# Estimation of Heavy Metals in Agricultural Soils Using Vis-NIR Spectroscopy with Fractional-Order Derivative and Generalized Regression Neural Network

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area and Sample Collection

^{2}, and its terrain is relatively undulating. Its climate is classified as continental monsoonal, with large temperature difference. The average annual precipitation and temperature are 759.1 mm and 5.1 °C, correspondingly. The accumulated sunshine hours in the area are between 2280 h and 2670 h, and the frost-free period is approximately 120 to 160 days. The soil types in the region include Cumulic Anthrosols, Gleyic Luvisols, Haplic Luvisols and Haplic Phaeozems [41]. The cultivated area accounts for about 13.05% of the study area, mostly distributed on both sides of the river. In this area, the crop types include rice, vegetables, and other economic crops. The upper reaches of the river basin are rich in mineral resources, with many small mines. Part of the mine tailings ponds are not designed according to standards, which leads to the leakage of heavy metal elements, thereby possibly leading to heavy metal accumulation in soils.

#### 2.2. Chemical Analysis and Statistic

#### 2.3. Spectral Measurement and Pretreatment

^{®}3 field spectroradiometer (Analytical Spectral Devices, Inc., Boulder, CO, USA). The device has a spectral resolution of 3 nm over the 350–1000 nm region and 10 nm over the 1000–2500 nm region, and records reflectance at 1 nm intervals. Spectral measurements were conducted in a dark room. The soil samples were uniformly put into black containers with a diameter of 10 cm, and the surface of soil samples was scraped flat. A 50 W halogen lamp was used as the light source, and the incident angle was 30°. The optical fiber probe was perpendicular to the surface of each sample at a distance of 5 cm. Before spectral measurements, the device was preheated for 30 min and optimized with a white panel. After every 10 soil samples, the whiteboard calibration was also carried out. Each sample was tested ten times, and the average value was obtained as the actual spectral reflection data.

#### 2.4. Spectral Feature Reduction

#### 2.5. Fractional-Order Derivatives

#### 2.6. Model Calculation and Accuracy Evaluation

#### 2.6.1. The Modeling Methods

#### 2.6.2. Model Accuracy Evaluation

^{2}), root mean square error (RMSE) and residual prediction deviation (RPD), were utilized as the indicators to evaluate model accuracy. The calculations of R

^{2}, RMSE and RPD are as Equations (21)–(23).

## 3. Results

#### 3.1. Descriptive Statistics for Heavy Metals

#### 3.2. Fractional-Order Derivative Spectrum

#### 3.3. Correlation between Heavy Metals and Optimal Spectral Indices

#### 3.4. Model Estimation and Comparisons

#### 3.4.1. Comparison of Results for Fractional-Order Derivatives

#### 3.4.2. Comparison of Results for Mathematical Models

## 4. Discussion

^{3+}, goethite, illite, organic matter, N-H, C-H, and O-H stretch [56,77]. The spectral characteristics of soil organic matter (SOM) and clay are closely related to the spectral response of N-H, C-H and O-H covalent bonds [78]. This is consistent with the absorption characteristics obtained. According to previous studies, due to the strong absorption capacity of SOM, iron oxides and clay, the Vis-NIR spectroscopy monitoring of heavy metals is through association with proxies (i.e., SOM, iron and clay) [79,80]. Therefore, Vis-NIR spectroscopy estimation for Hg, Cr and Cu can be performed indirectly through these components or functional groups.

