# Estimating Sub-Pixel Soybean Fraction from Time-Series MODIS Data Using an Optimized Geographically Weighted Regression Model

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{2}) of 0.80 and the root mean square error (RMSE) was 340.21 km

^{2}. Additionally, F-test results showed GWR model had better model goodness-of-fit and higher prediction accuracy than the traditional ordinary least squares (OLS) model. These promising results suggest crop spectral variations both at temporal and spatial scales should be considered when exploring its relationship with pixel-level crop acreage. The optimized GWR model by combining an automated feature selection strategy has great potential for estimating sub-pixel crop area at regional scale based on remote sensing time-series data.

## 1. Introduction

## 2. Study Area and Datasets

#### 2.1. Study Area

^{2}in 2005 to 24,761 km

^{2}in 2015, with corn replacing soybean as the dominant dry-land crop in this region [4,25]. A field survey in 2013 showed that the soybeans were mainly distributed in northwestern part of Heilongjiang Province, and soybean fields in regions near Songnen plain are almost continuous and large, while those in the northwest part close to Lesser Khingan Mountains are relatively small and fragment. Soybean in Heilongjiang is generally sowed in late-April-followed by emergence, three leaves, seven leaves, blooming, bearing pod, filling seed, senescence and harvested in early-October (Figure 2). The distinct difference in growing period length and biochemical properties between soybean and other crops in this region will be expressed as the variance of spectral signature (e.g., NDVI) across the growing seasons, which is the crucial characteristic to identify soybeans.

#### 2.2. Sample Data

#### 2.3. Feature Set

_{1}and Band

_{2}for each time point and used as the candidate input variables of GWR (shown as Equation (1)).

_{1}is the MODIS surface reflectance of red (620–670 nm) and Band

_{2}is the MODIS surface reflectance of near infrared (841–876 nm).

#### 2.4. Census Data

## 3. Methodology

#### 3.1. An Optimized Geographically Weighted Regression Model

#### 3.1.1. The Basic Framework of GWR Model

_{in}is the weight assigned to the observation at location i. The weights are chosen based on the assumption that those observations near the point in space have a more significant influence on the result than observations further away [16,29]. Therefore, near locations get high weights and far ones get low weights. There are usually two types of kernel functions, fixed kernel and adaptive kernel used to obtain weights. In this study, we used the adaptive kernel function, which can ensure a certain number of nearest neighbors as local samples and better represents the degree of spatial heterogeneity than the fixed kernel. This adaptive kernel function was based on a bi-square distance decay function as follows [30]:

#### 3.1.2. A Forward Stepwise Strategy for Selecting the Optimal Independent Variables

- Step 1: Start by calibrating all possible bivariate GW regressions by in turn regressing a single explanatory variable against the dependent variable. Calculate AICc in each case (31 runs in this study). Select the variable that produces the lowest AICc.
- Step 2: Sequentially introduce a variable from the remaining n − 1 features to construct new models with the permanently included independent variable in step 1. Calculate the change in AICc between step1 and step 2. Select variable yielding greatest reduction in AICc. Add this variable to the model.
- Step 3: Repeat step 2 until no independent variables among the candidate variables can be added into the model, and the model at this point is the final model.

#### 3.1.3. F-Test Statistics

^{2}and adjusted R

^{2}, were used to reflect the model fitting performance and the spatially inhomogeneous of the GWR. Then, an F-test statistical testing method proposed by Leung et al. [33], was utilized to test the improvements of GWR over OLS in fitting the model and to understand whether or not each variable in the GWR model vary significantly.

