=Paper=
{{Paper
|id=Vol-2753/paper18
|storemode=property
|title=Mathematical Modeling And Processing Of High Resolution Rhythmocardio Signal Based On A Vector Of Stationary And Stationary Related Random Sequences
|pdfUrl=https://ceur-ws.org/Vol-2753/short8.pdf
|volume=Vol-2753
|authors=Petro Onyskiv,Iaroslav Lytvynenko,Serhii Lupenko,Andriy Zozulia
|dblpUrl=https://dblp.org/rec/conf/iddm/OnyskivLLZ20
}}
==Mathematical Modeling And Processing Of High Resolution Rhythmocardio Signal Based On A Vector Of Stationary And Stationary Related Random Sequences==
https://ceur-ws.org/Vol-2753/short8.pdf
Mathematical Modeling and Processing of High Resolution
Rhythmocardio Signal Based on a Vector of Stationary and
Stationary Related Random Sequences
Petro Onyskiva, Iaroslav Lytvynenkoa, Serhii Lupenkoa, Andriy Zozuliaa
a
Ternopil Ivan Puluj National Technical University, Ternopil, Ukraine
Abstract.
Work is about substantiation of mathematical model of high resolution rhythmocardio
signal in the form of a vector of stationary and stationary related random sequences.
Investigated the structure of probabilistic characteristics of this model for analysis of
cardiac rhythm in modern cardiodiagnostic system. Based on a new mathematical
model of vector rhythmocardiosignals,was developed methods for statistical
evaluation of their spectral-correlation characteristics, which are used as diagnostic
features in automated diagnostic systems for functional diagnostics of the heart
condition and adaptive regulatory mechanisms of the human body
Keywords: mathematical model, cardiac rhythm, vector of stationary and stationary
related random sequences.
1. Introduction
Analysis of the heart rhythm makes it possible to evaluate not only the state of the cardiovascular
system, but also the state of the adaptive capacity of the whole human body. Most modern systems for
automated cardiac rhythm analysis are based on statistical analysis of rhythmocardio signal, which is
an ordered set of durations of R-R intervals in a registered electrocardiogram [1-8]. However, this
approach is uninformative, because the R-R intervals reflect only the change in the duration of the
cardiac cycles and not the totality of the time intervals between single-phase values of the
electrocardio signal for all its phases.
In [9, 10], a new approach to the analysis of cardiac rhythm on the basis of high resolution
rhythmocardio signal was developed. As noted in these works, the classical rhythmocardio signal is
embedded in the high resolution rhythmocardio signal, which is the basis for increasing the level of
informativeness of the analysis of cardiac rhythm in modern computer systems of functional
diagnostics of the human heart state.
In [9, 10], it is justified to use a vector of random variables as a mathematical model of high
resolution rhythmocardio signal. But, this model is a relatively bad mathematical model of high
resolution rhythmocardio signal, since it does not allow to study its temporal dynamics. To take into
account the temporal dynamics of the high resolution rhythmocardio signal, it is necessary to use a
mathematical apparatus of the theory of random sequences, namely, to consider it as a vector of
random sequences.
1.1. Setting objectives
In this work, we will develop a mathematical model of high resolution rhythmocardio signal in the
IDDM’2020: 3rd International Conference on Informatics & Data-Driven Medicine, November 19–21, 2020, Växjö, Sweden
EMAIL: rasegas21@gmail.com (P.Onyskiv); iaroslav.lytvynenko@gmail.com (I. Lytvynenko); lupenko.san@gmail.com (S. Lupenko);
bestguru@gmail.com (A. Zozulia)
ORCID: - [0000-0002-9717-4538] (P.Onyskiv); [0000-0001-7311-4103] (I. Lytvynenko); [0000-0002-6559-0721] (S. Lupenko); [0000-
0003-1582-3088] (A. Zozulia)
©️ 2020 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)
form of a vector of stationary and stationary related random sequences. Let's write down the structure
of probabilistic characteristics of this model for the analysis of cardiac rhythm in modern cardiac
diagnostics systems.