## 5. Conclusions

^{2}= 0.70, RPD = 1.86) and Cu (R

^{2}= 0.65, RPD = 1.73) was achieved by the 1.8-order and 1.2-order reflectance GRNN model, and the optimal estimated accuracy of Cr (R

^{2}= 0.69, RPD = 1.83) was achieved by the 1.6-order reflectance RF model. The optimal band combination algorithm is able to avoid the influence of spectral noise caused by high-order derivatives. Compared with conventional derivatives, FOD can identify the subtler spectral characteristics of heavy metals due to its gradual change in treatment of the spectrum. The high-order FOD is able to highlight hidden information and the separate minor absorbing peak. In addition, the incorporation of the optimal band combination algorithm and high-order FOD can further mine spectral information, ignoring noise. The optimal performance was achieved by the 1.8-order, 1.6-order, and 1.2-order spectra for Hg, Cr, and Cu, correspondingly. For estimation of heavy metals in soils, the modeling methods with the ability to solve non-linear problems are more suitable. When using the appropriate dimensionality reduction method, GRNN provides an obvious improvement to the estimation accuracy of all studied heavy metals compared to ANFIS, PLSR, and RF. Thus, the incorporation of the optimal band combination algorithm, FOD, and GRNN for the rapid spectral estimation of Hg, Cr, and Cu concentration was proven to be feasible. Additionally, this study is vital for the application of Vis-NIR spectroscopy technology to the investigation of other heavy metal contaminants in soils. In the future, we will conduct more studies to detect the influence of GRNN and FOD on Vis-NIR spectroscopy estimation of soil heavy metals through large soil spectral libraries.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Correlation coefficients between soil Hg and ratio index of the reflectance processed by (

**a**) 0-order, (

**b**) 0.2-order, (

**c**) 0.4-order, (

**d**) 0.6-order, (

**e**) 0.8-order, (

**f**) 1-order, (

**g**) 1.2-order, (

**h**) 1.4-order, (

**i**) 1.6-order, (

**j**) 1.8-order, (

**k**) 2-order.

**Figure A2.**Correlation coefficients between soil Hg and difference index of the reflectance processed by (

**a**) 0-order, (

**b**) 0.2-order, (

**c**) 0.4-order, (

**d**) 0.6-order, (

**e**) 0.8-order, (

**f**) 1-order, (

**g**) 1.2-order, (

**h**) 1.4-order, (

**i**) 1.6-order, (

**j**) 1.8-order, (

**k**) 2-order.

**Figure A3.**Correlation coefficients between soil Hg and normalized difference index of the reflectance processed by (

**a**) 0-order, (

**b**) 0.2-order, (

**c**) 0.4-order, (

**d**) 0.6-order, (

**e**) 0.8-order, (

**f**) 1-order, (

**g**) 1.2-order, (

**h**) 1.4-order, (

**i**) 1.6-order, (

**j**) 1.8-order, (

**k**) 2-order.

**Figure A4.**Correlation coefficients between soil Hg and product index of the reflectance processed by (

**a**) 0-order, (

**b**) 0.2-order, (

**c**) 0.4-order, (

**d**) 0.6-order, (

**e**) 0.8-order, (

**f**) 1-order, (

**g**) 1.2-order, (

**h**) 1.4-order, (

**i**) 1.6-order, (

**j**) 1.8-order, (

**k**) 2-order.

**Figure A5.**Correlation coefficients between soil Hg and sum index of the reflectance processed by (

**a**) 0-order, (

**b**) 0.2-order, (

**c**) 0.4-order, (

**d**) 0.6-order, (

**e**) 0.8-order, (

**f**) 1-order, (

**g**) 1.2-order, (

**h**) 1.4-order, (

**i**) 1.6-order, (

**j**) 1.8-order, (

**k**) 2-order.

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**Figure 1.**(

**a**) The location of study area (Suzi river basin) in Liaoning Province and the position of Liaoning Province in China, and (

**b**) the position of sampling sites (80) in study area with Landsat 8 OLI image after true color compositing by bands 4 (red), 3 (green), and 2 (blue).

