# Advancing Transportation Mode Share Analysis with Built Environment: Deep
Hybrid Models with Urban Road Network

Dingyi Zhuang<EMAIL_ADDRESS>Qingyi Wang<EMAIL_ADDRESS>Yunhan Zheng
<EMAIL_ADDRESS>Xiaotong Guo<EMAIL_ADDRESS>Shenhao Wang<EMAIL_ADDRESS>Haris N. Koutsopoulos<EMAIL_ADDRESS>Jinhua Zhao
<EMAIL_ADDRESS>Department of Civil and Environmental Engineering,
Massachusetts Institute of Technology, Cambridge, MA, USA  Department of Civil
and Environmental Engineering, Northeastern University, 360 Huntington Avenue,
Boston, MA, USA  Department of Urban and Regional Planning, University of
Florida, Gainesville, FL, USA

###### Abstract

Transportation mode share analysis is important to various real-world
transportation tasks as it helps researchers understand the travel behaviors
and choices of passengers. A typical example is the prediction of communities’
travel mode share by accounting for their sociodemographics like age, income,
etc., and travel modes’ attributes (e.g. travel cost and time). However, there
exist only limited efforts in integrating the structure of the urban built
environment, e.g., road networks, into the mode share models to capture the
impacts of the built environment. This task usually requires manual feature
engineering or prior knowledge of the urban design features. In this study, we
propose deep hybrid models (DHM), which directly combine road networks and
sociodemographic features as inputs for travel mode share analysis. Using
graph embedding (GE) techniques, we enhance travel demand models with a more
powerful representation of urban structures. In experiments of mode share
prediction in Chicago, results demonstrate that DHM can provide valuable
spatial insights into the sociodemographic structure, improving the
performance of travel demand models in estimating different mode shares at the
city level. Specifically, DHM improves the results by more than 20% while
retaining the interpretation power of the choice models, demonstrating its
superiority in interpretability, prediction accuracy, and geographical
insights.

###### keywords:

Mode Share Analysis; Graph Embedding; Machine Learning; Transportation
Planning

††journal:

grey!10 white

##  1 Introduction

Mode share analysis in transportation quantifies the usage distribution among
different transport modes, like cars, public transit, and bicycles, within a
certain area. It usually takes various factors into account, like age, income,
travel time, cost, etc., and applies travel demand models to analyze the mode
share. It aims to help researchers in planning and policy-making by
highlighting trends and priorities for sustainable and efficient mobility
solutions. (Li et al., 2017; Bucsky, 2020; Ben-Akiva et al., 1985b; Ben-Akiva
and Bierlaire, 1999)

In recent decades, the world has experienced a rapid surge in urbanization,
resulting in significant changes to the built environment. This change has a
direct impact on people’s daily commuting behavior, mode share, and trip
purpose, which should be properly incorporated into mode share analysis. For
instance, an increased number of rail/bus stops and a less extensive road
network can encourage people to use public transportation for commuting
purposes (Strano et al., 2017; Kalapala et al., 2006). Therefore,
comprehending the built environment is essential to studying the
infrastructure-side influence on travel demand.

As a consequence, mode share analysis that considers the built environment as
input is becoming increasingly popular in the transportation field. They
provide a comprehensive view of the relationship between the physical
environment and travel behavior, taking into account factors like land use
mix, road network characteristics, and public transportation accessibility,
which can affect travel behavior by influencing the availability and
accessibility of transportation options (Wang et al., 2020c, a; Koppa et al.,
2022). These models offer a more nuanced understanding of the relationship
between the physical environment and travel behavior than traditional utility-
based models that only include trip-based characteristics.

However, integrating the built environment into the analysis is a complex and
demanding task, which requires improvement through the technical framework.
Existing work that considers the built environment requires prior knowledge
and manual feature engineering to extract useful features for travel demand
models to analyze the mode share. This is because the built environment data
are unstructured, which provide non-tabular and non-quantitative information
(Lam and Huang, 2002; Gärling and Fujii, 2009). Take urban road network
structure as the example, researchers need to manually create accessibility,
design, diversity, and other features to represent its information in order to
study their effect on mode share analysis (Cervero and Kockelman, 1997;
Marshall and Garrick, 2010; Bettencourt et al., 2007; Arcaute et al., 2015; Xu
et al., 2020b). These process require prior knowledge, additional data (such
as land use data), and a well-designed feature set, which can make it
difficult to extend the knowledge to other cities.

Given these challenges, we propose the framework of Deep Hybrid Models (DHM)
which integrates deep learning (DL) and hybrid models to examine the built
environment impacts of urban road networks (Wang et al., 2023a). We utilize
the advantages of graph embedding (GE) techniques, Node2Vec, to directly learn
the latent variables from urban road networks as the key intermediate step to
build travel demand models. We develop a framework that can represent the
features of the road network without prior knowledge and reflect the interplay
between the road network structure, functionality, and urban demographics. We
open the gate of analyzing the impactful built environment factors by starting
from the urban road network structure, due to road networks’ high
accessibility to all transportation modes. Meanwhile, it is challenging to
design measurements and extract meaningful information due to its unstructured
data formats that cannot be directly processed by classic analytical models
(Wang et al., 2012; Zhan et al., 2017; Li et al., 2015; Saberi et al., 2020).

The contributions of this paper can be summarized into threefold:

  * 1.

We address the use of urban road structures as inputs for travel demand
models, an area that has been less studied before in mode share analysis. It
opens the gates to potentially integrate other types of unstructured built
environment data.

  * 2.

Our DHM framework, combining deep learning and (hybrid) choice models,
significantly enhances the regression power of travel demand models and
retains interpretability.

  * 3.

We interpret the impact of urban road networks, GE techniques, and patterns of
Graph Embedding Representations (GER), providing geographical and urban
planning insights for analyzing travel mode share and community structure.

The paper is structured as follows: in Section 2 we review the related works;
in Section 3 we introduce our GE methodology Node2Vec and the proof of concept
of the DHM; in Section 4 we implement a case study in Chicago to highlight the
regression performance; in Section 5 we give visual interpretations of GER and
its physical meaning; lastly, we conclude and discuss the future work in
Section 6.

##  2 Literature Review

###  2.1 Mode Share Analysis in Transportation Engineering

Mode share analysis is a critical component of transportation engineering,
profoundly influencing infrastructure planning, policy formulation, and system
optimization. Given a specific region or area, it focuses on understanding the
aggregated portions of people’s preferences across different transportation
modes—such as private cars, public transport, taxis, or non-motorized
options—and the consequent impact on urban planning and sustainability.
Traditionally, mode share analysis has relied on demand models like choice
models (CMs) to predict transportation mode shares, based on a mix of
observable and latent factors, emphasizing utility maximization (Ben-Akivai et
al., 1996; Cantarella and de Luca, 2005; Train, 2009; Koppa et al., 2022;
Small and Verhoef, 2007).

A remark of these traditional analyses was the emphasis on sociodemographic
data. Factors such as age, income, employment status, and household size were
considered vital in shaping travel behavior (Ben-Akiva et al., 1985b; Ben-
Akiva and Boccara, 1995; Salas et al., 2022). The methodology revolved around
the hypothesis that individuals with different socioeconomic backgrounds would
display discernibly varied transportation mode preferences (Abou-Zeid and Ben-
Akiva, 2010; Hasnine and Habib, 2018; Small and Winston, 1998; Smilkov et al.,
2017). For instance, communities with higher income levels might opt for
private cars, while those with limited financial resources might lean towards
public transportation or non-motorized modes. This reliance on structured
sociodemographic data enabled researchers and planners to derive patterns and
insights, albeit within the constraints of the data’s granularity and scope.

While structured sociodemographic data has been invaluable, there’s an
increasing recognition of the role of unstructured data in refining mode share
predictions (Yang et al., 2023; Snellen et al., 2002; Zhang, 2004). The
unstructured data are the data type that is non-tabular and non-quantitative
information to analyze directly, such as word-processing text documents,
images and video files, and so on (Blumberg and Atre, 2003). Among the
unstructured data, road network data provides critical insights into the
physical environment of transportation. By analyzing road network topologies,
junction densities, and connectivity indices, researchers can better
understand factors such as travel times, accessibility, and even the appeal of
non-motorized modes like walking or cycling in particular regions (Cooper,
2017; Scheepers et al., 2016).

