# A Pricing Model for Urban Rental Housing Based on Convolutional Neural Networks and Spatial Density: A Case Study of Wuhan, China

^{*}

## Abstract

**:**

^{2}) = 0.9097, root mean square error (RMSE) = 3.5126) in comparison with other commonly used pricing models.

## 1. Introduction

## 2. Literature Review

#### 2.1. Housing Price and Rental Price Models

#### 2.2. The Locational and Neighborhood Variables of Houses

## 3. Materials and Methodology

#### 3.1. Overall Framework

#### 3.2. Study Area

^{2}(Figure 2). The population of Wuhan was 12.45 million and the GDP was RMB 1562 billion in 2020 [47]. Among the major cities of China, Wuhan has had a high proportion of floating populations in recent years [4]. Since renting is the main way of living for floating populations, rental housing has a very large and active market in Wuhan.

#### 3.3. Data Collection

#### 3.3.1. POIs

#### 3.3.2. Rental Housing

#### 3.4. HPM and GWR

_{j}represents the change in the price y when the jth variable x

_{j}changes (namely, the marginal price), and m is the number of variables. The structural variables of housing are displayed in Table 2; the locational variables and neighborhood variables are discussed in the next section. HPM is a basis and fundamental framework for other housing price models. The MLR based HPM is usually implemented with OLS and is labeled as the “OLS” model in this paper.

_{i}, v

_{i}) denotes the spatial coordinate of the sample (housing) i, β

_{k}(u

_{i}, v

_{i}) denotes the regression coefficient of the kth influencing variable of the sample i, β

_{0}(u

_{i}, v

_{i}) denotes the spatial intercept, and ${\epsilon}_{i}$ denotes the error term. β

_{k}(u

_{i}, v

_{i}) varies with the coordinate (u

_{i}, v

_{i}), and can be estimated as follows:

_{ij}is the geographical weight of sample i and sample j, which denotes the geographical influence of the sample j on sample i

_{.}The most commonly adopted function for calculating W

_{ij}is the Gaussian function: ${W}_{ij}=\mathrm{exp}(-{d}_{ij}^{2}/{b}^{2})$, where d

_{ij}represents the distance between samples i and j, and b represents the bandwidth (nonnegative) indicating the degree of decaying effect related to the distance. Choosing an appropriate bandwidth (b) is an essential work for GWR and is usually based on the minimum Akaike information criterion (AICc) [52]. In this study, we use the AICc and the Gaussian function to determine the bandwidth and geographical weights of the GWR model. Since the factor of spatial heterogeneity is considered, the modeling accuracy of GWR is usually much better than that of the global OLS when the patterns and relationships of the data vary with geographic locations.

#### 3.5. Spatial Density and the Locational and Neighborhood Variables

#### 3.5.1. Modelling the Spatial Density of Geographic Objects

_{iSr}represents the production value of the industry S in the area with the ith enterprise as the center and radius r as the range (excluding the value of the ith enterprise itself), e

_{ir}represents the production value of all types of industries in the area with the ith enterprise as the center and r as the range (excluding the value of the ith enterprise itself), N

_{S}represents the number of enterprises belonging to the industry S, E

_{S|i}represents the total production value of industry S in the whole research area excluding the ith enterprise, and E

_{|i}represents the total production value of all types of industries in the whole area excluding the ith enterprise. The M function smooths the diminishing effect of the single element with the increase in the number of objects of the same type. Since this principle is homologous, if the M function is used to calculate the data of geographic elements such as POIs, housing, populations, it also measures the degrees of density of geographic elements within a certain range. Therefore, it is theoretically feasible to utilize the form of M function for the spatial density of POIs in this research. However, it is noteworthy that the calculations in the M function are based on simple quantitative accumulation, and do not consider the law that the influence between geographic objects gradually decays with their distance, which is included in the KDE and the GFM. Therefore, this indicator may need some improvements for calculating the influence of multiple geo-objects.

