# Site-selective-induced isomerization of formamide

S. Oberli1, J. González-Vázquez1, E. Rodríguez-Perelló2, M. Sodupe2, F.
Martín1,3,4, and A. Picón1 1Departamento de Química, Universidad Autónoma de
Madrid, 28049 Madrid, Spain  2Departament de Química, Universitat Autònoma de
Barcelona, 08193 Bellaterra, Spain  3Instituto Madrileño de Estudios Avanzados
en Nanociencia (IMDEA-Nanociencia), Cantoblanco, 28049 Madrid, Spain
4Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid,
28049 Madrid, Spain

###### Abstract

The new capacity of X-ray free-electron laser (XFEL) facilities to produce
multi-color X-ray femtosecond pulses paves the way to explore ultrafast
phenomena in matter induced by X-ray photons. In the present study, we exploit
the site selectivity and the high temporal resolution of a two-color X-ray
femtosecond pump-probe sequence to investigate the isomerization of formamide.
The pump pulse excites a particular atomic site in the molecule, while the
probe pulse captures changes in the chemical environment at a remote atomic
site.  The response of the system is found to strongly depends on the nature
of the excited site, and the core excitation provides a selective control on
chemical bond breaking. In particular, we show that the
$\mathrm{N1s}\rightarrow\pi^{*}$ transition favors the isomerization reaction
in formamide and demonstrate the possibility to observe in real time the
hydrogen migration by measuring the time-dependent chemical shifts with the
X-ray probe pulse. This work opens a unique perspective for X-ray-induced
photochemistry at XFEL facilities.

Time-resolved two-color X-ray spectroscopy, time-dependent chemical shifts,
ultrafast dynamics, core-excited/ionized molecular states, hydrogen transfer

##  I Introduction

With the revolution of X-ray sources and in particular the advent of X-ray
free-electron lasers (XFELs), the production of attosecond/few-femtosecond
X-ray pulses opens a new door to the investigation of ultrafast X-ray-induced
phenomena in matter. The dynamics is triggered by a high-energy photon that
interacts with core electrons that are well localized in the molecule, a
completely different regime than in traditional photochemistry. In this
emerging field, some unprecedented applications have already been demonstrated
Berrah (2017), such as ultrafast isomerization of acetylene dication Jiang
_et al._ (2010); Liekhus-Schmaltz  _et al._ (2015); Li  _et al._ (2017);
Osipov  _et al._ (2003), hetero-site-specific femtosecond intramolecular
dynamics Picón  _et al._ (2016) and charge migration Kuleff  _et al._ (2016);
Picón  _et al._ (2018); Inhester  _et al._ (2019), to name just a few.

In this context, previous studies of X-ray-induced isomerization Jiang  _et
al._ (2010); Liekhus-Schmaltz  _et al._ (2015); Li  _et al._ (2017); Osipov
_et al._ (2003) have not yet exploited the site-selective excitation of X
rays, which can be an additional knob to trigger and control the reaction in
the femtosecond time scale. In this work, we theoretically demonstrate a site-
selective-induced isomerization scheme that can be implemented in future XFEL
facilities.

Isomerization plays a key role in nature, for example to trigger the proton
transfer by the bacteriorhodopsin protein Mathies  _et al._ (1991). An other
molecule of biological interest is formamide, which contains the four most
abundant elements in the universe, and is involved in the formation of RNA
precursors through UV radiation, nucleobases and other pre-metabolic
components Saladino  _et al._ (2012); Ferus  _et al._ (2014). Previous studies
on formamide focus mainly on the ground state tautomerism Schlegel  _et al._
(1982) or the effect of solvation on the energetics of proton transfer Wang
_et al._ (1991); Barone  _et al._ (1995), while photo-dissociation dynamics of
the two isomers was investigated after valence excitation Spinlove  _et al._
(2018). In contrast with the usual photoinduced isomerization reactions
involving transitions between delocalized valence molecular orbitals, we
propose here a novel investigation of formamide tautomerism in the X-ray
regime. We take advantage of the unique capabilities of XFELs to produce a
two-color X-ray pump-probe sequence Picón  _et al._ (2016) to assess the
possibility to 1) trigger the isomerization reaction by exciting a core
electron of the molecule; 2) influence the hydrogen migration by exciting
different atomic sites; 3) observe a fingerprint of the isomerization dynamics
through core-ionization of an other atom in the same molecule. To these
purposes, we developed a full quantum model based on highly-correlated
electronic structure calculations and a nuclear propagation algorithm to
simulate the dynamics in the core-excited states. We demonstrate the
possibility to induce hydrogen migration by taking advantage of site-selective
core-excitation. By creating a core vacancy at the C, N, or O atoms, we show
that we can selectively weaken and strengthen some bonds in order to hinder or
favour hydrogen migration. We observe a clear signature of bond
breaking/forming as well as a footprint of the isomerization reaction in the
time-dependent X-ray photoelectron spectra. This study paves the way to the
investigation of site-selective X-ray-induced ultrafast photochemistry at XFEL
facilities.

##  II Hetero-site pump-probe scheme

Figure 1: Two-color femtosecond X-ray pump-probe scheme to investigate
formamide isomerization. a) Different responses of the molecule to core-
excitation by the X-ray pump pulse is expected depending on the nature of the
excited atom. The dynamics is probed by ionizing the core shell of an other
element with an X-ray probe pulse. b) Changes in the local chemical
environment of the ionized-atom may be reflected in the binding energy ($BE$)
of the core-electron.

The hydrogen migration in formamide is investigated using an hetero-site-
specific pump-probe scheme: A first X-ray pulse initiates the hydrogen
migration through core-excitation of a particular site of formamide, while a
second X-ray pulse ionizes a core electron either at the C, N or O atom, as
illustrated in Figure 1. The measurement of the photoelectron energy with
respect to the delay between the two pulses provides information on the
changes in the local chemical environment around the absorber atom and thus on
structural modifications with femtosecond resolution (see Figure 1) Lindgren
(2004). This is a well-established technique known as X-ray photoelectron
spectroscopy (XPS) Siegbahn (1982). So far XPS has been mainly used for static
spectroscopy but the recent developments in ultrashort-X-ray-pulse sources
have led to a renaissance of XPS for time-resolved studies. The capacity of
XFELs to deliver a sequence of two femtosecond X-ray pulses Lu  _et al._
(2018); Lutman  _et al._ (2013); Marinelli  _et al._ (2015); Lutman  _et al._
(2016); Hara  _et al._ (2013); Allaria  _et al._ (2013); Ferrari  _et al._
(2016); Penco  _et al._ (2018); Bozek  _et al._ (2015) allows for the
implementation of the proposed scheme.

Figure 2: Nuclear wavepacket dynamics in the a) $\mathrm{C1s}^{-1}\pi^{*}$ b)
$\mathrm{O1s}^{-1}\pi^{*}$ c) $\mathrm{N1s}^{-1}\pi^{*}$ core-excited states.
Upper panel: Comparison of the PECs of the ground state (dashed line) and the
core-excited states (solid line). The initial wavepacket in the ground
electronic state is also displayed (dash-dotted). Lower panel: The nuclear
wavepacket in the core-excited state is shown for some particular times during
the propagation.

##  III Site-selective-induced isomerization

In this section we explore the possibility to trigger and control the
isomerization reaction by specifically exciting a particular atom in the
molecule (see Figure 1). The excitation energy from the 1s orbital of the C, N
and O-sites to the $\pi^{*}$ orbital is around 292.5, 407.2, and 535.4 eV,
respectively. In the simulations we consider X-ray pulses of 1.67-fs full-
width at half maximum (FWHM) length, see more details in Section VI.1. Both
electronic and nuclear dynamics are treated in a fully quantum framework in
which only the translational motion of the hydrogen (along the $y$-coordinate)
is taken into account while the other nuclear degrees of freedom are frozen.
This approximation captures the main features of the dynamics under
investigation, see more details in Section VI.2.

We first discuss the response of the molecule when the X-ray pulse interacts
with the C site. The PECs of the ground state and the
$\mathrm{C1s^{-1}\pi^{*}}$ core-excited state are shown in the upper panel of
Figure 2a. We see that core-excitation of the C has only a small effect on the
shape of the PEC, which is very similar to that of the ground state especially
in the Franck-Condon region. Therefore the interaction with the pump pulse
mainly populates the ground vibrational state of $\mathrm{C1s^{-1}\pi^{*}}$,
i.e. $|\braket{\phi^{\mathrm{ce}}_{0}}{\phi^{\mathrm{gs}}_{0}}|^{2}=0.99$,
$\phi^{\mathrm{ce}}$ and $\phi^{\mathrm{gs}}$ being the vibrational states of
the core-excited and the ground electronic states, respectively. By exciting
the C site the stability of formamidic acid compared to formamide is increased
by 0.04 eV and the energy barrier is only lowered by 0.29 eV (see Table 1),
such that the nuclear wavepacket remains trapped in the region corresponding
to the formamide geometry (see lower panel of Figure 2a). The limited impact
of core excitation on the electronic structure is explained by the distances
between the carbon atom and those directly involved in the isomerization
reaction (the H, N and O atoms). Exciting the core electron in the $\pi^{*}$
orbital only weakens the delocalized $\pi$-orbitals but does not induce
breaking of the N-H bond, such that core excitation of the carbon does not
trigger the isomerization reaction.

Electronic state |  $E_{\mathrm{f^{\prime}}}-E_{\mathrm{f}}$ [eV] |  $E_{\mathrm{ts}}-E_{\mathrm{f}}$ [eV]
---|---|---
gs | 0.62 | 1.45
$\mathrm{C1s^{-1}\pi^{*}}$ | 0.58 | 1.16
$\mathrm{O1s^{-1}\pi^{*}}$ | 4.34 | 2.77
$\mathrm{N1s^{-1}\pi^{*}}$ | -2.68 | -0.97
Table 1: Difference in energy between the formamidic (f’) and formamide (f)
species, as well as the energy barrier height in the ground and the core-
excited states. $E_{\mathrm{f}}$, $E_{\mathrm{f^{\prime}}}$, and
$E_{\mathrm{ts}}$ correspond to the energy of the formamide, formamidic, and
transition state at the geometry of the corresponding species in the ground
electronic state.