However, despite the depth and nuance in mode share analysis provided by
traditional CMs, their inherent design has often been criticized for offering
limited flexibility in adapting to newer, unconventional data streams, like
unstructured data. This necessitates an evolved modeling framework that
synergizes the logical foundation of CMs with the adaptive prowess of advanced
computational techniques.

###  2.2 Hybrid Models in Transportation Engineering

Bridging the divide between the traditional CMs and the adaptability required
in various urban data formats, hybrid models emerge as the forerunners of
next-gen transportation analysis. These models, as the name suggests, blend
the strengths of two distinct analytical realms: CMs and machine learning (ML)
or DL techniques. While the former brings structure, hypothesis testing, and
behavioral insights, the latter adds the versatility to handle diverse data
forms and the capability to discern patterns in high-dimensional spaces. Such
models show potential in yielding more precise travel behavior patterns (de
Dios Ortúzar and Willumsen, 2011; Small and Verhoef, 2007; Zheng et al.,
2023). Traditional CMs, being theory-driven, struggle to capture multifaceted
relationships. In contrast, ML and DL represent rapidly advancing domains
within artificial intelligence capable of modeling intricate relationships in
data. This ability has sparked growing interest in their deployment in
transportation planning (Wang et al., 2020d, b).

The integration of ML/DL and choice models is defined as hybrid models, which
aims to leverage the advantages of both models. It can capture rational
passenger behavior and complex relationships among travel demand,
socioeconomic factors, and transportation network characteristics. According
to Van Cranenburgh et al. (2021), ML and DL can assist in finding utility
functions and identifying systematic and random heterogeneity based on the
original choice model. Hybrid models maintain the interpretability of the
choice model while overcoming the limitations of traditional methods using ML
techniques. For instance, Han et al. (2022) used a neural network to learn the
representation of taste heterogeneity while keeping the utility function the
same as the random utility function with heterogeneous taste. Hybrid models
are particularly useful in mode share analysis, which predicts the portions of
travelers choosing a particular mode of transportation. The combination of
traditional CMs and ML algorithms in hybrid choice models leads to more
accurate predictions, particularly in complex and rapidly changing
environments.

Latent variables are also an essential concept in choice models as they
capture psychometric features like individual preferences and traits, which
can explain the hidden correlations among different observed variables (Cho et
al., 2016; Greene and Hensher, 2003). The high-dimensional space in which
these latent variables reside is referred to as the latent space. Structural
equation models with indicators collected in surveys can estimate latent
variables, helping to formulate preference heterogeneity among the population
and explain behavior differences across population groups (Vij and Walker,
2016). By integrating the concepts of latent variables and random utility
theory, hybrid choice models are developed to describe the joint impacts of
latent variables and utilities (Walker and Ben-Akiva, 2002; Ben-Akiva et al.,
2002a). The concept of the latent variable as well as latent space is similar
to representation learning (e.g., embedding) in deep learning to reflect the
underlying correlation among the input variables as they both create
dimensions to formulate the observed data Wang et al. (2023a); Thorhauge et
al. (2019). However, current CMs do not consider other factors such as
infrastructure impacts. Therefore, there is a need to further enhance both
hybrid models and latent space concepts to address such limitations.

###  2.3 Graph Embedding of Urban Road Network Topology

In order to transform the road network structures into the latent space, we
introduce the Graph embedding technique. It is a machine learning method that
can transform graph-structured data into a low-dimensional Euclidean space. By
using vector representations of nodes (i.e. GER), these techniques can
facilitate various downstream tasks, including node classification, link
prediction, and clustering. The application of GE has shown promising
performance improvements in a range of areas, such as social network
embedding, human mobility analysis, and drug discovery, and so on (Cheng et
al., 2019; Jalili et al., 2017; Lerique et al., 2020; Ren et al., 2014; Teney
et al., 2017). In particular, GE has been widely applied in traffic flow
prediction for transportation planning. By embedding sensor networks and
historical traffic data into a high-dimensional space, machine learning
algorithms can learn patterns and correlations between different road segments
and traffic volumes (Wu et al., 2021a; Zhuang et al., 2022). This can be used
to predict traffic flow in real-time, which is valuable for traffic
management, safety control, and congestion reduction (Xu et al., 2019; Wu et
al., 2021b; Zhuang et al., 2020; Liu et al., 2022; Xu, 2021; Jiang et al.,
2023; Wang et al., 2023b; Wu et al., 2020).

Various GE techniques have been proposed, including matrix factorization,
random walk, and deep learning based methods. Matrix factorization-based
methods, such as graph factorization and matrix decomposition, map nodes to
vectors by approximating the adjacency matrix of the graph (Qiu et al., 2018;
Liu et al., 2019; Rozemberczki et al., 2019). Random walk-based methods, such
as DeepWalk and Node2Vec, map nodes to vectors by simulating random walks on
the graph and learning a representation based on the transitions between nodes
(Grover and Leskovec, 2016; Perozzi et al., 2014). Deep learning-based
methods, such as graph convolutional networks (GCN) and graph attention
networks (GAT), map nodes to vectors by incorporating the graph structure and
node features into a neural network architecture (Cai et al., 2018; Goyal and
Ferrara, 2018).

The built environment, especially the road network, is an important factor
that impacts travel behavior and is well suited for GE techniques. It is often
measured by the density, diversity, and design, known as the ”3Ds” (Ewing and
Cervero, 2010; Yin et al., 2020) in previous literatures. However, 3Ds are
difficult to measure, due the cost of time and manpower, making it challenging
to draw generalizable conclusions. GE can be introduced to directly capture
the geometric, topological, and semantic information of roads, including
location, length, direction, and category, as well as relationships between
different roads, such as intersections, connections, and proximity. The
learned embeddings can be used to support various urban planning and
management tasks, such as traffic prediction, route optimization, and
emergency response. For example, Xue et al. (2022) applied Node2Vec on road
networks across multiple cities worldwide to study spatial homogeneity
patterns and Xu et al. (2020a) applied DeepWalk on road networks to assist in
predicting the traffic states.

CMs and hybrid models are limited by the representation power of unstructured
data. Therefore, using advanced GE methodology to replace original handcrafted
features can be inspiring. By formulating urban road networks in a way that
can fit choice models, it may be possible to improve the accuracy and
effectiveness of such models.

##  3 Methodology

###  3.1 Problem Description

Our proposed framework aims to integrate deep learning and hybrid choice
modeling to improve the built environment impacts in mode share analysis.
Thus, the problem formulation consists of two major components: (1) how to
represent of urban road networks using GE, and (2) how to construct a mode
share model based on demand and sociodemographic data and learned GERs.

To abstractly represent a city’s road network, we treat intersections as nodes
and road segments as edges, where the sets of nodes and edges are represented
as $\mathcal{V}$ and $\mathcal{E}$, respectively. The adjacency matrix $A$,
encapsulating road topology and travel distance, is employed to construct a
road network graph, $\mathcal{G}=(\mathcal{V},\mathcal{E},A)$. The first
component of our problem involves deriving a GER of the road network,
expressed as $\mathcal{R}(K^{*})=Embedding(\mathcal{G})$. Here, $K^{*}$
represents the dimension of our tailored GER, and $Embedding$ symbolizes the
GE technique applied. For concise notation, we utilize
$\mathcal{R}\in\mathbb{R}^{N\times K^{*}}$ for the node embedding results
$\mathcal{R}(K^{*})$. Therefore, the representation of an individual node $u$
should be $\mathcal{R}_{u}\in\mathbb{R}^{1\times K^{*}}$. These embeddings are
treated as latent variables within the hybrid choice model framework, as they
are generated from observed variables and reflect inherent correlations.
Notice that in the classic mode share analysis, we might use features
$x\in\mathbb{R}^{N\times K}$ as the inputs, which is usually collected from
both supply and demand side of the traffic service. Note that $K$ denotes the
size of the sociodemographic features.

The second component involves constructing a travel demand model to predict
the mode shares based on $\mathcal{R}$ or $x$. We will demonstrate it in the
manner of CM. In a conventional CM task, each of the $N$ census tracts has the
mode set $C_{n}$. The mode set $C_{n}$ encompasses the driving, public
transit, walking, or other options, aggregating all individual surveys from
the $n$-th census tract. In earlier CM studies, the observed data for census
tract $n$ includes sociodemographic attributes of each alternative
$x_{in},\forall i\in C_{n}$, and the mode share $y_{n}$. The fraction of
census tract $n$ opting for alternative $i$ is $P(y_{n}=i|x_{jn},\forall j\in
C_{n})$. In the simplest form, we learn that the systematic utility $V_{in}$
of alternative $i$ for a census tract $n$ can be defined as a linear
combination of $K$ attributes (Equation 1), where $\beta_{ki}$ signifies the
parameters for attribute $x_{ki}$ and $\beta_{i0}$ denotes the alternative-
specific constant for alternative $i$.

