# Application of a Hybrid Interpolation Method Based on Support Vector Machine in the Precipitation Spatial Interpolation of Basins

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Study Basin and Data

#### 2.1. Research Basin

^{2}. The total length of the main stream is 658 km. It is located at east longitude between 106°36′00″ and 110°44′00″ and north latitude between 28°56′00″ and 31°44′18″. The terrain and geomorphologic conditions in the area are complex, and the higher western section of Fengjie is the low elevation area of the Sichuan Basin. The lower eastern section of Fengjie is the canyon alpine area, and the tributaries are relatively short. The climate in the basin area is in a transition zone from north temperate to subtropical monsoon. Because of the canyon terrain, the eastern and western climates are quite different. The south and north shores are located in the heavy rainstorm range of the southwestern Hubei and the Daba Mountains in the Yangtze River basin [30]. Heavy, high-intensity rain occurs frequently, with heavy rainstorms mostly moving from the west to the east, downstream along the main stream.

#### 2.2. Research Data

#### 2.2.1. Ground Observation Data

#### 2.2.2. TRMM 3B43 V7 Satellite Precipitation Data

#### 2.3. Interpolation Auxiliary Variable Selection

_{10}) were taken as interpolation auxiliary variables. Compared with the measured precipitation data, the TRMM satellite precipitation data better reflect the spatial distribution characteristics of the precipitation, although there is a certain degree of deviation in its accuracy. We also included the TRMM satellite precipitation data in the auxiliary variables, including the TRMM multi-year average annual precipitation (T) and multi-year average flood season precipitation (T

_{fs}), in order to improve the accuracy of site precipitation.

_{10}, T and T

_{fs}as interpolation auxiliary variables, but E and E

_{10}, T and T

_{fs}were not used as auxiliary variables at the same time.

## 3. Research Methods

#### 3.1. Commonly-Used Interpolation Methods

#### 3.2. Linear Regression Hybrid Interpolation Method

#### 3.3. Support Vector Machine Interpolation Model

#### 3.3.1. Methodology

#### 3.3.2. Modeling Steps

#### Input and Output Selection

_{fs}; ④ Y-E

_{10}; ⑤ Y-E

_{10}-T; ⑥ Y-E

_{10}-T

_{fs}.

#### Data Normalization

#### Parameter Initialization

#### Construct a Nonlinear Decision Function

#### Forecast Based on Nonlinear Decision Function

#### 3.4. Support Vector Machine Hybrid Interpolation Method

#### 3.5. Error Evaluation Index

#### 3.5.1. Root Mean Square Error

#### 3.5.2. Mean Relative Error

#### 3.5.3. Coefficient of Determination (R^{2})

^{2}is to 1, the better the model fits the data.

## 4. Analysis and Discussion of Results

#### 4.1. Results Analysis

_{fs}and E

_{10}-T

_{fs}combined interpolation accuracy with satellite precipitation data is significantly higher than that of the Y-E and E

_{10}without satellite precipitation data. The predictive results of the SVM model (from the first to the sixth line in Table 3) show that the SVM model based on latitude and elevation information only (Y-E, Y-E

_{10}) has poor interpolation accuracy, while adding the satellite precipitation data as the auxiliary variable greatly improves the interpolation accuracy, and the three combinations of Y-E-T, Y-E-T

_{fs}and Y-E

_{10}-T obtain better interpolation results than the inverse distance weighting and ordinary kriging method. The terrain of the Three Gorges Region is complex and varied, and the precipitation is obviously affected by latitude and elevation. However, using only the latitude, elevation information and precipitation to establish a linear regression equation or SVM model cannot provide high interpolation accuracy. The TRMM satellite precipitation data, despite a certain deviation, has good spatial distribution continuity and better reflects the basin precipitation trend. It provides an effective supplement to the latitude and elevation information, so adding the satellite precipitation data as auxiliary information makes the interpolation effect better.

