=Paper=
{{Paper
|id=Vol-3156/paper5
|storemode=property
|title=Computer Ontology of Mathematical Models of Cyclic Space-Time Structure Signals
|pdfUrl=https://ceur-ws.org/Vol-3156/paper5.pdf
|volume=Vol-3156
|authors=Christopher Nnamene,Serhii Lupenko,Oleksandr Volyanyk,Oleksandra Orobchuk
|dblpUrl=https://dblp.org/rec/conf/intelitsis/NnameneLVO22
}}
==Computer Ontology of Mathematical Models of Cyclic Space-Time Structure Signals==
https://ceur-ws.org/Vol-3156/paper5.pdf
Computer Ontology of Mathematical Models of Cyclic Space-
Time Structure Signals
Christopher Chizoba Nnamenea, Serhii Lupenkob,c, Oleksandr Volyanykd and Oleksandra
Orobchukе.
a
Alex Ekwueme Federal University Ndufu-Alike: Ikwo, Ebonyi, NG
b
Ternopil Ivan Puluj National Technical University, Ruska str., 56, Ternopil, 46001, Ukraine
c
Opole University of Technology, Opole, 45-758, Poland
d
Institute of Telecommunications and Global Information Space of National Academy of Sciences of Ukraine,
Кyiv, Ukraine
e
Ternopil Ivan Puluj National Technical University, Ruska str., 56, Ternopil, 46001, Ukraine
Abstract
The work is devoted to the development of formal and machine-interpretive models of
computer ontology of mathematical models of signals of cyclic space-time structure as
the core knowledge base of onto-oriented expert decision support system in solving
problems of reasonable choice of mathematical models of cyclic signals in the theory of
cyclic functional. Based on the obtained theoretical results, a procmdtotype of computer
ontology of mathematical models of signals of cyclic space-time structure is constructed,
which contains the set and taxonomy of names of classes of cyclic functional relations,
their attribute and definition vectors.
Keywords 1
computer ontology, expert system, ontological modeling, mathematical modeling, signal
processing, cyclic signals
1. Introduction
Many scientific works have been devoted to the development of computer systems for automated
analysis, forecasting, classification, clustering, regression, simulation (generation) of cyclic signals
[1-7]. The class of cyclic includes such types of signals as cyclic heart signals (electrocardiographic
signals, magnetocardiographic signals, phonocardiographic signals, sphygmocardiographic signals,
etc.), cyclic economic processes, cyclic processes of relief formation on the surface of materials,
cyclic processes of gas consumption, energy consumption and electricity consumption. water
consumption), cyclical processes in telecommunications systems and computer networks. The
accuracy, reliability, informativeness and computational complexity of the functioning of these
information systems significantly depends on and is determined by the properties of the relevant
mathematical models and methods of processing signals of cyclic structure, which underlie their
mathematical support.
In the last two decades, the theory of cyclic functional relations has developed significantly, which
is a fundamental theory of mathematical, computer modeling and processing of cyclic signals, and
significantly generalizes, expands, systematizes known mathematical models and methods of signal
processing and processes of cyclic structure [8, 9]. Within the framework of this theory, new
mathematical models, methods of processing (statistical estimation, spectral analysis, sampling) and
IntelITSIS’2022: 3rd International Workshop on Intelligent Information Technologies and Systems of Information Security, March 23–25,
2022, Khmelnytskyi, Ukraine
EMAIL: chrisnnamene@yahoo.com (C. Nnamene); lupenko.san@gmail.com (S. Lupenko); wonderage2018@gmail.com (O. Volyanyk);
orobchuko@gmail.com (O. Orobchuk).
ORCID: 0000-0002-1977-4490 (C. Nnamene); 0000-0002-6559-0721 (S. Lupenko); 0000-0001-9137-7580 (O. Volyanyk); 0000-0002-
8340-913X (O. Orobchuk).
©️ 2022 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)
methods of computer simulation of cyclic signals of biological, economic, and technical origin are
built. The theory of cyclic functional relations has the means of mathematical description and
processing of cyclic signals in the framework of deterministic, stochastic, fuzzy and interval
approaches to modeling cyclic signals. Within the framework of the stochastic approach mathematical
models in the form of cyclic random processes and vectors, as well as conditional cyclic random
processes are intensively applied and developed, which generalize periodic (periodically correlated
and periodically distributed) random processes and vectors and have formal means of considering the
variability and stochasticity of the rhythm of the studied cyclic signals. These mathematical models
and their corresponding methods of cyclic signal processing have become the core of mathematical
software for several software systems for analysis and prediction of cyclic signals (processes) in
medical cardio diagnostics, econometrics, biometric authentication, non-destructive diagnostics, and
energy consumption.
It is known that the development or reasonable choice of a mathematical model of the studied
cyclic signals is the first difficult but important step in building a whole set of mathematical tools for
automated processing and computer simulation of cyclic signals in modern information systems.
Given the large number of existing mathematical models of signals of spatiotemporal structure, which
describe them in the framework of deterministic, stochastic, fuzzy and interval approaches, there is a
need to develop an expert system to support decision-making in the problems of reasonable choice of
optimal (quasi-optimal) mathematical model of cyclic signals, within which they will be processed in
the appropriate automated information systems. The need to develop such an expert system becomes
even more obvious, because often the collective developers of appropriate automated information
systems do not have a direct highly qualified specialist in the field of mathematical modeling and
processing of cyclic signals. Users of such an expert system can be researchers, engineers who are not
direct highly qualified specialists in the field of modeling and processing of cyclic signals, but who
need to solve problems in this field using information system for modeling and processing of cyclic
signals.