We consider now the response of the molecule when the X-ray pulse excites the
O site. The PECs of the ground and core-excited states as well as the
evolution of the nuclear wavepacket in $\mathrm{O1s^{-1}\pi^{*}}$ are
displayed in Figure 2b. In this case there is a large energy barrier of 2.77
eV (see Table 1) in the core-excited state that prevents the isomerization to
take place. The equilibrium geometry is displaced by 0.08 Å and the nuclear
wavepacket evolves towards larger distances, i.e. the N-H bond stretches in
the early times of propagation. The formation of formamidic is not favorable
in the $\mathrm{O1s^{-1}\pi^{*}}$ state since the core electron is excited to
a delocalized $\pi^{*}$ orbital, leading to a depletion of the electron
density around the oxygen atom which becomes more basic. The strong repulsion
between the positively charged oxygen and the hydrogen prevents the
isomerization to occur.

The most interesting case is when the X-ray pulse excites the N site, as shown
in Figure 2c. The Franck-Condon distribution covers mainly the
$18-23^{\mathrm{th}}$ vibrational states of the core-excited state. We tune
the energy of the laser to the resonance corresponding to the population of
the $19^{\mathrm{th}}$ vibrational state that exhibits the largest Franck-
Condon factor, and lies above the small energy barrier of $\sim 0.02$ eV. The
excitation from the 1s to the $\pi^{*}$ increases the acidity of the
$\mathrm{NH_{2}}$ group, favors the breaking of the CO double bond and also
increases the O basicity: As a result the hydrogen migration is triggered as
shown in the lower panel of Figure 2c. Around 12 fs the nuclear wavepacket
reaches the region around $y=-0.43$ Å where the formamidic acid is formed
despite the fast Auger decay (5.77 fs) of the core-excited state.

In summary, the hydrogen migration after the absorption of an X-ray photon
strongly depends on the excited atom. This site selectivity provides the
possibility to trigger hydrogen migration in the core-excited state and
eventually to control the formation of formamidic acid. In the following, we
determine the chemical shifts with a second X-ray pulse to investigate these
scenarios and show that a clear signature of the hydrogen migration can be
obtained.

Figure 3: Time-dependent chemical shifts associated with core-excitation of
one element – a) C, b) O, c) N, d) N – followed by core-ionization of a remote
atom – a) O, b) N, c) O, d) C. The vertical dashed (dash-dotted) line
corresponds to the chemical shift at the optimized geometry of formamide
(formamidic) in the ground electronic state. The intensities are given in
arbitrary units.

##  IV Ultrafast chemical shifts

In this section, we use the femtosecond X-ray pump-probe scheme presented in
Figure 1 to calculate time-dependent chemical shifts, i.e. the change in the
binding energy of the core electron which is ejected from the molecule by the
second pulse as the system evolves in the core-excited state. For a direct
comparison with an experimental XPS spectrum, a convolution should be
performed to take into account the laser bandwidth ($\sim 1.1$ eV), the finite
lifetime of the double-core-hole states produced by the probe pulse
($\hat{\Gamma}_{\mathrm{NO}}\sim 272$ meV and $\hat{\Gamma}_{\mathrm{NC}}\sim
222$ meV), as well as the energy resolution of the photoelectron detector.
Details on the calculations can be found in Sections VI.1 and VI.3.

The time-dependent chemical shift associated with the C-site excitation
followed by the ionization of the O-site is shown in Figure 3a. A peak at
534.5 eV, which corresponds to the formamide configuration, appears in the
initial times of the dynamics, and the signal losses intensity with respect to
time due to the Auger decay. As expected, there is no signal at 537.4 eV which
would have been a signature of the formation of formamidic acid. Indeed, the
nuclear wavepacket does not have enough energy to overcome the energy barrier
that prevents the isomerization to take place in the core-excited state and
there are no clear changes in the chemical environment.

The time-dependent chemical shift associated with O-site excitation followed
by ionization of the N atom is displayed in Figure 3b. As expected from the
nuclear dynamics in the $\mathrm{O1s^{-1}\pi^{*}}$ state, there is no signal
at 402 eV in the spectrum since the large energy barrier prevents the hydrogen
migration to take place. The positive hole at the O-site has thus no influence
on the local chemical environment around the N-site.

The time-dependent chemical shift in which the N-site is excited first
followed by the ionization of the O- and C-sites are shown in Figures 3c and
3d, respectively. In the former case, in the earlier times of the dynamics a
peak appears at $\sim 532.6$ eV corresponding to the formamide geometry. The
intensity of this peak decreases in time due to the Auger decay on one hand,
and the evolution of the wavepacket towards a region where the chemical shift
is higher on the other hand. Indeed, after 12 fs a signal rises at $\sim
537.3$ eV which is a signature that the isomerization reaction has time to
occur before the Auger decay completely depopulates the core-excited state.
This large variation in the chemical shift ($\Delta E\sim 4.7$ eV) is clearly
observed and has an intuitive explanation. As the hydrogen approaches the
oxygen atom, the local positive charge increases around the oxygen and makes
more difficult the ionization of the O-1s core electron, i.e. the binding
energy increases. In the case of the C-site, the chemical shift is not as
strong ($\Delta E\sim 2$ eV): At early times, a peak appears around 289.6 eV
when the core-excited state is formed at the formamide geometry, while after
12 fs a small signal is observed around 291.6 eV when the hydrogen is attached
to the oxygen. In summary, the isomerization reaction is barrierless in the
$\mathrm{N1s^{-1}\pi^{*}}$ state and there is a probability that the hydrogen
migration occurs despite the fast Auger decay.

##  V Conclusions

The present study shed light on the ability of femtosecond X-ray pulses to
site-selectively induce and unravel the dynamics of the short-lived core-
excited molecular states, and opens up promising avenues to design reaction
routes inaccessible with standard photochemistry techniques. In particular, we
developed a simple and elegant model to demonstrate the possibility to trigger
an isomerization reaction and more generally to control bond breaking/forming
in molecules in the X-ray regime.

Using the prototypical formamide molecule, we show that the response of the
system to the excitation of a core electron is very different depending on the
element that absorbs the X-ray photon. In the present case, core-excitation of
the nitrogen atom triggers the isomerization reaction which is precluded in
the ground electronic state due to the energy barrier that separates the
formamide from the less stable formamidic species. Despite the fast Auger
decay that competes with isomerization in the core-excited state, we can
observe a strong shift in the binding energy of the carbon and oxygen core
electrons as the hydrogen migration takes place. Instead, core-excitation of
the carbon or oxygen atoms does not initiate hydrogen transfer because of the
significant energy barrier in these states.

We demonstrate the feasibility to track the hydrogen migration by using time-
resolved two-color X-ray photoelectron spectroscopy. This technique is very
sensitive to local dynamics and is exploited to detect the bonding of the
hydrogen atom to the oxygen in formamide. This work anticipates a unique
application that will be available at XFEL facilities with multi-pulse multi-
color capabilities.

##  VI Theoretical methods

###  VI.1 Quantum model

The core-excited state dynamics is investigated using a two-color X-ray pump-
probe scheme: The pump pulse induces the formation of a core-excited state,
whose dynamics is probed by resorting to the time-dependent chemical shifts,
treating the probe pulse in first-order perturbation theory. Atomic units are
used in the following equations.

The formation of core-excited species induced by the interaction of the
molecule with an X-ray pump pulse is described by solving the time-dependent
Schrödinger equation (TDSE):

$\displaystyle(\hat{H}+\hat{W}(t))\ket{\Psi(t)}=\mathrm{i}\frac{\mathrm{d}}{\mathrm{d}t}\ket{\Psi(t)},$
(1)

with $\hat{H}=\hat{T}_{\mathrm{n}}+\hat{h}_{\mathrm{e}}$ the unperturbed
Hamiltonian given by the sum of the nuclear kinetic energy
$\hat{T}_{\mathrm{n}}$ and the electronic Hamiltonian $\hat{h}_{\mathrm{e}}$.
The transition dipole operator $\hat{W}(t)$ describes the interaction of the
molecule with the pump pulse. The molecular wavepacket is given by the sum for
each electronic state $i$ of the product of the nuclear wavepacket
$\ket{\chi_{i}(t)}$ and the electron wavefunction $\ket{i}$:

$\displaystyle\ket{\Psi(t)}=\ket{\chi_{\mathrm{gs}}(t)}\otimes\ket{\mathrm{gs}}+\ket{\chi_{\mathrm{ce}}(t)}\otimes\ket{\mathrm{ce}},$
(2)

where $\mathrm{gs}$ and $\mathrm{ce}$ denote the ground and the core-excited
states, respectively. A two-state model is a good approximation in the present
case, as justified in Section VI.3. In order to derive the equations of motion
for the nuclei, we substitute the ansatz for the total molecular wavefunction
(2) in the TDSE (1) and take the scalar product with the different electronic
wavefunctions:

$\displaystyle\mathrm{i}\frac{\mathrm{d}}{\mathrm{d}t}\ket{\chi_{\mathrm{gs}}(t)}$
$\displaystyle=$
$\displaystyle\big{(}\hat{T}_{\mathrm{n}}+\hat{V}_{\mathrm{gs}}\big{)}\ket{\chi_{\mathrm{gs}}(t)}$
$\displaystyle-\braket{\mathrm{gs}}{\vec{r}}{\mathrm{ce}}\ket{\chi_{\mathrm{gs}}(t)}\vec{E}(t),$
$\displaystyle\mathrm{i}\frac{\mathrm{d}}{\mathrm{d}t}\ket{\chi_{\mathrm{ce}}(t)}$
$\displaystyle=$
$\displaystyle\big{(}\hat{T}_{\mathrm{n}}+\hat{V}_{\mathrm{ce}}-\mathrm{i}\frac{\hat{\Gamma}_{\mathrm{ce}}}{2}\big{)}\ket{\chi_{\mathrm{ce}}(t)}$
(3)
$\displaystyle-\braket{\mathrm{ce}}{\vec{r}}{\mathrm{gs}}\ket{\chi_{\mathrm{ce}}(t)}\vec{E}(t).$

Non-adiabatic couplings are not included in our model since they are
negligible in the region of interest, see Section VI.3 for more details. The
potential energy curves of the ground and the core-excited states are denoted
$\hat{V}_{\mathrm{gs}}$ and $\hat{V}_{\mathrm{ce}}$, respectively. The Auger
decay width $\hat{\Gamma}_{\mathrm{ce}}$ is inversely proportional to the
core-hole lifetime, i.e. $\hat{\Gamma}_{\mathrm{ce}}=1/\tau_{\mathrm{ce}}$,
and has been introduced phenomenologically to take into account the loss of
population due to the Auger decay of the excited species. The Auger decay
widths are taken as constant and are assumed to be identical to the atomic
ones for a similar core-excited state. Their values are taken from literature
and are summarized in Table 2. The Auger decay width appears naturally if the
Auger states are included in the ansatz (2) Cederbaum and Domcke (1981);
Oberli (2018).