_{fs}combined RMSE value is about 120 mm (from the second to the third line in Table 3), and as shown in Figure 10a,b, after correcting the residuals, the interpolation accuracy improves to a certain degree. The SVM model Y-E

_{10}-T combined RMSE value is about 100 mm (from the fifth line in Table 3); the Y-E-T

_{fs}combined RMSE value is about 140 mm (from the sixth line in Table 3); as shown in Figure 10c,d, the interpolation precision is reduced to a certain degree after the residuals are corrected. However, when the normalization range is (0.1, 0.9), while there is a decrease of the SVM model fitting accuracy, the SVM hybrid interpolation method has a certain degree of improvement in interpolation accuracy. In this study, we believe that, whether the SVM hybrid interpolation method can further improve the SVM model interpolation accuracy to a certain extent depends on the SVM model fitting accuracy. Choosing a suitable fitting accuracy, though difficult, is the key to ensuring the prediction accuracy of the SVM hybrid interpolation method.

#### 4.2. Discussion

- The insensitive loss parameter $\epsilon $, normalization range and other parameters of the SVM model have an impact on the final interpolation results. Rainfall has strong spatiotemporal distribution characteristics, and the rainfall in each region and at each time is not exactly the same, so it is necessary to adjust the parameters according to each set of data. The workload is large and restricted by personal experience and judgment, and the universality of the SVM model needs to be further strengthened.
- The SVM model can well fit the complex nonlinear relationship between the interpolation object and auxiliary variable. After the residual interpolation is superimposed, the prediction accuracy may be improved, or may be reduced, depending on the fitting degree of the SVM model. Choosing a suitable fitting accuracy so that the residuals retain enough precipitation feature information is the key to improving the prediction accuracy of the SVM hybrid interpolation method, but also its chief difficulty.
- Research data were divided into training samples and test samples randomly. During the training phase, SVM could reach the highest forecast accuracy by cross-validation; and in the test phase, the overall error indicator of test samples was used to verify models. Although test samples are random, the verification needs to be richer and more representative. Spatial interpolation calculation of precipitation with multiple time scales and different spatial scopes will result in more scientific and rational conclusions.

## 5. Conclusions

- TRMM 3B43 V7 data deviate from ground site precipitation. Overall, the value is too large, and the rainy center is too small. TRMM data detect the basin precipitation-rich center locations relatively accurately, showing a good spatial distribution continuity, to make up for the shortcomings that the number of local ground sites is insufficient or the distribution is unfavorable. When only the latitude, elevation information and precipitation are used to establish the linear regression equation, the SVM model has poor interpolation precision. Adding the satellite precipitation data as an auxiliary variable significantly improves the interpolation accuracy.
- The support vector machine and SVM hybrid interpolation method obtain better interpolation results than the inverse weight method and ordinary kriging method. The direct predictive result of the SVM model is overall better than that of the linear regression equation. The SVM hybrid interpolation method also obtains better interpolation results than the linear regression hybrid interpolation method.
- The SVM hybrid interpolation method depends on the SVM fitting degree, but it is not the case that the better SVM fits, the higher accuracy the SVM hybrid interpolation method has. The difficult task of choosing a suitable accuracy is the key to improving the prediction accuracy of the SVM.
- The linear regression equation has a greater degree of dependence on the TRMM data than the SVM. The SVM accepts the TRMM data information while maintaining its independence, taking into account that TRMM data linear regression and linear regression hybrid interpolation methods are not suitable for TRMM data evaluation.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**TRMM satellite data in the study area ((

**a**) TRMM multi-year average annual precipitation; (

**b**) TRMM multi-year average flood season precipitation).

**Figure 3.**Time series correlation coefficient ((

**a**) annual precipitation; (

**b**) flood season precipitation).

**Figure 4.**Precipitation relative errors ((

**a**) multi-year average annual precipitation; (

**b**) multi-year average flood season precipitation).

**Figure 7.**Test station multi-year average annual precipitation ((

**a**) linear regression and its hybrid interpolation method; (

**b**) support vector machine and its hybrid interpolation method).

**Figure 8.**Scatter diagrams of test station multi-year average annual precipitation ((

**a**) SVM; (

**b**) SVMRI; (

**c**) SVMRK; (

**d**) LR; (

**e**) LRI; (

**f**) LRK; (

**g**) IDW; (

**h**) OK).

**Figure 9.**Multi-year average annual precipitation interpolation results in the Three Gorges Region ((

**a**) IDW; (

**b**) OK; (

**c**) LR; (

**d**) SVM; (

**e**) LRRE-IDW; (

**f**) SVMRE-IDW; (

**g**) LRI; (

**h**) SVMRI; (

**i**) LRRE-OK; (

**j**) SVMRE-OK; (

**k**) LRK; (

**l**) SVMRK).

**Figure 10.**Comparison of normalization ranges ((

**a**) Y-E-T; (

**b**) Y-E-T

_{fs}; (

**c**) Y-E

_{10}-T; (

**d**) Y-E

_{10}-T

_{fs}).

