October 26, 2017

Advanced Robotics

supplementary˙v4

To appear in Advanced Robotics
Vol. 00, No. 00, Month 20XX, 1–14

FULL PAPER
Supplementary Material(Received 00 Month 20XX; accepted 00 Month 20XX)

1.

DatasetWe recorded a dataset [1] in a network of underground pedestrian streets in Umeda, the maincommercial and business district of Osaka, which is the second largest metropolitan area of Japan.
The entire complex connects four train stations and several department stores and shopping centers
and is visited by about 2.5 million people everyday.
We carried out our experiments on a weekday between 10 AM and 4 PM at an intersection of
two particular streets of this underground network, to which we refer to as the environment henceforth1 (see Figure 1). Since the experiment was performed on a weekday, students and colleagues
commuting together with their friends constitute a considerable portion of the crowd. Therefore,
the dataset contains an abundant number of social groups, among which social interaction takes
place at varying levels.

Figure 1. Experiment environment. The people marked with yellow move as a group and interact. Two of the laser range
finders are shown in red.

Two sorts of data are collected in the experiment; (i) range data and (ii) video data2 . These
data are utilized in the following manner:• Range data are incorporated with the particle filter based tracking algorithm of [2] for obtain1 The environment contains one main street, which intersects with a smaller street almost perpendicularly (see Figure 1). The

main street is 59.5 m long and 6.6 m wide and the small street 18 m long and 3.6 m wide.
2 The range data are obtained using 16 laser range finders equipped with Hokuyo UTM-30LX positioned along the walls (shown
on Figure 1 in red). The video data are recorded by two camcorders, where one camera is positioned at each end of the mainstreet.

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ing the pedestrian trajectories. By these means, a total of 12791 people are tracked during
the 6-hour experiment and the normalized density map illustrated in Figure 2 is obtained3 .
• The video data are used to establish the ground truth regarding group motion and interaction
relation. By positioning two camcorders at each end of the street, we get a good sight of thepedestrians walking in both directions as seen in the sample video frame shown in Figure 1.

Figure 2. Normalized density map of the environment.

After examining the data closely, we decided to focus on 2-people (2p) and 3-people (3p) groups
and the solitary individuals, i.e. non-groups. The reasons for this choice are the following:(1) Groups of 4p, 5p, 6p do not usually move in an abreast formation and break into subgroupsof 2p or 3p, making them the basic building blocks of all groups [3].
(2) The number of observations of 2p and 3p groups are enough to characterize their behavior,
whereas larger groups are observed seldom4 .

2.

Ground truth

Video data are used for establishing the ground truth regarding the following:
i Group relation: The possible states regarding this relation are “group” or “non-group”. We
term any two pedestrians who are observed simultaneously in the environment a pair and let
the coders label the pairs which constitute a group. The complementary case of non-group is
automatically attributed to all remaining pairs.
ii Interaction relation: Those pairs, which are labeled as “group” in the previous step, areexamined and one of the four possible labels for intensity of interaction I is assigned to eachpair. The possible labels are 0, 1, 2, and 3, where I = 0 indicates no-interaction, I = 1
indicates weak interaction, I = 2 indicates mild interaction and I = 3 indicates intenseinteraction. In the remaining of the paper, we will often use the shortening Ii for I = i.
2.1.

Method of annotation

In order to assess errors due to coding fatigue and coder’s bias, each relation (group/non-groupand intensity) is labeled by two different coders.
In the first stage of the coding (i.e. identification of the pedestrian groups), the coders watch
the walking patterns, age, gender, or clothing, and define which pedestrians constitute a group [4].
3 The higher density lane along the main street is due to the Braille blocks.
4 We have 22 4p groups, 2 5p groups, 3 6p groups and 1 7p group.

2

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supplementary˙v4

In the second stage (assessing intensity of interaction), we consider only the pedestrians which
are labeled to be a member of a social group at the first stage. This reduces the amount of datathat each coder has to view and thus provides a more efficient coding.
At this stage, the coders are asked to label the pedestrian groups with respect to their intensityof interaction by selecting one of the four options of 0, 1, 2, 3. Clearly, this is a highly subjective
measure. In order not to bias the coders’ assessment, we only defined the resolution (i.e. the numberof interaction levels as 4) but we did not give any guidelines on what can be considered as weak, mildor strong in terms of interaction intensity. Instead, we let the coders grasp a feeling about different
levels of interaction intensity through a free-viewing task. In other words, for understanding thevariations on interaction intensity, we let the coders initially watch for 3 hours the videos of thepedestrians which are labeled to be in a group relation at the previous stage.5
2.2.

