# Bayesian Networks for Raster Data (BayNeRD): Plausible Reasoning from Observations

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Bayesian Networks

_{1}, V

_{n}) to denote both a variable and its corresponding node, and the same but lower-case letters (e.g., v

_{1}, v

_{n}) to denote the state or value (defining a particular instantiation) of the variable. Then, the joint probability distribution for any particular instantiation of all n variables in a BN is given by:

_{i}represents the instantiation of variable V

_{i}and ϕ

_{i}represents the instantiation of its parents Φ

_{i}, with i varying from 1 to n. Parent variables are those whose instantiations directly influence other, descendent variables. The arcs (represented by arrows in the DAG) encode the conditional dependencies (i.e., which variables are parent/descendant of other variables) [9,11]. The joint probability of any instantiation of all the variables in a BN can be computed as the product of only n probabilities. Thus, we can determine any probability of the form:

_{i}are sets of variables with known values (v

_{i}, i.e., instantiated variables). This ability to compute posterior probabilities given some evidence is called inference. In the case of using Equation (2) for inferences about certain phenomena using BayNeRD, we named the variable that represents the phenomenon as the target variable and the variables that can be used to describe an outline of the phenomenon as context variables (i.e., those variables that are somehow related to the phenomenon).

## 3. Framework of the Implemented BayNeRD Algorithm in R Software

#### 3.1. Target Variable

#### 3.2. Context Variables

#### 3.3. Designing the Bayesian Network Graphical Model

#### 3.4. Discretization and Probability Functions

#### 3.5. Computing the Probability Image

_{1}, V

_{2}, … and V

_{n}individually influence the probability computed for Y by computing KL divergences between conditional and marginal probabilities in the BN model.

#### 3.6. Selecting the Target Probability Value

## 4. Case Study of Soybean Mapping in Brazil: Materials and Research Methods

^{2}[32]. Figure 2 shows the location of Mato Grosso State, highlighting thirty 30 × 30 km plots (and the Landsat path/row covering them) of reference data produced by Epiphanio et al. [33] for the crop year 2005/2006 (i.e., from August 2005 to July 2006) using visual interpretation of Landsat-5/TM images and field data. Additional data such as indigenous lands, conservation units, mapped forests and floodplains were used to mask out areas of no interest for mapping soybean (as will be described further).

#### 4.1. Variables

- (1)
- Target variable—soybean occurrence (S) corresponding to the studied phenomenon, represented by a thematic map with four classes for the crop year 2005/2006: (i) target presence observed (i.e., soybean); (ii) target absence observed (i.e., non-soybean); (iii) missing data (i.e., no observations); and (iv) pixels outside the study area. This thematic map, produced by Epiphanio et al. [33], was used as a reference in this study. In the BayNeRD modelling, S = s, where s = 1 for soybean presence and s = 0 for soybean absence. Two thirds of the pixels in each of the thematic class soybean and non-soybean were randomly selected from the reference map to compose the reference data for training. The remaining third of the reference map pixels was set aside to be used for accuracy assessment (reference data for testing).
- (2)
- Context variables—the selected and available variables to compose the model are listed in Table 1. From expert knowledge it is known that each context variable influences soybean occurrence (S).

#### 4.2. Bayesian Network Model

#### 4.3. Discretization and Probability Functions

#### 4.4. PI

## 5. Results and Discussion

#### 5.1. Probability Image (PI)

_{C}= 0.28 and KL

_{L}= 0.16, i.e., the KL divergence for C and L, respectively). It means that, as pointed out by Risso et al. [64], a proper vegetation index taken at key dates over the crop calendar can be used to identify specific crops such as soybean [69]. In fact, due to its ability and practicability to detect soybean areas, CEI is also used to monitor soybean plantations in the Brazilian Amazon Biome in the context of the Soy Moratorium [65,66]. For the remaining context variables A, T, W, and R, the KL divergences were 0.009, 0.002, 0.003, and 0.0001, respectively. This result means that soil type influenced more the calculated probability of soybean presence then terrain slope, water distance, and especially the distance to a road.

_{W}= 0.003), any decrease in the calculated probability of soybean presence is likely to be very small. However if the context variable has a strong relationship with soybean occurrence (for example C, which presented KL

_{C}= 0.28), any unfavorable condition of this variable is likely to decrease soybean probability values substantially. Additionally, the mixing within a pixel size of 250 × 250 m (defined as our nominal spatial resolution), especially over the boundaries of defined discretized intervals, could be noted in Figure 10, which presented both orange and light-green colored pixels surrounding green pixels in the PI.

