PROPERTIES OF DISCRETE-TIME CONDITIONAL LINEAR CYCLOSTATIONARY RANDOM PROCESSES IN THE PROBLEMS OF ENERGY INFORMATICS
Modern challenges in the energy industry require comprehensive research in the fieldof energy informatics, which combines computer science, control systems, and energy managementsystems within a single methodology. An important area of energy informatics is the study of problemsof systems and proces...
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General Energy Institute of the National Academy of Sciences of Ukraine
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System Research in Energy| _version_ | 1871103053598294016 |
|---|---|
| author | Mykhailo Fryz Leonid Scherbak |
| author_facet | Mykhailo Fryz Leonid Scherbak |
| author_institution_txt_mv | [
{
"author": "Mykhailo Fryz",
"institution": null
},
{
"author": "Leonid Scherbak",
"institution": null
}
] |
| author_sort | Mykhailo Fryz |
| baseUrl_str | https://systemre.org/index.php/journal/oai |
| collection | OJS |
| datestamp_date | 2026-07-18T12:57:30Z |
| description | Modern challenges in the energy industry require comprehensive research in the fieldof energy informatics, which combines computer science, control systems, and energy managementsystems within a single methodology. An important area of energy informatics is the study of problemsof systems and processes modeling in energy, including energy loads and consumption. Linear andconditional linear random processes (CLRP) are mathematical models of signals represented as the sumof a large number of random impulses occurring at random times. The energy consumption, vibrationsignals of energy objects, etc. can be modeled using this approach. A variant of the CLRP modelwith discrete time, taking into account the cyclic properties of energy consumption, has been investigatedin the paper. The goal is to justify the conditions for the discrete-time CLRP to be a periodicallycorrelated random process, as well as a cyclostationary process. It has been shown thatthe corresponding conditions depend on the periodicity of the probability distributions of the kernel andthe generating white noise of the CLRP representation. To achieve the goal, the propertiesof mathematical expectation and covariance function of CLRP, as well as the method of characteristicfunctions, have been used. The paper proves that the discrete-time CLRP is a periodically correlatedrandom sequence if the generating white noise has periodic mathematical expectation and variance, andthe kernel is a periodically correlated random field. Based on the analysis of the multivariatecharacteristic function, it has been proven that the discrete-time CLRP is cyclostationary ifthe generating white noise is a cyclostationary process and the kernel is a cyclostationary random field.The properties of discrete-time conditional linear cyclostationary random processes are importantfor mathematical modeling, simulation, statistical analysis, and forecasting of energy consumption. |
| doi_str_mv | 10.15407/srenergy2023.01.072 |
| first_indexed | 2026-03-24T02:00:44Z |
| format | Article |
| fulltext |
System Research in Energy. 2023. 1(72) 72
ISSN 2786-7102 (Online), ISSN 2786-7633 (Print)
https://doi.org/10.15407/srenergy2023.01.072
UDC 519.87:620.9
Mykhailo Fryz1, 2*, PhD (Engin.), Assoc. Prof., https://orcid.org/0000-0002-8720-6479
Leonid Scherbak2, Dr. Sci. (Engin.), Professor, https://orcid.org/0000-0002-1536-4806
1Ternopil Ivan Puluj National Technical University, 56, Ruska St., Ternopil, 46001,
Ukraine;
2General Energy Institute of NAS of Ukraine, 172, Antonovycha St., Kyiv, 03150,
Ukraine;
* Corresponding author: mykh.fryz@gmail.com
________________________________________________________________________________
PROPERTIES OF DISCRETE-TIME CONDITIONAL LINEAR
CYCLOSTATIONARY RANDOM PROCESSES IN THE PROBLEMS
OF ENERGY INFORMATICS
Abstract. Modern challenges in the energy industry require comprehensive research in the field
of energy informatics, which combines computer science, control systems, and energy management
systems within a single methodology. An important area of energy informatics is the study of problems
of systems and processes modeling in energy, including energy loads and consumption. Linear and
conditional linear random processes (CLRP) are mathematical models of signals represented as the sum
of a large number of random impulses occurring at random times. The energy consumption, vibration
signals of energy objects, etc. can be modeled using this approach. A variant of the CLRP model
with discrete time, taking into account the cyclic properties of energy consumption, has been investigated
in the paper. The goal is to justify the conditions for the discrete-time CLRP to be a periodically
correlated random process, as well as a cyclostationary process. It has been shown that
the corresponding conditions depend on the periodicity of the probability distributions of the kernel and
the generating white noise of the CLRP representation. To achieve the goal, the properties
of mathematical expectation and covariance function of CLRP, as well as the method of characteristic
functions, have been used. The paper proves that the discrete-time CLRP is a periodically correlated
random sequence if the generating white noise has periodic mathematical expectation and variance, and
the kernel is a periodically correlated random field. Based on the analysis of the multivariate
characteristic function, it has been proven that the discrete-time CLRP is cyclostationary if
the generating white noise is a cyclostationary process and the kernel is a cyclostationary random field.