_{1}-test statistics in Leung et al. [31]. Specifically, if the null hypothesis—H

_{0}: there is no significant difference between OLS and GWR models for the given data—is true, the quantity RSS

_{g}/RSS

_{o}is close to one. Otherwise, it tends to be small. The F

_{1}-value can be calculated as [33]:

_{g}and RSS

_{o}are the residual sum of squares of GWR and OLS model, respectively. ${\sigma}_{1}$ is the standard deviation of error and also equal to the mean of $RS{S}_{g}/{\sigma}^{2}$ where ${\sigma}^{2}$ is constant variance. n − p − 1 represents the degrees of freedom in the denominator. The distribution of F

_{1}-value may be approximated by an F-distribution with ${\sigma}_{1}{}^{2}/{\sigma}_{2}$ degrees of freedom in the numerator and n − p − 1 degrees of freedom in the denominator. If F

_{1}-value < ${F}_{1-\alpha}(\frac{{\sigma}_{1}{}^{2}}{{\sigma}_{2}},n-p-1)$, where $\alpha $ is the given significance level, the null hypothesis will be rejected and it is thus believed that the GWR model is significantly better than the OLS in describing the data. Otherwise, we will conclude that the GWR model cannot improve the model fitting significantly compared to the OLS model.

_{3}-test statistics in Leung et al. [31]. The null hypothesis is: ${H}_{0}:\text{}{\mathsf{\beta}}_{1k}={\mathsf{\beta}}_{1k}=\dots ={\mathsf{\beta}}_{nk}$ (for a given k), the F

_{3}-value can be calculated as:

_{k}

^{2}is a statistic that can reflect the spatial variation of the given set of parameters {${\mathsf{\beta}}_{ik};i=1,2,\dots ,n$}. ${\stackrel{\u2322}{\sigma}}^{2}$ is an unbiased estimator of ${\sigma}^{2}$. Its distribution can be approximated by an F-distribution with ${\gamma}_{1}{}^{2}/{\gamma}_{2}$ degrees of freedom in the numerator and ${\delta}_{1}{}^{2}/{\delta}_{2}$ degrees of freedom in the denominator. If ${F}_{3}\ge F({\gamma}_{1}{}^{2}/{\gamma}_{2},{\delta}_{1}{}^{2}/{\delta}_{2})$, reject H

_{0}; otherwise, accept H

_{0}. More details about the F-test statistical testing method are provided by Leung et al. [33].

#### 3.2. Accuracy Assessment

^{2}. They are defined as:

## 4. Results

#### 4.1. Descriptive Statistics

#### 4.2. The Optimal Independent Variables for Soybean Cultivation

#### 4.3. GWR Results of Sub-Pixel Soybean Area Estimation

^{2}, and adjusted R

^{2}) of GWR and OLS are summarized as Table 1. The adjusted overall mean local R

^{2}for GWR model is 0.4377, meaning 43.77% of the variability in the sub-pixel soybean proportion can be explained in the modeling dataset. Compared to OLS model, significant improvements for all descriptive statistics indicates GWR is superior in exploring the relationships between the time-series NDVI values and sub-pixel crop area. In addition, the non-stationarity F-test results including F

_{1}-test and F

_{3}-test between GWR and OLS is shown in Table 2. The F

_{1}-value indicates the null hypotheses is rejected and there is a significant difference between GWR and OLS model. Additionally, it is observed 21 variables of the total 31 NDVI variables as well as intercept show highly significant non-stationarity (α = 0.001 level), suggesting the response of the sub-pixel soybean area to those variables is not constant across the study area. By contrast, only six NDVI variables (i.e., NDVI of Julian day 81, 89,137, 153, 201 and 209) are not statistically significant, indicating they keep relatively stable relationships with the dependent variable. Not surprisingly, the four optimal variables, i.e., NDVI of Julian day 233, 241, 249, and 257 (Section 4.2) are found statistically significant, which confirms the importance of images acquired at filling seed stage for mapping sub-pixel soybean area. This F-test results also highlight the advantages of GWR over OLS model in expressing the spatially variant relationships between crop area and remote sensing images.

^{2}and adjusted R

^{2}. The higher accuracy derived by calculating great circle distance may be explained by the fact the spherical distance can characterize the real geographical correlations between different points more accurately than the plane distance.

