A wide variety of literature has aimed to integrate the road network into spatial predictive tasks by utilising non-Euclidean distance metrics. Many of these studies have concentrated solely on the length of the path between nodes in a pairwise fashion, which disregards important topological properties and structure of the road network as a whole. Given the increasing popularity of graph-based methodologies such as Graph Neural Networks, this work reflects on the conventional approaches and proposes a graph-based perspective to incorporate road network information in the context of geospatial prediction. Specifically, we propose a unified graph structure that incorporates both target nodes and road nodes and retains both geospatial (location) and topological (structure) information. We then conduct a comparative experiment including statistical models, machine learning techniques, and graph-specific models. The results of our experiment demonstrate that representing road-topology in a graph-based manner extends the range of available techniques in contexts where roads play an important role, i.e. real estate valuation modeling. Our approach is also highly interpretable and can be easily visualized to provide insightful results.
tance of spatial relationships amongst observations in contexts where observations are associated with a particular location [1]. When tasked with the construction of a predictive model for such observations, it is well known that spatial relationships should be derived in a diferent That is, the assumption of independent and identically distributed random variables typically does not apply, as spatial information comes with dependency structures that must be taken into account.
Throughout the past years, we have thus observed a variety of specialized techniques being applied to predictive tasks in the geospatial domain, which typically tween observation or the definition of a kernel function to define a neighborhood around observations. Many existing applications do so by expressing the notion of proximity between observations in Euclidean terms (“asthe-crow-flies”), based on an idealized Cartesian map.
Some works have challenged this assumption, and nEvelop-O Broucke) both traditional statistical and machine learning methrequire the construction of a pairwise distance matrix be- is to add “barriers”: obstacles that influence the shortest have argued for using non-Euclidean distance [2]. Es- tive task, namely a graph-based representation where ods, as well as graph-centric approaches can be easily rely on embedding approaches and dimensionality reapplied. As such, our goal is to convey the following duction techniques such as Isomap to tackle the issue contributions: of the non-Euclidean distance metric leading to invalid spatial covariance. Whilst rooted in solid statistical the• We present a graph-based representation for ory, one drawback of this group of techniques is that geospatial data which retains both location and ultimately, such approximations do remove some more road topology information; intricate details resulting from the road network connect• We conduct a comparative experiment including ing instances. Additionally, relying on pairwise distances well-known statistical and machine learning tech- between observations does cause that a more global inniques together with novel graph-based models sight in terms of the road topology (or road network) is adapted to our representation; lost. • We show how our representation leads to inter- Other scholars have also focused towards exploring pretability benefits; a graph-oriented perspective for geospatial prediction. • An implementation of our work is made publicly Notably in [11], the authors argue towards an integraavailable1. tion of network information with location information by directly leveraging the road network as a graph rep
The remainder of this work is structured as follows: resentation, which is most closely in line with what is Section 2 provides an overview of related work. Next, presented herein. Contrary to that work, our focus lies Section 3 presents our methodology towards construct- more on fine-tuning the manner how the graph is utilized ing a graph-based representation for geospatial data and to establish predictions rather than eficient query mechdescribes graph-centric models that can be applied on anisms. Recent work such as [12] has also confirmed the it. Section 4 then describes the experimental setup to viability of this approach. compare a number of geospatial models, including the Given the growing popularity of deep learning techgraph-centric models described previously. Section 5 niques that