# A Minimum Cross-Entropy Approach to Disaggregate Agricultural Data at the Field Level

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodological Approach

_{k}is the land cover class k; y

_{k}is the spectral signature of class k; and y

_{j}is the spectral signature of class j.

_{k}) is the probability that the correct class is ${C}_{k}$; $|{{\displaystyle \sum}}_{k}|$ is the determinant of the covariance matrix of the data in class ${C}_{k}$; and ${{{\displaystyle \sum}}_{k}}^{-1}$ is the inverse of the covariance matrix.

^{i}is the area weight of each disaggregated unit i; STAT

_{k}are the regional statistics for land-use k; $LAN{D}_{k}^{i}$ is the land use available for land-use k in disaggregated unit i; $H{M}_{k}^{i}$ are the minimum historical limits and $HM{X}_{k}^{i}$ are the maximum historical limits for each land-use by disaggregated unit i; and ${e}_{kn}$ refers to a parameterized error term [36].

^{i}indicator allows assessing the real deviation at the statistical unit c level and at the aggregate level and is obtained by the following:

^{2}, and the modeling efficiency (EF) were used to compare ${S}_{k}^{c}$ and ${\widehat{S}}_{k}^{c}$. The R coefficient is a measure of association among two variables while R

^{2}refers to how the variance of the dependent variable is explained by the independent variables and is used to measure the adjustment of a regression line. Thus, when R

^{2}is equal to 1, the estimated data are completely explained by the variance of real data. EF is a normalized measure to evaluate the model performance [6]. An EF indicator equal to 1 shows a total efficiency of the model, since there are complete information gains, while an indicator equal to 0 means the opposite. In cases where deviations between real and estimated data are high, this indicator may present negative values. These indicators were calculated as follows:

## 3. Data and Application Scenarios

^{2}and, in 2010, was composed of 16 municipalities and 84 parishes, which was reduced to 67 a few years later. The Mediterranean climate predominates and there are several biophysical contrasts between the coastal and inland areas with less fertile areas and higher slopes.

^{2}and, in 2009, was divided into 8 parishes which were later reduced to 6. It extends from the inland Algarve to the coast and in 2009, the agrarian census represented more than 17% of the permanent crop area and about 41% of the citrus area in the region.

## 4. Results and Analysis

^{i}lower than 30% can be observed in all simulations. The errors in other crops are of little importance and parishes with little relevance for the total area. Thus, the above values hide very satisfactory WPAD results.

^{2}indicators are above 0.5, except for fresh fruits, which present R

^{2}values of 43% in simulation SCMD2009 and of 41.4% in simulation SCML2009. In some crops, such as olive trees and citrus, the R

^{2}values are always higher than 0.7.

^{2}of 0.8 while the others presented values of R

^{2}between 0.40 and 0.45.