#### 4.4. Sub-Pixel Soybean Map and Accuracy Assessment

^{2}of 0.80 and RMSE of 340.21 km

^{2}and NRMSE of 0.1054 across 80 counties. A closer observation for this figure shows our GWR model based on MODIS data overestimated soybean areas of more than 750 km

^{2}, while almost underestimated those areas of less than 500 km

^{2}.

^{2}, such as Yichun, Hegan, Suiling, Hailun, Tangyuan and Baiquan. The high landscape heterogeneity and small field size are responsible for the relatively large confusion between soybean and non-soybean spectral signature in those regions. Overall, in the north part of Heilongjiang, our GWR model based on MODIS data overestimated soybean cultivation. In contrast, our GWR model slightly underestimated soybean acreage in northern districts, such as Shanzhi, Dongning, Mulin, Linkou and Hailin. The obvious regional similarity in the regression bias may be because the adjoining areas present similar spectral characteristics for soybean cultivation, and thereby they may have the similar or the same regression relationships between sub-pixel soybean fractions and time-series NDVI. It should be noted here some errors could be due to the uncertainty of the census data, since crop area information is acquired from sampling survey or communication with farmers which are not always accurate [13].

## 5. Discussion

## 6. Conclusions

^{2}of 0.80 and RMSE was 340.21 km

^{2}across 80 counties. The largest differences between these two datasets were observed in regions with high landscape heterogeneity and small field size. Additionally, the optimized GWR model had better performances with lower RSS and higher R

