# Spatially Explicit Mapping of Historical Population Density with Random Forest Regression: A Case Study of Gansu Province, China, in 1820 and 2000

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## Abstract

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^{2}) of 0.82, a positive reduction of error (RE, 0.72) and a coefficient of efficiency (CE) of 0.65. The RFRM-based reconstructions show that the population of Gansu Province in 1820 was mostly distributed in the Lanzhou, Gongchang, Pingliang, Qinzhou, Qingyang, and Ningxia prefecture. The macro-spatial pattern of the population density in 2000 kept approximately similar with that in 1820. However, fine differences could be found. The 79.92% of the population growth of Gansu Province from 1820 to 2000 occurred in areas lower than 2500 m. As a result, the population weighting in the areas above 2500 m was ~9% in 1820 while it was greater than 14% in 2000. Moreover, in comparison to 1820, the population density intensified in Lanzhou, Xining, Yinchuan, Baiyin, Linxia, and Tianshui, while it weakened in Gongchang, Qingyang, Ganzhou, and Suzhou.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

#### 2.2. Environment Factors and Data Resources

#### 2.3. Method

#### 2.3.1. Random Forest Regression Model

#### 2.3.2. Calibration and Verification of RFRM

- The training subsets were randomly extracted from the original dataset with replacement by using the bootstrap method, in which sizes were equal to the original dataset.
- When constructing the regression trees, the optimal split at each node was chosen from all the environmental factors or a random subset of them according to the lowest Gini Impurity Index. It can be calculated as Equation (1).$${I}_{G}\left({t}_{X\left(xi\right)}\right)=1-{{\displaystyle \sum}}_{j=1}^{m}f{\left({t}_{X\left(xi\right)},j\right)}^{2}$$
_{G}donates the Gini Impurity Index, f(t_{x(xi)}, j) donates the proportion of samples with the value x_{i}belonging to leave j as node t [46]. - Each regression tree grew recursively from top to bottom without pruning until a specified termination condition was reached [47].
- The final prediction result of the RFRM was determined by averaging the prediction results of all the individual decision trees.

^{2}), the relative error (E) (Equation (2)), the reduction of error (RE) (Equation (3)) and the coefficient of efficiency (CE) (Equation (4)) were used to evaluate the reliability and stability of RFRM. RE and CE were sensitive indicators ranging from negative infinity to 1. When they are greater than zero, the model is considered to be reliable [50].

_{i}

^{pre}and POP

_{i}

^{obs}denote the predicted and observed populations of town i, respectively, and n is the total number of towns in Gansu Province in 2000.

#### 2.3.3. Application of RFRM

_{ij}and P’

_{ij}denote the predicted population density and readjusted population density (persons per km

^{2}) for grid cell j within prefectural unit i, respectively, S

_{i}denotes the census data (persons) for prefectural unit i, D

_{ij}denotes the land area (km

^{2}) of grid cell j within prefectural unit i, and W

_{ij}donates the population distribution weight of grid cell j within prefectural unit i.

_{j}is the readjusted population density (persons per km

^{2}) for the grid cell j, which is shared by more than one prefectural unit; ${P}_{ij}^{\prime}$ denotes the population density (persons per km

^{2}), derived from Equation (4), for the grid cell j within prefectural unit i, D

_{ij}denotes the land area (km

^{2}) of grid cell j occupied by prefectural unit i, and D

_{j}denotes the total land area (km

^{2}) of grid cell j.

## 3. Results

#### 3.1. Evaluation of Model Performance

^{2}= 0.82) between the predicted population density and census data at the township level for 2000. This suggests that the RFRM driven by the abovementioned environmental factors was largely able to reproduce the spatial variability in the population distribution at the township level within Gansu Province. Nevertheless, it was found that errors exist and that the positive and negative errors always occurred in towns with low population density and high population density, respectively. Figure 6a confirms that the errors occur almost randomly and that the distributions of positive and negative errors were approximately symmetric with each other. In total, 81.58% of the towns had a relative error less than 50%, and only 8.55% of the towns had a relative error higher than 80%. Figure 6b shows that the positive and negative errors were evenly distributed in the large towns and small towns within Gansu Province, while the negative errors mainly occurred in the border towns of Gansu Province. Due to the complex natural and human factors of the border towns, the ability of the RFRM to predict the population density in those areas was weak. Another explanation may be the error of census data. Due to population mobility, the census coverage was usually higher in city and urban areas and lower in the rural areas, especially in the mountainous areas and remote districts. the actual population may be overestimated in the urban areas like cities and underestimated in the rural areas. Thus, the errors in those areas were relatively large. All of these findings, together with the positive reduction of error (RE = 0.72) and the coefficient of efficiency (CE = 0.65), suggest that the model is able to reproduce the spatial variability in population density and is likely stable.