## 5. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Hajkowicz, S.; Collins, E.; Cattaneo, E. Review of Agri-Environment Indexes and Stewardship Payments. Environ. Manag.
**2009**, 43, 221–236. [Google Scholar] [CrossRef] [PubMed] - Fritz, S.; You, L.; Bun, A.; See, L.; McCallum, I.; Schill, C.; Perger, C.; Liu, J.; Hansen, M.; Obersteiner, M. Cropland for sub-Saharan Africa: A synergistic approach using five land cover data sets. Geophys. Res. Lett.
**2011**, 38. [Google Scholar] [CrossRef] - Tan, J.; Yang, P.; Liu, Z.; Wu, W.; Zhang, L.; Li, Z. Spatio-temporal dynamics of maize cropping system in Northeast China between 1980 and 2010 by using spatial production allocation model. J. Geogr. Sci.
**2014**, 24, 397–410. [Google Scholar] [CrossRef] - Kempen, M.; Heckelei, T.; Britz, W.; Leip, A.; Koeble, R.; Marchi, G. Computation of a European Agricultural Land Use Map–Statistical Approach and Validation; Discussion Paper; Institute for Food and Resource Economics: Bonn, Germany, 2005. [Google Scholar]
- You, L.; Wood, S. An entropy approach to spatial disaggregation of agricultural production. Agric. Syst.
**2006**, 90, 29–347. [Google Scholar] [CrossRef] - You, L.; Wood, S.; Wood-Sichra, U. Generating plausible crop distribution maps for Sub-Saharan Africa using a spatially disaggregated data fusion and optimization approach. Agric. Syst.
**2009**, 99, 126–140. [Google Scholar] [CrossRef] - You, L.; Wood, S.; Wood-Sichra, U.; Wu, W. Generating global crop distribution maps: From census to grid. Agric. Syst.
**2014**, 127, 53–60. [Google Scholar] [CrossRef] - Chakir, R. Spatial downscaling of agricultural land use data: An econometric approach using cross–entropy. Land Econ.
**2009**, 85, 238–251. [Google Scholar] [CrossRef] - Xavier, A.; Costa Freitas, M.D.B.; Fragoso, R. Disaggregation of Statistical Livestock Data Using the Entropy Approach. Adv. Oper. Res.
**2014**, 397675. [Google Scholar] [CrossRef] - EUROSTAT. LUCAS 2012 (Land Use/Cover Area Frame Survey); EUROSTAT: Brussels, Belgium, 2013. [Google Scholar]
- Chakir, R.; Lungarska, A. Agricultural rent in land-use models: Comparison of frequently used proxies. Spatial Econ. Anal.
**2017**, 12, 279–303. [Google Scholar] [CrossRef] - Chakir, R.; Le Gallo, J. Predicting land use allocation in France: A spatial panel data analysis. Ecol. Econ.
**2013**, 92, 114–125. [Google Scholar] [CrossRef] - Chakir, R.; Parent, O. Determinants of land use changes: A spatial multinomial probit approach. Pap. Reg. Sci.
**2009**, 88, 327–344. [Google Scholar] [CrossRef] - Ferdous, N.; Bhat, C.R. A spatial panel ordered-response model with application to the analysis of urban land-use development intensity patterns. J. Geogr. Syst.
**2013**, 15, 1–29. [Google Scholar] [CrossRef] - Anselin, L. Spatial econometrics in RSUE: Retrospect and prospect. Reg. Sci. Urban Econ.
**2007**, 37, 450–456. [Google Scholar] [CrossRef] - Brady, M.; Irwin, E. Accounting for spatial effects in economic models of land use: Recent developments and challenges ahead. Environ. Resour. Econ.
**2011**, 48, 487–509. [Google Scholar] [CrossRef] - Howitt, R.; Reynaud, A.A. Spatial disaggregation of agricultural production data using maximum entropy. Eur. Rev. Agric. Econ.
**2003**, 30, 359–387. [Google Scholar] [CrossRef] - Fragoso, R.; Martins, M.B.; Lucas, M.R. Generate disaggregated soil allocation data using a Minimum Cross Entropy Model. WSEAS Trans. Environ. Dev.
**2008**, 9, 756–766. [Google Scholar] - Martins, M.B.; Fragoso, R.; Xavier, A. Spatial disaggregation of agricultural data in Castelo de Vide, Alentejo, Portugal: An approach based on maximum entropy. JP J. Biostat.
**2011**, 5, 1–16. [Google Scholar] - Louhichi, K.; Jacquet, F.; Butault, J.P. Estimating input allocation from heterogeneous data sources: A comparison of alternative estimation approaches. Agric. Econ. Rev.
**2012**, 13, 83–102. [Google Scholar] - Britz, W.; Verburg, P.H.; Leip, A. Modelling of land cover and agricultural change in Europe: Combining the CLUE and CAPRI-Spat approaches. Agric. Ecosyst. Environ.
**2011**, 142, 40–50. [Google Scholar] [CrossRef] - Xavier, A.; Freitas, M.B.; Fragoso, R.; Socorro Rosário, M. Agricultural data disaggregation at a local level: An approach using entropy and supervised classifications. In Proceedings of the 1st International Congress on Interdisciplinarity in Social and Human Sciences, Faro, Portugal, 5–6 May 2016; CIEO: Faro, Portugal, 2016. [Google Scholar]
- Congedo, L. Semi-Automatic Classification Plugin Documentation Release 4.8.0.1. 2015. Available online: https://semiautomaticclassificationmanual-v4.readthedocs.org/en/latest/ (accessed on 15 February 2016).