^{2}compared to OLS model, and their F-test results shown 23 variables of the total 31 time-series NDVI are statistically significant. These promising results indicate the spatial nonstationary within spectral-based variables should be considered when the relationship between crop sub-pixel fraction and remote sensing images is explored. Our presented GWR model shows great potential in mapping sub-pixel crop distribution over large region based on coarse spatial resolution imagery such as MODIS. However, if the presented GWR model is to be extended to other regions and land cover types, further development of the GWR method is possible, such as testing a variety of the input variables (e.g., vegetation indices and phenological metrics) and developing a mixed GWR model that allow some variables held constant while others vary spatially. In addition, future work could address the issues on how to use multi-source finer spatial resolution data to improve the selection of training and validation samples.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Zhong, L.; Gong, P.; Biging, G.S. Efficient corn and soybean mapping with temporal extendability: A multi-year experiment using landsat imagery. Remote Sens. Environ.
**2014**, 140, 1–13. [Google Scholar] [CrossRef] - Monfreda, C.; Ramankutty, N.; Foley, J.A. Farming the planet: 2. Geographic distribution of crop areas, yields, physiological types, and net primary production in the year 2000. Glob. Biogeochem. Cycles
**2008**, 22, GB1022. [Google Scholar] [CrossRef] - Zheng, H.; Chen, L.; Han, X.; Zhao, X.; Ma, Y. Classification and regression tree (cart) for analysis of soybean yield variability among fields in northeast china: The importance of phosphorus application rates under drought conditions. Agric. Ecosyst. Environ.
**2009**, 132, 98–105. [Google Scholar] [CrossRef] - Sun, J.; Wu, W.; Tang, H.; Liu, J. Spatiotemporal patterns of non-genetically modified crops in the era of expansion of genetically modified food. Sci. Rep.
**2015**, 5, 14180. [Google Scholar] [CrossRef] [PubMed] - Wardlow, B.; Egbert, S.; Kastens, J. Analysis of time-series MODIS 250 m vegetation index data for crop classification in the u.S. Central great plains. Remote Sens. Environ.
**2007**, 108, 290–310. [Google Scholar] [CrossRef] - Hu, Q.; Wu, W.; Song, Q.; Yu, Q.; Lu, M.; Yang, P.; Tang, H.; Long, Y. Extending the pairwise separability index for multicrop identification using time-series MODIS images. IEEE Trans. Geosci. Remote Sens.
**2016**, 54, 6349–6361. [Google Scholar] [CrossRef] - Hu, Q.; Wu, W.-B.; Song, Q.; Lu, M.; Chen, D.; Yu, Q.-Y.; Tang, H.-J. How do temporal and spectral features matter in crop classification in heilongjiang province, china? J. Integr. Agric.
**2017**, 16, 324–336. [Google Scholar] [CrossRef] - Chang, J.; Hansen, M.C.; Pittman, K.; Carroll, M.; DiMiceli, C. Corn and soybean mapping in the united states using MODIS time-series data sets. Agron. J.
**2007**, 99, 1654. [Google Scholar] [CrossRef] - Huang, X.; Schneider, A.; Friedl, M.A. Mapping sub-pixel urban expansion in china using MODIS and DMSP/OLS nighttime lights. Remote Sens. Environ.
**2016**, 175, 92–108. [Google Scholar] [CrossRef] - Pan, Y.; Li, L.; Zhang, J.; Liang, S.; Zhu, X.; Sulla-Menashe, D. Winter wheat area estimation from MODIS-evi time series data using the crop proportion phenology index. Remote Sens. Environ.
**2012**, 119, 232–242. [Google Scholar] [CrossRef] - Lobell, D.B.; Asner, G.P. Cropland distributions from temporal unmixing of MODIS data. Remote Sens. Environ.
**2004**, 93, 412–422. [Google Scholar] [CrossRef] - Atzberger, C.; Rembold, F. Mapping the spatial distribution of winter crops at sub-pixel level using avhrr ndvi time series and neural nets. Remote Sens.