Core-excited state |  $\hat{\Gamma}_{\mathrm{ce}}$ [meV] |  $\tau_{\mathrm{ce}}$ [fs]
---|---|---
$\mathrm{C1s^{-1}\pi^{*}}$ ($\mathrm{CO}$) | 108 | 6.09 Saito  _et al._ (2000)
$\mathrm{O1s^{-1}\pi^{*}}$ ($\mathrm{O_{2}}$) | 158 | 4.17 Saitoh  _et al._ (2001)
$\mathrm{N1s^{-1}\pi^{*}}$ ($\mathrm{N_{2}}$) | 114 | 5.77 Saitoh  _et al._ (2001)
Table 2: Auger decay widths and Auger lifetimes of the core-excited states
with a hole in the carbon, the nitrogen or the oxygen 1s orbital.

The Gaussian electric field is given by

$\displaystyle\vec{E}_{\mathrm{pump}}(t)=E_{0}\sin{(\omega_{0}(t-t_{0}))}e^{-\frac{(t-t_{0})^{2}}{2\sigma^{2}}}\vec{z},$
(4)

where the spatial dependence of the pulse envelope is neglected. $E_{0}$ is
the field strength, $\omega_{0}$ the central frequency and $t_{0}$ the
temporal center of the pulse. A linearly-polarized laser pulse along the
$\vec{z}$-axis is assumed. The pulse duration is given by
$t_{\mathrm{FWHM}}=2\sqrt{\ln{2}}\times\sigma$= 1.67 fs, corresponding to a
bandwidth of $\sim 1.1$ eV for a Fourier-transform-limited pulse. The
intensity of the pulse is $I=10^{14}~{}\mathrm{W/cm}^{2}$. The frequency
$\omega_{0}$ of the pump pulse is fixed at the resonance associated with the
largest Franck-Condon factor, i.e. $\omega_{0}=$ 292.5, 407.2 and 535.4 eV for
$\mathrm{C1s}\rightarrow\pi^{*}$ ($v^{\mathrm{gs}}_{0}\rightarrow
v^{\mathrm{ce}}_{0}$), $\mathrm{N1s}\rightarrow\pi^{*}$
($v^{\mathrm{gs}}_{0}\rightarrow v^{\mathrm{ce}}_{19}$) and
$\mathrm{O1s}\rightarrow\pi^{*}$ ($v^{\mathrm{gs}}_{0}\rightarrow
v^{\mathrm{ce}}_{0}$), respectively, where $v_{i}^{\mathrm{gs}}$ and
$v_{i}^{\mathrm{ce}}$ refer to the $i^{\mathrm{th}}$ vibrational level of the
ground and core-excited electronic states.

In our model, the time-dependent chemical shifts are determined by considering
the X-ray probe in first-order perturbation theory. In this approximation, we
calculate for each nuclear geometry the binding energy of the core electron
which is ejected in the core-excited state. The probability to get one
particular chemical shift $E_{i}$ is obtained from the nuclear distribution in
the core-excited state:

$\displaystyle\rho(E_{i},t)dE_{i}=\int dR|\chi_{\mathrm{ce}}(R,t)|^{2}$ (5)

with

$\displaystyle
E_{i}-\frac{dE_{i}}{2}<BE_{\mathrm{ce}}(R)<E_{i}+\frac{dE_{i}}{2}$

where $BE_{\mathrm{ce}}(R)=E_{\mathrm{ce/ci}}(R)-E_{\mathrm{ce}}(R)$ is given
by the difference in energy between the core-excited/core-ionized (ce/ci) and
the core-excited (ce) states. We select an energy resolution of $dE_{i}=0.054$
eV.

###  VI.2 Justification of the one-dimensional model for nuclear propagation

In this section, we assess the accuracy of our one-dimensional model to
describe formamide isomerization, where only the $y$-coordinate of the
hydrogen atom is considered as active in the dynamics. The geometries of the
formamide, the formamidic acid and the transition state in the ground
electronic state are optimized at the complete active space self-consistent
field CASSCF(12,10) level of theory using a 6-31g basis set with the Molcas
quantum chemistry package Karlström  _et al._ (2003).

Starting from the optimized geometry of the transition state, we perform an
internal reaction coordinate (IRC) calculation in full dimension at the CASSCF
level in Cs symmetry. The changes in energy and some molecular properties (CO
and CN bond distances and the angle between C, O and N atoms) along the IRC
coordinate are displayed in Figure 4. We see that the molecule is slightly
distorted during the hydrogen migration but remains practically planar. The
energy barrier and the difference in energy between the two isomers for the
full-dimensional case (1d-model) amount to 2.64 (1.45) eV and 1.15 (0.63) eV,
respectively. Although these energies are smaller in the 1d-model, their
relative ratio are maintained and the 1d-model grabs the relevant features,
describing then the main trends of the dynamics.

Figure 4: Properties of the molecule in its ground electronic state along the
internal reaction coordinate (IRC) in full dimension: a) Relative energy with
respect to the formamide species; b) CN internuclear distance
$R_{\mathrm{CN}}$; c) CO internuclear distance $R_{\mathrm{CO}}$; d) Angle
between the C, N and O atoms $\theta_{\mathrm{CNO}}$.

Therefore, to a first approximation one considers that the isomerization
reaction is mainly driven by the translational motion of the hydrogen between
the oxygen and nitrogen atoms (referred to as the $y$-coordinate in the text)
which does not differ qualitatively from the real dynamics, while the rest of
the molecule remains frozen Hirota  _et al._ (1974).

###  VI.3 Electronic structure calculations

Figure 5: Natural molecular orbitals included in the active space of the
ground state and core-excited states, optimized at the CASSCF level of theory.

The potential energy curves (PECs) are obtained from the optimized geometry of
the transition state (see Section VI.2). The PECs are calculated in Cs
symmetry using the correlation-consistent polarized valence quadruple-zeta
(cc-pVQZ) basis set of Dunning. The active space is prepared at the transition
state geometry. For the ground and the core-excited states, we perform a
CASSCF(12,10) calculation, where the active space contains 10 orbitals
occupied by 12 electrons. Except for the 1s orbital from which the electron is
excited, the 6 lowest molecular orbitals (MOs) are closed-shells. Figure 5
shows the natural MOs included in the active space, which is composed of 7 MOs
of $a^{\prime}$ symmetry (C1s or N1s or O1s, $\sigma_{\mathrm{CN}}$,
$\sigma_{\mathrm{CO}}$, $\sigma^{*}_{\mathrm{CO}}$,
$\sigma^{*}_{\mathrm{CN}}$, O lone pair, N lone pair + O lone pair + H1s) and
3 orbitals of $a^{\prime\prime}$ symmetry (two $\pi$ and one $\pi^{*}$). As
the reaction proceeds, the initial N-H $\sigma$ and $\sigma^{*}$ orbitals
together with the O lone pair will evolve to the O-H $\sigma$ and $\sigma^{*}$
while a lone pair is formed on the N. The delocalized $\pi$ network is
affected since the initial C-O double bond breaks to form a C-N double bond.
Concerning the core-excited/core-ionized states, we perform a CASSCF(13,11) to
additionally include in the active space the 1s orbital of the ionized
element.

The PECs of the ground electronic state is shown in the upper panel of Figure
2 (dashed line). The global minimum at $y=0.77$ Å corresponds to the formamide
species, while the local minimum at $y=-0.43$ Å is associated with the
formamidic compound. The formamidic acid is less stable by $\sim 0.63$ eV than
formamide. The transition state is located at $y=0.05$ Å. An energy barrier in
the order of 1.45 eV separates the formamide and formamidic species and
prevents the isomerization reaction to occur (see Table 1). There is another
local minimum at $-1.6$ Å where the hydrogen is at the equilibrium bond
distance from the oxygen. For $y>1$ Å the energy starts to increase due to the
steric interaction between the two hydrogen attached to the nitrogen atom,
while for $y<-2.5$ Å the energy decreases when the O-H bond dissociates.

The PECs of the core-excited states, where an electron from the 1s orbital of
either the C, the O or the N is promoted to the $\pi^{*}$ orbital are
displayed in the upper panel of Figure 2 (solid lines). Only singlet states
are considered in the calculations since their contributions dominate over the
triplet states. These PECs correspond to the lowest energetic state of a
state-average (SA) CASSCF calculation over 15 states. These close-by states
arise from shake-up processes, where core-excitation is accompanied by a
valence-to-valence excitation Lablanquie  _et al._ (2011). Non-adiabatic
couplings between the ground state of the core-excited state and the shake-up
states can be ignored since they play no role in the region where the dynamics
takes place. For the same reason, the contribution of shake-up states are not
important during the resonant excitation at the equilibrium geometry.

The PECs associated with core-excited/core-ionized doublet states, required
for the XPS spectrum calculations, are also obtained from a SA15-CASSCF
calculation.

The TDMs between the ground electronic state and the core-excited states are
calculated for each geometry where the dynamics will take place at the CASSCF
level.