Inter-coder reliability

For evaluating the consistency of the two annotations, we use inter-rater reliability measures ofCohen’s κ coefficient6 [5, 6], and Krippendorf’s α for ordinal data [7, 8] (see Table 1).
Table 1. Inter-rater reliability measures of Cohen’s κ and Krippendorf’s α for group relation and intensity of interaction,
respectively. Cohen’s κ is corrected for prevalence.

Annotation
Group relation
Interaction intensity

Statistics
Cohen’s κKrippendorf’s α

Value
0.96
0.67

For interpreting Cohen’s κ values, we use the guidelines of Fleiss et al. [9]., who consider κ < 0.4
as poor, 0.40 to 0.75 as fair to good, and over 0.75 as excellent agreement. Thus, the consistency
in labeling group relation is regarded to be almost perfect based on these principles. Here we also
note that it is common in the literature to rely on much lower values of κ ≈ 0.5 [10–12].
For interpreting the level of agreement in labeling intensity of interaction, we use the policy of
Krippendorf [7]. Krippendorff points out to the difficulty of establishing universal reliability bands
and states that 0.67 has been adopted as cut-off value for better or worse. Since our values arearound this widely accepted boundary, we consider the inter-rater agreement to be fairly good.
2.3.

Distribution of labels

Having confirmed that the coders have consistent labels, in the rest of the analysis, we pick the
annotations of one of the coders randomly and base our ground truth on her labels. According to
the annotations of the randomly picked coder, we have the number of pedestrians, pairs and groups
as given in Table 2.
As for group relation, there are 634 2p groups, 75 3p groups and 10892 non-groups (i.e. pedestrians who are not in any group). The second column of Table 2-(a) denotes the pairs relating 2p, 3pgroups and non-groups. The number of pairs in 2p and 3p groups are straightforward to computefrom the number of groups. However, for non-group pairs, we consider the pedestrians who satisfy
the following two conditions: (i) observations overlapping in time for at least 5 seconds, (ii) being atleast once at a distance of 3 meters or less7 . In this way, we eliminate the non-group pairs who areclearly not in group relation and consider only those which have a possibility of being in a group.
5 Here, we note that the total duration of the trajectories of 2p and 3p groups is less than 6 hours and thus a 3-hour free viewing

task corresponds to watching approximately half of the groups, which is considered sufficient to get an idea of variation of
intensity patterns.
6 with prevalence correction
7 Distances of more than 3 meters are extremely improbable in interacting groups, [1]3October 26, 2017

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supplementary˙v4Moreover, for the 2p and 3p groups, the distribution of interacting pairs at various intensities are
given in Table 2-(b).
Table 2. (a) The number of 2p, 3p groups and pedestrians who are not in any group (non-groups). (b) The number ofpedestrians interacting with varying intensities.

(a)
2p
3p
Non-groups

3.

N
634
75
10892

Npair
634
225
34243

2p
3p

(b)
Intensity
012
3
66 97 386 85
57 65 90 13

Motion models

This section presents details on the definition of the variables used in modeling of joint motion,
the preprocessing of trajectory points, analysis of variance of empirical distributions and expands
the description of corresponding motion model of each variable.
3.1.

Definition of joint motion attributes

Figure 3 illustrates an example of a 3p group composed of pedestrians pi , pj and pk . We define ourvariables on this sample 3p group, but they are defined in an analogous way for 2p groups as well.
~
vg
~
vi
ωjk

φij

~
vk

~
vj
pi
~
δo
ij

x
y

φik

v~j

pj
~
δo
jk

0

pk

~
δo
ik

Figure 3. Parameters used in modeling joint motion illustrated on a 3p group.

In examining the joint behavior, we focus particularly on three joint motion attributes, namely(i) interpersonal distance δ (ii) relative orientation φ with respect to motion direction and (iii)
absolute difference of velocity vectors ω.
(1) Although we will focus on the attribute δ, i.e. the absolute value of the distance between
pedestrians in a pair, in the development of some of our models it will be useful to takein consideration the components of the displacement vector ~δ, and in order to eliminate
the effects of changing motion direction due to taking curves or avoiding behavior, suchcomponents are better defined in relation to the group motion direction.
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supplementary˙v4

The interpersonal distance of pi with respect to his peer pj , is denoted with ~δij . Its explicit
components are computed projecting the displacement vector on group motion direction, v̂g ,which is the unit vector along the mean velocity of the group:

PN
~vg =

vi
i=0 ~
N

.