$V_{in}=\beta_{i0}+\sum_{k=1}^{K}\beta_{ik}x_{ikn}.$ (1)

For our road network-related choice task, it suffices to replace the
sociodemographic attributes $x_{ki}$ with our GER. Consequently, we can
readily restate the equation as Equation 2:

$V_{in}=\beta_{i0}+\sum_{k=1}^{K^{*}}\beta_{ik}\mathcal{R}_{ikn}.$ (2)

After that, we applied predictive travel demand models, like multinomial logit
(MNL) choice model or machine learning regressors, to estimate The probability
that an individual from census tract $n$ selects alternative $i$ as:

$P(y_{n}=i|V_{in})=g(V_{in})=\frac{e^{V_{in}}}{\sum_{j\in C_{n}}e^{V_{jn}}}.$
(3)

Since we only care about the portions of the travel modes, we keep the
obtained results in Equation 3 without discretizing them (Ben-Akiva et al.,
2002b, 1985a). It is worth noting that $g$ could be replaced by other choice
and machine learning models could be used in lieu of the linear utility form,
apart from MNL model. However, the central aim of this paper is to highlight
the potential of integrating unstructured data with traditional travel demand
modeling. Therefore, we use the simplest choice modeling form to underline the
advantages of employing GE.

###  3.2 Graph Embedding Method: Node2Vec

Our initial task is to learn the GER to serve as input for the travel demand
model. For this task, we employ the Node2Vec embedding model, as we found it
more suitable than GNN alternatives like graph auto-encoders (GAEs) (Grover
and Leskovec, 2016; Pan et al., 2018; Kipf and Welling, 2016; Wang et al.,
2017). While GAE, which includes encoder and decoder blocks, aims to
reconstruct the graph adjacency, its application requires the definition of
node features as inputs (e.g. traffic volume and speed), which necessitates
specific design considerations. Conversely, the Node2Vec model, based on
random walk sampling, requires only the adjacency matrix $A$ as input and
generates the embedding for each node as output. This approach is not only
more intuitive and flexible but also efficient, as the edge transition
probability, once calculated, remains fixed for a given road network of a
city. This consistency provides transferability and exploratory power when
applied to different cities.

The process of implementing Node2Vec encompasses three stages: computation of
edge transition probabilities, random walk sampling, and embedding
calculation. The neighborhood of a node $u$, represented as $N_{S}(u)$, is
determined based on the neighborhood sampling strategy $S$. We initiate a
$2^{nd}$ order random walk process by selecting a source node $u$ and
simulating a random walk of customized fixed length $l$. The $i$th node in the
walk, $c_{i}$, is generated based on the following distribution:

$P(c_{i}=x|c_{i-1}=v)=\left\\{\begin{array}[]{lc}\frac{\pi_{vx}}{Z}&\text{if}\quad(v,x)\in\mathcal{E}\\\
0&\text{otherwise}\end{array}\right.,$ (4)

where $\pi_{vx}$ is the unnormalized transition probability between nodes $v$
and $x$, and $Z$ is the normalizing constant. $\pi_{vx}$ is inversely
proportional to travel distance, resulting in more frequent visits to
proximate roads. Grover and Leskovec (2016) proposed a mixed Breadth-First
Search (BFS) and Depth-First Search (DFS) sampling strategy with two
parameters; $p$ and $q$, standing for the return and in-out parameters
respectively. These parameters control the probability of immediately
returning to a node just visited during the walk and the likelihood of the
walk visiting nodes further away from a source node $u$.

Assume a random walk has just traversed edge $(t,v)$ and arrived at node $v$.
The walk then evaluates the transition probability $\pi_{vx}$ on edges $(v,x)$
from $v$. The unnormalized transition probability can be reformulated as
$\pi_{vx}=\alpha_{pq}(t,x)\cdot w_{vx}$, where $w_{vx}$ is the edge weight
(i.e., road network length), and $\alpha_{pq}$ is the search bias, a function
of $p$ and $q$ defined as:

$\alpha_{pq}(t,x)=\left\\{\begin{array}[]{lc}\frac{1}{p}&\text{if}\quad
d_{tx}=0\\\ 1&\text{if}\quad d_{tx}=1\\\ \frac{1}{q}&\text{if}\quad
d_{tx}=2\end{array}\right.,$ (5)

where $d_{tx}$ is the shortest path distance between nodes $t$ and $x$ and
must be one of ${0,1,2}$ so that two parameters are sufficient and necessary
to guide the random walk. Parameters $p$ and $q$ control the exploration speed
of the random walk. The random walk process provides us with the sampling
strategy $S$ and the likelihood of visiting the neighboring node $u$. We aim
to make our node embedding as similar as possible to the embeddings of nodes
in its neighborhood. Thus, we optimize the objective function that maximizes
the log-probability of observing the neighborhood $N_{S}(u)$ of node $u$,
given as:

$\max_{{\mathcal{R}{u}}}\sum_{u\in\mathcal{V}}\log
Pr(N_{S}(u)|\mathcal{R}_{u}).$ (6)

To render the optimization problem tractable, we introduce two assumptions:

  * 1.

Conditional Independence: The log-sum operation is based on the assumption of
independence of the likelihood of observing a neighborhood from observing any
other neighborhood node, i.e., $Pr(N_{S}(u)|\mathcal{R}_{u})=\prod_{v_{i}\in
N_{S}(u)}Pr(v_{i}|\mathcal{R}_{u})$.

  * 2.

Symmetry in Feature Space: A source node and its neighboring node in
$N_{S}(u)$ have a symmetric effect in the feature space. In our context, as
our road network includes only the road connections between intersections, we
disregard the directions of the road. The conditional likelihood of each
source-neighborhood node pair can be parameterized as a softmax function:
$Pr(v_{i}|\mathcal{R}u)=\frac{\exp(\mathcal{R}{v_{i}}\cdot\mathcal{R}u)}{\sum{v\in\mathcal{V}}\exp(\mathcal{R}{v}\cdot\mathcal{R}_{u})}$,
where $v_{i}\in N_{S}(u)$.

Given these assumptions, Equation 6 can be simplified into its final form:

$\max_{{\mathcal{R}{u}}}\sum_{u\in\mathcal{V}}[-\log Z+\sum_{v_{i}\in
N_{S}(u)}\mathcal{R}_{v_{i}}\cdot\mathcal{R}_{u}],$ (7)

Parameter | Value
---|---
$K^{*}$ (Size of GER) | 128
# of walks for each node | 10
Walk length | 20
# of negative samples to use for each positive sample | 1
Return parameter $p$ | 1
In-out parameter $q$ | 1
# of epochs to run | 100
Learning rate | 0.01
Table 1: Parameter tables of our implemented Node2Vec model.

In the above equation, the per-node partition function,
$Z_{u}=\sum_{v\in\mathcal{V}}\exp(\mathcal{R}_{v}\cdot\mathcal{R}_{u})$, can
be computationally expensive for large networks. To improve efficiency, we
employ negative sampling, which reduces the number of training examples needed
by randomly sampling negative examples (i.e., node pairs not connected by an
edge) to train alongside positive examples (i.e., node pairs connected by an
edge). This strategy allows the model to learn to distinguish between positive
and negative examples more efficiently, without the need to generate all
possible negative examples. Equation 7 is learned via a two-layer neural
network and stochastic gradient ascent applied to the model parameters,
yielding the embedding $\mathcal{R}(k)$.

Note that many GE techniques, including Node2Vec, primarily focus on node-
level representation, which means they generate the embedding vector
$\mathcal{R}_{u}$ for node $u$ instead of the representation of the census
tract embedding $\mathcal{R}_{n}$ (Hamilton et al., 2017). To convert node-
level embedding into census tract (i.e., subgraph) level, we need an
additional aggregation step, known as ”readout” (Xu et al., 2018). For each
embedding vector dimension, the readout function is defined as:

$\mathcal{R}_{n,k}=\frac{1}{|\mathcal{G}n|}\sum_{u\in\mathcal{G}_{n}}\mathcal{R}_{u,k},$
(8)

where $\mathcal{G}_{n}$ is the subgraph comprising all nodes and their
interconnected edges within the census tract $n$, with $|\mathcal{G}_{n}|$
denoting the number of nodes within $\mathcal{G}_{n}$. In our implementation,
the parameters for Node2Vec are shown in Table 1.