##  VII Acknowledgments

S. Oberli and A. Picón acknowledge funding from Comunidad de Madrid through
TALENTO grant ref. 2017-T1/IND-5432. F. Martín and J. González-Vázquez
acknowledge the MINECO projects FIS2016-77889-R. F. Martín acknowledges the
Severo Ochoa Programme for Centres of Excellence in R&D (SEV-2016-0686) and
the María de Maeztu Programme for Units of Excellence in R&D (MDM-2014-0377).

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# Attitudes and Latent Class Choice Models using Machine Learning

Lorena Torres Lahoz1∗, Francisco Camara Pereira 1,
Georges Sfeir1, Ioanna Arkoudi1,
Mayara Moraes Monteiro1, Carlos Lima Azevedo1
1DTU Management, Technical University of Denmark
Building 116, Bygningstorvet, 2800 Kongens Lyngby
∗Corresponding author<EMAIL_ADDRESS>
## Abstract

Latent Class Choice Models (LCCM) are extensions of discrete choice models
(DCMs) that capture unobserved heterogeneity in the choice process by
segmenting the population based on the assumption of preference similarities.
We present a method of efficiently incorporating attitudinal indicators in the
specification of LCCM, by introducing Artificial Neural Networks (ANN) to
formulate latent variables constructs. This formulation overcomes structural
equations in its capability of exploring the relationship between the
attitudinal indicators and the decision choice, given the Machine Learning
(ML) flexibility and power in capturing unobserved and complex behavioural
features, such as attitudes and beliefs. All of this while still maintaining
the consistency of the theoretical assumptions presented in the Generalized
Random Utility model and the interpretability of the estimated parameters. We
test our proposed framework for estimating a Car-Sharing (CS) service
subscription choice with stated preference data from Copenhagen, Denmark. The
results show that our proposed approach provides a complete and realistic
segmentation, which helps design better policies.

### Keywords

Machine learning, Latent Class Choice Models, Car-Sharing, Psychometric
Indicators, Deep Learning.

##  1 Introduction

Latent Class Choice Models (LCCMs) are extensions of discrete choice models
(DCMs) that are typically implemented to capture unobserved heterogeneity in
the choice process by segmenting the population based on the assumption of
preference similarities. Traditionally, the latent classes are often defined
by the socio-economic characteristics of the decision-makers [1, 2, 3].
However, incorporating individuals’ attitudes and/or perceptions, typically
measured using psychometric indicators, into the specification of the latent
classes can offer additional behavioural insights, allowing for a more
realistic market segmentation that can help design more appropriate strategies
and effective policies [4, 5, 6].

In this study, we explore a new method of efficiently incorporating
attitudinal indicators in the specification of LCCM by relying on Machine
Learning (ML) techniques. All this while preserving the benefits of the
economic and behavioural interpretability of DCMs.

There are many examples in the literature of employing ML in DCMs, such as the
ones reviewed in section 2.3. However, there is a lack of effective use of ML
techniques for incorporating attitudinal variables into the model formulation.
Even though such variables have been included in traditional DCMs, a complex
interaction with the decision-making process should be expected [7]. We
hypothesise that ML could be a good starting point to explore it, given its
flexibility and power in capturing unobserved and complex interactions.
Therefore, we propose a model that includes latent classes and latent
variables, the latter being defined as an Artificial Neural Network (ANN).

Finally, the developed model follows the generalized random utility model [3]
theoretical structure, which provides us with a discrete representation of
heterogeneity, and, therefore, higher-level behavioural insights into the
composition of the latent classes. This has been shown to allow for more
accurate market segmentation and policies that account for the different
characteristics, preferences, and attitudes of individuals in each market
segment (see for example [8], [9]).

##  2 Literature Review

###  2.1 Latent Class Choice Models (LCCM)

Based on McFadden’s model [10], Walker and Ben-Akiva [3] presented a practical
generalized random utility model with extensions for latent variables and
classes. They extended basic Random Utility Model (RUM) to relax assumptions
and enrich the model’s capabilities. Latent classes refer to unobserved
population groups, in which each individual has an associated probability of
belonging to each group/class.
LCCM are a particular case of mixed logit models. Mixed logit models are
defined as logit models where the coefficients of the utility function vary
across decision-makers according to a given density $f(\beta)$ [11].
Therefore, unobserved heterogeneity is captured by assuming a parameter
distribution over the population, where each individual has a slightly
different behaviour toward the same attribute of the alternatives.
For the particular case of LCCM, the hypothesis is that there may be discrete
segments of decision-makers that are not immediately identifiable from the
data. Hence, the population can be probabilistically segmented into groups
with different preferences and decision protocols or that behave differently
towards a decision, signified by class-specific utility equations for each
class. This segmentation could lead to more practical results, such as more
efficient design policies [3].
Although mixed logit models are highly flexible and can approximate any random
utility model [12], they are more challenging to interpret and use in practice
as they do not identify the causes for the variations of taste among decision-
makers. On the other hand, evidence in the literature suggests that latent
class models are very convenient, flexible, and intuitive to account for taste
heterogeneity in discrete choice models [13].
A critical part of LCCMs is specifying the class membership model since it is
not defined beforehand. Traditionally, straightforward logit formulations are
employed since they allow for a closed-form expression of the probability to
belong to each class. Moreover, they typically base their segmentation on only
socio-demographic variables [1, 2, 3].
In the following sections, we will explore different class membership
formulations that include information about attitudinal variables. As we have
mentioned, incorporating individuals’ attitudes and/or perceptions into the
specification of the latent classes helps construct more realistic population
segments, allowing for more accurate and successful policies [13, 4].

###  2.2 Latent Psychosocial Constructs in DCM

Apart from the attributes of the alternatives and the characteristics of the
decision-maker, according to psychological theory, there exist more complex,
unobserved constructs that may have a relevant effect on how we make choices.
Some of these latent factors can be the decision maker’s lifestyle, attitudes,
or perceptions [14].
Psychometric indicators measure the effect of unobserved attributes on
individuals’ preferences on topics related to the choice. They are additional
information that can helps specify and estimate latent classes. Atasoy et al.
[15] estimated an LCCM where psychometric indicators are included in the
maximum likelihood estimation to improve the model’s accurancy. The
psychometric indicators were modelled as conditional on the latent class $k$.
Therefore, the item-response probability of observing indicator $I_{n}$ was
given by $P_{n}(I_{n}|k)$ that was defined as a parameter jointly estimated
with the choice and the class membership model. The model evidence that
includes the psychometric indicators allows for richer analysis and generates
significantly different estimates for the class membership model.
Another approach for including attitudinal variables in DCM models can be
found in work developed by Hurtubia et al. [13]. Psychometric indicators are
introduced by assuming that the probability of giving an agreement level to an
attitudinal question also depends on the respondent’s class. In this case,
they use an ordinal logit approach to model the response probability
$P_{n}(I_{k}|k)$, since the responses to the indicators consist of a few
ordered integer values corresponding with the level of agreement with the
statement on a Likert scale [16]. The advantage of this formulation is the
close form of the ordinal logit used to measure the indicators, which allows
for a more straightforward estimation procedure, where the choice and the
response to the indicators are estimated together.
Another study [17] has gone further by including latent variables and latent
classes, to model the relationship between normative beliefs, modality styles,
and travel behaviour. Normative beliefs refer to the individual’s perception
of the opinion of others concerning a specific behaviour, while modality style
describes the part of an individual’s lifestyle characterized by using a
particular set of travel modes. The study shows evidence of associations
between travel behaviour and different latent psychosocial constructs. Their
model exemplifies how the socio-psychological approach, which originated in
social psychology, can extend conventional travel behaviour analysis. However,
due to the model’s complexity, sequencial estimation procedure is employed,
encountering identification issues for some coefficients.
Finally, Alonso-González et al.[18] presented an exploratory factor analysis
and latent class cluster analysis of attitudinal variables performed to divide
individuals into groups concerning their inclination to adopt MaaS (Mobility
as a Service). Even though the model is enriched with a series of covariates
referring to socioeconomic, mobility, and technology-related characteristics,
they do not improve the model; they only help cluster identification.
Attitudinal variables are the ones defining the structure of the latent
classes.
The previous examples have highlighted the importance and relevance of
including attitudes, preferences, and beliefs in DCMs, especially in LCCM.
They have proved to be relevant for improving the model estimation and
creating more realistic and meaningful classes.

###  2.3 Machine Learning for Taste Heterogeneity in DCM

In recent years, the use of ML techniques has increased, mainly due to their
power to improve prediction accuracy. However, one of the main criticisms of
ML techniques vs. econometric models is that it provides results that are less
interpretable. Thus, in the transportation field, researchers have also
focused on providing interpretable and meaningful estimates from ML
applications, so that they can be more useful for travel analysis and policy
decisions.
Some interesting works are, for example, Sifringer et al. [19], which, based
on previous works [20],[21],[22], included an additional term in the utility
layer called the Learning Term, which was estimated separately with a Dense
Neural Network, using a priori disregard explanatory variables. This model
outperformed previous formulations while keeping the interpretability of the
other terms of the utility. Following this idea and based on Pereira [24]
previous work, Arkoudi et al [23] proposed an embedding encoding for the
socio-characteristic variables that provided a latent representation of these
variables in concordance with individuals’ choices. Moreover, they included a
Learning Term that increased the model’s accuracy. However, this embedding
formulation does not consider information about attitudinal indicators since
these variables cannot be directly included in the utility specifications.
Furthermore, none of these works provided latent classes or variables
formulations.
The work developed by Han et al [25], proposed a nonlinear LCCM by using a
neural network to specify the class membership model. In their application, a
model with eight latent classes outperformed the previous ones in prediction
accuracy. Due to non-linearity, the proposed LCCM was less transparent and
lost interpretability. The work of Sfeir et al. presented two model
formulations for the construct of latent class choice models using Gaussian
process [26] and Mixture models [27], while Wong et al. [28] employed a
generative non-parametric ML approach, a Boltzmann machine, to estimate latent
variables. Although these works allowed for more flexibility in the definition
of these latent constructs, none of them relied on attitudinal indicators’
information.
To summarize, the theory of LCCM has been widely explored and successfully
employed for many decades, and more recently, ML techniques have provided
practical solutions for situations where the traditional discrete choice
models were more constrained, improving their estimation and flexibility.
However, the employment of ML for incorporating attitudinal information in DCM
is still in the earliest steps of research, given its theoretical and
practical underlying complexity.