**Figure 2.**Mean reflectance (n = 80) processed by fractional-order derivatives in (

**a**) 0-order, (

**b**) 0.2-order, (

**c**) 0.4-order, (

**d**) 0.6-order, (

**e**) 0.8-order, (

**f**) 1-order, (

**g**) 1.2-order, (

**h**) 1.4-order, (

**i**) 1.6-order, (

**j**) 1.8-order, (

**k**) 2-order. The red areas represent the spectral standard deviations.

**Figure 3.**Line charts of RPD versus heavy metal concentration estimation results using fractional-order derivatives of (

**a**) Hg, (

**b**) Cr, (

**c**) Cu.

**Figure 4.**Error bar of RPD versus concentration estimation of Hg, Cr and Cu using fractional-order derivatives.

**Figure 5.**Boxplot of RPD versus heavy metal concentration estimation results using partial least squares regression (PLSR), random forest (RF), adaptive neural fuzzy inference system (ANFIS) and generalized regression neural network (GRNN) of (

**a**) Hg, (

**b**) Cr, (

**c**) Cu.

**Figure 6.**Scatterplots illustrating the optimal accuracy of (

**a**) Hg, 1.8-order reflectance generalized regression neural network (GRNN) model; (

**b**) Cr, 1.6-order reflectance random forest (RF) model; (

**c**) Cu, 1.2-order reflectance GRNN model. The black dotted lines represent 1:1 lines, and the red solid lines represent the regression lines.

Heavy Metal | Sample Sets | n ^{a} | Min | Max | Mean | Standard Deviation | CV ^{b} | Background Value ^{c} | Risk Screen Value ^{d} | Pollution Ratio ^{e} |
---|---|---|---|---|---|---|---|---|---|---|

Hg | Entire | 80 | 16.11 | 258.96 | 79.67 | 49.68 | 62.35% | 37 | 500 | 86.25% |

(ug/kg) | Training | 60 | 22.62 | 229.50 | 77.92 | 49.85 | 63.98% | 37 | 500 | 85.00% |

Validation | 20 | 16.11 | 258.96 | 78.33 | 51.24 | 65.42% | 37 | 500 | 86.67% | |

Cr | Entire | 80 | 26.70 | 130.30 | 80.13 | 27.60 | 34.45% | 57.9 | 150 | 75% |

(mg/kg) | Training | 60 | 27.90 | 129.10 | 79.46 | 28.07 | 35.33% | 57.9 | 150 | 75% |

Validation | 20 | 26.70 | 130.30 | 80.41 | 27.68 | 34.42% | 57.9 | 150 | 75% | |

Cu | Entire | 80 | 20.10 | 80.60 | 40.47 | 11.57 | 28.58% | 19.8 | 50 | 100% |

(mg/kg) | Training | 60 | 21.50 | 70.20 | 39.99 | 11.72 | 29.31% | 19.8 | 50 | 100% |

Validation | 20 | 20.10 | 80.60 | 40.30 | 11.61 | 28.82% | 19.8 | 50 | 100% |

^{a}Sample number.

^{b}Coefficient of variation.

^{c}The soil background values of Hg, Cr, and Cu in Liaoning Province [68].

^{d}Control standard for soil pollution risk of agricultural lands in China [67].

^{e}Percentage of contaminated samples (the soil background values are set as thresholds).

Order of FOD | Optimal Spectral Indices | Correlation Coefficients ^{a} |
---|---|---|

