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

^{1}

^{2}

^{3}

^{4}

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## 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

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**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