**Table 1.**Correlation coefficient between the precipitation of 41 stations and each auxiliary variable.

Correlation Coefficient | Geographic Location | Terrain Characteristics | Satellite Precipitation | |||||
---|---|---|---|---|---|---|---|---|

X | Y | S | A | E | E_{10} | T | T_{fs} | |

Pearson | 0.226 | 0.540 ** | 0.275 | 0.198 | 0.567 ** | 0.610 ** | 0.387 * | 0.528 ** |

Spearman | 0.338 * | 0.536 ** | 0.271 | 0.383 * | 0.524 ** | 0.552 ** | 0.453 ** | 0.497 ** |

Dependent Variable | Initial Independent Variable | Independent Variable Removed (Not Significant) | Final Independent Variable (Significant) | Regression Equation (Significant) | |
---|---|---|---|---|---|

P | 1 | Y-E-T | T | Y-E | P = −2819.38 + 123.69 × Y + 0.24 × E |

2 | Y-E-T_{fs} | Y | E-T_{fs} | P = −288.08 + 0.27 × E + 1.39 × T_{fs} | |

3 | Y-E_{10}-T | Y, T | E_{10} | P = 791.61 + 0.20 × Z_{10} | |

4 | Y-E_{10}-T_{fs} | Y | E_{10}-T_{fs} | P = −539.24 + 0.19 × Z_{10} + 1.40 × T_{fs} |

**Table 3.**Test station error indicators. SVMRI, SVM residual inverse distance weighting; SVMRK, SVM residual kriging; LRI, linear regression residual inverse distance weighting; LRK, linear regression residual kriging.

Method | Test Set RMSE (mm) | Test Set MRE | Test Set R^{2} | |||||||
---|---|---|---|---|---|---|---|---|---|---|

Worst | Mean | Best | Worst | Mean | Best | Worst | Mean | Best | ||

SVM | Y-E | 262.596 | 252.111 | 231.954 | 0.158 | 0.155 | 0.152 | 0.236 | 0.238 | 0.226 |

Y-E-T | 139.397 | 119.676 | 109.445 | 0.097 | 0.078 | 0.073 | 0.735 | 0.767 | 0.793 | |

Y-E-T_{fs} | 150.120 | 124.128 | 115.792 | 0.110 | 0.086 | 0.079 | 0.633 | 0.746 | 0.770 | |

Y-E_{10} | 230.890 | 222.975 | 203.216 | 0.127 | 0.127 | 0.125 | 0.427 | 0.439 | 0.459 | |

Y-E_{10}-T | 126.379 | 106.926 | 99.289 | 0.089 | 0.072 | 0.068 | 0.757 | 0.811 | 0.831 | |

Y-E_{10}-T_{fs} | 153.183 | 143.917 | 140.692 | 0.109 | 0.088 | 0.085 | 0.613 | 0.674 | 0.693 | |

SVMRI | Y-E-T | 120.502 | 109.988 | 88.255 | 0.092 | 0.079 | 0.066 | 0.801 | 0.830 | 0.883 |

Y-E-T_{fs} | 143.777 | 98.873 | 89.958 | 0.106 | 0.076 | 0.071 | 0.673 | 0.835 | 0.865 | |

Y-E_{10}-T | 119.379 | 116.727 | 114.794 | 0.085 | 0.082 | 0.081 | 0.761 | 0.769 | 0.773 | |

Y-E_{10}-T_{fs} | 154.225 | 151.522 | 147.850 | 0.098 | 0.096 | 0.099 | 0.610 | 0.623 | 0.656 | |

SVMRK | Y-E-T | 133.255 | 114.803 | 89.902 | 0.108 | 0.086 | 0.070 | 0.766 | 0.809 | 0.874 |

Y-E-T_{fs} | 139.153 | 105.924 | 97.733 | 0.103 | 0.082 | 0.077 | 0.704 | 0.811 | 0.838 | |

