Diverse movement patterns may be identified when we study a set of moving entities. One of these patterns is known as a Vformation for it is shaped like the letter V. Informally, a set of entities shows a Vformation if the entities are located on one of their two characteristic lines. These lines meet in a position where there is just one entity considered the leader of the formation. Another movement pattern is known as a circular formation for it is shaped like a circle. Informally, circular formations present a set of entities grouped around a center in which the distance from these entities to the center is less than a given threshold. In this paper we present a model to identify Vformations and circular formations with outliers. An outlier is an entity which is part of a formation but is away from it. We also present a model to identify doughnut formations, which are an extension of circular formations. We present formal rules for our models and an algorithm for detecting outliers. The model was validated with NetLogo, a programming and modeling environment for the simulation of natural and social phenomena.
Diverse movement patterns may be identified when we study a set of moving entities, for example, a flock of birds [
Vformation in birds. Source: [
Circular formation in a fish bank. Source: [
Informally, a set of entities shows a Vformation if the entities are located on one of their two characteristic lines. The lines meet in a position where there is just one entity considered the leader of the formation [
Other authors try to simulate Vformations at a computational level. For instance, Nathan and Barbosa [
On the other hand, a circular formation is a set of entities grouped around a common center and where the entities’ distance to the center is less than a given threshold. Regarding related works with circular formations, we identified the following.
In [
On the other hand, researchers in the field of robotics and in control theory, inspired by social grouping phenomena and by the patterns of birds and fish, have developed applications to coordinate the movement of multivehicle systems. Among these patterns are circular [
Robots in circular formation and planes in Vformation. Sources: [
On the other hand, regardless of the type of formation, Reynolds [
Although the previous works allow the simulation of a set of moving entities, they are not aimed at the explicit identification of Vformations and circular formations. The identification of these types of formations may be useful in fields as zoology, to analyze the movement of birds [
This paper is organized as follows. In Section
Next, we present the essential elements of Andersson’s model [
Consider a set
The coordinates of an entity at a timepoint are given by a pair of functions
Consider an entity
The front region of an entity
Front region of an entity
An entity
Entity
An entity
Consider the formation in Figure
An example of a Vformation at a timepoint
A Vformation: entities
In the following section, we extend Andersson’s model to identify leadership patterns in Vformations.
Let
There exist nonempty subsets
Entities in
Entities in
Straight lines
Applying our model to the formation in Figure
Characteristic line  Entities that make up the characteristic line  Characteristic line equation  Pearson’s coefficient  Entity coordinates: original and calculated using the characteristic line equation 




0.99 






0.97 

A formation of entities at a timepoint
In an analogous way to leadership patterns, we say that there is a V
In this section, we present a formal model to identify circular formations. We also present doughnut formations, which are an extension of circular formations.
Let
the minimum number of members in the formation is
A circular formation.
Let
for each entity of the formation its distance
the minimum number of members in the formation is
A doughnut formation.
Circular and doughnut patterns are also defined in an analogous way to a Vformation pattern.
Informally, for Vformations an outlier is an entity which is away from its characteristic lines, and for circular formations it is an entity found beyond the radius of the formation
There are sets of entities which tend to display a Vformation; they may have at a timepoint
There are numerous methods to detect outliers in different domains [
(1) Obtain the equation of the line
(2) Get the maximum number of outliers permitted on
(3) Find the
(4) Remove from
(5) Calculate Pearson’s coefficient using positions (
(6) If
A Vformation with two outliers.
Consider the set of entities in Figure
Characteristic line of a Vformation with two outliers.
Input algorithm parameters:
Equation of
The algorithm determines that entities
Let
A circular formation with two outliers.
Since the distancing of the outlier entity from the formation is temporal, an analyst may introduce a second parameter
An analyst can also specify a maximum permitted number of outliers
Let
Outliers in doughnut formations.
Analogously to circular formations,
For our experiment, we used Nathan’s model [
We worked with a population of 15 entities. The dimensions of the Euclidian plane where the entities move were
Parameters to generate Vformations in NetLogo using Nathan’s model.
Number of entities  15 
Vision parameters  
Vision distance  9 patches 
Vision cone  1.79 rad (103°) 
Obstruction cone  0.75 rad (43°) 
Movement parameters  
Base speed  0.2 
Speed change factor  0.15 
Updraft distance  9 patches 
Too close  3.1 
Max. turn  0.14 rad (8°) 
For outlier detection, we considered for our model the following parameters:
Vformations: results of the experiment in NetLogo.
Tick  Formations identified by our model (leader bolded, outliers in brackets)  Pearson’s coefficient straight line 1  Pearson’s coefficient straight line 2  Opening angle  Vformation  Formations identified by Andersson’s model (leader bolded) 