- Congalton, R.G. A review of assessing the accuracy of classifications of remotely sensed data. Remote Sens. Environ.
**1991**, 37, 35–46. [Google Scholar] [CrossRef] - 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] [CrossRef] - Xie, Y.; Sha, Z.; Yu, M. Remote sensing imagery in vegetation mapping: A review. J. Plant Ecol.
**2008**, 1, 9–23. [Google Scholar] [CrossRef] - Perumal, K.; Bhaskaran, R. Supervised classification performance of multispectral images. J. Comput.
**2010**, 2, 2151–9617. [Google Scholar] - Samaniego, L.; Schulz, K. Supervised classification of agricultural land cover using a modified k-NN technique (MNN) and landsat remote sensing imagery. Remote Sens.
**2009**, 1, 875–895. [Google Scholar] [CrossRef] - Bahadur, K.C. Improving Landsat and IRS image classification: Evaluation of unsupervised and supervised classification through band ratios and DEM in a mountainous landscape in Nepal. Remote Sens.
**2009**, 1, 1257–1272. [Google Scholar] [CrossRef] - Rozenstein, O.; Karnieli, A. Comparison of methods for land-use classification incorporating remote sensing and GIS inputs. Appl. Geogr.
**2011**, 31, 533–544. [Google Scholar] [CrossRef] - Shalaby, A.; Tateishi, R. Remote sensing and GIS for mapping and monitoring land cover and land-use changes in the Northwestern coastal zone of Egypt. Appl. Geogr.
**2007**, 27, 28–41. [Google Scholar] [CrossRef] - Jeon, Y.J.; Choi, J.G.; Kim, J.I. A study on supervised classification of remote sensing satellite image by bayesian algorithm using average fuzzy intracluster distance. In Combinatorial Image Analysis; Klette, R., Žunić, J., Eds.; Springer: Berlin/Heidelberg, Germany, 2004; pp. 597–606. ISBN 978-3-540-30503-3. [Google Scholar]
- Shannon, C.E. A Mathematical Theory of Communication. Bell Syst. Tech. J.
**1948**, 27, 379–423. [Google Scholar] [CrossRef] - Jaynes, E.T. Information theory and statistical methods I. Phys. Rev.
**1957**, 106, 620–630. [Google Scholar] [CrossRef] - Good, I. Maximum entropy for hypothesis formulation, especially for multidimensional contingency tables. Ann. Math. Stat.
**1963**, 34, 911–934. [Google Scholar] [CrossRef] - Golan, A.; Judge, G.; Miller, D. Maximum Entropy Econometrics: Robust Estimation with Limited Data; John Wiley & Sons: New York, NY, USA, 1996; ISBN 978-0-471-95311-1. [Google Scholar]
- Lence, H.L.; Miller, D. Estimation of Multi-Output Production Functions with Incomplete Data: A Generalized Cross Entropy Approach. Eur. Rev. Agric. Econ.
**1998**, 25, 188–209. [Google Scholar] [CrossRef] - Zhang, X.; Fan, S. Estimating crop-specific production technologies in Chinese agriculture: A generalized maximum entropy approach. Am. J. Agric. Econ.
**2011**, 83, 378–388. [Google Scholar] [CrossRef] - Howitt, R.E.; Msangi, S. Entropy estimation of disaggregate production functions: An application to northern Mexico. Entropy
**2014**, 16, 1349–1364. [Google Scholar] [CrossRef] - Aurbacher, J.; Dabbert, S. Generating crop sequences in land-use models using maximum entropy and Markov chains. Agric. Syst.
**2011**, 104, 470–479. [Google Scholar] [CrossRef] - Xavier, A.; Martins, M.B.; Fragoso, R. A mininum cross entropy model to generate disaggregated data at the local level. In Proceedings of the 122nd EAAE Seminar “Evidence-based agricultural and rural policy making: Methodological and empirical challenges of policy evaluation”, Ancona, Italy, 17–18 February 2011. [Google Scholar]
- Fragoso, R.M.; Carvalho, M.L. Estimation of joint costs allocation coefficients using the maximum entropy: A case of Mediterranean farms. J. Quant. Econ.
**2012**, 10, 91–111. [Google Scholar] - Fragoso, R.; Carvalho, M.L.D.S. Estimation of cost allocation coefficients at the farm level using an entropy approach. J. Appl. Stat.
**2013**, 40, 1893–1906. [Google Scholar] [CrossRef]