###  3.3 Deep Hybrid Model

We propose an innovative and comprehensive framework known as the Deep Hybrid
Models (DHMs), seeking to unify the principles of hybrid choice modeling and
the power of the machine and deep learning to accurately represent
unstructured data (Wang et al., 2023a). The DHM fuses different domains and
data types. As depicted in Figure 1, we provide a conceptual demonstration of
how DHM could be employed in the context of an urban road network.

Figure 1: A proof-of-concept demonstration of DHM applied to an urban road
network for mode share analysis.

For a more formal representation following the previous notations, DHM can be
mathematically defined as follows:

$P(y_{n}=i|V_{n})=g(V_{n})=g(z_{n})=g(\mathcal{M}(x_{n},\mathcal{R}_{n})),$
(9)

In the above equation, for the $n$-th census tract, $x_{n}$ signifies
numerical inputs, such as sociodemographics, and $\mathcal{R}_{n}$ represents
graph-embedded variables. The DHM framework is characterized by two main
components brought together as $\mathcal{M}(x_{n},\mathcal{R}_{n})$, while the
function $g(\cdot)$ acts as a prediction function, like MNL, estimating the
output probability. Therefore, the DHM framework essentially comprises a
mixing operator, denoted as $\mathcal{M(\cdot)}$, and a behavioral predictor,
denoted as $g(\cdot)$. Notably, the mixing operator could be a simple fusion
of the $x_{n}$ and $\mathcal{R}_{n}$, or even just a segment of it. The
behavioral predictor adopts a generalized linear form:
$g(z_{n})=\sigma(\beta^{\prime}z_{n})$, where $\sigma(\cdot)$ symbolizes the
link function and $\beta^{\prime}z_{n}$ is a linear transformation of the
latent variables ($z_{n}=\mathcal{M}(x_{n},\mathcal{R}_{n})$). It is worth
mentioning that we have intentionally simplified $g(\cdot)$ for the sake of
focusing on the mixing operator. However, $g(\cdot)$ possesses the flexibility
to handle a diverse range of output categories, such as single variable
outputs, soft choice probabilities, and discrete choices.

In a similar vein to the hybrid choice model, the DHM leverages a latent
variable $z_{n}$ within a latent space to encapsulate complex alternative
information — in this case, unstructured road network data. Our proposed GE
architecture adeptly transforms the road network structures into a high-
dimensional latent space. The concept of a latent space and latent variables
remains relevant and advantageous. DHM, in fact, enhances this concept with
its versatility; the latent space can also incorporate and interact with
supplementary data such as sociodemographics. Moreover, the variables residing
in the latent space can directly serve as inputs for travel demand models,
enhancing the predictive power of the model.

Another key feature of DHM is its adaptability. Both the input and output
components of DHM can be configured to cater to different tasks, making it a
highly flexible solution that can be employed across various problem domains.
Further extending its versatility, DHM can process other forms of unstructured
data, including images and texts, by substituting the GNN encoder with other
suitable embedding techniques, like Convolutional Neural Networks (CNNs),
Recurrent Neural Networks (RNNs), or Transformers. This substitution
capability makes DHM a scalable and robust framework, able to handle a wide
array of data types and tasks.

##  4 Experiments

###  4.1 Data Resources

We use Chicago to demonstrate the proposed model and include multiple data
sources. Notice that our research scope is at census tract level, which means
our inputs $x_{n}$ and $\mathcal{R}$ come from the sociodemographics and road
network structure from the census tract.

####  4.1.1 Road Network Data Description

We leveraged the utility of the OSMnx package to streamline and download road
network data from the open-source platform, OpenStreetMap (OSM) (Boeing, 2017;
Haklay and Weber, 2008). OSMnx operates by constructing a simplified and
topologically corrected street network, effectively filtering out certain
problematic elements such as edge dead-ends, edge self-loops, and complicated
intersections where multiple streets intersect and at least one street
continues beyond the intersection (Boeing, 2017, 2020; Kirkley et al., 2018;
Ganin et al., 2017; Xue et al., 2022). This process of simplification is
important due to the inherent complexity of raw urban road network data
sourced from OSM, which contains an overwhelming volume of information that
often misrepresents the true topological relationship between intersections.
For instance, OSM data employs multiple nodes and edges to represent a
singular freeway, which can inadvertently compromise the performance of the GE
model. Furthermore, additional complications arising from elements like dead-
ends or intricately intertwined roads are also systematically eliminated
through the process of simplification. Consequently, this transformative
process significantly reduces the number of nodes (intersections) and edges
(road segments) in the dataset. Following the implementation of OSMnx, the
network was reduced from 390,642 nodes and 1,121,620 edges to 28,701 nodes and
76,174 edges. Figure 2 visually depicts the simplified road network of Chicago
after the implementation of OSMnx.

Figure 2: The simplified road network of Chicago using OSMnx. Nodes represent
intersections and edges are road segments. The simplified road network shows
the topology, but not the real road shape.

####  4.1.2 Sociodemographic Data Description and Analysis

The dataset central to this research is primarily derived from the American
Community Survey (ACS) conducted over the span of 2017-2018. The ACS provides
a wealth of sociodemographic information such as income brackets, age
demographics, racial group compositions, prevailing modes of commuting, and
average commuting durations, all specific to each of the 811 distinct census
tracts located within the boundaries of Chicago.

To further broaden the scope and depth of our analysis, we have also
considered the potential benefits of manually engineered features derived from
both road and transit networks, such as the number of rail stations and bus
stops. These data have the potential to shed light on the capacity of
Chicago’s road network to manage varying volumes of traffic. Our inclusion of
these road network features is based on precedents set by previous
investigations that have successfully applied manual feature engineering to
both the built environment and sociodemographic data in the context of mode
share analysis (Wang et al., 2012; Zhan et al., 2017; Li et al., 2015; Saberi
et al., 2020). For a comprehensive overview of the specific variables utilized
in this research, refer to Table 2.

We can categorize the sociodemographic inputs into the following
classifications:

  1. 1.

Demographics (e.g., population, age, gender)

  2. 2.

Income and employment (e.g., income levels, employment status)

  3. 3.

Education (e.g., education levels, specific age and gender groups with certain
educational attainments)

  4. 4.

Race and ethnicity

  5. 5.

Housing (e.g., property values, rent, housing units)

  6. 6.

Area and density (e.g., area of the census tract, number of nodes per area)

  7. 7.

Transportation infrastructure (e.g., number of vehicles, vehicle per capita)

  8. 8.

Road Network features (e.g., road density, number of intersections)

In our research, the travel mode shares under consideration encompass driving,
public transit (PT), taxi, cycling, and walking. Shares of these modes can be
derived from the travel_ratio variables present in the sociodemographic data
in Table 2, representing the choice set $C_{n}$. In Table 2, we detail a
collection of 80 sociodemographic variables. Given the extensive list of
variables, we employed a Pearson correlation coefficient with a threshold of
0.05 to refine our selection process. Only variables that demonstrated
significant correlations ($\geq 0.05$) with all travel modes were retained to
ensure their pertinence in our model. Variables that satisfy this threshold
are colored in grey within the table.

