# Analysis of Factors Influencing the Lake Area on the Tibetan Plateau Using an Eigenvector Spatial Filtering Based Spatially Varying Coefficient Model

^{*}

## Abstract

**:**

## 1. Introduction

^{2}, adjusted R

^{2}, pseudo-R

^{2}, RMSE, AIC, and Moran’s coefficient. The results show that the ESF-SVC model improved the goodness of fit of the regression model and could fit the relationship between each factor and the change in lake area well. On this basis, the spatial and temporal characteristics of the coefficients of independent variables and the degree of influence of different regions by each independent variable were analyzed.

## 2. Data and Methods

#### 2.1. Study Area and Datasets

#### 2.2. Methodology

#### 2.2.1. Yearly Lake Area Extraction

#### 2.2.2. Extraction and Processing of Influencing Factors

#### 2.2.3. Construction of ESF-based Spatially Varying Coefficient Structure

- (1)
- The first step is to construct the spatial weight matrix based on the spatial relationships of each small watershed delineated in the HydroSHEDS dataset. If two watersheds share one or more boundary point, the corresponding value in the spatial weight matrix is 1; if not, the value is 0. The spatial weight matrix was constructed using the spdep package in R language.
- (2)
- Centralize the spatial weight matrix $\begin{array}{c}{\mathrm{C}}_{0}\end{array}$ using the following method [55]:

- (3)
- Extract eigenvectors and perform preliminary screening. The spatial weight matrix is eigen-decomposed using linear variation, the eigenvalues and eigenvectors are calculated using the “spmoran” package in R language, and the eigenvectors are initially filtered using a threshold of 0.25 [56]; thus, the filtered eigenvectors correspond to eigenvalues equal to or greater than one-fourth of the largest eigenvalue.
- (4)
- Select the appropriate eigenvectors to be used in the model. ESF-SVC regression uses a set of eigenvectors as new variables. The model introduces two new parameters α, ${{\displaystyle \sigma}}_{\gamma}^{2}$ [21], and the eigenvectors that contribute more to the regression are selected by the method of great likelihood estimation from the centralized spatial weight matrix C in the previous step and added to the model as independent variables [55]. However, the introduction of the two new parameters leads to increased complexity of the ESF-SVC model.
- (5)
- Construct the ESF-SVC model. The ESF-SVC model is built on the basis of the ESF model. In the ESF model, the eigenvectors are added to the model as follows:

#### 2.2.4. Variable Selection and Model Validation

^{2}, adjusted R

^{2}, pseudo-R

^{2}, RMSE, AIC, and the global Moran coefficient of residuals (RMC). R

^{2}represents the ratio of the dependent variable fitted from the model and can be computed from Formula (8):

^{2}takes the increase into account by adding additional explanatory variables, as defined in Formula (9):

^{2}and adjusted R

^{2}were used to assess the goodness of fit of the regression models. SEM and SLM do not have R

^{2}or adjusted R

^{2}, and use pseudo-R

^{2}to represent the goodness of fit. The closer the value of these three indicators to 1, the better the model accuracy.

## 3. Results

#### 3.1. Pre-Analysis of Lake Area and Variables

^{2}, with an average growth rate of 358.67 km

^{2}/y. The annual total area of lakes is shown in Figure 6.

^{2}/y. Then, 2008 to 2010 showed little change or even a decrease in lake area, followed by continued expansion from 2010 to 2013, but at a slower rate of about 457.56 km

^{2}/y. However, a decrease in lake area occurred in 2013 to 2015.

#### 3.2. Variable Selection

#### 3.3. Model Accuracy Assessment

^{2}, adjusted R

^{2}, RMSE, and AIC. In addition, the ESF-SVC model eliminated the spatial autocorrelation of the residuals, and the p-values of the Moran coefficient of residuals are all greater than 0.05. The average R

^{2}of ESF-SVC reached 0.97, which was 28.55, 28.55, 29.23, 19.00, and 9.41% higher than that of OLS, SEM, SLM, ESF, and GWR, respectively. The ESF-SVC model also showed a very large advantage in the index of adjusted R

^{2}, which was 29.10, 20.05, and 10.91% higher than that of OLS, ESF, and GWR, respectively, and the difference with R

^{2}was very small. In contrast, the results of ESF and GWR models showed some differences between R

