# Ecological Data Reveal Imbalances in Collision Avoidance Due to Groups’
Social Interaction

Adrien Gregorj  Okayama University, Okayama, Japan  Zeynep Yücel  Okayama
University, Okayama, Japan  Advanced Telecommunication Research Institute
International, Kyoto, Japan  Francesco Zanlugo  Okayama University, Okayama,
Japan  Advanced Telecommunication Research Institute International, Kyoto,
Japan  Osaka International Professional University, Osaka, Japan  Takayuki
Kanda  Advanced Telecommunication Research Institute International, Kyoto,
Japan  Kyoto University, Kyoto, Japan

###### Abstract

The relative dynamics in collision avoidance between individual pedestrians
and dyads has been recently studied, and it was shown that individuals may
intrude and disrupt those dyads that are not socially interacting, but not the
ones that are engaged in interaction. Building on this, our current study
examines how much each party contributes to collision avoidance in the
absolute sense, i.e. by measuring deviations from their intended paths.

Our findings suggest that individuals prioritise trajectory efficiency in
undisturbed situations, but prioritise safety when encountering dyads, by
deviating more from their intended path. Not socially interacting dyads
present a similar behavior, although their trajectories appear to be even more
efficient than those of individuals in undisturbed situations, and their
deviations during encounters less pronounced. On the other hand, socially
interacting dyads are not very efficient in undisturbed situations, and their
behavior is mostly unaffected by encounters.

These results strongly suggest that group dynamics affects in two ways the
behavior of pedestrians, namely it has a dynamical and a social effect. The
dynamical effect, i.e. the necessity to keep a spatial vicinity to one’s
partner, stabilises their trajectory, while the social one, by redirecting
one’s attention to the partner, decreases the ability to focus on the external
environment, and thus leads to reduced efficiency and safety.

Another finding concerns the tendency of individuals to avoid in a more
prominent way the interacting dyads as compared to non-interacting ones. This
suggests that individuals not only observe their surroundings to anticipate
the future paths of others, but may also assess others’ contribution to
collision avoidance.

An impact parameter analysis reveals that collision risk influences path
deviations in pedestrian encounters. For individuals, larger behavioral
differences between low and high interaction levels of the dyad occur both
when the collision risk is high and during less critical encounters. For
dyads, the deviation differences between low and high interaction levels are
most pronounced when the individual is on course to pass close to the dyad.

##  1 Introduction

Human walking motion has been a research subject in various disciplines, each
with distinct scopes and objectives. Crowd dynamics primarily studies
collective human movement [1], often with the goal of ensuring safety in
large-scale event management (e.g. Olympic games, Hajj) [2], prevention of
crowd accidents (e.g. during festivals) [3], and evacuation planning (e.g.
during fire, terror attacks etc.) [4], among others. On the other hand,
architecture and urban design extend this focus to everyday life settings [5],
prioritizing safety and comfort. This includes architectural designs for
enhancing business in malls [6], landscape designs [7, 8] for recreational
spaces, and urban infrastructure planning for efficient travel and minimal
detours [9]. In addition, robotics directs its attention towards individuals
[10] with the objective of achieving natural and comfortable motion at the
actuator and/or navigation levels. Applications range from assisting human
limbs [11] to seamless integration into human crowds [12] for services such as
museum guiding or companionship [13]. Furthermore, kinesiology examines motion
from a biomechanical standpoint, aiming to explain the fundamentals of motor
control [14]. In contrast, cognitive science predominantly explores the
neurophysiological perspective, with a specific focus on such aspects as
perception, visuo-motor coordination, and embodied cognition [15]. Despite its
brevity, we believe that the above overview already provides a glimpse into
the multifaceted nature of research on human walking motion and the vast
variety of scopes, objectives and methodologies.

In this study, we choose to view human walking motion in its simplest form,
namely as the most fundamental means of transportation [16] and as an
essential activity for an independent life and daily tasks (e.g. trips to the
market, the bank etc.). In such settings, individuals maneuver among fellow
pedestrians and obstacles with a focus on safety and fluidity, a concept known
as collision avoidance. These localised interactions are thought to give rise
to self-organization within a local-to-global framework [17]. For capturing
the properties of this sort of navigation behaviour, a large variety of
computational models have been proposed [18], where the early models were
inspired by physics and used repulsive forces to reproduce collision avoidance
[19]. While these models have been successful in generating coherent patterns
at the collective level and have greatly contributed to our understanding of
pedestrian motion [19], they often fall short in accurately capturing the
realistic attributes of human trajectories [17].

In that respect, this study will focus on human walking behavior in an
ordinary public space aiming to unveil nuanced patterns in decision-making and
path selection. Specifically, we will focus on collision avoidance between
individual pedestrians and social groups of two people. By dissecting such
encounters according to the group’s level of engagement in social interaction,
we aim to illustrate the emergence of an intriguing pattern, namely a degree
of involvement in collision avoidance and possibly an understanding, by
participants not absorbed in social interaction, of the others’ involvement.

In a previous study, we studied dyad-individual collision avoidance in a
relative sense —specifically, in terms of how closely they approach each other
during collision avoidance [20]— and demonstrated that individuals tend to
intrude and disrupt a dyad when its members are not socially interacting, but
they refrain from doing so when the dyad is socially interacting or appears to
have a strong social bond (e.g. couples). Additionally, we have shown that the
collision avoidance dynamics between the dyad and individual can be modeled
using an “interaction potential”, obtained through an analogy with the two-
body scattering problem, increasing rapidly as the distance decreases, and
depending on the level of interaction or the nature of the social relation of
the dyad.

In the current study, we aim to build upon our previous research by examining
individual-dyad collision avoidance in an absolute sense, i.e. quantifying
each peer’s contribution to collision avoidance based on their deviations from
intended paths. Firstly, we will illustrate, using ecological data, that
social interaction within groups leads to less efficient path adjustments.
Specifically, when groups move without other pedestrians nearby, their
deviations from a straight path increase with higher levels of interaction,
resulting in longer and less economical trajectories. Conversely, when groups
encounter individuals, their deviations decrease with higher interaction
levels, indicating reduced responsiveness in collision avoidance.

On the other hand, individual pedestrians clearly modify their behavior in
presence of dyads. Additionally, individuals show sensitivity to the level of
interaction, albeit not significantly so. This result, when combined with the
statistically significant result reported in [20], namely that relative
collision avoidance is stronger between individuals and socially interacting
dyads with respect to the one between individuals and not socially interacting
dyads, suggests that pedestrians anticipate the diminished involvement of
groups in collision avoidance and adjust their deviations from intended paths.

Additionally, we will investigate the role of collision risk in shaping path
deviations during pedestrian encounters. The concept of impact parameter,
borrowed from physics and used in [20], will be employed to quantify the
collision risk between individuals and dyads. We will show that collision risk
influences path deviations in pedestrian encounters, with larger behavioral
differences between low and high interaction levels of the dyad occurring both
when the collision risk is high and during less critical encounters. The
particularly straight trajectories of individuals encountering non-interacting
dyads echoes the findings of [20], suggesting that are more susceptible to
pass through such dyads. We will also demonstrate that the deviation
differences between low and high interaction levels are most pronounced when
the individual is on course to pass close to the dyad.

Furthermore, a comparison between the behavior of individuals, dyads with and
without social interaction shows that group dynamics presents both a dynamical
and a social effect, suggesting that this aspect should be introduced also in
crowd models to properly asses the presence of groups and their effect on
overall dynamics.

The adaptive behavior observed suggests a nuanced understanding of social cues
and an ability to adjust behavior accordingly. Moreover, it sheds light on the
complexity of human interaction within crowded environments, where individuals
must constantly explore path choices/adjustments and negotiate space alongside
others. Understanding these dynamics not only enhances our comprehension of
pedestrian behavior but also has implications for urban planning, crowd
management, and the design of intelligent systems aimed at facilitating smooth
and safe movement in public spaces.

In the following section, we will frame our work and the scenario in focus and
explain related literature concerning the factors relevant for shaping human
motion in those settings.

##  2 Background

The concept of “community ambulation” as described in the literature refers to
individuals’ ability to move independently in public spaces [21]111A more
precise definition given in [22] specifies that community ambulation entails
walking a defined distance (e.g., 800 m) and navigating stairs without
assistance. . Community ambulation, by definition, necessitates the capacity
to integrate walking with a diverse range of demands arising from the dynamic
nature of the surrounding public environment. Patla et al. precisely identify
factors contributing to this integration as walking distance and speed,
ambient conditions, physical load, terrain variations, postural transitions,
traffic density, and attentional demands [23].

In order to meet these demands, individuals must constantly navigate decision-
making processes while engaged in ongoing activities [24]. These embodied
decisions necessitate rapid and continuous processing of multi-modal sensory
information [25] as well as evaluating all possibilities in parallel,
culminating in the execution of the anticipated optimal choice. Although there
is ongoing debate regarding the exact mechanisms underlying these decision-
making processes [26, 27], the complexity of this cognitive process is widely
accepted.

We consider a typical urban setting to be characterised by low to medium
density, plain geometry, and even terrain. Such an environment is often
populated by individuals from diverse backgrounds, spanning various ages,
occupations, and other demographic factors. In this study, as an adequate
example of urban environment, we use the DIAMOR pedestrian trajectory data set
[28] (see Sec. 3.1 for details). The use of ecological data is essential in
our study, as it captures real-world complexities and nuances inherent in
human behaviour within urban environments. Unlike simulated or laboratory-
based data, ecological data reflect the genuine interactions and responses of
individuals navigating through authentic urban settings. By leveraging such
data, we hope to better understand the intricacies of pedestrian dynamics,
including the influence of social interactions, and individual behaviours.

Even though there is an inherent lack of control over the factors influencing
navigation choices of pedestrians in this naturalistic environment [29], we
can assess extrinsic factors in accordance with Patla and Shumway-Cook’s
guidelines [23]. Regarding walking speed, the environment imposes no
significant constraints, as evidenced by empirical velocity distributions that
align with ranges observed in other ecological studies on community ambulation
[28, 30] (see Sec. 3.2). Analyzing a sufficiently large sample size allows for
the proper representation of interpersonal variations. Regarding postural
transitions, the absence of curves, slopes, or stairs in the environment
indicates minimal need for significant adjustments, a conclusion further
supported by video recordings. In terms of physical load, analysis of the same
data reveals that pedestrians predominantly carry either no load or a
seemingly light handbag or backpack, typical for urban travel222Human
annotations of video data show that pedestrians mainly consist of students,
workers, or shoppers commuting to schools, workplaces, or commercial centers..

Furthermore, the terrain and ambient conditions within the data set remain
constant, whereas density undergoes minute changes (see Sec. 3.2 and Sec.
3.3). Regarding distance, as detailed in Sec. 3.3, specific preprocessing
steps are employed to ensure the comparability of path lengths across
pedestrians333However, we cannot comment on how long the pedestrians might
have walked before entering our observation space.. This leaves us with the
last extrinsic factor of community ambulation, namely, attentional demands,
which may actually be quite challenging to assess.

While navigating in public environments, pedestrians need to allocate
attentional resources for the observation of their surroundings to identify
potential hazards (e.g. stairs, static obstacles) and remain vigilant to
changes [31]. Nevertheless, the underlying principles of anticipatory
locomotor adaptations used to circumvent such challenges is not completely
understood [32]. Shumway-Cook et al. identify three key factors influencing
attentional demands in community ambulation: familiarity with the trip
location, environmental distractions, and the presence or absence of travel
companions [22]. In our case, information regarding pedestrians’ familiarity
with the environment is unavailable. Nevertheless, given the straightforward
nature of the corridor shown in Fig. 1, it is unlikely that familiarity with
such an environment would significantly affect route finding. Additionally,
with no visual or auditory distractions such as shops or music present,
environmental distractors are expected to be minimal and consistent across all
observed pedestrians. This allows us to isolate the effect of the presence or
absence of travel companions and analyze its specific impact.

The distinction between physical crowds, comprised of individuals in close
spatial proximity but lacking a shared psychological identity, and
psychological crowds, characterised by a collective sense of identity or
purpose among its members, has been introduced in [33, 34]. Pedestrians
observed in the DIAMOR data set are likely to belong to the former category,
as they do not appear to have a collective identity. Nevertheless, our data
set contains instances of small social groups and individuals. Here, by groups
we refer to 2 or more pedestrians travelling together towards a shared
destination and engaged in a social relationship [35, 36]. Conversely,
pedestrians who are not part of a group are termed individuals444It is
important to acknowledge that while members of a group are individuals, in the
context of this research, their group affiliation is considered to
significantly influence their navigation and, thus, they are identified by
this aspect of their social dynamics throughout the study.. Additionally, we
use the term interaction, in a specific context, referring exclusively to
social interaction characterised by verbal communication, gestures, gaze,
physical contact, etc., among members of a group. In this respect, we
emphasise that we strictly separate social interaction from collision
avoidance.

Note that while individuals need to allocate a certain portion of their
attentional resources for the inspection of their surroundings and collision
avoidance, they do not need to allocate any resources for coordinating their
motion with a partner or for social interaction. Consequently, we posit that
cognitive load is likely to be lower for individuals, as well as relatively
consistent across different individuals.

On the other hand, groups are likely to experience higher mental workload
compared to individuals, since they need to invest effort in managing group’s
internal dynamics as well as social interaction. Concerning the former, group
members need to maintain a cohesive pace and orientation with their partners
while ensuring group consistency [37, 38]. In addition, if they also carry out
social interaction, they need to ensure its smoothness through practices like
mutual gaze (on a common target or as eye contact), facial expressions, hand
gestures, head/body pose, back-channeling etc. Furthermore, the level of
effort may vary depending on factors such as group size, hierarchy intricacy,
and the level of group members’ engagement in social interaction [39].
Therefore, we cannot simply assume comparable attentional demands for all
groups.

At this point, in order to dissociate the attentional demands of groups from
their size and hierarchy, we choose to focus on two-people groups, known as
dyads. We argue that this approach is not oversimplified, as the majority of
groups in crowds consist of two people [40] and larger groups are shown to
break down into sub-groups of two or three people, making dyads a fundamental
building block of crowds [30, 41]. In addition, by breaking down these dyads
into sub-categories according to their level of involvement in social
interaction [39], we can also address the non-uniformity of attentional
demands and approximate the gradation of mental workload.