Census tract variables |  Description |  Variable level
---|---|---
pop_total |  total number of population |  Integer values $\in[347,20087]$
sex_total |  total number of population, same as pop_total |  Integer values $\in[347,20087]$
sex_male |  number of males |  Integer values $\in[113,9179]$
sex_female |  number of females |  Integer values $\in[201,11373]$
age_median |  median age |  Continuous values $\in[16,67.4]$
households |  number of households |  Integer values $\in[113,12017]$
race_total |  total number of population, same as pop_total |  Integer values $\in[347,20087]$
race_white |  number of white people |  Integer values $\in[0,11933]$
race_black |  number of black people |  Integer values $\in[0,6448]$
race_native |  number of native American |  Integer values $\in[0,294]$
race_asian |  number of asians |  Integer values $\in[0,6166]$
inc_total_pop |  total number of population recoding incomes |  Integer values $\in[263,17976]$
inc_no_pop |  population without income |  Integer values $\in[20,8718]$
inc_with_pop |  population with income |  Integer values $\in[180,17076]$
inc_pop_10k |  number of people with less than 10K income per year |  Integer values $\in[21,1926]$
inc_pop_1k_15k |  number of people with more than 10K and less than 15K income per year |  Integer values $\in[11,904]$
inc_pop_15k_25k |  number of people with more than 15K and less than 25K income per year |  Integer values $\in[15,1412]$
inc_pop_25k_35k |  number of people with more than 25K and less than 35K income per year |  Integer values $\in[5,980]$
inc_pop_35k_50k |  number of people with more than 35K and less than 50K income per year |  Integer values $\in[12,1377]$
inc_pop_50k_65k |  number of people with more than 50K and less than 65K income per year |  Integer values $\in[0,1830]$
inc_pop_65k_75k |  number of people with more than 65K and less than 75K income per year |  Integer values $\in[0,895]$
inc_pop_75k |  number of people with more than 75K income per year |  Integer values $\in[0,8823]$
inc_median_ind |  median individual income per year |  Integer values $\in[4494,96667]$
travel_total_to_work |  total number of population traveling to work |  Integer values $\in[53,14332]$
travel_driving_to_work |  total number of population driving to work |  Integer values $\in[32,6407]$
travel_pt_to_work |  total number of population taking transit to work |  Integer values $\in[8,4169]$
travel_taxi_to_work |  total number of population taking taxi to work |  Integer values $\in[0,498]$
travel_cycle_to_work |  total number of population riding bicycles to work |  Integer values $\in[0,608]$
travel_walk_to_work |  total number of population walking to work |  Integer values $\in[0,3756]$
edu_total_pop |  total number of population (based on education) - not sure why it is different from pop_total |  Integer values $\in[200,17976]$
bachelor_male_25_34 |  number of males with bachelor degree between 25 and 34 years old |  Integer values $\in[0,1382]$
master_phd_male_25_34 |  number of males with master and PhD degree between 25 and 34 years old |  Integer values $\in[0,1104]$
bachelor_male_35_44 |  number of males with bachelor degree between 35 and 44 years old |  Integer values $\in[0,443]$
master_phd_male_35_44 |  number of males with master and PhD degree between 35 and 44 years old |  Integer values $\in[0,977]$
bachelor_male_45_64 |  number of males with bachelor degree between 45 and 64 years old |  Integer values $\in[0,864]$
master_phd_male_45_64 |  number of males with master and PhD degree between 45 and 64 years old |  Integer values $\in[0,1098]$
bachelor_male_65_over |  number of males with bachelor degree older than 65 years old |  Integer values $\in[0,286]$
master_phd_male_65_over |  number of males with master and PhD degree older than 65 years old |  Integer values $\in[0,566]$
bachelor_female_25_34 |  number of females with bachelor degree between 25 and 34 years old |  Integer values $\in[0,1175]$
master_phd_female_25_34 |  number of females with master and PhD degree between 25 and 34 years old |  Integer values $\in[0,2007]$
bachelor_female_35_44 |  number of females with bachelor degree between 35 and 44 years old |  Integer values $\in[0,639]$
Census tract variables |  Description |  Variable level
---|---|---
master_phd_female_35_44 |  number of females with master and PhD degree between 35 and 44 years old |  Integer values $\in[0,1154]$
bachelor_female_45_64 |  number of females with bachelor degree between 45 and 64 years old |  Integer values $\in[0,788]$
master_phd_female_45_64 |  number of females with master and PhD degree between 45 and 64 years old |  Integer values $\in[0,996]$
bachelor_female_65_over |  number of females with bachelor degree older than 65 years old |  Integer values $\in[0,369]$
master_phd_female_65 |  number of females with master and PhD degree older than 65 years old |  Integer values $\in[0,575]$
edu_total |  total number of population, a bit different from pop_total |  Integer values $\in[144,17171]$
edu_bachelor |  number of people with bachelor degree |  Integer values $\in[0,5322]$
edu_master |  number of people with master degree |  Integer values $\in[0,4085]$
edu_phd |  number of people with PhD degree |  Integer values $\in[0,1053]$
inc_median_household |  median household income |  Integer values $\in[11146,194167]$
inc_per_capita |  average income per capita |  Integer values $\in[1801,134796]$
employment_total_labor |  total number of population (based on employment) |  Integer values $\in[216,17976]$
employment_employed |  number of employed people |  Integer values $\in[98,14680]$
employment_unemployed |  number of unemployed people |  Integer values $\in[97,9362]$
housing_units_total |  total number of housing units |  Integer values $\in[129,12660]$
housing_units_occupied |  total number of occupied housing units |  Integer values $\in[113,12017]$
housing_units_vacant |  total number of vacant housing units |  Integer values $\in[0,1580]$
rent_median |  median ret |  Integer values $\in[274,2563]$
property_value_total |  total property values |  Integer values $\in[1,6612]$
property_value_median |  median property value |  Integer values $\in[9999,1122700]$
vehicle_total_imputed |  total number of vehicles |  Integer values $\in[53,14332]$
household_size_avg |  average household size |  Continuous values $\in[1.3,35.6]$
sex_male_ratio |  ratio of males |  Continuous values $\in[0,1]$
race_white_ratio |  ratio of white people |  Continuous values $\in[0,1]$
race_black_ratio |  ratio of black people |  Continuous values $\in[0,1]$
race_native_ratio |  ratio of native American |  Continuous values $\in[0,1]$
race_asian_ratio |  ratio of asians |  Continuous values $\in[0,1]$
travel_driving_ratio |  ratio of people driving to work |  Continuous values $\in[0,1]$
travel_pt_ratio |  ratio of people taking public transit to work |  Continuous values $\in[0,1]$
travel_taxi_ratio |  ratio of people taking taxi to work |  Continuous values $\in[0,1]$
travel_cycle_ratio |  ratio of people riding bicycles to work |  Continuous values $\in[0,1]$
travel_walk_ratio |  ratio of people walking to work |  Continuous values $\in[0,1]$
travel_work_home_ratio |  ratio of people working from home |  Continuous values $\in[0,1]$
edu_bachelor_ratio |  ratio of people with bachelor degree |  Continuous values $\in[0,1]$
edu_master_ratio |  ratio of people with master degree |  Continuous values $\in[0,1]$
edu_phd_ratio |  ratio of people with PhD degree |  Continuous values $\in[0,1]$
edu_higher_edu_ratio |  ratio of people with bachelor, master, or PhD degree |  Continuous values $\in[0,1]$
vehicle_per_capita |  vehicle per capita |  Continuous values $\in[0,1]$
vehicle_per_household |  vehicle per household |  Continuous values $\in[0,1]$
vacancy_ratio |  rato of vacant houses |  Continuous values $\in[0,1]$
area |  Area of the census tract |  Continuous values $\in[0.1,38.8]$ (100sq km)
num_bus_stop |  # of bus stops within each census tract |  Integer values $\in[0,63]$
num_rail_stop |  # of rail stations within each census tract |  Integer values $\in[0,18]$
Road network variables |  Description |  Variable level
road_density |  Road density defined by road length per area |  Continuous values $\in[170.7,18975.2]$ (km per 100sq km)
num_node_per_area |  # of intersection per area |  Continuous values $\in[1.6,97.3]$ (# per 100sq km)
num_road_per_area |  # of road segments per area |  Continuous values $\in[0.8,189.5]$ (# per 100sq km)
sub_sum_cent |  Summation of centrality for all intersections within each census tract |  Continuous values $\in[1.0,3.5]$
sum_deg |  Summation of degrees for all intersections within each census tract |  Integer values $\in[2,858]$
sub_sum_nodes |  Total # of intersections within each census tract |  Integer values $\in[2,166]$
Table 2: Description of variables from the Chicago sociodemographic data and
feature engineered road network variables.

We also include the features from network science and transit services.
Variables such as $road\\_density$ and $sub\\_sum\\_cent$ have been defined
based on the frameworks proposed by Hawbaker et al. (2005) and van den Heuvel
and Sporns (2013) respectively. It is worth noting the presence of several
features that exhibit substantial correlation with each other. For example,
the calculation of $num\\_node\\_per\\_area$ is based on $sub\\_sum\\_nodes$
and $area$. Moreover, racial group proportions are exclusive to each other,
which underscores the necessity of exploring the degree of correlation between
different features to accurately determine the inputs for the regression
model. From Table 2, we can find that variables related to education and
income exert more pronounced influences than other factors. Concurrently,
younger age groups, specifically those aged 25-34 and 35-44, display a
heightened impact on their travel mode decisions. Furthermore, distinct racial
groups manifest varied influences on these choices, an issue that aligns with
broader discussions on fairness and social equity within transit systems as
elucidated by Zheng et al. (2023).