#### 5.2. Creating Thematic Maps from the PI

## 6. Conclusion

## Acknowledgments

## Conflicts of Interest

## References

- Melesse, A.M.; Weng, Q.; Thenkabail, P.S.; Senay, G.B. Remote sensing sensors and applications in environmental resources mapping and modelling. Sensors
**2007**, 7, 3209–3241. [Google Scholar] - Donner, R.; Barbosa, S.; Kurths, J.; Marwan, N. Understanding the earth as a complex system—Recent advances in data analysis and modelling in Earth sciences. Eur. Phys. J. Spec. Top
**2009**, 174, 1–9. [Google Scholar] - Li, Z.; Chen, J.; Baltsavias, E. (Eds.) Advances in Photogrammetry, Remote Sensing and Spatial Information Sciences: 2008 ISPRS Congress Book, 1st ed; CRC Press: Trowbridge, UK, 2008; Volume 23, p. 546.
- Lee, C.A.; Gasster, S.D.; Plaza, A.; Chang, C.-I.; Huang, B. Recent developments in high performance computing for remote sensing: A review. IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens
**2011**, 4, 508–527. [Google Scholar] - Lu, D.; Weng, Q. A survey of image classification methods and techniques for improving classification performance. Int. J. Remote Sens
**2007**, 28, 823–870. [Google Scholar] - Richards, J.A. Analysis of remotely sensed data: The formative decades and the future. IEEE Trans. Geosci. Remote Sens
**2005**, 43, 422–432. [Google Scholar] - Jaynes, E.T. Probability Theory: The Logic of Science; Cambridge University Press: Cambridge, UK, 2003; p. 727. [Google Scholar]
- McGrayne, S.B. The Theory that would not Die: How Bayes’ Rule Cracked the Enigma Code, Hunted down Russian Submarines, and Emerged Triumphant from Two Centuries of Controversy; Yale University Press: New Haven, CT, USA, 2011; p. 336. [Google Scholar]
- Pearl, J. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference, 1st ed; Morgan Kaufmann: San Francisco, CA, USA, 1988; p. 552. [Google Scholar]
- Jensen, F.V.; Nielsen, T.D. Bayesian Networks and Decision Graphs, 2nd ed; Springer: New York, NY, USA, 2007; p. 447. [Google Scholar]
- Neapolitan, R.E. Learning Bayesian Networks; Prentice Hall: Upper Saddle River, NJ, USA, 2003; p. 674. [Google Scholar]
- Darwiche, A. Modeling and Reasoning with Bayesian Networks; Cambridge University Press: New York, NY, USA, 2009; p. 560. [Google Scholar]
- Heckerman, D. Bayesian networks for data mining. Data Min. Knowl. Discov
**1997**, 1, 79–119. [Google Scholar] - Uusitalo, L. Advantages and challenges of Bayesian networks in environmental modelling. Ecol. Model
**2007**, 203, 312–318. [Google Scholar] - Aguilera, P.A.; Fernández, A.; Fernández, R.; Rumí, R.; Salmerón, A. Bayesian networks in environmental modelling. Environ. Model. Softw
**2011**, 26, 1376–1388. [Google Scholar] - Garrett, R.D.; Lambin, E.F.; Naylor, R.L. Land institutions and supply chain configurations as determinants of soybean planted area and yields in Brazil. Land Use Policy
**2013**, 31, 385–396. [Google Scholar] - Park, S.; McSweeney, K.; Lowery, B. Identification of the spatial distribution of soils using a process-based terrain characterization. Geoderma
**2001**, 103, 249–272. [Google Scholar] - Cooper, G.F.; Herskovits, E. A Bayesian method for the induction of probabilistic networks from data. Mach. Learn
**1992**, 9, 309–347. [Google Scholar] - Mello, M.P.; Rudorff, B.F.T.; Adami, M.; Rizzi, R.; Aguiar, D.A.; Gusso, A.; Fonseca, L.M.G. A Simplified Bayesian Network to Map Soybean Plantations. Proceedings of 2010 IEEE International Geoscience and Remote Sensing Symposium, Honolulu, HI, USA, 25–30 July 2010; pp. 351–354.
- R Core Team. R: A Language and Environment for Statistical Computing; R Foundation for Statistical Computing: Vienna, Austria, 2013. [Google Scholar]
- Crawley, M.J. The R Book; John Wiley & Sons: Chichester, UK, 2007; p. 950. [Google Scholar]
- Bivand, R.S.; Pebesma, E.J.; Gómez-Rubio, V. Applied Spatial Data Analysis with R; Springer: New York, NY, USA, 2008; p. 378. [Google Scholar]
- Albert, J. Bayesian Computation with R, 2nd ed; Springer: New York, NY, USA, 2009; p. 298. [Google Scholar]