0 | RI_{475,473},DI_{1279,1278},NDI_{475,473},PI_{1912,1912},SI_{1912,1912} | 0.34,0.36,0.34,0.21,0.20 |

0.2 | RI_{1022,1021},DI_{1011,1010},NDI_{1022,1021},PI_{1904,1904},SI_{1905,1905} | 0.36,0.41,0.36,0.21,0.21 |

0.4 | RI_{1683,1682},DI_{1012,1009},NDI_{1683,1682},PI_{1897,1891},SI_{1897,1891} | 0.40,0.43,0.40,0.22,0.22 |

0.6 | RI_{1024,1018},DI_{1012,1008},NDI_{1024,1018},PI_{1407,1407},SI_{1407,1407} | 0.41,0.42,0.41,0.24,0.24 |

0.8 | RI_{1374,2147},DI_{2147,1374},NDI_{2162,1902},PI_{2321,1364},SI_{2321,1370} | 0.43,0.43,0.57,0.36,0.36 |

1 | RI_{1377,2286},DI_{2146,1375},NDI_{1583,1410},PI_{1364,475},SI_{1364,475} | 0.46,0.44,0.51,0.43,0.43 |

1.2 | RI_{1961,1978},DI_{1841,1011},NDI_{1522,677},PI_{733,707},SI_{2139,1430} | 0.54,0.45,0.55,0.43,0.43 |

1.4 | RI_{925,944},DI_{1642,1011},NDI_{1649,495},PI_{896,609},SI_{2255,1459} | 0.55,0.45,0.55,0.44,0.46 |

1.6 | RI_{1683,1826},DI_{2138,1011},NDI_{2228,722},PI_{1710,1173},SI_{1710,1173} | 0.51,0.45,0.57,0.47,0.45 |

1.8 | RI_{1022,1325},DI_{2087,1295},NDI_{1520,658},PI_{1969,1173},SI_{1710,1173} | 0.58,0.49,0.64,0.49,0.47 |

2 | RI_{1396,1458},DI_{2087,1295},NDI_{2357,864},PI_{1969,1173},SI_{1710,1173} | 0.56,0.49,0.58,0.49,0.47 |

^{a}For a more visual representation, the correlation coefficients are expressed as absolute values.

Order of FOD | Optimal Spectral Indices | Correlation Coefficients ^{a} |
---|---|---|

0 | RI_{2259,2262},DI_{955,954},NDI_{2262,2259},PI_{1008,1008},SI_{1020,1020} | 0.39,0.40,0.39,0.11,0.10 |

0.2 | RI_{2258,2262},DI_{955,954},NDI_{2262,2258},PI_{955,955},SI_{979,979} | 0.39,0.39,0.39,0.11,0.11 |

0.4 | RI_{2242,2266},DI_{1004,985},NDI_{2266,2242},PI_{955,955},SI_{955,955} | 0.41,0.41,0.41,0.12,0.12 |

0.6 | RI_{841,849},DI_{849,841},NDI_{849,841},PI_{893,893},SI_{955,955} | 0.45,0.43,0.45,0.15,0.14 |

0.8 | RI_{1211,1135},DI_{1211,1135},NDI_{1211,1135},PI_{2321,2319},SI_{2320,2320} | 0.47,0.46,0.47,0.27,0.26 |

1 | RI_{464,2261},DI_{713,464},NDI_{2261,464},PI_{2089,955},SI_{955,2089} | 0.51,0.51,0.51,0.45,0.46 |

1.2 | RI_{505,1368},DI_{1137,866},NDI_{1368,497},PI_{2259,2231},SI_{2346,2262} | 0.50,0.49,0.49,0.46,0.46 |

1.4 | RI_{972,2390},DI_{1137,866},NDI_{1142,714},PI_{2334,843},SI_{866,813} | 0.47,0.49,0.46,0.46,0.48 |

1.6 | RI_{2345,1847},DI_{2249,1312},NDI_{1047,714},PI_{2342,1831},SI_{2346,547} | 0.48,0.48,0.52,0.50,0.48 |

1.8 | RI_{2228,2087},DI_{2152,693},NDI_{2224,955},PI_{2342,1565},SI_{2346,547} | 0.49,0.47,0.47,0.51,0.48 |

2 | RI_{981,1821},DI_{814,626},NDI_{1629,1017},PI_{2342,1967},SI_{2346,547} | 0.48,0.46,0.51,0.50,0.47 |

^{a}For a more visual representation, the correlation coefficients are expressed as absolute values.