Figure 3 shows the correlation matrix among variables. Notice that there is an
additional variable denoted as embd_readout, which represents the average
value of $\mathcal{R}_{n}$:
$embd\\_readout=\frac{1}{K}(\mathcal{R}_{n,1}+\mathcal{R}_{n,2}+\dots+\mathcal{R}_{n,K})$.
Given that all other variables constitute single values for each census tract,
the computation of the average value is crucial for the correlation
calculation. Meanwhile, number of rail stations and bus stops are also
included to provide the insight from the transit service supply side.

Figure 3: Correlation matrix of variables from the census survey, road network
metrics, and graph embedding readouts.

Moreover, Figure 3 reveals different levels of correlations in the input
variables:

  1. 1.

Demographic variables tend to have a varied impact on travel mode shares.
Specifically, education level, especially for younger males and females, and
race seem to play significant roles in influencing the choice of travel mode.
Younger and middle-aged males with a bachelor’s or master’s degree tend to
drive less and prefer public transit, taxis, and cycling. Areas with a higher
black population have a slight correlation with a preference for walking and a
negative correlation with cycling, and areas with a higher Asian population
show a strong correlation with using taxis and walking.

  2. 2.

Income and employment variables significantly influence travel mode shares.
Lower income brackets (lower than 35k) tend to drive more and use public
transit less. Areas with higher median individual incomes tend to prefer
public transit, taxis, cycling, walking, and working from home over driving.
Employed individuals show a preference for taxis and walking, whereas a higher
unemployment ratio correlates with a preference for driving and a reduced
preference for cycling.

  3. 3.

Housing-related variables, especially concerning vacancy rates and rents,
influence travel mode shares. Areas with more vacant housing units tend to
prefer taxis and walking over driving, while Areas with higher median rents
have a negative correlation with driving and a positive correlation with
public transit, taxis, cycling, and walking.

  4. 4.

Higher education levels show a distinct preference for alternative travel
modes to driving. Higher education ratios, including bachelor, master, and
Ph.D., have a negative correlation with driving and a positive correlation
with public transit, taxis, cycling, and walking.

  5. 5.

The availability of vehicles in an area has a clear impact on driving mode
share. It is interesting to see the vehicle_per_capita has negative
correlations with the driving modes.

  6. 6.

The built environment of an area, including its size and infrastructure like
road density and nodes, influence travel mode preferences. Larger areas show a
slight preference for driving and a negative correlation with public transit,
taxis, cycling, and walking. Areas with more intersections per area have a
positive correlation with public transit, taxis, cycling, and walking.
Conversely, areas with higher road densities (more roads and intersections)
tend to prefer driving and show a negative correlation with public transit and
cycling.

  7. 7.

Our averaged graph embedding readout value, $embd\\_readout$ has the strongest
positive correlation with driving and the strongest negative correlation with
public transit. The exact nature and implications of this variable might be
clearer with more context about what $embd\\_readout$ represents in the
dataset. A critical clarification is that variables with lower correlations
with the embedding readout values do not necessarily indicate the inadequacy
of the readout for their estimation. The variable $embd\\_readout$ is
essentially an aggregation of the GER, while the original GER vector with
length $K$ encapsulates a greater amount of information useful in the
regression.

###  4.2 Mode Share Prediction Performance

Using the collected and refined feature data, our study aims to evaluate the
numerical prediction capabilities of various models in relation to regressing
the proportions associated with distinct travel modes. This assessment not
only delves into the specifics of the inputs furnished to the model but also
provides a comprehensive overview of the selected models for representation.
Furthermore, to offer a holistic understanding, we will delineate the
comparative outcomes derived from comparing the performance metrics of these
models.

####  4.2.1 Inputs and Travel Demand Models

In order to evaluate the performance of DHMs, we measure the regression
accuracy by employing different travel demand models, along with varying input
parameters. Specifically, these inputs encompass the use of the 37 highlighted
variables from Table 2, the sole employment of our urban road network GER, and
a vector amalgamation of both. For ease of reference and clarity, we denote
these three experimental conditions as the baseline, GER, and concatenated
inputs, respectively.

The reason why we need to include the concatenated inputs is that the
objective of our DHM is to facilitate a comprehensive understanding of urban
dynamics by concurrently utilizing both road network and sociodemographic data
as inputs. As part of this study, we have previously expounded on the
performance of GE as a standalone predictor. However, to enhance the model’s
predictive capability, we now propose a Concatenated input travel demand
model, a model that concatenates these two distinct yet interrelated sources
of data to enhance the prediction power.

We differentiate different input types. We initiate our experimentation with
the baseline inputs, representing a traditional mode share modeling
methodology employed using sociodemographic and manually created network
attributes as the inputs. In the context of our computational formulation,
these values align with the $x_{ikn}$ parameters in Equation 1, embodying a
linear utility structure. A granular breakdown of the 37-dimensional baseline
input can be gleaned from Table 2, wherein the highlighted variables have been
detailed discussed in Section 4.1.2. The graph embedding readouts, denoted as
$\mathcal{R}_{n}\in\mathbb{R}^{1\times K^{*}},\forall n=1,2,\dots,N$ is
subsequently labeled as the GER input for models. The concatenation of the
baseline and GER inputs is denoted as the concatenated input.

Note that our behavioral predictor, denoted by $g(\cdot)$, boasts of inherent
versatility, accommodating an array of prediction functions. Contextualizing
this within our research spectrum, our predictive model $g$ integrates the
following regression methodologies:

MNL is a sophisticated statistical tool tailored for analyzing choices among
multiple discrete options, widely used in travel mode share analysis to
determine the probability of selecting various transportation options like
car, bus, or bike. Distinguished by its capacity to handle multiple
categorical outcomes, it employs a range of predictors—socioeconomic profiles,
trip details, and transportation system features—to compute the probabilities
of each mode being chosen. This model yields insights into how changes in
predictors affect the odds of selecting each travel mode, thereby aiding in
predicting mode share and guiding transportation policy decisions (Hausman and
McFadden, 1984; Anas, 1981).

Random Forests, transitioning from traditional regression to ensemble methods,
it employ multiple decision trees to yield predictions with heightened
accuracy and resilience (Biau and Scornet, 2016). In the context of travel
mode share analysis, their potency is particularly evident. They capably
manage extensive predictor sets, account for intricate variable interactions,
and mitigate potential overfitting inherent to a singular decision tree. When
forecasting mode shares, the decision trees collectively vote for a travel
mode based on input features, with the mode securing the majority being the
conclusive prediction. The breadth of these features mirrors those in logistic
regression, encompassing socio-economic traits and transportation specifics. A
salient advantage of random forests is their capacity to prioritize
predictors, guiding researchers and planners to discern paramount factors
steering travel mode decisions. T his knowledge becomes foundational in
sculpting potent transportation policies and interventions.

XGBoost, standing for ”Extreme Gradient Boosting”, represents the cutting edge
in gradient boosting frameworks, distinguished by its prowess and efficiency
in both competitive and practical machine learning scenarios (Chen et al.,
2015). Rooted in gradient boosting principles, XGBoost creates a cumulative
ensemble of decision trees, with succeeding trees remedying the predecessors’
errors. In the sphere of travel mode selection and mode share prediction,
XGBoost excels by capturing intricate non-linear associations and variable
interactions, often outclassing many conventional algorithms in predictive
accuracy. Key attributes such as adeptness at navigating missing data,
intrinsic regularization to counteract overfitting, and scalability make it a
frontrunner for extensive transportation data analytics. Complementing its
predictive capabilities, XGBoost’s feature importance metrics furnish
researchers and planners with invaluable insights into pivotal determinants
shaping travel mode inclinations, thereby informing strategic transport policy
crafting and infrastructure development endeavors.

We fine-tune both the Random Forest and XGBoost model parameters based on
their performance in regressing each of the travel modes. That means we try
multiple combinations of parameter sets and always present the best model
performances when regressing the travel mode share.