- Balov, N.; Salzman, P. catnet: Categorical Bayesian Network Inference. R Package Version 1.14.2. Available online: http://CRAN.R-project.org/package=catnet (accessed on 18 August 2013).
- Kullback, S.; Leibler, R.A. On information and sufficiency. Ann. Math. Stat
**1951**, 22, 79–86. [Google Scholar] - Zweig, M.H.; Campbell, G. Receiver-operating characteristic (ROC) plots: A fundamental evaluation tool in clinical medicine. Clin. Chem
**1993**, 39, 561–577. [Google Scholar] - Altman, D.G.; Bland, J.M. Statistics notes: Diagnostic tests 1: Sensitivity and specificity. BMJ
**1994**, 308, 1552–1552. [Google Scholar] - Cohen, J. A coefficient of agreement for nominal scales. Educ. Psychol. Measur
**1960**, 20, 37–46. [Google Scholar] - Hudson, W.D. Correct formulation of the Kappa coefficient of agreement. Photogramm. Eng. Remote Sens
**1987**, 53, 421–422. [Google Scholar] - Congalton, R.G.; Green, K. Assessing the Accuracy of Remotely Sensed Data: Principles and Pratices, 2nd ed; CRC Press: Boca Raton, FL, USA, 2009; p. 183. [Google Scholar]
- CONAB. Séries Históricas Relativas às Safras 1976/77 a 2011/2012 de Área Plantada, Produtividade e Produção. Available online: http://www.conab.gov.br/conteudos.php?a=1252&t= (accessed on 21 March 2013).
- BRASIL. Resolução da Presidência do IBGE de n° 5 (R.PR-5/02) de 10 de outubro de 2002; Diário Oficial da União: Brasília, DF, Brazil, 2002; pp. 48–69. [Google Scholar]
- Epiphanio, R.D.V.; Formaggio, A.R.; Rudorff, B.F.T.; Maeda, E.E.; Luiz, A.J.B. Estimating soybean crop areas using spectral-temporal surfaces derived from MODIS images in Mato Grosso, Brazil. Pesquisa Agropecuária Brasileira
**2010**, 45, 72–80. [Google Scholar] - FAO. FAOSTAT: FAO Statistical Database. Available online: http://faostat.fao.org (accessed on 9 April 2012).
- Atzberger, C. Advances in remote sensing of agriculture: Context description, existing operational monitoring systems and major information needs. Remote Sens
**2013**, 5, 949–981. [Google Scholar] - Rudorff, B.F.T.; Aguiar, D.A.; Silva, W.F.; Sugawara, L.M.; Adami, M.; Moreira, M.A. Studies on the rapid expansion of sugarcane for ethanol production in São Paulo State (Brazil) using Landsat data. Remote Sens
**2010**, 2, 1057–1076. [Google Scholar] - Rizzi, R.; Rudorff, B.F.T. Estimativa da área de soja no Rio Grande do Sul por meio de imagens Landsat. Revista Brasileira de Cartografia
**2005**, 57, 226–234. [Google Scholar] - Vieira, M.A.; Formaggio, A.R.; Rennó, C.D.; Atzberger, C.; Aguiar, D.A.; Mello, M.P. Object based image analysis and data mining applied to a remotely sensed Landsat time-series to map sugarcane over large areas. Remote Sens. Environ
**2012**, 123, 553–562. [Google Scholar] - Mello, M.P.; Vieira, C.A.O.; Rudorff, B.F.T.; Aplin, P.; Santos, R.D.C.; Aguiar, D.A. STARS: A new method for multitemporal remote sensing. IEEE Trans. Geosci. Remote Sens
**2013**, 51, 1897–1913. [Google Scholar] - Asner, G.P. Cloud cover in Landsat observations of the Brazilian Amazon. Int. J. Remote Sens
**2001**, 22, 3855–3862. [Google Scholar] - Sano, E.E.; Ferreira, L.G.; Asner, G.P.; Steinke, E.T. Spatial and temporal probabilities of obtaining cloud-free Landsat images over the Brazilian tropical savanna. Int. J. Remote Sens
**2007**, 28, 2739–2752. [Google Scholar] - Arvor, D.; Jonathan, M.; Meirelles, M.S.P.; Dubreuil, V.; Durieux, L. Classification of MODIS EVI time series for crop mapping in the state of Mato Grosso, Brazil. Int. J. Remote Sens
**2011**, 32, 7847–7871. [Google Scholar] - Macedo, M.N.; DeFries, R.S.; Morton, D.C.; Stickler, C.M.; Galford, G.L.; Shimabukuro, Y.E. Decoupling of deforestation and soy production in the southern Amazon during the late 2000s. Proc. Natl. Acad. Sci. USA
**2012**, 109, 1341–1346. [Google Scholar] - Morton, D.C.; DeFries, R.S.; Shimabukuro, Y.E.; Anderson, L.O.; Arai, E.; del Bon Espirito-Santo, F.; Freitas, R.M.; Morisette, J. Cropland expansion changes deforestation dynamics in the southern Brazilian Amazon. Proc. Natl. Acad. Sci. USA
**2006**, 103, 14637–14641. [Google