# Estimating Loess Plateau Average Annual Precipitation with Multiple Linear Regression Kriging and Geographically Weighted Regression Kriging

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## Abstract

**:**

## 1. Introduction

## 2. Study Area and Data

#### 2.1. Study Area

^{4}km

^{2}. The Loess Plateau has a diverse topography. The Luliang Mountains and Taihang Mountains are located in the eastern Loess Plateau and the Liupan Mountains are situated in the western part. The Yellow River flows through the western and northern boundaries, generating the Ningxia Plain and Hetao Plain, where it then crosses the central and southern part of the Loess Plateau, generating the Guanzhong Plain. The middle of the Loess Plateau is covered with highly erodible aeolian silt deposits and has become one of the most severely eroded regions in the world. The northern Loess Plateau contains Mu Us Sandy Land and Kubuqi Desert. A continental monsoon climate is the major climate type of this area. Under its influence, dry and cold winds in winter are followed by frequent and intense rainfall in summer [27]. Annual precipitation recorded by meteorological stations ranges from 200 mm to 800 mm and decreases from the southeast to the northwest of the Loess Plateau [28,29]. In order to mitigate the prediction error of “edge effects” in interpolation [15], a buffer area with a 100 km bandwidth was considered around the Loess Plateau.

#### 2.2. Data Sets and Data Processing

## 3. Methods

#### 3.1. Logit Transformation and Exploratory Data Analysis Methods

^{++}is logit-transformed precipitation; z

^{+}is the precipitation standardized to the 0 to 1 range:

_{min}and z

_{max}are the physical minimum and maximum of precipitation in this region.

_{min},z

_{max}) based on prior knowledge [22]. According to previous studies, average annual rainfall in the study area is generally within 100–1000 mm [29,30]. Therefore, z

_{min}was set to 100 mm and z

_{max}was set to 1000 mm. The logit-transformed precipitation can be reversed to the original scale by using the back transformation:

#### 3.2. Interpolation Techniques: Multiple Linear Regression Kriging (MLRK) and Geographically Weighted Regression Kriging (GWRK)

#### 3.2.1. Regression Kriging

#### 3.2.2. Multiple Linear Regression Kriging

#### 3.2.3. Geographically Weighted Regression Kriging

#### 3.3. Validation Techniques

^{2}) were calculated for the deterministic part by MLR and GWR, and five-fold cross-validation was implemented for interpolated residuals by OK [15]. The whole prediction error of back-transformed precipitation corresponding to MLRK and GWRK was evaluated by comparing estimated values with actual observations at validation points. The following indexes were used to verify prediction accuracy:

_{r}):

_{r}):

^{2}) were calculated to indicate the amount of variance explained by MLRK and GWRK [5].

## 4. Results

#### 4.1. Spatial Autocorrelation

#### 4.2. Exploratory Data Analysis

#### 4.3. Diagnosis and Evaluation of Regression

^{2}of 0.93 was much higher than that of MLR (0.44, Table 2), which indicates that the variance explained by GWR was much larger than by MLR. However, the degree of local variance explanations was uneven (Figure 6f) and especially poor in the Taihang and Luliang Mountains.

#### 4.4. Regression Residuals Interpolation

_{0}/(C

_{0}+ C)), called the nugget effect, represents the spatial dependence structure, which can be explained as the proportion of spatial heterogeneity caused by random factors. The higher the ratio, the more variations determined by random factors [35]. In general, a ratio of less than 25% indicates that there is a strong spatial dependence structure in the regression residuals; if the ratio was between 25% and 75%, the residuals had a moderately intense spatial dependence structure; and until the ratio reached more than 75%, the spatial dependence structure was very weak, i.e., the regression residuals variability consists of unexplained or random variations. In this study, the nugget effect of MLR residuals was 22.16%, i.e., less than 25% (Table 4), which shows that the spatial variability was predominately caused by structural factors. A nugget effect of GWR residuals of 36.58% signifies that spatial variability caused by random factors in GWR residuals was greater than in MLR residuals. These different degrees of spatial dependence structure in the two regression residuals could be induced by different degrees of trend elimination in residuals of the two regression models.

^{2}) were calculated. The adjusted R

^{2}of Kriging prediction for the residuals of MLR and GWR corresponded to 0.91 and 0.24, which indicates that the degree of variance explanation of ordinary Kriging was much higher for MLR than for GWR.

