=Paper=
{{Paper
|id=Vol-3101/Paper7
|storemode=property
|title=Risk modeling during complex dynamic system evolution using abstract event network model
|pdfUrl=https://ceur-ws.org/Vol-3101/Paper7.pdf
|volume=Vol-3101
|authors=Volodymyr Sherstjuk,Maryna Zharikova,Ibraim Didmanidze,Irina Dorovskaja,Svitlana Vyshemyrska
|dblpUrl=https://dblp.org/rec/conf/citrisk/SherstjukZDDV21
}}
==Risk modeling during complex dynamic system evolution using abstract event network model==
https://ceur-ws.org/Vol-3101/Paper7.pdf
Risk Modeling During Complex Dynamic System Evolution
Using Abstract Event Network Model
Volodymyr Sherstjuk1, Maryna Zharikova2, Ibraim Didmanidze3, Irina Dorovskaja1 and
Svitlana Vyshemyrska1
1Kherson National Technical University, Beryslavske shose 24, Kherson, 73008, Ukraine
2Universität der Bundeswehr, Werner-Heisenberg-Weg 393GEB, München, Neubiberg, 85579, Germany
3Batumi Shota Rustaveli State University, Ninoshvili/Rustaveli str. 35/32, Batumi, 6010, Georgia
Abstract
The paper is devoted to the issues of knowledge representation about a plurality of processes of
various intensities that arise unexpectedly and evolve simultaneously in complex dynamical systems in
a wide range of domains. A knowledge representation model based on events that are referenced to
certain time intervals and spatial areas is proposed. Since some information about events is inaccurate
or blurred, the uncertainty of the observed information about the evolving processes is represented
using gray numbers and soft sets. Events are connected in an abstract network, where arcs express
possible transitions from one event to another. Within the network, transitions between events are
driven by impulses and correspond to transitions of a dynamic system from state to state. The energy
accumulated in the nodes is considered to generate impulses, which express the achievement of a
certain threshold by the energy. The proposed abstract event network can be used to model the
evolution of dynamic systems that can be expressed by events that occurred inside the system but
driven by impacts from the outside (environmental effects). Such evolution is considered in a wide
range of rates, from the slowest processes associated with climate change to the most rapid processes
associated with the effects of natural forces of a destructive nature. Connections between nodes
allow representing sequential and parallel streams of events concerning cascade and triggering
effects, which makes it possible to study complex interactions between separate processes within a
complex system of a random structure.
Keywords 1
knowledge representation model, dynamic system evolution, energy-driven abstract event network,
uncertainty model, temporal and spatial referencing
1. Introduction
There are many domains of technical, socio-technical, or socio-economical nature that evolve in
space and time within the natural environment. Usually, they can be represented by complex
CITRisk’2021: 2nd International Workshop on Computational & Information Technologies for Risk-Informed Systems, September
16–17, 2021, Kherson, Ukraine
EMAIL: vgsherstyuk@gmail.com (V.Sherstjuk); maryna.zharikova@unibw.de (M.Zharikova); ibraim.didmanidze@bsu.edu.ge
(I.Didmanidze); irina.dora07@gmail.com (I.Dorovskaja); vyshemyrska.svitlana@kntu.net.ua (S.Vyshemyrska)
ORCID: 0000-0002-9096-2582 (V.Sherstjuk); 0000-0001-6144-480X (M.Zharikova); 0000-0001-6695-4980 (I.Didmanidze); 0000-
0001-9280-8098 (I.Dorovskaja); 0000-0002-6343-7512 (S.Vyshemyrska)
© 2021 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
dynamic systems containing multitudes of interacting dynamic objects. Due to such interactions,
a plurality of processes of various intensity arises unexpectedly and evolve simultaneously
within dynamic systems [1]. Typically, these processes are transient, non-linear, and non-
stationary. The dynamic objects have specific states at specific times, which are represented by
certain points within the state space of the dynamic system, so their behavior is usually
represented by their trajectories within this space. Clearly, if their trajectories touch or intersect
each other (and even in the case of some proximity of the trajectories within the state space),
most often these objects are exposed to a variety of dangers, threats, and risks.