Y-E_{10}-T | 125.083 | 116.418 | 111.540 | 0.089 | 0.083 | 0.081 | 0.739 | 0.769 | 0.785 | |

Y-E_{10}-T_{fs} | 156.121 | 151.206 | 147.150 | 0.105 | 0.095 | 0.088 | 0.626 | 0.631 | 0.645 | |

L | Y-E | 170.976 | 0.106 | 0.504 | ||||||

E-T_{fs} | 145.495 | 0.108 | 0.681 | |||||||

E_{10} | 191.842 | 0.116 | 0.488 | |||||||

E_{10}-T_{fs} | 145.771 | 0.092 | 0.775 | |||||||

LRI | Y-E | 108.666 | 0.081 | 0.816 | ||||||

E-T_{fs} | 90.303 | 0.071 | 0.868 | |||||||

E_{10} | 144.099 | 0.098 | 0.719 | |||||||

E_{10}-T_{fs} | 128.143 | 0.085 | 0.768 | |||||||

LRK | Y-E | 111.001 | 0.087 | 0.796 | ||||||

E-T_{fs} | 90.987 | 0.072 | 0.864 | |||||||

E_{10} | 147.082 | 0.097 | 0.661 | |||||||

E_{10}-T_{fs} | 138.954 | 0.098 | 0.729 | |||||||

IDW | 127.092 | 0.090 | 0.869 | |||||||

OK | 121.424 | 0.090 | 0.824 |

Input Vector | SVM Normalization Range | SVMRI Normalization Range | SVMRK Normalization Range | |||
---|---|---|---|---|---|---|

The Worst RMSE | The Best RMSE | The Worst RMSE | The Best RMSE | The Worst RMSE | The Best RMSE | |

Y-E | (0, 50) | (0.1, 0.9) | / | / | / | / |

Y-E-T | (0.1, 0.9) | (0, 50) | (0.1, 0.9) | (0, 50) | (0.1, 0.9) | (0, 50) |

Y-E-T_{fs} | (0, 50) | (0, 4) | (0, 50) | (0, 20) | (0, 50) | (0, 20) |

Y-E_{10} | (0, 10) | (0.1, 0.9) | / | / | / | / |

Y-E_{10}-T | (0.1, 0.9) | (0, 50) | (0, 1) | (0, 50) | (0, 1) | (0, 50) |

Y-E_{10}-T_{fs} | (0.1, 0.9) | (0, 4) | (0, 4) | (0.1, 0.9) | (0.1, 0.9) | (0, 50) |

Support Vector Machine Hybrid Interpolation | Linear Regression Hybrid Interpolation | ||||
---|---|---|---|---|---|

Auxiliary Variable | SVMRI | SVMRK | Auxiliary Variable | LRI | LRK |

Y-E-T | −19.36% | −17.86% | Y-E | −36.44% | −35.08% |

Y-E-T_{fs} | −22.31% | −15.60% | E-T_{fs} | −35.11% | −37.46% |

Y-E_{10}-T | 15.62% | 12.34% | E_{10} | −24.89% | −23.33% |

Y-E_{10}-T_{fs} | 5.09% | 4.59% | E_{10}-T_{fs} | −17.92% | −4.68% |

Index | IDW | OK | LRI | SVMRI |
---|---|---|---|---|

Gauge precipitation (mm) | 1085.811 | 1094.046 | 1165.304 | 1109.604 |

TRMM (mm) | 1267.984 | 1267.984 | 1267.984 | 1267.984 |

Relative error | 16.78% | 15.90% | 8.81% | 14.27% |

© 2017 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**

Zhang, X.; Liu, G.; Wang, H.; Li, X.
Application of a Hybrid Interpolation Method Based on Support Vector Machine in the Precipitation Spatial Interpolation of Basins. *Water* **2017**, *9*, 760.
https://doi.org/10.3390/w9100760

**AMA Style**

Zhang X, Liu G, Wang H, Li X.
Application of a Hybrid Interpolation Method Based on Support Vector Machine in the Precipitation Spatial Interpolation of Basins. *Water*. 2017; 9(10):760.
https://doi.org/10.3390/w9100760

**Chicago/Turabian Style**

Zhang, Xiaoxiao, Guodong Liu, Hantao Wang, and Xiaodong Li.
2017. "Application of a Hybrid Interpolation Method Based on Support Vector Machine in the Precipitation Spatial Interpolation of Basins" *Water* 9, no. 10: 760.
https://doi.org/10.3390/w9100760