## 1. Introduction

^{2}), mean absolute error (MAE), and root mean square error (RMS) are better for the ANN model than for MLR. Similarly, in [8], the ANN and MLR methods were used to produce a model of safflower yield (Carthamus tinctorius L.). The results of analyses (R

^{2}, MAE, RMS) also confirm better results for ANN models than MLR models. Therefore, crop yield models are used to develop forecasting tools, which can be an important element of high-precision agriculture and the main element of the decision-making support systems [9]. Precision agriculture can help in managing crop production inputs in an environmentally friendly way. By using site-specific knowledge, precision agriculture can target rates of fertilizer, seed and chemicals for soil, and other conditions. The concepts of precision agriculture and sustainability are inextricably linked. From the first time a global positioning system was used on agricultural equipment, the potential for environmental benefits has been discussed. Intuitively, applying fertilizers and pesticides only where and when they are needed should reduce the environmental burden [10].

## 2. Materials and Methods

#### 2.1. Method of Construction of Neural Models

#### 2.2. Methodology for Validating the Neural Models

- RAE—relative approximation error:$$RAE=\sqrt{\frac{{{\displaystyle \sum}}_{i=1}^{n}{\left({y}_{i}-{\widehat{y}}_{i}\right)}^{2}}{{{\displaystyle \sum}}_{i=1}^{n}{\left({y}_{i}\right)}^{2}}}$$
- RMS—root mean square error:$$RMS=\sqrt{\frac{{{\displaystyle \sum}}_{i=1}^{n}{\left({y}_{i}-{\widehat{y}}_{i}\right)}^{2}}{n}}$$
- MAE—mean absolute error:$$MAE=\frac{1}{n}{\displaystyle \sum}_{i=1}^{n}\left|{y}_{i}-{\widehat{y}}_{i}\right|$$
- MAPE—mean absolute percentage error:$$MAPE=\frac{1}{n}{\displaystyle \sum}_{i=1}^{n}{\left|\frac{{y}_{i}-{\widehat{y}}_{i}}{{y}_{i}}\right|}^{}\cdot 100\%$$
- n—number of observations;
- ${y}_{i}$—actual values obtained during research; and
- ${\widehat{y}}_{i}$—values given by the model.

#### 2.3. Neural Network Sensitivity Analysis

## 3. Results

^{−1}. For field no. 29, the observed yield was 2.14 t∙ha

^{−1}, and 4.63 t∙ha

^{−1}for field no. 33. The average value of the forecast yield for models WR15_04, WR31_05, and WR30_06 amounted to 3.92 t∙ha

^{−1}, 3.58 t∙ha

^{−1}, and 3.87 t∙ha

^{−1}respectively.

#### Network Sensitivity Analysis

^{2}was 0.6286. The other models slightly deviate from this result. In model WR15_04, the determination coefficient, R

^{2}, of 0.6187 was achieved, while in model WR30_06, the determination coefficient, R

^{2}, of 0.5976 was obtained.

## 4. Discussion

^{−1}of cultivated area was forecasted.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**Graphical presentation of the observed and predicted yield by neural model: (

**a**) WR15_04, (

**b**) WR31_05, and (

**c**) WR30_06.

**Figure 4.**Relation between the observed and predicted yield by models (

**a**) WR15_04, (

**b**) WR31_05, and (

**c**) WR30_06.

Set A | Set B | |||||||
---|---|---|---|---|---|---|---|---|

Year | 2008 | 2009 | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 |

Number of fields | 32 | 49 | 48 | 50 | 45 | 28 | 40 | 36 |

Symbol | Unit of Measure | Variable Name | Model WR15_04 | Model WR31_05 | Model WR30_06 | The Scope of Data |
---|---|---|---|---|---|---|

R9-12_LY | mm | The sum of precipitation from 1 September to 31 December of the previous year | + | + | + | 63–234 |

T9-12_LY | °C | The average air temperature from 1 September to 31 December of the previous year | + | + | + | 4.9–9.4 |

R1-4_CY | mm | The sum of precipitation from 1 January to 15 April of the current year | + | + | + | 59–185 |

T1-4_CY | °C | The average air temperature from 1 January to 15 April of the current year | + | + | + | −0.4–4.9 |

R4_CY | mm | The sum of precipitation from 1 April to 30 April of the current year | - | + | + | 8.7–60.4 |

T4_CY | °C | The average air temperature from 1 April to 30 April of the current year | - | + | + | 5.9–12.2 |

R5_CY | mm | The sum of precipitation from 1 May to 31 May of the current year | - | + | + | 14.2–132.5 |

T5_CY | °C | The average air temperature from 1 May to 31 May of the current year | - | + | + | 11.8–16.2 |

R6_CY | mm | The sum of precipitation from 1 June to 30 June of the current year | - | - | + | 15–121 |

T6_CY | °C | The average air temperature from 1 June to 30 June of the current year | - | - | + | 14.2–19.6 |