Some processes such as climatic changes proceed very slowly, others, on the contrary, very
quickly, so they can be dangerous and cause deaths, injuries, and huge damage [2]. Since the
processes, which evolve and interact within complex dynamic systems, are principally
stochastic, the dynamic system has a random structure and is difficult to control. The evolution
of the dynamic systems is mainly driven by the impacts of humans and nature [3]. The
manifestation of such evolution can be ex-pressed by events that occurred inside the dynamic
system but associated with im-pacts from the outside (environmental effects). Due to the wide
dispersion of a plural-ity of the evolving processes over space and time, events should be
properly referenced within space and time.
At the same time, people must be concerned about the prevention or elimination of dangerous
events to minimize losses, so there is a need to ensure the safety of the dynamic system [4]. To
make the dynamic system safe, people need first to observe the system itself and the
environment. Although there are a lot of technical means of observation (i. e. sensor networks,
unmanned vehicles, etc.), all of them are based on sensors, which usually provide ambiguous,
imprecise, incomplete, inconsistent, and doubtful information. Such a wide range of
uncertainties distorts observations and introduces unpredictability of the states of the dynamic
objects and the dynamic system as a whole. Naturally, this requires the development of special
safety-enabled methods to overcome uncertainty both in the spatial and temporal aspects [5].
Second, since a human cannot directly control the flow of processes within the complex
dynamic system, he must be able to make decisions, the implementation of which allows him to
change the flow of processes indirectly. Thus, people need to use decision support systems [6].
However, due to the above-mentioned reasons, decision support is a complex and non-trivial
task.
The spatial aspects of the dynamic system can be typically expressed by a geo-graphic
information system (GIS), which contains a spatial model and outlines an area of interest (AOI)
critical for decision making. Thus, the events distributed over the AOI can be properly referred
to spatially, but their references are usually limited by measurement inaccuracy.
The temporal aspects are more complicated due to their stochastic nature and simultaneous
impacts. Moreover, it is very difficult to refer to events temporally be-cause of the different rates
of their occurrence. Unlike spatial references, which usual-ly are vague or blur due to the
sensors’ imprecision, temporal references are inaccurate and can be primarily represented by
time intervals.
Since decision-makers usually operate under the conditions of high responsibility and a lack
of time, the decision support system must work in real-time. Thus, considering the above
reasons, the most topical and important issue for today is the development of GIS-based real-
time decision-support systems (DSS) aimed at danger and risk assessment for the considered
class of complex dynamic systems.
Unfortunately, today there is a lack of knowledge representation models and methods that
would be operable in real-time and take into account a wide range of measurement uncertainties
in data captured by sensors during the observation of the dynamic system. Thus, the research of
knowledge representation methods based on events and adequate uncertainty models that allow
describing dynamic systems by spatially and temporally referred trajectories is a topic of our
interest. The problem addressed in this paper relates to the representation of knowledge about
dynamic system evolution using abstract event networks from the point of view of developing
real-time DSS.
2. Recent Works
Representation of uncertain knowledge about events that occur simultaneously and jointly has
been studied in many fields of knowledge. A wide range of models of knowledge representation
about events has been proposed in the field of natural language processing [7]. Mainly, such
models are focused on event-to-event relations and processing of events to discover their causal
relationships and detect anomalies within language structures [8].
In existing risk assessment decision support systems, the most frequently observed
information is represented by event streams, which describe sequences of time-referred events
[9]. Mainly, the event stream model is a subject of study in event sequence analysis, which
focuses on the time gaps between events and their order in the sequence [10]. Another trend is
the use of knowledge structures based on events as certain building blocks to build and update
situation models [11]. Thus, a most used definition of an event has been proposed in the
literature as “a segment of time at a given location that is conceived by an observer to have a
beginning and an end” [12].
The above-mentioned approaches are mainly semantic based, so they use strict definitions of
events, time, and space and take little into account the uncertainty of the information, based on
which the events must be determined [13].
Many non-semantic methods have been proposed including Causal events, Force Dynamics,
Stochastic Context-Free Grammars [14] that represent complex event structures based on
hierarchical definitions, which are hard for decision-maker interpretation and do not meet the
requirements for real-time DSSs.