**2013**, 5, 1335–1354. [Google Scholar] [CrossRef] [Green Version] - Peña-Barragán, J.M.; Ngugi, M.K.; Plant, R.E.; Six, J. Object-based crop identification using multiple vegetation indices, textural features and crop phenology. Remote Sens. Environ.
**2011**, 115, 1301–1316. [Google Scholar] [CrossRef] - Löw, F.; Michel, U.; Dech, S.; Conrad, C. Impact of feature selection on the accuracy and spatial uncertainty of per-field crop classification using support vector machines. ISPRS J. Photogramm. Remote Sens.
**2013**, 85, 102–119. [Google Scholar] [CrossRef] - Drumetz, L.; Chanussot, J.; Jutten, C. Variability of the endmembers in spectral unmixing: Recent advances. In Proceedings of the 8th IEEE Workshop on Hyperspectral Image and Signal Processing: Evolution in Remote Sensing, Los Angeles, CA, USA, 21–24 August 2016. [Google Scholar]
- Zhang, L.; Wei, Y.; Meng, R. Spatiotemporal dynamics and spatial determinants of urban growth in Suzhou, China. Sustainability
**2017**, 9, 393. [Google Scholar] [CrossRef] - Propastin, P.A. Spatial non-stationarity and scale-dependency of prediction accuracy in the remote estimation of lai over a tropical rainforest in sulawesi, indonesia. Remote Sens. Environ.
**2009**, 113, 2234–2242. [Google Scholar] [CrossRef] - Wu, J.; Yao, F.; Li, W.; Si, M. Viirs-based remote sensing estimation of ground-level PM
_{2.5}concentrations in Beijing–Tianjin–Hebei: A spatiotemporal statistical model. Remote Sens. Environ.**2016**, 184, 316–328. [Google Scholar] [CrossRef] - Fotheringham, A.S.; Brunsdon, C.; Charlton, M. Geographically Weighted Regression: The Analysis of Spatially Varying Relationships; John Wiley & Sons Inc.: Hoboken, NJ, USA, 2002. [Google Scholar]
- You, W.; Zang, Z.; Zhang, L.; Li, Z.; Chen, D.; Zhang, G. Estimating ground-level PM
_{10}concentration in northwestern china using geographically weighted regression based on satellite aod combined with calipso and MODIS fire count. Remote Sens. Environ.**2015**, 168, 276–285. [Google Scholar] [CrossRef] - Fang, X.; Zou, B.; Liu, X.; Sternberg, T.; Zhai, L. Satellite-based ground PM
_{2.5}estimation using timely structure adaptive modeling. Remote Sens. Environ.**2016**, 186, 152–163. [Google Scholar] [CrossRef] - Propastin, P. Modifying geographically weighted regression for estimating aboveground biomass in tropical rainforests by multispectral remote sensing data. Int. J. Appl. Earth Obs. Geoinf.
**2012**, 18, 82–90. [Google Scholar] [CrossRef] - See, L.; Schepaschenko, D.; Lesiv, M.; McCallum, I.; Fritz, S.; Comber, A.; Perger, C.; Schill, C.; Zhao, Y.; Maus, V.; et al. Building a hybrid land cover map with crowdsourcing and geographically weighted regression. ISPRS J. Photogramm. Remote Sens.
**2015**, 103, 48–56. [Google Scholar] [CrossRef] [Green Version] - Hu, Q.; Wu, W.; Xia, T.; Yu, Q.; Yang, P.; Li, Z.; Song, Q. Exploring the use of google earth imagery and object-based methods in land use/cover mapping. Remote Sens.
**2013**, 5, 6026–6042. [Google Scholar] [CrossRef] - Agriculture, C.M.O. China Agricultural Statistics Yearbook; China Statistics Press: Beijing, China, 2015. [Google Scholar]
- Rui, X.; Zhongjun, L.; Yang, L.; Bin, F.; Kebao, L. Comparison on linear feature real width and interpretation width using landsat tm8 images and gf-1 images. Trans. Chin. Soc. Agric. Eng.