^{2}and adjusted R

^{2}. The observed values of the ESF-SVC model showed the smallest deviation from the true values, and the average RMSE was 73.72, 72.59, 71.54, 59.25, and 38.32 km² lower than that of the OLS, SEM, SLM, ESF, and GWR models, respectively. This indicates significantly higher fitting accuracy of the ESF-SVC model than the other models. The AIC values of the ESF-SVC model also remained the lowest among all experiments; the average AIC of the ESF-SVC model was 503.67, 498.51, 489.55, 405.26, and 184.96 lower than that of the OLS, SEM, SLM, ESF, and GWR models, respectively, which indicates that the ESF-SVC model not only had improved accuracy but also maintained very low complexity after adding eigenvectors and considering both spatial autocorrelation and spatial heterogeneity. From the perspective of eliminating spatial autocorrelation, the ESF-SVC model and the SEM, ESF, and GWR models can eliminate the autocorrelation of residuals, and the p-values of the Moran coefficient residuals were all greater than 0.05, whereas the OLS and SLM models exhibited the phenomenon of spatial aggregation of residuals.

#### 3.4. ESF-SVC Model Coefficients

## 4. Discussion

#### 4.1. Analysis of Influencing Factors

#### 4.2. Limitations and Future Enhancement

## 5. Conclusions

^{2}, adjusted R

^{2}, pseudo-R

^{2}, RMSE, and AIC criteria and was able to eliminate spatial autocorrelation. The advantage of the method in our paper compared with previous studies is the ability to quantitatively analyze the effects of multiple factors on lake areas in different spatial regions. Compared with other spatial and non-spatial models, the ESF-SVC model took better account of spatial autocorrelation and spatial heterogeneity by incorporating the eigenvectors selected from the spatial weight matrix into the regression, which not only improved the accuracy and reduced the fitting error, but also maintained a very low complexity. In addition, the model results showed spatial differences in the degree of influence of each factor, thus allowing us to explore the differences in the drivers of lake expansion in different regions.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Basin boundaries of TP. Twelve watersheds delineated by yellow lines are from dataset of river basins map over the TP (2016), and sub-watersheds in red are from HydroSHEDS.

**Figure 5.**Data after preprocessing (PREC, EVAP, DAYT, SME, SMO, and NDVI are yearly values in 2003).

**Figure 7.**Lake area changes (

**A**) and the lake area change rate (

**B**) in inner TP from 2000 to 2015. (* represents no lakes in the watershed in 2000).

**Figure 8.**Average residuals of the (

**A**) OLS, (

**B**) SEM, (

**C**) SLM, (

**D**) ESF, (

**E**) GWR, and (

**F**) ESF-SVC models for 2000 to 2015.

**Figure 9.**Average relative errors of the (

**A**) OLS, (

**B**) SEM, (

**C**) SLM, (

**D**) ESF, (

**E**) GWR, and (

**F**) ESF-SVC models for 2000 to 2015.

**Figure 10.**Average coefficients of (

**A**) PREC, (

**B**) EVAP, (

**C**) DAYT, (

**D**) SME, (

**E**) SMO, (

**F**) VAREA, and (

**G**) ELEV in the ESF-SVC model for each region from 2000 to 2015.

Variable | Resolution | Products | Source |
---|---|---|---|

Surface reflectance | 500 m | MOD09A1 | https://search.earthdata.nasa.gov (accessed on 18 December 2020) |

DEM | 90 m | SRTM DEM | https://srtm.csi.cgiar.org/ (accessed on 4 November 2020) |

EVAP | 0.1° | Monthly mean evapotranspiration data set of the Tibet Plateau | https://data.tpdc.ac.cn/ (accessed on 29 September 2021) |

PREC | 1000 m | 1-km monthly precipitation dataset for China | https://data.tpdc.ac.cn/ |

SME | 0.25° | GLDAS_NOAH025_M | https://ldas.gsfc.nasa.gov/gldas/ (accessed on 14 September 2020) |

DAYT | 1000 m | TRIMS LST-TP | https://data.tpdc.ac.cn/ |

SMO | 0.25° | SMsmapTE | https://data.tpdc.ac.cn/ |

NDVI | 1000 m | MOD13A3 | https://search.earthdata.nasa.gov |

DAYT (Days) | EVAP (mm/Month) | SMO (cm ^{3}/cm^{3}) | PREC (mm/Month) | SME (kg/m ^{2}) | VAREA (km ^{2}) | ELEV (masl) | |
---|---|---|---|---|---|---|---|

