Geographically Weighted Regression Effects on Soil Zinc Content Hyperspectral Modeling by Applying the Fractional-Order Differential
"> Figure 1
<p>Spatial distribution of the 67 soil samples and their Zn contents. The colors denote the different ranges of soil Zn content, (<b>a</b>) the 46 modeling samples and (<b>b</b>) the 21 verification samples.</p> "> Figure 2
<p>The original hyperspectral data and the associated four transformations: (<b>a</b>) original reflectance, (<b>b</b>) square root, (<b>c</b>) logarithm, (<b>d</b>) reciprocal of logarithm, and (<b>e</b>) reciprocal.</p> "> Figure 3
<p>The correlation coefficients of hyperspectral reflectance data and the four transformations and different orders of differential operations applied: (<b>a</b>) reflectance, (<b>b</b>) square root, (<b>c</b>) logarithm, (<b>d</b>) reciprocal of logarithm, and (<b>e</b>) reciprocal. The x- and y-axes represent the wavelengths and fractional-orders (FOs) in differential operations from 0 to 2, respectively. The colors denote the correlation coefficients. For the soil sample size of 67, the absolute values of correlation coefficients greater than 0.3125 are considered to be significant (α = 0.01), which are denoted by deep blue and red in the color bar.</p> "> Figure 4
<p>Numbers of sifted candidates for geographically weighted regression (GWR) and ordinary least squares (OLS) modeling in 55 types of spectral data. The x-axis denotes reflectance and the four transformations. We sifted 304 candidates from the 55 types of spectral data.</p> "> Figure 5
<p>Flowchart showing the methods of this study.</p> "> Figure 6
<p>Modeling and verification results for 46 modeling and 21 verification samples for an OLS model (M<sub>1</sub>) and two GWR models (M<sub>2</sub> and M<sub>3</sub>). The left, middle, and right columns denote M<sub>1</sub>, M<sub>2</sub>, and M<sub>3</sub>, respectively. The top and bottom rows represent the associated modeling and verification results, respectively. The associated R<sup>2</sup> and R<sub>adj</sub><sup>2</sup> are displayed on the top-left of each panel.</p> "> Figure 7
<p>The results of the 60-repetition random processes by repeatedly selecting 57 samples for modeling and 10 samples for validation. The top and bottom panels display the 60 repetitions of the modeling results from the 57 randomly-picked samples and verification results from the left 10 samples, respectively. Values in the legend represent the associated mean R<sup>2</sup> of the 60-repetition random process. (<b>a</b>) The GWR models with two variates (D/4) in Method 1 and with bandwidths changing from D/4 to 3D/4; (<b>b</b>) the GWR models with four variates (3D/4) in Method 2 and with bandwidths changing from D/4 to 3D/4; (<b>c</b>) the GWR models with four variates obtained by OLS stepwise regression and with bandwidths changing from D/4 to 3D; (<b>d</b>) the GWR models with variates obtained by GWR stepwise regression for different bandwidths from D/4 to 3D/4 as shown in <a href="#remotesensing-11-00636-t006" class="html-table">Table 6</a>; (<b>e</b>) the GWR models with the superior mean R<sub>v</sub><sup>2</sup> in Methods 1–4; (<b>f</b>) the GWR model with the superior mean R<sub>v</sub><sup>2</sup> in Method 4, and those with one of the two variates x<sub>3</sub> and x<sub>4</sub> removed; and (<b>g</b>) the GWR models with variates of IO derivatives obtained by GWR stepwise regression for different bandwidths from D/4 to 3D/4 as shown in <a href="#remotesensing-11-00636-t007" class="html-table">Table 7</a>. In (<b>a</b>)–(<b>e</b>), the OLS models in Method 3 are compared with the GWR models, whereas in (<b>f</b>) and (<b>g</b>), the GWR models with the superior mean R<sub>v</sub><sup>2</sup> in Method 4 are compared to the other GWR models.</p> "> Figure 7 Cont.
