Bayesian Modeling for Nonstationary Spatial Point Process via Spatial Deformations
<p>Intensity function given by expression (<a href="#FD12-entropy-26-00678" class="html-disp-formula">12</a>) exhibited in different angles.</p> "> Figure 2
<p>Estimation of the intensity function of expression (<a href="#FD12-entropy-26-00678" class="html-disp-formula">12</a>): (<b>a</b>) True intensity function on the defined grid along with generated point process (dots), (<b>b</b>) estimated intensity function not considering and (<b>c</b>) considering the deformation for <span class="html-italic">probit</span> link function, (<b>d</b>) estimated intensity function not considering and (<b>e</b>) considering the deformation for <span class="html-italic">log</span> link function.</p> "> Figure 3
<p>Further details of estimation of the intensity function (<a href="#FD12-entropy-26-00678" class="html-disp-formula">12</a>). Scatter plot of true and estimated intensity function for different estimating scenarios: (<b>a</b>) intensity function not considering and (<b>b</b>) considering the deformation for <span class="html-italic">probit</span> link function, (<b>c</b>) intensity function not considering and (<b>d</b>) considering the deformation for <span class="html-italic">log</span> link function. The dots represent the pair of true and estimated intensity function values for all pixels. Full and dashed lines represent the identity and the exploratory regression lines, respectively. The latter is obtained by performing the fit of the estimated values based on the assumed true values as a covariate.</p> "> Figure 4
<p>For the scenario of expression (<a href="#FD12-entropy-26-00678" class="html-disp-formula">12</a>): (<b>a</b>) IS for the model not considering and (<b>b</b>) considering the deformation for <span class="html-italic">probit</span> link function, (<b>c</b>) IS for the model not considering and (<b>d</b>) considering the deformation for <span class="html-italic">log</span> link function.</p> "> Figure 5
<p>Intensity function estimation for the four selected locations: true intensity function (<b>top</b>) and density plot for the selected locations with their respective credibility interval of 95% (<b>bottom</b>) for the four models considered.</p> "> Figure 6
<p>Estimation of the deformation <span class="html-italic">d</span>: (<b>a</b>) estimated mean deformation for the <span class="html-italic">probit</span> link function; (<b>b</b>) mean and (<b>c</b>) standard deviation for the posterior distribution of the distance between each centroid <math display="inline"><semantics> <msub> <mi>s</mi> <mi>z</mi> </msub> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>(</mo> <msub> <mi>s</mi> <mi>z</mi> </msub> <mo>)</mo> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>∈</mo> <mo>{</mo> <mn>1</mn> <mo>,</mo> <mspace width="0.166667em"/> <mo>…</mo> <mo>,</mo> <mspace width="0.166667em"/> <mi>Z</mi> <mo>}</mo> </mrow> </semantics></math> for the <span class="html-italic">probit</span> link function.</p> "> Figure 7
<p>Point pattern for the corn plants infected by the <span class="html-italic">S. frugiperda</span> with (<b>a</b>) original and (<b>b</b>) rotated latitude and longitude and (<b>c</b>) a satellite image of the experimental area with the polygon of the study area. Adapted from Nava et al. [<a href="#B37-entropy-26-00678" class="html-bibr">37</a>].</p> "> Figure 8
<p>Estimation of the intensity function for <span class="html-italic">Spodoptera frugiperda</span> pest data: (<b>a</b>) not considering and (<b>b</b>) considering the deformation for probit link function, (<b>c</b>) not considering and (<b>d</b>) considering the deformation for log link function.</p> "> Figure 9
<p>Standard deviation for the intensity function for the model for the <span class="html-italic">Spodoptera frugiperda</span> pest data: (<b>a</b>) not considering and (<b>b</b>) considering the deformation for <span class="html-italic">probit</span> link function, (<b>c</b>) not considering and (<b>d</b>) considering the deformation for <span class="html-italic">log</span> link function.</p> "> Figure 10
<p>Estimated intensity function map for the <span class="html-italic">Spodoptera frugiperda</span> pest data showing the positions of intensity function selected for posterior density comparison across models (<b>top</b>) and density plot for the selected positions with their respective credibility interval of 95% (<b>bottom</b>).</p> "> Figure 11