In general, considering the nature of the events, we emphasize two main approaches to
represent events - probabilistic and non-probabilistic. The first one includes such methods as
Hidden Markov Models, dynamic Bayesian networks, Monte Carlo sampling, Variance
propagation, etc. [15]. The second one includes methods mainly based on fuzzy sets and
possibility theory [16]. Obviously, a lack of sufficient statistical data for probabilistic approach
as well as a lack of well-known possibility degrees or membership functions for non-
probabilistic approach complicates their efficient use and leads to high computational
complexity. Detailed overviews of such approaches concerning a considered class of DSSs have
been presented in [17].
The event tree approach enables the modeling of a sequence of events, which constitute the
structures of any level of complexity adapted to various uncertainty models (probabilistic, fuzzy,
rough, etc.) [18]. Despite the flexibility of this approach and its potential for evolution, the
existing event tree models refer events to time only, so there is a lack of spatial localization of
events. A method for representing hierarchical structures of events referred both to time and
space and equipped with a complex uncertainty model has been proposed in [19]. However, the
computational complexity of the proposed method is significant, which complicates its use in
real-time DSSs.
Thus, we conclude that existing event-based knowledge representation models can be weakly
applied to DSSs of the considered class. We need not only to refer events to a certain time and
spatial locations but also use a relatively simple but effective model for the representation of
uncertainty that could satisfy both the requirements for the representation of incomplete and
inaccurate information obtained from the observation of a dynamic system and requirements to
efficiency that makes a model suitable for real-time GIS-based DSS.
3. Uncertainty Models
Last years, researchers have directed sufficient efforts towards improving the above-mentioned
issues. The classical approach [20] is that a behavior of a dynamic system can be represented by
high-dimensional nonlinear equations, which describe complex processes arising within the
considered space and time. However, such continuous modeling of dynamic systems contradicts
a lack of data of required quality and accuracy caused by uncertainty and imprecision of sensor
data as well as its discontinuous measurements. Furthermore, systems of high-dimensional
equations cannot be solved in a reasonable time due to a significant computational complexity
[21]. The statistical approaches [22] also cannot provide justified assessments primarily be-cause
of a lack of reliable statistics and weak observability of a dynamic system.
Since the use of the continuous space, time, and correspondent models of the dynamic system
leads to a huge computational complexity, it is advisable to discretize time and space (e.g. AOI)
[23] for better compatibility with discrete measurements provided by sensors and to speed up the
calculations. At the same time, the use of a discrete model reduces the accuracy and credibility
of assessments [24]. Therefore, based on the domain features, it is necessary to choose such a
sampling discrete that provide both the sufficient performance of assessments and the required
accuracy of its results.
The main question is how to manage uncertainties. The most commonly used approach to
take into account uncertainties is probabilistic, which represents different aspects of uncertainty
in terms of chances [25]. However, typically there are few repeated occurrences of events under
the same conditions, especially considering their spatial and temporal references. Thus, the
probabilistic approach deals only with stochastically stable data and represents uncertainty
inadequately [26].
In this regard, various non-probabilistic methods of uncertainty modeling have been
developed such as fuzzy, rough, vague, soft, grey sets, etc.
Obviously, real data cannot be represented as crisp and well-determined. Instead, sensors
provide data that can be not clearly known, undetermined, problematic, varying, vague, can have
many interpretations but not certain information, and, of course, can be no reliable [27].
Discretization of space and time will also lead to various sampling errors, delays in time,
outdated data, etc. [28].
Thus, there are a lot of types of uncertainties that should be modeled and processed within
event-based knowledge representation about complex dynamic systems. Let us consider the
available tools for modeling uncertainty.
Zadeh introduced the concept of the fuzzy set [29] that has been used in a wide range of
various fields and proposes a convenient tool to represent vague data. Its drawbacks are that
fuzzy sets are computationally hard, and their membership functions are subjective and can be
difficult to found [30].
The long-term study of many researchers has led to the emergence of dozens of fuzzy
extensions and additions both in theoretical and practical areas [31], however, they do not allow
to overcome the above disadvantages, and further increase the computational complexity, which
hinders their use.