#### 3.5.2. Locational and Neighborhood Variables Based on Synthetic Spatial Density

_{iSr}can be expressed by the kernel density estimation (or the GFM effect score) of the S-type POIs in the area within a range of r (excluding the ith POI itself), e

_{ir}represents the kernel density estimation (or the GFM effect score) of all types of POIs within a range of r (excluding the ith POI itself); N

_{S}represents the number of the S-type POIs; E

_{S|i}represents the total kernel density estimation (or the total GFM effect score) of the S-type POIs (excluding the ith POI) in the whole area; and E

_{|i}represents the total kernel density estimation (or the total GFM effect score) of all types of POIs (excluding the ith POI) in the whole area. From this perspective, the model can include both the law that the influence decays with the distances of geographic objects and the fact that the actual influence of a single geographic object gradually diminishes with the increase in the number of objects of the same type. The locational and neighborhood variables based on this approach may provide a more comprehensive generalization of the aggregated geographic information and may enable a more accurate analysis of related issues.

_{j,k}represents the kth POI in the j-type POIs; for the j-type POIs, λ

_{j}(h) is their density estimated value at the house h, Distance(h, p

_{j,k}) is the distance between the house h and the POI p

_{j,k}, and N

_{j}is the number of the j-type POIs; K(·) is the kernel function of KDE, and the Epanechnikov kernel is adopted as the kernel function in this research; b is the bandwidth of the KDE, which means only points within b are effective for calculating the KDE value. The bandwidth of each variable is determined by the condition that the correlation coefficient of this KDE-generated variable with the housing rental price is maximized.

#### 3.6. The 2-Dimensional Housing Price Variables and the CNN Model

#### 3.6.1. The CNN Deep-Learning Model for the Rental Housing Price

#### 3.6.2. Transforming Rental Housing Price Variables into Two Dimensions

_{1}, x

_{2}, …, x

_{n}are aimed to be mapped into a low-dimensional space Y = y

_{1}, y

_{2}, …, y

_{n}(2-dimensional in this study). At first, t-SNE calculates the similarity of high-dimensional values x

_{i}and x

_{j}, which is represented by p

_{j|i}. The similarity p

_{j|i}is the conditional probability that x

_{i}picks x

_{j}as a neighbor in the case that neighbors are picked in proportion to a Gaussian density centered at x

_{i}:

_{i}represents the variance of Gaussian function, which is centered at the high-dimensional location x

_{i}. The similarity is defined in a symmetrized form, that is, p

_{i,j}= (p

_{j|i +}p

_{i|j})/2n, where n is the number of data points. For the target low-dimensional Y, the definition is extended and the similarity of them is modeled as:

_{me}, Y

_{me}) can be calculated, which can represent the central point of the “image” of the 2-dimensional variables. Second, according to the central point, the 4 directions around it (the upper left, lower left, upper right and lower right) compose 4 quadrants. For the “coordinates” of every variable, it is easy to know which direction to the central point is, so as to know which quadrant they should be in. Third, the points (variables) in each quadrant can be sorted by their “x-coordinates” and equally separated by the quantiles of the “x-coordinates”; then, what row should be in the “image” can be determined for each variable. Last, the points (variables) in each row can be sorted by their “y-coordinates”, and what column should be in can be determined.

## 4. Results and Discussion

#### 4.1. Experimental Groups and Model Accuracy Assessment

^{2}), the root mean squared error (RMSE) and its percentage (%RMSE) are adopted as the indicators for the accuracy evaluation of the models, which are commonly used indicators in existing studies [20,40]:

_{i,o}and y

_{i,s}are the observed and predicted value of the ith housing, n represents the number of samples in the dataset, m represents the number of variables, and ${\stackrel{-}{y}}_{o}$ represents the mean observed value.