The process of building its knowledge base is central to the process of developing any expert
system. Given the rapid development of knowledge engineering technologies of ontological modeling
of various subject areas, it is appropriate to use the computer ontology of the corresponding
mathematical models as the core of the knowledge base of the expert decision support system in the
problems of choosing mathematical models of cyclic signals. The expediency of the ontological
approach is also justified by the fact that ontology allows to specify the knowledge and automate the
procedures of logical inference (proof), which are contained in the theory of modeling and processing
of cyclic signals, enables the presentation of the theory of cyclic functional relations in machine-
interpretive form as a basis for the development of onto-oriented information systems for modeling,
generation, processing (analysis, forecasting, decision-making) of cyclic signals. In addition, the
ontological approach is well coordinated with the axiomatic-deductive strategy of modeling theory
and processing of cyclic signals based on cyclic functional relations, which significantly increases its
structure, rigor, and formalism, facilitates the identification of new directions and regions of cyclical
functional theory.
2. Review of literature sources
Ontologies as a means of presenting and preserving knowledge have been actively used in various
subject areas. Thus, in the field of medicine, well-known computer ontologies are GO [10], a
systematized multilingual thesaurus with the properties of SNOMED CT ontology [11], domain
ontology FMA Ontology [12], ontology-like reference terminology dictionary NCI [13], ontology
Malaria Ontology, which covers all aspects of malaria and measures to combat it; Hypertension
Ontology to present clinical data on hypertension; Antibiotic Resistance Ontology, which describes
genes and mutations that are resistant to antibiotics [14].
Computer ontologies are actively used for folk and alternative medical systems in projects such as
CLINMED, TCM-Grid, TCMLARS, TCMID, Database of Chinese Medical Formula, TCM
Electronic Medical Record Database, China Hospitals Database, Venomous Chinese Herb Database
Platform, TCDBASE, Database of Chemical Composition from Chinese Herbal Medicine, TCM
Herbal Medicine Database, Chinese Materia Medica, TCMGeneDIT, Chinese Herbal Medicine
Ontology and others. [15-19].
Ontology has also found wide application in the problems of materials design and computational
materials science. An example of such an ontology is the MDO (Materials Design Ontology) ontology
[20]. In the field of economics, in the field of textual data in economics, computer ontologies are also
actively used. An example of such a well-known ontology is the Ontology-based multi-label
classification of economic articles, which is designed to automatically classify documents in the field
of economics, allowing their automatic description and search [21]. For problems in mathematics and
mathematical modeling, the OntoMath ontology is fruitfully used, and in the subject area of signal
and image processing - the computer ontology ODSPTB (Ontology-Directed Signal Processing
Toolbox) [22].
In [23, 24] a generalized conceptual model of ontology and a formal model of ontology of the
subject area "Modeling and processing of cyclic signals" were developed. The ontology studied in
these works includes the ontology of mathematical models of cyclic signals as its subontology. Also,
an important result of research, the results of which are presented in [23, 24] is the application of the
method of induction to form the main components of the computer ontology of mathematical models
of cyclic signals for the formation of many names and definitions of classes of cyclic functional
relations, their taxonomy. This method consists of a combinatorial ordered combination of names of
types of regions of values, types of attributes of cyclicity, types of domains of definition and types of
rhythm functions in the definition of abstract cyclic functional relation as a generalized mathematical
model of signals of cyclic space-time structure. mathematical models of cyclic signals.
Given the above material, the aim of this work is to develop the known results obtained in [23, 24],
namely, the development of a formal model of ontology of mathematical models of cyclic signals and
the development of a prototype of the corresponding computer ontology, which is the core knowledge
base of the expert system of support of model decision-making in problems of modeling and
processing of signals of cyclic space-time structure.
3. Main part
3.1. Formal model of computer ontology of cyclic signal models
Let us concentrate on the construction of computer ontology of mathematical models of signals of
cyclic space-time structure, namely, ontology O. Ontology O - ontology of models of cyclic signals
(ontology of cyclic functional relations) at the formal level is given by the following relational
system:
𝑨 = 𝑩 ⋃ 𝑪 , 𝑹 = {𝐀𝐊𝐎, 𝐈𝐒 − 𝐀, 𝑷 ̅ = (p1 , p2 , … , pn ), ̅̅̅̅̅̅
𝑳𝑖𝑚𝑝 = (l1 , l2 , … , l8 )}, (1)
𝑶={ }
𝐅 = {f(⋅)}
where: A is a finite set of terms (concepts), which defines the lexical stock of the ontology O.
B is a finite set (names) of classes of cyclic functional relations. C is a finite set of 4-element
vectors, each element of which is a term, which derive their values from the corresponding
sets𝑿𝜳 ,𝑿𝑨, 𝑿 𝑇(𝑡,𝑛) , 𝑿𝑾 , and has the domain of predicate 𝑃(𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 ). Sets X Ψ , is a set of
predefined classes (types) of linear spaces 𝚿 in which the corresponding cyclic functional relations
gain the values; plural 𝑿𝑨 is a set of____ predefined classes (types) of possible attributes 𝑝: 𝜳 → 𝑨 or sets
𝑛𝑘
of attributes {𝑝𝑘 : 𝜳 → 𝑨𝑘 , 𝑘 = 1, 𝐾 }, in which the cyclical structure of the functional relation is
postulated (reflected); Sets 𝑿 𝑇(𝑡,𝑛) is a set of predefined classes (types) of rhythm functions 𝑇(𝑡, 𝑛)
cyclic functional relations, and the set 𝑿𝑾 is a set of predefined types of definition areas 𝑾 cyclic
functional relationship. Identically true 4-local predicate 𝑃(𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 ) is a function-expression,
which is set on sets𝑿𝜳 ,𝑿𝑨,𝑿 𝑇(𝑡,𝑛),𝑿𝑾 (𝑥1 ∈ 𝑿𝜳,𝑥2 ∈ 𝑿𝑨 ,𝑥3 ∈ 𝑿 𝑇(𝑡,𝑛),𝑥4 ∈ 𝑿𝑾 ) and which derives
its meanings from the set 𝑫𝑒𝑓𝑐𝑓 all possible definitions of specific subclasses of cyclic functional
relations.
