Machine learning approach to residential valuation: a convolutional neural network model for geographic variation

The hedonic price model (HPM) is the most widely used model in residential valuation because of its consistency and straightforwardness (Bartholomew and Ewing 2011). The hedonic price model is extended from Alonso’s (1964) bid-rent theory, which shows the trade-off relationship between accessibility to the central business district (CBD) and expected rental yields (Evans 1995). Rosen (1974) proposed the HPM to estimate the impact of various housing characteristics on residential prices beyond the impact of accessibility to the CBD. The traditional HPM is based on ordinary least squares (OLS) regression, and its coefficients show how much value changes by each unit of an attribute increasing or decreasing. House price in the HPM is generally regarded as a set of the characteristics of houses that contributes to an equilibrium price between sellers and buyers (Seo 2019). In particular HPMs need statistical assumptions on data distribution and linear relationships between dependent and independent variables. Statistical bias and incorrect inferences can be produced when these assumptions are violated (Li et al. 2012). HPMs also have the risk of multicollinearity, which exists when there is a high correlation between several independent variables (Grover 2016). Multicollinearity issues can lead to unstable hedonic prices of independent variables, which reduces the advantage of the HPM in interpretability (Powe et al. 1995).

Advanced HPMs have been used to control spatial effects such as spatial dependency and heterogeneity. Conventional HPMs have spatial autocorrelation among residuals, and it violates the assumption of independent residuals among observations and may lead to inaccurate results (D’amato 2017). Spatial autocorrelation occurs because relationships between neighbouring areas are spatially dependent as neighbouring zones are likely to have similar external factors such as neighbourhood and attributes (Bourassa et al. 2007). To overcome the issue of spatial dependency, spatial lag or spatial error terms have been introduced in HPMs (Feng and Humphreys 2012; Brennan et al. 2014).

Another spatial effect is spatial heterogeneity which refers to the spatially non-stationary distribution of the coefficients of HPMs (D’Amato 2017). To deal with spatial heterogeneity, moving window regression (MWR), geographical weighted regression (GWR), and eigenvector spatial filtering (ESP) have been used. MWR estimates local regression and coefficients of attributes using neighbouring observations. In MWR, neighbouring observations are included with equal weight (Páez et al. 2008). GWR, as proposed by Fotheringham et al. (2003), added distance-based weights to MWR to more effectively address the heterogeneity issue (Helbich and Griffith 2016) and has been applied in residential valuation studies (Hu et al. 2016; Mulley et al. 2016). However, the reliability of coefficients due to multicollinearity among independent variables is consistently noted as a weakness of GWR (LeSage 2004).

Determinants of residential value can be characterised into three categories: 1) structural attributes, 2) location attributes, and 3) neighbourhood attributes (Dubin 1988; Lieske et al. 2019). Structural attributes are variables related to the structure of the house itself and often include variables such as the number of bedrooms, bathrooms, garages, floor area, built year, and presence of guest parking.

Location and neighbourhood attributes can be grouped as external factors related to the location of the house and its surroundings (Bartholomew and Ewing 2011). Location attributes are typically defined as accessibility to the amenities that considerably impact residential value. Most studies have used one of the measures of accessibility: 1) distance to the closest amenity (AlQuhtani and Anjomani 2019; Hu et al. 2019), 2) existence of the amenity within a certain distance (McIntosh et al. 2014; Mulley and Tsai 2017), and 3) density of the amenities (Li et al. 2019; Yang et al. 2019) (see Fig. 1). These studies found a significant relationship between residential price and accessibility to amenities, including transportation, shopping centres, medical services, education facilities, and the city centre (AlQuhtani and Anjomani 2019, Hu et al. 2019; Yang et al. 2019).

Despite this, some studies have criticised the limitations of quantifying location attributes as numeric and categorical variables. Sadayuki (2018) and Hu et al. (2019) pointed out that each measurement of locational attributes represents different information about amenities such as distance to and distribution of amenities. Therefore, they can have a divergent impact on the hedonic price model. Similarly, Yuan et al. (2020) addressed that these measurements of location attributes could not provide the comprehensive effects of amenities on residential price but only part of them. For example, in measuring the distance to the closest amenity, additional amenities beyond the closest one are assumed to have no contribution to residential price. Furthermore, buffers for density or using dummy variables for the existence of amenities is usually determined in an arbitrary and heuristic manner (Sadayuki 2018; Lieske et al. 2019).

Furthermore, predefined geographical boundaries such as jurisdictional or statistical areas may misinterpret the magnitude of neighbourhood attributes (e.g. household income, employment, education and crime) to the property price. Neighbourhood features used in the previous studies often consider only the neighbourhood the house is in, even though residential value can be affected by the characteristics of surrounding neighbourhoods. For a house located at the edge of a neighbourhood, this problem is more prominent. Some studies on residential valuations considered neighbourhood characteristics such as socioeconomic and demographic status using a spatial Durbin model (Mussa et al. 2017; Copiello 2020).