Order of FOD | Optimal Spectral Indices | Correlation Coefficients ^{a} |
---|---|---|

0 | RI_{2261,2266},DI_{2266,2264},NDI_{2266,2261},PI_{998,403},SI_{403,998} | 0.44,0.39,0.44,0.08,0.01 |

0.2 | RI_{980,979},DI_{980,979},NDI_{980,979},PI_{955,955},SI_{979,979} | 0.49,0.45,0.49,0.08,0.08 |

0.4 | RI_{2245,2266},DI_{980,979},NDI_{2266,2245},PI_{939,939},SI_{939,939} | 0.49,0.46,0.49,0.09,0.10 |

0.6 | RI_{2238,2263},DI_{2262,2227},NDI_{2263,2238},PI_{403,403},SI_{900,403} | 0.54,0.49,0.54,0.12,0.13 |

0.8 | RI_{980,932},DI_{1106,1028},NDI_{980,932},PI_{2281,862},SI_{862,403} | 0.55,0.54,0.55,0.23,0.24 |

1 | RI_{1084,890},DI_{1084,890},NDI_{1084,890},PI_{1779,1743},SI_{2347,2253} | 0.59,0.58,0.58,0.49,0.49 |

1.2 | RI_{980,1093},DI_{2248,980},NDI_{2263,656},PI_{980,842},SI_{2300,980} | 0.57,0.56,0.60,0.59,0.56 |

1.4 | RI_{667,1030},DI_{980,558},NDI_{667,631},PI_{2294,980},SI_{980,868} | 0.58,0.54,0.58,0.60,0.54 |

1.6 | RI_{667,1628},DI_{1129,1015},NDI_{1428,1304},PI_{1562,667},SI_{693,564} | 0.58,0.54,0.59,0.62,0.53 |

1.8 | RI_{2070,2103},DI_{1129,1015},NDI_{1047,544},PI_{1562,1554},SI_{923,667} | 0.59,0.53,0.62,0.61,0.52 |

2 | RI_{667,1880},DI_{1129,1015},NDI_{1815,1616},PI_{1966,1936},SI_{923,667} | 0.57,0.52,0.58,0.68,0.55 |

^{a}For a more visual representation, the correlation coefficients are expressed as absolute values.

Order | ANFIS | PLSR | GRNN | RF | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

RMSE ^{a} | R^{2 b} | RPD ^{c} | RMSE | R^{2} | RPD | RMSE | R^{2} | RPD | RMSE | R^{2} | RPD | |

0 | 48.43 | 0.01 | 1.03 | 48.27 | 0.01 | 1.03 | 47.51 | 0.04 | 1.05 | 47.10 | 0.06 | 1.06 |

0.2 | 45.68 | 0.12 | 1.09 | 48.08 | 0.02 | 1.04 | 47.51 | 0.04 | 1.05 | 45.99 | 0.10 | 1.08 |

0.4 | 46.05 | 0.10 | 1.08 | 45.77 | 0.11 | 1.09 | 47.50 | 0.04 | 1.05 | 46.27 | 0.09 | 1.08 |

0.6 | 48.37 | 0.01 | 1.03 | 47.81 | 0.03 | 1.04 | 47.51 | 0.04 | 1.05 | 47.08 | 0.06 | 1.06 |

0.8 | 40.62 | 0.30 | 1.23 | 44.10 | 0.18 | 1.13 | 47.06 | 0.06 | 1.06 | 42.41 | 0.24 | 1.18 |

1 | 44.68 | 0.15 | 1.12 | 47.26 | 0.05 | 1.05 | 46.59 | 0.08 | 1.07 | 47.16 | 0.06 | 1.06 |

1.2 | 46.39 | 0.09 | 1.07 | 47.59 | 0.04 | 1.05 | 47.30 | 0.05 | 1.05 | 45.69 | 0.12 | 1.09 |

1.4 | 38.66 | 0.37 | 1.29 | 45.81 | 0.11 | 1.09 | 30.84 | 0.60 | 1.62 | 43.81 | 0.19 | 1.14 |

1.6 | 43.60 | 0.19 | 1.14 | 45.29 | 0.13 | 1.10 | 36.83 | 0.43 | 1.35 | 43.82 | 0.19 | 1.14 |

1.8 | 27.92 | 0.67 | 1.79 | 36.63 | 0.43 | 1.36 | 26.81 | 0.70 | 1.86 | 30.06 | 0.62 | 1.66 |

2 | 33.63 | 0.52 | 1.48 | 39.53 | 0.34 | 1.26 | 29.78 | 0.62 | 1.67 | 34.96 | 0.48 | 1.43 |

^{a}Root mean square error.