#### 4.2. Results of 1-Dimensional and 2-Dimensional Models

^{*}represents the true value. The learning parameters of the fully connected layers are set as follows: the L2 regularization is used with a regularization weight of 0.00005; the batch size for each training step is 32; the initial learning rate is 0.5; the decay rate of the learning rate is 0.99996; and the moving average decay is 0.99996. After the training process is completed, the models are run on the test set to estimate the fitting accuracy and predictive power for unknown samples. The locational and neighborhood variables adopted in this section are kept the same, which are the synthetic spatial density-based (GFM) locational and neighborhood variables obtained by combining the M function and the GFM approach. We preferentially combine the M function with GFM rather than with KDE because GFM usually performs better than KDE for the model in our research, which is demonstrated in Section 4.3. The results of other kinds of variables are also discussed in Section 4.3.

#### 4.3. Results Based on Different Kinds of Locational and Neighborhood Variables

#### 4.4. Results of Different Combinations of 2-Dimensional Rental Housing Price Variables

## 5. Conclusions

^{2}= 0.9097, RMSE = 3.5126), since this combination contains relatively massive information and not too much redundance. The proposed model may provide individuals and enterprises with suitable decision-making information for their rental housing transactions; it may also provide the government with a valuable decision-support reference about the locations and prices of public rental housing.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**The range of the study area: Wuhan, China (The municipal districts are: ① Jiang’an, ② Jianghan, ③ Qiaokou, ④ Qingshan, ⑤ Wuchang, ⑥ Hanyang and ⑦ Hongshan).

**Figure 3.**The local influence of “the number of supermarkets within 2 km” on the rental housing price (based on Shapley value analysis).

**Figure 6.**Parameters observed during the training processes of the: (

**a**) FCNN model, and (

**b**) CNN model (for an average group).

**Figure 7.**The basic framework of the neurons in the neural network of (

**a**) an example of FCNN model and (

**b**) an example of CNN model.

Primary Category | Secondary Category | Number |
---|---|---|

Food | Chinese restaurant, foreign restaurant, snack shop, cake dessert shop, coffee shop, tea shop, bar, etc. | 86,443 |

Hotel | Star hotel, fast hotel, apartment hotel, etc. | 12,817 |

Shopping | Shopping mall, supermarket, convenience store, household building material, digital appliance, shop, market, etc. | 139,893 |

Life and service | Communication business hall, post office, logistics company, ticket office, laundry, photo shop, real estate intermediary, public utility, maintenance point, housekeeping service, funeral service, lottery sales point, pet service, newspaper booth, public toilet, etc. | 55,793 |

beauty | Beauty, hairdressing, manicure, body beautification | 13,339 |

Scenic spot | Park, zoo, botanical garden, museum, aquarium, beach bath, church, scenic spot, etc. | 3398 |

Recreation and entertainment | Holiday village, farmhouse, cinema, KTV, theatre, song and dance hall, internet cafe, playground, bath massage, leisure square, etc. | 14,698 |

Sports and fitness | Stadium, extreme sports venue, fitness center, etc. | 3127 |

Education and training | Institution of higher learning, secondary school, primary school, kindergarten, adult education, parent–child education, special education school, scientific research institution, training institution, library, science and technology museum, etc. | 21,219 |

Cultural media | Press and publishing, radio and television, art group, galleries, exhibition, cultural palace, etc. | 3227 |

Medical care | General hospital, specialized hospital, clinic, pharmacy, medical institution, sanatorium, emergency center, etc. | 10,973 |

Automobile service | Automobile sale, automobile maintenance, automobile beauty, automobile parts, car rental, automobile testing ground, etc. | 13,958 |

Traffic facility | Railway station, long-distance bus station, port, parking lot, gas station, service area, toll station, bridge, etc. | 29,265 |

Finance | Bank, ATM, credit cooperative, investment and financing, pawnbroker, etc. | 7138 |

