# Statistical inference for random T-tessellations models: application to agricultural landscape modeling.

Katarzyna Adamczyk-Chauvat, Mouna Kassa, Julien Papaïx, Kiên Kiêu, Radu S. Stoica

## ► To cite this version:

Katarzyna Adamczyk-Chauvat, Mouna Kassa, Julien Papaïx, Kiên Kiêu, Radu S. Stoica. Statistical inference for random T-tessellations models: application to agricultural landscape modeling. 11th International Conference on Spatio-Temporal Modelling METMA XI, Lancaster University, Jul 2024, Lancaster, United Kingdom. [hal-04669756](https://hal.inrae.fr/hal-04669756v1)

HAL Id: hal-04669756

https://hal.inrae.fr/hal-04669756v1

Submitted on 9 Aug 2024

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Distributed under a Creative Commons Attribution 4.0 International License
Statistical inference for random T-tessellations models:
application to agricultural landscape modeling

Katarzyna Adamczyk-Chauvat¹

Kiên Kiêu¹

Mouna Kassa²

Radu S. Stoica³

Julien Papaïx¹

¹INRAE, Dep. MathNum, Jouy-en-Josas, ²INSA, Rennes, ³Université de Lorraine-CNRS-Inria, Nancy

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# Outline

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## 1 Random T-tessellation models

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## 2 Statistical inference

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## 3 Agricultural landscape model

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# T-tessellation

## Definition

A planar tessellation of a bounded window $W$ is called a *T-tessellation of W* if it satisfies two conditions:

(i) all the vertices in $\text{int}(W)$ are of degree 3,

(ii) two of the three edges connected by the same vertex are collinear.

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Introducing randomness

$$l_{(\theta,p)} \longleftrightarrow (\theta, p)$$

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$$\mathcal{L} = \{(\theta, p) : l_{(\theta,p)} \cap W \neq \emptyset\}$$

T-tessellations supported by the same line set $L \subset \mathcal{L}$:

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Poisson lines as a skeleton to define a random T-tessellation

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# Random T-tessellation models

## Definition (Completely Random T-tessellation model)

Let $\mathbf{L}$ be a unit-rate Poisson line process on $\mathcal{L}$. Let $\mathcal{T}(\mathbf{L})$ be the set of $T \in \mathcal{T}$ with segments lying on the lines of $\mathbf{L}$ and such that for each such line $l_i, l_i \cap T$ consists of exactly one segment. The CRTT model is defined by:

$$ \mu(A) = P(T \in A) = \frac{1}{Z} \mathbb{E} \left( \sum_{T \in \mathcal{T}(\mathbf{L})} \mathbb{1}_A(T) \right) $$

for $A \subseteq \mathcal{T}$ and $Z$ - normalising constant.

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# Random T-tessellation models

## Definition (Gibbs model)

Let $h: T \to [0, \infty[$ be the non-negative function, integrable with respect to the measure $\mu$. The Gibbs random T-tessellation model is defined by:

$$P(dT) \propto h(T)\mu(dT).$$

The function $U(T) = -\log h(T)$ is called the energy of the model.

Exponential family:

$$h_{\theta}(T) = \exp(t(T)^{\theta})$$

$t(T)$: vector of tessellation statistics, $\theta \in \Theta$: vector of model parameters.

<figure><img src="image_36.png" /></figure>
# Sampling from the Gibbs model

Metropolis-Hastings algorithm based on splits, merges and flips

**Required:** initial tessellation $T_0$, density $h(T)$, $(p_0, p_n, p_m)$, uniform measures $(q_S^T(S), q_M^T(M), q_F^T(F))$.

1: The current T-tessellation is $T_n$.

2: Draw the update type $t$ from $(s, m, f)$ with probabilities $(p_0, p_n, p_m)$.

3: If there is no update of type $t$ applicable to $T_n$ **then**

4: $T_{n+1} = T_n$

5: **else**

6: Draw the update $U$ of type $t$ from the distribution $q_T^U(U)$.

7: Compute the Hastings ratio

$$r_i(T, U) = \frac{h(U(T)) p_{i-1} q_{i-1}^T (U^{-1})}{h(T) p_i q_i^T (U)}$$

8: Accept the tessellation $T_{n+1} = U(T_n)$ with probability

$$\min(1, r_i(T_n, U)).$$

9: end if

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$\mathcal{S}_T, \mathcal{M}_T, \mathcal{F}_T$: sets of splits, merges, flips applicable to $T$

$q_S^T, q_M^T, q_F^T$: uniform measures on $\mathcal{S}_T, \mathcal{M}_T, \mathcal{F}_T$

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# Sampling from the Gibbs model

Metropolis-Hastings algorithm based on splits, merges and flips

Require: initial tessellation $T_0$, density $h(T)$, $(p_0, p_n, p_m)$, uniform measures $(q_S^T(S), q_M^T(M), q_F^T(F))$.