The properties of discrete-time conditional linear cyclostationary random processes are important
for mathematical modeling, simulation, statistical analysis, and forecasting of energy consumption.
Keywords: mathematical model, energy informatics, conditional linear random process, cyclostationary
process, white noise, characteristic function.
1. Introduction
Problems and challenges in the modern energy industry are related to the volatility and partial
controllability of renewable energy sources, the uncertainty of the energy consumers' behavior (which no
longer necessarily corresponds to standard load profiles), decentralization, modification of the dynamic
characteristics of sustainable energy systems, etc. [1]. There is a strong need for essential contributions from
=++=the computer science community to overcome the above challenges. This can be done within
the energy informatics framework, a highly interdisciplinary and dynamic field of research and development.
Energy informatics combines computer science, control systems, and energy management systems in
a single methodology [1–3]. Important directions of energy informatics are associated with the collection,
analysis, deployment, and exploitation of energy status data, modeling, simulation, and prediction of
the behavior of energy systems and processes [1], including energy loads and consumption mathematical
modeling and computer simulation. Energy resources consumption (e.g., electric, gas, water consumption)
analysis and simulation are also important tools for the problems of energy consumption behavior-based user
segmentation, electricity consumption pattern (load profiles) analysis, energy consumption forecasting,
System Research in Energy. 2023. 1(72) 73
development of information measurement and information control systems in the electric power
industry [4, 5].
The mathematical model, being the theoretical foundation of the structural, algorithmic, and technical
implementation of the developed systems and technologies, should be adequate to the measuring process,
represent the physical mechanism of its generation, and also be suitable for performing its theoretical
analysis, solving problems of monitoring and diagnostics of energy facilities based on the results of
the experiments. Linear and conditional linear random processes (CLRP) are the mathematical models
satisfying the above demands, representing the investigated signals and processes as the sum of many
random stochastically dependent impulses occurring at random Poisson times. The energy consumption,
vibration signals of energy objects, etc. can be modeled using this approach.
The concept of a "conditionally linear random process" has been developed by Percy A. Pierre [6]
in the context of his research on the problem of mathematical modeling of radar clutter. Continuous-time
CLRP [6–8] is defined in the form of a stochastic integral of a random kernel driven by the process with
independent increments. Different cases of the general approach have been investigated in [9, 10].
In the papers [7, 8] and others, the characteristic function method has been applied to continuous-time CLRP
to study its probability distribution properties, including the conditions of its cyclostationarity (that is,
the periodicity of finite-dimensional distribution functions, characteristic functions, or moment functions
with respect to their time arguments), which is obviously important properties in the context of energy
informatics because of cyclic nature of energy loads and consumption processes. This paper deals with
the same properties but for the discrete-time variant of the model.
The discrete-time CLRP, represented as a stochastic sum with a stationary generative white noise has
been introduced and analyzed in [6]. The general case of discrete-time CLRP has been defined in [11], and
the properties of its mathematical expectation and covariance function have been analyzed. There
are theoretical papers where the central limit problem in relation to randomly weighted sums of random
variables [12], linear processes with random coefficients [13], etc. have been investigated. In the literature,
however, there is no analysis of the properties of multidimensional probability distributions or moment
functions for the general case of a discrete-time conditional linear random process, which can be used in
the applied problems of mathematical modeling of cyclostationary (or periodically correlated) [14] signals
and processes.
The concept of periodically correlated random sequences has been introduced in [15] and
their spectral properties have been investigated. The statistical analysis methods of such processes have been
considered in [14, 16–18]. The relationships between periodically correlated, cyclostationary, and linear
random sequences (in the form of autoregressive moving average models with periodic coefficients) have
been studied by many authors, including [14, 16, 19]. The corresponding applications for modeling
the signals in energy and diagnostics of energy equipment have been represented in [14, 16, 18].
The main goal of the paper is to obtain the conditions for discrete-time CLRP to be periodically
correlated using the properties of its moment functions, and to characterize the conditions for discrete-time
CLRP to be cyclostationary using the characteristic function method.