##  3 Methodology

In this section, we present our model formulation following the generalized
random utility model structure (Figure 1) presented by Walker et al., 2002 [3]
and the added modifications for the incorporation of latent variables through
the employment of ANN in the latent class model.
In Figure 1, solid arrows represent structural equations (cause-effect
relationships), and dashed arrows represent measurement equations
(relationship between observable indicators and the underlying latent
variable).

Figure 1: Generalized random utility model [3]

###  3.1 Latent Class Choice Models (LCCM)

LCCMs are composed of two sub-models: a class membership model and a class-
specific choice model. The former computes the probability of an individual
$n$ belonging to a certain class, while the latter assigns the probability of
choosing each alternative, given that individual $n$ belongs to a certain
class $k$.
Typically, in the class membership model is define as logit, where the utility
of an individual $n$ belonging to class $k$ can be expressed as follows:

$U_{nk}=ASC_{k}+Q_{n}\gamma_{k}+\upsilon_{nk}$ (1)

where $ASC_{K}$ is the alternative-specific value for each class $k$, $Q_{n}$
is a vector containing the characteristics of individual n, $\gamma_{k}$ is a
vector of the unknown parameters that need to be estimated and $\upsilon_{nk}$
is an error term that is assumed to be independently and identically
distributed Extreme Value Type I over individuals and classes. Thus, the
probability that individual $n$ belongs to class $k$ given their
characteristics is the logit formula:

$P(q_{nk}|Q_{n},\gamma_{k})=\frac{e^{Q_{n}\gamma_{k}}}{\sum^{K}_{k^{\prime}=1}e^{Q_{n}\gamma_{k^{\prime}}}}$
(2)

where $q_{nk}=1$ if the individual belongs to class $k$ and 0 otherwise.
As for the class-specific choice model, it returns the probability that an
individual $n$ chooses a specific alternative, given that he/she belongs to a
certain class $k$.
The utility of individual n choosing alternative i, given that he/she belongs
to class k is formulated as:

$U_{ni|k}=V_{ni|k}+\epsilon_{ni|k}=X_{ni}\beta_{k}+\epsilon_{ni|k}$ (3)

where $V_{nit|k}$ is the observed part of the utility, and $\epsilon_{nit|k}$
is an independent and normally distributed Extreme Value Type I error term
distributed over individuals, alternatives and classes. $X_{ni}$ is the vector
of observed attributes of alternative i and normally includes an alternative-
specific constant, and $\beta_{k}$ is the vector of unknown parameters that
need to be estimated.
Therefore, the probability that individual $n$ chooses alternative $i$,
conditioned on class $k$, is formulated as:

$P(y_{ni}|X_{ni},q_{nk},\beta_{k})=\frac{e^{V_{ni|k}}}{\sum^{J}_{j^{\prime}=1}e^{V_{nj^{\prime}|k}}}$
(4)

where $y_{ni}$ is 1 if individual $n$ chooses alternative i and 0 otherwise,
and $J$ is the total number of alternatives.
Conditional on the class, the probability of observing $y_{n}$ is formulated
as:

$P(y_{n}|X_{n},q_{nk},\beta_{k})=\prod^{J}_{j=1}(P(y_{nj}|X_{nj},q_{nk},\beta_{k}))^{y_{nj}}$
(5)

where $y_{n}$ is a $J$ vector of all choices $y_{nj}$ of an individual $n$ and
$X_{n}$ is a $J$ vector of $X_{nj}$.
The unconditional probability of observing $y_{n}$ can be calculated by mixing
the previous conditional choice probability with the probability of belonging
to each class $k$, as follows:

$P(y_{n})=\sum^{K}_{k=1}P(q_{nk}|Q_{n},\gamma_{k})P(y_{n}|X_{n},q_{nk},\beta_{k})$
(6)

Finally, the likelihood over all individuals $N$, considering independence
between them, is:

$P(y)=\prod^{N}_{n=1}\sum^{K}_{k=1}P(q_{nk}|Q_{n},\gamma_{k})P(y_{n}|X_{n},q_{nk},\beta_{k})$
(7)

In section 5.1, we employ this LCCM formulation as a baseline for our new
model.

###  3.2 Latent Variables and Latent Classes with ANN for Incorporation of
Attitudinal Indicators

The straightforward solution for incorporating the attitudinal indicators in
DCMs would include their responses directly into the utility of the latent
classes. However, there are three main reasons to avoid this approach.
Firstly, this type of data is not always available for forecasting since these
types of surveys are usually expensive and time-consuming. Therefore, the
model can predict the class membership model probability even when no
information about attitudinal indicators is provided. The second reason is
that the multicollinearity inherent in the responses makes it difficult to
estimate the unknown parameters correctly and can lead to inconsistent
estimates. In this sense, even a low-dimensional representation of the
indicators, e.g., factor analysis, cannot be included directly in the utility
unless the correlated error terms are added. Lastly, and more importantly,
there is no apparent justification that such attitudinal information can
uncover some causal relationship directly with behaviour. Instead, the
literature points out that these indicators seem to be the consequences of
some latent constructs, which combined can represent individuals’ attitudes
[29].
We follow the generalized random utility model structure presented by [3] to
address the previously mentioned issues. Moreover, we include the information
on the attitudinal indicators by employing an ANN to formulate the latent
variables constructs, which provides more flexibility in how the latent
variables are formulated. The graphical representation of the proposed
formulation is shown in Figure 2.
Finally, we decided not to include the arrow between the latent variables and
the utilities, and the revealed preference indicators, since we did not find
it relevant to our concrete application. However, these relations can be
included for other cases by following the same principle as in Figure 1 [3].

Figure 2: Graphical representation of the model formulation

####  3.2.1 Latent classes with latent variables formulation

The utility of the class membership model can be written as:

$\begin{split}U_{nk}=V_{nk}+\upsilon_{nk}\end{split}$ (8)

where $V_{nk}$ is the representative utility of individual $n$ belonging to
class $k$ and, as defined in the traditional LCCM, $\upsilon_{nk}$ is the
error term that is assumed to be independent and identically distributed
Extreme Value Type I over individuals and classes ans defined in 1.
In this case, we define $V_{nk}$ as:

$\begin{split}V_{nk}=ASC_{k}+Q_{n}\gamma_{k}+r_{n}\delta_{k}+\omega_{n}b_{k}\end{split}$
(9)

where $ASC_{k}$ is the alternative-specific value for class $k$, $Q_{n}$ is
the vector containing socio-characteristics of individual n, and $\gamma_{k}$,
the vector of unknown parameters that need to be estimated for each class $k$.
In addition, $r_{n}$ is the vector of latent variables for individual $n$ and
$\delta_{k}$ the corresponding vector of unknown parameters specific to class
$k$, that also need to be estimated. Finally, $\omega_{n}$ is an individual-
specific constant with its corresponding coefficient $b_{k}$ for each class
$k$.
The term $\omega_{n}$ represents the individual variation of all the latent
variables caused by the variance of their underlying distributions. It is
formulated as a one-layer ANN that gets activated by the ID of each decision
maker in the train set ($Id_{n}$ is 1 for decision-maker $n$ and 0 otherwise)

$\omega_{n}=\sum_{1}^{N}w_{1n}^{(1)}Id_{n}$ (10)

where $w_{1n}^{(1)}$ are the weights of the layer. Since only the difference
between classes matters, we introduce the corresponding $b_{k}$ term for each
class, which accounts for the effect of the latent variables variation for
each of the latent classes.
Given the distribution of the error term ($\upsilon_{nk}$), the probability
$P(q_{nk}|Q_{n},\gamma_{k},r_{n},b_{k})$ can be expressed as:

$P(q_{nk}|Q_{n},\gamma_{k},r_{n},b_{k})=\frac{e^{V_{nk}}}{\sum^{K}_{k^{\prime}=1}e^{V_{nk^{\prime}}}}$
(11)

As it is standard practice in class membership models, the utility of one
class is set to zero due to the parameters’ identification since only the
differences in utility matter.

####  3.2.2 Latent variables formulation as ANN

The novelty of this work is the employment of ANN for the construction of
latent variables. We propose a non-linear relationship between the socio-
characteristics of the individuals and the latent constructs by employing two
densely connected layers. It can be expressed as follows:

$r_{zn}=a_{2}(\sum_{h=0}^{H}w_{zh}^{(2)}a_{1}(\sum_{m=0}^{M}w_{hm}^{(1)}Q_{mn}))$
(12)

where $M$ is the number of socio-demographic variables used to predict the
answer to the indicators, and $H$ is the number of hidden units in the hidden
layer. $w_{hm}^{(1)}$ are the weights of the first layer, and $a_{1}$
represents the first activation function defined as a Rectified Linear Unit
(ReLU) ($a_{1}(x)=max(0,x)$); for the second layer, a linear activation
function is applied $a_{2}(x)=x$, and the weights are represented by
$w_{zh}^{(2)}$. By adding an extra input $Q_{0n}$, which is set to one and
extending the sum to go from zero, we avoid writing the intercept term.
The number of latent variables, defined as $Z$, the number of hidden neurons
in the hidden layer, $H$, and the number of densely connected layers are
dimensions that the researcher can tune since they are not observed in the
data. Employing different values for these hyperparameters for a different
model application does not invalidate the proposed formulation and can help
achieve higher model prediction for the test data.
The formulation presented is based on the hypothesis that the socio-
characteristics of the individuals define the latent variables. Moreover,
these latent constructs influence the response to specific attitudinal
indicators.