Recent studies have applied machine learning models to residential valuation. Generally, machine learning models do not need strict statistical assumptions and are more capable than HPMs of analysing the nonlinear relationship between residential price and its attributes (Li et al. 2012; Chen et al. 2017). In particular, several studies showed the improved performance of the residential valuation model by employing a convolutional neural network (CNN) which is one type of deep neural network. A CNN specialises in capturing spatial patterns (LeCun et al. 2015) and has an advantage in the efficient processing of 2-dimensional data such as images. Studies using CNNs used outdoor and indoor images of houses (Kang et al. 2020, Kostic and Jevremovic 2020), street view images of the neighbourhood (Law et al. 2019; Kang et al. 2020) and birds-eye view images of the surrounding area (Bency et al. 2017, Kostic and Jevremovic 2020) as input for the neural network.

In these studies, CNN has two purposes. First, most of these studies use a CNN to extract visual features of the house and neighbourhood, or specific concepts such as neighbourhood safety (De Nadai and Lepri 2018) and luxury home features (e.g. landscaping) (Poursaeed et al. 2018), which are usually not considered in a hedonic price model. For example, Kang et al. (2020) extracted the visual features from internal house photos and street view images using a CNN and included them in their residential valuation model. Poursaeed et al. (2018) showed that the performance of residential valuation is improved by including visual luxury home features measured by a CNN using house photos from Zillow, a leading online real estate valuation company in USA.

Secondly, other studies used a CNN to gain information on more comprehensive features of the surrounding area of the house. Bency et al. (2017) included features extracted from a CNN using satellite images of the surroundings in their valuation model of residential property, rent and sales price in UK cities, arguing that satellite images can provide contextual information on the neighbouring area. They showed the model is superior to the spatial autoregressive (SAR) model, and using a wider surrounding area of the target house improved model performance. Similarly, Bin et al. (2019) included street maps in their CNN-based valuation model to consider latent geographical features from the open street map and a birds-eye view image of the neighbourhood.

In summary, previous studies on residential valuation have shown significant relationships between housing price and external factors, including location attributes and neighbourhood attributes. However, information on these external factors might be lost while measuring them as numeric or categorical variables in conventional valuation models, and thus lead to a less accurate valuation model. Meanwhile, despite CNNs being recently applied to residential valuation models, the models have focused on including visual features extracted from images, such as outdoor and indoor images of houses, street view images and birds-eye view images rather than geographical layers, especially the surrounding amenities which are closely related to house prices. Our research approach focused on the application of CNN for modelling geographically relationships for improving residential value prediction.

The MLP is a type of neural network that has multiple hidden layers. It has an input layer, multiple hidden layers, which are fully connected layers, and output layers. Each hidden layer consists of multiple hidden nodes, which are neurons. Each neuron multiplies values from previous layers with weights and feeds them to the next layer. The function of each neuron can be described as follows:where \(a\) is the output of the neuron, \(f\) is the weight, \(x\) is the input of the neuron from the previous layer, \(b\) is bias, and \(\sigma \left( \cdot \right)\) denotes the activation function.

Compared to other neural network algorithms, a CNN is characterised by convolutional layers and pooling layers. In convolutional layers, neurons compute the output value that is connected to the spatial location \(\left( {i,j} \right)\) of the input. The output of neurons in a convolutional layer \(a_{i,j}\) can be defined as follows:\(F\) is a kernel with weights, \(X\) is the input of the neuron, which are input geographical data or feature maps from the previous convolutional layer, \(b\) is bias, and \(\sigma \left( \cdot \right)\) denotes the activation function, which is a nonlinear function. The sigmoid function is applied as an activation function throughout this study. The notation of the sigmoid function is as follows:

The pooling layer performs down sampling feature maps from convolutional layers to reduce competitive cost and over-fitting. Max pooling, which is the most common, is applied in this study. Max pooling divides a features map into a local rectangle region and makes a summarised feature map with maximum values of each region.

Neural network models, including CNNs and MLP, minimise loss function \({\mathcal{L}}\) by updating weights and bias of neurons by repeating training. In this study, the following loss function was applied to reduce the errors of residential price and standard deviation of residential price errors:where \(\widehat{{Y_{i} }}\) is predicted price, \(Y_{i}\) is the actual price, and \(N\) is the number of observations.

In each iteration of training, weight and bias are updated to reduce the loss by gradient descent optimisation algorithm and backpropagation algorithm, which calculate gradient descent backwards through the network to minimise the cost of the network (Goodfellow et al. 2016). Adaptive moment estimation (Adam) optimiser, which is the most frequently used optimisation algorithm, is applied throughout this study. Adam is an optimiser that applies momentum and an adaptive learning rate to prevent falling into a local minimum, which is a limitation of the basic gradient descent optimiser (Roodposhti et al. 2019).