2. Discrete-time conditional linear periodically correlated random processes
We start our analysis with the definition of continuous-time CLRP [7, 8]. A real-valued continuous-
time conditional linear random process ( , ), , ( , )t tξ ω ω∈Ω ∈ −∞ ∞ (where { }, ,PΩ F is some probability
space) is defined in the following form:
( , ) ( , , ) ( , ),t t d
∞
−∞
ξ ω = ϕ ω t η ω t∫ (1)
where ( , , ), , ( , )t tϕ ω t t ∈ −∞ ∞ is a real-valued stochastic kernel of CLRP; ( , ), ( , )η ω ττ ∈ −∞ ∞ is
a stochastically continuous Hilbert process with independent increments, satisfying the following conditions:
( , ) ( )E aη ω τ = τ < ∞ and [ ]Var ( , ) ( )bη ω τ = τ < ∞ , ∀τ ; random functions ( , , )tϕ ω t and ( , )η ω τ
are stochastically independent.
System Research in Energy. 2023. 1(72) 74
A real-valued discrete-time conditional linear random process ( )tξ ω , Zt∈ , ω∈Ω is defined as
a random sequence in the following form [11]:
,( ) ( ) ( )t t
∞
t t
t=−∞
ξ ω = ϕ ω ζ ω∑ , (2)
where , ( )ttϕ ω , , Ztt ∈ is a real-valued random function (kernel), which can be considered a random matrix
as well as a two-dimensional random field on 2Z ;
( )τζ ω , Zτ∈ is a sequence of infinitely divisible independent random variables (infinitely divisible discrete-
time white noise);
random functions , ( )ttϕ ω and ( )τζ ω are stochastically independent.
Let us denote the mathematical expectation and variance of the above white noise ( )τζ ω as follows:
( )E aττ ζ ω = < ∞ , [ ] 2Var ( )ττ ζ ω = σ < ∞ , ∀τ .
The sum (2) is assumed to be exist in the mean-square convergence sense [11]. The relationship
between models (1) and (2) has been also analyzed in [11].
The mathematical expectation ( )E tξ ω and covariance function
1 2, 1 2, , Zt tR t t ∈ of a discrete-time
conditional linear random process (2) is represented as:
,( )E t ta
∞
t t
t=−∞
ξ ω = φ∑ , (3)
( )1 2 1 2 1 2
2
, , ; , , ,( ) ( ) ,Et t s t t s t t
s
R R a a
∞ ∞ ∞
ϕ
t t t t t
t=−∞ =−∞ t=−∞
= + ϕ ω ϕ ω s∑ ∑ ∑ (4)
where , , ( )=Et tt tφ ϕ ω is the mathematical expectation of the kernel of discrete-time CLRP;
1 21 2 , ,, ; , ( ) ( )=E t s ts t tRϕ
tt
ϕ ω ϕ ω
is the covariance function of the kernel of a discrete-time conditional linear
random process ( , , ,( ) ( ) ( )Et t tt t tϕ ω = ϕ ω − ϕ ω
is the centered kernel).
Let there exist the least integer number (period) 1Т > such that white noise ( )τζ ω has periodic
mathematical expectation and variance, that is,
( )E Ta aτττ +ζ ω = = and [ ] 2 2Var ( ) Tτττ +ζ ω = σ = σ , (5)
and mathematical expectation and covariance function of the kernel has the following properties:
1 2 1 2, , , ; , , ; ,, ,= =t T t T s t t T s T t T t TR Rϕ ϕ
t t+ + t t+ + + +φ φ (6)
then discrete-time CLRP (2) is periodically correlated in a random sequence.
To prove this, we should analyze the properties of mathematical expectation and covariance function
of discrete-time conditional linear random process under conditions (5) and (6). Thus, we have the following
property of mathematical expectation:
, , ,( ) ( )E Et t T t T T s t T s t T
s
a a a
∞ ∞ ∞
t t t+ + t+ + +
t=−∞ t=−∞ =−∞
ξ ω = φ = φ = φ = ξ ω∑ ∑ ∑ .