#### 3.2. Modeling the Population Density in the 2000

^{2}. Due to the restriction of the Qilian Mountains, the population in the Hexi corridor was zonal distribution and exhibited an extension from southeast to the northwest. To the northeast, i.e., the Ningxia Plain, the high population density mainly occurred in the urban areas of cities and surrounding areas such as Yinchuan, Wuzhong, and Shizuishan. To the southwest, i.e., the northeastern portion of the Tibetan Plateau, the highest population density occurred in Xining city, with a population density of more than 1500 persons per km

^{2}.

^{2}. The errors are likely random and have an approximately normal distribution. Both positive errors and negative errors exist, and moreover, there is a high frequency of small errors and low frequency of large errors. Positive errors mostly occurred in the areas with low population density, while negative errors mostly occurred in the areas with high population density. This finding suggests that grid cell-based RFRM predictions could not reproduce the areas of relatively low and high density of the population well.

#### 3.3. Modeling the Population Density in the 1820

^{2}. In the Hexi Corridor, a high population density was mainly found in the Suzhou, Ganzhou, and Liangzhou city, with a population density of more than 100 persons per km

^{2}. Moreover, the population in the Ningxia plain and northeastern Tibetan Plateau was densely distributed in Ningxia and Xining city, respectively. The population density in Ningxia reached more than 200 persons per km

^{2}.

^{2}. The weakened population density mainly occurred in Gongchang, Ganzhou, Suzhou, and Qinyang. The population density reductions in those areas can exceed 100 persons per km

^{2}.

^{2}in 1820 to 31 persons per km

^{2}in 2000. In addition, affected by the climate and river change, the living condition in some extremely arid areas of Gansu Province became worse, it might have a certain impact on the population distribution.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Map of Gansu Province in year 1820 (the bottom-left insert shows the location of the study area in China).

**Figure 5.**The RFRM predictions with the leave-one-out method plotted against the 2000 Census data at the township level. Abbreviations: RMSE, root mean square error.

**Figure 6.**Histogram (

**a**) and spatial distributions (

**b**) of relative errors of the RFRM predictions with the leave-one-out method at the township level for the 2000.

**Figure 8.**Comparison of the RFRM grid cell-based predictions aggregated into township with the township level census data for 2000.

**Figure 10.**Population density differences in Gansu Province between 1820 and 2000 at cell size of 10 by 10 km.

**Figure 11.**Population proportion of the townships to the prefectures from the RFRM grid cell-based predictions (

**a**) and from the China 1 km Gridded Population (CnPop) dataset (

**b**) plotted against that from the census and comparison of the RFRM grid cell-based predictions aggregated into county with the county level census data for 1990 (

**c**) and 2010 (

**d**), respectively.

Altitude (Meters) | Population in 1820 (10 ^{4} Persons) | Population in 2000 (10 ^{4} Persons) | Change (10 ^{4} Persons) | Proportion of Population Growth (%) |
---|---|---|---|---|

<1500 | 520.11 | 906.74 | 386.63 | 23.60 |

1500–2000 | 684.54 | 1196.88 | 512.34 | 31.27 |

2000–2500 | 416.87 | 827.44 | 410.57 | 25.06 |

2500–3000 | 126.90 | 329.82 | 202.92 | 12.38 |

3000–3500 | 28.99 | 126.12 | 97.13 | 5.93 |

>3500 | 7.15 | 36.07 | 28.92 | 1.76 |

Total | 1784.56 | 3423.06 | 1638.51 | 100 |

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**MDPI and ACS Style**

Wang, F.; Lu, W.; Zheng, J.; Li, S.; Zhang, X.
Spatially Explicit Mapping of Historical Population Density with Random Forest Regression: A Case Study of Gansu Province, China, in 1820 and 2000. *Sustainability* **2020**, *12*, 1231.
https://doi.org/10.3390/su12031231

**AMA Style**

Wang F, Lu W, Zheng J, Li S, Zhang X.
Spatially Explicit Mapping of Historical Population Density with Random Forest Regression: A Case Study of Gansu Province, China, in 1820 and 2000. *Sustainability*. 2020; 12(3):1231.
https://doi.org/10.3390/su12031231

**Chicago/Turabian Style**

Wang, Fahao, Weidong Lu, Jingyun Zheng, Shicheng Li, and Xuezhen Zhang.
2020. "Spatially Explicit Mapping of Historical Population Density with Random Forest Regression: A Case Study of Gansu Province, China, in 1820 and 2000" *Sustainability* 12, no. 3: 1231.
https://doi.org/10.3390/su12031231