####  4.2.2 Model Comparison

Before introducing the comparison results, we firstly specify our data split
approach. We then describe the chosen evaluation metrics and present the
numerical analysis of mode share regression results.

We split the 811 census tracts of Chicago by random 70-30 train-test split.
The different input formats are divided accordingly. The metrics we use are
in-sample R-square (ISR2) and out-of-sample R-square (OSR2), which are the
linear regression models run on the train and test set respectively.

The R-square value, often termed the coefficient of determination, quantifies
the proportion of the variance in the dependent variable that is predictable
from the independent variables. It is defined as:

$R^{2}=1-\frac{\sum_{i}(y_{i}-f_{i})^{2}}{\sum_{i}(y_{i}-\frac{1}{N}\sum_{n=1}^{N}y_{i})^{2}}$
(10)

where $y_{i}$ and $f_{i}$ are the true values and regression results
accordingly. The ISR2 value provides insight into the model’s performance on
the data it was trained on. A higher ISR2 value suggests that the model can
explain a significant portion of the variability in the dependent variable
based on the training dataset. However, an excessively high ISR2 can be
indicative of potential overfitting, suggesting the model might be excessively
tailored to the training data and could underperform when exposed to new,
unseen data. On the other hand, the OSR2 value evaluates the model’s
predictive capacity on previously unseen data, in this context, the test set.
This metric is pivotal as it assesses the model’s ability to generalize to new
data. A pronounced disparity between ISR2 and OSR2 values can signal
overfitting or underfitting. This underlines the significance of assessing
models on both training and testing datasets to guarantee their robustness and
predictive precision.

Table 3 provides a comprehensive comparison of three different modeling
techniques applied to travel mode share analysis using three different inputs:

Models & Modes |  | Baseline input | GER input | Concatenated input
---|---|---|---|---
|  ISR2 |  OSR2 |  ISR2 |  OSR2 |  ISR2 |  OSR2
MNL |  Driving |  0.752 |  0.679 |  0.814 |  0.708 |  0.858 |  0.747
PT |  0.546 |  0.388 |  0.731 |  0.553 |  0.785 |  0.588
Taxi |  0.241 |  0.203 |  0.441 |  0.366 |  0.503 |  0.330
Cycling |  0.368 |  0.231 |  0.616 |  0.279 |  0.656 |  0.390
Walking |  0.642 |  0.512 |  0.731 |  0.624 |  0.855 |  0.646
Random Forest |  Driving |  0.939 |  0.559 |  0.959 |  0.697 |  0.955 |  0.686
PT |  0.904 |  0.470 |  0.931 |  0.562 |  0.938 |  0.574
Taxi |  0.515 |  0.166 |  0.684 |  0.292 |  0.587 |  0.242
Cycling |  0.837 |  0.380 |  0.877 |  0.389 |  0.882 |  0.466
Walking |  0.873 |  0.344 |  0.915 |  0.496 |  0.937 |  0.519
XGBoost |  Driving |  0.961 |  0.622 |  0.995 |  0.727 |  0.997 |  0.753
PT |  0.935 |  0.549 |  0.972 |  0.576 |  0.996 |  0.639
Taxi |  0.534 |  0.200 |  0.693 |  0.318 |  0.736 |  0.35
Cycling |  0.852 |  0.144 |  0.964 |  0.168 |  0.993 |  0.332
Walking |  0.975 |  0.435 |  0.986 |  0.533 |  0.991 |  0.553
Table 3: Travel demand model comparisons on different travel modes using
different input features. Bold fonts indicate the best out-of-sample
regression results for a specific travel mode across all the travel demand
models. The underlining indicates the best ISR2 or OSR2 for a particular
travel model using a particular model.

From the analysis, it’s evident that models utilizing concatenated inputs
consistently outshine others, particularly in predictions pertaining to unseen
data. This underscores the value of blending conventional data with insights
from urban layouts to more accurately anticipate travel behaviors. On average,
the introduction of GER inputs yields a 20% enhancement in OSR2 relative to
baseline models. Furthermore, the top-performing results derived from either
the GER-input travel demand models or the Concatenated travel demand model
demonstrate a remarkable 40% improvement in OSR2 when contrasted with the
baseline models.

When we look at individual travel modes:

  * 1.

For Driving, models using concatenated inputs consistently outperformed the
others, suggesting that understanding both personal factors and city layout is
crucial for predicting driving behavior.

  * 2.

For PT and Walking, there was a clear benefit from using GER and concatenated
inputs, underlining the importance of the urban environment in influencing
these choices.

  * 3.

Taxi and Cycling predictions also saw improvements with GER and concatenated
inputs, indicating that the city’s road structure can impact decisions to hail
a taxi or hop on a bike.

As for the predictive models, MNL didn’t predict as accurately as the ensemble
models, particularly when enriched with GER and concatenated inputs. Ensemble
models like Random Forest and XGBoost, especially when input with the
concatenated dataset, achieve higher OSR2. However, a key observation was the
occasional gap between a model’s predictions for its training data and its
predictions for new data. This was especially noticeable for the Random Forest
model using just the baseline inputs. This might mean the model is too
tailored to its training data, which could make it less accurate in real-world
scenarios.

(a) PT Mode Share

(b) Driving Mode Share

(c) Taxi Mode Share

Figure 4: Comparison of different mode share regression outcomes (part 1):
Ground truth vs. baseline and GER-input demand models using MNL

(a) Cycling Mode Share

(b) Walking Mode Share

Figure 5: Comparison of different mode share regression outcomes (part 2):
Ground truth vs. baseline and GER-input travel demand models using MNL.

##  5 Visual Interpretation

###  5.1 Spatial Patterns of Regression Results

The findings presented in Table 3 shed light on a rather interesting finding.
It is evident that when the road network topology is employed solely as the
defining input, the GER-input demand model substantially outstrips the
performance of the Baseline demand model, registering a commendable
improvement margin of approximately 20%. This pivotal outcome underscores the
importance of delving deeper into the inherent patterns and discernible
physical implications that the embeddings might elucidate. In pursuit of a
comprehensive comparative analysis of the spatial distributions associated
with regression outcomes across the entire city, we opted to re-conduct the
fitted linear regression model, this time leveraging the entirety of the
available data.

The regression results for each of the travel modes are visualized spatially
in Figure 4 and 5. We present the results derived from MNL, as predicted
either by the Baseline Demand Models or the GER-input Demand Models.

In examining the spatial distributions of various transportation mode shares
across Chicago, several distinct patterns emerge when comparing the ground-
truth data with predictions from the baseline and GER models. From Figure
4(a), it is clear that public transit mode shares exhibit a lack of spatial
contiguity in relation to their neighboring regions in numerous locales. While
the Baseline demand model predominantly tends towards underfitting the data,
it remarkably retains the capability to assimilate the overarching spatial
contour. Contrarily, our GER-input demand model augments the linear model’s
capacity to differentiate nuances and, in doing so, unveils a pronounced
spatial continuity.

Conversely, Driving mode share patterns in Figure 4(b), both models exhibit
broadly congruent patterns, capturing the prevalent driving behaviors of
Chicago’s inhabitants. However, subtle variations, particularly in the
intensity and localization of driving patterns, accentuate the GER model’s
heightened sensitivity. For Cycling mode share pattern in Figure 5(a), the GER
model aligns superiorly with the ground-truth data, capturing urban transit
dynamics effectively. Specifically, the GER’s spatial representation closely
matches the actual distribution, while the baseline model appears less
connected to these dynamics. The Taxi and Walking Mode Shares present
intriguing contrasts, shown in Figure 4(c) and 5(b). For taxi usage, the GER
model portrays a more dispersed and pronounced distribution, suggesting its
potential to discern intricate transit choices. In terms of walking, while
both models convey analogous distributions, the GER model presents a more
resonant depiction, especially in pedestrian-dominant regions.

This naturally leaves us with the question: how exactly does the GER model’s
performance interface with sociodemographic data and the intricate constructs
of road networks? We therefore further interpret the patterns of $\mathcal{R}$
and its correlations with other features.