Taking into account ( )1 2 1 2 1 2, , , ; , , ,( ) ( )E t t t t t tRϕ
t t t t t tϕ ω ϕ ω = + φ φ , and denoting 1 1,T s T sτ + = τ + = , we
obtain the following property of the covariance function:
( )
( )
1 2 1 2 1 2 1 2
1 1 1 2 1 1 1 1 1 2 1 1 1 2 1 1 2
1 1 1
2
, , ; , , ; , , ,
2
, ; , , ; , , , , .
t t T s T t T t T T s T T T t T t T T t T T t T T
s
s t T t T s t T t T t T t T t T t T
s
R R a a R
R a a R R
∞ ∞ ∞
ϕ ϕ
t+ + + + t+ + t+ t+ + + t+ + t+ + t+
t=−∞ =−∞ t=−∞
∞ ∞ ∞
ϕ ϕ
t + + t t t + + t + t + t + +
t =−∞ =−∞ t =−∞
= + + φ φ s =
= + + φ φ s =
∑ ∑ ∑
∑ ∑ ∑
The mathematical expectation and covariance function of discrete-time CLRP is periodical functions
with respect to their time arguments. Thus, the investigated process is periodically correlated [14, 15] if
conditions (5) and (6) hold.
System Research in Energy. 2023. 1(72) 75
3. Discrete-time conditional linear cyclostationary random processes
Let ϕ ⊂F F be σ -subalgebra, generated by a random matrix , ( )ttϕ ω , satisfying with probability 1
the following conditions (which are important for the convergence of (2) by distribution):
, ( )t a
∞
t t
t=−∞
ϕ ω < ∞∑ and
2 2
, ( )t
∞
t t
t=−∞
ϕ ω σ < ∞∑ , t∀ .
The m -dimensional characteristic function of discrete-time CLRP (2) is represented as:
1 2 1 2 1 2 1 2
1
( , ,..., ; , ,..., ) exp ( ) ( , , ,..., ; , ,..., ),E E
k
m
m m k t m m
k
f u u u t t t i u f u u u t t tϕ
xx
=
= x ω = ω
∑ F
where 1 2 1 2
1
( , , ,..., ; , ,..., ) exp ( )E
k
m
m m k t
k
f u u u t t t i uϕ
x ϕ
=
ω = x ω
∑F F is conditional with respect to σ -
subalgebra ϕF characteristic function of discrete-time CLRP (2).
Taking into account that elements , ( ) ( )tt tϕ ω ζ ω in the sum (2) are conditionally ϕF -independent
infinitely divisible random variables (with respect to σ -subalgebra ϕF ), the random function
1 2 1 2( , , ,..., ; , ,..., )m mf u u u t t tϕ
ξ ωF is represented as:
,
1
1 2 1 2
,
1
( )
, 2
1
( , , ,..., ; , ,..., )
exp ( )
( ; )1 ( ) ,
, , 1, ,Z R
k
m
k tk
k
k
m m
m
k t
k
mix u
x
k t
k
k k
f u u u t t t
i u a
d K xe ix u
x
t u k m
ϕ
t
=
x
∞
t t
= t=−∞
∞∞ ϕ ω
t
t=−∞ =−∞
ω =
= ϕ ω +
∑ t + − − ϕ ω
∈ ∈ =
∑ ∑
∑ ∑∫
F
(7)
where ( ; ), ( , ), ZK x xτ ∈ −∞ ∞ τ∈ is a Poisson jump spectrum in Kolmogorov’s form of the infinitely
divisible white noise ( )τζ ω .
Probability distribution of discrete-time conditional linear random process belongs to the class
of mixtures of infinitely divisible distributions.
Let there exist the least integer number (period) 1Т > such that the following holds:
- random matrices , ( )ttϕ ω , , Ztt ∈ and , ( )T t Tt+ +ϕ ω are stochastically equivalent in the wide sense, that is,
their finite-dimensional cumulative distribution functions are equal:
{ } { }, ,
1 1 1 1
: ( ) : ( ) ,P P R
j j j j
n m n m
t ij T t T ij ij
i j i j
x x xt t + +
= = = =
ω j ω < = ω j ω < ∈
11 11 , (8)
- infinitely divisible white noise ( ), Zτζ ω τ∈ is stochastically periodic in the sense of
Ta aττ += , ( ; ) ( ; )x xd K x d K x Tτ = τ + . (9)
Then the characteristic function 1 2 1 2( , ,..., ; , ,..., )m mf u u u t t tξ is periodic by its time arguments, that is,
1 2 1 2 1 2 1 2( , ,..., ; , ,..., ) ( , ,..., ; , ,..., )m m m mf u u u t t t f u u u t T t T t Tξ ξ= + + + . (10)
The random process satisfying the property (10) is called cyclostationary of the order m [14]. Thus,
under the conditions (8) and (9) the sequence (2) is discrete-time conditional linear cyclostationary random
processes. Obviously, the white noise ( )τζ ω , satisfying (9) is also cyclostationary, that is, its one-
dimensional characteristic function is T -periodic by time argument. It should be also noted that if 1T =
then (2) is strict-sense stationary conditional linear random process.