Regardless of whether they travel as an individual or as part of a group, to
ensure safe community ambulation, humans need to attentively monitor the
continuously changing environment. To that end, among the five sensory
information channels, the visual one emerges as being the most important.
Generally speaking, humans have high quality perceptual access to the world
[25], although not flawless [42]. In community ambulation, such visual
information is processed online [14] under the influence of visual conspicuity
[43, 25] and task demands [44]. The principal role of visual channel in
locomotion is providing an understanding of location of the self, the goal,
and the environment (e.g. dimensions, terrain features etc.), which are
essential in the control of adaptive locomotion [45] as demonstrated by
studies contrasting open loop obstacle avoidance to full visual sampling [14].

Studies [46, 47] have demonstrated pedestrians’ reliance on visual cues for
efficient path planning around static obstacles. This process entails
utilizing visual information to locate the target destination and assess the
surrounding path area. Additionally, pedestrians evaluate the necessary
magnitude of the deviation to navigate obstacles effectively, ensuring a
seamless progression towards their intended destination.

In addition to static obstacles, pedestrians must also be aware of moving
targets (e.g. other pedestrians, bicycles, cars etc.) and anticipate potential
collisions, which present greater challenges compared to stationary obstacles
due to their momentum and unpredictable motion [48]. In this study, we focus
on human-human collision avoidance and thus consider as moving obstacles the
other pedestrians in the environment.

Previous research on gaze analysis of pedestrians encountering other moving
people has revealed several noteworthy findings. It was observed that gaze
fixations occur at consistent frequencies regardless of increases in
pedestrian density, albeit with a narrower scanning range of the street [49].
This suggests that pedestrians tend to focus more on people directly in front
of them, particularly those in closer proximity. Moreover, pedestrians in the
central and right positions were fixated at greater distances compared to
those on the left, indicating a modulation of gaze behaviour based on the
location and direction of pedestrians in the community environment [50].
Additionally, studies have found that during encounters, pedestrians tend to
focus around the chest of the oncoming pedestrian [51], and initially scanning
various parts of their body, providing evidence that body motion cues serve as
a significant source of visual information during such situations [52, 53].
Hessels et al. also concluded that walkers tend to look at different body
parts based on the behaviour of the other pedestrian, with more attention
directed towards individuals who address or direct themselves towards them,
with the exception of pairs engaged in conversation, who are likely to be
looked at despite not being directed towards the observer [54].

Pedestrians use visual information not only for anticipating others’ path, but
also for estimating certain personal features [55, 56]. Some person-specific
information which can be observed by visual inspection (even without detailed
scrutiny) involve height, gender, and group relation. Uninstructed pedestrians
circumventing around standing people are found to maintain a greater distance
from males than females, from groups than individuals, and from an attractive
person than an unattractive one [55]. In addition, role dependent strategies
in collision avoidance (i.e. who passes first and who gives way) are shown to
depend on gender and height [56]555Knorr et al. failed to observe any
significance on gender and height, possibly due to the lack of cognitive load
and oblique (i.e. not frontal) crossing situation [57].. Other person-specific
information which cannot be directly observed but which can be estimated
visually involve age, personality and mood. Interestingly, it has been shown
that people can estimate such traits as domination, boldness, easygoingness,
happiness, youthfulness with quite high agreement, even by watching only
walkers’ point light displays [58, 59]. To the best of our knowledge, no study
has explored whether pedestrians perceive social interactions within the
groups they encounter. However, given our findings, we believe it is probable
that they do notice such interactions.

In addition, eye gaze is shown to serve not only for collecting information
from one’s surroundings, but also for delivering information about one’s
intended path to others [54, 60]. In particular, Nummenmaa et al. [60]
interestingly shed light on the communicative efficacy of “gaze aversion” and
demonstrated that diverting gaze to one side can effectively signal a planned
path to others, suggesting steering in that direction to avoid collision,
thereby prompting the other pedestrian to maneuver in the opposite direction.
Generally speaking, studies on the Theory of Mind Model (ToMM) suggest that
eye gaze plays a crucial role in understanding others’ intentions [61], and
the absence of gaze alternation can result in a failure to anticipate others’
attentional focus and intentions in static scenarios, referred to as “mind
blindness” [62]. In addition to gaze direction, head orientation, known to
indicate focus of attention [63], serves as a cue for path selection, while
body orientation is associated with a “potential to move” [12]. Yet, such
findings imply that the brain, responsible for orchestrating swift and
successful behavior in such settings, relies not only on spontaneous sensory
inputs but also on internal representations of surrounding pedestrians to
fulfill this role [15].

On the other hand, the auditory channel can be considered as the second most
important sensory channel during community ambulation. Although dual-task
studies suggest that environmental sounds (e.g. construction noise, music
etc.) are likely to be distractive rather than supportive [21], holistically
speaking, the audio channel plays a significant role in shaping the overall
perception of the surroundings. Specifically, task load on the auditory
channel has been shown to interfere with locomotion, leading to lower walking
speed or collisions, and a preference for maintaining a larger personal space
[22, 32]666The concept of personal space can be interpreted as a zone
pedestrians maintain for themselves and others, ensuring a comfortable
distance [64]. More recently, researchers have proposed an alternative
interpretation of personal space as a buffer zone that enables people to
perceive risks and plan trajectory adaptations [32, 65]. Some recent works
advocate for a combined approach, incorporating both a social influence field
and a collision avoidance field [12].

Furthermore, in dual-task scenarios, individuals tend to move at lower speeds,
and the increased workload on the auditory channel may lead to changes in
visual scanning patterns. Specifically, eye gaze is seen to be directed more
frequently towards pedestrians posing a higher risk of collision, and
fixations are observed more commonly on upper body segments, possibly since
the upper body is more informative regarding future walking directions [21,
12].

In addition to securing a certain degree of awareness about the environment,
group members may also need to observe their partners for ensuring group
cohesion and possibly for orchestrating social interactions. This monitoring
process often involves utilizing primarily visual and auditory channels, akin
to a dual-task mobility scenario. Specifically, social interaction in a group
is likely to require visual resources for tracking focus of attention (e.g.
gaze on partner or a mutual gaze on a target) and elicitation of emotions
(e.g. facial expression, gestures) etc. [66]. In addition, audio channel is
essential for comprehension of speech, detection of adjustments on pitch,
loudness, intonation etc. of the interlocutor, as well as turn management
(signaling turn taking, holding, giving or skipping), communicating feedback
(e.g. agreement, surprise etc.) or acknowledgement (i.e. back-channeling).
Note that since humans have limited attentional resources and group members
may already need to devote part of those to the afore-mentioned commitments
towards their partners, they are likely to be left with less resources for
navigation planning than individuals777One may also expect them to require a
larger personal space to make the trajectory adjustments necessary for
collision avoidance [32].. Intuitively, assessing the likelihood of collision
with moving obstacles, such as other pedestrians, is more challenging compared
to navigating around stationary obstacles [31] and requires an anticipation of
others’ trajectory. At this point, it is highly likely that humans consider
other pedestrians around them not simply as obstacles that can move, but as
humans like themselves [61]. Therefore, they possibly also monitor and assess
whether others are engaged in the same visual exploration and this meta-
awareness can play a crucial role in their own planning and, in turn, visual
sampling.

The current study aims to contribute beyond existing research by showcasing
how social interaction affects the ability of pedestrians to focus on the
surrounding environment, with possible effects on efficiency and safety.
Furthermore, it attempts to demonstrate that humans possess awareness not only
of others’ planned paths inferred from subtle actions, but also of the absence
of such indications. Specifically, we believe individuals are capable of
recognizing the lack of these subtle communicative actions or understanding
that they are not directed towards an implication of a planned path (but, for
instance, addressed at a travel companion accompanying their social
interaction). In such cases, they proactively anticipate a limited
contribution to collision avoidance from those pedestrians and take
compensatory measures on their part.

##  3 Methods

###  3.1 Data set

The data set utilised in this study is the DIAMOR data set [28], which was
previously employed in [67, 39] for the purpose of group recognition and
pedestrian dynamics modeling. The recordings were made in an underground
pedestrian street network located in a commercial district of Osaka, Japan.
Specifically, the data set comprises recordings from two straight corridors
within this street network, and our focus centers on one of these. With
several train stations, business centers, and shopping malls accessible from
the recording location, there is a diversity in pedestrian profiles. The
recording area is roughly 200 m2 and allows continuous tracking along
approximately 50 m and the recording spans eight hours in a weekday.

We would like to highlight that this data set is collected from uninstructed
pedestrians in their ecological environment, which we consider a valuable
asset for observing naturalistic behaviour888Experimentation has been reviewed
and approved by ATR ethics board with document number 10-502-1. Posters
explaining that an experiment concerning pedestrian tracking was being held
were present in the environment. The data are publicly available [68] and
contain anonymous trajectories derived from range data [69].. Notably, studies
involving non-human animals have revealed intriguing distinctions and
qualitative differences in behaviour between constrained tasks and
uncontrolled settings [70, 71, 72]. Similarly, in human studies, the
phenomenon of modifying of one’s behaviour in response to the awareness of
being observed has even been given a name, the “Hawthorne effect”999Although
it is now largely agreed that such effect was less significant than originally
thought in the scenario from which it takes its name. [73, 74]. This effect
has been observed in various settings, including when assessing the quality of
care provided by trained practitioners [75], or the energy awareness of
consumers [76]. In the specific context of human locomotion, Farhan et al.
have shown that observed participants exhibit lower variability in gait
parameters [77], and Friesen et al. have demonstrated that locomotion
parameters (e.g. speed, step length) were impacted by the number of
researchers present in the room [78, 77]. In this respect, the ecological data
studied in the upcoming sections are considered to be largely devoid of
experimental or behavioural bias, or subconscious alterations in behaviour, or
at least minimally affected by such factors 101010 Note that with the above
arguments, we by no means intend to assert that the outcomes derived from
traditional, meticulously controlled experimental paradigms are inaccurate or
invalid. These approaches, proficient at dissecting intricate behaviours into
their individual components, have significantly contributed to our
understanding of the fundamental processes that govern behaviour. Nonetheless,
their reductionist approach is likely to constrain their capacity to elucidate
naturalistic behaviour comprehensively, since pristine experiences are more of
an exception than the norm in real-world settings and it is ideal to study
humans in naturalistic settings for ultimately explaining their real-world
behaviour [29]. .

The data include both depth and video information. The depth information is
utilised to derive the trajectories of pedestrians [69], which can be freely
downloaded from [68]. As a result of this tracking process, we obtain the
normalised cumulative density map shown in Fig. 1 (refer also to [20]). The
map is obtained by dividing the recording area into a grid of 10 cm $\times$
10 cm cells and counting the number of pedestrians that have been in each cell
at any point in time. The counts are then normalised by dividing by the
maximum count in the grid. Darker areas indicate higher pedestrian density.

The video data served as the basis for establishing the ground truth regarding
dyads and the intensity of interaction. To assess errors arising from coding
fatigue and coder bias, each relation (belonging to a group and intensity) was
labeled by two different coders. In the first stage of coding, coders observed
walking patterns, age, gender, and clothing etc. to determine which
pedestrians formed a group. In the second stage, focusing solely on
pedestrians labeled as dyads in the first stage, coders assessed the intensity
of interaction. This reduced the amount of data each coder had to view, thus
enhancing coding efficiency.

Coders were asked to label the intensity of interaction for pedestrian groups
using a 0-3 subjective scale. To avoid biasing the assessment, only the
resolution (i.e. the number of interaction levels) was predefined (four
levels), with no guidelines provided on what constituted weak, mild, or strong
interaction intensity. Instead, coders carried out free-viewing, watching
three hours of video footage of groups to intuitively grasp variations in
interaction intensity, before the actual coding task began.

The agreement between coders for group relation labeling was evaluated using
Cohen’s $\kappa$ coefficient, resulting in a high value of $\kappa$ = 0.96,
indicating strong agreement [79]. For interaction intensity labeling, the
reliability was assessed using Krippendorff’s $\alpha$ coefficient, yielding a
value of $\alpha$ = 0.67, which is usually considered sufficiently high [80].

Figure 1: The normalised cumulative density map for the DIAMOR data set. It is
obtained by dividing the recording area into a grid of 10 cm $\times$ 10 cm
cells and counting the number of pedestrians that have been in each cell at
any point in time. The counts are then normalised by dividing by the maximum
count in the grid. Darker areas indicate higher pedestrian density.

###  3.2 Metadata

To further clarify the context of the study, we provide some metadata about
the studied corridor. The density of pedestrians in the recording area is
shown in Fig. 2. We compute it by counting the number of pedestrians in the
recording area during 1 min time windows and dividing it by the area of the
recording area ($40$ m $\times$ $7$ m $=280$ m2). The density of pedestrians
in the recording area varies between 0 and 0.06 pedestrians/m2, with an
average value of 0.04 pedestrians/m2, which can be considered as a low
density.

The probability density function of the velocity of individuals and dyads in
the recording area is shown in Fig. 3. The dyads are categorised based on
their level of interaction, as defined in Sec. 3.1. We observe the effect of
interaction previously reported in [39], namely that strongly interacting
dyads have a lower average velocity compared to weakly interacting dyads. In
addition, we observe that the velocity of non-interacting dyads is comparable
to that of individuals.

(a)

(b)

Figure 2: Density of pedestrians in the DIAMOR data set. (a) Density of
pedestrians in the recording area over time. (b) Histogram of the density of
pedestrians. Density is calculated as the number of pedestrians in the
recording area during a 1 min time window, divided by the area of the
recording area. Figure 3: The probability density function of the velocity of
individuals and dyads.

###  3.3 Data preparation

The raw trajectories in the DIAMOR data set are sampled at a non-uniform rate,
with a frequency varying between 20 and 50 Hz [68] due to holes in the data
caused by occlusions or tracking errors. To ensure that the analysis is not
affected by this, we resample the trajectories at a constant rate $f_{s}$ of
33 Hz with cubic spline interpolation [57].

As mentioned in Sec. 1, the focus of this study lies on understanding the
dynamics of typical individual-dyad collision avoidance in public settings. To
achieve this, we choose to keep only the trajectories that align with typical
walking speeds in public spaces and disregard those that do not. For
establishing the common velocity range of urban pedestrians, we refer to
literature on human locomotion and, basing our reasoning on the findings of
[28], we consider the trajectories with an average velocity falling within the
range of $[0.5,3]$ m/sec to belong to typical urban walking motion and the
rest to be associated with other states (e.g. standing/running or tracking
artifacts).

Following this, to address the impact of sensing noise and natural swaying
resulting from human gait, we applied filtering to the trajectories. Commonly,
low-pass filtering is employed for this purpose [81, 82, 17], and in this
study we opted for a Savitzky-Golay filter [83], which is known to be well-
suited for smoothing noisy data. Specifically, we adjusted the polynomial
order used to fit the samples to 2 and the length of filter window to 3 s.
This decision was guided by the observation that typical gait cycles last
between 1 and 2 seconds [84, 85].