work directly on graph-based data in recent discusses the results, and illustrates the interpretability years, an increasing number of researchers are also exbenefits of our approach. Finally, the paper is concluded ploring potential solutions for house price prediction by in Section 6. applying related techniques. For example, [13] make use of geometric learning and apply an attention mechanism in the context of real estate appraisal. [14] uses a graph 2. Related Work neural network on a graph representation constructed using point of interest information. Nonetheless, these apTraditional statistical approaches towards modeling re- proaches sufer once more from a disconnect between the gression tasks in a geospatial setting are grounded on geospatial and graph perspective: distance-based methexploring the correlation between geographical instances ods abstract away the road topology and have to rely using Euclidean distance as a basis [5]. Various interpo- on approximations, whereas graph-based methods have lation techniques, such as Kriging [6], geographically been mainly centered on incorporating the instances as weighted regression (GWR) [7], and inverse distance vertices and connecting them—likewise—based on Euweighting (IDW) [8] have been widely utilized in this clidean or road distance and have seldomly considered regard. the notion that a direct graph structure follows from
As was elaborated in the introductory section, it is a given road topology. Notably, this disconnect is also possible to utilize another distance metric (e.g. to calcu- apparent from an interpretability angle, e.g. when visulate pairwise distances in the case of Kriging) or kernel alizing results of a given predictive model, where most function (in the case of geographically weighted regres- displays exhibit e.g. predictions over distinct segments sion), which in theory allows the user to break away on the road network without being able to provide more from the Euclidean assumption. For example, [9] investi- granular predictions over, say, a long singular street seggated this approach by applying a GWR model calibrated ment. with a non-Euclidean metric that included road network In this work, we take a new look at the graph-oriented distance and travel time metrics. In [3], similar metrics perspective and propose an adjusted representation where utilized and combined with a Kriging model. To which shows that both road topology as well as locasatisfy the underlying assumptions of this method, the tion information can be easily incorporated in a singular authors present a Minkowski approximation to make view. Based on this representation, we show that novel the resulting metric well-behaved in terms of symme- graph-centric approaches can be easily applied by means try and the triangular inequality property. In [10, 4], of a comparative experiment. We also show that our repsimilar adjustments were proposed, where the authors resentation leads to interpretability benefits by means of 1See: https://github.com/ArmonGo/knnroadgraph naturally-understandable visualization and retaining the ability to make predictions on a point level rather than street or street-segment level only.
In this section, we describe the two key contributions of this work. First, we detail our graph-based representation and how it can be constructed given a set of spatial observations and a road network. Second, we describe an adjusted k-Nearest Neighbor approach as a graph-centric technique that leverages this representation.
{ , } A graph is a structure over a set of objects detailing how pairs of objects are in some sense “related”. The objects are typically denoted as vertices (or nodes) whereas the pairs are denoted as edges (or links, arcs). Formally, a graph is a pair = ( , )
with the vertices and ⊆ {( , , ) ∈ 2 × ℝ
0+ ∧ ≠ }, where , ∈ and is the weight of the edge. Each element ∈ is defined as a tuple (, , , )
with and spatial coordinates, ∈ indicating the vertex type (road segment node or node carrying a target value) and ∈ ℝ the target value (i.e. estate price). Only target nodes carry target values, so ∀(, , , ) ∈ ∧ = ∶ = ∅ and target nodes need to be connected to a road node, so ) = ∧ ( ) = ) with ∀( , , ) ∈ ∶ ¬( (
a function returning the type of a vertex.
The end result is a graph structure with two node types, geospatial coordinates for all nodes, target values for one node type, and edges inter-connecting road nodes as well as target nodes to the road structure. This structure is suficient to present a straightforward view of a graph and geospatial representation. Note that traditional tech, ,