####  3.2.3 The ordinal measurement model

We focus on the case where indicators take the form of questions that receive
an ordered response, as in Likert scales [16]. Each option of the scale
represents a level of agreement with a particular statement. Thus, we define
the utility of individual $n$ for indicator $p$, as a measurement of the level
of agreement with its statement, and we formulated it as:

$U_{pn}=V_{pn}+\nu_{pn}=r_{n}\alpha_{p}+c_{p}\omega_{n}+\nu_{pn}$ (13)

where $V_{pn}$ is the representative utility of individual $n$ to indicator
$p$ and $\nu_{pn}$ is the error term that is assumed to be independently and
identically distributed Extreme Value Type I over individuals and indicators.
$r_{n}$ is a vector of length $Z$ containing the latent variables of
individual $n$, $\alpha_{p}$ is also a vector of size $Z$ with its
corresponding parameters to be estimated. $\omega_{n}$ is the individual-
specific parameter estimated together with the latent class model, and $c_{p}$
is its corresponding coefficient for each indicator $p$ .
Therefore, the probability that individual $n$ answers with a certain level of
agreement $l$ to indicator $p$ is expressed as:

$P(I_{pln}=1|r_{n},\alpha_{p},c_{p},\omega_{n})=P(\tau^{p}_{l-1}<U_{pn}<\tau^{p}_{l})$
(14)

where we define $I_{pln}$ as 1 if individual $n$ answers with a level of
agreement $l$ to indicator $p$ and 0 otherwise. $\tau^{p}_{l}$ are strictly
increasing class-specific thresholds that define an ordinal relation between
the utility $U_{pn}$ and the level of agreement to indicator $p$.
Finally, the probability of individual $n$ providing an answer $l$ to
indicator $p$ can be computed as an ordinal softmax:

$\begin{split}P(I_{pln}=1|r_{n},\alpha_{p},c_{p},\omega_{n})=P(\tau^{p}_{l-1}<U_{pn}<\tau^{p}_{l})=&P(\tau^{p}_{l-1}<V_{pn}+\nu_{vp}<\tau^{p}_{l})=\\\
=Prob(\nu_{vp}<\tau^{p}_{l}-V_{pn})-P(\nu_{vp}<\tau^{p}_{l-1}-V_{pn})=&=\frac{e^{\tau^{p}_{l}-V_{pn}}}{1+e^{\tau^{p}_{l}-V_{pn}}}-\frac{e^{\tau^{p}_{l-1}-V_{pn}}}{1+e^{\tau^{p}_{l-1}-V_{pn}}}\end{split}$
(15)

where one threshold per indicator is set to zero, as only the difference
between them matters.

The final model formulation is presented in Figure 2. Maximizing the log-
likelihood with respect to the unknown parameters will not lead to a closed-
form solution. Thus, the estimation becomes more complex and challenging as
empirical singularity issues could appear at some iterations [30]. To overcome
these problems, we employ the Expectation-Maximization (EM) algorithm
(Dempster et al., 1977 [31]), a powerful method used for maximizing the
likelihood in models with latent variables.

Figure 3: Model architecture222The expressions without simplifying for
$P(k=1)$ and $P(k=K)$ can be found in equation (11), for $P(I_{pnl}=1)$ and
$P(I_{pnl}=L)$ in equation (15) and for $P(y_{nj}=1|k=K)$ and similars in
equation (5).

####  3.2.4 EM algorithm

We estimate all components of the proposed model together since simultaneous
estimation usually leads to more efficient estimates than sequential
estimation [32], [14].
For this purpose, we employ the EM algorithm, which combines an expectation
step with a maximization one until convergence is reached. We start by
randomly initializing the unknown parameters. After that, we estimate the
expected values of the latent variables (E-step) using Bayes’ theorem. Then,
we update the unknown parameters’ values using log-likelihood maximisation
(M-step). Finally, we evaluate the log-likelihood with the updated values of
the unknown parameters and check for convergence. We return to the E-step
until convergence is reached. In this case, the convergence is defined by the
number of iterations.
We can rewrite the joint likelihood assuming that the clusters are observed as
follows:

$\begin{split}P(y,q,Q,X)=&\prod^{N}_{n=1}\prod_{k=1}^{K}\left[\frac{e^{V_{nk}}}{\sum^{K}_{k^{\prime}=1}e^{V_{nk^{\prime}}}}\right]^{q_{nk}}\\\
&X\prod^{N}_{n=1}\prod^{K}_{k=1}\prod^{J}_{j=1}\left[\frac{e^{V_{nj|k}}}{\sum^{J}_{j^{\prime}=1}e^{V_{nj^{\prime}|k}}}\right]^{y_{nj}q_{nk}}\end{split}$
(16)

The function is divided into two parts when we take the logarithm of the
likelihood:

$\begin{split}LL=\sum^{N}_{n=1}\sum_{k=1}^{K}q_{nk}log\left[\frac{e^{V_{nk}}}{\sum^{K}_{k^{\prime}=1}e^{V_{nk^{\prime}}}}\right]+&\sum^{N}_{n=1}\sum^{K}_{k=1}\sum^{J}_{j=1}y_{nj}q_{nk}log\left[\frac{e^{V_{nt|k}}}{\sum^{J}_{j^{\prime}=1}e^{V_{n^{\prime}t|k}}}\right]\end{split}$
(17)

this equation can be solved by equating its derivatives to zero with respect
to each of the unknown parameters of the choice model if $q_{nk}$ is assumed
to be known.
In order to find the initial values of $q_{nk}$, we estimate the expectation
of $q_{nk}$ (E-step) using Bayes’s theorem:

$\begin{split}E[q_{nk}]=\gamma_{qnk}=\frac{\frac{e^{V_{k}}}{\sum^{K}_{k^{\prime}=1}e^{V_{k^{\prime}}}}\prod^{J}_{j=1}\left[\frac{e^{V_{nj|k}}}{\sum^{J}_{j^{\prime}=1}e^{V_{nj^{\prime}|k}}}\right]^{y_{nj}}}{\sum_{k^{\prime}=1}^{K}\left[\frac{e^{V_{k}}}{\sum^{K}_{k^{\prime}=1}e^{V_{k^{\prime}}}}\prod^{J}_{j=1}\left[\frac{e^{V_{nj|k}}}{\sum^{J}_{j^{\prime}=1}e^{V_{nj^{\prime}|k}}}\right]^{y_{nj}}\right]}\end{split}$
(18)

where $\gamma_{qnk}$ is considered as the posterior probability of the
classes.
Combining equations (17) and (18), we can formulate the expected value of the
log-likelihood as:

$E(LL)=\sum^{N}_{n=1}\sum^{K}_{k=1}\sum^{J}_{j=1}y_{nj}\gamma_{qnk}log\left[\frac{e^{V_{nj|k}}}{\sum^{J}_{j^{\prime}=1}e^{V_{nj^{\prime}|k}}}\right]++\sum^{N}_{n=1}\sum_{k=1}^{K}\gamma_{qnk}log\left[\frac{e^{V_{k}}}{\sum^{K}_{k^{\prime}=1}e^{V_{k^{\prime}}}}\right]$
(19)

By setting the derivatives of the expected log-likelihood to zero with respect
to the unknown parameters $\beta_{k}$ of the choice model, we can estimate
them with the following equation:

$\beta_{k}=argmax_{\beta_{k}}\sum_{n=1}^{N}\sum_{j=1}^{J}y_{nj}\gamma_{q_{nk}}log\left[\frac{e^{V_{nj|k}}}{\sum_{j^{\prime}=1}^{J}e^{V_{nj^{\prime}|k}}}\right]$
(20)

where no closed-form solution can be obtained. Thus, we employ the gradient-
based numerical optimization method BFGS [33].
For the unknown parameters of the class membership, back-propagation with
cross-entropy loss is applied since the parameters are encoded inside a neural
network. As presented in [19], minimizing the cross entropy loss is equivalent
to maximizing the log-likelihood function, which allows us to compute the
Hessian matrix and, therefore, provides useful post-estimation indicators such
as the standard deviation or the confidence intervals of the parameters of the
model.
After convergence is reached, we can calculate the probability of observing a
vector of choices $y$ for all individuals as follows:

$P(y)=\prod_{n=1}^{N}\sum_{k=1}^{K}P(q_{nk}|Q_{n},\gamma_{k},r_{n},b_{k})\prod_{j=1}^{J}\left(P(y_{nj}|X_{nj},q_{nk},\beta_{k})\right)^{y_{nj}}$
(21)

To benchmark our proposed model we use the formulation presented in [3] as
baseline. One can represent it in graphical representation, similarly as in 2,
with the difference that the relation between the indicators’ utility and the
decision-makers’ socio-demographic characteristics is linear.

##  4 Data

The data used in this study comes from a survey carried out for the Share-More
project 333https://eitum-sharemore.net.technion.ac.il/, which aimed to analyse
individual preferences towards car-sharing (CS) services under different
personalized incentive schemes.
The data was collected through a tailor-made online survey (available in both
web and mobile versions), from July $16^{th}$ to August $6^{th}$ 2020, in
Copenhagen (CPH). Respondents were recruited through panels, and the
eligibility criteria were being more than 18 years old and having a valid
driver’s license. The survey yielded a total of 542 complete answers and was
available in English and Danish. The average duration was around 15 minutes.
It included five relevant parts:

  1. 1.

A brief introduction to the project and its objective.

  2. 2.

A survey on the subject’s socio-demographic characteristics ($Q_{n}$),
including experience with CS.

  3. 3.

A survey addressing questions regarding the respondents’ attitudes toward
private and CS. The respondents’ perceptions in regard to car ownership and CS
usage were measured by their level of agreement with relevant statements using
a 5-point Likert scale. The statements employed are presented in Table 1.

  4. 4.

A survey with questions related to preferences about CS incentives.

  5. 5.

A Stated Preference (SP) experiment with different options for CS plans,
taking into account possible incentive schemes for CPH.