Mean | 189.70 | 36.12 | 0.11 | 129.30 | 14.71 | 818.71 | 951.75 |

Max | 276.25 | 67.10 | 0.20 | 400.06 | 90.81 | 22,669.03 | 3194.80 |

Min | 102.05 | 11.78 | 0.06 | 10.08 | 0.69 | 0.00 | 113.98 |

Std.dev | 32.11 | 13.28 | 0.02 | 96.49 | 10.24 | 1962.52 | 517.47 |

Skewness | 0.52 | 0.25 | 1.02 | 1.19 | 4.12 | 6.25 | 0.79 |

Kurtosis | −0.53 | −1.03 | 1.13 | 0.72 | 21.74 | 51.90 | 0.79 |

**Table 3.**Pearson correlation coefficient of each variable with lake area for each year from 2000 to 2015.

DAYT | EVAP | SMO | PREC | SME | VAREA | ELEV | |
---|---|---|---|---|---|---|---|

2000 | 0.145 ** | \ | 0.311 ** | 0.248 ** | 0.227 ** | 0.841 ** | 0.406 ** |

2001 | 0.169 ** | 0.278 ** | \ | 0.230 ** | 0.266 ** | 0.847 ** | 0.409 ** |

2002 | −0.003 | 0.298 ** | 0.273 ** | 0.259 ** | 0.251 ** | 0.845 ** | 0.410 ** |

2003 | 0.156 ** | 0.190 ** | 0.285 ** | 0.246 ** | 0.160 ** | 0.848 ** | 0.413 ** |

2004 | 0.137 ** | 0.218 ** | 0.293 ** | 0.247 ** | 0.238 ** | 0.847 ** | 0.411 ** |

2005 | 0.165 ** | 0.178 ** | 0.283 ** | 0.247 ** | 0.273 ** | 0.854 ** | 0.411 ** |

2006 | 0.124 * | 0.258 ** | 0.284 ** | 0.244 ** | 0.214 ** | 0.846 ** | 0.412 ** |

2007 | 0.136 ** | 0.201 ** | 0.283 ** | 0.250 ** | 0.123 * | 0.850 ** | 0.415 ** |

2008 | 0.147 ** | 0.180 ** | 0.289 ** | 0.252 ** | 0.286 ** | 0.837 ** | 0.418 ** |

2009 | 0.145 ** | 0.239 ** | 0.289 ** | 0.237 ** | 0.317 ** | 0.861 ** | 0.415 ** |

2010 | 0.076 | 0.183 ** | 0.290 ** | 0.270 ** | 0.169 ** | 0.864 ** | 0.417 ** |

2011 | 0.134 ** | 0.223 ** | 0.294 ** | 0.246 ** | 0.145 ** | 0.855 ** | 0.419 ** |

2012 | 0.084 | 0.135 ** | 0.312 ** | 0.250 ** | 0.282 ** | 0.863 ** | 0.420 ** |

2013 | 0.092 | 0.281 ** | 0.339 ** | 0.253 ** | 0.193 ** | 0.850 ** | 0.421 ** |

2014 | 0.136 ** | 0.283 ** | 0.333 ** | 0.268 ** | 0.141 ** | 0.842 ** | 0.421 ** |

2015 | 0.111 * | 0.272 ** | 0.370 ** | 0.241 ** | 0.168 ** | 0.868 ** | 0.422 ** |