Pawlak introduced the concept of rough sets [32] as a mathematical tool to deal with
imprecise or noisy data based on equivalence (e.g., indiscernibility) relation, that fits well to
discrete-valued and nominal data. Rough sets are easy to understand, suitable for inconsistent
data, and do not need any additional information about data. Their drawbacks are a problem with
dirty or noisy data, depending on complete information, and a lack of membership values. In
general, in many fields of application, the rough set algorithms are much more efficient than
fuzzy.
Clearly, fuzzy sets and rough sets provide models for two different types of uncertainties.
Fuzzy sets can be represented by their membership values, but rough sets can be approximated
through partitions. Fuzzy sets highlight the vagueness of information while rough sets focus on
incomplete information. Therefore, fuzzy sets mainly represent subjective uncertainties while
rough sets represent objective uncertainties.
Grey sets and grey numbers have been proposed by Deng Julong [33]. A grey number is a
number, whose exact value is not known but its interval is known. In general, the grey number
introduces a certain set of values and represents only one number, which is not clearly identified
among the elements of the set. It can be reduced to a white number or black number, the first one
is an exact or crisp value while the second is a number, whose exact value or value’s interval is
not known. Thus, grey numbers can be an adequate model in the case when the exact value is not
completely known.
Molodtsov introduced the concept of the soft sets [34] as a relatively new approach to model
both uncertainties and vagueness, which is free from the difficulties of existing methods. Soft set
membership can be determined through a certain parametrization given by real numbers,
functions, mappings, and others, even words or sentences. Thus, a membership function problem
cannot arise in the soft sets. They are a convenient and easy tool for model both objective and
subjective uncertainty, their main advantage is that they are free from the inadequacy of
parameterization tool. The drawback is that the soft set does not assign any membership values.
Soft sets are rarely used independently, but often become the basis for complex models for
representing uncertainty.
Obviously, in the real world, objective uncertainty and subjective uncertainty may exist at the
same time. Accordingly, researchers try to build complex models that consider different types of
uncertainty at once either by extending existing models or by combining them. Often, the
uncertainty models presented above are related and complementary to each other. Moreover,
they can also be reduced to one another [35]. This makes it possible to combine them, obtaining
complex models of uncertainty.
The most effective uncertainty models are soft and gray. In this work, we propose to use gray
numbers to represent the values of the observed parameters of the dynamic system as well as soft
sets to represent events concerning their classes, temporal and spatial locations that make them
applicable to the analysis of dynamic processes.
Let us consider a model of a dynamic process at two levels: at the micro level considering
individual spatial elements, and at the macro-level considering spatial areas of a larger or smaller
scale.
4. Micro-Model of Dynamic Processes
4.1. Cells and their states
Consider an AOI as a two-dimensional Euclidean space discretized uniformly by a metrical grid
D . Using the spatial discrete δ , the grid D outlines the two-dimensional array D = {d xy }x , y = 0 of
N
square cells d xy of size δ × δ , where x and y are the array indices corresponding to the
coordinate axes. Suppose a certain cell d xy ∈ D is an object of consideration that represents a
minimal homogeneous area within the AOI.
Let us imagine that a certain non-empty set of parameters A = {ai }i =1 can be obtained as a
m
result of the observation and associated with the cell d xy ∈ D . Let Va be a domain of each ai ∈ A ,
i
V = ∪a ∈ AVa , and f be a value function f : A × D → V that returns a value of a certain parameter for
i i
the cell d xy ∈ D .
If the values of some parameters ai ∈ A are unchanged over time, such parameters belong to a
subset AS of static parameters, AS ⊆ A . In this way, if the values of certain parameters are
varying over time, such parameters constitute a subset AD of dynamic parameters, AD ⊆ A . There
can also be such parameters whose values change over time, but slow enough (e.g.,
environmental parameters), they constitute a subset AE of slowly changing parameters.
Obviously, A = AS ∪ AD ∪ AE .
Each cell d xy ∈ D can be associated with a certain subset A ( d xy , t ) ⊆ A of parameters’ values at
a certain time t ; some of them can be imprecise while others can be unobservable at the time t .
It should also be noted that some of the parameters cannot be directly measured or estimated, so
they require to use of indirect methods.