<p>The results of the 60-repetition random processes by repeatedly selecting 57 samples for modeling and 10 samples for validation. The top and bottom panels display the 60 repetitions of the modeling results from the 57 randomly-picked samples and verification results from the left 10 samples, respectively. Values in the legend represent the associated mean R<sup>2</sup> of the 60-repetition random process. (<b>a</b>) The GWR models with two variates (D/4) in Method 1 and with bandwidths changing from D/4 to 3D/4; (<b>b</b>) the GWR models with four variates (3D/4) in Method 2 and with bandwidths changing from D/4 to 3D/4; (<b>c</b>) the GWR models with four variates obtained by OLS stepwise regression and with bandwidths changing from D/4 to 3D; (<b>d</b>) the GWR models with variates obtained by GWR stepwise regression for different bandwidths from D/4 to 3D/4 as shown in <a href="#remotesensing-11-00636-t006" class="html-table">Table 6</a>; (<b>e</b>) the GWR models with the superior mean R<sub>v</sub><sup>2</sup> in Methods 1–4; (<b>f</b>) the GWR model with the superior mean R<sub>v</sub><sup>2</sup> in Method 4, and those with one of the two variates x<sub>3</sub> and x<sub>4</sub> removed; and (<b>g</b>) the GWR models with variates of IO derivatives obtained by GWR stepwise regression for different bandwidths from D/4 to 3D/4 as shown in <a href="#remotesensing-11-00636-t007" class="html-table">Table 7</a>. In (<b>a</b>)–(<b>e</b>), the OLS models in Method 3 are compared with the GWR models, whereas in (<b>f</b>) and (<b>g</b>), the GWR models with the superior mean R<sub>v</sub><sup>2</sup> in Method 4 are compared to the other GWR models.</p> "> Figure 7 Cont.
<p>The results of the 60-repetition random processes by repeatedly selecting 57 samples for modeling and 10 samples for validation. The top and bottom panels display the 60 repetitions of the modeling results from the 57 randomly-picked samples and verification results from the left 10 samples, respectively. Values in the legend represent the associated mean R<sup>2</sup> of the 60-repetition random process. (<b>a</b>) The GWR models with two variates (D/4) in Method 1 and with bandwidths changing from D/4 to 3D/4; (<b>b</b>) the GWR models with four variates (3D/4) in Method 2 and with bandwidths changing from D/4 to 3D/4; (<b>c</b>) the GWR models with four variates obtained by OLS stepwise regression and with bandwidths changing from D/4 to 3D; (<b>d</b>) the GWR models with variates obtained by GWR stepwise regression for different bandwidths from D/4 to 3D/4 as shown in <a href="#remotesensing-11-00636-t006" class="html-table">Table 6</a>; (<b>e</b>) the GWR models with the superior mean R<sub>v</sub><sup>2</sup> in Methods 1–4; (<b>f</b>) the GWR model with the superior mean R<sub>v</sub><sup>2</sup> in Method 4, and those with one of the two variates x<sub>3</sub> and x<sub>4</sub> removed; and (<b>g</b>) the GWR models with variates of IO derivatives obtained by GWR stepwise regression for different bandwidths from D/4 to 3D/4 as shown in <a href="#remotesensing-11-00636-t007" class="html-table">Table 7</a>. In (<b>a</b>)–(<b>e</b>), the OLS models in Method 3 are compared with the GWR models, whereas in (<b>f</b>) and (<b>g</b>), the GWR models with the superior mean R<sub>v</sub><sup>2</sup> in Method 4 are compared to the other GWR models.</p> "> Figure 7 Cont.
<p>The results of the 60-repetition random processes by repeatedly selecting 57 samples for modeling and 10 samples for validation. The top and bottom panels display the 60 repetitions of the modeling results from the 57 randomly-picked samples and verification results from the left 10 samples, respectively. Values in the legend represent the associated mean R<sup>2</sup> of the 60-repetition random process. (<b>a</b>) The GWR models with two variates (D/4) in Method 1 and with bandwidths changing from D/4 to 3D/4; (<b>b</b>) the GWR models with four variates (3D/4) in Method 2 and with bandwidths changing from D/4 to 3D/4; (<b>c</b>) the GWR models with four variates obtained by OLS stepwise regression and with bandwidths changing from D/4 to 3D; (<b>d</b>) the GWR models with variates obtained by GWR stepwise regression for different bandwidths from D/4 to 3D/4 as shown in <a href="#remotesensing-11-00636-t006" class="html-table">Table 6</a>; (<b>e</b>) the GWR models with the superior mean R<sub>v</sub><sup>2</sup> in Methods 1–4; (<b>f</b>) the GWR model with the superior mean R<sub>v</sub><sup>2</sup> in Method 4, and those with one of the two variates x<sub>3</sub> and x<sub>4</sub> removed; and (<b>g</b>) the GWR models with variates of IO derivatives obtained by GWR stepwise regression for different bandwidths from D/4 to 3D/4 as shown in <a href="#remotesensing-11-00636-t007" class="html-table">Table 7</a>. In (<b>a</b>)–(<b>e</b>), the OLS models in Method 3 are compared with the GWR models, whereas in (<b>f</b>) and (<b>g</b>), the GWR models with the superior mean R<sub>v</sub><sup>2</sup> in Method 4 are compared to the other GWR models.</p> "> Figure 8
<p>Spatial distributions of parameter estimates for β<sub>0</sub>, β<sub>1</sub>, β<sub>2</sub>, β<sub>3</sub>, and β<sub>4</sub>; the associated t-values; and the contributions of C<sub>ik</sub> in predicted Zn content for intercept and the four variates of the representative GWR model. The left, middle, and right columns denote the parameter estimates, t-values, and the contributions C<sub>ik</sub>, respectively. (<b>a</b>) The intercept and (<b>b</b>)–(<b>e</b>) for the four variates x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>, and x<sub>4</sub>, respectively.</p> "> Figure 8 Cont.
