The procedure is repeated until there is no X-vertex left in the tessellation. Figure 17 illustrates the modifications of I-, L- and X-vertices.

B. Numerical examples of T-tessellation construction

1.1 B.1 Sampling from the CRTT model

This section gives an example of source code for sampling from the CRTT model with scale parameter $\theta$ (see Example 1) with Algorithm 1 implemented in the RLiTe package (see Adamczyk-Chauvat and Kiêu (2015)):

1.2 B.2 Random splits

In this section, the algorithm for simulating the T-tessellation of a bounded window W by a sequence of random splits is provided. It can be applied to create a tessellation of an empty window or to modify the non-empty tessellation, e.g., to add dummy splits to the existing tessellation in the pseudolikelihood optimization algorithm.

The following notation is used:

• $e=(p_1,p_2)$: an oriented edge of a tessellation T (halfedge for short) with the starting point $p_1$ and the ending point $p_2$;

• c(e): tessellation cell bounded by e and located on the left side of e

• ${{{\mathcal {E}}}}(T)$: ordered list of halfedges of T

Two auxiliary algorithms, 3 and 4, are first introduced and applied in the main algorithm 5.

Algorithm 3

Draw at random a half-edge bounding an internal face of T

Algorithm 4

Make a split starting from the selected halfedge of T

Algorithm 5

Random splits