#### 4.5. Validation of MLRK and GWRK

^{2}of the MLR model was substantially less than the geographically weighted regression GWR model; whereas the adjusted R

^{2}of interpolated regression residuals by Kriging in MLRK was far more than in GWRK. Furthermore, owing to the logit-transformation of precipitation data, the entire prediction error could not be directly evaluated but required complex conversions with the regression models’ errors and the ordinary Kriging interpolations [22]. Therefore, it was difficult to judge which method ultimately obtained more accurate precipitation predictions. As such, the validation data set was used to calculate the entire errors for two regression Kriging interpolations. The means of back-transformed precipitations (MLRK: 455.3 mm/m

^{2}; GWRK: 466.4 mm) were lower than the mean of precipitation validation data (476.2 mm). The values of MAE, MAEr, and RMSE of MLRK were better than those of GWRK, but the values of ME and MEr of MLRK were worse than that of GWRK. This implies that MLRK prediction errors on several validation points could be larger than GWRK, but the whole prediction error over all validation points was less than GWRK (Table 5). The degree of variance explanation of MLRK was slightly better than that of GWRK, but in reality, the adjusted determination coefficients (adjusted R

^{2}) of the two methods showed little difference.

## 5. Discussion

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

CV | Cross-Validation |

DEM | Digital Elevation Model |

DEM_std | Standardized Normal Variable of Digital Elevation Model |

E | Dummy Variable of East Aspect |

GWR | Geographically Weighted Regression |

GWRK | Geographically Weighted Regression Kriging |

MAE | Mean Absolute Error |

MAE_{r} | Mean Absolute Relative Error |

ME | Mean Error |

ME_{r} | Mean Relative Error |

MLR | Multiple Linear Regression |

MLRK | Multiple Linear Regression Kriging |

N | Dummy Variable of North Aspect |

NE | Dummy Variable of Northeast Aspect |

NW | Dummy Variable of Northwest Aspect |

NDVI | Normalized Difference Vegetation Index |

NDVI_std | Standardized Normal Variable of Normalized Difference Vegetation Index |

OK | Ordinary Kriging |

OLS | Ordinary Least Squares |

PRE | Annual Average Precipitation from meteorological stations |

PreT | Logit-Transformed Precipitation |

rad_std | Standardized Normal Variable of Solar Radiation |

RK | Regression Kriging |

RMSE | Root Mean Square Error |

slope_std | Standardized Normal Variable of Slope |

S | Dummy Variable of South Aspect |

SE | Dummy Variable of Southeast Aspect |

Ste. | Matern with Stein’s Parameterization |

SW | Dummy Variable of Southwest Aspect |

W | Dummy Variable of West Aspect |

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**Figure 1.**Underlying surface features of the Loess Plateau (

**a**); and location of meteorological and hydrologic stations (

**b**).

**Figure 2.**Structural characteristics of annual average precipitation at meteorological stations. (

**a**) Distribution of annual average precipitation; and (

**b**) correlation of precipitation for different stations at certain lag distance ranges.

**Figure 4.**Relationship between logit-transformed precipitation (PreT) and main environmental variables, including the standardized normal variable of NDVI (

**a**); the standardized normal variable of DEM (

**b**); the standardized normal variable of Slope (

**c**); and the standardized normal variable of solar radiation (

**d**).

**Figure 5.**Diagnostic plots for multiple linear regression analysis. (

**a**) is the scatter plot for the regression-predicted values versus the corresponding residuals; (

**b**) is the normal Q-Q diagram; (

**c**) is the scale-location graph; and (

**d**) is the Cook’s distance diagram.

**Figure 6.**Maps of GWR coefficients and local adjusted R

^{2}. (

**a**) is the distribution of DEM coefficient; (

**b**) is the distribution of slope coefficient; (

**c**) is the distribution of NDVI coefficient; (

**d**) is the distribution of north aspect coefficient; (

**e**) is the distribution of northwest aspect coefficient; and (

**f**) is the distribution of local R

^{2}.

**Figure 7.**Comparison of regression coefficients between GWR and MLR. The orange points represent the mean of predicator coefficients in GWR, and the blue points represent the regression coefficients of MLR.

**Figure 9.**Precipitation climatology map of the 30 years (1980–2010) at Loess Plateau predicted by MLRK (

**a**) and GWRK (

**b**).

**Table 1.**Pearson correlation matrix between logit-transformed precipitation (PreT) and environmental variables.

Variables | PRE | DEM_std | NDVI_std | rad_std | slope_std | N | NE | E | SE | S | W | NW | SW |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

PRE | 1 | – | – | – | – | – | – | – | – | – | – | – | – |

DEM_std | −0.331 ** | 1 | – | – | – | – | – | – | – | – | – | – | – |

NDVI_std | 0.600 ** | −0.386 ** | 1 | – | – | – | – | – | – | – | – | – | – |

rad_std | −0.349 ** | 0.968 ** | −0.385 ** | 1 | – | – | – | – | – | – | – | – | – |

slope_std | 0.315 ** | 0.286 ** | 0.097 * | 0.172 ** | 1 | – | – | – | – | – | – | – | – |