Projecting the displacement vector from pi to pj , i.e. ~δij on v̂g , the x-component δx of interpersonal distance δij is found to be (see Figure 3) as follows,δx = ~δij · v̂g .
Similarly, the y-component δy is obtained by a projection onto a unit vector orthogonal tov̂g (obtained through a π/2 clockwise rotation). The interpersonal distance relation of a pair{pi , pj } is then given by distance of each pedestrian to their peer, nexpressed in theo projection
coordinate frame introduced above, namely by the set of vectors ~δij , ~δji = −~δij .
(2) The relative orientation φij , of pj with respect to the motion direction of pi , is defined as
the angle between the velocity vector of pi and the displacement vector from pi to pj , i.e.
~vij , δij . To represent relative orientation of a pair {pi , pj }, we use the pair {φij , φji } 8 .
(3) The third joint motion attribute of the pair {pi , pj }, namely their absolute difference ofvelocity vectors ωij , is simply defined as the magnitude of the difference vector
ωij = |~
vi − v~j | .
Figure 3 demonstrates the absolute difference of velocity for the pair {pj , pk }, ωjk .
3.2.

Preprocessing

The trajectories are obtained by processing the raw range data. Subsequently, they are first exposed to several preprocessing operations and then tested against several criteria, where only thosetrajectories which satisfy these criteria are used in the analysis.
The preprocessing of the trajectories is composed of a smoothing operation of the data points
to obtain samples at uniform time intervals. The reason for this is that the range sensors return
readings in an irregular manner in time and at a very high frequency, which makes some samples
redundant. 9
To obtain readings over uniform time intervals, we average the values falling in the same 100 msec
time window as follows:~x(t0 ) = (xt0 , yt0 ) = {(x̄t , y¯t ) | t0 − 0.1 ≤ t < t0 } .
and then compute the velocity as~v (t0 ) = (~x(t0 + 0.1) − ~x(t0 − 0.1))/0.2
The joint motion attributes of interpersonal distance δ, relative orientation φ and absolute differenceof velocities ω are derived from the trajectory points exposed to the smoothing operation described
8 Each angle is defined using the individual velocity ~vij . Nevertheless, when the pedestrians are part of a group and moving

together, we may expect to have ~vij ≈ ~vji and thus φij ≈ φji + π.
9 This sampling is particularly useful in the computation of pedestrian velocity, since it provides a smoother trajectory andthus a more reliable and stable velocity.

5October 26, 2017

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above.
Subsequently, we test the trajectories against several conditions. The first condition relates thecontinuity of locomotion. Namely, we focus on the behavior of the pedestrians during their active
motion and skip the parts where they keep still (to wait, meet, greet etc). Therefore, we set a
velocity threshold at 0.5 m/sec and consider the data points where the pedestrians are moving at
a higher velocity than this threshold value10 .
The second condition concerns the length of the trajectory. Specifically, we consider trajectories
which are sufficiently long, in order to comment on group relation or interaction in a plausible way.
We set the threshold value for trajectory length to 10 meters and omit the pedestrians who covera shorter distance (for instance, by only taking a curve from the small street to the main street,
see Figure 2).
The third condition, which applies only to 3p groups, relates degree of neighborhood. Since
pedestrians in 3p groups may switch their position resulting in varying degrees of neighborhood,which affects distance relation drastically, we exclude the 3p groups that are subject to position
switches and consider the ones which keep a stable neighborhood structure.
3.3.

Analysis of variance of empirical distributions

We confirm the qualitative observations of Section 4 in the Main Text through a quantitative way
using analysis of variance as illustrated in Table 3 11 . As expected, in detection of group relation
all three attributes have quite small p-values.
Furthermore, also in detection of intensity of interaction in 2p groups, the p-values suggest astatistically significant difference for all three joint motion attributes δ, φ and ω.
Finally, based on the p-values for 3p groups, we may notice that ω appears to be the bestindicator in distinction of intensity of interaction, whereas δ achieves low scores even though thecurves in Figure 4 in the Main Text suggest otherwise, particularly for second neighbors. This isprobably due to the fact that the number of samples at each intensity level is not well-balanced asshown in Table 2.
Table 3. Analysis of variance for empirical distributions. The
joint motion attributes, which are considered to serve useful indistinction of group relation and intensity of interaction are illustrated in bold face.
δφωGroup relation< 10−4< 10−4< 10−4
Intensity relation in 2p< 10−4< 10−4< 10−4
Intensity relation in 3p
0.60 and 0.42
0.95
0.03

3.4.