Let us consider a state S ( d xy , t ) of the cell d xy at the time t such that S ( d xy , t ) = {am ( t )}m =1 ,
z
∀am ( t ) ∈ A ( d xy , t ) , and a state function υ : D × A → S , which returns the state of the cell d xy diagnosed
at the time t based on the observed subset of the cell’s parameters.
Let C = {c j } j =1 be a set of cell’s statuses and ϑ be a status function ϑ : D × AS → C . Thus, we can
q
correlate a cell with a particular status based on information about the values of a certain subset
of its static parameters. For example, a cell can have a status of “water”, “soil”, “sand”, or
“rocky” based on the value of the observed parameter "ground". Obviously, the meaning of the
cell’s status can vary but it should be based on a selected set of static parameters. Similarly, the
state of the cell can be defined with respect to the different points of view.
Let W = {wk }k =1 be a set of state classes. Suppose each state class wk ∈ W contains a finite set of
n
micro-states, wk = {wkl }l =1 , which constitute an ordered sequence of transitions wk1 ,...wkf , where
f
wk1 is an initial micro-state and wkf is a final micro-state. Concerning the dynamic process, we
assume that the dynamic process covers the cell when the last being in the micro-state wk1 and
ends within the cell when the cell enters the state wkf . Obviously, each micro-state wkl can be
defined as a certain subset of the cell’s state at the time t determined by a specific subset of
parameters
= Akl {aklm }m =1 ⊆ A .
u
Fig. 1 shows the example of two classes of states, namely “moisture” and “burning”, defined
by sequences of micro-states w1 and w2 . Some microstates can be observed simultaneously, for
example, the cell can be defined by micro-states "dry" and "pre-ignited" at the same time.
However, micro-states "wet" and "pre-ignited" cannot coexist. Another peculiarity of the
proposed micro-model is that micro-states of a certain class are not always compatible with
every possible cell status. For example, the cell statuses "water" or "rocky" is not compatible
with the state class "moistening" for obvious reason. Clearly, we need to define a state class
function ω : D × A → W that returns a micro-state of the cell based on the values of its parameters
that belong to Akl .
We assume that each transition of the cell d xy from one micro-state wkl to another micro-state
wkm is a micro-event : wkl → wkm (Fig. 2).
Cell d xy
moistening w1 w2 burning
overdried w11 w21 free
w12 w22 pre-ignited
dried
w13 w23 ignited
moistened
w24
wet w14 burned
watery w15 w25 burnt
dissolved w16
Figure 1: Representation of micro-states of the cell
1 : wkl → wk ( l +1)
wkl
wk ( l +1)
2 : wk ( l +1) → wkl
Figure 2: Example of direct and reverse transition of a cell from one micro-state to another
Thus, the dynamic process can be modeled by sequences of dynamic changes of micro-states of
cells covered by the process spatially. During the process, cells pass through a sequence of
micro-states. Obviously, some transitions from one micro-state to another in the context of a
certain state class can entail transitions within other state classes. Moreover, the transitions of
micro-states of one cell can spread to neighboring cells, which makes it possible to simulate the
propagation of the dynamic processes within AOI.
4.2. Propagation of the Dynamic Processes
We assume that all changes in the micro-states of cells, as well as all conditions for the
propagation of a dynamic process between cells, are associated with the transfer of certain
contingent energy.
Suppose that each class of states has its class of energy. Energy can be generated because of
changes in the values of the cell parameters and accumulated in the specific energy storage
inside the cell.
There are a set I of energy transfer channels to transfer energy between cells as shown in
Fig. 3. Eight channels {I + , + , I + ,0 , I + , − , I −,0 , I 0, + , I −, − , I 0, − , I −, + } ∈ I reflect the relative position of cells in
space and therefore are denoted relative to the considered cell.
These channels can be unidirectional or bidirectional.