**2015**, 16. [Google Scholar] [CrossRef] - Wardlow, B.D.; Egbert, S.L. Large-area crop mapping using time-series MODIS 250 m ndvi data: An assessment for the u.S. Central great plains. Remote Sens. Environ.
**2008**, 112, 1096–1116. [Google Scholar] [CrossRef] - Chen, J.; Chen, J.; Liao, A.; Cao, X.; Chen, L.; Chen, X.; He, C.; Han, G.; Peng, S.; Lu, M.; et al. Global land cover mapping at 30 m resolution: A pok-based operational approach. ISPRS J. Photogramm. Remote Sens.
**2015**, 103, 7–27. [Google Scholar] [CrossRef] - Lesiv, M.; Moltchanova, E.; Schepaschenko, D.; See, L.; Shvidenko, A.; Comber, A.; Fritz, S. Comparison of data fusion methods using crowdsourced data in creating a hybrid forest cover map. Remote Sens.
**2016**, 8, 261. [Google Scholar] [CrossRef] - Salas, C.; Ene, L.; Gregoire, T.G.; Næsset, E.; Gobakken, T. Modelling tree diameter from airborne laser scanning derived variables: A comparison of spatial statistical models. Remote Sens. Environ.
**2010**, 114, 1277–1285. [Google Scholar] [CrossRef] - Guo, L.; Ma, Z.; Zhang, L. Comparison of bandwidth selection in application of geographically weighted regression: A case study. Can. J. Forest Res.
**2008**, 38, 2526–2534. [Google Scholar] [CrossRef] - Gollini, I.; Lu, B.B.; Charlton, M.; Brunsdon, C.; Harris, P. Gwmodel: An r package for exploring spatial heterogeneity using geographically weighted models. J. Stat. Softw.
**2015**, 63, 1–50. [Google Scholar] [CrossRef] - Leung, Y.; Mei, C.L.; Zhang, W.X. Statistical tests for spatial nonstationarity based on the geographically weighted regression model. Environ. Plan. A
**2000**, 32, 9–32. [Google Scholar] [CrossRef] - Maxwell, S.K.; Nuckols, J.R.; Ward, M.H.; Hoffer, R.M. An automated approach to mapping corn from landsat imagery. Comput. Electron. Agric.
**2004**, 43, 43–54. [Google Scholar] [CrossRef] - Xiao, X.; Boles, S.; Frolking, S.; Li, C.; Babu, J.Y.; Salas, W.; Moore, B. Mapping paddy rice agriculture in south and southeast asia using multi-temporal MODIS images. Remote Sens. Environ.
**2006**, 100, 95–113. [Google Scholar] [CrossRef] - Colditz, R.R.; López Saldaña, G.; Maeda, P.; Espinoza, J.A.; Tovar, C.M.; Hernández, A.V.; Benítez, C.Z.; Cruz López, I.; Ressl, R. Generation and analysis of the 2005 land cover map for mexico using 250 m MODIS data. Remote Sens. Environ.
**2012**, 123, 541–552. [Google Scholar] [CrossRef] - Mei, C.L.; He, S.Y.; Fang, K.T. A note on the mixed geographically weighted regression model. J. Reg. Sci.
**2004**, 44, 143–157. [Google Scholar] [CrossRef] - Huang, B.; Wu, B.; Barry, M. Geographically and temporally weighted regression for modeling spatio-temporal variation in house prices. Int. J. Geogr. Inf. Sci.
**2010**, 24, 383–401. [Google Scholar] [CrossRef] - Schepaschenko, D.; See, L.; Lesiv, M.; McCallum, I.; Fritz, S.; Salk, C.; Moltchanova, E.; Perger, C.; Shchepashchenko, M.; Shvidenko, A.; et al. Development of a global hybrid forest mask through the synergy of remote sensing, crowdsourcing and fao statistics. Remote Sens. Environ.
**2015**, 162, 208–220. [Google Scholar] [CrossRef] - Torbick, N.; Chowdhury, D.; Salas, W.; Qi, J. Monitoring rice agriculture across myanmar using time series sentinel-1 assisted by landsat-8 and palsar-2. Remote Sens.
**2017**, 9, 119. [Google Scholar] [CrossRef] - Radoux, J.; Chomé, G.; Jacques, D.; Waldner, F.; Bellemans, N.; Matton, N.; Lamarche, C.; d’Andrimont, R.; Defourny, P. Sentinel-2’s potential for sub-pixel landscape feature detection. Remote Sens.
**2016**, 8, 488. [Google Scholar] [CrossRef] - Roy, D.P.; Wulder, M.A.; Loveland, T.R.; Woodcock, C.E.; Allen, R.G.; Anderson, M.C.; Helder, D.; Irons, J.R.; Johnson, D.M.; Kennedy, R.; et al. Landsat-8: Science and product vision for terrestrial global change research. Remote Sens. Environ.
**2014**, 145, 154–172. [Google Scholar] [CrossRef] - Li, Z.; Shen, H.; Li, H.; Xia, G.; Gamba, P.; Zhang, L. Multi-feature combined cloud and cloud shadow detection in gaofen-1 wide field of view imagery. Remote Sens. Environ.
**2017**, 191, 342–358. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) Landsat-8 OLI images and field data (green dots) used to derive the 30-m soybean reference map; (