<p>Spatial distributions of parameter estimates for β<sub>0</sub>, β<sub>1</sub>, β<sub>2</sub>, β<sub>3</sub>, and β<sub>4</sub>; the associated t-values; and the contributions of C<sub>ik</sub> in predicted Zn content for intercept and the four variates of the representative GWR model. The left, middle, and right columns denote the parameter estimates, t-values, and the contributions C<sub>ik</sub>, respectively. (<b>a</b>) The intercept and (<b>b</b>)–(<b>e</b>) for the four variates x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>, and x<sub>4</sub>, respectively.</p> "> Figure 9
<p>Spatial distributions of contributions C<sub>ik</sub> in predicted Zn content for intercept and the four variates in the representative OLS model. The columns from left to right denote the contributions C<sub>ik</sub> of the intercept and the four variates x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>, and x<sub>4</sub>, respectively.</p> "> Figure 10
<p>Scatter plots of the representative GWR (left panel) and OLS (right panel) models constructed using the 67 samples. The R<sub>adj</sub><sup>2</sup>, R<sup>2</sup>, RMSE, and average relative error δ calculated using the 67 samples are shown in the top of two panels. The red dashed vertical line indicates the measured Zn content of 25 mg/kg.</p> ">
Abstract
:1. Introduction
2. Materials and Methods
2.1. Study Site
2.2. Data Collection and Processing
2.3. Data Transformations and Fractional-Order Differentials
2.4. Correlation Analyses and Data Sifting
2.5. Geographically Weighted Regression
3. Results and Discussion
3.1. Modeling Results of the 46 Samples
3.2. Variate Selections from 67 Samples
3.3. Model Assessments
3.4. Analyses of Parameter Estimates
3.5. Performance of the Representative Models
3.6. Summary
4. Conclusions and Recommendations
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Heavy Metal Element | Min (mg/kg) | Max (mg/kg) | x (mg/kg) | SD (mg/kg) | cv (%) | Kurtosis | Skewness |
---|---|---|---|---|---|---|---|
Zn | 4 | 141 | 63.180 | 30.529 | 48.321 | 0.161 | 0.080 |
M1 (OLS) | M2 (GWR) | M3 (GWR) | |||||
---|---|---|---|---|---|---|---|
D/4 | D/2 | 3D/4 | D/4 | D/2 | 3D/4 | ||
Rm2 | 0.288 | 0.545 | 0.396 | 0.342 | 0.610 | 0.430 | 0.403 |
Radj,m2 | 0.255 | 0.524 | 0.368 | 0.311 | 0.591 | 0.389 | 0.360 |
Rv2 | –0.074 | –0.186 | –0.075 | –0.067 | –0.204 | –0.181 | –0.217 |
Radj,v2 | –0.194 | –0.318 | –0.194 | –0.186 | –0.337 | –0.390 | –0.432 |
x1 | reciprocal, FO = 0.8, 1560 nm | logarithm, FO = 1, 1140 nm | reciprocal, FO = 0.8, 1560 nm | ||||
x2 | sqrt, FO = 1.2, 1140 nm | reciprocal, FO = 1, 2020 nm | reciprocal, FO = 0.8, 1010 nm | ||||