Real estate | Office building, residential area, dormitory, etc. | 38,771 |

Company and business | Company, park, agriculture, forestry, horticulture, factory and mine, etc. | 78,328 |

Government | Government of all levels, administrative unit, public prosecution and law institution, foreign-related institution, party group, welfare institution, political and educational institution, etc. | 21,478 |

Variable | Variable Definition and Measurement Method | Mean | Std. | Expected Effect |
---|---|---|---|---|

Area | The area of the housing unit (m^{2}) | 86.32 | 36.51 | Negative |

TotalFloor | Total number of floors in the building | 20.76 | 12.24 | Unknown |

Level | The rank of the floor level on which the room is situated. (1: “low-level”, in the bottom third of floors in the building; 2: “middle level”, in the middle third of total floors, 3: “high level”, in the top third of floors. This information is provided by the Lianjia website without the actual house floors.) | 2.14 | 0.76 | Unknown |

Year | The year the structure was built | 2008.96 | 7.51 | Positive |

Room | Number of bedrooms | 2.06 | 0.85 | Positive |

Hall | Number of halls | 1.51 | 0.67 | Negative |

Toilet | Number of toilets | 1.13 | 0.48 | Unknown |

South | Whether the room faces south (1: when the description text of the housing direction contains “south”, 0: otherwise) | * | * | Positive |

North | Whether the room faces north (1: when the description text of the housing direction contains “north”, 0: otherwise) | * | * | Unknown |

East | Whether the room faces east (1: when the description text of the housing direction contains “east”, 0: otherwise) | * | * | Positive |

West | Whether the room faces west (1: when the description text of the housing direction contains “west”, 0: otherwise) | * | * | Negative |

PlotRatio | Plot ratio of the belonging community | 3.51 | 1.94 | Unknown |

Green | Greening rate of the belonging community | 0.28 | 0.11 | Positive |

ParkSpace | Parking space numbers in the belonging community | 725.27 | 1173.32 | Positive |

Fee | Property management fee of the housing (RMB/month/m^{2}) | 1.77 | 0.99 | Positive |

Adj R^{2} | RMSE | %RMSE | |
---|---|---|---|

OLS | 0.7498 | 5.4674 | 16.633% |

GWR | 0.7962 | 5.1121 | 15.574% |

FCNN | 0.8797 | 3.6983 | 11.262% |

Yao | 0.8513 | 4.1980 | 12.778% |

Yu | 0.8754 | 3.6765 | 11.191% |

Bin | 0.8847 | 3.6469 | 11.102% |

CNN (5, 2, P) ^{1} | 0.8866 | 3.6176 | 11.010% |

CNN (5, 2, N) | 0.8969 | 3.5678 | 10.858% |

CNN (5, 3, P) | 0.8870 | 3.6156 | 11.006% |

CNN (5, 3, N) | 0.8958 | 3.5706 | 10.866% |

CNN (3, 2, P) | 0.8913 | 3.5918 | 10.930% |

CNN (3, 2, N) | 0.9018 | 3.5439 | 10.791% |

CNN (3, 3, P) | 0.8911 | 3.5967 | 10.949% |

CNN (3, 3, N) | 0.9001 | 3.5513 | 10.807% |

^{1}CNN (5, 2, P) denotes that the size of the convolution kernel in the CNN model is 5, there are 2 convolutional layers, and pooling layers are included in the neural network; CNN (3, 2, N) indicates that the size of the convolution kernel in the CNN is 3, there are 2 convolutional layers, and there are NO pooling layers in the network. The same form is also used in the following tables and text.