Indicator | Description
---|---
$I_{1}$ | I feel stressed when driving
$I_{2}$ | It is difficult to find parking
$I_{3}$ | Owning and using a car is a big expense
$I_{4}$ | It’s easy for me to conduct my daily trips without a private car
$I_{5}$ | Driving a car is the most convenient way to move around
$I_{6}$ | For me the car is a status symbol
$I_{7}$ | I am worried about the environmental footprints of my car
$I_{8}$ | Car sharing is more affordable than owning my own car
$I_{9}$ | Car sharing is a more environmentally friendly alternative to car ownership
$I_{10}$ | By using car-sharing, I do not have to deal with vehicle maintenance and repair
$I_{11}$ |  | Thanks to car sharing, I can save from the fuel, taxes, insurance and parking
---
expenses associated with private vehicle ownership
$I_{12}$ | Having access to different types of vehicles is an important advantage
$I_{13}$ | Using car-sharing service can lower my transport expenses
$I_{14}$ | Using car-sharing whenever I need it can make my life easier
$I_{15}$ | I would not need to buy a car because I have car-sharing
$I_{16}$ | Car-sharing is more convenient than public transport
$I_{17}$ | I wouldn’t mind sharing my personal car with other people
Table 1: Attitudinal indicators employed in the survey

The sample socio-characteristics ($Q_{n}$) are described in Table 2. More than
90% of respondents stated to be aware of CS services, which indicates that
they are well-known. The sample is quite balanced regarding gender and
proportionally representative of the population in terms of age. Most of the
respondents live in the city centre and are employed. For the level of
education, more than 60% of the population have at least a bachelor’s degree.
Moreover, most respondents live in households of 1 or 2 members, up to one car
and have two or more bikes at home. Given the income level, most respondents
earn around average (350.000 kr./year) or above, which is likely related to
the high level of education of the sample.

| Total | %
---|---|---
Car -sharing membership status | Car-sharing member | 96 | 17.68
Past-car sharing member | 64 | 11.79
Non-car-sharing member | 383 | 70.53
Car-sharing awareness | Yes | 490 | 90.24
No | 53 | 9.76
Gender | Man | 267 | 49.17
Woman | 275 | 50.64
Prefer not to say | 1 | 0.18
Age | 18-30 | 146 | 26.89
31-40 | 88 | 16.21
41-50 | 97 | 17.86
51-60 | 88 | 16.21
More than 60 | 124 | 22.84
Place of residence | City center | 235 | 43.28
Suburbs | 190 | 34.99
Another city in the metropolitan region | 71 | 13.08
Outside the metropolitan region | 47 | 8.66
Employment status | Student | 74 | 13.63
Employed | 354 | 65.19
Unemployed | 12 | 2.21
On leave | 7 | 1.29
Retired | 100 | 18.42
Other | 8 | 1.47
Level of education | Less than high school | 39 | 7.18
High school diploma or equivalent | 150 | 27.62
Bachelor’s degree | 169 | 31.12
Master’s degree | 134 | 24.68
Doctoral degree | 8 | 1.47
Other | 17 | 3.13
Did not answer | 26 | 4.79
Size of the household | 1 | 152 | 27.99
2 | 223 | 41.07
3 | 80 | 14.73
4 | 68 | 12.52
>4 | 20 | 3.68
Number of cars in the household | 0 | 139 | 25.60
1 | 304 | 55.99
2 | 91 | 16.76
>2 | 9 | 1.66
Number of bicycles in the household | 0 | 40 | 7.37
1 | 128 | 23.57
2 | 156 | 28.73
>2 | 219 | 40.33
Income | Low (Up to 250.000 kr.) | 82 | 15.1
Medium (251-500.000 kr.) | 140 | 25.8
High (Over 500.000 kr.) | 221 | 40.7
Did not answer | 100 | 18.4
Table 2: Copenhagen Sample characteristics

For further details on the survey and the CPH data collection and sampling
processes, the reader is referred to [34, 35].
Our analysis focuses on the data from parts 2, 4 and 5 of the questionnaire,
using the latter for modelling the choice of CS subscription plans. In this
SP, each respondent answered 3 experiment tasks which presented 4 CS
alternative plans with 8 associated attributes $X_{n}$, (one-time subscription
cost, usage cost, walking time to access the vehicle, probability to get a
shared vehicle, CS vehicle type, CS vehicle engine type, walking time from
parking location to destination and extra features) and the option to select
’None of the alternatives.’ Figure 4 shows an example of a choice task
presented to the respondents. To minimize response bias, the order of
appearance of the attributes was random for each individual but was the same
for the three tasks presented to the same individual.

Figure 4: Example of choice task presented to respondents

As the name suggests, in one-way CS (station-based), or OWSB, the cars can be
found in fixed locations (stations) around the city. The user can take a
vehicle from one of the fixed locations available and return it to the same
place or another fixed location available in the service. In one-way CS (free-
floating), or OWFF, the users can pick up or return the car in any available
parking spot in the city within a delimited coverage area of the service
(thus, no fixed stations), giving extra freedom to users. In roundtrip CS
(RT), one can pick up a car at fixed stations scattered around the city, and
need to drop it off at the same location. Finally, peer-to-peer (P2P) CS
services are those which make it possible for vehicle owners to rent out their
cars to other users for a defined time period, also relying on fixed (and
identical) pick-up and drop-off locations for a given ride.
The attributes of the alternatives included in the tasks and their
corresponding levels were selected by considering the published literature and
qualitative surveys previously conducted during the Share-More project [34,
35]. The cost of usage was presented in cost per minute, hour or day to test
the influence of these units on respondents’ choices.

##  5 Results

###  5.1 Baseline Model Results

To benchmark the proposed model, we tried to use the formulation presented in
the generalized random utility model [3] as a baseline. One can represent it
in graphical representation, similarly as in Figure 2, with the difference
that the relation between the indicators’ utility and the decision-makers
socio-demographic characteristics is linear instead of a densely connected
layer. However, the class membership formulation’s simplicity made the model
unable to converge for our data, making the Hessian matrix for the variance
computation not invertible.
To have a benchmark model with comparable magnitude for the likelihood, we
have estimated the model with a traditional LCCM as a baseline without
including the attitudinal variables. We have employed 20% of the data for
testing, dividing the data into train and test with 433 and 109 individuals,
respectively. The results are presented in Table 2:

Model | Nº Classes | Nº parameters | Null LL | LL | AIC | BIC | R-squared | Test null LL | Test LL
---|---|---|---|---|---|---|---|---|---
LCCM | 2 | 30 | -2047.21 | -1599.41 | 3258.81 | 3413 | 0.22 | -515.02 | -400.52
LCCM | 3 | 43 | -2047.21 | -1568.14 | 3234.29 | 3487 | 0.23 | -515.02 | -398.73

Table 3: LCCM results without attitudinal variables

For the models above, we have used the same socio-characteristics for
constructing the class membership as in our proposed formulation. More
specifically, we employed age, binary variables that indicate if the decision-
makers have a bike or car at home, their CS membership status, kids at home,
or if they are students or retired.
However, we found that having a car at home, being retired or being a student
is not statistically significant under this formulation. When applying the
proposed formulation, we can effectively include this information in the
model, which improve our characterization of the latent space.
Given the probability of each individual in the sample and its corresponding
socio-characteristics, we have represented the classes in Figure 5, by using
the Bayes Theorem to compute:

$P_{n}(socio-characteristic|K=k)$ (22)

where $k$ is the corresponding class number.

Figure 5: Representation of the class membership of the LCCM model

Figure 5 is compared in the next subsection with the proposed model results.

###  5.2 Proposed Model Results

We have explored our proposed formulation for different numbers of latent
classes and latent variables. We have employed the same train/test split as
for the baseline model. It is important to note that our EM process has been
replicated five times with random initializations. We have computed the
likelihood variance between the different obtained model estimates to check
for stability. The results are summarized in the following table:

Nº Classes |  | Nº latent
---
variables
| EM
---
iterations
Null LL | LL |  | Variance
---
LL
R-squared |  | Test null
---
LL
Test LL |  | Variance
---
Test LL
2 | 2 | 15 | -2047.21 | -1575.32 | 14.62 | 0.23 | -515.02 | -404.72 | 0.28
3 | 2 | 30 | -2047.21 | -1539.56 | 22.01 | 0.25 | -515.02 | 402.72 | 9.8
3 | 3 | 25 | -2047.21 | -1531.41 | 26.36 | 0.25 | -515.02 | -400.73 | 9.36

Table 4: Model results

The model with three latent classes and two latent variables has given the
best overall estimation. Even though the model with three classes and three
latent classes has a slightly better fit, when another latent variable is
included, its corresponding estimates parameters were not statistically
significant, which made us reject this model formulation.
Comparing the results from Tables 3 and 4, we observe an increase in the
training likelihood for our formulation. On the other hand, we do not provide
better results for the test data, but just comparable ones. This could be due
to the small size of the test sample or to the fact that we don’t have access
to attitudinal information or $w_{n}$ values in the test stage, which can
affect the prediction accuracy.
We now analyse the parameters of the best model, where all the data has been
used for the estimation.