In what follows, we will use the notation $\mathbf{p}(t)$ to denote the
position of a pedestrian at time $t$, and $\mathbf{v}(t)$ to denote the
velocity of the pedestrian at time $t$. We will also use the notation
$\mathbf{p}_{i}(t)$ and $\mathbf{v}_{i}(t)$ to denote the position and
velocity of the individual $i$ at time $t$, and $\mathbf{p}_{d}(t)$ and
$\mathbf{v}_{d}(t)$ to denote the position and velocity of the dyad $d$ at
time $t$111111Note that when this notation is used, we consider the dyad as a
single entity and use the average position and velocity of the dyad members.
In other situations, we will consider the dyad members separately to avoid
biasing the results with an artificial smoothing caused by averaging..

A trajectory $T$ is defined as the sequence of positions $\mathbf{p}(t_{k})$
and velocities $\mathbf{v}(t_{k})$ of a pedestrian, where $t_{k}$ is the time
at which the positions are recorded, with $k\in[0,N-1]$ and $N$ being the
number of samples.

The velocity $\mathbf{v}(t_{k})$ is derived from the positions using a simple
forward Euler difference, i.e.

$\mathbf{v}(t_{k})=\begin{cases}\frac{\mathbf{p}(t_{k+1})-\mathbf{p}(t_{k})}{t_{k+1}-t_{k}}&\text{if
}k<N-1\\\ \mathbf{v}(t_{k-1})&\text{if }k=N-1\end{cases}$ (1)

$T=\left[(\mathbf{p}(t_{0}),\mathbf{v}(t_{0})),(\mathbf{p}(t_{1}),\mathbf{v}(t_{1})),\ldots,(\mathbf{p}(t_{N-1}),\mathbf{v}(t_{N-1}))\right]$
(2)

###  3.4 Intended direction of motion

Our primary assumption in assessing the trajectory deviation is that
pedestrians aim to minimise the distance traveled and will therefore select
the straightest path to reach their destination whenever possible. This
assumption, notably introduced in [86] and adopted by numerous other studies
[21, 87], posits that at the tactical level, where pedestrians make decisions
about their desired area and route, they do so by minimizing a cost function.
This function takes into account factors such as distance traveled, trajectory
comfort, or anticipated encounters with other pedestrians [86]. In the context
of a straight corridor, such as the one examined in our study, a straight line
is reasonably expected to be the optimal route to cross the
corridor121212Assuming that the pedestrian is not aiming to exit the corridor
through a side passage.,131313It is worth noting that if the corridor is
sufficiently wide, the optimal path may still be straight, but not perfectly
aligned with the corridor axis, as pedestrians may cross it diagonally..

To compute the intended straight line trajectory, we first need to identify
the intended direction of motion of the pedestrian. We define it as the line
going through $\mathbf{p}(t_{0})$ and guided by $\mathbf{v_{0}}$, which is the
average velocity vector over a 0.5 s window starting at $t_{0}$. At the
sampling frequency of 33 Hz, this corresponds to $N_{e}=\lfloor 33\times
0.5\rfloor=16$ samples and141414Since the velocity vectors are computed using
a forward Euler difference, the velocity time at $t_{k}$ is a vector pointing
from $\mathbf{p}(t_{k})$ to $\mathbf{p}(t_{k+1})$. Therefore, the average
velocity vector $\mathbf{v_{0}}$ is a vector pointing from $\mathbf{p}(t_{0})$
to $\mathbf{p}(t_{N_{e}})$.

$\mathbf{v_{0}}=\frac{1}{N_{e}}\sum_{k=0}^{N_{e}-1}\mathbf{v}(t_{k}).$ (3)

We believe that $0.5$ s is a reasonable window size for this purpose, as,
since it is approximately the time it takes for a pedestrian to take one step,
it is large enough to capture the general direction of motion, while being
small enough to avoid capturing the effects of the collision avoidance
behavior.

The intended direction of motion $L_{0}$ is therefore formally defined as

$L_{0}=\\{\mathbf{p}(t_{0})+\lambda\mathbf{v_{0}}\mid\lambda\in\mathbb{R}\\}.$
(4)

We argue that this line better represents the intended motion of the
pedestrian than taking the line going through $\mathbf{p}(t_{0})$ and
$\mathbf{p}(t_{N-1})$, as the latter would be influenced by the trajectory
deviations that we aim to quantify, particularly if the pedestrian does not
return to its original intended path after the deviation.

In addition, the straight line trajectory $T_{0}$ is defined as the trajectory
that the pedestrian would follow, should she maintain her desired motion
direction walking at a constant speed. The points of the straight line
trajectory are all on the line $L_{0}$ and verify

$\left\\{\begin{array}[]{@{}l@{}}\mathbf{\tilde{p}}(t_{k})=\mathbf{p}(t_{0})+\mathbf{v_{0}}t_{k}\\\
\mathbf{\tilde{v}}(t_{k})=\mathbf{v_{0}}\end{array}\right.\quad\forall
k\in[0,N-1]$ (5)

We emphasise that the straight line trajectory, composed of a sequence of $N$
discrete positions and velocities, differs from the desired direction of
motion, which represents a line with an infinite number of points.

###  3.5 Situations of interest

Once again, drawing from the terminology introduced in [86], at the
operational level -where pedestrians execute their selected route- they
deviate from the straight line trajectory, possibly influenced by such factors
as gait characteristics [88] and the presence of other pedestrians or
obstacles. In this study, since we aim to quantify and compare trajectory
deviations of individuals and dyads, magnifying on the impact of groups’
social interaction (and the amount of attentional resources available for
navigation planning), we examine two types of situations: (1) encounters,
where a dyad and an individual are on a frontal collision or close-to-
collision course, and (2) undisturbed segments, where neither the individual
nor the dyad encounters any other pedestrian in a reasonably large area around
them, allowing them to move freely without needing to perform avoidance
behaviour. In the following discussion, we begin by providing detailed
insights into the undisturbed case, since it serves as a baseline representing
the deviation in the absence of specific collision avoidance maneuvers.

####  3.5.1 Undisturbed situations

As explained in Sec. 3.4, we argue that although pedestrians aim to minimise
the traveled distance and will therefore select the straightest path to reach
their destination when possible, it is unrealistic to anticipate them to walk
on a perfectly straight line, even when there are no other pedestrians
present. This meandering [88] can be explained by the natural swaying
resulting from the human gait and the impact of factors such as the cognitive
load [89] and was modeled using a Langevin-like model in [90].

In this respect, we wish to obtain a baseline of straightness for both
individuals and dyads, when they are not forced to perform collision avoidance
behavior. To that end, we define undisturbed segments as the portions of a
trajectory where no other pedestrian is located at less than 4 m away [91,
52], the same window size adopted in the computation of trajectory deviation
during encounters.

However, although the encounters defined in Sec. 3.5.2 are by definition
spatially bounded151515Usually these portions measure 3 to 4 m, depending on
the lateral distance and speed of the two parties., undisturbed segments can
be arbitrarily long161616In less densely populated environments, longer
undisturbed segments can be observed.. To enable comparability of segments
from the two scenarios, we must select undisturbed segments of similar lengths
to those in the spatially constrained encounter cases. To that end, we extract
undisturbed segments of 4 m, ensuring there are no overlaps between
them171717In addition, we require that the direction of motion at the
beginning and end of the segment are aligned with the horizontal axis. This is
ensured by verifying that the absolute value of the angles of the velocity
vector in 0.5 s windows at the beginning and end of the segment, wrapped in
the range $[-\pi,\pi]$ are less than $\frac{\pi}{8}$ or greater than
$\frac{7\pi}{8}$ in more than 90% of all $N_{e}$ time steps in these windows.
This is to ensure that the pedestrian is not turning at the beginning or end
of the segment..

According to the above, one individual or one dyad may have multiple
undisturbed segments (in particular if they are observed for a long time and
if there are few other pedestrians around). Since these segments might not be
independent, given that the pedestrians’ behavior in one segment may be
influenced by the behavior in the previous one, we will consider the average
deviation over all undisturbed segments of an individual or dyad as a single
data point in the analysis. In Tab. 1-(a), we show the number of individuals
and dyads in undisturbed situations and in Tab. 1-(b), we show a breakdown of
the number of undisturbed dyads according to the intensity of interaction of
the dyad.

Table 1:  (a) Number of individuals and dyads in undisturbed situations and number of encounters. (b) Breakdown of the number of undisturbed dyads and encounters according to the intensity of interaction of the dyad.  (a) | (b)
---|---
| Undisturbed | Encounter
---|---
Individuals | Dyads |
1966 | 457 | 609
| Intensity of interaction | Undisturbed | Encounter
---|---|---
0 | 18 | 45
1 | 60 | 88
2 | 299 | 380
3 | 80 | 96

####  3.5.2 Encounters

We define encounters as situations where a dyad $d$ and an individual $i$ are
moving towards each other and are on a collision or close-to-collision course.
In such scenarios, it is likely that one or both parties will engage in
collision avoidance behaviour to ensure a comfortable passage.

Although the superposition assumption, commonly employed in many models
(e.g.[19]) suggests that the collective effects on a pedestrian from multiple
neighbors can be linearly combined [17], there is ongoing debate regarding
whether neighborhood is determined by metric or topological distances (e.g.
degree of neighborhood). In this work, we choose to use metric distance, since
the density associated with our data set is not high enough to result in
collective behaviour [92]. In that respect, we consider only those dyads and
individuals, who approach each other sufficiently close. Specifically,
sufficiency is assessed by the condition $\exists t\mid d_{di}(t)\leq 4$m,
where $t$ represents time and $d_{di}$ denotes the instantaneous distance
between the dyad and the individual.

The choice of a 4 m threshold is grounded in prior research on collision
avoidance. Cinelli and Patla found that the “safety zone,” which is the
distance individuals allow a moving object to approach before initiating an
avoidance behaviour, averages around 3.73 m [91]. In addition, Gérin-Lajoie et
al. showed that anticipatory locomotor phase starts with an initial path
deviation which occurs about 4.5 m from the obstacle [32]. Moreover, Kitazawa
et al. demonstrated that pedestrians focus their gaze most intensely on
approaching individuals when they are, on average, approximately 3.97 m away,
seldom directing their attention to pedestrians at greater distances [52].

Among dyad-individual pairs that approach sufficiently close, we solely
consider those engaging in frontal encounters, namely moving in opposite
directions. There are two main reasons for this selective approach. Firstly,
given our focus on collision avoidance inside an environment with
predominantly bi-directional flow, we contend that encounters involving
pedestrians from opposite flows are more pertinent than those within the same
flow. Secondly, since we aim to discern the implications of social interaction
level (and in turn speculate on the allocation of attentional demands), only
those instances where involved parties can visually examine each other
(particularly in which the individual can judge groups relation and assess
dyad’s level of interaction) are considered [57, 56]. In contrast, non-frontal
encounters are omitted, since collision avoidance is less prominent within
low-density bi-directional flow settings, and individuals are not likely to
react to dyads’ characteristics due to limited observation capabilities.

To address this, we compute the predominant relative motion direction of $d$
and $i$ at the beginning of the encounter by calculating the cosine of the
angle between their velocity vectors,

$c_{di}(t)=\frac{\mathbf{v}_{d}(t)\cdot\mathbf{v_{i}}(t)}{||\mathbf{v}_{d}(t)||||\mathbf{v}_{i}(t)||}.$
(6)

We classify an encounter as frontal if, during a 0.5 second window starting at
$t_{0}$, the cosine of the angle between the velocity vectors is smaller than
$-\cos(\frac{\pi}{8})$ for at least $90\%$ of all $N_{e}$ time steps
(indicating an angle range of $[\frac{7\pi}{8},\frac{9\pi}{8}]$). This
condition is formally expressed as

$\frac{1}{N_{e}}\sum_{k=0}^{N_{e}-1}\mathbbm{1}_{\\{c_{di}(t_{k})<-\cos(\frac{\pi}{8})\\}}\geq
0.9,$ (7)

where $\mathbbm{1}$ is the indicator function that is equal to 1 if the
condition inside the brackets is true and 0 otherwise.

For further ensuring anticipatory locomotor adjustments during frontal
encounters, we calculate extrapolated straight line trajectories (see Eq. 5)
at the initial instant of the encounter and require the closest approach
distance on such paths to be less than 2 m181818Note that despite starting the
encounter at a distance of 4 m, it is possible that the dyad and the
individual have sufficient lateral distance to comfortably clear each other,
rendering such encounters irrelevant for the scope of this work..
Specifically, we derive the approach distance using the average velocity of
the dyad and the individual over a 0.5 s window (the same time window used for
determining the relative motion direction, see also Eq. 7), employing a method
akin to computing the impact parameter for examining the interaction of
charged particles.

Finally, we ensure that both the dyad and the individual’s behaviour is
captured even after they have laterally passed each other. This conditioning
is motivated by the findings reported in [93], where the authors noted that,
in cases of very close encounters, lateral distance kept increasing even after
ensuring avoidance. Hence, we necessitate that both the individual and the
dyad clear each other entirely, with adequately long trajectories preceding
and following the clearance. To be precise, we specify that they must maintain
a distance of at least 3 m apart at the beginning and end of the encounter,
demonstrating an initial approach succeeded by subsequent distancing.

After applying the conditions described above, the number of encounters which
will be subject to an analysis in the upcoming sections, turn out to be as
illustrated in Tab. 1-(b).

###  3.6 Measures of deviation

In this section, we provide the definitions of a set of measures for
quantifying the deviation of an actual trajectory from an intended straight
line trajectory (or, equivalently, its dissimilarity to such path) together
with a discussion on their specifications. While most of the measures are
compiled from existing literature on collision avoidance and trajectory
clustering [94, 95], we also introduced several original measures to capture
various aspects of deviation. Additionally, while some measures from the
literature are directly applied to our data, others are adapted to suit our
specific case.

When computing the deviation of a dyad, we consider the deviation of both
members separately. This ensures that the deviation of the dyad is not
artificially reduced by averaging the positions of the two members.

####  3.6.1 Position based measures

In collision avoidance literature, it is common to assess avoidance behaviour
as deviation from a straight line. In studies with handcrafted experimental
settings, such as frontal encounters in a corridor [96, 97, 98], such straight
line is considered directly as the environment axis and deviation is measured
along the direction orthogonal to that (typically denoted as $y$-axis). In our
case, rather than using the environment (i.e. corridor) axis, we derive an
intended direction of motion for each individual and group (see Eq. 4 and Eq.
5 and $L_{0}$, for instance, in Fig. 7). We compare an observed trajectory to
a straight line along that intended direction of motion in various ways as
explained below.