Modelling interpersonal distance

Since interpersonal distance depends to a great extent on the degree of neighborhood, we treat 2pgroups and the first neighbors in 3p groups in the same manner, whereas second neighbors in 3p
groups and non-groups are handled in different ways.
3.4.1.

Interpersonal distance for groups

The work of [1] introduces a potential that describes the position dynamics of two pedestrians ina socially interacting group. Such a discomfort potential, i.e. the function that pedestrians try to
10 For a justification of this value, refer to [13].
11 For the φ attribute, we use in the Anova only the values φ ∈ [0, π/2] to be able to highlight differences in spread between
distributions with the same average value ≈ π/2.

6October 26, 2017

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supplementary˙v4minimize in order to maximize their comfort in social interaction, depends on the sum of two terms,
one of them being the distance between pedestrians δ and the other one the relative orientation
φ 12 . Using methods from statistical physics, [1] shows that δ and φ result to be statisticallyindependent variables, and in particular they show that the probability distribution for δ is givenby
p(δ|β, r0 ) = δ exp−β(δ − r0 )2
δr0


(1)

.

Here the parameter r0 is the preferred distance for social interaction, while β is, in the statisticalphysics terminology, the inverse temperature (namely, the larger β is the more the pedestriansystem will be ordered and the distribution narrowly centered on r0 ).
According to [1], the interaction of first neighbors in a 3p group is given by the same potential,
and thus, ignoring second order terms, their distance distribution will be given by the same model
used in the 2p situation. For second neighbors, they do not provide a closed form for the probability
distribution of distance, since the non-trivial structure (V shaped formation) of the group allows
only for a numerical solution. Nevertheless, if the angle in the V formation is assumed to be close
to 180 degrees, namely if the pedestrians are assumed to walk abreast (which results to be a good
approximation in most cases, as discussed also by Moussaı̈d et al., [14]), the distance between second
neighbors can be modeled as the convolution of the unimodal model with itself. This multi-modal
framework is expressed as follows:p(δ|β, r0 ) = (p ∗ p) δ exp
3.4.2.

−β(δ − r0 )2
δr0


.

(2)

Interpersonal distance for non-groups

Proximity models for pedestrians who are not engaged in social interaction usually refer to theguidelines defined by Hall (see Figure 4-(a)). Although these may make sense for standing pedestrians, they do not apply to actively moving pedestrians, in particular when there is a prominent
flow direction along a street as in our case. Figure 4 contrasts the definitions of Hall with theempirical distribution of distances between non-groups in our dataset. This proves that simple ring
like structures do not suffice to model the actual behavior of mobile non-groups.
Examining Figure 4-(b) closely, we may conclude that when the environment presents a prominent flow direction, there is an area, on the front and back of the pedestrian (i.e., along the y
direction, according to our previous definition), and with a width comparable to body size, inwhich the probability of finding another pedestrian is extremely low. As a simplified, but nevertheless quite effective, model, we may assume (recalling that we are considering only pedestrians
with a distance δ ≤ R, with Rq= 3 meters) that the variables δx and δy have uniform distributions

subject to the conditions δ =δx2 + δy2 ≤ R and δx ≥ r, where r is the “human body size” (Hall’s

intimate distance, for which we assume the value r=0.5 meters). Geometrically, this means thatthe relative pedestrian distance will be uniformly distributed inside two circular segments, locatedsymmetrically with respect to the y axis and at an r distance between them, and thus the probability of finding the pedestrians at a distance r ≤ δ ≤ R is proportional to δ arccos(r/δ). Throughan elementary integration, the normalized formula for the explicit probability distribution function
results to be 13
12 Such angle is actually defined in a slightly different way by [1], namely with respect to the group goal direction (empiricallyassumed to be aligned to the group velocity).
13 On the other hand in an isotropic environment we could assume uniform distribution between r and R and, by trivial

7

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supplementary˙v4Social(3.6m)
Personal(1.2m)
Intimate(0.45m)(a)(b)

Figure 4. (a) Hall’s definitions for proxemics. (b) Empirical distribution of interpersonal distance on cumulative data for
non-groups.

p(δ|R, r) =3.5.