###  5.2 Relation Between Graph Embedding Readouts and Road Network Structures

In an effort to visualize the spatial implications and insights unveiled by
the graph embedding readouts, we employ the value of $embd\\_readout$ to
investigate the correlations with the geographical characteristics of the
city. The geographical distribution of $embd\\_readout$ demonstrates spatial
continuity, depicted in Figure 6. It is intriguing to observe that elevated
values of $embd\\_readout$ are predominantly found in the Northern and
Southern regions of Chicago, areas recognized for their prosperity and
popularity within the city’s confines
111https://en.wikipedia.org/wiki/Community_areas_in_Chicago. This observation
intriguingly aligns with the robust negative correlation between
$embd\\_readout$ and the public transit mode share, as elucidated in Figure 3.

(a) Value of $embd\\_readout$

(b) Chicago boroughs

Figure 6: Visualization of averaged GER values along and the boroughs of
Chicago. The North Side, South Side, and Central parts of Chicago are
typically considered popular areas.

As we delve deeper into the interpretation of the graph embedding readouts, we
aspire to visualize the structure of the road network within each census tract
in correspondence with the sorted graph embedding readouts. Given the 811
census tracts, each is assigned a graph embedding value to facilitate the
visualization as in Figure 6. This enables us to sort these values,
facilitating the identification of the road network that corresponds to the
${5\%,25\%,50\%,75\%,95\%}$ quantiles of the graph embedding readouts. By
doing this, we aim to elucidate the correlations between the numerical graph
embedding readouts and the inherent network topology of the inputs. The result
of this exploration is illustrated in Figure 7:

Figure 7: Quantiles of graph embedding readouts and their corresponding road
network structures.

Figure 7 presents an evolution from smaller to larger quantiles, wherein the
depicted road network structure transitions from a sparse, irregular, and non-
rectangular form to a denser, organized, and grid-like structure. This
evolution is concordant with the underlying logic of the GE technique. The
technique relies on formulating the information passing in the network,
necessitating more weight/values on popular nodes to encapsulate the
complexities of network structures.

In the context of real-world applications, denser and more organized road
structures typically signify areas of high travel demand, such as downtown
districts or central business sectors. Thus, along with the insights gleaned
from Figure 6, it becomes apparent that GER have substantial potential in
estimating aspects such as the density, popularity, prosperity, and other
sociodemographic features of a city.

###  5.3 Clustering Analysis of the GER

The GE technique possesses a potent capability to effectively discern the
relationships between sociodemographic factors and census tracts, utilizing
only road network topology as its primary input. This process is notable for
its capacity to distill the heterogeneous information derived from different
sociodemographics into a more coherent and concise representation of census
tracts. Consequently, it unveils intricate correlations among these tracts,
obviating the need for laborious collection and analysis of various
sociodemographic factors.

To further elucidate the spatial correlation patterns of these embeddings, we
employ clustering algorithms to visually demarcate areas of similar
characteristics. Specifically, for this study, we have utilized the K-Means
Clustering Algorithm (Krishna and Murty, 1999) on the graph embedding
readouts. The decision to opt for 30 clusters was dictated by the fact that it
constitutes roughly 30% of the total number of census tracts, thereby ensuring
a sufficient level of granularity in the spatial division. Of course, it’s
worth mentioning that other clustering algorithms, as well as different
numbers of clusters, could also be employed depending on the specific
analytical requirements and data characteristics.

Figure 8: 30 clusters generated by applying Gaussian Mixture Models Clustering
Algorithm on GER. Red lines represent the distribution of rail lines.

The spatial distribution of the resulting clusters is depicted in Figure 8.
Each unique color in Figure 8 symbolizes a distinct cluster, with the largest
cluster manifesting itself in light green. Upon examining these clusters, it’s
apparent that they closely reflect the geographical proximity of neighboring
census tracts. The GERs possess the capacity to learn not only the intricacies
of network structures but also their spatial distances, thereby demonstrating
a comprehensive understanding of the spatial relationships within the data.
Interestingly, the clusters are generally observed to align with the city’s
rail lines, implying a strong correlation between public transit
infrastructure and the spatial distribution of these tracts.

Such a finding significantly emphasizes the profound impact that the transit
system can exert on various aspects of urban life. Not only does it shape the
physical structure of the road network, but it also influences a myriad of
social characteristics within its sphere of influence. These may encompass
income levels, age demographics, and transit usage patterns among the resident
population. Thus, this observation offers valuable insights into the interplay
between infrastructure planning and sociodemographic development, further
attesting to the applicability and efficacy of our approach in urban studies.

###  5.4 Interpretation of the GER variables

Understanding the relative importance of distinct attributes is important in
mode share analysis. Typically, elasticity analysis, which measures the effect
of a 1% change in an independent variable on a dependent variable, has been
the standard in traditional choice models. This relies on evaluating and
interpreting each input dimension. Within the ambit of DHMs, however, the
normalization which bounds the embedded values to the [0,1] range intimates
that changes in embedded values can’t be directly attributed to fluctuations
in a singular dimension. The crux of this limitation lies in the embedded
values’ representation: they are but a specific output from the GE model and
lack the tangible significance synonymous with input values in baseline demand
models. Thus, a mere 1% perturbation in embedding values might be devoid of
substantive interpretation.

Figure 9: Interpretations of latent variables.

Yet, understanding the relative weight of each dimension is achievable. We
investigate the correlations between each embedded dimension and assorted
sociodemographic indicators. Referring to the provided Figure 9, the
correlations between the initial three dimensions of the embedded values and
various sociodemographic factors are highlighted. These correlations provide
us with the analytical perspectives of how each dimension might resonate with
the influences of the embedded value dimension.

From the visual data shown in Figure 9, Dimension 1 appears to have a negative
association with attributes such as sum_deg and num_node_per_area, whereas it
establishes a positive correlation with indicators like pop_total and
sex_female. Such correlations suggest that this dimension may primarily be
inclined towards understanding population metrics and gender distribution. In
contrast, Dimension 2 presents strong correlations with factors like
race_black, race_asian, and inc_pop_1k_15k, signifying its sensitivity towards
racial demographics and certain income brackets. Lastly, Dimension 3, while
exhibiting less pronounced associations with racial metrics, underscores its
relevance with economic parameters like inc_median_ind and
travel_total_to_work, pointing towards an economic or occupational-oriented
theme.

While this interpretation approach echoes the methodology of topic modeling,
it is invariably constrained by the scope of human prior knowledge and
understanding. Future efforts to enhance interpretation could consider
incorporating generative models to reverse the GE process into graph
generation. This approach would utilize the GER values as inputs and the road
networks as outputs, which would foster a more nuanced understanding of how
each dimension influences the learning of road network structures.

##  6 Conclusion

In this paper, we have developed a novel framework by integrating graph-
embedded urban road networks to train sophisticated travel demand models for
mode share analysis, obviating the need for exhaustive feature engineering or
reliance on a priori knowledge of built environment. This approach entails the
introduction of the DHM, a computational framework that fuses the principles
of deep learning and hybrid choice models, underpinned by graph embedding
techniques. Specifically, the GE method, Node2Vec, is utilized to predict the
individual portions of choosing different travel modes. Our empirical results
substantiate the assertion that the implementation of GE can considerably
amplify the effectiveness of mode share analysis. Furthermore, the
amalgamation of GERs with sociodemographic variables demonstrably enhances the
overall performance of the regression model. The DHM framework is adaptive to
other machine learning predictive methods. We further recognize that the GE
technique offers profound spatial insights, which are inherently correlated
with the popularity and affluence of various census tracts. This correlation
further extends to the density and morphological characteristics of road
networks within respective regions. Furthermore, the GER clusters evince
robust spatial contiguity and a notable convergence around railway lines.

Looking ahead to future research directions, we aspire to bolster the
interpretability and generalizability of the DHM. A promising direction
involves incorporating the graph generation process, an innovative approach
that could potentially enhance interpretability. This will answer the
question: what should the road network be like if we know the features? This
could be achieved by systematically manipulating the GE and subsequently
scrutinizing its resultant impact on the graph structure through graph
generation techniques. In addition, we posit that a comparative analysis of
patterns across diverse geographical locations and historical time periods
worldwide may shed light on whether changes in the structure of urban road
networks resonate with alterations in community structures. Such comparative
studies have the potential to provide broader insights into the complex
interplay of social and infrastructural factors in urban environments.

##  7 Acknowledgement

This material is based upon work supported by the U.S. Department of Energy’s
Office of Energy Efficiency and Renewable Energy (EERE) under the Vehicle
Technology Program Award Number DE-EE0009211. The views expressed herein do
not necessarily represent the views of the U.S. Department of Energy or the
United States Government.

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