#####  3.6.1.1 Euclidean deviation $\delta_{E}$

The Euclidean distance $\delta_{E}$ [99] is probably one of the most intuitive
and straightforward measures of deviation. It is computed as the average
distance between the points of the observed trajectory $T$ and the points on
the straight line trajectory $T_{0}$, i.e.

$\delta_{E}=\frac{1}{N}\sum_{k=0}^{N-1}||\mathbf{p}(t_{k})-\mathbf{\tilde{p}}(t_{k})||.$
(8)

In Fig. 4, we illustrate the Euclidean deviation $\delta_{E}$ for some
hypothetical $T$ and $T_{0}$.

Figure 4: Illustration of the Euclidean deviation $\delta_{E}$, which is the
average of the distances between the observed trajectory of the pedestrian $T$
and the straight line trajectory $T_{0}$.

#####  3.6.1.2 Lockstep maximum deviation $\delta_{max}$

The lockstep maximum deviation $\delta_{max}$ is defined as the maximum
distance between simultaneous pairs of points of the trajectory and the
straight line trajectory. While the Euclidean deviation $\delta_{E}$ provides
an average measure of deviation across the entire trajectory, the lockstep
maximum deviation $\delta_{max}$ captures the maximum deviation, at a certain
time, between the observed trajectory $T$ of the pedestrian and her straight
line trajectory $T_{0}$.

Formally,

$\delta_{max}=\max_{k\in[0,N-1]}||\mathbf{p}(t_{k})-\mathbf{\tilde{p}}(t_{k})||.$
(9)

Note that this measure is sometimes called the lockstep Euclidean distance in
the literature [99].

In Fig. 5, we show an example for the computation of lockstep maximum
deviation $\delta_{max}$.

Figure 5: Illustration of the lockstep maximum deviation $\delta_{max}$, which
is the maximum distance between simultaneous pairs of points of the trajectory
$T$ and the straight line trajectory $T_{0}$.

#####  3.6.1.3 Discrete Fréchet deviation $\delta_{F}$

We derive the discrete Fréchet deviation from the standard discrete Fréchet
distance between two trajectories. Originally, Fréchet distance is a measure
of similarity between two continuous curves and is often colloquially referred
to as the “dog leash” distance, as it can be interpreted as the length of the
shortest leash that would allow a dog to walk along one trajectory, while its
owner walks along the other trajectory (without backtracking, but allowing to
stop). The discrete Fréchet distance is its discrete version and assesses the
dissimilarity of two discrete curves [100]191919In the figures, we illustrate
$T$ and $T_{0}$ as piecewise continuous for making them easier to understand.
However, we use only the end points of the line segments in evaluating the
measures.. To that end, an optimal mapping between the points of the two
discrete curves is identified as the one minimizing the maximum distance
between the matched points. Given such mapping, discrete Fréchet deviation
$\delta_{F}$ is the maximum of the distances between each pair of matched
points along $T$ and $T_{0}$.

In Fig. 6, we show the discrete Fréchet distance $\delta_{F}$ for some
hypothetical $T$ and $T_{0}$ and illustrate the two points in space that
correspond to the $\delta_{F}$ of such $T$ and $T_{0}$.

Figure 6: Illustration of the discrete Fréchet deviation $\delta_{F}$, which
is the maximum distance in the optimal mapping (shown in black dashed lines)
between the observed trajectory of the pedestrian $T$ and the straight line
trajectory $T_{0}$.

#####  3.6.1.4 Maximum lateral deviation $d_{max}$

The maximum lateral deviation $d_{max}$ is defined as the maximum distance
between a point on the trajectory of the pedestrian and the intended direction
of motion $L_{0}$ (see Fig. 7).

Formally,

$d_{max}=\max_{k\in[0,N-1]}|d_{k}|$ (10)

where $d_{k}$ is the signed distance between the point $\mathbf{p}(t_{k})$ and
its projection $\mathbf{h}(t_{k})$ on $L_{0}$ computed as

$d_{k}=\left(\left(\mathbf{p}(t_{k})-\mathbf{p}(t_{0})\right)\times\frac{\mathbf{v_{0}}}{||\mathbf{v_{0}||}}\right)_{z}$
(11)

where $\times$ denotes the cross product202020The cross product is defined in
3D space, while we are working in 2D space. However, we can consider the cross
product of two 2D vectors $\mathbf{a}=(a_{x},a_{y})$ and
$\mathbf{b}=(b_{x},b_{y})$ as
$(\mathbf{a}\times\mathbf{b})_{z}=a_{x}b_{y}-a_{y}b_{x}$, i.e. the
$z$-component of the 3D cross product of the two 3D vectors $(a_{x},a_{y},0)$
and $(b_{x},b_{y},0)$. It can also be interpreted as the determinant of the
matrix $\begin{pmatrix}a_{x}&b_{x}\\\ a_{y}&b_{y}\end{pmatrix}$..

The maximum lateral deviation $d_{max}$ and variables involved in computing it
are illustrated in Fig. 7.

We draw the reader’s attention to the fact that the maximum lateral deviation
is not the same as the lockstep maximum deviation $\delta_{max}$, which is the
maximum distance between simultaneous pairs of points of the observed
trajectory and the straight line trajectory. Although it is slightly more fine
grained, as it does not require the discretization of the intended direction
of motion, it also does not take into consideration velocity (magnitude)
information.

Figure 7: Illustration of the maximum lateral deviation $d_{max}$. The
intended direction of motion $L_{0}$ is shown with a solid line. The maximum
lateral deviation $d_{max}$ is the maximum distance between a point on $T$ and
$L_{0}$.

#####  3.6.1.5 Integral of lateral deviation $\Delta$

The integral of lateral deviation is defined as the integral of the distance
between a pedestrian’s observed trajectory $T$ and the intended direction of
motion $L_{0}$. Using the definitions of $d_{k}$ and $\mathbf{h}(t_{k})$ given
above, it can be approximated with the trapezoidal rule as

$\Delta=\sum_{k=0}^{N-2}||\mathbf{h}(t_{k+1})-\mathbf{h}(t_{k})||\frac{d_{k}+d_{k+1}}{2}.$
(12)

One notable difference of the integral of lateral deviation with the maximum
lateral deviation is that deviation on both sides of the intended direction of
motion will tend to cancel each other out (since $d_{k}$ is signed, see also
Eq. 11). This measure is therefore more sensitive to the overall deviation of
the trajectory of the pedestrian from the intended direction of motion (over
the entire course), rather than to the maximum deviation (at a single point in
space).

In Fig. 8, we show an example for the computation of integral of lateral
deviation $\Delta$, which corresponds to area of the shaded region.

Figure 8: Illustration of the integral of lateral deviation $\Delta$. The
intended direction of motion $L_{0}$ is shown with a solid line. The integral
of lateral deviation $\Delta$ is the sum of the areas of the shaded region.

#####  3.6.1.6 Dynamic time warping deviation $\delta_{DTW}$

The dynamic time warping (DTW) deviation is computed using the time warped
distance between a pedestrian’s observed trajectory $T$ and her straight line
trajectory $T_{0}$. Notably, it allows for non-linear alignment of points
[101], where each point on $T$ is matched to a point on $T_{0}$, ensuring that
the mapping is monotonically increasing in time (i.e. not going back in time).
Additionally, the first (resp. last) point of one trajectory has to be matched
to the first (resp. last) point of the other one. The cost of such mapping is
computed as the sum of the distances between the matched points, and the DTW
deviation $\delta_{DTW}$ is simply the cost of the optimal mapping and can
efficiently be computed using dynamic programming.

In Fig. 9, we show a hypothetical optimal mapping between a pedestrian’s
trajectory $T$ and her straight line trajectory $T_{0}$. For this example,
$\delta_{DTW}$ is the sum of the lengths of the dashed lines.

Figure 9: Illustration of the dynamic time warping deviation $\delta_{DTW}$.
The optimal mapping between the trajectory of the pedestrian and the straight
line trajectory is shown with black dashed lines. The dynamic time warping
deviation $\delta_{DTW}$ is the sum of the distances between the matched
points.

#####  3.6.1.7 Longest common subsequence deviation $\delta_{LCSS}$

Originally, longest common subsequence is defined for assessing the similarity
of two sequences of symbols (e.g. text strings). Specifically, it is computed
as the length of the longest subsequence that is common to these sequences,
where a subsequence can be derived from a sequence by deleting some elements
without changing the order of the remaining ones.

In the context of trajectories, the longest common subsequence [102]
concerning two trajectories $T,T^{\prime}$ denoted as $l_{LCSS}(T,T^{\prime})$
is defined as the length of the longest subsequence of points that are
considered close enough, i.e. the distance between the points is less than a
given threshold $\epsilon$.

The choice of threshold is crucial. Namely, a threshold that is too small
would make the measure overly sensitive to any residual noise that is not
filtered out during preprocessing, while a threshold that is too large would
result in considering all points as close enough, rendering
$l_{LCSS}(T,T^{\prime}))$ equal to the length of the trajectories, which would
not be informative. Given that deviations of pedestrians are usually of the
order of a few tens of centimeters, we consider that two points are close
enough, if the distance between them is less than 5 cm. This value was
empirically selected to be small enough to capture the deviations of
pedestrians, while being large enough to remain robust to noise (a position
accuracy of 34 mm was reported for the tracking system used in this work in
[67]).

Finally, we turn this measure of similarity into a distance with the
normalization proposed in [99], and apply it on a pedestrian’s observed
trajectory $T$ and her corresponding straight line trajectory $T_{0}$,

$\delta_{LCSS}(T,T_{0})=1-\frac{l_{LCSS}(T,T_{0})}{N}.$ (13)

In Fig. 10, we illustrate the computation of the longest common subsequence
deviation. The circles of radius $\epsilon$ represent the threshold under
which two points are considered close enough. Points matched when computing
the longest common subsequence are shown with dashed lines.

Figure 10: Illustration of the longest common subsequence deviation
$\delta_{LCSS}$. The circles of radius $\epsilon$ represent the threshold on
vicinity. Points connected with black dashed lines are considered close
enough.

#####  3.6.1.8 Levenshtein deviation $\delta_{Lev}$

Levenshtein distance [103] $d_{Lev}$ (also referred to as edit distance on
real sequence, or simply as edit distance, even though the latter actually
refers to a family of distance metrics) is a measure of similarity between two
trajectories. It is defined as the minimum number of edit operations that
transforms one trajectory into the other, with possible edit operations being
insertions or deletions of points. Similar to the longest common subsequence
distance, we consider that two points can be mapped, provided that the
distance between them is less than $\epsilon=5$ cm.

Levenshtein distance and LCSS are closely related and in some situations
complementary (when no points is mapped to more than one point,
$N=l_{LCSS}+d_{Lev}$, as the number of deletion/insertion is the number of
points that are not matched by the longest common subsequence).

The Levenshtein deviation $\delta_{Lev}$ is defined as the Levenshtein
distance between a pedestrian’s observed trajectory $T_{0}$ and her straight
line trajectory, i.e. $\delta_{Lev}=d_{Lev}(T,T_{0})$.

#####  3.6.1.9 Deviation index $\tilde{\tau}$

The straightness index [104] of a discrete trajectory is defined as the ratio
of the net displacement to the gross displacement (thus, also referred to as
the net-to-gross displacement ratio) [105], where the net displacement $D$, is
simply

$D=\left\|\mathbf{p}(t_{N-1})-\mathbf{p}(t_{0})\right\|,$ (14)

and the gross displacement $L$ is the sum of the distances between consecutive
positions:

$L=\sum_{k=1}^{N-1}\left\|\mathbf{p}(t_{k})-\mathbf{p}(t_{k-1})\right\|.$ (15)

The straightness index is then defined as

$\tau=\frac{D}{L}.$ (16)

From Eq. 14 and Eq. 15, one can see that $\tau$ takes values between 0 and 1.
In particular, for an infinitely tortuous trajectory ($D$ fixed, $L\to\infty$)
or a closed curve ($D=0$, $L$ fixed), $\tau$ assumes the value of 0, whereas
for a perfectly straight trajectory (i.e. $D=L$), it leads to the value of 1.
In order to be consistent with other measures of deviation, i.e. to have a
lower value for straighter trajectories, we define the deviation index
$\tilde{\tau}$ as $1-\tau$.

Fig. 11 illustrates the computation of the deviation index. The net
displacement $D$ is shown in orange and all the distances between consecutive
positions are shown in green. The gross displacement $L$ is the sum of these
distances.

Figure 11: Illustration of deviation index $\tilde{\tau}$. The net
displacement $D$ is shown in orange and the gross displacement $L$ is the sum
of the distances between consecutive positions show in green. The deviation
index is $1-\frac{D}{L}$.

####  3.6.2 Orientation based measures

In addition to the position based measures, we introduce a set of measures
that quantify the deviation of the trajectory of a pedestrian from a straight
line trajectory based on the orientation of the velocity vectors, i.e. the
direction of motion.

#####  3.6.2.1 Maximum cumulative turning angle $\theta_{max}$

To deviate from an intended trajectory, pedestrians naturally have to turn. We
can therefore quantify the deviation of the trajectory by looking at the
amount of turning performed. The cumulative turning angle until time $t_{k}$
is defined as the sum of the turning angles between consecutive velocity
vectors until time $t_{k}$212121Note that is not the same as computing the
angle between $\mathbf{v}(t_{0})$ and $\mathbf{v}(t_{k})$, since it is
theoretically possible to obtain an angle larger than $2\pi$ if the pedestrian
makes a full turn. Nonetheless, this situation should not occur in practice..

Formally, it is defined as

$\theta_{k}=\sum_{j=0}^{k-1}d\theta_{j},$ (17)

where $d\theta_{j}$ is the signed angle between the velocity vectors
$\mathbf{v}(t_{j})$ and $\mathbf{v}(t_{j+1})$,
$d\theta_{j}=\angle(\mathbf{v}(t_{j}),\mathbf{v}(t_{j+1}))$.

The turning angles being signed, the cumulative turning angle can be positive
or negative, depending on the direction of the turning (see Fig. 12). We are
interested in the maximum cumulative turning angle $\theta_{max}$, which is
the maximum of the absolute value of the cumulative turning angles over the
trajectory.