40200 



1.5 rad (86.49°)  Yes 




1.33 rad (75.92°)  Yes 





0.52 rad (29.97°)  Yes 



NA 

NA  No 




40220 



1.51 rad (86.49°)  Yes 




1.33 rad (75.92°)  Yes 





0.52 rad (29.97°)  Yes 



NA 

NA  No 




 


40380 



0.79 rad (45.13°)  Yes 




1.33 rad (75.92°)  Yes 





2.68 rad (153.74°)  Yes  No  

NA 

NA  No 



NA  NA  NA  NA  NA  


40400 



0.86 rad (49.5°)  Yes 




1.33 rad (75.92°)  Yes 



NA 

NA  No  No  

NA 

NA  No 



NA 

NA  No  No 
Vformations. Position of entities at ticks (a) 40200, (b) 40220, (c) 40380, and (d) 40400.
According to the results shown in Table
We also applied Andersson’s model. We considered
For experiments with circular and doughnut formations, we also worked in NetLogo and used Wilensky’s model [
We worked with a population of 102 entities. The dimensions of the Euclidian plane where the entities move were
Parameters to generate formations in NetLogo using Wilensky’s model.
Number of entities  102 
Vision parameters  
Vision distance  3 patches 
Minimum Separation  1 patch 
Movement parameters  
Maximum angle of rotation  0.08 rad (4.75°) 
Maximum following angle  0.04 rad (2.50°) 
Maximum angle of separation  0.06 rad (3.5°) 
The model was executed on 1200 consecutive ticks (one run) and we conducted an analysis of circular formations and doughnut formations every 400 ticks. We considered that a set of entities showed a circular/doughnut pattern if the set showed a circular/doughnut formation during all the run (1200 ticks). A total of 100 runs were conducted. The parameters used to detect circular and doughnut formations are shown in Tables
Parameters for circular formation detection in NetLogo.
Minimum number of entities ( 
5 
Maximum distance ( 
15 patches 
Maximum percentage of outliers permitted ( 
30% 

30 patches 

400 ticks 
Parameters for doughnut formation detection in NetLogo.
Minimum number of entities ( 
5 
Maximum distance ( 
30 patches 
Maximum distance ( 
10 patches 
Maximum percentage of outliers permitted  30% 

50 patches 

400 ticks 
Circular and doughnut formations: results of the experiments in NetLogo.
Circular  Doughnut  Andersson’s model  

Total number of patterns identified during the 100 runs  332  198  752 


Average number of patterns identified in each run (1200 ticks)  3  2  8 


Average number of entities in each pattern  13  22  4 


Average number of outliers  2  Internal: 3 
NA 
Circular and doughnut formations. Position of entities at ticks (a) 0, (b) 400, (c) 800, and (d) 1200.
According to the results shown in Table
With respect to Andersson’s model, we considered
In this paper, we propose two models:
a model to identify Vformations with outliers. The model considers the location of entities to determine if they form this type of formation during a time interval;
a model to identify circular formations with outliers. The model considers the location of entities to determine if they form this type of formation during a time interval. In addition, we proposed an extension to identify doughnut formations with outliers.
Regarding future work, we plan to conduct a series of experiments in the stock market where Vformations usually appear [
The authors declare that there is no conflict of interests regarding the publication of this paper.
This paper presents preliminary results of the project “Apoyo al Grupo de Sistemas Inteligentes WebSINTELWEB” with Quipú code 205010011129, developed at the Universidad Nacional de Colombia, Sede Medellín.