Based on this model, we develop methods for statistical analysis of the rhythmocardiosignal with
increased resolution in the framework of the spectral correlation theory of random processes
2. Research results
In the most general form stochastic models that takes into account the dynamics of changes high
resolution rhythmocardio signal is vector Ξ L (ω' , m) = Tl (ω' , m), ω' Ω' , l = 1,L , m Z ____
random
sequences. In this vector, the index m indicates the cycle number of the electrocardio signal, and the
index l indicates the reference number of the electrocardio signal within its cycle. The number L of
intervals per cycle of the electrocardio signal determines the resolution of the rhythmocardio signal
and sets the number of phases in the cycle of the electrocardio signal that can be separated by methods
of segmentation and detection in solving the problem of automatic formation of the rhythm cardio
signal from the electrocardio signal.
Justify of probabilistic characteristics of the vector Ξ L (ω , m) random sequences. One of the
'
simplest stochastic models that takes into account the dynamics of high resolution of rhythmocardio
signal is the vector Ξ L (ω' , m) = Tl (ω' , m), ω' Ω' , l = 1,L , m Z
____
stationary and stationary
related random sequences. First of all, note that the vector Ξ L (ω , m) stationary and stationary
'
related random sequences, in the particular case, if its components are stationary sequences with
independent values, that is, white noises given on the set of integers, is a known model of high
resolution rhythmocardio signal in the form of a vector of random variables, which was developed in
[9, 10]. But in practice, the hypothesis of the independence or non-correlation of the time intervals
between single-phase values of the electrocardio signal is not true, requiring a stochastic dependence
between the rhytocardiogram intervals with higher resolution, and hence the use of a more complex
and more general mathematical model as a vector Ξ L (ω , m) stationary and stationary related
'
random sequences.
The defining property of a vector Ξ L (ω , m) of stationary and stationary related random
'
sequences is the invariance of its family of distribution functions to time shifts by an arbitrary integer
k Z . For any distribution function Fp
Tl ...Tl
x1 ,..., x p , m1 ,..., m p order p ( p N ) from the
1 p
family of vector Ξ L (ω , m) distribution functions of stationary and stationary related random
'
sequences there must be such equality:
Fp
Tl ...Tl
x ,..., x , m ,...,m = F ...T x ,..., x , m + k,...,m + k ,
1 p 1 p p
Tl
1 p 1 p
lp
1 p 1
____
x1 ,..., x p R, m1 ,...,mp Z, l1 ,...,l p 1,L , k Z . (1)
Distribution function Fp
Tl ...Tl
x ,..., x , m ,...,m in the case, when l = l = ... = l = l is a
1 p 1 p 1 2 p
1 p
distribution function Fp
Tl
x ,..., x , m ,..., m l - stationary components T (ω , m) of vector
1 p 1 p l
'
Ξ L (ω' , m) - so it must to be p -dimensional auto-function of distribution for stationary random
'
sequence Tl (ω , m) , that describing the time distances between single-phase electrocardiogram for it
l -phase. So, if p = 1 , then we will have one-dimensional F1T x, m auto-function distribution of
l
'
stationary random sequence Tl (ω , m) .
In the case where equality l1 = l2 = ... = l p = l is not executed then the distribution function
Fp
Tl ...Tl
x ,..., x , m ,...,m is a p -dimensional compatible distribution function for several (at
1 p 1 p
1 p
least two) stationary components of a vector Ξ L (ω , m) , what describing the time distances between
'
single-phase intervals of an electrocardio signal generally for its various phases.
The distribution functions family of vector Ξ L (ω , m) of stationary and stationary related random
'
sequences most fully describes its probabilistic structure, however, methods for statistically
estimating the distribution function Fp
x1 ,..., x p , m1 ,...,m p have too high computational
Tl ...Tl
1 p
complexity for their practical use in the computer diagnostic systems of the functional state of the
cardiovascular system of the human body. We can use not just vector distribution functions
p
Ξ L (ω' , m) but we can use momentary functions by s = s j order, which, if they are also invariant
j=1
to time offsets (offsets by argument m ).
m ,..., m order s = s vector`s
p
So, if there is a mixed initial momentary function сs 1 p j
Tl ...T lp j=1
1
Ξ L (ω' , m) stationary and stationary related random sequences, then it has equality:
s
m1 ,..., m p = M Tl 1 ω' , m1 ... Tl p ω' , m p = сs m1 + k,..., m p + k ,
s
сs
Tl ...Tl 1 p
Tl ...Tl
, k Z .