Scholar] - Justice, C.; Townshend, J.R.G.; Vermote, E.; Masuoka, E.; Wolfe, R.; Saleous, N.; Roy, D.; Morisette, J. An overview of MODIS land data processing and product status. Remote Sens. Environ
**2002**, 83, 3–15. [Google Scholar] - Rizzi, R.; Risso, J.; Epiphanio, R.D.V.; Rudorff, B.F.T.; Formaggio, A.R.; Shimabukuro, Y.E.; Fernandes, S.L. Estimativa da área de Soja no Mato Grosso por meio de Imagens MODIS. Proceedings of the 14th Brazilian Remote Sensing Symposium, Natal, RN, Brazil, 25–30 April 2009; INPE: São José dos Campos, SP, Brazil, 2009; pp. 387–394. [Google Scholar]
- Huete, A.R.; Didan, K.; Miura, T.; Rodriguez, E.; Gao, X.; Ferreira, L.G. Overview of the radiometric and biophysical performance of the MODIS vegetation indices. Remote Sens. Environ
**2002**, 83, 195–213. [Google Scholar] - Risso, J. Diagnóstico Espacialmente Explícito da Expansão da Soja no Mato Grosso de 2000 a 2012; National Institute for Space Research: São José dos Campos, SP, Brazil, 2013; p. 110. [Google Scholar]
- SEPLAN-MT. Sistema Interoperável de Informações Geoespaciais do Estado do Mato Grosso (SIIGEO). Available online: http://www.siigeo.mt.gov.br/ (accessed on 14 April 2012).
- Palmiere, F.; Santos, H.G.; Gomes, I.A.; Lumbreras, J.F.; Aglio, M.M.D. The Brazilian Soil Classification System. In Soil Classification: A Global Desk Reference; Rice, T., Eswaran, H., Stewart, B., Ahrens, R., Eds.; CRC Press: Boca Raton, FL, USA, 2002; pp. 127–146. [Google Scholar]
- Sistema Brasileiro de Classificação de Solos, 2nd ed; Santos, H.G.; Oliveira, J.B.; Lumbrelas, J.F.; Anjos, L.H.C.; Coelho, M.R.; Jacomine, P.K.T.; Cunha, T.J.F.; Oliveira, V.Á. (Eds.) Embrapa Solos: Rio de Janeiro, RJ, Brazil, 2006; p. 306.
- Rabus, B.; Eineder, M.; Roth, A.; Bamler, R. The shuttle radar topography mission—A new class of digital elevation models acquired by spaceborne radar. ISPRS J. Photogramm. Remote Sens
**2003**, 57, 241–262. [Google Scholar] - Shaxson, F. New Concepts and Approaches to Land Management in the Tropics with Emphasis on Steeplands; FAO: Rome, Italy, 1999; p. 125. [Google Scholar]
- Seeruttun, S.; Crossley, C.P. Use of digital terrain modelling for farm planning for mechanical harvest of sugar cane in Mauritius. Comput. Electron. Agric
**1997**, 18, 29–42. [Google Scholar] - ANEEL. Sistema de Informações Georeferenciadas do Setor Elétrico (SIGEO). Available online: http://sigel.aneel.gov.br (accessed on 20 April 2012).
- Silva, J.A.A.; Nobre, A.D.; Joly, C.A.; Nobre, C.A.; Manzatto, C.V.; Rech Filho, E.L.; Skorupa, L.A.; May, P.H.; Cunha, M.M.L.C.; Rodrigues, R.R.; et al. Brazil Forest Code and Science: Contributions to the Dialogue, 2nd ed; The Brazilian Society for the Advancement of Science—SBPC: São Paulo, SP, Brazil, 2012; p. 147. [Google Scholar]
- IBGE. Maps. Available online: http://mapas.ibge.gov.br/en/ (accessed on 29 April 2012).
- Fearnside, P.M. Soybean cultivation as a threat to the environment in Brazil. Environ. Conserv
**2002**, 28, 23–38. [Google Scholar] - INPE. PRODES: Projeto de Monitoramento do Desflorestamento na Amazônia Legal. Available online: http://www.obt.inpe.br/prodes/index.php (accessed on 20 January 2012).
- Shimabukuro, Y.E.; Batista, G.T.; Mello, E.M.K.; Moreira, J.C.; Duarte, V. Using shade fraction image segmentation to evaluate deforestation in Landsat Thematic Mapper images of the Amazon Region. Int. J. Remote Sens
**1998**, 19, 535–541. [Google Scholar] - FUNAI. Maps. Available online: http://mapas.funai.gov.br (accessed on 20 January 2013).
- MMA. Download de Dados Geográficos. Available online: http://mapas.mma.gov.br/i3geo/datadownload.htm (accessed on 20 January 2013).
- Galford, G.L.; Mustard, J.F.; Melillo, J.; Gendrin, A.; Cerri, C.C.; Cerri, C.E. Wavelet analysis of MODIS time series to detect expansion and intensification of row-crop agriculture in Brazil. Remote Sens. Environ