F – a one-element set that contains an interpretation function f(𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 ), the domain of which
is the set C, and the domain of values is the set B. Interpretation function 𝑓(𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 ) identical 4-
local predicate 𝑃(𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 ) and, in fact, for specific sets 𝑥1 ∈ 𝑿𝜳 ,𝑥2 ∈ 𝑿𝑨 ,𝑥3 ∈ 𝑿 𝑇(𝑡,𝑛),𝑥4 ∈ 𝑿𝑾
sets the definition of the corresponding classes of cyclic functional relations with B, forming a
glossary of ontology O - the set of all definitions of cyclic functional relations.
R - a finite set of relations{𝐀𝐊𝐎, 𝐈𝐒 − 𝐀, 𝑷 ̅ = (p1 , p2 , … , pn ), ̅̅̅̅̅̅
𝑳𝑖𝑚𝑝 = (l1 , l2 , … , l8 )}, namely:
1. AKO genus-species subordination relationship, which connects a set (class) and a subset
(subclass) of cyclic functional relationships, defining the 𝑻𝒂𝒙_𝒐𝒇_𝑪𝒇 taxonomy (hierarchy) between
these classes as a taxonomic tree.
2. The IS-A membership relationship, which corresponds to a specific cyclic functional
relationship (specific model of cyclic signals) to their corresponding class (class of mathematical
models of cyclic signals).
3. 𝑷 ̅ - vector = 𝑷 ̅ = (p1 , p2 , … , pn ) unary relations (p1 , p2 , … , pn ), which define the properties
(features, attributes) of the corresponding class of cyclic functional relations. The domain of the unary
relation 𝑷i is the set of models A, and the domain of values is the set 𝑷i of the values of the
corresponding property.
4. ̅̅̅̅̅̅
𝑳𝑖𝑚𝑝 – vector ̅̅̅̅̅̅
𝑳𝑖𝑚𝑝 = (l1 , l2 , … , l8 ), components, which characterize the state (level) of
elaboration (implementation, development) of the corresponding information technologies of
processing and computer simulation (generation) of cyclic signals within the framework of this
mathematical model.
3.2. Substantiation of the choice of language and software for the
development of the ontology of mathematical models of cyclic signals
The formal ontology model developed above requires a reasonable choice of machine-
implemented languages and development environments for their presentation in modern onto-oriented
information systems. Currently, we have so many ontology management tools, such as Protégé,
OntoEdit, the Ontolingua ontology development environment, Chimaera and OntoGen. Given the
results of a comparative analysis of ontology development tools, it is appropriate to use the OWL
ontology description language and the Protégé environment [25].
OWL (Web Ontology Language) is a standard in the World Wide Web Consortium and is
currently the world's most widely used ontology description language. There are three types of OWL.
The OWL DL language is the most acceptable for solving the problems of developing the ontology of
the subject area "Modeling and processing of cyclic signals", which allows, on the one hand,
achieving the maximum expressiveness of the descriptive logic that underlies it, and on the other - to
ensure the solvability of the system of logical inferences with its use. OWL DL language is provided
with standardized language constructions for adequate expression of terms-concepts of the theory of
cyclic functional relations, their taxonomies and other relations, logical operations on classes of
models, methods and means of modeling and processing of cyclic signals.
Due to the inconvenient for human perception syntax of the OWL language in the development of
ontologies, it is necessary to use the graphical tools of specialized software systems for the
development of ontologies. These software tools include a graphics editor and a reasoner machine,
which allows both automation of logical derivation based on ontology knowledge and automatic
verification of the correctness of the developed ontology. Protégé is the most popular ontology editor
that supports the OWL DL language. This graphics editor is a freely distributable Java application that
contains many plug-ins. In general, the Protégé editor allows you to design, view, edit, integrate,
populate, and adapt ontologies to different data formats (text, XML, RDf (s), OWL, etc.).
The formation of the set B of names of classes of cyclic functional relations must be unified. The
names of the classes of cyclic functions are derived from the corresponding names of the elements of
the sets 𝑿𝜳 , 𝑿𝑨, 𝑿 𝑇(𝑡,𝑛) , 𝑿𝑾 according to the rules of the relevant national language (Ukrainian,
English, etc.), ie from the names of the relevant types of areas of definition, attributes, functions of
rhythm and areas of definition of the cyclical functional relationship. To simplify the names of cyclic
functional relationships, we will use a shorter version instead of the name "Cyclic functional
relationship" - "Cyclic function".
The name of the cyclic functional relation can be conditionally divided into the following four
parts. The first part of the name of the cyclic functional relationship reflects the type of attribute
𝑛
cyclicity of this function, namely, contains the ____ name of the attribute 𝒑: 𝜳 → 𝑨 (чи 𝑝: 𝜳 → 𝑨) or a
set of attribute names {𝑝𝑘 : 𝜳𝑛𝑘 → 𝑨𝑘 , 𝑘 = 1, 𝐾 }) from a certain, predefined set of names of possible
attributes of cyclicity (sets of names of elements from 𝑿𝑨). The second part of the name of the cyclic
functional relation derives its verbal meanings from the set of possible, pre-defined, names of general
approaches to considering (or not considering) uncertainties in modeling the studied signals, namely,
from the set {deterministic, random, fuzzy, interval}, as well as from the names of the types of
domains of values of the cyclic functional relation (sets of names of elements from𝑿𝜳 ). The third part
of the name of the cyclic functional relationship reflects the type of domain of the cyclic function, i.e
gets its values from the set of names of elements from𝑿𝑾 . The fourth part of the name of the cyclic
functional relation reflects the type of rhythm of the cyclic function, that is, it derives its values from
the set of names of elements from 𝑿 𝑇(𝑡,𝑛).