Variable | Class specific choice model
---|---
Class 1 | Class 2 | Class 3
$ASC_{CS\ free-floating}$ | 3.50(0.46) | -2.71(1.34) | -1.68(0.89)
$ASC_{CS\ station-based}$ | 3.04(0.48) | -2.64(1.34) | 0.07(0.80)
$ASC_{CS\ peer\ to\ peer}$ | 4.12(0.52) | 0.47(1.45) | 1.50(0.83)
$ASC_{roundtrip}$ | 2.97(0.48) | -3.50(0.40) | 0.15(0.80)
$\beta_{One\ time\ subscription\ cost}$ | -1.00(0.17) | -1.12(0.51) | -0.35(0.22)
$\beta_{Usage\ cost(OWFF,OWST,RT)}$ | 0.05(0.04) | -0.16(0.09) | -0.34(0.08)
$\beta_{Usage\ cost(P2P)}$ | -1.30(0.39) | -5.57(1.31) | -3.86(0.80)
$\beta_{Usage\ cost\ per\ day}$ | -0.35(0.24) | -2.40(0.74) | -1.09(0.39)
$\beta_{Usage\ cost\ per\ hour}$ | -0.10(0.21) | -0.99(0.50) | -0.43(0.34)
$\beta_{Only\ combustion\ cars}$ | -0.23(0.12) | 0.09(0.33) | -0.66(0.20)
$\beta_{Probability\ of\ finding\ a\ shared\ car}$ | 0.13(0.44) | 1.38(1.35) | 2.70(0.75)
$\beta_{Walking\ time\ from\ parking\ to\ destination}$ | -0.05(0.01) | 0.02(0.04) | 0.03(0.02)
Table 5: Estimate and standard deviation of the parameters of the class-specific choice model Variable | Parameter | St error | P-value
---|---|---|---
$ASC_{class_{1}}$ | 0.97 | 0.45 | 0.031
$ASC_{class_{2}}$ | -1.40 | 0.51 | 0.0057
$\gamma_{kidsathome,class_{1}}$ | 0.56 | 0.29 | 0.050
$\gamma_{kidsathome,class_{2}}$ | -0.45 | 0.37 | 0.22
$\delta_{r_{1},class_{1}}$ | 0.18 | 0.13 | 0.17
$\delta_{r_{1},class_{2}}$ | -0.48 | 0.14 | 0.0009
$\delta_{r_{2},class_{1}}$ | 0.12 | 0.10 | 0.27
$\delta_{r_{2},class_{2}}$ | -0.44 | 0.11 | 0.00
$b_{class_{1}}$ | 0.61 | 0.44 | 0.17
$b_{class_{2}}$ | -4.014 | 0.54 | 0.00
Table 6: Parameters of the class membership model

Table 5 shows the estimated parameters of the class-specific choice model with
its corresponding standard deviation. The utility for not choosing any of the
CS services offered is set to zero for parameter identification.
Based on the values and signs of the estimated beta parameters, we observe
that class 1 and class 2 are more negatively affected by the subscription
cost, while class 3 is less influenced by this cost, but more negatively
affected by the cost usage. In general, P2P presents a higher alternative-
specific value than the other alternatives for all the classes. However, this
is compensated by the $\beta_{Usage\ cost(P2P)}$ values being also more
damaging in the three classes. Class 2 is the one less inclined about the P2P
CS type since it has the lowest ASC-P2P and the more negative $\beta_{Usage\
cost(P2P)}$ estimator. On the other hand, individuals with a high probability
of belonging to class 3 are the most concerned if the type of engine is just
combustion. Thus, CS could possibly be seen as an electric alternative to
those who are more worried about the environmental footprint of their trips.
Given the beta values for displaying the cost in hours ($\beta_{Usage\ cost\
per\ hour}$) or days ($\beta_{Usage\ cost\ per\ day}$), Table 5 also shows a
bias towards displaying the price per minute (baseline) instead of per day or
hour across all the classes, related to the fact that CS users tend to drive
it for short time periods. Regarding the probability of finding a car, it is a
more important feature for classes 2 and 3, which make them more dependent on
the availability of the service.
Overall, class 2 seems to be less prone to use any CS, given all its estimated
parameters.

The parameters of the class membership model are summarised in Table 6. Given
the probability of each individual in the sample and its corresponding socio-
characteristics, we have represented the classes in Figure 6. Individuals with
a higher probability of belonging to class 1 have around 20% probability of
being a CS member, a bit above the sample average (17.5%). They also tend to
have more kids at home, as well as bikes than the other classes. Studies like
[37] have shown that when there are significant life changes, such as the
birth of a child, people become more inclined to use CS. On the other hand,
class 2 presents the lowest probability of being a CS member and having kids
or bikes at home. Retired people tend to have more predisposition for this
class, while students have less. This is aligned with evidence in the
literature [38] suggesting that young people are more prone to use this
service. Finally, class 3 has the same probability of being a CS member as
class 1, but it is also the class with less probability of having a car at
home which could make them more dependent on the availability of the service.

Figure 6: Representation of the class membership of our proposed model

Comparison between Figure 5 and Figure 6, shows that the configuration of the
classes with respect to the socio-characteristics changes when we include
attitudinal information, as is expected. However, bearing in mind that the
betas values of the two models are slightly different and classes cannot be
directly compared.
Continuing with the proposed formulation results, in order to understand the
implications of the coefficient of the latent variables in the utility of the
class membership model, we have plotted the latent constructs for each
individual in the sample and characterised them with their socio-
characteristics. Figure 7 provides us with a visual understanding of how these
latent variables are constructed and distributed.

Figure 7: Latent variables representation characterised by socio-
characteristics

By analysing the parameters for the latent variables in Table 6 and looking at
their distributions in Figure 7, we notice that the values of the first latent
variable ($r_{1}$) are always negative. The more negative value of r1, the
more probable is to belong to class 2, and therefore, the less inclined people
are to use CS services. A negative value of $r_{2}$ seems to have the same
effect. Thus, individuals with a more negative combination of $r_{1}$ and
$r_{2}$ tend to be less inclined about CS and the other way around. Figure 7,
suggests that students are more prone to use the service while retired people
are the least predisposed. Moreover, having or not having a car seems to
determine the clusters in which the $r_{s}$ values are structured, as we can
see in the upper right plot of Figure 7.
The coefficients of the $\omega_{n}$ parameter are a representation of the
heterogeneity present in the classes. Therefore, class three seems to be more
heterogeneous. Finally, the parameters of the utility for each of the
indicators are presented in Table 7. The mean accuracy for all the indicators
given the ordinal logit probability is 0.44.

Indicator | $\alpha_{1}$ | $\alpha_{2}$ | $c$
---|---|---|---
$I_{1}$ | 0.24 | 0.42 | 0.65
$I_{2}$ | 0.43 | 0.34 | -0.14
$I_{3}$ | -0.14 | 0.28 | 0.65
$I_{4}$ | -0.22 | 0.27 | 0.87
$I_{5}$ | -0.40 | 0.24 | 0.56
$I_{6}$ | 0.19 | -0.38 | -0.62
$I_{7}$ | -0.026 | 0.23 | 1.59
$I_{8}$ | -0.40 | 0.14 | 2.59
$I_{9}$ | 0.09 | 0.045 | 1.72
$I_{10}$ | -0.070 | 0.19 | 2.56
$I_{11}$ | -0.40 | -0.083 | 2.28
$I_{12}$ | -0.33 | -0.016 | 2.97
$I_{13}$ | -0.040 | 0.044 | 1.80
$I_{14}$ | 0.019 | -0.10 | 2.40
$I_{15}$ | -0.095 | 0.45 | 2.42
$I_{16}$ | -0.17 | 0.43 | 2.35
$I_{17}$ | 0.17 | 0.34 | 1.60
Table 7: Parameters of the measurement model

Finally, it is interesting to see how the answers for some indicators are
distributed in the latent space. Figures 7 and 8, show that people with a car
at home agree more with the statement that the car is a status symbol. For
indicator 15, people with a more positive value of $r_{2}$ seem to agree more
with the statement that they wouldn’t need a car if they have CS, as we would
expect given the $r_{2}$ coefficients of Table 6 and the positive $\alpha_{2}$
coefficient for the utility of indicator 15 in Table 7.

Figure 8: Latent variables representation characterized by the answers to
indicators 6 and 15

##  6 Conclusions

We focused on the relevance and impact of adding attitudinal information to
the models, evidencing its importance in configuring the latent classes for
heterogeneity representation. We propose a new way to account for complex (and
non-linear) constructs of latent classes by extending the general random
utility model with ML frameworks for latent space representation.
We have tested our proposed framework for estimating a car-sharing (CS)
service subscription choice with SP data from Copenhagen, Denmark. Our results
confirm that the inclusion of attitudinal variables provides a DCM more
behaviorally realistic. For example, individuals who are more inclined towards
the concept of CS tend to be grouped together in classes with higher parameter
estimates of the utility of choosing different CS plans. Also, as expected,
such classes are sensitive to different attributes of the CS plans, but, more
importantly, the complex structure of such classes can be modelled by
considering non-linear formulations learned by an ANN framework. This confirms
that complex beliefs and attitudes play a key role in CS subscription
decision-making, and including this information allows for more accurate
estimation and a better understanding of the classes.
From the application perspective, it provides us with extra insights to help
us design better policies by having a more realistic population segmentation.
In our CS application, when creating new approaches to attract and retain CS
members, attitudinal information allows us to focus on changing beliefs and
mindsets that we know are closely related to individual choices. Finally, it
also helps to understand better how the decision-making process works from a
theoretical point of view. We have also improved our understanding of how the
features of the CS business can maintain and attract new members. We have
noted the importance of the subscription and usage cost throughout all the
models. In addition, the price format is relevant to the decision, and the
price per minute display is the most attractive, possibly driven by the past
experience in the case-study at stake. We have also observed that the
availability of cars is essential in attracting new members, which can be seen
as an improvement opportunity for CS companies. Choosing an electric car is
more favourable for those more inclined to use CS, so having a fleet with
electric vehicles can be essential to maintain CS members.
Our study also has a few limitations. Regarding our proposed model at the
estimation level, the values of the latent variables change between
iterations, even though the underlying structure and model probabilities are
kept between runs. This is the main difference between an ML learning approach
and a traditional DCM estimation, where all the parameters need to be
identifiable. On the other hand, convergence is defined empirically by setting
the number of iterations due to small fluctuations in the convergence of the
EM algorithm. Also, given the small sample size, we could not divide the
dataset in training, validation, and testing; therefore, the hyperparameters
referring to the number of nodes and layers in the NN were not tuned according
to the validation samples. Regarding our overall model architecture, and in
order to improve the prediction performance, further work can be done, trying
to apply other types of explainable AI (XAI) and other latent space
representations (word embeddings) [23].
Although this work has its limitations, we are optimistic that this analysis
has opened the door to future research on integrating attitudinal variables in
DCMs through ML techniques. This investigation could improve the overall model
fit and prediction accuracy, thanks to the representation of heterogeneity due
to ML techniques’ ability to capture complex unobserved patterns. However, we
always have to bear in mind that the transparency and interpretability of new
modelling frameworks are of utmost importance.

###  6.1 Acknowledgements

We thank Share-More project for providing the data used in this study and
eMOTIONAL CitiesGrant agreement ID: 945307, funding from EU’s Horizon 2020.
444https://emotionalcities-h2020.eu/

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