**2008**, 112, 576–587. [Google Scholar] - Risso, J.; Rizzi, R.; Rudorff, B.F.T.; Adami, M.; Shimabukuro, Y.E.; Formaggio, A.R.; Epiphanio, R.D.V. Índices de vegetação Modis aplicados na discriminação de áreas de soja. Pesquisa Agropecuária Brasileira
**2012**, 47, 1317–1326. [Google Scholar] - Rudorff, B.F.T.; Adami, M.; Risso, J.; de Aguiar, D.A.; Pires, B.; Amaral, D.; Fabiani, L.; Cecarelli, I. Remote sensing images to detect soy plantations in the amazon biome—The soy moratorium initiative. Sustainability
**2012**, 4, 1074–1088. [Google Scholar] - Rudorff, B.F.T.; Adami, M.; Aguiar, D.A.; Moreira, M.A.; Mello, M.P.; Fabiani, L.; Amaral, D.F.; Pires, B.M. The soy moratorium in the Amazon biome monitored by remote sensing images. Remote Sens
**2011**, 3, 185–202. [Google Scholar] - IBGE. Sistema IBGE de Recuperação Automática (SIDRA)—Produção Agrícola Municipal (PAM) 2012. Avaiable online: http://www.sidra.ibge.gov.br (accessed on 2 July 2012).
- Jepson, W. Producing a modern agricultural frontier: Firms and cooperatives in Eastern Mato Grosso, Brazil. Econ. Geogr
**2009**, 82, 289–316. [Google Scholar] - Rizzi, R.; Rudorff, B.F.T.; Shimabukuro, Y.E.; Doraiswamy, P.C. Assessment of MODIS LAI retrievals over soybean crop in Southern Brazil. Int. J. Remote Sens
**2006**, 27, 4091–4100. [Google Scholar] - Jasinski, E.; Morton, D.; DeFries, R.; Shimabukuro, Y.; Anderson, L.; Hansen, M. Physical landscape correlates of the expansion of mechanized agriculture in Mato Grosso, Brazil. Earth Interact
**2005**, 9, 1–18. [Google Scholar] - Soares-Filho, B.S.; Nepstad, D.C.; Curran, L.M.; Cerqueira, G.C.; Garcia, R.A.; Ramos, C.A.; Voll, E.; McDonald, A.; Lefebvre, P.; Schlesinger, P. Modelling conservation in the Amazon basin. Nature
**2006**, 440, 520–523. [Google Scholar] - Foody, G.M. Status of land cover classification accuracy assessment. Remote Sens. Environ
**2002**, 80, 185–201. [Google Scholar] - Liu, C.; Frazier, P.; Kumar, L. Comparative assessment of the measures of thematic classification accuracy. Remote Sens. Environ
**2007**, 107, 606–616. [Google Scholar] - Smits, P.C.; Dellepiane, S.G.; Schowengerdt, R.A. Quality assessment of image classification algorithms for land-cover mapping: A review and a proposal for a cost-based approach. Int. J. Remote Sens
**1999**, 20, 1461–1486. [Google Scholar] - Foody, G.M. Assessing the accuracy of land cover change with imperfect ground reference data. Remote Sens. Environ
**2010**, 114, 2271–2285. [Google Scholar] - Hanley, J.A.; McNeil, B.J. The meaning and use of the area under a receiver operating characteristic (ROC) curve. Radiology
**1982**, 143, 29–36. [Google Scholar] - Krug, L.A.; Gherardi, D.F.M.; Stech, J.L.; Leão, Z.M.A.N.; Kikuchi, R.K.P.; Hruschka, E.R.; Suggett, D.J. The construction of causal networks to estimate coral bleaching intensity. Environ. Model. Softw
**2013**, 42, 157–167. [Google Scholar] - Silvestrini, R.A.; Soares-Filho, B.S.; Nepstad, D.; Coe, M.; Rodrigues, H.; Assunção, R. Simulating fire regimes in the Amazon in response to climate change and deforestation. Ecol. Appl. Public. Ecol. Soc. Am
**2011**, 21, 1573–1590. [Google Scholar] - Aragão, L.E.O.C.; Malhi, Y.; Barbier, N.; Lima, A.; Shimabukuro, Y.; Anderson, L.; Saatchi, S. Interactions between rainfall, deforestation and fires during recent years in the Brazilian Amazonia. Philos. Trans. R. Soc. Lond. Ser. B Biol. Sci
**2008**, 363, 1779–1785. [Google Scholar] [Green Version] - Fell, R.; Corominas, J.; Bonnard, C.; Cascini, L.; Leroi, E.; Savage, W.Z. Guidelines for landslide susceptibility, hazard and risk zoning for land-use planning. Eng. Geol
**2008**, 102, 99–111. [Google Scholar] - Oliveira, F.S.C.; Gherardi, D.F.M.; Stech, J.L. The relationship between multi-sensor satellite data and Bayesian estimates for skipjack tuna catches in the South Brazil Bight. Int. J. Remote Sens
**2010**, 31, 4049–4067. [Google Scholar] - Li, L.; Wang, J.; Leung, H.; Jiang, C. Assessment of catastrophic risk using Bayesian network constructed from domain knowledge and spatial data. Risk Anal
**2010**, 30, 1157–1175. [Google Scholar] - Rodrigues, E.C.; Assunção, R. Bayesian spatial models with a mixture neighborhood structure. J. Multivar. Anal
**2012**, 109, 88–102. [Google Scholar]