1: The current T-tessellation is $T_n$.

2: Draw the update type $t$ from $(s, m, f)$ with probabilities $(p_0, p_n, p_m)$.

3: If there is no update of type $t$ applicable to $T_n$ **then**

4: $T_{n+1} = T_n$

5: **else**

6: Draw the update $U$ of type $t$ from the distribution $q_U^T(U)$.

7: Compute the Hastings ratio

$$r_i(T, U) = \frac{h(U(T)) p_{i-1} q_{i-1}^T(U^{-1})}{h(T) p_i q_i^T(U)}$$

8: Accept the tessellation $T_{n+1} = U(T_n)$ with probability

$$\min(1, r_i(T_n, U)).$$

9: end if

<figure><img src="image_53.png" /></figure>

$\mathcal{S}_T, \mathcal{M}_T, \mathcal{F}_T$: sets of splits, merges, flips applicable to $T$

$q_S^T, q_M^T, q_F^T$: uniform measures on $\mathcal{S}_T, \mathcal{M}_T, \mathcal{F}_T$

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# Sampling from the Gibbs model

Metropolis-Hastings algorithm based on splits, merges and flips

**Required:** initial tessellation $T_0$, density $h(T)$, $(p_0, p_n, p_m)$, uniform measures $(q_S^T(S), q_M^T(M), q_F^T(F))$.

1: The current T-tessellation is $T_n$.

2: Draw the update type $t$ from $(s, m, f)$ with probabilities $(p_0, p_n, p_m)$.

3: If there is no update of type $t$ applicable to $T_n$ **then**

4: $T_{n+1} = T_n$

5: **else**

6: Draw the update $U$ of type $t$ from the distribution $q_U^T(U)$.

7: Compute the Hastings ratio

$$r_i(T, U) = \frac{h(U(T))}{h(T)} \frac{p_{i-1} q_{i-1}^T(U^{-1})}{p_i q_i^T(U)}$$

8: Accept the tessellation $T_{n+1} = U(T_n)$ with probability

$$\min(1, r_i(T_n, U)).$$

9: end if

<figure><img src="image_55.png" /></figure>

$\mathcal{S}_T, \mathcal{M}_T, \mathcal{F}_T$: sets of splits, merges, flips applicable to $T$

$q_S^T, q_M^T, q_F^T$: uniform measures on $\mathcal{S}_T, \mathcal{M}_T, \mathcal{F}_T$

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# Sampling from the Gibbs model

Metropolis-Hastings algorithm based on splits, merges and flips

Require: initial tessellation $T_0$, density $h(T)$, $(p_0, p_n, p_m)$, uniform measures $(q_S^T(S), q_M^T(M), q_F^T(F))$.

1: The current T-tessellation is $T_n$.

2: Draw the update type $t$ from $(s, m, f)$ with probabilities $(p_0, p_n, p_m)$.

3: If there is no update of type $t$ applicable to $T_n$ **then**

4: $T_{n+1} = T_n$

5: **else**

6: Draw the update $U$ of type $t$ from the distribution $q_U^T(U)$.

7: Compute the Hastings ratio

$$r_i(T, U) = \frac{h(U(T)) p_{i-1} q_{i-1}^T(U^{-1})}{h(T) p_i q_i^T(U)}$$

8: Accept the tessellation $T_{n+1} = U(T_n)$ with probability

$$\min(1, r_i(T_n, U)).$$

9: end if

<figure><img src="image_71.png" /></figure>

$\mathcal{S}_T, \mathcal{M}_T, \mathcal{F}_T$: sets of splits, merges, flips applicable to $T$

$q_S^T, q_M^T, q_F^T$: uniform measures on $\mathcal{S}_T, \mathcal{M}_T, \mathcal{F}_T$

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# Sampling from the Gibbs model

Metropolis-Hastings algorithm based on splits, merges and flips

**Required:** initial tessellation $T_0$, density $h(T)$, $(p_0, p_n, p_m)$, uniform measures $(q_S^T(S), q_M^T(M), q_F^T(F))$.

1: The current T-tessellation is $T_n$.

2: Draw the update type $t$ from $(s, m, f)$ with probabilities $(p_0, p_n, p_m)$.