Here are some examples of the names of cyclical functional relationships formed in accordance
with the above unified approach, namely:
• cyclic value-determined deterministic real function of a real argument with a constant rhythm;
• cyclic values of deterministic real-valued function of a valid argument with variable
piecewise-cubic rhythm;
• cyclic value-determined significant function of many real arguments with variable piecewise-
linear rhythm;
• cyclic modulo deterministic real function of a valid argument with a variable periodic rhythm;
• cyclic squared value determined deterministic real function of a real argument with a variable
piecewise-square rhythm;
• cyclic for many autocorrelations and intercorrelation functions vector random function of a
valid argument with a constant rhythm;
• cyclic in the trapezoidal membership function fuzzy function of a discrete argument with a
variable rhythm;
• cyclic with respect to the system of intervals interval function of a discrete argument with a
variable piecewise linear rhythm.
Each class of cyclic functional relations is defined by its definition, which should be contained in
the glossary of ontology O mathematical models of cyclic signals. Therefore, the next stage in the
formation of the glossary of ontology of mathematical models of cyclic signals is the formation of a
set of definitions of classes of cyclic functional relations, which according to the ontology model O,
the interpretation function is set 𝑓1 (𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 ), which is identical 4-local predicate 𝑃(𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 )
and for specific sets 𝑥1 ∈ 𝑿𝜳,𝑥2 ∈ 𝑿𝑨 ,𝑥3 ∈ 𝑿 𝑇(𝑡,𝑛),𝑥4 ∈ 𝑿𝑾 defines the corresponding classes of
cyclic functional relations, forming a glossary of ontology O.
As stated above, the definition of the cyclic functional relation is considered as an identically true -
local predicate, namely, as a function-expression𝑃(𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 ), which is given on sets
𝑿𝜳 ,𝑿𝑨,𝑿 𝑇(𝑡,𝑛),𝑿𝑾 (𝑥1 ∈ 𝑿𝜳 ,𝑥2 ∈ 𝑿𝑨 ,𝑥3 ∈ 𝑿 𝑇(𝑡,𝑛),𝑥4 ∈ 𝑿𝑾 ),and which derives its meanings from
the set 𝑫𝑒𝑓𝑐𝑓 all possible definitions of specific subclasses of cyclic functional relations. Because the
predicate 𝑃(𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 ) is identically true, then for any set of values 𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 from sets
𝑿𝜳 ,𝑿𝑨,𝑿 𝑇(𝑡,𝑛),𝑿𝑾 , it always turns into a true statement, namely, the definition of a particular
subclass of cyclical functional relations. Also, given that with classes 𝑿𝜳 ,𝑿𝑨,𝑿 𝑇(𝑡,𝑛),𝑿𝑾 related
taxonomies are related 𝑻𝜳 ,𝑻𝑨 ,𝑻𝑇(𝑡,𝑛),𝑻𝑾 , then it can be argued that for any set of values 𝑥1 , 𝑥2 , 𝑥3 , 𝑥4
from any nodes (classes, sets) 𝑿𝜳 ,𝑿𝑨,𝑿 𝑇(𝑡,𝑛),𝑿𝑾 taxonomies 𝑻𝜳 ,𝑻𝑨 ,𝑻𝑇(𝑡,𝑛),𝑻𝑾 , predicate
𝑃(𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 ) always turns into a true statement, which will be automatically generated (generated)
definition of a particular subclass of cyclical functional relations, and will be the appropriate node of
the taxonomy (classification tree) 𝑻𝒂𝒙_𝒐𝒇_𝑪𝒇 models of cyclic signals within the theory of cyclic
functional relations. Figures 1 - 4 show fragments of the glossary of cyclic functional relations as
components of the ontology of mathematical models of cyclic signals, built in the Protégé
environment.
Figure 1: Graphical representation in the Protégé environment of a glossary fragment - the definition
of a cyclic numerical function
Figure 2: Graphical representation in the Protégé environment of a glossary fragment - definition of
a cyclic random process of a continuous argument
Figure 3: Graphical representation in the Protégé environment of a glossary fragment - the definition
of cyclic relative to the set of interests of the function
Figure 4: Graphical representation in the Protégé environment of a glossary fragment - definition of
cyclic relative to the set of function intervals
3.3. Method of forming vector of properties of classes of cyclic functional
relations in the framework of computer ontology of mathematical models of
cyclic signals
In addition to outlining the set of names and definitions of classes of mathematical models of
cyclic signals in the theory of cyclic functional relations, an important component of the ontology of
mathematical models of cyclic signals is the vector 𝑷 ̅ , namely, the vector 𝑷 ̅ = (p1 , p2 , … , pn ) unary
relations (p1 , p2 , … , pn ), which specify the properties (features, attributes) of the corresponding class
of cyclic functional relations, which specify the properties (features, attributes) of the corresponding
class of cyclic functional relations. As mentioned in the second section of the dissertation, the area of
definition of the unary ratio pi , 𝑖 = ̅̅̅̅̅
1, 𝑛 is a set 𝐁1 of the names of mathematical models of cyclic
signals, and the range of values is the set 𝑷i values of the corresponding property.