The above statement can be proven analyzing the properties of conditional characteristic function (7)
of discrete-time conditional linear random process under the conditions (8) and (9).
System Research in Energy. 2023. 1(72) 76
First of all, it can be noted that
, ,
1 1
,
1
Law ( ) Law ( )
Law ( ) ,
k k
k
m m
k t k T t T T
k k
m
k s t T s
k
u a u a
u a
∞ ∞
t t t+ + t+
= t=−∞ = t=−∞
∞
+
= t=−∞
ϕ w = ϕ w =
= ϕ w
∑ ∑ ∑ ∑
∑ ∑
where s T= τ + , and also we use the notation Law( ( )) Law( ( ))ξ w = η w if given random variables ( )ξ ω and
( )η ω have the same probability distributions.
Further, the second part of the expression (7) has the same property, that is,
,
1
,
1
,
1
( )
, 2
1
( )
, 2
1
( )
( ; )Law 1 ( )
( ; )Law 1 ( )
Law 1
m
k tk
k
k
m
k T t Tk
k
k
m
k s t Tk
k
mix u
x
k t
k
mix u
x
k T t T
k
ix u
d K xe ix u
x
d K x Te ix u
x
e ix
t
=
t+ +
=
+
=
∞∞ ϕ w
t
t=−∞ =−∞
∞∞ ϕ w
t+ +
t=−∞ =−∞
ϕ w
∑ t − − ϕ w =
∑ t + = − − ϕ w =
∑
= − −
∑ ∑∫
∑ ∑∫
, 2
1
( ; )( ) .
k
m
x
k s t T
k
d K x su
x
∞∞
+
t=−∞ =−∞
ϕ w
∑ ∑∫
From the above analysis it follows that probability distribution of the random ϕF -conditional
characteristic function 1 2 1 2( , , ,..., ; , ,..., )m mf u u u t t tϕ
ξ ωF is periodic by its time arguments, that is,
1 2 1 2 1 2 1 2Law( ( , , ,..., ; , ,..., )) Law( ( , , ,..., ; , ,..., ))m m m mf u u u t t t f u u u t T t T t Tϕ ϕ
ξ ξw = w + + +F F
Taking into account that unconditional characteristic function of discrete-time CLRP is equal to
1 2 1 2 1 2 1 2( , ,..., ; , ,..., ) ( , , ,..., ; , ,..., )Em m m mf u u u t t t f u u u t t tϕ
ξ ξ= ωF , we can state that (10) holds, and investigated
process is cyclostationary of the order m.
4. Discussion
As it was already mentioned above the key element of the methodology of application of continuous-
time linear and conditional linear random processes for the mathematical modelling of electricity loads is
a representation of investigated process as a sum of a large amount of random impulses (loads related
to individual consumers) occurring at random Poisson time moments. In context of linear random processes,
the impulses (with random duration and amplitude) are independent. The authors of the paper [20]
investigated electricity consumption data at the level of individual households, measured by modern
information systems based on smart meters. According to the results of experiments it has been shown [20]
that the processes of electricity consumption of individual households are stochastically dependent. That is
why, we state that mathematical modelling of electricity loads process ( , )tξ ω in the form of conditional
linear random process (considering the stochastic dependency between the individual impulses-loads)
is more adequate to investigated object.
The resulting representation follows from (1) (where process with independent increments
is nonhomogeneous Poisson counting process) and has the following form:
( , ) ( , ( ), )k
k
t t
∞
=−∞
ξ ω = ϕ ω t ω∑ ,
where 1 1... ( ) ( ) ( ) ...k k k− +< τ ω < τ ω < τ ω < are the times of a Poisson process, which are equal to the times
of random impulses ( , ( ), )k tϕ ω t ω occurrence ( ( , ( ), ) 0k tϕ ω t ω = if ( )kt < t ω ), and random functions
1..., ( , , )k t−ϕ ω t , ( , , )k tϕ ω t , 1( , , ), ...k t+ϕ ω t are stochastically dependent (at fixed nonrandom times
1 1... ...k k k− +< τ < τ < τ < ).
System Research in Energy. 2023. 1(72) 77
Analyzing the cyclic behaviour of energy consumers of residential areas or enterprises, it can be
shown that the investigated electricity loads process ( , )tξ ω will be cyclostationary [7, 8, 14] with a period
of 0 24T = hours, that is, its m -dimensional ( 1m ≥ ) characteristic function is periodic by its time arguments.