<p>Estimation of the deformation <span class="html-italic">d</span> for the <span class="html-italic">Spodoptera frugiperda</span> pest data: (<b>a</b>) estimated mean deformation for the <span class="html-italic">probit</span> link function; (<b>b</b>) mean and (<b>c</b>) standard deviation for the posterior distribution of the distance between each centroid <math display="inline"><semantics> <msub> <mi>s</mi> <mi>z</mi> </msub> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>(</mo> <msub> <mi>s</mi> <mi>z</mi> </msub> <mo>)</mo> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>z</mi> <mo>∈</mo> <mo>{</mo> <mn>1</mn> <mo>,</mo> <mspace width="0.166667em"/> <mo>…</mo> <mo>,</mo> <mspace width="0.166667em"/> <mi>Z</mi> <mo>}</mo> </mrow> </semantics></math> for the <span class="html-italic">probit</span> link function.</p> "> Figure 12
<p>Estimated (<b>a</b>) intensity function and (<b>b</b>) DP for the model considering the <span class="html-italic">probit</span> link function for unrotated and rescaled <span class="html-italic">Spodoptera frugiperda</span> pest data.</p> ">
Abstract
:1. Introduction
2. Methodology
3. Results
3.1. Simulating the Process with a Deterministic Intensity
3.2. Real Data Application—Spodoptera frugiperda Pest in a Corn-Producing Agricultural Area
3.3. Comparative Analysis of Spatial Deformation and Geometric Anisotropy
4. Discussion
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
IS | Interval score |
AIS | Averaged interval score |
CI | Credibility interval |
CPO | Conditional predictive ordinate |
DIC | Deviance information criterion |
DP | Deformation process |
GP | Gaussian process |
IS | Interval score |
HMC | Hamiltonian Monte Carlo |
LGCP | Log–Gaussian Cox process |
MAE | Mean absolute error |
MCMC | Markov Chain Monte Carlo |
PP | Poisson process |
slCPO | Sum of the log of the conditional predictive ordinate |
SD | Standard deviation |
Appendix A. Stan Codes
Appendix A.1. Isotropic Scenario with Log Link Function
Appendix A.2. Deformation Process with Log Link Function
Appendix A.3. Isotropic Scenario with Probit Link Function
Appendix A.4. Deformation Process with Probit Link Function
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Link | Deformation | MAE | AIS | slCPO | DIC |
---|---|---|---|---|---|
probit | without | 180.76 | 1463.83 | −507.68 | 988.25 |
with | 133.5 | 1163.06 | −479.31 | 947.21 | |
log | without | 199.43 | 1588.27 | −508.09 | 1001.09 |
with | 154.67 | 1198.28 | −484.63 | 966.1 |
Deformation | (CI) | (CI) |
---|---|---|
without | 1646.7 (1464; 1889.6) | 0.51 (0.21; 1.11) |
with | 1697.4 (1518.5; 1933.2) | 0.65 (0.28; 1.36) |
Link | Deformation | slCPO | DIC |
---|---|---|---|
probit | without | −550.584 | 1075.75 |
with | −537.145 | 1055.69 | |
log | without | −553.976 | 1087.62 |
with | −543.871 | 1069.05 |
Deformation | Link | Mean Intensity Function(loc[1]) | lCPO[loc[1]] | Mean Intensity Function(loc[2]) | lCPO[loc[2]] |
---|---|---|---|---|---|
without | probit | 1743.21 | − 4.03 | 2048.23 | −3.54 |
log | 1973.68 | −3.72 | 2144.00 | −3.93 | |
with | probit | 2187.37 | −2.68 | 2285.29 | −2.90 |
log | 2205.25 | −2.94 | 2392.56 | −2.99 | |
Deformation | Link | mean intensity function(loc[3]) | lCPO[loc[3]] | mean intensity function(loc[4]) | lCPO[loc[4]] |
without | probit | 1840.86 | −2.56 | 1754.79 | −3.35 |
log | 1821.05 | −2.75 | 1882.98 | −3.27 | |
with | probit | 2048.01 | −2.28 | 1846.86 | −2.91 |
log | 1939.08 | −2.43 | 1876.19 | −3.00 |
Link | Deformation | (CI) | (CI) |
---|---|---|---|
probit | without | 3648.1 (2895.8; 4526.0) | 1.88 (0.77; 4.01) |
with | 3698.7 (2993.7; 4506.2) | 1.71 (0.68; 3.59) |
Link | Anisotropy | MAE | AIS | slCPO | DIC |
---|---|---|---|---|---|
probit | Geometric | 198.82 | 1529.01 | − 505.94 | 985.27 |
Deformation | 133.5 | 1163.06 | −479.31 | 947.21 | |
log | Geometric | 205.71 | 1639.04 | −507.24 | 997.5 |
Deformation | 154.67 | 1198.28 | −484.63 | 966.1 |
Link | Anisotropy | slCPO | DIC |
---|---|---|---|
probit | Geometric | −548.351 | 1072.98 |
Deformation | −537.145 | 1055.69 | |
log | Geometric | −554.508 | 1087.49 |
Deformation | −543.871 | 1069.05 |
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Gamerman, D.; Quintana, M.d.S.B.; Alves, M.B. Bayesian Modeling for Nonstationary Spatial Point Process via Spatial Deformations. Entropy 2024, 26, 678. https://doi.org/10.3390/e26080678
Gamerman D, Quintana MdSB, Alves MB. Bayesian Modeling for Nonstationary Spatial Point Process via Spatial Deformations. Entropy. 2024; 26(8):678. https://doi.org/10.3390/e26080678
Chicago/Turabian StyleGamerman, Dani, Marcel de Souza Borges Quintana, and Mariane Branco Alves. 2024. "Bayesian Modeling for Nonstationary Spatial Point Process via Spatial Deformations" Entropy 26, no. 8: 678. https://doi.org/10.3390/e26080678