Thus, each class of cyclic functional relations must have its own vector 𝑷 ̅ = (p1 , p2 , … , pn )
properties (features, attributes), which contains the properties of the corresponding class of cyclic
functions, and which fully characterizes it and underlies the procedure of classification of cyclic
functional relations. To each such vector of properties 𝑷 ̅ = (p1 , p2 , … , pn )we put in correspondence
the vector of numerical variables 𝑰̅ = (𝑖1 , 𝑖2 , … , 𝑖n ), which take values from finite subsets of natural
numbers. Vector 𝑰̅ = (𝑖1 , 𝑖2 , … , 𝑖n ) encodes in numerical format the vector of properties 𝑷 ̅=
(p1 , p2 , … , pn ) and underlies the numerical coding of classes of cyclic functional relations.
Based on the constructed taxonomies 𝑻𝜳 ,𝑻𝑨 ,𝑻𝑇(𝑡,𝑛) ,𝑻𝑾 and their defined priorities, as well as
applying the approach that underlies the method of induction of classes of cyclic functional relations,
let us construct a property vector 𝑷 ̅ = (p1 , p2 , … , pn ) for each class of cyclic functional relations as
mathematical models of cyclic signals. Note that the type of ordering of properties p1 , p2 , … , pn in the
vector 𝑷 ̅ is determined by the order of nodes and priorities of taxonomies 𝑻𝜳 ,𝑻𝑨 ,𝑻𝑇(𝑡,𝑛) ,𝑻𝑾 .
Firstly, highest priority property in the vector of properties 𝑷 ̅ = (p1 , p2 , … , pn ) of classes of cyclic
functional relations concerns the type of uncertainty of values of cyclic signal. Namely, this property
p1 can take values from the set {𝑑, 𝑠, 𝑓, 𝑖}, where d denotes a deterministic approach (determined); s -
stochastic approach (stochastic); f - fuzzy approach (fuzzy); i - interval approach (interval).
The number 𝑖1 can take values from one to four. One (𝑖1 = 1, p1 = 𝑑)will encode a deterministic
approach to the description of cyclic signal values, two (𝑖1 = 2, p1 = 𝑠) will encode a stochastic
approach to the description of cyclic signal values, three (𝑖1 = 3, p1 = 𝑓)will encode something
approach to the description of cyclic signal values, and four (𝑖1 = 4, p1 = 𝑖) will encode an interval
approach to the description of cyclic signal values.
The second priority property p2 (encoded by the variable 𝑖2 ) indicates such a property of the class
of cyclic functional relations as the type of value of the cyclic signal. Namely, this property p2 can
take values from the set
𝑟𝑒𝑎𝑙, 𝑐𝑜𝑚𝑝𝑙𝑒𝑥, 𝑣𝑒𝑐𝑡𝑜𝑟𝑟𝑒𝑎𝑙 , 𝑣𝑒𝑐𝑡𝑜𝑟𝑐𝑜𝑚𝑝𝑙𝑒𝑥 , 𝑚𝑎𝑡𝑟𝑖𝑥𝑟𝑒𝑎𝑙 , 𝑚𝑎𝑡𝑟𝑖𝑥𝑐𝑜𝑚𝑝𝑙𝑒𝑥 , 𝑡𝑒𝑛𝑠𝑜𝑟𝑟𝑒𝑎𝑙 ,
{ 𝑡𝑒𝑛𝑠𝑜𝑟𝑐𝑜𝑚𝑝𝑙𝑒𝑥 , … }
and the number 𝑖2 can take values from 1 to 12. One (𝑖2 = 1, p2 = 𝑟𝑒𝑎𝑙) will encode cyclic signals
whose values are real numbers; two (𝑖2 = 2, p1 = 𝑐𝑜𝑚𝑝𝑙𝑒𝑥) will encode cyclic signals, whose
values are complex numbers, three (𝑖2 = 3, p2 = 𝑣𝑒𝑐𝑡𝑜𝑟_𝑟𝑒𝑎𝑙) will encode cyclic signals whose
values are vectors of real numbers, four (𝑖2 = 4, p2 = 𝑣𝑒𝑐𝑡𝑜𝑟_𝑐𝑜𝑚𝑝𝑙𝑒𝑥) will encode cyclic signals
whose values are vectors of complex numbers, five (𝑖2 = 5, p2 = 𝑚𝑎𝑡𝑟𝑖𝑥_𝑟𝑒𝑎𝑙, ) will encode cyclic
signals whose values are matrices of real numbers, six (𝑖2 = 6, p2 = 𝑚𝑎𝑡𝑟𝑖𝑥𝑐𝑜𝑚𝑝𝑙𝑒𝑥 ) will encode
cyclic signals whose values are matrices of complex numbers, seven (𝑖2 = 7, p2 = 𝑡𝑒𝑛𝑠𝑜𝑟) will
encode cyclic signals whose values there is a tensor, and so on.
The third priority property p3 (encoded by the variable 𝑖3 ) indicates the type of cyclicity attribute
of this class of cyclic functional relations. As a rule, 𝑖3 = 1, when the cyclicity attribute of the
function is its value. For other cyclicity attributes, 𝑖3 is set to 2 or more.
The fourth priority property p4 (encoded by the variable 𝑖4 ) indicates the type of uncertainty in the
description of the elements of the domain of the cyclic function. As in the case of property p1 ,
property p4 can take values from the set {d, s, f, i}, and the number 𝑖4 can take values from one to
four. One (𝑖4 = 1, p4 = 𝑑) will encode a deterministic approach to the description of the elements of
the domain of cyclic signals; two (𝑖4 = 2, p4 = 𝑠) will encode a stochastic approach to the description
of the elements of the domain of cyclic signals, three (𝑖4 = 3, p4 = 𝑓) will encode a fuzzy approach
to the description of the elements of the cyclic signal definition area, and four (𝑖4 = 4, p4 = 𝑖) will
encode an interval approach to the description of the elements of the cyclic signal definition area.