Since the propagation of the process is usually influenced by a significant number of factors
of a stochastic nature located outside the system under consideration, it is necessary to have a
specific tool for modeling such effects. Many dynamic systems are influenced by such external
processes associated with the environment. For example, the speed and direction of propagation
of a forest fire are most influenced by the speed and direction of the wind. Clearly, the wind is a
separate dynamic process observable by dynamic parameters, which should be taken into
account in the model.
d ( x −1)( y +1) d x ( y +1) d ( x +1)( y +1)
I −,+ I 0, + I +,+
d ( x −1) y d xy d ( x +1) y
I − ,0 I + ,0
I −,− I 0, − I +,−
d ( x −1)( y −1) d x ( y −1) d ( x +1)( y −1)
Figure 3: Channels for energy transfer between cells
Thus, it is proposed to use a matrix Ω of coefficients of energy transfer through channels, where
each coefficient can slow down or accelerate the process of energy transfer, so the speed of the
dynamic process propagation can change accordingly.
The matrix Ω can be defined as
α + , − α + ,0 α + , +
Ω =α − ,0 0 α + ,0 ,
α − , − α 0, − α + , −
where each coefficient α ∗,∗ corresponds to a certain energy transfer channel I∗,∗ and takes values
in the range [ −1,1] . Direct determination of the values of the coefficient matrix Ω is beyond the
scope of this paper, but its source is obviously the observations.
4.3. Micro-Events
In this work, each micro-event is referenced in space and time. The impulse paradigm is used to
model micro-events, which are considered as the direct effects of the change of values of specific
observable parameters.
A unique descriptor y j can be defined as a tuple:
=yj a j , λ j ,δ j , ∆ j , (1)
where a j is a certain parameter, a j ∈ Akl , ∆ j is an absolute change of the value a j , λ j is a
susceptibility for the attribute a j , and δ j is a threshold value.
Due to the inaccuracy of observations, the estimated values of parameters are mainly
inaccurate. This requires adequate representation. To ensure correct representation, we can
define intervals that contain the confidence degrees from the minimum to the maximum possible
value. Since such intervals can be narrowed during further observations, we propose to represent
the value of the observed parameter as a gray number. Moreover, it is also necessary to represent
the threshold value as a gray number to achieve a trigger effect, which is often observed in real
natural systems.
Thus, the value ∆ j can be described as the interval ∆ ±j = ∆ −j , ∆ +j that represents a grey number
and δ j can be “grayed” in the same way. The descriptor y j also turns gray and is denoted by y j .
An integral descriptor y can be defined as a tuple:
{β ⋅ y }
m
=y j j j =1
, (2)
where β j is a certain coefficient for each corresponding descriptor y j . It represents the direct
effect of the simultaneous change of several parameters’ values within the exposed cell.
Consider a micro-event as a consequence of the transition of a cell’s micro-state of a certain
class, which can be generated as an energy impulse ε based on y .
The impulse generation scheme is shown in Fig. 4.
γ −,−
I −,−
Channels …
γ +,+ Correlator
I +,+
⊕ ( γ ⋅ I )∗,∗
ι εk
ι = ⊕
∗, ∗ ∗, ∗
Impulse
⊕ (λ ⋅ ∆± )
δ +
j j generator
λ1 1
y1
δ1−
Descriptors …
δ m+
λm
ym
δ m−
Figure 4: The impulse generation schemes
It considers the fact that the change in the values of the parameters is influenced by the impulses
received through the channels from the neighboring cells. The correlator (Fig. 4) accumulates
inputs and ensures that the control stimulus is found, and the impulse generator produces an
energy impulse of an amount ε k that depends on the magnitude of the control stimulus ι . The
descriptors’ values are compared with a threshold value. The result of the descriptors’ triggering
helps us to determine whether an event will occur.
Thus, let us describe the micro-event k as a couple
t j , d xy , wk , wkm , ε k , (3)
where t j is a time reference of , d xy is a spatial reference of within AOI, wkm is a new
d xy micro-state of state class wk , and ε k is an amount of generated energy of class k that
represents an event magnitude.
5. Macro-Model of Dynamic Processes
At the higher level, the grid D can be divided into disjoint objects, which describe the
homogeneous areas of the AOI in terms of their parameters’ values.
Suppose Akl ⊆ A is a non-empty finite subset of parameters. Let us define an Akl -
indiscernibility relation on the grid D =
, RDA kl
{( d , d ) ∈ D × D ∀a ∈ A , f ( d=
xy mn , a ) f ( d , a )} . Using
j kl xy j mn j
this relation, we can describe homogeneous spatial areas, which are uniform concerning the
values of the parameters belonging to the subset Akl , represented by the approximating set of
cells, and denoted by h . All cells that belong to the spatial area h are Akl -indiscernible.