Now, let us consider the discrete-time random process
0
( 1)
( ) ( , ) , , ,Z N
th
t
t h
Ts ds t h T
T−
ξ ω = ξ ω ∈ = ∈∫ ,
which for each Zt∈ equals to electricity consumption during time interval [ ]( 1) ,t h th− . If 1h = hour, then
( ), Zt tξ ω ∈ is hourly electricity consumption, which can be modelled as discrete-time conditional linear
cyclostationary random processes (or discrete-time conditional linear periodically correlated random
processes when we need to consider only its moment functions up to second order) with the period T .
The characteristics of representation (2), that is probability properties of the kernel and generating
white noise, as well as moment (3), (4) and characteristic (7) functions can be used in the different areas
in energy informatics, such as electricity consumption monitoring and forecasting, computer simulation,
identification of the electricity consumption profile features etc.
The perspective theoretical research should be related to the analysis of random matrix , ( )ttϕ ω from
the point of view of periodically correlated or cyclostationary random field on 2Z [14]. In this context,
the theoretical results of the current paper can be extended analyzing the different combinations
of components , ( )ttϕ ω and ( )τζ ω of discrete-time conditional linear cyclostationary random process. That
is, the following cases can be considered:
- , ( )ttϕ ω is cyclostationary random field (in general, with different periods by τ and t ) and ( )τζ ω is
cyclostationary white noise;
- , ( )ttϕ ω is cyclostationary random field and ( )τζ ω is stationary white noise;
- , ( )ttϕ ω is cyclostationary random field with the same period 1T = by both time arguments and ( )τζ ω is
cyclostationary white noise.
The perspective applied research is based on using the random coefficient periodic autoregressive
model (as a particular case of discrete-time conditional linear cyclostationary random process) for
the problems of estimation, identification, computer simulation, and forecasting in energy.
5. Conclusions
The properties of discrete-time conditional linear random processes have been analyzed in the context
of applications for mathematical modelling of electricity consumption and other important problems
of energy informatics.
Using the analysis of the expressions of mathematical expectation and covariance function of discrete-
time CLRP it has been shown that process is periodically correlated if the kernel of its representation
is periodically correlated random field and generating white noise is periodically correlated random process.
Using the method of characteristic functions, it has been proven that discrete-time CLRP
is cyclostationary if the kernel of its representation is cyclostationary random field and generating white
noise is cyclostationary random process.
The results can be used for the mathematical modelling, theoretical analysis of probability
characteristics of electricity, gas, water, and other energy resources consumptions, vibration signals
of energy facilities, etc.
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https://doi.org/10.1111/jtsa.12610
https://doi.org/10.1016/j.sigpro.2005.06.016
https://doi.org/10.1007/978-3-030-44443-3_2
https://doi.org/10.1016/j.sigpro.2009.07.031
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System Research in Energy. 2023. 1(72) 79
ВЛАСТИВОСТІ УМОВНИХ ЛІНІЙНИХ
ЦИКЛОСТАЦІОНАРНИХ ВИПАДКОВИХ ПРОЦЕСІВ
З ДИСКРЕТНИМ ЧАСОМ У ЗАДАЧАХ ЕНЕРГЕТИЧНОЇ
ІНФОРМАТИКИ
Михайло Фриз1, 2*, к.т.н., доцент, https://orcid.org/0000-0002-8720-6479
Леонід Щербак2, д.т.н., професор, https://orcid.org/0000-0002-1536-4806
1Тернопільський національний технічний університет імені Івана Пулюя,
вул. Руська, 56, м. Тернопіль, 46001, Україна;
2Інститут загальної енергетики НАН України, вул. Антоновича, 172, м. Київ, 03150,
Україна;
* Автор-кореспондент: mykh.fryz@gmail.com
Анотація. Сучасні проблеми та виклики в енергетиці вимагають проведення комплексних
досліджень у галузі енергетичної інформатики, що об’єднує в рамках єдиної методології
комп’ютерні науки, системи керування та системи енергоменеджменту. Важливим напрямом
енергетичної інформатики є вивчення проблем моделювання систем та процесів в енергетиці,
включно з процесами енергонавантаження та енергоспоживання. Лінійні та умовні лінійні
випадкові процеси (УЛВП) є математичними моделями сигналів, представлених у вигляді суми
великого числа випадкових імпульсів, що виникають у випадкові моменти часу. Саме таким чином
можна моделювати процеси споживання енергоресурсів (електро-, газо-, водоспоживання),
вібраційні сигнали енергооб’єктів та ін. У роботі досліджена модель УЛВП з дискретним часом,
що дозволяє враховувати циклічні властивості енергоспоживання. Метою роботи є
обґрунтування умов, за яких УЛВП з дискретним часом буде періодично корельованим процесом,
а також циклостаціонарним випадковим процесом. Показано, що відповідні умови залежать
від властивостей періодичності ймовірнісних розподілів ядра та породжуючого білого шуму
в зображенні УЛВП. Для досягнення мети використано властивості математичного сподівання
та кореляційної функції УЛВП, а також метод характеристичних функцій. У роботі доведено,
що УЛВП з дискретним часом є періодично корельованою випадковою послідовністю, якщо
породжуючий білий шум має періодичні математичні сподівання та дисперсію, а ядро є
періодично корельованим випадковим полем. На основі аналізу багатовимірної характеристичної
функції доведено, що УЛВП з дискретним часом є циклостаціонарним, якщо породжуючий білий
шум є циклостаціонарним процесом, а ядро є циклостаціонарним випадковим полем.