The fifth priority property p5 (encoded by the variable 𝑖5 ) indicates the dimension of the domain of
the cyclic function, namely, whether the cyclic function is a function of one or many (several)
arguments (cyclic field).
The sixth priority property p6 (encoded by the variable 𝑖6 ) indicates the type of power (continuous
or discrete) of the domain of the cyclic function, namely, whether the cyclic function is a function of
valid arguments or a function of discrete arguments.
The seventh priority property p7 (encoded by the variable 𝑖7 ) indicates the type of rhythm
(constant or variable) of the cyclic function, namely, whether the cyclic function is a cyclic function
with a variable rhythm or with a constant rhythm (periodic function).
The eighth priority property p8 (encoded by the variable 𝑖8 ) specifies the type of variable rhythm
of the cyclic function, namely, whether the cyclic function is with a piecewise-linear type of rhythm
or with a piece-square type of rhythm or a piece-cubic type of rhythm or with a periodic type of
rhythm, etc. That is, this property specifies the type of rhythm variability of the cyclic function.
In general, the number of properties n (i.e., the dimension of the vector ̅ 𝑷 = (p1 , p2 , … , pn )) that
characterizes the class of cyclic functional relations may be different for different classes. However, it
is necessary to follow the rule according to which all the properties of a super-class of cyclic
functional relations, which are components of its property vector, are part of the property vectors of
all its subclasses.
Given that four possible approaches to uncertainty, namely, deterministic, stochastic, fuzzy and
interval, are used to describe the values and elements of the domain (and, accordingly, rhythm
functions) of cyclic functions, namely, a total of 16 approaches to mathematical description of the
uncertainty of cyclic signals. A summary of these 16 approaches is given in Table 1.
Table 1
Approaches to the mathematical description of the uncertainty of cyclic signals
№ The name of the The name of the The name of a
approach approach to the approach to the generalized approach
description of the description of the to describing the
uncertainty of the uncertainty of the uncertainty of cyclic
values of cyclic signals elements of the domain signals
and the rhythm of
cyclic signals
1 Deterministic Deterministic Deterministic
(p1 = 𝑑, 𝑖1 = 1) (p4 = 𝑑, 𝑖4 = 1)
2 Deterministic Stochastic Deterministic-
(p1 = 𝑑, 𝑖1 = 1) (p4 = 𝑠, 𝑖4 = 2) stochastic
3 Deterministic Vague Deterministic-
(p1 = 𝑑, 𝑖1 = 1) (p4 = 𝑓, 𝑖4 = 3) fuzzy
4 Deterministic Interval Deterministic-interval
(p1 = 𝑑, 𝑖1 = 1) (p4 = 𝑖, 𝑖4 = 4)
5 Stochastic Deterministic Stochastic-
(p1 = 𝑠, 𝑖1 = 2) (p4 = 𝑑, 𝑖4 = 1) deterministic
6 Stochastic Stochastic Stochastic
(p1 = 𝑠, 𝑖1 = 2) (p4 = 𝑠, 𝑖4 = 2)
7 Stochastic Vague Vague
(p1 = 𝑠, 𝑖1 = 2) (p4 = 𝑓, 𝑖4 = 3)
8 Stochastic Interval Stochastic-interval
(p1 = 𝑠, 𝑖1 = 2) (p4 = 𝑖, 𝑖4 = 4)
9 Vague Deterministic Vague-
(p1 = 𝑓, 𝑖1 = 3) (p4 = 𝑑, 𝑖4 = 1) Deterministic
10 Vague Stochastic Vague-Stochastic
(p1 = 𝑓, 𝑖1 = 3) (p4 = 𝑠, 𝑖4 = 2)
11 Vague Vague Vague
(p1 = 𝑓, 𝑖1 = 3) (p4 = 𝑓, 𝑖4 = 3)
12 Vague Interval Vague-interval
(p1 = 𝑓, 𝑖1 = 3) (p4 = 𝑖, 𝑖4 = 4)
13 Interval Deterministic Interval-
(p1 = 𝑖, 𝑖1 = 4) (p4 = 𝑑, 𝑖4 = 1) Deterministic
14 Interval Stochastic Interval-stochastic
(p1 = 𝑖, 𝑖1 = 4) (p4 = 𝑠, 𝑖4 = 2)
15 Interval Vague Interval-Vague
(p1 = 𝑖, 𝑖1 = 4) (p4 = 𝑓, 𝑖4 = 3)
16 Interval Interval Interval
(p1 = 𝑖, 𝑖1 = 4) (p4 = 𝑖, 𝑖4 = 4)
Figures 5-10 show examples of the vector of properties for a few classes of cyclic functional
relations, namely, as attributes of the corresponding classes of ontology of mathematical models of
cyclic signals built in Protégé.