Each spatial area cannot overlap or cover one another, but they can be adjacent or adjoin to
one another. They have such features as continuity and connectivity (spatial concentration of the
underlying cells).
Let H be a set of spatial areas, H = {h1 ,...hk } . To represent a plurality of spatial areas that have
not the property of the continuity, but describe a set of separate areas spatially distributed on the
set D, we can also use a certain Alm -indiscernibility relation defined over H,
{ }
RHAlm = ∀hl , hq ∈ H , ∀d m , d n ∈ D, ∃d m ∈ hl , d n ∈ hq ∀ak ∈ Alm , f ( d m , ak ) =f ( d n , ak ) . Obviously, all areas
belonging to R are Alm -indiscernible.
Aj
H
5.1. Energy transfer
The accumulation of energy inside the cell cannot continue indefinitely. As soon as the amount
of energy reaches a certain predetermined level, it "overflows", forming an impulse, which
should be distributed between the channels under the matrix Ω of coefficients and transmitted to
other cells.
Energy transfer is seen both as a basis of causal relationships between events and as a means
of organizing them into cascading structures.
Obviously, energy can be of different types, each of which corresponds to the class of the
observed event. The class of observed events, in turn, corresponds to the class of state that
changed and raised the event. Usually, specific events can be driven only by the appropriate type
of energy.
When the event of a certain class occurs, the impulse generator must eject some quantum of
energy of a corresponding class with an appropriate amount. In general, the event model should
consider the possibility that a quantum of energy contains interrelated portions of energy of
different classes, which do not just correspond to the event class. The key role in understanding
the dynamics of the ongoing processes is played by the dynamic parameters of the generated
impulses. Depending on the impulse duration, its amplitude, the pressure it exerts on the input of
the event, a different picture of the reaction of one event to the other event that has occurred can
be observed.
Thus, the proposed model can transfer different types of energy by short bursts (as impulses)
or by long potential inputs. The event can be only triggered if a sufficient amount of appropriate
energy is received, which should be determined by the integral of the received energy over time.
Consider the portions of energy transfer. Let ε k be an energy portion of class k that
corresponds to wk . Suppose energy portion ε k can be described as
ε k = ( k ,τ , q1 , q2 ) , (4)
where k is an energy class, τ is a duration, q1 is an initial amplitude, and q2 is a final amplitude
(Fig. 5).
q1
q2
Impulse
τ
Figure 5: Dynamic parameters for energy transfer impulse
The energy quantum can be represented by a tuple of energy portions:
Ε = ε1 ,...ε m . (5)
The released energy can directly impact events sensitive to this type of energy through certain
connections between them. The proposed model allows us to express not only the direct but also
indirect effects of events through the energy transfer.
5.2. Connectors
Let ϕl be a connector that connects events j and k transferring energy portions ε j1 ,...ε jm from
the first one to the second. Such transfer for each energy portion can be labeled by certain
confidence represented by a gray number µl± and a certain time tl± , which is also grayed. The
connector ϕl can also have a sensitivity point σ l , which controls the transfer process, if
necessary. Suppose the impact of a certain energy portion ε l on the connector allows it to block,
break, or amplify the energy transfer from j to k (depending on the type and amount of ε l ).
Thus, the connector ϕl can be defined as a tuple
ϕl = {ε j1 ,...ε jm } , j , k , µl± , tl± ,σ l : {ε l1 ,...ε lw } , (6)
where j and k are events, {ε j1 ,...ε jm } is a subset of energy portions permitted to transfer
through ϕl , µl± is a likelihood, tl± is the time to receive energy portion, and σ l is an optional
sensitivity point with the energy portions {ε l1 ,...ε lw } permitted to receive.
Let ψ k be a meta-connector that connects the event j and the sensitivity point σ l of a
certain connector. It allows transferring energy portions of given classes {ε f 1 ,...ε fm } with a certain
degree of acceleration χ k as follows:
ψ k = {ε f 1 ,...ε fm } , j ,σ l , χ k . (7)
5.3. Abstract Events
Suppose an abstract event η consists of a set of inputs { xk1 , xk 2 ,...xkm } , each of which is sensitive
to a class k of energy and receive only the correspondent energy potions ε k , a set of outputs
{ yk1 ,... ykn } , each of which ejects an energy portion ε p of a class p, and a set of accumulators
Ξ ={π k 1 ,...π km } , where each π kj accumulates energy portions ε j of a class j through the input xkj .