3: If there is no update of type $t$ applicable to $T_n$ **then**

4: $T_{n+1} = T_n$

5: **else**

6: Draw the update $U$ of type $t$ from the distribution $q_U^T(U)$.

7: Compute the Hastings ratio

$$r_i(T, U) = \frac{h(U(T)) p_{i-1} q_{i-1}^T(U^{-1})}{h(T) p_i q_i^T(U)}$$

8: Accept the tessellation $T_{n+1} = U(T_n)$ with probability

$$\min(1, r_i(T_n, U)).$$

9: end if

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$\mathcal{S}_T, \mathcal{M}_T, \mathcal{F}_T$: sets of splits, merges, flips applicable to $T$

$q_S^T, q_M^T, q_F^T$: uniform measures on $\mathcal{S}_T, \mathcal{M}_T, \mathcal{F}_T$

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Examples of model simulations

<figure><img src="image_90.png" /><figcaption>CRTT</figcaption></figure>

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# Additional tools

## Definition

Let $\mathcal{C}_s = \{(s, T) : T \in \mathcal{T}, s \in \mathbb{S}_T\}$. The split Campbell measure of $\mathbf{T} \sim P$ is defined by:

$$C_s^!(B) = \mathbb{E} \sum_{s \in \mathbb{M}_T} \mathbb{I}_B(s, \mathbf{T} \setminus \{s\}) \quad \forall B \subset \mathcal{C}_s$$

$$C_s^!(ds, dT) = \underbrace{\frac{h(T \cup \{s\})}{h(T)}}_{\text{split Papangelou intensity } \lambda_s(s, T)} \, dsP(dT)$$

Analogous definitions hold for flips.

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## Outline

1. Random T-tessellation models

2. Statistical inference

3. Agricultural landscape model

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Minimum contrast estimation

$$
\begin{align*}
\rho(\theta; T) = & - \sum_{m \in \mathbb{M}_T} \log \lambda_{s,\theta}(m^{-1}, m(T)) + \int_{\mathbb{F}_T} \lambda_{s,\theta}(s, T) ds \\
& - \sum_{f \in \mathbb{F}_T} \log \lambda_{f,\theta}(f^{-1}, f(T)) + \sum_{f \in \mathbb{F}_T} \lambda_{f,\theta}(f, T)
\end{align*}
$$

Proposition

If $\mathbf{T} \sim P_{\theta^*}$ then for every $\theta \in \Theta$:

$$
\begin{align*}
\mathbb{E}_{\theta^*} \rho(\theta; \mathbf{T}) = & - \int_{\mathcal{C}_s} \log \lambda_{s,\theta}(s, T) C_{s,\theta^*}^1 (ds, dT) + \int_{\mathcal{C}_s} \lambda_{s,\theta}(s, T) ds P_{\theta^*}(dT) \\
& - \int_{\mathcal{C}_f} \log \lambda_{f,\theta}(s, T) C_{f,\theta^*}^1 (df, dT) + \int_{\mathcal{C}_f} \lambda_{f,\theta}(f, T) df P_{\theta^*}(dT)
\end{align*}
$$

$\mathbb{E}_{\theta^*} \rho(\theta; \mathbf{T})$ has a global minimum at $\theta = \theta^*$.

$$
\hat{\theta}_{MPL} = \arg \max_{\theta} (-\rho(\theta; T))
$$

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# Statistical tools extended to T-tessellations

* Monte Carlo Maximum Likelihood to refine the pseudolikelihood estimation

$$ \underbrace{\hat{L}(\theta; T)}_{\text{MC log likelihood}} = - \left\langle t(T); \theta \right\rangle - \log \frac{1}{n} \sum_{i=1}^{n} \exp \left( - \left\langle t(T_i); \theta - \theta_0 \right\rangle \right) + \text{const} $$

* Model assessment based on global envelope tests:

$H_0$: observed T-tessellation $\sim P_\theta$

$F = (F(r_1), \dots, F(r_d))$: functional statistic of tessellation

$\underbrace{F_1}_{\text{data}}$, $\underbrace{F_2, \dots, F_s}_{\text{model simulations}}$

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## Outline

1. Random T-tessellation models

2. Statistical inference

3. Agricultural landscape model

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Landscapes approximated by T-tessellations

<figure><img src="image_121.png" /><figcaption>$L_1$: Selommes</figcaption></figure>

<figure><img src="image_122.png" /><figcaption>$L_2$: Kervidy</figcaption></figure>

<figure><img src="image_123.png" /><figcaption>$L_3$: BVD</figcaption></figure>

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Landscapes approximated by T-tessellations