**Figure 1.**Directed Acyclic Graph (DAG) representing a hypothetical BN graphical model where the target variable soybean occurrence (S) is influenced by two context variables: terrain slope (T) and soil aptitude (A). Since soil formation processes are strongly influenced by terrain slope, T is also parent of A. Variables are represented by nodes and dependences are represented by arcs between pairwise nodes.

**Figure 2.**Study area corresponding to Mato Grosso State, Brazil. The analysis was only performed in areas that were not masked out.

**Figure 3.**Summary of the procedures used in the case study of applying BayNeRD to identify soybean plantations in Mato Grosso State, Brazil. Table 1 provides a description of the variables used.

**Figure 4.**Directed Acyclic Graph (DAG) encoding assertions of conditional (in)dependence among the variables and representing the designed Bayesian Network graphical model for the case study of soybean occurrence in Mato Grosso.

**Figure 5.**Discretization of context variable terrain slope (T) into three intervals. The percentage at the top of each bar represents the probability of finding a pixel within the defined interval limits, e.g., P(−∞ ≤ T = t < 0.06) = 82.9%; and the percentage at the bottom of each bar represents the conditional probability of soybean presence given the defined interval limits for T, e.g., P(S = 1 | −∞ ≤ T = t < 0.06) = 53.6%.