Охарактеризовані властивості умовних лінійних циклостаціонарних випадкових процесів
з дискретним часом є важливими для вирішення задач математичного, комп’ютерного
моделювання, статистичного аналізу та прогнозування споживання енергоресурсів.
Ключові слова: математична модель, енергетична інформатика, умовний лінійний випадковий
процес, циклостаціонарний процес, білий шум, характеристична функція.
Надійшла до редколегії: 21.02.2023
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| id | systemreorg-article-17 |
| institution | System Research in Energy |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-07-19T01:02:05Z |
| publishDate | 2023 |
| publisher | General Energy Institute of the National Academy of Sciences of Ukraine |
| record_format | ojs |
| resource_txt_mv | systemreorg/ec/abfb079a46527fad4335676fc616f4ec.pdf |
| spelling | systemreorg-article-172026-07-18T12:57:30Z PROPERTIES OF DISCRETE-TIME CONDITIONAL LINEAR CYCLOSTATIONARY RANDOM PROCESSES IN THE PROBLEMS OF ENERGY INFORMATICS ВЛАСТИВОСТІ УМОВНИХ ЛІНІЙНИХ ЦИКЛОСТАЦІОНАРНИХ ВИПАДКОВИХ ПРОЦЕСІВ З ДИСКРЕТНИМ ЧАСОМ У ЗАДАЧАХ ЕНЕРГЕТИЧНОЇ ІНФОРМАТИКИ Mykhailo Fryz Leonid Scherbak mathematical model, energy informatics, conditional linear random process, cyclostationary process, white noise, characteristic function. математична модель, енергетична інформатика, умовний лінійний випадковий процес, циклостаціонарний процес, білий шум, характеристична функція. Modern challenges in the energy industry require comprehensive research in the fieldof energy informatics, which combines computer science, control systems, and energy managementsystems within a single methodology. An important area of energy informatics is the study of problemsof systems and processes modeling in energy, including energy loads and consumption. Linear andconditional linear random processes (CLRP) are mathematical models of signals represented as the sumof a large number of random impulses occurring at random times. The energy consumption, vibrationsignals of energy objects, etc. can be modeled using this approach. A variant of the CLRP modelwith discrete time, taking into account the cyclic properties of energy consumption, has been investigatedin the paper. The goal is to justify the conditions for the discrete-time CLRP to be a periodicallycorrelated random process, as well as a cyclostationary process. It has been shown thatthe corresponding conditions depend on the periodicity of the probability distributions of the kernel andthe generating white noise of the CLRP representation. To achieve the goal, the propertiesof mathematical expectation and covariance function of CLRP, as well as the method of characteristicfunctions, have been used. The paper proves that the discrete-time CLRP is a periodically correlatedrandom sequence if the generating white noise has periodic mathematical expectation and variance, andthe kernel is a periodically correlated random field. Based on the analysis of the multivariatecharacteristic function, it has been proven that the discrete-time CLRP is cyclostationary ifthe generating white noise is a cyclostationary process and the kernel is a cyclostationary random field.The properties of discrete-time conditional linear cyclostationary random processes are importantfor mathematical modeling, simulation, statistical analysis, and forecasting of energy consumption. Сучасні проблеми та виклики в енергетиці вимагають проведення комплекснихдосліджень у галузі енергетичної інформатики, що об’єднує в рамках єдиної методологіїкомп’ютерні науки, системи керування та системи енергоменеджменту. Важливим напрямоменергетичної інформатики є вивчення проблем моделювання систем та процесів в енергетиці,включно з процесами енергонавантаження та енергоспоживання. Лінійні та умовні лінійнівипадкові процеси (УЛВП) є математичними