**Figure 3.**The supervised classification results for 2009 in the Algarve region; (source: model results).

**Figure 5.**The examples of the data disaggregation results in the Silves municipality; source: model results.

**Table 1.**The examples of the prior estimates i at a pixel level using the Maximum Likelihood algorithm in the Algarve region in 2009.

Territorial Unit i | Nuts | Vineyards | Olive Trees | Citrus | Fresh Fruits | Permanent Crops | Other Areas |
---|---|---|---|---|---|---|---|

i1003 | 0.201 | 0.001 | 0.057 | 0.007 | 0.025 | 0.029 | 0.680 |

i1004 | 0.050 | 0.005 | 0.034 | 0.043 | 0.007 | 0.030 | 0.832 |

i1005 | 0.135 | 0.002 | 0.096 | 0.005 | 0.009 | 0.053 | 0.701 |

i1006 | 0.018 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.982 |

i1007 | 0.080 | 0.000 | 0.044 | 0.005 | 0.000 | 0.015 | 0.857 |

i1008 | 0.000 | 0.000 | 0.069 | 0.008 | 0.000 | 0.000 | 0.922 |

i1009 | 0.058 | 0.000 | 0.034 | 0.004 | 0.005 | 0.023 | 0.877 |

i1010 | 0.097 | 0.002 | 0.017 | 0.011 | 0.000 | 0.040 | 0.832 |

i1011 | 0.055 | 0.000 | 0.007 | 0.042 | 0.001 | 0.005 | 0.890 |

i1012 | 0.101 | 0.000 | 0.037 | 0.019 | 0.002 | 0.009 | 0.834 |

i1013 | 0.039 | 0.003 | 0.039 | 0.007 | 0.006 | 0.025 | 0.881 |

i1014 | 0.067 | 0.012 | 0.010 | 0.024 | 0.006 | 0.015 | 0.866 |

i1015 | 0.000 | 0.000 | 0.150 | 0.158 | 0.005 | 0.143 | 0.544 |

i1016 | 0.038 | 0.000 | 0.006 | 0.000 | 0.000 | 0.038 | 0.918 |

i1017 | 0.021 | 0.013 | 0.035 | 0.031 | 0.000 | 0.068 | 0.832 |

**Table 2.**The average and median PAD indicator per crop type and simulation in the Algarve region and Silves municipality.

Scenario | Nuts | Vineyards | Olive Trees | Citrus | Fresh Fruits | Other Permanent Crops | |
---|---|---|---|---|---|---|---|

Algarve Region | |||||||

SCMD2009 | Average | 87.7 | 87.1 | 110.6 | 77.8 | 90.4 | 26.1 |

Median | 53.8 | 44.6 | 31.3 | 56.5 | 72.6 | 0.0 | |

SCML2009 | Average | 77.8 | 84.0 | 114.1 | 90.6 | 89.6 | 27.2 |

Median | 47.8 | 46.4 | 35.3 | 52.1 | 67.6 | 0.0 | |

Silves Municipality | |||||||

SCMD2009 | Average | 51.3 | 91.0 | 65.9 | 44.7 | 69.9 | 31.9 |

Median | 43.6 | 93.1 | 21.9 | 34.9 | 87.6 | 5.0 | |

SCML2009 | Average | 51.6 | 91.0 | 62.4 | 78.4 | 82.6 | 31.9 |

Median | 50.1 | 93.1 | 19.0 | 32.6 | 93.6 | 5.0 | |

SCMD2009WR | Average | 65.3 | 70.9 | 201.6 | 1217.4 | 50.4 | 259.8 |

Median | 47.2 | 38.6 | 61.0 | 222.9 | 46.9 | 4.8 | |

SCML2009WR | Average | 61.2 | 75.2 | 217.6 | 1290.4 | 53.1 | 274.4 |

Median | 42.7 | 50.4 | 58.9 | 230.6 | 49.0 | 6.8 | |

SCMD2013 | Average | 50.1 | 50.6 | 57.7 | 184.5 | 47.4 | 21.8 |

Median | 43.2 | 33.6 | 19.0 | 38.3 | 45.2 | 5.0 | |

SCMD2013WR | Average | 50.3 | 75.9 | 50.3 | 84.4 | 69.3 | 16.2 |

Median | 50.1 | 91.0 | 57.7 | 44.7 | 69.9 | 15.0 |

Simulation | WPAD (%) |
---|---|

Algarve Region | |

SCMD2009 | 42.80 |

SCML2009 | 41.10 |

Silves Municipality | |

SCMD2009 | 25.60 |

SCML2009 | 26.12 |

SCMD2009WR | 34.99 |

SCML2009WR | 35.87 |

SCMD2013 | 20.86 |

SCMD2013WR | 33.38 |

**Table 4.**The correlation and determination coefficients R and R

^{2}in the Algarve region and Silves municipality.