$\theta_{max}=\max_{k\in[0,N-1]}|\theta_{k}|$ (18)

In Fig. 12, we illustrate the computation of the maximum cumulative turning
angle. The left part of the figure shows the turning angles $d\theta_{j}$ and
the segment of the trajectory where the cumulative turning angle is maximum.
In the right part of the figure, we show a graph of the turning angles
$d\theta_{j}$, the cumulative turning angles $\theta_{k}$, $|\theta_{k}|$ and
the maximum cumulative turning angle $\theta_{max}$.

We note that, in contrast with the position based measures, the maximum
cumulative turning angle is an early indicator of the deviation of the
trajectory. Typically, this maximum value is not reached at the same position,
for instance, as the maximum lateral deviation. Specifically, by the time the
pedestrian has begun turning back towards their intended direction of motion,
the cumulative turning angle starts to decrease.

(a)

(b)

Figure 12: Illustration of the variables used in computing maximum cumulative
turning angle. (a) The turning angles $d\theta_{j}$ are shown in red. The
segment of the trajectory for which the cumulative turning angle is maximum is
drawn in purple. (b) Corresponding values of turning angles $d\theta_{j}$,
cumulative turning angles $\theta_{k}$, their absolute values $|\theta_{k}|$
and the maximum cumulative turning angle $\theta_{max}$.

#####  3.6.2.2 Average cumulative turning angle $\Theta$

In the same way that we defined the Euclidean deviation as the average of the
deviations, we can define the average cumulative turning angle $\Theta$ as the
average of the cumulative turning angles over the trajectory. Formally,

$\Theta=\frac{1}{N}\sum_{k=0}^{N-1}|\theta_{k}|.$ (19)

#####  3.6.2.3 Sinuosity $S$

Sinuosity was introduced in [106] as a way to quantify the randomness of an
animal’s path (especially in the case of foraging patterns, for instance when
studying how ants locate and collect food). It is a measure proportional to
the standard deviation of the distribution of the turning angles for a given
trajectory and is formally defined as

$S=1.18\frac{\sigma_{q}}{\sqrt{q}},$ (20)

where $q$ is the step size of the trajectory. According to the definition of
[106], the step size has to be constant across the trajectory. Thus, $S$
requires the trajectory $T$ to be re-discretised into another trajectory
$\tilde{T}$ with equal spacing between all the points. $\sigma_{q}$ is the
(circular) standard deviation of the turning angles $\theta$. The value $1.18$
has been numerically computed by the authors of [106] using simulations of
correlated random walks.

This measure is based on the assumption that the turning angles are drawn from
a normal distribution wrapped on a circle (with some correlation between
consecutive draws, to account for the forward tendency of the locomotion).

In Fig. 13, we illustrate the re-discretization of the trajectory and the
computation of the turning angles.

Figure 13: Illustration of sinuosity $S$. The trajectory is re-discretised to
obtain a trajectory $\tilde{T}$ with equal spacing $q$ between all the points.
The turning angles are then computed.

#####  3.6.2.4 Energy curvature $E_{\kappa}$

In two dimensions, curvature is a measure of how much a curve deviates from a
straight line. It describes the rate of change of the tangent angle of the
curve and can be computed as

$\kappa(t_{k})=\frac{||\mathbf{v}(t_{k})\times\mathbf{a}(t_{k})||}{||\mathbf{v}(t_{k})||^{3}},$
(21)

where $\mathbf{v}(t_{k})$ and $\mathbf{a}(t_{k})$ are the velocity and
acceleration vectors of the trajectory at times $t_{k}$ and
$\mathbf{a}(t_{k})$ is derived in a similar way to $\mathbf{v}(t_{k})$ as in
Eq. 1.

Another way to understand curvature is to consider the circle that best
approximates the curve at each point, i.e. the osculating circle. The
curvature at a point is then defined as the reciprocal of the radius of the
osculating circle.

In Fig. 14, we illustrate the computation of curvature. The color of the point
on the trajectory corresponds to the signed curvature at that point. We also
show the osculating circle at a point of the trajectory.

Figure 14: Illustration of curvature $\kappa$. The color of a point on the
trajectory correspond to the signed curvature at that point. The osculating
circle at a point of the trajectory is shown (the curvature at that point is
the reciprocal of the radius of the circle).

The energy curvature is defined as the integral of the square of the curvature
over the trajectory. The integration is performed using the trapezoidal rule,
and the energy curvature is given by

$E_{\kappa}=\sum_{k=0}^{N-2}(t_{k+1}-t_{k})\frac{\kappa(t_{k})^{2}+\kappa(t_{k+1})^{2}}{2}.$
(22)

In path planning, minimising the energy curvature is often considered as a way
to ensure smooth trajectories [107].

####  3.6.3 Mixed measures

Some works have proposed measures that combine position and orientation
information to quantify the deviation of a trajectory. In this section, we
present two such measures: turn intensity and suddenness of turn.

#####  3.6.3.1 Turn intensity $i$

The turn intensity was introduced in [108]. In the authors’ experiment,
participants were moving and crossing each other along a straight elongated
path ($x$-axis) and they consider instants at which the motion along the
orthogonal axis ($y$-axis) changes direction, which they refer to as “turning”
instants. A “step” is defined as the motion between two consecutive turning
instants. The “step length” is defined as the $y$ component of a step and the
“step angle” is defined as the absolute value of the angle deviation of a step
from the horizontal axis. The turn intensity is then defined as the product of
the step length and step angle.

In our case, since pedestrians might not be moving along the environment axis
(i.e. $x$ axis, as detailed in Sec. 3.4), we adapt the definitions of the
turning instants and steps to consider the deviation from the intended
direction of motion. In particular, we consider the signed angle $\psi_{k}$
between the initial velocity vector $\mathbf{v_{0}}$ and the velocity vector
$\mathbf{v}(t_{k})$ at each time step $t_{k}$,
$\psi_{k}=\angle(\mathbf{v_{0}},\mathbf{v}(t_{k}))$. The turning instants
$t_{s}$ are then defined as the instants at which $\psi_{k}$ changes sign.

We then consider the $s$-th step angle $\omega_{s}$ to be the absolute value
of the angle deviation of a step from the intended direction of motion and the
step length $\lambda_{s}$ to be the orthogonal distance between the position
of the pedestrian at the end of the step and the intended direction of motion.

$\omega_{s}=\Big{|}\angle(\mathbf{v_{0}},(\mathbf{p_{s+1}}-\mathbf{p_{s}}))\Big{|},$
(23)

$\lambda_{s}=\frac{||(\mathbf{p_{s+1}}-\mathbf{p_{s}})\times\mathbf{v_{0}}||}{||\mathbf{v_{0}}||}.$
(24)

The turn intensity is then defined as average value of the product of the step
lengths and step angles.

$i=\frac{\sum_{s=0}^{N_{S}-1}\omega_{s}\lambda_{s}}{N_{S}},$ (25)

where $N_{S}$ is the number of steps.

In Fig. 15, we illustrate the variables used in computing turn intensity $i$.

Figure 15: Illustration of the variables used in computing turn intensity $i$.
The steps are shown in orange, step angles $\omega$ are shown in green and
step lengths $\lambda$ are shown in purple.

#####  3.6.3.2 Suddenness of turn $\sigma$

The suddenness of turn was introduced in [108]. It is defined as the absolute
value of the product of the angle between the velocity vector at a given time
$t$ and the horizontal axis (since the participants were moving along the
$x$-axis in the experiments of [108]) and the change in speed, i.e. the
difference between the speed at time $t$ and the speed at time $t-dt$, $dt$
being the sampling period.

Similar to the previous measures, we adapt this definition to consider the
intended direction of motion rather than the environment (i.e. $x$) axis. In
particular, we use the angles $\psi_{k}$ defined in the previous section and
the suddenness of turn $\sigma$ is defined as

$\sigma=\frac{1}{N-2}\sum_{k=1}^{N-2}\left|\psi_{k}(||\mathbf{v}(t_{k})||-||\mathbf{v}(t_{k-1})||)\right|.$
(26)

###  3.7 Toy trajectories

To demonstrate how various deviation measures evaluate different trajectory
shapes, we examine a set of 7 artificially generated toy trajectories (see
Fig. 16). These trajectories, designed to be comparable with the actual
undisturbed and encounter paths studied in this work, were generated by
defining handcrafted waypoints and interpolating between them using a cubic
spline, such that a constant velocity of 1.2 m/s is maintained at a sampling
frequency of 33 Hz.

a. Straight line:

perfectly straight line.

b. Small deviation without recovery:

the trajectory starts straight, then deviates to the left (up to 30 cm) and
does not recover (i.e. the trajectory does not return to the straight line).

c. Small deviation with recovery:

the trajectory starts straight, then deviates to the left (up to 30 cm) and
recovers (i.e. the trajectory returns to the straight line).

d. Big deviation without recovery:

the trajectory starts straight, then deviates to the left (up to 60 cm) and
does not recover.

e. Big deviation with recovery:

the trajectory starts straight, then deviates to the left (up to 60 cm) and
recovers.

f. Fast deviation with recovery:

the trajectory starts straight, then deviates to the left (up to 30 cm) and
recovers, but the deviation is performed over a shorter distance than the
previous cases.

g. Deviation on both sides:

the trajectory starts straight, then deviates to the left (up to 30 cm), then
to the right (up to 60 cm) and recovers.

Figure 16: Toy trajectories, representing different types and levels of
deviation.

We calculate the value of deviation measures outlined in Sec. 3.6 for each of
these toy trajectories and present the results in Tab. 2. As anticipated, we
observe that all deviation measures yield a 0 or very small value for a,
identifying it as the one with least deviation222222While this information may
seem trivial, it serves as a sort of sanity check. To facilitate the
comparison of deviation values, in each column of Tab. 2 we underline the
second smallest value and highlight the highest value in bold. Additionally,
the rank of each value in a column is provided in parentheses on its right.

Just by observing the underlined and bold values, significant discrepancies
between different deviation measures become apparent. For example, “fast
deviation with recovery” f is identified to have very small deviation by
measures such as $\delta_{F}$ and $d_{max}$, but it is identified as having
the largest deviation by measures like $S$, $E_{\kappa}$, and $\sigma$. Upon
closer examination of the ranks and comparison across different columns, we
notice that trajectories other than a can be evaluated quite differently by
various deviation measures.

In particular, three trajectories (d, f, and g) are identified as having the
largest deviation by our deviation measures. Since distance-based measures are
primarily influenced by the distance from the straight-line trajectory and the
number of points significantly deviating from it, they consequently rank “big
deviation without recovery” d and “deviation on both sides” g as the most
deviated. Furthermore, suddenness of turn $\sigma$ is highest for “fast
deviation with recovery” f. In addition, since angular measures are mainly
affected by the number of turning points and the magnitude of the turning
angles, they rank “deviation on both sides” g as having the largest deviation.
Moreover, deviation index $\tilde{\tau}$ also ranks it as the most deviated
due to its highest gross displacement among all toy trajectories.

By examining the second smallest (underlined) value in each column, we notice
that position-based measures (with the exception of $\delta_{E}$, albeit with
a small difference between second and third places) rank “fast deviation with
recovery” f as the second least deviated. Since these measures are primarily
influenced by the distance to the straight line trajectory and the number of
points deviating, they evaluate f to be relatively straight, as it deviates
over the shortest distance. In contrast, position based measure
$\tilde{\tau}$, orientation based measures such as $\theta_{max}$,
$E_{\kappa}$, $S$, and mixed measures $i$ and $\sigma$ rank “small deviation
without recovery” b as the second least deviated. This is consistent with the
fact that b has the smallest gross displacement and undergoes a slower
deviation.

We highlight the fact that “fast deviation with recovery” f is ranked as both
the most deviated and the least deviated (ignoring the straight line) by
different measures. This shows that the measures capture different aspects of
the deviation, and no single measure is expected to fully capture the
complexity of deviations of real (human) trajectories. In the specific
scenarios investigated in this study, we nevertheless anticipate the
deviations not to be excessively abrupt (i.e. with low acceleration and jerk)
due to the density and geometry of the environment.

Table 2: Measures of deviation for the toy trajectories. The maximum value for
each measure is highlighted in bold and the second smallest value is
underlined (since the straight line trajectory is the smallest value for all
the measures). The ranking of the trajectories (from the straightest to the
most deviated) is also indicated in parenthesis.