N | −0.134 ** | 0.022 | −0.042 | −0.009 | −0.090 | 1 | – | – | – | – | – | – | – |

NE | −0.019 | 0.069 | −0.080 | −0.011 | 0.067 | −0.138 ** | 1 | – | – | – | – | – | – |

E | 0.051 | 0.011 | 0.074 | 0.010 | −0.029 | −0.133 ** | −0.172 ** | 1 | – | – | – | – | – |

SE | 0.028 | −0.086 | 0.061 | −0.039 | −0.096 * | −0.148 ** | −0.191 ** | −0.185 ** | 1 | – | – | – | – |

S | 0.080 | −0.127 ** | 0.061 | −0.068 | −0.035 | −0.127 ** | −0.164 ** | −0.158 ** | −0.176 ** | 1 | – | – | – |

W | 0.014 | 0.052 | −0.068 | 0.114 * | 0.095 * | −0.123 * | −0.159 ** | −0.153 ** | −0.170 ** | −0.146 ** | 1 | – | – |

NW | −0.033 | 0.052 | −0.027 | −0.014 | 0.091 | −0.103 * | −0.133 ** | −0.128 ** | −0.142 ** | −0.122 * | −0.118 * | 1 | – |

SW | 0.014 | 0.052 | −0.068 | 0.114 * | 0.095 * | −0.123 * | −0.159 ** | −0.153 ** | −0.170 ** | −0.146 ** | 1.000 ** | −0.118 * | 1 |

PreT | 0.985 ** | −0.306 ** | 0.578 ** | −0.324 ** | 0.309 ** | −0.133 ** | −0.011 | 0.052 | 0.023 | 0.080 | 0.021 | −0.051 | 0.021 |

Models | R^{2} | Adjusted R^{2} | Residuals SE | F-statistic | p-Value |
---|---|---|---|---|---|

a * | 0.4481 | 0.4338 | 0.6046 | 31.23 | <2.2 × 10^{−16} |

b ** | 0.4465 | 0.4400 | 0.6012 | 69.21 | <2.2 × 10^{−16} |

Coefficients | Estimate | Std. Error | t Value | P(>|t|) |
---|---|---|---|---|

a | −0.2110 | 0.0404 | −5.224 | 2.73 × 10^{−7} *** |

b | 0.3903 | 0.0466 | 8.376 | 7.89 × 10^{−16} *** |

c | 0.5449 | 0.0477 | 11.414 | <2 × 10^{−16} *** |

d | −0.2341 | 0.0986 | −2.375 | 0.0180 * |

e | −0.1856 | 0.1019 | −1.821 | 0.0692 |

f | −0.2903 | 0.0414 | −7.022 | 8.64 × 10^{−12} *** |

Residuals | Model | Nugget (C_{0}) (km^{2}/m^{4}) | Partial Sill (C) (km^{2}/m^{4}) | Range (m) | Kappa | C_{0}/(C_{0} + C) (%) |
---|---|---|---|---|---|---|

MLR | Ste. | 0.1,182 | 0.4151 | 590,950.4 | 1.9 | 22.16 |

GWR | Ste. | 0.01,537 | 0.02,665 | 66,963.64 | 10 | 36.58 |

Method | Adjusted R^{2} | ME (mm/m^{2}) | ME_{r} (%) | MAE (mm/m^{2}) | MAE_{r} (%) | RMSE (mm/m^{2}) |
---|---|---|---|---|---|---|

MLRK | 0.87 | −20.88 | −3.82 | 30.85 | 6.58 | 40.05 |

GWRK | 0.85 | −9.80 | −1.29 | 35.75 | 7.78 | 43.24 |

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## Share and Cite

**MDPI and ACS Style**

Jin, Q.; Zhang, J.; Shi, M.; Huang, J.
Estimating Loess Plateau Average Annual Precipitation with Multiple Linear Regression Kriging and Geographically Weighted Regression Kriging. *Water* **2016**, *8*, 266.
https://doi.org/10.3390/w8060266

**AMA Style**

Jin Q, Zhang J, Shi M, Huang J.
Estimating Loess Plateau Average Annual Precipitation with Multiple Linear Regression Kriging and Geographically Weighted Regression Kriging. *Water*. 2016; 8(6):266.
https://doi.org/10.3390/w8060266

**Chicago/Turabian Style**

Jin, Qiutong, Jutao Zhang, Mingchang Shi, and Jixia Huang.
2016. "Estimating Loess Plateau Average Annual Precipitation with Multiple Linear Regression Kriging and Geographically Weighted Regression Kriging" *Water* 8, no. 6: 266.
https://doi.org/10.3390/w8060266