N_LY | kg ha^{−1} | The sum of N fertilization—autumn in the previous year | + | + | + | 0–41 |

N_CY | kg ha^{−1} | The sum of N fertilization—spring in the current year | + | + | + | 0–175 |

P2O5_CY | kg ha^{−1} | The sum of P_{2}O_{5} fertilization in the current year | + | + | + | 0–104 |

K2O_CY | kg ha^{−1} | The sum of K_{2}O fertilization in the current year | + | + | + | 0–234 |

MGO_CY | kg ha^{−1} | The sum of MgO fertilization in the current year | + | + | + | 0–298 |

SO3_CY | kg ha^{−1} | The sum of sulfate ions (VI) fertilization in the current year | + | + | + | 14–115 |

B_CY | g ha^{−1} | The sum of B fertilization in the current year | + | + | + | 0–3.66 |

CU_CY | g ha^{−1} | The sum of Cu fertilization in the current year | + | + | + | 10–487 |

MN_CY | g ha^{−1} | The sum of Mn fertilization in the current year | + | + | + | 70–600 |

MO_CY | g ha^{−1} | The sum of Mo fertilization in the current year | + | + | + | 0–60 |

ZN_CY | g ha^{−1} | The sum of Zn fertilization in the current year | + | + | + | 10–560 |

WR15_04 | WR31_05 | WR30_06 | |
---|---|---|---|

Neural Network Structure | MLP 15:15-18-11-1:1 | MLP 19:19-15-18-1:1 | MLP 21:21-15-14-1:1 |

Learning error | 0.1229 | 0.1053 | 0.0924 |

Validation error | 0.0625 | 0.1051 | 0.1053 |

Test error | 0.1283 | 0.1258 | 0.1277 |

Mean | 3.3626 | 3.3626 | 3.3626 |

Standard deviation | 1.0703 | 1.0703 | 1.0703 |

Average error | 0.0762 | −0.0401 | −0.0245 |

Deviation error | 0.7183 | 0.6264 | 0.6098 |

Mean Absolute error | 0.5819 | 0.5021 | 0.4771 |

Quotient deviations | 0.6711 | 0.5852 | 0.5697 |

Correlation | 0.7413 | 0.8127 | 0.8218 |

Model | RAE [-] | RMS [t ha^{−1}] | MAE [t ha^{−1}] | MAPE [%] |
---|---|---|---|---|

WR15_04 | 0.075 | 0.337 | 0.282 | 7.51 |

WR31_05 | 0.091 | 0.401 | 0.344 | 7.85 |

WR30_06 | 0.081 | 0.341 | 0.306 | 8.12 |

Variable | Model | |||||
---|---|---|---|---|---|---|

WR15_04 | WR31_05 | WR30_06 | ||||

Quotient | Rank | Quotient | Rank | Quotient | Rank | |

R9-12_LY | 1.1669 | 3 | 1.3693 | 1 | 1.1508 | 1 |

T9-12_LY | 1.1933 | 2 | 1.0366 | 10 | 1.0529 | 9 |

R1-4_CY | 1.0224 | 6 | 1.0599 | 6 | 1.0252 | 16 |

T1-4_CY | 1.0250 | 5 | 1.0689 | 5 | 1.1034 | 3 |

R4_CY | - | - | 1.0698 | 4 | 1.0906 | 4 |

T4_CY | - | - | 1.0403 | 9 | 1.0376 | 12 |

R5_CY | - | - | 1.1412 | 3 | 1.0881 | 5 |

T5_CY | - | - | 1.2512 | 2 | 1.0340 | 14 |

R6_CY | - | - | - | - | 1.0527 | 10 |

T6_CY | - | - | - | - | 1.0471 | 11 |

N_LY | 0.9966 | 14 | 1.0031 | 17 | 1.0296 | 15 |

N_CY | 1.0124 | 10 | 1.0167 | 12 | 1.0163 | 19 |

P2O5_CY | 1.0000 | 12 | 0.9796 | 19 | 0.9854 | 21 |

K2O_CY | 0.9975 | 13 | 1.0043 | 15 | 1.0086 | 20 |

MGO_CY | 1.0143 | 8 | 1.0550 | 7 | 1.1308 | 2 |

SO3_CY | 1.0340 | 4 | 1.0295 | 11 | 1.0655 | 6 |

B_CY | 1.0034 | 11 | 0.9911 | 18 | 1.0357 | 13 |

CU_CY | 0.9905 | 15 | 1.0103 | 14 | 1.0200 | 18 |

MN_CY | 1.0138 | 9 | 1.0033 | 16 | 1.0589 | 8 |

MO_CY | 1.1935 | 1 | 1.0540 | 8 | 1.0650 | 7 |

ZN_CY | 1.0220 | 7 | 1.0135 | 13 | 1.0239 | 17 |

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