4
πR2 − 2R2 arcsin


r
R

√

− 2r R2 − r2δ arccos

r
.
δ

(3)

Modelling relative orientation

For φ, the analytical model of [1] leads to a Gaussian distribution. Their model presents nevertheless, for reasons of analytical tractability, a cuspidal point in the potential at φ = 0, π, which is thereason they obtain a Gaussian distribution instead of a von Mises14 . Since we do not need in thiswork to derive the distribution from first principles, we will directly use the more appropriate vonMises distribution without paying regard to the groups size or the degree of neighborhood15 . The
model is given as follows,p(φ|ν, κ) =exp(κ cos(φ − ν)),
2πI0 (κ)(4)

where ν denotes the mean value, κ is analogous of 1/σ 2 in the normal distribution, and I0 is themodified Bessel function of the first kind with order zero [15].
Since we do not pay regard to be positioned on the right or left side with respect to motion
direction, both pedestrians in a pair are treated in the same manner, which leads to a distribution
with two peaks. Thus, we use a linear combination of two von Mises distributions to represent therelative orientation of a pair,Φ(φ|ν1,2 κ1,2 ) = α1 (φ|ν1 , κ1 ) + α2 (φ|ν2 , κ2 ),(5)which, due to the obvious left-right symmetry of the problem (for every person on the right side,these has to be another person on the left), can be simplified toΦ(φ|ν, κ) = 0.5p(φ|ν, κ) + 0.5p(φ|ν + π, κ).

(6)geometrical considerations, obtain p = 2δ/(R2 − r2 ).
14 Note that Von Mises distribution is the generalization of a Gaussian describing a smooth circular (angle) distribution.
15 Again, according to the analysis of [1], a different model for the first and second neighbor distribution should be introduced
to asses the presence of a V formation. For the purpose of this work, as we did in the development of the convolution modelfor the second neighbor distance, we may follow [14] and assume that 3p groups walk roughly in an abreast formation.

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3.6.

supplementary˙v4Modelling absolute difference of velocities

Pedestrians in a group move towards the same target point independent of the groups size andneighborhood. Even though there may be slight deviations from the target direction due to various
reasons such as avoidance behavior or interaction, the expected value is always around 0. Therefore,
we approximate the x and y components of ω with a Gaussian with 0 mean, which makes ω comefrom a Rayleigh distribution,p(ω|σ) =

−ω 2
ω
exp
.
σ2
2σ 2

(7)

For non-groups, we may expect the distributions of the the x and y components of the velocity difference to have non-zero mean value ν, and thus it is necessary to use the corresponding
generalization of the Rayleigh distribution, namely the Rice distributionp(ω|ν, σ) =

ω
exp
σ2



−ω 2 − ν 2
2σ 2


I0

ω ν 
σ2(8),where I0 is again the modified Bessel function of the first kind with order zero.
Furthermore, we need to distinguish the case of pedestrians moving in the same direction (forwhich we expect ν  v, v being the pedestrian average velocity) and pedestrians moving in opposite
directions (for which we expect ν ≈ 2v). The proposed model thus readsωp(ω|C1,2 , ν1,2 , σ1,2 ) =Ci 2 exp
σi
i=1,2

X

4.



−ω 2 − νi2
2σi2




I0

ω νi
σi2


.

(9)

Calibration of models

The observed distributions are expressed in terms of the histograms relating each set, whereas thecorresponding model parameters are solved minimizing the squared error between the observations
and the relating models through a golden section search.
In case the models are governed by more than a single parameter, we adopt an alternating
strategy (for instance, as in Equation 1 or Equation 6). For solving these systems, we minimize
the squared error in an iterative way alternating between the parameters. In other words, wefirst optimize for one parameter keeping the other(s) fixed and repeat this procedure for otherparameters until satisfactory convergence is achieved.
In addition, we shuffle each observation set of interest and randomly select 20% of the samples
and calibrate the model parameters on this subset. Repeating this procedure 50 times16 , we computethe mean values and standard deviations regarding the model parameters, which gives an idea on
the sensitivity ( i.e. dependence) of the parameters on training set.
4.1.