Measures $\delta_{E}$ $\delta_{max}$ $\delta_{F}$ $d_{max}$ $\Delta$
$\delta_{DTW}$ $\delta_{LCSS}$ $\delta_{Lev}$ $\tilde{\tau}$ $\theta_{max}$
$\Theta$ $S$ $E_{\kappa}$ $i$ $\sigma$ a $7.61\times 10^{-17}$ (1) $4.55\times
10^{-16}$ (1) $4.55\times 10^{-16}$ (1) $0$ (1) $0$ (1) $7.61\times 10^{-17}$
(1) $8.93\times 10^{-3}$ (1) $0$ (1) $0$ (1) $0$ (1) $0$ (1) $0$ (1) $0$ (1)
$0$ (1) $0$ (1) b $1.65\times 10^{-1}$ (4) $3.37\times 10^{-1}$ (4)
$3.36\times 10^{-1}$ (4) $3.36\times 10^{-1}$ (4) $6.50\times 10^{-1}$ (5)
$1.65\times 10^{-1}$ (4) $7.14\times 10^{-1}$ (4) $7.90\times 10$ (4)
$\underline{1.76\times 10^{-3}}$ (2) $\underline{1.97\times 10^{-1}}$ (2)
$\underline{1.32\times 10^{-1}}$ (2) $\underline{1.32\times 10^{-1}}$ (2)
$\underline{4.21\times 10^{-5}}$ (2) $\underline{6.58\times 10}$ (2)
$\underline{7.31\times 10^{-2}}$ (2) c $\underline{1.35\times 10^{-1}}$ (2)
$\underline{3.18\times 10^{-1}}$ (2) $3.16\times 10^{-1}$ (3) $3.16\times
10^{-1}$ (3) $4.85\times 10^{-1}$ (4) $1.24\times 10^{-1}$ (3) $5.75\times
10^{-1}$ (3) $6.60\times 10$ (3) $1.89\times 10^{-2}$ (4) $3.77\times 10^{-1}$
(4) $1.74\times 10^{-1}$ (5) $5.18\times 10^{-1}$ (4) $6.48\times 10^{-4}$ (4)
$1.20\times 10^{2}$ (3) $2.22$ (4) d $2.98\times 10^{-1}$ (5) $6.38\times
10^{-1}$ (5) $\mathbf{6.34\times 10^{-1}}$ (7) $\mathbf{6.34\times 10^{-1}}$
(7) $\mathbf{1.17}$ (7) $\mathbf{2.98\times 10^{-1}}$ (7) $7.35\times 10^{-1}$
(5) $8.20\times 10$ (5) $7.10\times 10^{-3}$ (3) $2.62\times 10^{-1}$ (3)
$1.44\times 10^{-1}$ (3) $2.56\times 10^{-1}$ (3) $1.62\times 10^{-4}$ (3)
$1.81\times 10^{2}$ (5) $5.83\times 10^{-1}$ (3) e $3.10\times 10^{-1}$ (6)
$\mathbf{6.45\times 10^{-1}}$ (7) $6.28\times 10^{-1}$ (6) $6.28\times
10^{-1}$ (6) $9.41\times 10^{-1}$ (6) $2.55\times 10^{-1}$ (6) $7.48\times
10^{-1}$ (6) $9.50\times 10$ (6) $6.81\times 10^{-2}$ (6) $6.93\times 10^{-1}$
(5) $3.28\times 10^{-1}$ (6) $9.41\times 10^{-1}$ (5) $2.36\times 10^{-3}$ (5)
$4.60\times 10^{2}$ (6) $1.40\times 10$ (5) f $1.37\times 10^{-1}$ (3)
$3.23\times 10^{-1}$ (3) $\underline{3.00\times 10^{-1}}$ (2)
$\underline{3.00\times 10^{-1}}$ (2) $\underline{1.00\times 10^{-1}}$ (2)
$\underline{4.76\times 10^{-2}}$ (2) $\underline{1.78\times 10^{-1}}$ (2)
$\underline{2.60\times 10}$ (2) $5.91\times 10^{-2}$ (5) $9.11\times 10^{-1}$
(6) $1.64\times 10^{-1}$ (4) $\mathbf{2.84}$ (7) $\mathbf{2.71\times 10^{-2}}$
(7) $1.70\times 10^{2}$ (4) $\mathbf{6.09\times 10}$ (7) g $\mathbf{3.39\times
10^{-1}}$ (7) $6.39\times 10^{-1}$ (6) $5.43\times 10^{-1}$ (5) $5.43\times
10^{-1}$ (5) $2.04\times 10^{-1}$ (3) $2.39\times 10^{-1}$ (5)
$\mathbf{7.70\times 10^{-1}}$ (7) $\mathbf{1.04\times 10^{2}}$ (7)
$\mathbf{1.27\times 10^{-1}}$ (7) $\mathbf{9.97\times 10^{-1}}$ (7)
$\mathbf{4.40\times 10^{-1}}$ (7) $1.57$ (6) $7.32\times 10^{-3}$ (6)
$\mathbf{7.98\times 10^{2}}$ (7) $4.20\times 10$ (6)

###  3.8 Impact parameter

In this section, we briefly introduce the concept of the impact parameter,
which we previously used in [20].

In the scattering of particles in physics, the impact parameter is the
distance between the path of an incoming particle and the target particle. To
apply this concept to the study of pedestrian trajectories, we treat the
deviation of an individual from a dyad as a scattering event. We start by
transforming the trajectories of the dyad $d$ and the individual $i$ into a
reference frame that moves with the dyad. Specifically, at each time instant
the positions of the dyad and the individual are translated so that the dyad’s
center of mass is positioned at the origin, and their velocities are rotated
such that the dyad’s velocity is aligned with the positive x-axis232323In
physical scattering theory there is no need to build a special frame, since
only relative positions and velocities are used. Nevertheless in our
computation, since we use a box (see Fig. 17) which is not rotation invariant,
the definition of frame became important. The reason we use a box is that our
environment has walls and preferred directions.. In this reference frame, the
dyad remains fixed at the origin, while the individual moves towards it,
analogous to a particle approaching a target in a scattering event.

We denote the position of the individual in the reference frame as
$\mathbf{\hat{p}}_{i}$ and its velocity as $\mathbf{\hat{v}}_{i}$.

In this reference frame, the impact parameter $r_{b}$ is computed as the
distance from the dyad (positioned at the origin) to the line guided by the
individual’s velocity vectors at the beginning of the encounter (as in Sec.
3.4, we average the velocity vectors over $N_{e}$ time instants to alleviate
the impact of orientation noise).It is a measure of how close the individual
and the dyad would have passed each other if there were no collision
avoidance. We illustrate the computation of the impact parameter in Fig. 17.

Let $\mathbf{\hat{v}}_{i_{0}}$ be the average velocity vector of the
individual over the first $N_{e}$ time instants of the encounter,

$\mathbf{\hat{v}}_{i_{0}}=\frac{1}{N_{e}}\sum_{k=0}^{N_{e}-1}\mathbf{\hat{v}}_{i}(t_{k}),$
(27)

where $N_{e}$ is the same as in the computation of the desired direction of
motion (see Sec. 3.4).

We can compute $r_{b}$ as

$r_{b}=\frac{||\mathbf{\hat{v}}_{i_{0}}\times\mathbf{\hat{p}}_{i}(t_{0})||}{||\mathbf{\hat{v}}_{i_{0}}||}.$
(28)

As detailed in [20], we scale the impact parameter by the width of the dyad
(i.e. the average distance between the members of dyads with that level of
interaction) to obtain a dimensionless measure $\bar{r}_{b}$ that better
captures the relative distance between the individual and the member of the
dyad. In particular, a value of $\bar{r}_{b}$ smaller than $0.5$ indicates
that the individual would have passed through the dyad.

Figure 17: Illustration of the impact parameter. The individual’s trajectory
is transformed into a reference frame that moves with the dyad. The impact
parameter $r_{b}$ is the distance from the dyad to the line guided by the
individual’s velocity vectors at the beginning of the encounter.

##  4 Results

###  4.1 Undisturbed situations

We begin by comparing the deviation of individuals and dyads in undisturbed
situations, which we argue to constitute a baseline for the amount of
deviation that can be expected in the absence of any encounter. As noted in
Sec. 3.5.1, undisturbed situations occur when individuals or dyads are walking
without any other pedestrian within a distance of at least 4 m along or
perpendicular to their direction of motion.

In Tab. 3 $\sim$ Tab. 17, we present the average deviation for dyads
categorised by levels of interaction ranging from 0 to 3, all dyads, and
individuals, with respect to each measure introduced in Sec. 3.6.

For convenient comparison with individuals, we also display the ratio of the
average deviation of the dyads to the average deviation of the individuals.
Additionally, we present the Kruskal-Wallis test $p$-value for the difference
of means between dyads with various level of interaction, and the Welch T-test
$p$-value for the difference of means between individuals and all dyads [109,
110].

The initial noteworthy finding is that, across the vast majority of measures,
deviation is significantly higher for dyads (averaged across all levels of
interaction) compared to individuals. However, to accurately interpret this
observation, it is essential to delve into the allocation of attentional
resources during social interaction, achieved by contrasting the deviations of
dyads with varying levels of interaction. Examining the breakdown of
normalised deviation values according to the level of interaction of the dyad,
we observe a trend where the level of interaction correlates with an increase
in deviation. Interestingly, the deviations of non-interacting dyads appear to
be smaller to those of individuals, while interacting dyads which exhibit
comparable or higher deviations.

###  4.2 Encounters

We now turn to the encounter situations, which correspond to the situations in
which dyads and individuals pass each other frontally at a distance less than
4 m [32, 91, 52]. We first contrast deviations during encounters with
undisturbed situations. Subsequently, we conduct a closer examination of
encounters, comparing deviations of individuals and dyads during these cases.
Finally, we provide insights into the effect of dyads’ level of interaction on
both their own deviation and the deviation of the individuals involved.

####  4.2.1 Deviations during encounters

In Tab. 18 $\sim$ Tab. 32 we show the amount of deviation of individuals and
dyads for each measure introduced in Sec. 3.6. It is crucial to emphasise that
when reporting deviations related to encounter situations, we categorise not
only the values of dyads but also those of individuals, with respect to the
level of social interaction of the dyad involved in the encounter.

We notice that individuals have a tendency to deviate more when encountering
dyads with high or medium interaction levels (i.e. levels 3, 2) compared to
when they encounter non- or weakly interacting ones (i.e. levels 0, 1). In
particular, the deviation when encountering a strongly interacting dyad (level
3) is almost always242424Except for $\Delta$, $\delta_{LCSS}$ an $\sigma$.
larger than for other levels of interaction. However, upon subjecting the data
to a Kruskal-Wallis test, we see that the differences in deviation for the
individual when encountering dyads with varying levels of interaction are not
statistically significant for most metrics (all except energy curvature
$E_{\kappa}$ and sinuosity $S$). Conversely, the variation in dyad deviation
is statistically significant but does not exhibit a clear pattern, with dyads
of interaction levels 0 and 3 deviating more than those with interaction
levels of 1 and 2. Finally, averaging across all interaction levels,
individuals deviate more than dyads in a statistically significant manner for
all measures.

####  4.2.2 Comparison of deviations during encounters and undisturbed
situations

The results from Tab. 18 $\sim$ Tab. 32 can be better understood when
contrasted with undisturbed situations. To facilitate this comparison, Tab. 33
$\sim$ Tab. 47 provide the ratio of average deviations during encounters to
the corresponding values observed in undisturbed scenarios.

One notable observation regarding individuals is the consistent and steady
increase in deviation during encounters across the majority of measures (see
ratios of individuals in Tab. 33 $\sim$ Tab. 47). Specifically, the ratios are
consistently greater than 1 for all measures except suddenness of turn
$\sigma$ and energy curvature $E_{\kappa}$, which we believe to be caused by
the noise introduced by further differentiation when deriving the
accelerations.

A similar increase is noted for dyads with 0 interaction levels, exhibiting
generally large ration ($>1.5$), whereas higher interaction levels show a
ratio close to 1, suggesting minimal change. Furthermore, we observe a
decrease in the ratios with increasing interaction level, with statistically
significant differences between various interaction levels. Additionally, a
statistically significant difference is evident between the ratios of
individuals and dyads after averaging across all interaction levels (see
T-test $p$-values in the bottom line of Tab. 33 $\sim$ Tab. 47)

These findings are further clarified by Tab. 48 $\sim$ Tab. 62, which provide
$p$-values for the difference between undisturbed situations and encounters
for both individuals and dyads across all interaction levels. Upon averaging
over interaction levels, both individuals and dyads exhibit a statistically
significant difference between undisturbed and encounter scenarios for all
measures. However, while this disparity holds true for most measures for
individuals even when considering the interaction level of the encountered
dyad, it appears that only dyads with 0 interaction levels demonstrate a
distinct behavior in encounters compared to undisturbed situations.

Finally, we show in Tab. 63 $\sim$ Fig. 77 the $p$-values for the difference
of means between dyads with various level of interaction and individuals
encountering these dyads. We see that the differences in the deviation between
the dyads and the individuals are often significant for interaction (levels 1,
2 and 3), but not for the non-interacting dyads.

The results presented thus far do not take into account the initial risk of
collision, as we only set a threshold of 4 m to qualify an encounter. Among
these encounters, some might have been more critical than others, with
individual and dyad even facing a risk of collision. In the following section,
we investigate the impact of the initial risk of collision on the deviation of
individuals and dyads, by considering the impact parameter.

###  4.3 Effect of the impact parameter

As detailed in Sec. 3.8, the impact parameter is a measure of how close the
individual would have passed the dyad if there were no collision avoidance
behavior involved. In particular, $\bar{r}_{b}$ is a dimensionless measure
that indicates the initial risk of collision, since it measures the distance
between the individual and the dyad relative to the dyad’s width252525The
width of the dyad is the average distance between the members of dyads with
that level of interaction..

We chose to bin the values of $\bar{r}_{b}$ into 4 bins of equal size, which
we interpret as follows: a value of $\bar{r}_{b}$ smaller than 1 indicates
that the individual is on track to pass through or collide with the dyad.
Between 1 and 2, the individual is close to the dyad, but not on track to pass
through it. Between 2 and 3, the individual is further away from the dyad and
may not need to deviate significantly to pass comfortably. Finally, a value of
$\bar{r}_{b}$ larger than 3 indicates that the individual is far from the dyad
and does not need to deviate.

In Fig. 20 $\sim$ Fig. 34, we illustrate the ratio of the average deviation
during encounters to the average deviation in undisturbed situations for
individuals and dyads with respect to the normalised impact parameter
$\bar{r}_{b}$ for each measure. We also provide the T-test $p$-values for the
difference in the ratio between dyads with low interaction levels (0 and 1)
and dyads with high interaction levels (2 and 3) for each bin of
$\bar{r}_{b}$.

Because the binning necessarily reduces the amount of data, in particular for
interaction with already limited data (e.g. levels 0 and 3), we chose to
separate encounters into two categories based on the level of interaction of
the dyad: one with interaction levels 0 and 1, and another with levels 2 and
3. This helps balancing the number of data points in each and to get
comparable sizes of samples for both classes. We argue that this
categorisation is not unreasonable, as it allows to contrast low and high
interaction levels.

For individuals, we observe that the ratio is generally262626Except for the
suddenness of turn $\sigma$. higher when encountering dyads with higher
interaction levels (2 and 3) compared to those with lower interaction levels
(0 and 1). This is consistent with the results presented in Sec. 4.2.2, where
we observed that the deviation of individuals was higher when encountering
dyads with higher interaction levels. We observe that the associated T-test
$p$-values generally follow a similar pattern for the position-based
measures272727Except for $\delta_{LCSS}$ and $\delta_{Lev}$., where the
difference in the ratio between the two classes of dyads is statistically
significant (or close to being significant) in the first and third bins of
$\bar{r}_{b}$ (i.e. when the individual is on track to pass through the dyad
or when the individual is relatively far from the dyad).

For the dyads, in accordance with the results of Sec. 4.2.2, we observe that
the ratio of the lower interaction levels (0 and 1) is systematically higher
than that of the higher interaction levels (2 and 3) for all measures and all
values of $\bar{r}_{b}$. For all measures except $\sigma$, the difference in
the ratio between the two classes of dyads is statistically significant in the
second bin of $\bar{r}_{b}$ (i.e. when the individual is close to the dyad but
not on track to pass through it).

###  4.4 Correlation between measures

In Sec. 3.7, we demonstrated that different measures of deviation may capture
distinct aspects of deviation. Some measures may be more sensitive to the
magnitude of deviation, while others may be more sensitive to its abruptness.
In this section, our objective is to assess the consistency of deviation
measures to some extent.

To that end, we illustrate Spearman correlation between the measures of
deviation for all trajectories (dyads and individuals) in both undisturbed and
encounter situations in Fig. 18. It is evident from this figure that the
deviation measures generally exhibit correlation, with coefficients exceeding
0.5 for most measures.