x3 | reciprocal, FO = 0.6, 1010 nm |
Bandwidth = D/4 | ||||
Variates | I1 | tnew | Radj,m2 | Radj,v2 |
reciprocal, FO = 1.6, 1510 nm | 0.0458 | 1.0826 | 0.2822 | 0.1498 |
reciprocal, FO = 1.6, 1510 nm*; reciprocal, FO = 1, 1560 nm | 0.1421 | 1.8641 | 0.3916 | 0.1947 |
reciprocal, FO = 1, 1560 nm; reciprocal, FO = 0.8, 2310 nm | 0.1634 | 1.743 | 0.4477 | 0.2094 |
reciprocal, FO = 1, 1560 nm; reciprocal, FO = 0.8, 2310 nm*; reciprocal, FO = 0.8, 2020 nm | 0.1147 | 1.4718 | 0.4997 | 0.1559 |
reciprocal, FO = 1, 1560 nm; reciprocal, FO = 0.8, 2020 nm; reciprocal, FO = 1.2, 970 nm | 0.0817 | 1.0291 | 0.5489 | 0.1446 |
Bandwidth = D/2 | ||||
Variates | I1 | tnew | Radj,m2 | Radj,v2 |
logarithm, FO = 1, 1560 nm | 0.0236 | 2.6336 | 0.1794 | 0.0500 |
logarithm, FO = 1, 1560 nm; reciprocal, FO = 2, 2200 nm | 0.0786 | 2.0766 | 0.2387 | 0.1587 |
logarithm, FO = 1, 1560 nm; reciprocal, FO = 2, 2200 nm; sqrt, FO = 0.6, 1380 nm; | 0.0673 | 1.4093 | 0.2783 | 0.1717 |
logarithm, FO = 1, 1560 nm; reciprocal, FO = 2, 2200 nm; sqrt, FO = 0.6, 1380 nm; reciprocal, FO = 1, 2200 nm | 0.1106 | 2.0686 | 0.3555 | 0.1504 |
logarithm, FO = 1, 1560 nm; reciprocal, FO = 2, 2200 nm; sqrt, FO = 0.6, 1380 nm; reciprocal, FO = 1, 2200 nm; logarithm, FO = 1, 2200 nm | 0.0754 | 1.6947 | 0.3979 | 0.1119 |
logarithm, FO = 1, 1560 nm; reciprocal, FO = 2, 2200 nm; sqrt, FO = 0.6, 1380 nm*; reciprocal, FO = 1, 2200 nm; logarithm, FO = 1, 2200 nm; reciprocal of logarithm, FO = 0.2, 1930 nm | 0.0659 | 1.0264 | 0.4062 | 0.1580 |
logarithm, FO = 1, 1560 nm; reciprocal, FO = 2, 2200 nm; reciprocal, FO = 1, 2200 nm; logarithm, FO = 1, 2200 nm; reciprocal of logarithm, FO = 0.2, 1930 nm; logarithm, FO = 1.6, 2170 nm | 0.0885 | 1.1712 | 0.4224 | 0.1790 |
Bandwidth = 3D/4 | ||||
Variates | I1 | tnew | Radj,m2 | Radj,v2 |
reciprocal, FO = 0.8, 2220 nm | 0.0160 | 2.6938 | 0.1473 | 0.0404 |
reciprocal, FO = 0.8, 2220 nm; reciprocal, FO = 1, 1560 nm | 0.0525 | 2.1102 | 0.2116 | 0.1176 |
reciprocal, FO = 0.8, 2220 nm; reciprocal, FO = 1, 1560 nm; sqrt, FO = 0.6, 1380 nm | 0.0399 | 1.2801 | 0.2269 | 0.1374 |
reciprocal, FO = 0.8, 2220 nm; reciprocal, FO = 1, 1560 nm; sqrt, FO = 0.6, 1380 nm; reciprocal, FO = 1.2, 2020 nm | 0.0669 | 1.8941 | 0.2791 | 0.1265 |
reciprocal, FO = 0.8, 2220 nm; reciprocal, FO = 1, 1560 nm; sqrt, FO = 0.6, 1380 nm; reciprocal, FO = 1.2, 2020 nm; logarithm, FO = 1.2, 2020 nm | 0.1016 | 1.8254 | 0.3257 | 0.1709 |
reciprocal, FO = 0.8, 2220 nm*; reciprocal, FO = 1, 1560 nm; sqrt, FO = 0.6, 1380 nm *; reciprocal, FO = 1.2, 2020 nm; logarithm, FO = 1.2, 2020 nm; reciprocal of logarithm, FO = 1.8, 2200 nm | 0.0918 | 2.1128 | 0.3804 | 0.1143 |