Algorithm 1 Geospatial graph construction from road lines and target points Input: = {( 1, 1, 2, 2)} set of road lines Input: , , vectors of target coordinates and values Output: = ( , )
a graph with road and target nodes Add road nodes and edges for ( 1, 1, 2, 2) ∈ do ∶= ( 1, 1, , ∅) ∶= ( 2, 2, , ∅) ∶= ∪ { ∶= ∪ ( , } , , ( end for
for ∈ do
, )) ∶= { ∈ | ≠ ∧ ( , ) < } Remove edges from between and a ∈ Reconnect edges in : replace endpoint ∈ with ∶= ⧵ end for
for (, , ) ∈ ((
, ) × ) do ∶= (, , , ) Find closest edge = ( , , ) ∈
to with both
Find closest point ( , ) along line of edge to endpoints road nodes ∶= ( , , , ∅) ∶= ∪ { ∶= ∪ {( ∶= ∪ {( ∶= ⧵ { } , } , , ( , , ( end for , )), ( , , ( , ))} , ))} niques can simply utilize the geospatial coordinates and = {( targets the vertices. Also, we define
to represent the length of geodesic path between , ∈ , the geodesic path being the shortest path between two vertices utilizthe line. ing the edge weights, e.g. using Dijkstra’s algorithm [15]. lined in Figure 1 and further detailed in Algorithm 1. The
We now detail the construction of such a graph given a series of instances with coordinates and associated target values, e.g. house prices. We also assume access ifrst step consists of defining the main road topology by adding in road nodes and edges (
is returns the Euclidean distance between two vertices using their spatial to the road topology or geometry2. Common geospa- location). This is, in practical terms, not directly ready tial notation describes roads as a set of “LineString” objects (a sequence of points) or “MultiLineString” objects (a collection of the former though with no guarantees on disjointedness amongst the members), i.e. through for use, i.e. numerical stability inconsistencies as well as measurement errors typically lead to a graph with many disconnected edges. Hence, the set of road vertices are merged based on their spatial location as described the “Well-Known Text” (WKT) markup language for rep- in [17] ( here is a small threshold value). Next, target resenting vector geometry objects, a format originally defined by the Open Geospatial Consortium (OGC) and now by the ISO/IEC 13249-3:2016 standard [16]. For sake
2E.g. through OpenStreetMap as is provided in our reference implementation. nodes are added in and connected to the road graph. For the sake of computational eficiency, one could opt here to add in edges so that each target node is connected to their closest road node directly. Given the distance-based edge weights which are used to derive geodesics, it is of brevity, we do not formalise these objects in detail, but will assume simply a set of roads segments as a set of corresponding coordinates for the two points defining
1, 1, 2, 2)} lines, where ( 1, 1), ( 2, 2) are the
The construction of the graph representation is outGiven road segments and target nodes
Construct road graph
Simplify graph: merge nearby vertices
Add target nodes and calculate closest points on road graph
Add connecting edges more appropriate to connect target nodes based on spa- Table 1 known k-Nearest Neighbor (K-NN) algorithm [18] to con- node positioned at a given location. For instances present
In general, K-NN works as follows: given a set of train- a new position , it is likely no matching node can be ℝ, where is the target value of , the feature vec- if the set of nodes to be predicted on during inference tial closeness to the road network, as we cannot make strong assumptions in terms of granularity of road segmentation. Hence, we connect each target node to their nearest found points on the road network (spatially) by creating a road node at that nearest location, add in an edge to the target node, add in two edges from that road node to the original endpoints of the nearest edge (also road nodes), and remove the original edge.
To emphasize the eficacy our graph-centric representation, we present a series of modifications for the well struct four diferent variations that will be contrasted to alternative techniques in the next section. ing instances ( 1, 1), ( 2, 2), … , ( , ) each ∈ ℝ × tor describing instances.
Given a distance function (, ) ∶ ℝ × ℝ
→ ℝ and a given point ∈ ℝ , let ( (1), (1)), ( (2), (2)), … , ( () , () ) be the reordering of the training instances so that ( (1), ) ≤ ⋯ ≤ ( () , ) .
Given a value ≥ 1 , predictions for new points can be made as follows: =1 (̂ ) ∶ ℝ → ℝ = ∑ () × () (1) where = ⟨ (1), … , () ⟩ a vector weighting the neighbors. By default, is uniform so that = 1 for each . Often, the inverse distance is also used, i.e. = (a small constant can be added to prevent division by zero; alternatively, in the presence of zero distances, = 1 if ( () , ) = 0 or 0 otherwise).