Simulations | Nuts | Vineyards | Olive Trees | Citrus | Fresh Fruits | Other Permanent Crops | |
---|---|---|---|---|---|---|---|

Algarve Region | |||||||

SCMD2009 | R | 0.838 | 0.716 | 0.871 | 0.960 | 0.656 | 0.741 |

R^{2} | 0.703 | 0.513 | 0.758 | 0.922 | 0.430 | 0.549 | |

SCML2009 | R | 0.896 | 0.732 | 0.863 | 0.946 | 0.643 | 0.743 |

R^{2} | 0.802 | 0.536 | 0.744 | 0.896 | 0.414 | 0.552 | |

Silves Municipality | |||||||

SCMD2009 | R | 0.937 | 0.863 | 0.985 | 0.992 | 0.686 | 0.873 |

R^{2} | 0.877 | 0.744 | 0.969 | 0.983 | 0.471 | 0.762 | |

SCML2009 | R | 0.938 | 0.863 | 0.980 | 0.989 | 0.455 | 0.873 |

R^{2} | 0.880 | 0.744 | 0.960 | 0.979 | 0.207 | 0.762 | |

SCMD2009WR | R | 0.909 | 0.771 | 0.769 | 0.985 | 0.525 | 0.324 |

R^{2} | 0.827 | 0.594 | 0.591 | 0.971 | 0.276 | 0.105 | |

SCML2009WR | R | 0.923 | 0.709 | 0.705 | 0.988 | 0.484 | 0.262 |

R^{2} | 0.852 | 0.503 | 0.497 | 0.977 | 0.234 | 0.069 | |

SCMD2013 | R | 0.938 | 0.766 | 0.991 | 0.995 | 0.760 | 0.963 |

R^{2} | 0.880 | 0.587 | 0.982 | 0.991 | 0.577 | 0.928 | |

SCMD2013WR | R | 0.780 | 0.743 | 0.695 | 0.994 | 0.621 | 0.423 |

R^{2} | 0.608 | 0.552 | 0.483 | 0.988 | 0.385 | 0.179 |

Scenario | Nuts | Vineyards | Olive Trees | Citrus | Fresh Fruits | Other Permanent Crops |
---|---|---|---|---|---|---|

Algarve Region | ||||||

SCMD2009 | 0.614 | 0.455 | 0.724 | 0.885 | 0.369 | 0.489 |

SCML2009 | 0.758 | 0.493 | 0.707 | 0.811 | 0.362 | 0.492 |

Silves Municipality | ||||||

SCMD2009 | 0.544 | −2.172 | 0.967 | 0.983 | 0.095 | 0.476 |

SCML2009 | 0.551 | −2.172 | 0.954 | 0.979 | −0.573 | 0.476 |

SCMD2009WR | 0.782 | 0.541 | 0.440 | 0.915 | 0.019 | 0.088 |

SCML2009WR | 0.822 | 0.373 | 0.375 | 0.910 | −0.039 | 0.065 |

SCMD2013 | 0.552 | 0.285 | 0.960 | 0.988 | 0.535 | 0.900 |

SCMD2013WR | 0.387 | 0.067 | 0.430 | 0.972 | 0.343 | 0.165 |

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## Share and Cite

**MDPI and ACS Style**

Xavier, A.; Fragoso, R.; De Belém Costa Freitas, M.; Do Socorro Rosário, M.; Valente, F.
A Minimum Cross-Entropy Approach to Disaggregate Agricultural Data at the Field Level. *Land* **2018**, *7*, 62.
https://doi.org/10.3390/land7020062

**AMA Style**

Xavier A, Fragoso R, De Belém Costa Freitas M, Do Socorro Rosário M, Valente F.
A Minimum Cross-Entropy Approach to Disaggregate Agricultural Data at the Field Level. *Land*. 2018; 7(2):62.
https://doi.org/10.3390/land7020062

**Chicago/Turabian Style**

Xavier, António, Rui Fragoso, Maria De Belém Costa Freitas, Maria Do Socorro Rosário, and Florentino Valente.
2018. "A Minimum Cross-Entropy Approach to Disaggregate Agricultural Data at the Field Level" *Land* 7, no. 2: 62.
https://doi.org/10.3390/land7020062