reciprocal, FO = 1, 1560 nm; reciprocal, FO = 1.2, 2020 nm; logarithm, FO = 1.2, 2020 nm; reciprocal of logarithm, FO = 1.8, 2200 nm; logarithm, FO = 1.8, 700 nm | 0.0827 | 1.4927 | 0.4124 | 0.1343 |
Bandwidth = D/4 | |||||
Variates | I2 | tnew | tmin | Radj,m2 | Radj,v2 |
reciprocal, FO = 1, 1560 nm | 0.0174 | 1.9250 | 0.6434 | 0.3303 | 0.0424 |
reciprocal, FO = 1, 1560 nm; reciprocal, FO = 2, 640 nm | 0.0415 | 1.8741 | 0.3729 | 0.4261 | 0.1395 |
reciprocal, FO = 1, 1560 nm; reciprocal, FO = 2, 640 nm; reciprocal of logarithm, FO = 2, 2200 nm | 0.0179 | 1.3117 | 0.2945 | 0.4441 | 0.1041 |
reciprocal, FO = 1, 1560 nm; reciprocal, FO = 2, 640 nm; reciprocal of logarithm, FO = 2, 2200 nm; logarithm, FO = 0.6, 2380 nm | 0.0076 | 1.1330 | 0.1056 | 0.5363 | 0.1182 |
reciprocal, FO = 1, 1560 nm; reciprocal, FO = 2, 640 nm; reciprocal of logarithm, FO = 2, 2200 nm; logarithm, FO = 0.6, 2380 nm; reciprocal, FO = 1.8, 820 nm | 0.0125 | 1.3054 | 0.0673 | 0.6454 | 0.2198 |
reciprocal, FO = 1, 1560 nm; reciprocal, FO = 2, 640 nm; reciprocal of logarithm, FO = 2, 2200 nm; logarithm, FO = 0.6, 2380 nm; reciprocal, FO = 1.8, 820 nm; reciprocal, FO = 1.2, 1970 nm | 0.1091 | 1.6182 | 0.7625 | 0.6897 | 0.1281 |
Bandwidth = D/2 | |||||
Variates | I2 | tnew | tmin | Radj,m2 | Radj,v2 |
logarithm, FO = 1, 1560 nm | 0.0505 | 2.6336 | 2.1384 | 0.1794 | 0.05 |
logarithm, FO = 1, 1560 nm; logarithm, FO = 2, 2200 nm; | 0.1249 | 2.1617 | 1.6863 | 0.2399 | 0.1428 |
logarithm, FO = 1, 1560 nm; logarithm, FO = 2, 2200 nm; reflectance, FO = 1, 1560 nm | 0.064 | 1.9158 | 1.5799 | 0.2943 | 0.0719 |
logarithm, FO = 1, 1560 nm; logarithm, FO = 2, 2200 nm; reflectance, FO = 1, 1560 nm; reciprocal, FO = 1.2, 2020 nm | 0.0467 | 1.7515 | 1.2311 | 0.3474 | 0.0623 |
logarithm, FO = 1, 1560 nm; logarithm, FO = 2, 2200 nm; reflectance, FO = 1, 1560 nm; reciprocal, FO = 1.2, 2020 nm; logarithm, FO = 1.2, 2020 nm | 0.0410 | 1.1916 | 0.8381 | 0.3508 | 0.1170 |
Bandwidth = 3D/4 | |||||
Variates | I2 | tnew | tmin | Radj,m2 | Radj,v2 |
reciprocal, FO = 0.8, 2220 nm | 0.0405 | 2.6938 | 2.5271 | 0.1473 | 0.0404 |
reciprocal, FO = 0.8, 2220 nm; reciprocal, FO = 1, 1560 nm | 0.1010 | 2.1102 | 1.9241 | 0.2116 | 0.1176 |
reciprocal, FO = 0.8, 2220 nm*; reciprocal, FO = 1, 1560 nm; reciprocal, FO = 1.2, 2020 nm | 0.0578 | 1.8003 | 1.5415 | 0.2590 | 0.0804 |
reciprocal, FO = 1, 1560 nm; reciprocal, FO = 1.2, 2020 nm; reciprocal of logarithm, FO = 2, 2200 nm | 0.1100 | 2.0887 | 1.9006 | 0.3049 | 0.0909 |
reciprocal, FO = 1, 1560 nm; reciprocal, FO = 1.2, 2020 nm; reciprocal of logarithm, FO = 2, 2200 nm; logarithm, FO = 1.2, 2020 nm | 0.2670 | 2.1087 | 1.8366 | 0.3591 | 0.1920 |