Each input xkj is connected to the accumulator π kj with a weight (multiplication factor) ζ kj .
T
Thus, the energy portion ε j received through input xkj adds the amount of energy ζ kj ⋅ ∫ qτ of
0
type j to the accumulator π kj . If accumulated π kj energy exceeds a threshold value β kj , a certain
quantum of energy Ε should be released and ejected to the connectors. The classes, amplitudes,
and durations of energy portions released by energy quantum depend on a certain multiplication
factor γ kj .
Thus, the abstract event ηk can be represented as
ηk X k , Yk , Ξ k , (8)
where X k is an input part, X k = {( xk1 , ζ k1 ) ,...( xkm , ζ km )} , Yk is an output part, Yk = { yk1 ,... ykn } , and Ξ k is
an accumulation part, Ak = {(π k1 , β k1 , γ k1 ) ,...(π km , β km , γ km )} . The proposed model of the event is
flexible and dynamic, since the weights ζ kj , threshold values λkj , and factors γ kj can
dynamically change in time.
5.4. Abstract Event Network
An abstract event network can be represented as a time-ordered event structure
{ηk }k 1 ,τ ,ν= ,{ψ k }k 1 , where {η k }k =1 is a set of abstract events, τ : η → T is a mapping that
,{ϕl }l 1 =
n w v n
G ==
expresses sequential order of the events, ν : η → H is a mapping that expresses the spatial
reference of events, {ϕl }l =1 is a set of connectors between events, and {ψ k }k =1 is a set of meta-
w v
connectors, which connect events and corresponding sensitivity point of the connectors. This
formalization allows the use of soft sets to represent different sequences of events, based on the
indiscernibility relation between events by class, spatial position, or time intervals.
Fig. 6 shows the abstract event network, in which nodes represent events and the arcs
represent connectors and meta-connectors.
Figure 6: Abstract Event Network
6. Implementation
The proposed model has been implemented using Visual C++ based on the double indexed lists
and approbated on the simulated area. The AOI is the Lower Dnieper Sands (Oleshky Sands) in
the Kherson region, in Southern Ukraine. The sands are surrounded by very dense artificial
coniferous forests that prevent the sands from moving during strong winds. Global warming
leads to the loss of forests in this area. As a result of global warming, we can observe chains of
cascading effects. Due to warming, the groundwater levels are decreased, which further
increases fire danger, rapid destruction of forests in large areas, desertification of the territory,
and the revival of sand movement. Due to warming, forests are also being affected by invasions
of insects, and also become more prone to forest fires (Fig. 7).
The results of the conducted simulation show that the proposed model provides enough
performance to real-time modeling of a wide range of natural processes from climate change to
forest fires and adequate knowledge representation about cascading events taking into account
the uncertainty of the observations.
Figure 7: Cascading chains of events in Kherson Region
7. Conclusion
The proposed event-based model enables knowledge representation about an observed plurality
of dynamic processes of various intensities that arise unexpectedly and evolve simultaneously in
a wide range of domains. It is based on events that are referenced to time intervals and spatial
areas and take into account the uncertainty of the observed information about the evolving
processes. Uncertainty is represented by gray numbers and soft sets. Events are connected in an
abstract network, where arcs represent connectors and meta-connectors that transfer energy by
impulses to model transitions of a dynamic system from state to state. The proposed abstract
event network can be used to model the evolution of dynamic systems that can be expressed by
events that occurred inside the system but driven by impacts from the outside (environmental
effects). Such evolution is considered in a wide range of rates, from the slowest processes
associated with climate change to the most rapid processes associated with the effects of natural
forces of a destructive nature. The proposed model also makes it possible to adequately represent
sequential, parallel, and cascade chains of events with a trigger effect, information about which
is incomplete and inaccurate. Future research will be devoted to the study of the coefficient
matrices and formalization of the process of generating energy quantum.
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