Similar to [11], we will use as a distance function the length of the geodesic path on the constructed graph, so ∑=1 ( (
() ,) () ,) that ( 1, 1, 2, 2) ∶ ℝ2 × ℝ2 → ℝ = ( 1, 1), ( 2, 2) (2)
Diferent variations of the proposed K-NN approach Name KNN-Power KNN-PowerDecay KNN-Softmax KNN-SoftmaxDecay
Decay
Minmax
Weighting function 7 3 7 3 7 7 3 3
Power inverse Power inverse Softmax Softmax with returning the length of the geodesic path between two nodes , ∈ as introduced earlier, the geodesic path being the shortest path between two vertices utilizing the edge weights and (, ) = ⋃{ ∈ | ( ) = ∧ ( ) = ∧ ( ) = } returning the in the given training set, it is guaranteed by construction that they can be found in the graph. However, given found. Two approaches are possible to resolve this. First, is known, they can be simple added in as target nodes with their target set blank. They are then to be ignored during the retrieval of neighbors (as neighbors should only consist of nodes considered to be in the train set). This procedure allows that the full graph and distances can be pre-calculated and is hence more computationally eficient. In case inference over unknown nodes is necessary, one has to modify the graph at inference time, i.e. add as an target node to the graph, find the closest connection point, add it as a road node to the graph, add an edge to connect the node to the graph, add two edges from the connection point to the original endpoints of the closest edge, remove the original edge, and then infer the closest neighbors only using target nodes diferent from .
Next, we introduce the following methods to establish the weight vector . First, weights are set equal to the retrieved distances = (
() , ) . Naturally, they do not yet sum to one. First, a decay efect can be optionally applied as such: = 1 − exp(− ∗ ) (3) (4) (5) (7) =
− ( ) ( ) − ( ) = ( ( )
= exp( − = ∑ =1 ) ) with a hyperparameter to adjust the distribution of neighboring distances. Alternative, a softmax weighting can be applied:
with a hyperparameter to adjust the distribution of neighboring distances. Finally, the weights are normalized so that:
Not all scaling steps are necessary depending on the weighting function chosen: four variations of the steps above are viable and are summarized in Table 1.
In this section, we describe an empirical experiment to compare our presented K-NN based models with other well-known geospatial predictive modeling techniques, using a collection of real-life datasets.
Four diferent datasets were collected, encompassing different regional location and predictive tasks across the country of Belgium. To retrieve the road network, oficial and open road data provided by the Flemish geospatial institute3. Regarding instances, properties where sourced for the cities of Brussels and Antwerp respectively. An open Airbnb dataset4 was used to provide a rental-based regression target (price per property per person per night per num. bedrooms). We also obtained properties with a valuation based sale regression target (i.e. house price per square meters) for both cities from a Belgian real estate platform. 3See: https://www.vlaanderen.be/datavindplaats/ 4See: http://insideairbnb.com/get-the-data/ with the decay amount. Afterwards, minmax scaling (6) sale-based datasets exhibit an even distribution across the map, whilst the rental-based data is more concentrated.
We include the following models in our comparison. First, GWR [7] and Kriging [6], both traditional geospatial statistical techniques, using Euclidean distances for both and GWR fitting an intercept only using an adaptive kernel. We also include K-means [19], by first clustering based on the (normalized) geospatial coordinates using Euclidean distance and then using the average target per cluster to predict on unseen observations. K-means++ was used as the initialization scheme. A standard CART regression tree (“Reg. Tree”) [20] as well as Random Forest [21] was also considered, as well as default K-NN performed using each node’s spatial coordinates. Target nodes are colored according to their target value (lighter is more expensive). Rows: Antwerp, Brussels. Columns: sale, rental using Euclidean distances with uniform and inverse dis- Table 3 tance weights (“KNN-EuclUni” and “KNN-EuclInv”) [18]. Overview of the hyperparameter grid for the experimental With our graph representation, we include the four K-NN setup. Non-listed hyperparameters were kept to their default adaptations presented above as well as another graph- values based technique, namely Label Propagation [22], a semi- Model supervised algorithm that assigns labels to unlabeled GWR × 2] nodes in a graph by spreading labels from labeled nodes. Kriging Since Label Propagation does not guarantee proper con- RKe-mg.eTarnese vergence for regression tasks, we first discretize the target Random Forest by binning it and using the average price per bin as the KNN-EuclUni target when predicting on new observations. KNN-EuclInv
Hyperparameters for all models were systematically KNN-Power tuned on the validation set using mean absolute error (MAE) before deciding upon a final model which was KNN-PowerDecay then evaluated on the unseen test set. Evaluation re- KNN-Softmax sults are reported using MAE and root mean square error (RMSE). The full hyperparameter tuning grid is KNN-SoftmaxDecay provided in Table 3. All experiments were executed using Python and the scikit-learn, PyKrige, mgwr, and PyTorch Geometric packages. Label Propagation
In this section, we present the results of our experiments and highlight interpretability benefits ofered by our approach. avenues for future work which will be discussed in the concluding section.