reciprocal, FO = 1, 1560 nm; reciprocal, FO = 1.2, 2020 nm; reciprocal of logarithm, FO = 2, 2200 nm; logarithm, FO = 1.2, 2020 nm; logarithm, FO = 1.8, 680 nm | 0.1119 | 2.1072 | 1.7310 | 0.4132 | 0.0742 |
reciprocal, FO = 1, 1560 nm; reciprocal, FO = 1.2, 2020 nm; reciprocal of logarithm, FO = 2, 2200 nm; logarithm, FO = 1.2, 2020 nm; logarithm, FO = 1.8, 680 nm; reciprocal of logarithm, FO = 0.4, 2140 nm | 0.0141 | 1.1922 | 1.0272 | 0.4263 | 0.0269 |
reciprocal, FO = 1, 1560 nm; reciprocal, FO = 1.2, 2020 nm; reciprocal of logarithm, FO = 2, 2200 nm; logarithm, FO = 1.2, 2020 nm; logarithm, FO = 1.8, 680 nm; reciprocal of logarithm, FO = 0.4, 2140 nm; reciprocal, FO = 0.8, 1010 nm | 0.0204 | 1.3752 | 1.2141 | 0.4493 | 0.0271 |
Method 1 (46 samples) | variates | reciprocal, FO = 1, 1560 nm; | reciprocal, FO = 0.8, 2310 nm | ||
t | 1.657 | 1.743 | |||
tmin | 0.119 | 0.001 | |||
Method 1 (67 samples) | t | 2.305 | 2.312 | ||
tmin | 0.142 | 0.060 | |||
Method 2 (46 samples) | variates | reciprocal, FO = 1, 1560 nm | reciprocal, FO = 1.2, 2020 nm | reciprocal of logarithm, FO = 2, 2200 nm | logarithm, FO = 1.2, 2020 nm |
t | 2.458 | 2.396 | 2.607 | 2.109 | |
tmin | 2.234 | 2.101 | 2.449 | 1.837 | |
Method 2 (67 samples) | t | 3.682 | 3.345 | 3.364 | 3.002 |
tmin | 3.507 | 2.927 | 3.212 | 2.589 | |
Method 3 (67 samples) | variates | reciprocal, FO = 0.8, 1560 nm | reciprocal, FO = 1.2, 2020 nm | logarithm, FO = 1.2, 2020 nm | reciprocal of logarithm, FO = 1.6, 2200 nm |
t | 3.479 | 4.166 | 3.889 | 2.467 | |
tmin | 3.219 | 3.938 | 3.667 | 2.405 | |
OLS (67 samples) | t | −3.558 | −4.272 | −4.001 | 2.607 |
Bandwidth | Variates | Rm2 | Radj,m2 |
---|---|---|---|
D/4 | reciprocal, FO = 0.8, 2020 nm | 0.3673 | 0.3576 |
3D/8 | reciprocal, FO = 1, 1560 nm; reciprocal of logarithm, FO = 2, 2200 nm; reciprocal, FO = 1.2, 2020 nm; logarithm, FO = 1.2, 2020 nm | 0.4644 | 0.4299 |
D/2 | reciprocal, FO = 1, 1560 nm; reciprocal of logarithm, FO = 1.6, 2200 nm; reciprocal, FO = 1.2, 2020 nm; logarithm, FO = 1.2, 2020 nm | 0.4634 | 0.4288 |
5D/8 | reciprocal, FO = 1, 1560 nm; reciprocal of logarithm, FO = 1.6, 2200 nm; reciprocal, FO = 1.2, 2020 nm; logarithm, FO = 1.2, 2020 nm | 0.4572 | 0.4221 |
3D/4 | reciprocal, FO = 1, 1560 nm; reciprocal of logarithm, FO = 1.6, 2200 nm; reciprocal, FO = 1.2, 2020 nm; logarithm, FO = 1.2, 2020 nm | 0.4542 | 0.4190 |
Bandwidth | Variates | Rm2 | Radj,m2 |
---|---|---|---|
D/4 | reciprocal, FO = 1, 2020 nm | 0.337 | 0.326 |
3D/8 | reciprocal, FO = 1, 2020 nm; reciprocal, FO = 1, 1560 nm; reciprocal of logarithm, FO = 2, 2200 nm; logarithm, FO = 1, 2020 nm | 0.457 | 0.422 |