1 p 1 p
____
m1 ,...,mp Z, l1 ,...,l p 1,L (2)
m ,..., m order s = s vector`s
p
If there is a mixed central momentary function rs 1 p j
Tl ...Tl j=1
1 p
Ξ L (ω' , m) stationary and stationary related random sequences, then it has equality:
s1
sp
m1 ,..., m p = M Tl ω' , m1 с1 ... Tl ω' , m p с1 =
rs
Tl ...Tl p Tl
1
1 Tl
p
1 p
= rs
Tl ...Tl
m1 + k,..., m p + k ,
1 p
____
m1 ,...,mp Z, l1 ,...,l p 1,L , k Z . (3)
Where с1 ,..., с1 is the set of first-order initial moments (mathematical expectations) of
Tl1
Tl
p
stationary random sequences from the set Tl (ω' , m),..., Tl (ω' , m) .
1 p
In practice, for analysis of high resolution rhythmocardio signal, it is reasonable to use mixed
high-order momentary functions, namely, mixed second-order initial momentary functions -
covariance functions and mixed second-order central momentary functions - correlation functions. In
this case, the initial second-order momentary functions for the vector Ξ L (ω , m) stationary and
'
stationary related random sequences are presented as a matrix of covariance functions:
СT = с2 m1 , m2 с2 m1 , m2 с2 m1 , m2
T1T1 T
T1 2 T
T1 p
с2 m1 , m2 с2 m1 , m2 с2 m1 , m2 , (4)
T2T1 T
T2 2 T2T p
с2T T m1 , m2 с2 m1 , m2 с2 m1 , m2
p 1 Tp 2 T T pT p
which can be presented more compactly as:
СT = с2 m1 , m2 , l1 ,l2 = ___
1,L , (5)
Tl1Tl2
where each of its elements is a covariance function сs m1 , m2 , which is given as:
Tl Tl
m1 ,m2 = M Tl1 ω' , m1 Tl2 ω' , m2 , m1 ,m2 Z, l1 , l2 ___
1,L .
1 2
с2 (6)
T
Tl l
1 2
Vector`s components Ξ L (ω , m) random sequences are stationary and stationary related
'
sequences, then their covariance functions are functions of only one integer argument u , which is
equal to u = m1 m2 . Therefore, the covariance matrix of this random vector can be represented as
follows:
СT = с2 u , l1 ,l2 = ___
1,L , (7)
Tl1Tl2
where each of its elements is a covariance function с2 u , which is equal to:
Tl Tl
m1 m2 , u, m1 ,m2 Z, l1 , l2 ___
1,L .
1 2
с2 u = с2 (8)
Tl lT Tl l T
1 2 1 2
Provided that l1 = l2 = l , the covariance function сs u is an auto-covariance function l -
Tl lT
stationary components Tl (ω , m) of vector Ξ L (ω , m) , which describes the time distances between
' '
single-phase intervals of electrocardiogram for l -phase. If l1 l2 , that means the covariance function
с2 u is the mutual covariance function for two stationary components of a vector Ξ L (ω' , m) ,
Tl lT
1 2
they describe the time distances between single-phase intervals of electrocardiogram l1 and l 2 -
phase.
Mixed central second-order momentary functions for a vector Ξ L (ω , m) stationary and stationary
'
related random sequences are presented as a matrix of correlation functions:
RT = r2 m1 , m2 r2 m1 , m2 r2 m1 , m2
T1T1 T1 2T T
T1 p
r2 m1 , m2 r2 m1 , m2 r2 m1 , m2 , (9)
T2T1 T2 2T T2T p
r2T T m1 , m2 r2 m1 , m2 r2 m1 , m2
p 1 Tp 2T T pT p
which can be presented more compactly as:
RT = r2 m1 , m2 , l1 ,l2 = ___
1,L , (10)
Tl1Tl2
where each of its elements is a correlation function rs m1 , m2 , which is given as:
Tl Tl
1 2
с2 m1 , m2 = M Tl1 ω' , m1 с1T Tl2 ω' , m2 с1T ,
T
Tl l l1 l2 (11)
1 2
___
m1 , m2 Z, l1 , l2 1,L .