Sensitivity of model parameters in group identification

The models proposed in Sections 3.4, 3.5 and 3.6 are calibrated as explained in Section 4. The
resulting model parameters are shown to differ between groups and non-groups in a significant
way, which proves that the proposed models can be used in estimation of group relation. Moreover,
16 By randomly selecting 20% of the set and repeating this process 50 times, the probability that a particular observation is

not used in estimating the model parameters in any epoch, falls below 10−4 , which is quite small.

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the parameters are shown not to be sensitive to the changes in the set of observation which is used
to estimate them. This indicates that the models can stably be constructed from any subset ofobservations (of a reasonable size). In what follows, the details of these statements are presented.
4.1.1.

Sensitivity of models of interpersonal distance regarding group relation

The dashed curves in Figure 2-(a) in the Main Text demonstrate the models constructed at asample run. This figure proves that the models approximate the observations in a quite effective
way, capturing general tendency with a good accuracy.
Table 4. Sensitivity of model parameters for δ with respect togroup relation. Calibration is repeated 50 times by on a random
subset of observations composed of 20% of the entire observation
set.

2pFirst neighbors in 3p
Second neighbors in 3p

NT otal
634
150
75

β
18 ± 1.7
14.7 ± 0.2
10.4 ± 0.3

r0748 ± 8788 ± 1716 ± 3

In addition, Table 4 presents the variation on the 2p and 3p group motion models in relation tochanges in the subset of observations used in estimating them 17 18 . These values prove that thestandard deviations of β and r0 are small enough to lead to similar models when different training
sets are used19 .
4.1.2.

Sensitivity of models of relative orientation

In Figure 2-(b) in the Main Text, we illustrate the relative orientation models for 2p, 3p groups
and non-groups with dashed lines. This figure ascertains that the three empirical distributions aswell as the models differ significantly between groups and non-groups, which is already determined
quantitatively by the p-values presented in Table 3.
Table 5. Sensitivity of model parameters for φ withrespect to group relation. Calibration is repeated 50
times on a different randomly chosen subset of observations composed of 20% of the entire observation set.
NT otalνκ2p634 1.56 ± 0.002
7.06 ± 0.62
3p
225
1.57 ± 0.01
2.53 ± 0.81
Non-groups
10892
0.15 ± 0.02
5.31 ± 0.93

Besides, the variation on the relative orientation model parameters given in Table 5 prove thatthe dependence of the parameters on the chosen subset of observations is quite limited, suggesting
the estimation process to be quite robust. In addition, it ascertains that the variation on theparameters is considerably smaller than the variation between different (competing) categories(i.e. groups vs non-groups).
4.1.3.

Sensitivity of models of absolute difference of velocities

Figure 2-(c) in the Main Text shows the three models (in dashed lines) for absolute differenceof velocities for 2p, 3p groups and non-groups. These models approximate the observations in an
17 Note that the interpersonal distance relating non-groups does not involve any parameters.
18 The lower value assumed by r for 3p groups is consistent with the fact that pedestrians are actually not abreast but in a V0formation, and thus the distance of second neighbors is smaller than twice the distance of first neighbors.
19 In particular, we may notice that standard deviations are small with respect to both the absolute values of parameters andto the difference between the values that such parameters assume in different categories.

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accurate manner and in a good agreement with the analysis of Table 3. They are also distinct
enough to serve useful in identification of group relation.
Table 6. Sensitivity of model parameters for ω with respect to group relation. Calibration is repeated 50 times
on a different randomly chosen subset of observations
composed of 20% of the entire observation set.
NT otalσ2p
634 0.23 ± 0.004
3p
225 0.24 ± 0.008
NT otal
ν1,2
σ1,20.30 ± 0.01
0.28 ± 0.01
Non-groups
10892
2.40 ± 0.01
0.48 ± 0.009

Moreover, Table 6 demonstrates that the sensitivity of the model parameters to changes incalibration subsets is limited.
4.2.

Sensitivity of model parameters in estimation of intensity of interaction

We present the sensitivity of the model parameters relating each level of intensity of interaction
using different calibration subsets, similar to Section 4.1.
4.2.1.