^{b}Determination coefficient.

^{c}Residual prediction deviation.

Order | ANFIS | PLSR | GRNN | RF | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

RMSE ^{a} | R^{2 b} | RPD ^{c} | RMSE | R^{2} | RPD | RMSE | R^{2} | RPD | RMSE | R^{2} | RPD | |

0 | 24.81 | 0.18 | 1.13 | 25.18 | 0.15 | 1.11 | 24.37 | 0.21 | 1.15 | 25.33 | 0.14 | 1.11 |

0.2 | 25.92 | 0.10 | 1.08 | 24.82 | 0.18 | 1.13 | 25.45 | 0.14 | 1.10 | 25.06 | 0.16 | 1.12 |

0.4 | 24.92 | 0.17 | 1.13 | 25.89 | 0.10 | 1.08 | 24.25 | 0.21 | 1.16 | 25.31 | 0.14 | 1.11 |

0.6 | 21.97 | 0.35 | 1.28 | 23.10 | 0.29 | 1.22 | 21.43 | 0.39 | 1.31 | 21.16 | 0.40 | 1.33 |

0.8 | 24.29 | 0.21 | 1.16 | 23.68 | 0.25 | 1.19 | 24.01 | 0.23 | 1.17 | 23.67 | 0.25 | 1.19 |

1 | 22.72 | 0.31 | 1.24 | 22.36 | 0.33 | 1.26 | 21.13 | 0.40 | 1.33 | 24.39 | 0.21 | 1.15 |

1.2 | 21.03 | 0.41 | 1.33 | 21.08 | 0.41 | 1.33 | 20.65 | 0.43 | 1.36 | 23.50 | 0.26 | 1.19 |

1.4 | 21.06 | 0.41 | 1.33 | 20.33 | 0.45 | 1.38 | 19.10 | 0.51 | 1.47 | 20.93 | 0.42 | 1.34 |

1.6 | 18.65 | 0.54 | 1.51 | 21.81 | 0.36 | 1.29 | 16.92 | 0.62 | 1.66 | 15.32 | 0.69 | 1.83 |

1.8 | 21.60 | 0.38 | 1.30 | 22.64 | 0.32 | 1.24 | 20.63 | 0.43 | 1.36 | 23.02 | 0.29 | 1.22 |

2 | 23.79 | 0.24 | 1.18 | 23.97 | 0.23 | 1.17 | 20.75 | 0.42 | 1.35 | 24.84 | 0.18 | 1.13 |

^{a}Root mean square error.

^{b}Determination coefficient.

^{c}Residual prediction deviation.

Order | ANFIS | PLSR | GRNN | RF | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

RMSE ^{a} | R^{2 b} | RPD ^{c} | RMSE | R^{2} | RPD | RMSE | R^{2} | RPD | RMSE | R^{2} | RPD | |

0 | 10.00 | 0.23 | 1.17 | 10.15 | 0.21 | 1.15 | 9.01 | 0.38 | 1.30 | 9.98 | 0.24 | 1.17 |

0.2 | 9.16 | 0.36 | 1.28 | 10.84 | 0.10 | 1.08 | 10.31 | 0.19 | 1.14 | 9.83 | 0.26 | 1.19 |

0.4 | 9.71 | 0.28 | 1.21 | 9.68 | 0.28 | 1.21 | 8.25 | 0.48 | 1.42 | 9.85 | 0.26 | 1.19 |

0.6 | 9.00 | 0.38 | 1.30 | 9.02 | 0.38 | 1.30 | 8.04 | 0.50 | 1.46 | 9.82 | 0.26 | 1.19 |

0.8 | 9.25 | 0.34 | 1.27 | 8.55 | 0.44 | 1.37 | 7.66 | 0.55 | 1.53 | 9.60 | 0.29 | 1.22 |