Adj R^{2} | RMSE | %RMSE | ||
---|---|---|---|---|

OLS | distance-based | 0.7015 | 5.8527 | 17.822% |

GFM-based | 0.7370 | 5.5660 | 16.953% | |

KDE-based | 0.7283 | 5.6325 | 17.156% | |

synthetic spatial density-based (GFM) | 0.7498 | 5.4674 | 16.633% | |

synthetic spatial density-based (KDE) | 0.7447 | 5.5135 | 16.786% | |

GWR | distance-based | 0.7751 | 5.2572 | 16.003% |

GFM-based | 0.7867 | 5.1758 | 15.767% | |

KDE-based | 0.7834 | 5.2138 | 15.878% | |

synthetic spatial density-based (GFM) | 0.7962 | 5.1121 | 15.574% | |

synthetic spatial density-based (KDE) | 0.7928 | 5.1408 | 15.644% | |

FCNN | distance-based | 0.8673 | 3.8121 | 11.601% |

GFM-based | 0.8753 | 3.7594 | 11.450% | |

KDE-based | 0.8727 | 3.7767 | 11.498% | |

synthetic spatial density-based (GFM) | 0.8797 | 3.6983 | 11.262% | |

synthetic spatial density-based (KDE) | 0.8763 | 3.7451 | 11.407% | |

CNN (3, 2, N) | distance-based | 0.8883 | 3.6102 | 10.995% |

GFM-based | 0.8978 | 3.5664 | 10.853% | |

KDE-based | 0.8939 | 3.5783 | 10.893% | |

synthetic spatial density-based (GFM) | 0.9018 | 3.5439 | 10.791% | |

synthetic spatial density-based (KDE) | 0.9004 | 3.5548 | 10.819% |

Adj R^{2} | RMSE | %RMSE | ||
---|---|---|---|---|

OLS | 0.7498 | 5.4674 | 16.633% | |

GWR | 0.7962 | 5.1121 | 15.574% | |

FCNN | 0.8797 | 3.6983 | 11.262% | |

the proposed CNN with different combinations of rental housing price variables | distance-based | 0.8883 | 3.6102 | 10.995% |

GFM-based | 0.8978 | 3.5664 | 10.853% | |

KDE-based | 0.8939 | 3.5783 | 10.893% | |

synthetic spatial density based (GFM) | 0.9018 | 3.5439 | 10.791% | |

distance-based + GFM-based | 0.9068 | 3.5231 | 10.723% | |

distance-based + synthetic spatial density-based (GFM) | 0.9097 | 3.5126 | 10.692% | |

GFM-based + synthetic spatial density-based (GFM) | 0.9051 | 3.5311 | 10.754% | |

distance-based + GFM-based + synthetic spatial density-based (GFM) | 0.9042 | 3.5347 | 10.752% |

Distance-Based | GFM-Based | Synthetic Spatial Density-Based (GFM) | |
---|---|---|---|

Distance-Based | - | −0.6087 | −0.5732 |

GFM-Based | −0.6087 | - | 0.8829 |

Synthetic Spatial Density-based (GFM) | −0.5732 | 0.8829 | - |

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## Share and Cite

**MDPI and ACS Style**

Shen, H.; Li, L.; Zhu, H.; Li, F.
A Pricing Model for Urban Rental Housing Based on Convolutional Neural Networks and Spatial Density: A Case Study of Wuhan, China. *ISPRS Int. J. Geo-Inf.* **2022**, *11*, 53.
https://doi.org/10.3390/ijgi11010053

**AMA Style**

Shen H, Li L, Zhu H, Li F.
A Pricing Model for Urban Rental Housing Based on Convolutional Neural Networks and Spatial Density: A Case Study of Wuhan, China. *ISPRS International Journal of Geo-Information*. 2022; 11(1):53.
https://doi.org/10.3390/ijgi11010053

**Chicago/Turabian Style**

Shen, Hang, Lin Li, Haihong Zhu, and Feng Li.
2022. "A Pricing Model for Urban Rental Housing Based on Convolutional Neural Networks and Spatial Density: A Case Study of Wuhan, China" *ISPRS International Journal of Geo-Information* 11, no. 1: 53.
https://doi.org/10.3390/ijgi11010053