In this study, our proposed framework, which integrates MLP and CNNs, is compared to the single MLP methodology. The single MLP is used as a benchmark model, referred to as Model 1, and the proposed fused model is referred to as Model 2. Comparing Model 1 and Model 2 in terms of the accuracy of residential price estimation quantifies the contribution of additional information of location and neighbourhood features that are not included in numeric and categorical variables. During the experiments, the dataset is firstly randomly split into training and testing dataset (75:25), and the cross-validation technique is used for hyperparameter tuning. To avoid the issue of spatial autocorrelation in using spatial data, and to train a generalizable model, spatially stratified split is usually recommended in splitting training and validation datasets over random split. A spatially stratified split diminishes the probability that the observations which have autocorrelation with observation in the training set are included in the validation set. It is because observations in training set which are spatially correlated observations in validation set can provide “sneak preview” and it can result in an over-optimistic validation result and poor performance in testing (Lovelace et al. 2019; Salazar et al. 2022). However, the purpose of AVM in most cases is estimation of current price of houses in existing housing market rather than estimating housing prices in new market (e.g. other countries or cities). Also, most of related studies randomly split training and validation datasets (Bency et al. 2017, Bin et al. 2019, Hu et al. 2019, Kang et al. 2020, Kostic and Jevremovic 2020, Gao et al. 2022). In this context, we randomly split train and validation data instead of using spatially stratified split.

Both Model 1 and 2 are trained in Python using TensorFlow and Keras which are Python libraries for neural networks modelling.

Figures 4 and 5 illustrate the structure of the MLP (Model 1) and the fused model (Model 2). The applied MLP has a total of four hidden layers, and each layer has 32, 64, 32, and 16 hidden nodes. The CNN has three convolutional layers and pooling layers, and each convolutional layer includes 16, 32 and 64 kernels with the size of 3\(\times\)3 for each kernel. Also, 2\(\times { }\)2 max-pooling layers are used between convolutional layers along with batch normalisation to prevent ‘gradient vanishing’, which may downgrade the performance of the neural network (Ioffe and Szegedy 2015).

The sigmoid activation function is applied, and Adam optimiser is employed as a gradient descent algorithm throughout the two models in common. The learning rate of both models is 0.001, and the model was trained using mini-batch sizes 32.

Table 3 shows the modelling results using the testing set. Model 2 performs better than Model 1 in measurements of residential valuation accuracy, based on MAPE (changes from 11.59 to 8.71%), R² (changes from about 0.83 to 0.88), PPE5 (changes from 33.57 to 49.89%), PPE10 (changes from 55.92 to 69.71%) and PPE20 (changes from 82.16 to 88.53%). In terms of accuracy of the models, Model 2 shows 2.88% less MAPE which is an improvement in accuracy compared to Model 1. Although it could be perceived as a minor improvement, it is an improvement that can reduce significant costs in the housing market considering the size of the residential real estate market. Also, recent studies related to this study show a similar level of improvement in their results. Bin et al. (2019) suggested an improvement of 2.6% in MAPE between the baseline model, a boosting regression model (20.1% in MAPE), and Attention-based Multi-Modal Fusion (AMMF) that a proposed method in this study. Poursaeed et al. (2018) trained CNN to measure the luxury level of the area at the house such as the bedroom, bathroom, living room, and kitchen using pictures of the house. In the results, by adding the CNN to the existing model, the median absolute percentage error (MdAPE) was decreased by 2.4% from 8.0 to 5.6%. The results of the related studies above can underpin that the improvement in this study is significant.

In terms of uniformity, both models show a COD value lower than 15%, which is the COD value for single-family homes and condominiums advised by the International Association of Assessing Officers (IAAO) (International Association of Assessing Officers 2017). COD in Model 2 (8.74%) outperforms Model 1 (11.67%). Furthermore, PRD in Model 2 (1.02) also had a smaller PRD than Model 1 (1.04), which means the results of Model 2 have smaller variability than Model 1 and the PRD value of Model 2 falls in the acceptable range for PRD of 0.98–1.03 that provided by IAAO. On the other hand, the PRD value of Model 1 is higher than acceptable range. It means Model 1 has a tendency that its assessment ratios decline with higher price.

Figure 6 shows the distribution of the actual price and predicted price of each transaction using Models 1 and 2. In this figure, transactions are categorised into four groups by their absolute percentage error (APE). The distribution of results using Model 2 (Fig. 6b) is more correlated with actual price than the result using Model 1 (Fig. 6a). A correlation test between prediction and actual price shows Model 2 has a correlation of 0.9384 with the actual price, which is 0.0234 higher than the result from Model 1 (0.9150). Model 2 has 49.89% of the observations with absolute percentage error less than 5%, while Model 1 has 33.57%.

Figure 7 shows the relationship between the number of observations in the training set and the valuation performance of Models 1 and 2. Both models offer the best performance in houses valued under Aus$2 million, which is 88.36% of the validation observations, and MAPE of both models steadily increased in houses over A$2 million. It might be because the determinants of prices of high-priced houses and their magnitudes are different from those of relatively inexpensive houses, and more observations are needed to train the model. Comparing Models 1 and 2, Model 2 outperformed Model 1 in terms of MAPE, especially in houses valued under $4.5 million. However, for houses over A$4.5 million, the performance of Model 2 is similar to Model 1.