D/2 | reciprocal, FO = 1, 2020 nm; reciprocal, FO = 1, 1560 nm; reciprocal of logarithm, FO = 2, 2200 nm; logarithm, FO = 1, 2020 nm; reciprocal of logarithm, FO = 0, 2200 nm; sqrt, FO = 2, 640 nm | 0.518 | 0.469 |
5D/8 | reciprocal, FO = 1, 2020 nm; reciprocal, FO = 1, 1560 nm; reciprocal of logarithm, FO = 2, 2200 nm; logarithm, FO = 1, 2020 nm; reciprocal of logarithm, FO = 0, 2200 nm; sqrt, FO = 2, 640 nm | 0.506 | 0.456 |
3D/4 | reciprocal, FO = 2, 2200 nm; reciprocal, FO = 1, 1560 nm; | 0.294 | 0.272 |
Method 1 (D/4) | Method 2 (3D/4) | Method 3 (3D/4) | Method 4 (3D/4) | OLS | IO (5D/8) | IO (3D/4) | OLS-2 | OLS-3 | |
---|---|---|---|---|---|---|---|---|---|
Mean Rm2 | 0.480 | 0.438 | 0.432 | 0.459 | 0.423 | 0.519 | 0.510 | 0.452 | 0.488 |
Mean Radj,m2 | 0.461 | 0.395 | 0.389 | 0.418 | 0.378 | 0.461 | 0.451 | 0.410 | 0.427 |
Mean Rv2 | 0.082 | 0.205 | 0.163 | 0.242 | 0.175 | 0.220 | 0.228 | 0.253 | 0.244 |
Mean Radj,v2 | −0.180 | −0.431 | −0.508 | −0.364 | −0.486 | −1.340 | −1.316 | −0.346 | −1.270 |
Parameter | β0 | β1 | β2 | β3 | β4 |
---|---|---|---|---|---|
β | 5.922 | –7.460 × 104 | 1.551 × 104 | –2.515 × 105 | –1.018 × 106 |
SE | 9.264 | 1.799 × 104 | 3.989 × 103 | 5.961 × 104 | 2.604 × 105 |
t-value | 0.639 | −4.148 | 3.888 | −4.219 | −3.909 |
Representative GWR model | Representative OLS model | |||||
---|---|---|---|---|---|---|
67 samples | 9 samples (<25 mg/kg) | 58 samples (>25 mg/kg) | 67 samples | 9 samples (<25 mg/kg) | 58 samples (>25 mg/kg) | |
Radj2 | 0.419 | –73.868 | 0.396 | 0.413 | –76.057 | 0.399 |
R2 | 0.454 | –36.434 | 0.438 | 0.449 | –37.529 | 0.441 |
RMSE | 22.338 | 40.977 | 17.773 | 22.452 | 41.571 | 17.725 |
δ | 89.2% | 537.2% | 19.7% | 90.2% | 545.2% | 19.6% |
© 2019 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 (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
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Lin, X.; Su, Y.-C.; Shang, J.; Sha, J.; Li, X.; Sun, Y.-Y.; Ji, J.; Jin, B. Geographically Weighted Regression Effects on Soil Zinc Content Hyperspectral Modeling by Applying the Fractional-Order Differential. Remote Sens. 2019, 11, 636. https://doi.org/10.3390/rs11060636
Lin X, Su Y-C, Shang J, Sha J, Li X, Sun Y-Y, Ji J, Jin B. Geographically Weighted Regression Effects on Soil Zinc Content Hyperspectral Modeling by Applying the Fractional-Order Differential. Remote Sensing. 2019; 11(6):636. https://doi.org/10.3390/rs11060636
Chicago/Turabian StyleLin, Xue, Yung-Chih Su, Jiali Shang, Jinming Sha, Xiaomei Li, Yang-Yi Sun, Jianwan Ji, and Biao Jin. 2019. "Geographically Weighted Regression Effects on Soil Zinc Content Hyperspectral Modeling by Applying the Fractional-Order Differential" Remote Sensing 11, no. 6: 636. https://doi.org/10.3390/rs11060636