Notably, sinuosity, curvature, and suddenness of turn display lower
correlations with other measures. We attribute this observation to the fact
that their computation involves some form of differentiation, such as deriving
turning angles from velocity vectors for $S$, and calculating $\sigma$ and
$E_{\kappa}$ using velocity and acceleration values.

We also note that the correlation between the measures is generally larger for
the undisturbed situations than for the encounters. We believe this result to
be partly due to the fact that the number of encounters is smaller than the
number of undisturbed situations, which might make the correlation less
reliable.

To further investigate the correlation between the measures, we visualise the
dependence of the different measures against the lockstep maximum deviation in
Fig. 19. The deviation $\delta_{max}$ is binned in 16 bins of equal size, and
the average value of the different measures is computed for each bin. We can
see that the measures are all increasing with the lockstep maximum deviation,
which is consistent with the idea that the measures are correlated. We see
that although most measures seem to be linearly correlated with the lockstep
maximum deviation, the rate of increase for $\delta_{LCSS}$ and $\delta_{Lev}$
slows down for large values of $\delta_{max}$. The main reason is that these
two measures are dimensionless values (see Sec. 3.6) which are impacted by the
number of points in the trajectory that are far away from the undisturbed
trajectory rather than the actual amplitude of the deviation.

Figure 18: Spearman correlation between the measures of deviation. Figure 19:
Average value of the different measures of deviation as a function of the
lockstep maximum deviation. Values are binned in 16 bins of equal size. The
scale of the y-axis is logarithmic for $E_{\kappa}$.

##  5 Discussion

We interpret the presented results considering two kinds of effects: the
dynamic stability of the considered entity (individual or dyad) and the amount
of awareness regarding the environment. We argue that the amount of deviation
observed is a result of the interplay between these two factors.

The deviation of the non- and weakly interacting dyads is seen to be
significantly smaller than the deviation of the individuals, which implies
that dyads keep a more straight trajectory than individuals. This suggests
that dyads maintain a more consistent trajectory compared to individuals,
potentially due to their inclination to remain physically close, thereby
constraining deviations from the intended path. Using a physical analogy, the
inertia of the system composed of the two members of the dyad can be expected
to be larger than the inertia of the individual, which would make the dyad
more stable.

In addition, awareness about the changes in the environment is argued to be
directly related to the level of interaction of the dyad. Namely, it is
reasonable to assume that non interacting dyads in undisturbed situations are
more attentive to their surroundings, since they do not need to allocate
attention towards their social interaction. Consequently, in undisturbed
scenarios, these dyads are more aware of the availability of economical
straight paths and are more likely to adopt them compared to interacting
dyads, which may deviate due to their internal -social interaction- dynamics.

During encounters, the fact that all dyads deviate significantly less than the
individuals suggests that they contribute less to the avoidance process. In
addition, the deviation of non-interacting dyads is higher than that of
interacting dyads, which suggests that the level of interaction affects the
dyad’s ability to focus on the environment. This is further supported by the
fact that the deviation of individuals increases significantly during
encounters, regardless of the level of interaction of the dyad. This indicates
that individuals anticipate (or react to) the diminished involvement of dyads
in collision avoidance and adjust their deviations accordingly282828Despite
their simplicity, force based models such as the traditional social force
model [19] might predict such results, since the repulsive force perceived by
the individual would be the sum of the repulsive forces perceived by the two
members of the dyad.

All the used measures show an increase in individuals’ deviation when
encountering dyads with higher interaction levels, although no measure
suggests statistical significance. Nevertheless, our prior investigation
focusing on relative dynamics [20] revealed an interaction-level dependent
statistical difference in encounters, particularly when such encounters could
lead to a collision without deviation. By combining these findings with those
of the current study, which show that in encounters between an interacting
dyad and an individual, the latter bears the burden of collision avoidance, we
can deduce that the individual’s behavior is influenced by the dyad’s level of
interaction.

The impact parameter analysis provides further insights into the dynamics of
the encounters. The observed patterns in the ratios of deviation during
encounters to undisturbed situations for individuals and dyads with respect to
the impact parameter suggest that the initial risk of collision influences the
deviation of both individuals and dyads.

Concerning the individual, the difference between high and low interaction
levels is most pronounced when the individual is on track to pass close to the
dyad, with the deviation of the high interaction levels being significantly
higher than that of the low interaction levels. We believe that is an effect
of the intrusion phenomena that we observed in [20], where the individual is
more likely to pass through the dyad when the dyad is less reactive,
essentially non-deviating and maintaining a straight trajectory. In situations
where the encounter is expected to be close but not on a collision course, the
deviation of the individual is less dependent on the dyad’s level of
interaction, which might be due to the fact that the individual has to deviate
regardless of the dyad’s involvement in social interaction. For less critical
situations, the deviation of the individuals is again more dependent on the
dyads’ level of interaction. We believe that this is due to the fact that in
such situations the individuals mostly do not need to avoid low interacting
dyads, since they walk on straight paths, while they more often need to avoid
high interacting ones, due to their wandering behaviour. Finally, for high
values of the impact parameter, the deviation of the individual does not
depend on the dyad’s level of interaction, since there is not much risk of
collision and the individual essentially doesn’t need to deviate.

For the dyad, the difference between high and low interaction levels is most
pronounced when the individual and dyad are on track to pass close to each
other, with the deviation of the low interaction levels being significantly
higher than that of the high interaction levels. We hypothesise that this is
where the effect of the dyad’s awareness of the environment is most
pronounced, as the dyad with low interaction levels is more likely to be
focused on the environment and react to the individual’s presence. For very
low values of the impact parameter, there is no difference between the
deviation of the dyads with different interaction levels, which might arise
from the fact that the situation is so critical that the dyad has to react
regardless of its involvement in social interaction by either opening up or
deviating. At higher values of the impact parameter, the deviation of the
dyads with different interaction levels is also similar, which might be due to
the fact that the situation is less critical and the dyad can afford to be
less reactive and the stability takes over.

While previous research has suggested that utilizing angular variables [111,
108] or velocity adjustments [97, 112] to assess deviations from the intended
path could offer a promising approach, our study faced limitations in this
regard. As discussed in Sec. 4.4, the trajectory data we utilised may not
adequately support differentiation, resulting in certain measures (such as
$\sigma$, $E_{\kappa}$, and $S$) being less effective in capturing deviations
compared to others. While controlled experiments can employ various wearable
sensors to accurately register position or acceleration [113, 114],
implementing such methods in real-world settings poses challenges.
Environmental sensors may be sparse or susceptible to measurement noise and
clutter, making it difficult to derive accurate values in naturalistic
environments. Although higher-quality tracking data can be obtained in
laboratory settings using wearable sensors, it remains uncertain whether the
collected data accurately reflects the behaviour of human subjects in
naturalistic settings, as discussed in Sec. 1 [29]. Moreover, with more
detailed data, such as limb orientations, it could become feasible to
investigate how pedestrians manage deviations. This could involve analyzing
adjustments in step lengths or cephalocaudal reorientations [32, 115].

Evidence from ethology suggests that interactions among collectively moving
non-human animals, such as bird flocks, are primarily influenced by
topological distance rather than metric distance [92]. While there is ongoing
debate regarding whether neighborhood in human motion is determined in a
topological or metric manner [17], it is important to consider that the
conclusions drawn from ethological studies may not directly apply to our
scenario. This is because collective behaviour, which is characterised by
emergent complex patterns, typically arises in situations involving a large
number of individuals, whereas our scenario involves low density.

Research on collision avoidance often examines metrics such as the time or
distance to the predicted point of collision [116, 117]. For instance, Olivier
et al. define the Minimal Predicted Distance (MDP) as the anticipated closest
distance between participants at a given time, assuming no adjustments are
made to their velocity vectors [87]. Similarly, Bhojwani et al. introduce the
Theoretical Point of Collision (TPC), representing the point where a collision
with an approaching virtual pedestrian would occur if no locomotor adjustments
are made [21]. These metrics, akin to $d_{max}$, primarily focus on the point
of collision rather than the deviation from the intended path. They also
consider the involved parties in the encounter in a more integrated manner
compared to our isolated approach. In future research, it would be beneficial
to explore how deviation metrics evolve over time and their relationship with
these collision distance measures.

Another aspect of pedestrian dynamics that should be investigated is the
behaviour of overtaking or following others, whether they are individuals or
groups. Given the typically faster and more flexible nature of individuals
compared to groups, it is reasonable to expect that individuals would engage
in overtaking more frequently than groups. Moreover, individuals may tend to
overtake groups more frequently than other individuals, again due to their
greater speed and maneuverability. Interestingly, the decision of individuals
to overtake groups may not necessarily depend on the social relation or
interaction within the group. Since individuals approaching from behind may
have limited visibility of the group’s social features or intentions, their
overtaking behaviour may primarily be driven by considerations such as speed
and convenience rather than social dynamics.

##  6 Conclusion

Over the last few decades, numerous pedestrian collision avoidance models have
emerged, offering insights into various aspects of pedestrian behavior [118,
117]. However, despite this extensive body of work, most microscopic models
predominantly focus on one-on-one collision avoidance scenarios, neglecting
the dynamics involving groups [119]. This oversight is notable considering
that groups represent a substantial portion of crowds in real-world settings
[120, 40]. Thus, understanding the mechanisms underlying group-individual
collision avoidance has remained a gap in pedestrian dynamics research.

Addressing this gap, our study aims to contribute to the understanding of how
groups and individuals navigate and avoid collisions in crowded environments,
shedding light on this understudied aspect of pedestrian behavior. In
particular, by analyzing ecological data and considering various deviation
measures, this study presents insights into the stability and deviation
dynamics of dyads and individuals.

In particular, concerning individuals, we demonstrate a heightened inclination
to avoid interacting dyads compared to non-interacting ones. This suggests
that individuals not only anticipate the future paths of others but also
assess their contribution to collision avoidance. It is probable that upon
detecting a lack of involvement, individuals adjust their behavior based on
their internal model of human navigation.

Regarding dyads, our findings suggest that, in undisturbed situations, non-
interacting ones, exhibit small deviations from their intended trajectories,
indicating a high level of efficiency and stability. Conversely, interacting
dyads show larger deviations, possibly attributable to being more consumed
with their internal dynamics. Additionally, interacting dyads are less
impacted by encounters compared to individuals, while non-interacting dyads
display a similar pattern to individuals.

Based on these observations, we believe that being part of a group affects the
behavior of a pedestrian in two, possibly contrasting, ways. A dynamical
effect, i.e. the tendency to move together, appears to increase the stability
and economy of motion. On the other hand, a social effect diverts attention
from the environment, resulting in a less economical movement, in a way
similar to how the usage of smart phones affects pedestrian behavior [108].

We believe that this study not only provides valuable insights into the
complex dynamics of group-individual collision avoidance, but it also has
tangible implications and has the potential to open up new avenues of
technical improvement. In particular, it holds practical significance in
various real-world scenarios, such as urban planning and crowd management.
Namely, with such insights, we can inform the development of more effective
pedestrian flow management strategies, improve the design of urban spaces to
leverage safety and mobility efficiency, and enhance the planning of events or
gatherings to reduce congestion and potential hazards. Additionally, our
results can help the development of intelligent systems for autonomous
vehicles, robots, or ubiquitous technology, providing safer and more seamless
interactions with human pedestrians in diverse smart environments.