DAYT | EVAP | SMO | PREC | SME | VAREA | ELEV | |
---|---|---|---|---|---|---|---|

2000 | 1.367 | \ | 2.798 | 3.197 | 1.168 | 1.391 | 1.352 |

2001 | 1.516 | 3.050 | \ | 2.728 | 1.776 | 1.386 | 1.441 |

2002 | \ | 2.145 | 2.846 | 3.393 | 1.369 | 1.362 | 1.408 |

2003 | 1.518 | 2.382 | 2.358 | 3.683 | 1.235 | 1.384 | 1.403 |

2004 | 1.444 | 2.360 | 2.581 | 4.100 | 1.280 | 1.401 | 1.389 |

2005 | 1.313 | 3.382 | 2.679 | 5.379 | 1.279 | 1.389 | 1.371 |

2006 | 1.328 | 2.600 | 2.214 | 3.923 | 1.140 | 1.381 | 1.495 |

2007 | 1.450 | 1.708 | 2.311 | 3.140 | 1.315 | 1.395 | 1.461 |

2008 | 1.432 | 1.739 | 2.475 | 2.867 | 1.224 | 1.387 | 1.465 |

2009 | 1.277 | 3.661 | 2.766 | 5.785 | 1.398 | 1.393 | 1.523 |

2010 | \ | 2.307 | 2.599 | 3.893 | 1.182 | 1.408 | 1.319 |

2011 | 1.407 | 2.370 | 2.612 | 4.101 | 1.325 | 1.428 | 1.544 |

2012 | \ | 1.641 | 2.991 | 3.458 | 1.171 | 1.414 | 1.414 |

2013 | \ | 2.134 | 2.120 | 3.001 | 1.149 | 1.381 | 1.446 |

2014 | 1.413 | 2.882 | 2.882 | 4.359 | 1.252 | 1.412 | 1.653 |

2015 | 1.282 | 1.891 | 1.949 | 2.333 | 1.396 | 1.440 | 1.622 |

Criteria | OLS | SEM | SLM | ESF | GWR | ESF-SVC |
---|---|---|---|---|---|---|

R^{2}/Pseudo R^{2} (*) | 0.760 | 0.760 | 0.756 | 0.821 | 0.893 | 0.977 |

Adjust R^{2} | 0.756 | * | * | 0.813 | 0.880 | 0.976 |

RMSE | 105.631 | 104.499 | 103.452 | 91.165 | 70.233 | 31.912 |

AIC | 4892.236 | 4887.080 | 4878.113 | 4793.832 | 4573.523 | 4388.568 |

RMC | 0.093 | −0.009 | 0.067 | −0.061 | −0.037 | −0.083 |

RMC’s p-value | 0.020 | 0.587 | 0.012 | 0.836 | 0.800 | 0.980 |

^{2}and adjusted R

^{2}; pseudo R

^{2}is used instead.

DAYT | EVAP | SMO | PREC | SME | VAREA | ELEV | |
---|---|---|---|---|---|---|---|

2000 | −13.043 | \ | 13.715 | −0.599 | 27.435 | 83.037 | 20.665 |

2001 | −21.206 | −5.129 | \ | 17.189 | 19.997 | 92.865 | 20.933 |

2002 | \ | −4.091 | 6.745 | 7.994 | 18.513 | 70.368 | 25.275 |

2003 | −17.226 | 3.877 | 8.882 | 31.129 | 23.958 | 91.716 | 25.980 |

2004 | −27.322 | 3.436 | 7.688 | −1.694 | 49.162 | 97.793 | 21.582 |

2005 | −22.745 | −5.016 | 2.055 | 10.478 | 25.976 | 102.070 | 34.046 |

2006 | −9.397 | −4.500 | 9.784 | 12.378 | 37.994 | 105.679 | 30.591 |

2007 | −8.623 | −12.112 | 6.778 | 13.220 | 21.150 | 115.889 | 33.024 |

2008 | −8.790 | −15.598 | 4.563 | 22.838 | 21.656 | 94.174 | 32.156 |

2009 | −12.429 | −1.498 | 10.184 | 1.637 | 10.163 | 120.882 | 35.164 |

2010 | \ | 2.295 | 6.289 | 11.000 | 49.994 | 99.399 | 24.477 |

2011 | 2.103 | −3.512 | 19.913 | 0.017 | 22.417 | 123.165 | 32.207 |

2012 | \ | 5.790 | 14.622 | −5.420 | 26.959 | 114.631 | 25.676 |

2013 | \ | 4.558 | 18.209 | −5.134 | 24.344 | 121.842 | 28.033 |

2014 | −11.514 | −3.972 | 16.660 | 5.434 | 24.462 | 121.960 | 37.158 |

2015 | −16.758 | −3.500 | 9.375 | 5.218 | 20.407 | 120.320 | 29.268 |

16-year average | −13.913 | −2.598 | 10.364 | 7.855 | 26.537 | 104.737 | 28.515 |

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## Share and Cite

**MDPI and ACS Style**

Xiong, Z.; Chen, Y.; Tan, H.; Cheng, Q.; Zhou, A.
Analysis of Factors Influencing the Lake Area on the Tibetan Plateau Using an Eigenvector Spatial Filtering Based Spatially Varying Coefficient Model. *Remote Sens.* **2021**, *13*, 5146.
https://doi.org/10.3390/rs13245146

**AMA Style**

Xiong Z, Chen Y, Tan H, Cheng Q, Zhou A.
Analysis of Factors Influencing the Lake Area on the Tibetan Plateau Using an Eigenvector Spatial Filtering Based Spatially Varying Coefficient Model. *Remote Sensing*. 2021; 13(24):5146.
https://doi.org/10.3390/rs13245146

**Chicago/Turabian Style**

Xiong, Zhexin, Yumin Chen, Huangyuan Tan, Qishan Cheng, and Annan Zhou.
2021. "Analysis of Factors Influencing the Lake Area on the Tibetan Plateau Using an Eigenvector Spatial Filtering Based Spatially Varying Coefficient Model" *Remote Sensing* 13, no. 24: 5146.
https://doi.org/10.3390/rs13245146