моделями сигналів, представлених у вигляді сумивеликого числа випадкових імпульсів, що виникають у випадкові моменти часу. Саме таким чиномможна моделювати процеси споживання енергоресурсів (електро-, газо-, водоспоживання),вібраційні сигнали енергооб’єктів та ін. У роботі досліджена модель УЛВП з дискретним часом,що дозволяє враховувати циклічні властивості енергоспоживання. Метою роботи єобґрунтування умов, за яких УЛВП з дискретним часом буде періодично корельованим процесом,а також циклостаціонарним випадковим процесом. Показано, що відповідні умови залежатьвід властивостей періодичності ймовірнісних розподілів ядра та породжуючого білого шумув зображенні УЛВП. Для досягнення мети використано властивості математичного сподіваннята кореляційної функції УЛВП, а також метод характеристичних функцій. У роботі доведено,що УЛВП з дискретним часом є періодично корельованою випадковою послідовністю, якщопороджуючий білий шум має періодичні математичні сподівання та дисперсію, а ядро єперіодично корельованим випадковим полем. На основі аналізу багатовимірної характеристичноїфункції доведено, що УЛВП з дискретним часом є циклостаціонарним, якщо породжуючий білийшум є циклостаціонарним процесом, а ядро є циклостаціонарним випадковим полем.Охарактеризовані властивості умовних лінійних циклостаціонарних випадкових процесівз дискретним часом є важливими для вирішення задач математичного, комп’ютерногомоделювання, статистичного аналізу та прогнозування споживання енергоресурсів. General Energy Institute of the National Academy of Sciences of Ukraine 2023-04-07 Article Article application/pdf https://systemre.org/index.php/journal/article/view/17 10.15407/srenergy2023.01.072 System Research in Energy; No. 1 (72) (2023): System Research in Energy; 72-79 Системні дослідження в енергетиці; № 1 (72) (2023): Системні дослідження в енергетиці; 72-79 2786-7102 2786-7633 en https://systemre.org/index.php/journal/article/view/17/15 Copyright (c) 2023 Mykhailo Fryz, Leonid Scherbak https://creativecommons.org/publicdomain/zero/1.0 |
| spellingShingle | mathematical model energy informatics conditional linear random process cyclostationary process white noise characteristic function. Mykhailo Fryz Leonid Scherbak PROPERTIES OF DISCRETE-TIME CONDITIONAL LINEAR CYCLOSTATIONARY RANDOM PROCESSES IN THE PROBLEMS OF ENERGY INFORMATICS |
| title | PROPERTIES OF DISCRETE-TIME CONDITIONAL LINEAR CYCLOSTATIONARY RANDOM PROCESSES IN THE PROBLEMS OF ENERGY INFORMATICS |
| title_alt | ВЛАСТИВОСТІ УМОВНИХ ЛІНІЙНИХ ЦИКЛОСТАЦІОНАРНИХ ВИПАДКОВИХ ПРОЦЕСІВ З ДИСКРЕТНИМ ЧАСОМ У ЗАДАЧАХ ЕНЕРГЕТИЧНОЇ ІНФОРМАТИКИ |
| title_full | PROPERTIES OF DISCRETE-TIME CONDITIONAL LINEAR CYCLOSTATIONARY RANDOM PROCESSES IN THE PROBLEMS OF ENERGY INFORMATICS |
| title_fullStr | PROPERTIES OF DISCRETE-TIME CONDITIONAL LINEAR CYCLOSTATIONARY RANDOM PROCESSES IN THE PROBLEMS OF ENERGY INFORMATICS |
| title_full_unstemmed | PROPERTIES OF DISCRETE-TIME CONDITIONAL LINEAR CYCLOSTATIONARY RANDOM PROCESSES IN THE PROBLEMS OF ENERGY INFORMATICS |
| title_short | PROPERTIES OF DISCRETE-TIME CONDITIONAL LINEAR CYCLOSTATIONARY RANDOM PROCESSES IN THE PROBLEMS OF ENERGY INFORMATICS |
| title_sort | properties of discrete-time conditional linear cyclostationary random processes in the problems of energy informatics |
| topic | mathematical model energy informatics conditional linear random process cyclostationary process white noise characteristic function. |
| topic_facet | mathematical model energy informatics conditional linear random process cyclostationary process white noise characteristic function. математична модель енергетична інформатика умовний лінійний випадковий процес циклостаціонарний процес білий шум характеристична функція. |
| url | https://systemre.org/index.php/journal/article/view/17 |
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