**Figure 6.**(

**a**) Histogram of context variable CEI value in the last crop year (L); (

**b**) Discretization of L into four intervals. The percentage at the top of each bar represents the probability of finding a pixel within the defined interval limits, e.g., P(0.26 ≤ L = l < +∞) = 7.0%; and the percentage at the bottom of each bar represents the conditional probability of soybean presence given the defined interval limits for L, e.g., P(S = 1 | 0.26 ≤ L = l < +∞) = 95.4%.

**Figure 7.**Histogram of CEI values observed in the current crop year (C) and boxplot showing the strong relationship between soybean presence (S = 1) and C greater than 0.2.

**Figure 9.**Probability Image (PI) of soybean presence for the entire Mato Grosso State, Brazil. Main soybean producer centers and the capital, Cuiabá, are highlighted. The color indicates the calculated probability of soybean presence in 2005/2006 given the observations made for the context variables, as expressed by Equation (6).

**Figure 10.**Probability Image (PI) of soybean presence and six context variables (described in Table 1) zoomed in on the central part of the Sapezal municipality. The legend for the context variables followed the intervals stated in Table 2. Regions labeled 1, 2, and 3 show respectively, ideal, intermediate and poor conditions for soybean cultivation.

**Figure 11.**Receiver Operating Characteristic (ROC) curve, depicting sensitivity and specificity indices associated with thematic maps generated from the Probability Image (PI) by varying the Target Probability Value (TPV) from 0% to 100%. The circle points out the best TPV according to the chosen criterion.

**Figure 12.**Accuracy indices associated with thematic maps generated from the Probability Image (PI) by varying the Target Probability Value (TPV) from 0% to 100%. The vertical line identifies the best TPV, according to the chosen criterion, highlighting the accuracy achieved according to each index (described in the legend).

Variable | Description |
---|---|

C | CEI^{*} value in the Current crop year (2005/2006) |

L | CEI^{*} value in the Last crop year (2004/2005) |

A | Soil Aptitude |

T | Terrain slope (given in %) |

W | Distance to the nearest Water body (given in km) |

R | Distance to the nearest Road (given in km) |

^{*}Crop Enhancement Index [46].

**Table 2.**Summary of the intervals limits defined for each of the six context variables, described in Table 1.

Interval # | C | L | A | T | W | R |
---|---|---|---|---|---|---|

1 | [−∞; 0.05) | [−∞; 0.05) | low | [−∞; 0.06) | [−∞; 0.5) | [−∞; 3.0) |

2 | [0.05; 0.20) | [0.05; 0.20) | high | [0.06; 0.12) | [0.5; 1.0) | [3.0; 8.0) |

3 | [0.20; 0.26) | [0.20; 0.26) | [0.12; +∞) | [1.0; 2.0) | [8.0; +∞) | |

4 | [0.26; +∞) | [0.26; +∞) | [2.0; +∞) | |||

# of intervals | 4 | 4 | 2 | 3 | 4 | 3 |

© 2013 by the authors; licensee MDPI, Basel, Switzerland This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license ( http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Mello, M.P.; Risso, J.; Atzberger, C.; Aplin, P.; Pebesma, E.; Vieira, C.A.O.; Rudorff, B.F.T.
Bayesian Networks for Raster Data (BayNeRD): Plausible Reasoning from Observations. *Remote Sens.* **2013**, *5*, 5999-6025.
https://doi.org/10.3390/rs5115999

**AMA Style**

Mello MP, Risso J, Atzberger C, Aplin P, Pebesma E, Vieira CAO, Rudorff BFT.
Bayesian Networks for Raster Data (BayNeRD): Plausible Reasoning from Observations. *Remote Sensing*. 2013; 5(11):5999-6025.
https://doi.org/10.3390/rs5115999

**Chicago/Turabian Style**

Mello, Marcio Pupin, Joel Risso, Clement Atzberger, Paul Aplin, Edzer Pebesma, Carlos Antonio Oliveira Vieira, and Bernardo Friedrich Theodor Rudorff.
2013. "Bayesian Networks for Raster Data (BayNeRD): Plausible Reasoning from Observations" *Remote Sensing* 5, no. 11: 5999-6025.
https://doi.org/10.3390/rs5115999