1 | 7.95 | 0.52 | 1.47 | 8.42 | 0.46 | 1.39 | 7.33 | 0.59 | 1.60 | 8.27 | 0.48 | 1.42 |

1.2 | 7.69 | 0.55 | 1.52 | 7.94 | 0.52 | 1.48 | 6.78 | 0.65 | 1.73 | 7.92 | 0.52 | 1.48 |

1.4 | 7.56 | 0.56 | 1.55 | 8.18 | 0.49 | 1.43 | 7.51 | 0.57 | 1.56 | 7.62 | 0.56 | 1.54 |

1.6 | 7.80 | 0.53 | 1.50 | 8.15 | 0.49 | 1.44 | 7.22 | 0.60 | 1.62 | 7.82 | 0.53 | 1.50 |

1.8 | 7.73 | 0.54 | 1.52 | 8.60 | 0.43 | 1.36 | 7.51 | 0.57 | 1.56 | 8.20 | 0.48 | 1.43 |

2 | 8.19 | 0.49 | 1.43 | 8.75 | 0.41 | 1.34 | 7.71 | 0.54 | 1.52 | 8.18 | 0.49 | 1.43 |

^{a}Root mean square error.

^{b}Determination coefficient.

^{c}Residual prediction deviation.

Source of Variation | Degree of Freedom | Sum of the Squares | Mean Square | F-Value | p-Value | F_{critical} |
---|---|---|---|---|---|---|

FOD | 10 | 1.51 | 0.15 | 18.13 | 4.2 × 10^{−10} | 2.16 |

Modeling methods | 3 | 0.08 | 0.03 | 3.34 | 0.03 | 2.92 |

Residuals | 30 | 0.25 | 0.01 | |||

Sums | 43 | 1.85 |

Source of Variation | Degree of Freedom | Sum of the Squares | Mean Square | F-Value | p-Value | F_{critical} |
---|---|---|---|---|---|---|

FOD | 10 | 0.66 | 0.07 | 16.22 | 1.6 × 10^{−9} | 2.16 |

Modeling methods | 3 | 0.05 | 0.02 | 3.97 | 0.02 | 2.92 |

Residuals | 30 | 0.12 | 0.00 | |||

Sums | 43 | 0.83 |

Source of Variation | Degree of Freedom | Sum of the Squares | Mean Square | F-Value | p-Value | F_{critical} |
---|---|---|---|---|---|---|

FOD | 10 | 0.56 | 0.06 | 16.93 | 9.7 × 10^{−10} | 2.16 |

Modeling methods | 3 | 0.11 | 0.04 | 11.00 | 4.9 × 10^{−5} | 2.92 |

Residuals | 30 | 0.10 | 0.00 | |||

Sums | 43 | 0.76 |

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## Share and Cite

**MDPI and ACS Style**

Xu, X.; Chen, S.; Ren, L.; Han, C.; Lv, D.; Zhang, Y.; Ai, F.
Estimation of Heavy Metals in Agricultural Soils Using Vis-NIR Spectroscopy with Fractional-Order Derivative and Generalized Regression Neural Network. *Remote Sens.* **2021**, *13*, 2718.
https://doi.org/10.3390/rs13142718

**AMA Style**

Xu X, Chen S, Ren L, Han C, Lv D, Zhang Y, Ai F.
Estimation of Heavy Metals in Agricultural Soils Using Vis-NIR Spectroscopy with Fractional-Order Derivative and Generalized Regression Neural Network. *Remote Sensing*. 2021; 13(14):2718.
https://doi.org/10.3390/rs13142718

**Chicago/Turabian Style**

Xu, Xitong, Shengbo Chen, Liguo Ren, Cheng Han, Donglin Lv, Yufeng Zhang, and Fukai Ai.
2021. "Estimation of Heavy Metals in Agricultural Soils Using Vis-NIR Spectroscopy with Fractional-Order Derivative and Generalized Regression Neural Network" *Remote Sensing* 13, no. 14: 2718.
https://doi.org/10.3390/rs13142718