Sensitivity of models of interpersonal distance

The parameters of interpersonal distance models presented in Tables 7 and 8 are in agreement with
our hypothesis described in the Main Text. Nevertheless, the variation of parameters are comparable(although always smaller) to the differences that such parameters assume between neighboring
categories, namely between Ii and Ii+1 . This is not completely surprising, given the arbitrary andsubjective nature of these labels. Nevertheless, the difference between the two extremes, I0 and I3 ,is much larger than the parameter variation
Table 7. Model parameters of [1] for δ in 2p groups
with varying intensity of interaction. Calibration is repeated 50 times on a different randomly chosen subset
of observations composed of 20% of the entire observation set.
Intensity
NT otalβr0066
10.14 ± 3.33
863.74 ± 37.28
197
14.22 ± 3.31
791.56 ± 28.50
2386
21.65 ± 2.17
749.17 ± 7.96
385
26.21 ± 3.80
723.56 ± 14.08

Table 8. Model parameters of[1] for δ in 3p groups with varying intensity ofinteraction. At every calibration round, 20% of the total number of relating
samples is used.
First neighbors
Second neighbors
12Intensity
N3p
β
r0
N3p
β
r0
0
10
14.2 ± 8.9
851 ± 139921.2 ± 8.1
750 ± 631
23
15.2 ± 5.9
828 ± 5710
17.8 ± 9.5
718 ± 742
36
25.2 ± 4.9
767 ± 1815
15.2 ± 10.2
691 ± 1093
3
28.4 ± 1.6
791 ± 1182
22.8 ± 3.4
658 ± 194.2.2.

Sensitivity of models of relative orientation

The evolution of the parameters in Table 9 with increasing intensity of interaction supports the
hypotheses presented in Section 1 of the Main Text. Moreover, the standard deviation values are
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supplementary˙v4quite smaller than the separation of the means for 2p groups (at least when the extremes arecompared), which proves not only the distinction of the models but also the resilience of thecalibration against varying parameter estimation subsets. However, for 3p groups although the
expected evolution in κ is observed through I = 0 to I = 2, the patterns is broken for I = 3
probably due to small number of samples (5).
Table 9. Model parameters for φ with varying intensity of interaction. Von Mises modelis used for all pairs. At every calibration round, 20% of the total number of relating
samples is used.
2p groups
3p groups
Intensity
N2p
µ
κ
N3p
µ
κ
0
66
1.53 ± 0.01
3.24 ± 1.42
19
1.71 ± 0.76
1.04 ± 0.91
197
1.56 ± 0.007
5.85 ± 1.81
33
1.61 ± 0.20
1.85 ± 1.29
2386
1.56 ± 0.002
7.63 ± 0.70
51
1.56 ± 0.01
4.47 ± 1.85
385
1.56 ± 0.003
8.86 ± 2.68
5
1.92 ± 0.60
2.65 ± 1.94

4.2.3.

Sensitivity of models of absolute difference of velocities

The effect of interaction is grasped by the proposed models since the σ values get smaller withincreasing intensity of interaction as seen in Table 10. In addition, the variation on the calibrated
parameters with shuffling and random selection, suggests that the distinction is significant for both
2p and 3p groups. Nevertheless, as stated also for the other variables, the variances in parameters are comparable to the differences in the same parameters between neighboring categories,
suggesting again the use of the extremes I0 and I3 as benchmarks for intensity estimation.
Table 10. Rayleigh model parameters for ω with varying
intensity of interaction.
Intensity
2p groups
3p groups
NT ot
σ
NT ot
σ
0
66
0.28 ± 0.01
19
0.26 ± 0.01
197
0.25 ± 0.01
33
0.25 ± 0.01
2386
0.23 ± 0.005
51
0.23 ± 0.009
385
0.21 ± 0.008
50.21 ± 0.02

5.

Observations on model parameters in estimation of intensity of interaction

As briefly explained in Section 1 in the Main Text, our hypothesis is that when two people interact,a more regular pattern is observed in their joint behavior. In other words, non-interacting pedestrians have a larger flexibility on their motion patterns, whereas interacting peers have (i) a stable
and short distance, (ii) an abreast formation, and (iii) a smaller difference between velocities. In
addition, for increasing intensities of interaction, we expect to see a shift towards a more regularbehavior in terms of all joint motion attributes.
Therefore, for increasing intensities of interaction we expect to observe the following:(1) Interpersonal distance δ should decrease with increasing intensity, which defines a smaller
preferred distance between peers (namely, we expect r0 to decrease with intensity). Moreover, its variation should drop pointing to a more steady group structure (β increasing withdensity).
(2) Relative orientation φ must be distributed around ±π/2 for all cases but with decreasing
dispersion (increasing κ), suggesting a more abreast formation.
(3) Absolute difference of velocities ω should decrease and adopt a more compact distribution(decreasing σ) as intensity of interaction grows.