**b**) 30 m soybean reference map, from which 4000 training points (blue dots) were selected as training data for GWR model; (

**c**) An example to illustrate how soybean fraction was calculated at 250-m grid based on the 30-m soybean reference map.

**Figure 3.**Overview of the optimized GWR model for estimating the sub-pixel soybean fractions and the evaluation methods. The equation of GWR here is simplified and the detailed equation of GWR and associated parameters are described in Section 3.1.1.

**Figure 4.**The iteration process of the forward stepwise strategy for selecting the optimal NDVI features (independent variables) from the thirty-one candidate NDVI variables. The dependent variable is located at the center of the vortex. Different symbol in different shape represents different NDVI temporal variable. The iteration process of the forward stepwise selection strategy starts from the first vortex circle (

**inside**) and ends at the last vortex (

**outside**), where the symbol with the largest amount in each vortex circle was the optimal NDVI variable of this iteration.

**Figure 5.**Histograms and descriptive statistics for the independent variables (

**a**–

**e**) and dependent variable (

**f**). These five NDVI data acquired from different Julian day respond to different phenological phases and are shown as examples to represent the candidate independent variables for the 31 NDVI variables.

**Figure 6.**Time-series NDVI curves of pure soybean pixels (

**a**–

**c**) and mixed pixels (

**d**,

**e**) during the entire crop growing period in Heilongjiang. The specific spatial locations of those five pixels were described in Figure 1. These five pixels were presented here to show the time-series curve characteristics of different soybean fractions and to examine whether the distance of spatial locations is related to the variations of crop time-series spectral curves.

**Figure 7.**AICc values of different feature combinations derived from the forward stepwise strategy in Section 3.1.2. Each point corresponds to one vortex circle in Figure 4.

**Figure 8.**Sub-pixel proportion map of soybean cultivation in Heilongjiang province derived from the optimized GWR model.

**Figure 9.**MODIS versus Landsat sub-pixel soybean fractions in Heilongjiang based on 2000 validation samples. Results are summarized for each 0.1 bin (0–0.1, 0.1–0.2, 0.2–0.3, 0.3–0.4, 0.4–0.5, 0.5–0.6, 0.6–0.7, 0.7–0.8, 0.8–0.9, 0.9–1 increase). The dots indicate median soybean proportion at each bin. The bars correspond to the standard deviation of sub-pixel soybean fractions derived from Landsat (horizontal direction) and MODIS (vertical direction), which can reflect the regression bias.

**Figure 10.**Regression-based comparisons between the MODIS-based soybean area estimates and census data respectively at county level. ** Significant at the α = 0.001 level.

**Figure 11.**The spatial distribution of agreement between MODIS-based result and census data on soybean cultivation area for each county in Heilongjiang Province. The colors show the percentage (%) differences between the two estimates and the accompanying bar plots show the actual area comparison of the two estimates (km

^{2}).

Model | AICc | RSS | R^{2} | Adjusted R^{2} |
---|---|---|---|---|

GWR | −3087.201 | 85.8289 | 0.5144 | 0.4377 |

OLS | −2220.059 | 132.1226 | 0.2524 | 0.2466 |

Variables | F-Value | NDF | DDF | p-Value (Significance) | |
---|---|---|---|---|---|