<figure><img src="image_125.png" /><figcaption>$L_1$: Selommes</figcaption></figure>

<figure><img src="image_126.png" /><figcaption>$L_2$: Kervidy</figcaption></figure>

<figure><img src="image_127.png" /><figcaption>$L_3$: BVD</figcaption></figure>

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# Model construction

Guidelines:

* accounts for typical landscape features

* highlights the main differences between patterns

$$U_{\theta}(T) = \theta_{1} \text{ number\_of\_segments}(T) \leftarrow \text{scale statistic } t_{1}(T)$$

$$+ \theta_{2} \sum_{\text{faces } f} \text{area}(f)^{2} \leftarrow \text{area statistic } t_{2}(T)$$

$$+ \theta_{3} \sum_{\text{vertices } v} \left( \frac{\pi}{2} - \alpha(v) \right) \leftarrow \text{angle statistic } t_{3}(T)$$

$$+ \theta_{4} \sum_{\text{faces } f} g\left(\frac{\text{length}}{\text{width}}(f)\right) \leftarrow \text{elongation statistic } t_{4}(T)$$

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Estimation results

<table><thead><tr><th>Model parameter</th><th>Landscape</th><th>θ<sub>MPL</sub></th><th>θ<sub>MCML</sub></th></tr></thead><tbody><tr><td rowspan="3">θ<sub>1</sub> (scale)</td><td>Selommes</td><td>1.62</td><td>1.99 (0.12)</td></tr><tr><td>Kervidy</td><td>1.43</td><td>1.70 (0.11)</td></tr><tr><td>BVD</td><td>2.32</td><td>2.11 (0.13)</td></tr><tr><td rowspan="3">θ<sub>2</sub> (areas)</td><td>Selommes</td><td>68.36</td><td>181.19 (66)</td></tr><tr><td>Kervidy</td><td>23.73</td><td>79.19 (56)</td></tr><tr><td>BVD</td><td>1361.46</td><td>2561.46 (679)</td></tr><tr><td rowspan="3">θ<sub>3</sub> (angles)</td><td>Selommes</td><td>0.95</td><td>2.23 (0.15)</td></tr><tr><td>Kervidy</td><td>0.77</td><td>1.26 (0.12)</td></tr><tr><td>BVD</td><td>1.72</td><td>2.21 (0.16)</td></tr><tr><td rowspan="3">θ<sub>4</sub> (elongation)</td><td>Selommes</td><td>0.48</td><td>0.28 (0.17)</td></tr><tr><td>Kervidy</td><td>0.31</td><td>0.07 (0.15)</td></tr><tr><td>BVD</td><td>-0.64</td><td>-0.63 (0.08)</td></tr></tbody></table>

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Model checking

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$t_1(T)$

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$t_2(T)$

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$t_3(T)$

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$t_4(T)$

Distributions of energy statistics calculated over 500 model runs.

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Observed and simulated landscape pattern.

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# Goodness-of-fit

GET for tessellation and/or landscape metrics c.d.f.

* features of a typical cell: the area $a(c)$, the perimeter $p(c)$, the width $w(c)$, the length $l(c)$ of the smallest rectangle encompassing c and the isoperimetric quotient $r(c)$;

* spatial pattern of cell centroids: $G(r)$, $F(r)$, $L(r)$, $D(o)$ (density of the nearest neighbor vector orientation)

<figure><img src="image_152.png" /><figcaption>Cell centroid pattern</figcaption></figure>

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GET for the features of a typical cell

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GET for the summary statistics of cell centroids

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Concluding remarks

* Gibbs T-tessellation model: feature-driven model built on Poisson line skeleton;

* statistical methods: MPL estimation, adaptation of point process methods;

* application: characterization of agricultural landscapes... and other patterns that can be approximated by T-tessellations.

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References

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[1] K. Kiêu, K. Adamczyk-Chauvat, H. Monod, R.S. Stoica. A completely random T-tessellation model and Gibbsian extensions. *Spatial Statistics*, 6, 118-138 (2013).

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[2] K. Kiêu and K. Adamczyk-Chauvat. Pseudolikelihood inference for Gibbsian T-tessellations... and point processes. Preprint HAL, hal-01234119, 2015.

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[3] K. Adamczyk-Chauvat and K. Kiêu http://kien-kieu.github.io/lite.

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[4] K. Adamczyk-Chauvat, M. Kassa, J. Papaïx, K. Kiêu, R.S. Stoica. Statistical inference for random T-tessellations models. Application to agricultural landscape modeling. *Ann Inst Stat Math*, 76, 447-479 (2024).

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