Figure 5: Graphical representation in the Protégé environment of the property vector ̅ P=
(p1 , p2 , … , pn ) for the class "Cyclic values of a deterministic real-valued function of a real argument
with a constant rhythm"
Figure 6: Graphical representation in the Protégé medium of the property vector P ̅=
(p1 , p2 , … , pn ) for the class "Cyclic values of a deterministic real-valued function of a real argument
with a variable piecewise-cubic rhythm"
Figure 7: Graphical representation in Protégé medium of the vector of properties ̅ P=
(p1 , p2 , … , pn ) for the class "Cyclic modulo deterministic real function of a valid argument with
variable periodic rhythm"
Figure 8: Graphical representation in Protégé medium of the vector of properties P ̅=
(p1 , p2 , … , pn ) for the class "Cyclic in the set of autocorrelations and intercorrelation functions
vector random function of a real argument with a constant rhythm"
Figure 9: Graphical representation in Protégé medium of the vector of properties P ̅=
(p1 , p2 , … , pn ) for the class "Cyclic trapezoidal membership function fuzzy function of a discrete
argument with variable rhythm"
Figure 10: Graphical representation in Protégé medium of the vector of properties ̅ P=
(p1 , p2 , … , pn ) for the class "Cyclic with respect to the system of intervals interval function of a
discrete argument with variable piecewise linear rhythm"
3.4. Method of forming a vector of levels of development of mathematical
and software for processing and simulation of cyclic signals within the
corresponding class of cyclic functional relations
Since each class of cyclic functional relations is associated with the relevant information
technologies (methods and means) of processing and computer simulation (generation) of cyclic
̅=
signals, mathematical model of which is this class of cyclic functions, in addition to the vector 𝑷
(p1 , p2 , … , pn ), which contains the properties of the corresponding class of cyclic functions, it is
correct to enter the vector ̅̅̅̅̅̅
𝑳𝑖𝑚𝑝 = (l1 , l2 , … , l8 ), components, which characterize the state (level) of
elaboration (implementation, development) of the corresponding information technologies of
processing and computer simulation (generation) of cyclic signals within the framework of this
mathematical model. That is, the vector ̅̅̅̅̅̅ 𝑳𝑖𝑚𝑝 = (l1 , l2 , … , l8 ) is a vector of levels of development of
mathematical and software for processing and imitation of cyclic signals within the corresponding
class of cyclic functional relations. The information contained in the components of the vector ̅̅̅̅̅̅ 𝑳𝑖𝑚𝑝 =
(l1 , l2 , … , l8 ) is important in solving problems of reasonable choice of mathematical model and
processing technologies (transformation, analysis (evaluation of cyclicity attributes and rhythm
attributes), clustering, classification, forecasting, regression), technologies of simulation (generation)
of cyclic signals.
Let us briefly consider each component of the vector ̅̅̅̅̅̅ 𝑳𝑖𝑚𝑝 = (l1 , l2 , … , l8 ).
The first component l1 of the vector ̅̅̅̅̅̅
𝑳𝑖𝑚𝑝 = (l1 , l2 , … , l8 ) takes its values from a subset of integers
from 0 to 3 and characterizes the level of development of the class of cyclic functional relations as a
mathematical model of cyclic signals, namely:
l1 = 0, when there are no definitions, inference properties (lemmas and theorems) that relate to
this class of cyclic functional relations, and there are no examples of successful application of this
class of cyclic functions as mathematical models of cyclic signals (processes, phenomena);
l1 = 1, When there is a clear mathematically correct definition of the class of cyclic functional
relations as a subclass of abstract cyclic functional relations, but there are no derivational properties
(lemmas and theorems), relating to this class of cyclic functional relations and there are no examples
of successful application of this class of cyclic functions as mathematical models of cyclic signals
(processes, phenomena);
l1 = 2, When there is a clear mathematically correct definition of the class of cyclic functional
relations as a subclass of abstract cyclic functional relations, there are derivational properties (lemmas
and theorems), relating to this class of cyclic functional relations, but there are no examples of
successful application of this class of cyclic functions as mathematical models of cyclic signals
(processes, phenomena);
l1 = 3, When there is a clear mathematically correct definition of the class of cyclic functional
relations as a subclass of abstract cyclic functional relations, there are inference properties (lemmas
and theorems) related to this class of cyclic functional relations and there are examples of successful
application of this class of cyclic functions as mathematical models of cyclic signals phenomena).
Components l2 , … , l7 characterize the level of development of technologies (methods and
tools) for processing cyclic signals to solve typical problems within their mathematical model in the
form of a corresponding class of cyclic functional relations. These components derive their values
from a subset of integers from 0 to 2, namely:
l𝑗 = 0 (𝑗 ∈ [2,7̅̅̅̅]), When there is no method for solving the corresponding typical problem within
their mathematical model in the form of this class of cyclic functional relations;
l𝑗 = 1 (𝑗 ∈ [2,7̅̅̅̅]), When there is a method for solving the corresponding typical problem within
their mathematical model in the form of this class of cyclic functional relations, but there is no
software (hardware, hardware, hardware) that implements this method;
l𝑗 = 2 (𝑗 ∈ [2,7̅̅̅̅]), When there is a method for solving the corresponding typical problem within
their mathematical model in the form of a given class of cyclic functional relations, as well as
available software (hardware, software, and firmware) that implements this method.
The component l2 of the vector ̅̅̅̅̅̅ 𝑳𝑖𝑚𝑝 = (l1 , l2 , … , l8 ) characterizes the level of development of
technologies (methods and means) of cyclic signal processing to solve the problem of estimating
cyclicity attributes (morpho analysis problem) of cyclic signals within their mathematical model in
the form of the corresponding class of cyclic functional relations.
The component l3 of the vector ̅̅̅̅̅̅ 𝑳𝑖𝑚𝑝 = (l1 , l2 , … , l8 )characterizes the level of development of
technologies (methods and means) of cyclic signal processing to solve the problem of estimating
rhythm attributes (rhythm analysis problem) of cyclic signals within their mathematical model in the
form of a corresponding class of cyclic functional relations.
The component l4 of the vector (𝑳 ̅̅̅̅̅̅
𝑖𝑚𝑝 = (l1 , l2 , … , l8 ) characterizes the level of development of
technologies (methods and means) of cyclic signal processing to solve the problem of classification
(recognition) of cyclic signals within their mathematical model in the form of a corresponding class of
cyclic functional relations.
The component l5 of the vector ̅̅̅̅̅̅ 𝑳𝑖𝑚𝑝 = (l1 , l2 , … , l8 ) characterizes the level of development of
technologies (methods and means) of cyclic signal processing to solve the problem of clustering
(construction of diagnostic space) of cyclic signals within their mathematical model in the form of the
corresponding class of cyclic functional relations.