F_{1}-test | / | 0.75 | 3606.49 | 3966 | <2.20 × 10^{−16} *** |

F_{3}-test | Intercept | 6.85 | 348.37 | 3606.5 | <2.20×10^{−16} *** |

NDVI_65 | 1.81 | 523.21 | 3606.5 | <2.20 × 10^{−16} *** | |

NDVI_73 | 2.59 | 457.76 | 3606.5 | <2.20 × 10^{−16} *** | |

NDVI_81 | 1.78 | 445.29 | 3606.5 | <2.20 × 10^{−16} *** | |

NDVI_89 | 0.72 | 353.16 | 3606.5 | 0.999973 | |

NDVI_97 | 4.75 | 332.68 | 3606.5 | <2.20 × 10^{−16} *** | |

NDVI_105 | 2.08 | 258.34 | 3606.5 | <2.20 × 10^{−16} *** | |

NDVI_113 | 3.61 | 269.81 | 3606.5 | <2.20 × 10^{−16} *** | |

NDVI_121 | 1.15 | 484.77 | 3606.5 | 0.019104 * | |

NDVI_129 | 1.02 | 485.38 | 3606.5 | 0.354737 | |

NDVI_137 | 1.15 | 736.19 | 3606.5 | 0.006462 ** | |

NDVI_145 | 1.32 | 589.97 | 3606.5 | 2.18 × 10^{−6} *** | |

NDVI_153 | 0.53 | 666.26 | 3606.5 | 1 | |

NDVI_161 | 0.86 | 907.46 | 3606.5 | 0.998103 | |

NDVI_169 | 2.46 | 904.98 | 3606.5 | <2.20 × 10^{−16} *** | |

NDVI_177 | 1.25 | 610.68 | 3606.5 | 8.12 × 10^{−5} *** | |

NDVI_185 | 1.04 | 407.12 | 3606.5 | 0.296108 | |

NDVI_193 | 1.40 | 629.78 | 3606.5 | 6.44 × 10^{−9} *** | |

NDVI_201 | 0.85 | 292.13 | 3606.5 | 0.969006 | |

NDVI_209 | 0.77 | 226.61 | 3606.5 | 0.995028 | |

NDVI_217 | 1.64 | 252.09 | 3606.5 | 4.06 × 10^{−9} *** | |

NDVI_225 | 4.43 | 160.93 | 3606.5 | <2.20 × 10^{−16} *** | |

NDVI_233 | 7.19 | 227.22 | 3606.5 | <2.20 × 10^{−16} *** | |

NDVI_241 | 13.18 | 336.43 | 3606.5 | <2.20 × 10^{−16} *** | |

NDVI_249 | 2.70 | 632.45 | 3606.5 | <2.20 × 10^{−16} *** | |

NDVI_257 | 11.21 | 711.35 | 3606.5 | <2.20 × 10^{−16} *** | |

NDVI_265 | 2.93 | 772.44 | 3606.5 | <2.20 × 10^{−16} *** | |

NDVI_273 | 3.30 | 955.55 | 3606.5 | <2.20 × 10^{−16} *** | |

NDVI_281 | 2.40 | 737.06 | 3606.5 | <2.20 × 10^{−16} *** | |

NDVI_289 | 2.13 | 622.15 | 3606.5 | <2.20 × 10^{−16} *** | |

NDVI_297 | 1.80 | 418.44 | 3606.5 | <2.20 × 10^{−16} *** | |

NDVI_305 | 1.55 | 634.59 | 3606.5 | 1.28 × 10^{−14} *** |

**Table 3.**A comparison of model-fitting performances for GWR based on different distance calculation ways.

Distance Calculation Ways | Bandwidth | AICc | RSS | R^{2} | Adjusted R^{2} |
---|---|---|---|---|---|

GCS/WGS84 (Great circle) | 736 | −3087.201 | 85.8289 | 0.5144 | 0.4377 |

Sinusoidal (Euclidean distance) | 687 | −2928.25 | 88.4218 | 0.4997 | 0.4177 |

© 2018 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/).

## Share and Cite

**MDPI and ACS Style**

Hu, Q.; Ma, Y.; Xu, B.; Song, Q.; Tang, H.; Wu, W.
Estimating Sub-Pixel Soybean Fraction from Time-Series MODIS Data Using an Optimized Geographically Weighted Regression Model. *Remote Sens.* **2018**, *10*, 491.
https://doi.org/10.3390/rs10040491

**AMA Style**

Hu Q, Ma Y, Xu B, Song Q, Tang H, Wu W.
Estimating Sub-Pixel Soybean Fraction from Time-Series MODIS Data Using an Optimized Geographically Weighted Regression Model. *Remote Sensing*. 2018; 10(4):491.
https://doi.org/10.3390/rs10040491

**Chicago/Turabian Style**

Hu, Qiong, Yaxiong Ma, Baodong Xu, Qian Song, Huajun Tang, and Wenbin Wu.
2018. "Estimating Sub-Pixel Soybean Fraction from Time-Series MODIS Data Using an Optimized Geographically Weighted Regression Model" *Remote Sensing* 10, no. 4: 491.
https://doi.org/10.3390/rs10040491