Table 3: Average and standard deviation of the Euclidean distance $\delta_{E}$ (in m), for dyads and individuals in undisturbed situations. The Kruskal-Wallis $p$-value for the difference of means between dyads with various level of interaction and the Welch T-test $p$-value for the difference of means between all individuals against all dyads are also shown. Intensity of interaction | $\delta_{E}$ | Normalized |  $p$-value
---|---|---|---
0 | $6.74\times 10^{-2}\pm 4.54\times 10^{-2}$ | $0.80$ | $\mathbf{3.21\times 10^{-4}}$
1 | $8.71\times 10^{-2}\pm 5.71\times 10^{-2}$ | $1.03$
2 | $9.93\times 10^{-2}\pm 8.97\times 10^{-2}$ | $1.18$
3 | $1.22\times 10^{-1}\pm 7.54\times 10^{-2}$ | $1.44$
All dyads | $1.00\times 10^{-1}\pm 8.31\times 10^{-2}$ | $1.19$ | $\mathbf{2.18\times 10^{-4}}$
Individuals | $8.44\times 10^{-2}\pm 7.86\times 10^{-2}$ | $1.00$
Table 4: Average and standard deviation of the lockstep maximum deviation $\delta_{max}$ (in m), for dyads and individuals in undisturbed situations. The Kruskal-Wallis $p$-value for the difference of means between dyads with various level of interaction and the Welch T-test $p$-value for the difference of means between all individuals against all dyads are also shown. Intensity of interaction | $\delta_{max}$ | Normalized |  $p$-value
---|---|---|---
0 | $1.89\times 10^{-1}\pm 1.32\times 10^{-1}$ | $0.77$ | $\mathbf{1.01\times 10^{-3}}$
1 | $2.49\times 10^{-1}\pm 1.66\times 10^{-1}$ | $1.01$
2 | $2.69\times 10^{-1}\pm 2.07\times 10^{-1}$ | $1.09$
3 | $3.21\times 10^{-1}\pm 1.83\times 10^{-1}$ | $1.31$
All dyads | $2.72\times 10^{-1}\pm 1.97\times 10^{-1}$ | $1.11$ | $\mathbf{9.37\times 10^{-3}}$
Individuals | $2.46\times 10^{-1}\pm 1.92\times 10^{-1}$ | $1.00$
Table 5: Average and standard deviation of the Fréchet deviation $\delta_{F}$ (in m), for dyads and individuals in undisturbed situations. The Kruskal-Wallis $p$-value for the difference of means between dyads with various level of interaction and the Welch T-test $p$-value for the difference of means between all individuals against all dyads are also shown. Intensity of interaction | $\delta_{F}$ | Normalized |  $p$-value
---|---|---|---
0 | $1.89\times 10^{-1}\pm 1.32\times 10^{-1}$ | $0.77$ | $\mathbf{1.21\times 10^{-3}}$
1 | $2.48\times 10^{-1}\pm 1.66\times 10^{-1}$ | $1.01$
2 | $2.65\times 10^{-1}\pm 1.98\times 10^{-1}$ | $1.08$
3 | $3.17\times 10^{-1}\pm 1.80\times 10^{-1}$ | $1.29$
All dyads | $2.69\times 10^{-1}\pm 1.91\times 10^{-1}$ | $1.10$ | $\mathbf{1.60\times 10^{-2}}$
Individuals | $2.45\times 10^{-1}\pm 1.92\times 10^{-1}$ | $1.00$
Table 6: Average and standard deviation of the maximum lateral deviation $d_{max}$ (in m), for dyads and individuals in undisturbed situations. The Kruskal-Wallis $p$-value for the difference of means between dyads with various level of interaction and the Welch T-test $p$-value for the difference of means between all individuals against all dyads are also shown. Intensity of interaction | $d_{max}$ | Normalized |  $p$-value
---|---|---|---
0 | $1.52\times 10^{-1}\pm 1.30\times 10^{-1}$ | $0.75$ | $\mathbf{2.27\times 10^{-2}}$
1 | $1.99\times 10^{-1}\pm 1.50\times 10^{-1}$ | $0.98$
2 | $2.02\times 10^{-1}\pm 1.36\times 10^{-1}$ | $1.00$
3 | $2.37\times 10^{-1}\pm 1.50\times 10^{-1}$ | $1.17$
All dyads | $2.06\times 10^{-1}\pm 1.41\times 10^{-1}$ | $1.02$ | $6.76\times 10^{-1}$
Individuals | $2.03\times 10^{-1}\pm 1.49\times 10^{-1}$ | $1.00$
Table 7: Average and standard deviation of the integral of lateral deviation $\Delta$ (in m2), for dyads and individuals in undisturbed situations. The Kruskal-Wallis $p$-value for the difference of means between dyads with various level of interaction and the Welch T-test $p$-value for the difference of means between all individuals against all dyads are also shown. Intensity of interaction | $\Delta$ | Normalized |  $p$-value
---|---|---|---
0 | $2.07\times 10^{-1}\pm 1.76\times 10^{-1}$ | $0.79$ | $\mathbf{1.39\times 10^{-2}}$
1 | $2.62\times 10^{-1}\pm 1.95\times 10^{-1}$ | $0.99$
2 | $2.81\times 10^{-1}\pm 2.02\times 10^{-1}$ | $1.07$
3 | $3.37\times 10^{-1}\pm 2.26\times 10^{-1}$ | $1.28$
All dyads | $2.85\times 10^{-1}\pm 2.07\times 10^{-1}$ | $1.08$ | $\mathbf{4.22\times 10^{-2}}$
Individuals | $2.63\times 10^{-1}\pm 2.05\times 10^{-1}$ | $1.00$
Table 8: Average and standard deviation of the dynamic time warping deviation $\delta_{DTW}$ (in m), for dyads and individuals in undisturbed situations. The Kruskal-Wallis $p$-value for the difference of means between dyads with various level of interaction and the Welch T-test $p$-value for the difference of means between all individuals against all dyads are also shown. Intensity of interaction | $\delta_{DTW}$ | Normalized |  $p$-value
---|---|---|---
0 | $5.88\times 10^{-2}\pm 4.70\times 10^{-2}$ | $0.78$ | $\mathbf{6.02\times 10^{-3}}$
1 | $7.58\times 10^{-2}\pm 5.55\times 10^{-2}$ | $1.00$
2 | $8.23\times 10^{-2}\pm 6.35\times 10^{-2}$ | $1.09$
3 | $9.71\times 10^{-2}\pm 6.17\times 10^{-2}$ | $1.28$
All dyads | $8.31\times 10^{-2}\pm 6.21\times 10^{-2}$ | $1.10$ | $\mathbf{2.64\times 10^{-2}}$
Individuals | $7.57\times 10^{-2}\pm 7.36\times 10^{-2}$ | $1.00$
Table 9: Average and standard deviation of the longest common subsequence deviation $\delta_{LCSS}$ (dimensionless), for dyads and individuals in undisturbed situations. The Kruskal-Wallis $p$-value for the difference of means between dyads with various level of interaction and the Welch T-test $p$-value for the difference of means between all individuals against all dyads are also shown. Intensity of interaction | $\delta_{LCSS}$ | Normalized |  $p$-value
---|---|---|---
0 | $3.08\times 10^{-1}\pm 2.10\times 10^{-1}$ | $0.77$ | $\mathbf{3.63\times 10^{-2}}$
1 | $4.01\times 10^{-1}\pm 1.63\times 10^{-1}$ | $1.01$
2 | $4.16\times 10^{-1}\pm 1.70\times 10^{-1}$ | $1.05$
3 | $4.53\times 10^{-1}\pm 1.65\times 10^{-1}$ | $1.14$
All dyads | $4.17\times 10^{-1}\pm 1.72\times 10^{-1}$ | $1.05$ | $\mathbf{4.17\times 10^{-2}}$
Individuals | $3.98\times 10^{-1}\pm 1.77\times 10^{-1}$ | $1.00$
Table 10: Average and standard deviation of the Levenshtein deviation $\delta_{Lev}$ (dimensionless), for dyads and individuals in undisturbed situations. The Kruskal-Wallis $p$-value for the difference of means between dyads with various level of interaction and the Welch T-test $p$-value for the difference of means between all individuals against all dyads are also shown. Intensity of interaction | $\delta_{Lev}$ | Normalized |  $p$-value
---|---|---|---
0 | $3.31\times 10\pm 2.38\times 10$ | $0.80$ | $\mathbf{4.81\times 10^{-4}}$
1 | $4.33\times 10\pm 1.99\times 10$ | $1.04$
2 | $5.03\times 10\pm 2.65\times 10$ | $1.21$
3 | $5.84\times 10\pm 2.76\times 10$ | $1.41$
All dyads | $5.01\times 10\pm 2.64\times 10$ | $1.21$ | $\mathbf{<10^{-4}}$
Individuals | $4.14\times 10\pm 2.19\times 10$ | $1.00$
Table 11: Average and standard deviation of the deviation index $\tilde{\tau}$ (dimensionless), for dyads and individuals in undisturbed situations. The Kruskal-Wallis $p$-value for the difference of means between dyads with various level of interaction and the Welch T-test $p$-value for the difference of means between all individuals against all dyads are also shown. Intensity of interaction | $\tilde{\tau}$ | Normalized |  $p$-value
---|---|---|---
0 | $8.40\times 10^{-4}\pm 8.84\times 10^{-4}$ | $0.58$ | $\mathbf{5.46\times 10^{-3}}$
1 | $1.30\times 10^{-3}\pm 1.59\times 10^{-3}$ | $0.89$
2 | $1.73\times 10^{-3}\pm 5.65\times 10^{-3}$ | $1.19$
3 | $1.77\times 10^{-3}\pm 1.63\times 10^{-3}$ | $1.21$
All dyads | $1.65\times 10^{-3}\pm 4.66\times 10^{-3}$ | $1.13$ | $4.19\times 10^{-1}$
Individuals | $1.46\times 10^{-3}\pm 3.85\times 10^{-3}$ | $1.00$
Table 12: Average and standard deviation of the maximum cumulative turning angle $\theta_{max}$ (in rad), for dyads and individuals in undisturbed situations. The Kruskal-Wallis $p$-value for the difference of means between dyads with various level of interaction and the Welch T-test $p$-value for the difference of means between all individuals against all dyads are also shown. Intensity of interaction | $\theta_{max}$ | Normalized |  $p$-value
---|---|---|---
0 | $1.18\times 10^{-1}\pm 6.98\times 10^{-2}$ | $0.81$ | $\mathbf{3.92\times 10^{-3}}$
1 | $1.37\times 10^{-1}\pm 6.87\times 10^{-2}$ | $0.95$
2 | $1.56\times 10^{-1}\pm 1.92\times 10^{-1}$ | $1.08$
3 | $1.83\times 10^{-1}\pm 1.38\times 10^{-1}$ | $1.27$
All dyads | $1.57\times 10^{-1}\pm 1.69\times 10^{-1}$ | $1.08$ | $1.44\times 10^{-1}$
Individuals | $1.45\times 10^{-1}\pm 1.24\times 10^{-1}$ | $1.00$
Table 13: Average and standard deviation of the integral cumulative turning angle $\Theta$ (in rad), for dyads and individuals in undisturbed situations. The Kruskal-Wallis $p$-value for the difference of means between dyads with various level of interaction and the Welch T-test $p$-value for the difference of means between all individuals against all dyads are also shown. Intensity of interaction | $\Theta$ | Normalized |  $p$-value
---|---|---|---
0 | $5.12\times 10^{-2}\pm 3.93\times 10^{-2}$ | $0.77$ | $\mathbf{9.48\times 10^{-3}}$
1 | $6.52\times 10^{-2}\pm 4.58\times 10^{-2}$ | $0.97$
2 | $6.91\times 10^{-2}\pm 6.78\times 10^{-2}$ | $1.03$
3 | $8.53\times 10^{-2}\pm 8.10\times 10^{-2}$ | $1.27$
All dyads | $7.07\times 10^{-2}\pm 6.74\times 10^{-2}$ | $1.06$ | $2.58\times 10^{-1}$
Individuals | $6.69\times 10^{-2}\pm 5.37\times 10^{-2}$ | $1.00$
Table 14: Average and standard deviation of the sinuosity $S$ (in rad/m${}^{\frac{1}{2}}$), for dyads and individuals in undisturbed situations. The Kruskal-Wallis $p$-value for the difference of means between dyads with various level of interaction and the Welch T-test $p$-value for the difference of means between all individuals against all dyads are also shown. Intensity of interaction | $S$ | Normalized |  $p$-value
---|---|---|---
0 | $1.41\times 10^{-1}\pm 4.63\times 10^{-2}$ | $1.15$ | $\mathbf{1.03\times 10^{-2}}$
1 | $1.40\times 10^{-1}\pm 3.75\times 10^{-2}$ | $1.15$
2 | $1.73\times 10^{-1}\pm 2.42\times 10^{-1}$ | $1.42$
3 | $1.83\times 10^{-1}\pm 1.02\times 10^{-1}$ | $1.50$
All dyads | $1.69\times 10^{-1}\pm 2.01\times 10^{-1}$ | $1.39$ | $\mathbf{<10^{-4}}$
Individuals | $1.22\times 10^{-1}\pm 1.04\times 10^{-1}$ | $1.00$
Table 15: Average and standard deviation of the energy curvature $E_{\kappa}$ (in s/m2), for dyads and individuals in undisturbed situations. The Kruskal-Wallis $p$-value for the difference of means between dyads with various level of interaction and the Welch T-test $p$-value for the difference of means between all individuals against all dyads are also shown. Intensity of interaction | $E_{\kappa}$ | Normalized |  $p$-value
---|---|---|---
0 | $5.79\times 10^{-8}\pm 4.68\times 10^{-8}$ | $5.85\times 10^{-5}$ | $\mathbf{2.53\times 10^{-3}}$
1 | $5.64\times 10^{-8}\pm 4.55\times 10^{-8}$ | $5.71\times 10^{-5}$
2 | $1.99\times 10^{-3}\pm 3.43\times 10^{-2}$ | $2.01$
3 | $4.48\times 10^{-4}\pm 2.95\times 10^{-3}$ | $0.45$
All dyads | $1.38\times 10^{-3}\pm 2.78\times 10^{-2}$ | $1.40$ | $7.80\times 10^{-1}$
Individuals | $9.88\times 10^{-4}\pm 2.34\times 10^{-2}$ | $1.00$
Table 16: Average and standard deviation of the turn intensity $i$ (in deg$\times$cm), for dyads and individuals in undisturbed situations. The Kruskal-Wallis $p$-value for the difference of means between dyads with various level of interaction and the Welch T-test $p$-value for the difference of means between all individuals against all dyads are also shown. Intensity of interaction | $i$ | Normalized |  $p$-value
---|---|---|---
0 | $3.38\times 10\pm 5.04\times 10$ | $0.59$ | $\mathbf{6.51\times 10^{-3}}$
1 | $5.59\times 10\pm 1.06\times 10^{2}$ | $0.98$
2 | $5.50\times 10\pm 8.71\times 10$ | $0.96$
3 | $7.42\times 10\pm 9.49\times 10$ | $1.30$
All dyads | $5.77\times 10\pm 9.04\times 10$ | $1.01$ | $9.29\times 10^{-1}$
Individuals | $5.72\times 10\pm 1.05\times 10^{2}$ | $1.00$
Table 17: Average and standard deviation of the suddenness of turn $\sigma$ (in deg$\times$cm/s), for dyads and individuals in undisturbed situations. The Kruskal-Wallis $p$-value for the difference of means between dyads with various level of interaction and the Welch T-test $p$-value for the difference of means between all individuals against all dyads are also shown. Intensity of interaction | $\sigma$ | Normalized |  $p$-value
---|---|---|---
0 | $5.04\times 10^{3}\pm 6.02\times 10^{3}$ | $0.75$ | $1.38\times 10^{-1}$
1 | $5.10\times 10^{3}\pm 4.30\times 10^{3}$ | $0.76$
2 | $5.15\times 10^{3}\pm 4.30\times 10^{3}$ | $0.76$
3 | $5.29\times 10^{3}\pm 3.30\times 10^{3}$ | $0.79$
All dyads | $5.17\times 10^{3}\pm 4.22\times 10^{3}$ | $0.77$ | $\mathbf{<10^{-4}}$
Individuals | $6.74\times 10^{3}\pm 6.39\times 10^{3}$ | $1.00$

Table 18: Average and standard deviation of the Euclidean distance $\delta_{E}$ (in m), for dyads and individuals during encounters. Kruskal-Wallis $p$-values for the difference of means between dyads (and individuals encountering dyads) with various level of interaction and the Welch T-test $p$-value for the difference of means between all individuals against all dyads are also shown. | Individual | Dyad
---|---|---
Intensity of | $\delta_{E}$ | Kruskal-Wallis | $\delta_{E}$ | Kruskal-Wallis
interaction |  |  $p$-value |  |  $p$-value
0 | $1.25\times 10^{-1}\pm 8.73\times 10^{-2}$ | $9.58\times 10^{-1}$ | $1.02\times 10^{-1}\pm 5.90\times 10^{-2}$ | $\mathbf{2.61\times 10^{-2}}$
1 | $1.16\times 10^{-1}\pm 7.02\times 10^{-2}$ | $8.91\times 10^{-2}\pm 6.56\times 10^{-2}$
2 | $1.30\times 10^{-1}\pm 1.14\times 10^{-1}$ | $9.65\times 10^{-2}\pm 7.05\times 10^{-2}$
3 | $1.31\times 10^{-1}\pm 1.00\times 10^{-1}$ | $1.09\times 10^{-1}\pm 8.43\times 10^{-2}$
All | $1.28\times 10^{-1}\pm 1.05\times 10^{-1}$ |  | $9.78\times 10^{-2}\pm 7.17\times 10^{-2}$ |
T-test $p$-value | $\mathbf{<10^{-4}}$