The component l6 of the vector ̅̅̅̅̅̅ 𝑳𝑖𝑚𝑝 = (l1 , l2 , … , l8 ) characterizes the level of development of
technologies (methods and tools) for processing cyclic signals to solve the problem of predicting
cyclic signals within their mathematical model in the form of a corresponding class of cyclic
functional relations.
The component l7 of the vector ̅̅̅̅̅̅ 𝑳𝑖𝑚𝑝 = (l1 , l2 , … , l8 ) characterizes the level of development of
technologies (methods and means) of cyclic signal processing to solve the problem of regression of
cyclic signals within their mathematical model in the form of a corresponding class of cyclic
functional relations.
The component l8 of the vector ̅̅̅̅̅̅ 𝑳𝑖𝑚𝑝 = (l1 , l2 , … , l8 ) characterizes the level of development of
technologies (methods and tools) for processing cyclic signals to solve the problem of computer
simulation (generation) of cyclic signals within their mathematical model in the form the
corresponding class of cyclic functional relations.
Thus, the vector 𝑷 ̅ = (p1 , p2 , … , pn ) (or the vector (𝑰̅ = (𝑖1 , 𝑖2 , … , 𝑖n ))) together with the vector
̅̅̅̅̅̅
𝑳𝑖𝑚𝑝 = (l1 , l2 , … , l8 ) completely characterize both each specific class of cyclic functional relations
and the level of development of technologies (methods and tools) for processing and simulation of
cyclic signals within their mathematical model in the form of a corresponding class of cyclic
functional relations.
3.5. Coding of classes of cyclic functional relations and their taxonomy
Based on the vectors ̅ 𝑷 = (p1 , p2 , … , pn ), 𝑰̅ = (𝑖1 , 𝑖2 , … , 𝑖n ) and 𝑳
̅̅̅̅̅̅
𝑖𝑚𝑝 = (l1 , l2 , … , l8 ), we form a
coding system for classes of cyclic functional relations and information technologies (methods and
means) of processing (analysis (estimation of cyclicity attributes and rhythm attributes)), clustering,
classification, forecasting, regression), technologies of simulation (generation) of cyclic signals within
this mathematical model. Such a coding system enables a unified numerical representation of each
class of cyclic functional relations and makes it possible to organize all these classes into their
taxonomy 𝑻𝒂𝒙_𝒐𝒇_𝑪𝒇.
Thus, for each class of cyclic functional relations from the vector ̅𝑰 = (𝑖1 , 𝑖2 , … , 𝑖n ) it is possible
to form a unique code vector that has the form: 𝑖1 . 𝑖2 . , … , . 𝑖n , i.e., which is a sequence of components
of the vector 𝑰̅ = (𝑖1 , 𝑖2 , … , 𝑖n ), separated by a dot (".").
To organize (structure) the set of definitions 𝑫𝑒𝑓𝑐𝑓 classes of cyclic functional relations), as well as
to develop a general classification of cyclic functional relations, we construct a taxonomy 𝑻𝒂𝒙_𝒐𝒇_𝑪𝒇
definitions of classes of cyclic functional relations, elements (nodes) of which are predicate
definitions from 𝑫𝑒𝑓𝑐𝑓 . To do this, we apply the method of induction of taxonomy 𝑻𝒂𝒙_𝒐𝒇_𝑪𝒇 from
taxonomies 𝑻𝜳 ,𝑻𝑨 ,𝑻𝑇(𝑡,𝑛) ,𝑻𝑾 .
It should be noted that the type of ordering of different classes of cyclic functional relations within
their taxonomy is determined by the types of ordering of the respective sets within the
taxonomies 𝑻𝜳 ,𝑻𝑨 ,𝑻𝑇(𝑡,𝑛) ,𝑻𝑾 , and the ordering of mutual priorities between these taxonomies.
Figures 11 - 16 show examples of taxonomy fragments of cyclic deterministic functional relations
constructed in the Protégé environment.
Figure 11: Graphical representation in the Protégé environment for the class "Cyclic value-
determined real-value function of one argument"
Figure 12: Graphical representation in the Protégé environment for the class "Cyclic value-
determined real-valued function of a discrete argument"
Figure 13: Graphical representation in the Protégé environment for the class "Cyclic value-
determined real-valued function of many discrete arguments"
Figure 14: Graphical representation in the Protégé environment for the class "Cyclic modulo
deterministic real function of a valid argument"
Figure 15: Graphical representation in the Protégé environment for the class "Cyclic modulo
deterministic complex function of a valid argument"
Figure 16: Graphical representation in the Protégé environment for the class "Cyclic value-
determined matrix real-component function of many discrete arguments"
4. Conclusions
A formal model of computer ontology of cyclic signal modeling is developed in the work. This
formal model is presented as a relational system, which includes a finite set of names of classes of
cyclic functional relations; the function of interpretation, which defines the corresponding classes of
cyclic functional relations as components of the glossary; the relationship of genus-species
subordination, which determines the taxonomy (hierarchy) between different classes of cyclical
functional relations; vector of unary relations that define the properties (features, attributes) of the
corresponding class of cyclic functional relations, and eight-component vector, the elements of which
characterize the state (level) of elaboration (implementation, development) of relevant information
technologies for processing and computer simulation (generation) of cyclic signals within the
corresponding class of cyclic functional relations.
Based on the developed formal model of computer ontology of mathematical models of cyclic
signals in the Protégé environment, a prototype of this ontology was built, which lays a solid
technological foundation for creating an onto-oriented knowledge base of expert decision support
system in correct and reasonable choice solving specific problems of research of cyclic signals within
the framework of deterministic, stochastic, fuzzy and interval model approaches (paradigms).
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