The components of the vector Ξ L (ω , m) random sequences are stationary and stationary related
'
sequences, their correlation functions are functions of only one integer argument u , which is equal to
u = m1 m2 . This correlation matrix of this random vector can be represented as:
RT = r2 u , l1 ,l2 = ___
1,L , (12)
Tl1Tl2
where each of its elements is a correlation function r2 u , which is equal to:
Tl Tl
m1 m2 , u, m1 ,m2 Z, l1 , l2 ___
1,L .
1 2
r2 u = r2 (13)
T
Tl l T
Tl l
1 2 1 2
Provided that l1 = l2 = l , correlation function r2 u is auto-correlation function l -stationary
T
Tl l
components Tl (ω , m) of vector Ξ L (ω , m) , which describes the time distances between single-phase
' '
intervals of electrocardiogram for l -phase. If l1 l2 , then the correlation function r2 u is a
T
Tl l
1 2
mutual correlation function for two stationary components of a vector Ξ L (ω , m) , description of time
'
distances between single-phase intervals of electrocardiogram l1 and l 2 -phase.
Figures 1-4 show the results of statistical processing of the high resolution rhythmocardio signal,
by statistical evaluation of its corresponding probability characteristics.
120 (ti )
100
80
60
40
20
0
-20
-40 ti
0 500 1000 1500 2000 2500
Figure 1: Several cycles of the investigated electrocardio signal
19 T1 (m) 31 T2 (m)
18 30
29
17
28
16 27
15 26
25
14
24
13 23
12 22
21
11 m 20 m
10
0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 80 100 120 140 160 180 200
а) b)
Figure 2: Schedule of T1 (m) , T2 (m) realizations of the first component T1(ω' , m) and second
ω' ω'
'
component T2 (ω , m) of the vector rhythmocardiogram, that describes duration: а) P -intervals of
electrocardio signal; b) R - intervals of electrocardio signal
а) b)
'
Figure 3: Histograms of T1 (m) , T2 (m) realizations of the first component T1(ω , m) and second
ω' ω'
'
component T2 (ω , m) , of the vector rhythmocardiogram describing the duration accordingly: а) P -
intervals of electrocardio signal; b) R - intervals of electrocardio signal
rˆ2 u
T1T1 rˆ2 u
5 T1T1
4
2
3
1 2
1
0 -0
-1
-1
-2
-3
-2 u u
-4
0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70
а) b)
Figure 4: Schedule of implementation rˆ2 u statistical estimates of autocorrelation functions
T
T1 1
r2 u ( l1 = l2 = 1 ) first component T1(ω' , m) and second component T2 (ω' , m) , what describing
T
T1 1
the duration accordingly: а) P - intervals of electrocardio signal; b) R - intervals of electrocardio
signal.
3. Conclusions
The mathematical model of high resolution rhythmocardio signal in the form of a vector of
stationary and stationary related random sequences is substantiated. The structure of probabilistic
characteristics of this model for analysis of cardiac rhythm in modern cardiodiagnostic systems is
investigated. Unlike the existing model of high resolution rhythmocardio signal in the form of a
vector of random variables, new model take into account the temporal dynamics of the high resolution
rhythmocardio signal, which is the basis for increasing the level of informativeness of the analysis of
cardiac rhythm in modern computer systems of functional diagnostics. Based on a new mathematical
model of a rhythmocardiosignal with increased resolution, a statistical estimation of its probabilistic
characteristics is carried out within the framework of the spectral correlation theory of random
processes.
In the future scientific researches it is planned to justify the choice of the minimum number of
diagnostic features necessary for carrying out the diagnosis in the analysis of heart rhythm on the
basis of the obtained statistical estimates.
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