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supplementary˙v4

In what follows, we discuss on agreement of the empirical observations with the above listed
statements. Moreover, for each case, we demonstrate the concerning models and comment on theiraccuracy in modeling the distinctive characteristics.
5.1.

Observations on models of interpersonal distance

Figure 3-(a) in the Main Text presents the empirical distributions of δ regarding 2p groups withvarying intensities of interaction. Our hypothesis implies that as intensity of interaction grows,
peers keep a stable and reasonably close distance throughout their interaction, which leads ~δ tobe distributed around a shorter preferred distance with a smaller deviation, as clearly supportedby these empirical distributions. Also note that the p-value for analysis of variance of these four
distributions is found to be smaller than 10−4 , which ascertains that the difference between these
distributions is statistically significant.
In addition, the models given in dashed lines follow the actual distributions to a considerable
degree. Moreover, parameters of these models presented in Table 7 are in agreement with our
hypothesis. In explicit terms, with growing intensity, β increases pointing out to the increasing
concentration of the distribution, whereas r0 decreases showing a smaller preferred distance between
peers. In addition, the variation on the parameters are concluded to be insignificant in comparison
to their mean values.
Figures 4-(a) and (b) in the Main Text demonstrate the behavior of δ for first and second
neighbors, respectively, in 3p groups for varying intensities of interaction. As seen in Figure 4-(a)in the Main Text, δ has a peak around a smaller value and with a smaller variation with increasing
intensity of interaction from 0 to 3, as expected. However, only for I = 3 the variation is larger than
expected, which is probably due to the fact that the number of observations are not distributed ina balanced manner and the samples from second neighbors with I = 3 are quite smaller than theothers. This phenomenon is observed in terms of model parameters as well (see Table 8).
Figure 4-(b) in the Main Text demonstrates that δ does not present any distinction from I = 0
to I = 2 but for I = 3 it is distributed around smaller values in a more concentrated way. Theseobservation are confirmed by the calibrated parameters as well. In other words, while the expectedvalues of β do not seem to present any distinction, r0 decreases with increasing intensity, but thedeviation on the estimated variables are quite high. Therefore, a similar conclusion is drawn also
for second neighbors, that more samples are necessary to build stable models.
5.2.

Observations on models of relative orientation

Figure 3-(b) in the Main Text presents the empirical distribution of φ regarding 2p groups forvarying intensities of interaction. As suggested by our hypothesis, the motion of pedestrians become
more abreast with growing intensity of interaction and the decreasing dispersion of the curves in
Figure 3-(b) in the Main Text support this statement.
In addition, this tendency is captured in terms of the model parameters as well. As expected,
the mean values are around π/2 for all cases. Nevertheless, the increasing values of κ in Table 9
indicate that the distributions adopt a more concentrated pattern as intensity increases, provingthat the proposed models reflect the tendency in an efficient manner.
Figure 4-(c) in the Main Text presents the relative orientation pattern for 3p groups. From thesecurves we conclude that φ for 3p groups presents a similar pattern to that of 2p groups but thedistinction is not as clear as in dyads. However, this pattern is broken for I = 3, probably due toinsufficient number of samples.
This can be observed in terms of the κ parameter which increases from 1.04 to 4.47 as in Table 9.
Nevertheless, comparing Figure 4-(c) and Figure 3-(b) in the Main Text, and the κ parameters inright and left columns of Table 9, one may conclude that the distinction between different patterns
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supplementary˙v4of φ with varying levels of intensity are not as profound as the one for 2p groups.
5.3.

Observations on models of absolute difference of velocities

Figure 3-(c) and Figure 4-(d) in the Main Text present the empirical distributions and concerningmodels for ω of 2p and 3p groups, respectively, with varying intensity of interaction. As expected,
the peak of the absolute difference of velocities get smaller and the distribution less sparse withincreasing intensities of 2p and 3p groups, the former being more prominent.
In addition, the effect of interaction is grasped by the proposed models since the σ values get
smaller with increasing intensity of interaction as seen in Table 9. In addition, the variation on
the calibrated parameters with shuffling and random selection, suggests that the distinction issignificant for both 2p and 3p groups.

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