5.1. Results In many settings, it is important for constructed predictive models (geospatial or otherwise) to be interpretable.
The results of the experiments on the four datasets are For instance, geospatial models are often used to make provided in Table 4. With regards to MAE, we notice decisions that can have significant impacts on the envithat the graph-centric methods (bottom half in the ta- ronment, public health, and the economy. For geospatial ble) outperform the traditional geospatial and machine models, an obviously appealing avenue towards getting learning techniques for three out of four datasets. With insights in predictions ofered by the model is to visualregards to RMSE, the results are more in favor of tradi- ize the results on a map. Unlike previous visualization tional techniques though here too we notice that one of methodologies that rely on community segmentation or the graph-based K-NN techniques obtains the advantage. heatmaps, our approach quite naturally leads to a vi
Models leveraging the graph structure hence show sualization where the road network is used in a direct an interesting alternative compared to those based on manner. location only with promising results. These findings Given a graph as detailed above and one of the K-NN highlight the importance of incorporating both location based models, a visualization can be set up by interpoand topological structure and suggest the potential for lating across each edge and querying the model for its utilizing graph structures in predictive modeling tasks. prediction for this new point. Note that this potentially We must, however, highlight the various limitations still leads to a large amount of predictions to be generated, present. First of all, executing the experiment over more which can be computationally expensive. As such, we datasets and over repeated train-validation-test splits (e.g. also provide an approximation where predictions are genby means of a more complete (nested) cross-validation) erated for each road node present in the graph, which is would allow to make stronger statements by means of then used to interpolate values across each edge. Visue.g. significance tests. Also, only geospatial coordinates alizations for the four datasets are provided by Figure 3, were used as regressors. For e.g. house valuation tasks, overlaid on a terrain background. As can be observed, it is common to set up hedonic pricing models where central areas of both cities display higher prices compared location information is combined with other attributes to the suburban areas. In Brussels, the South-Eastern part describing characteristics of the property. Merging this has higher prices (as is known), while in Antwerp, the with our graph based representation leads to interesting highest prices spread out from the central city along the river, indicating a discernible trend. Also interesting is the variation between the rental and sale targets. Streets located near tourist destinations, universities, and embassies demonstrate the higher prices in the rental map, whilst on the sale map, the prices of streets exhibit a smoother tendency.
The primary aim of this work was to showcase the possibility of representing road networks and location-bound observations as graphs and to examine the potential usefulness of graph-based techniques in predictive modeling tasks. A comparative experiment was conducted to illustrate the predictive and interpretative capabilities of graph-oriented methods.
The results provide some initial insights towards utilizing graph-based representations and techniques which retain both location information (coordinates) as well as topological information (road-house network structure). The limitations should also be acknowledged: an exhaustive study should consider more datasets and modeling techniques, stemming from diferent regions exhibiting difering population densities (our datasets stemmed from dense urban areas). This, together with a repeated experiment, would allow to make stronger statements using statistical hypothesis testing. Moreover, our setup was limited to comparatively small datasets. When considering large datasets and graphs, efort should be spend on the inclusion of eficient graph construction and neighbor retrieval mechanisms such as outlined in [23, 24, 25, 26].
Nevertheless, we do believe that there is strong potential for future work along this direction. For instance, the graph structure has not yet been fully exploited: feature engineering based on e.g. centrality metrics could be an efective strategy to easily enhance model performance.
Currently, only geospatial coordinates were used as regressors. For e.g. house valuation tasks, also attributes describing characteristics of the property are typically available. Merging this with our graph based representation also leads to avenues towards future work, such as leveraging geometric learning based approaches based on message passing over graphs.
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