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
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 [
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In this work, we will develop a mathematical model of high resolution rhythmocardio signal in the 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
In the most general form stochastic models that takes into account the dynamics of changes high resolution rhythmocardio signal is vector ΞL(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 ΞL(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 ΞL(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 ΞL(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
[
The defining property of a vector ΞL(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 F x1 ,..., x p , m1 ,...,m order p ( p N ) from the
p pTl ...Tl 1
p pTl ...Tl 1 p pTl ...Tl 1 p family of vector ΞL (ω' , m) distribution functions of stationary and stationary related random sequences there must be such equality:
F x1 ,..., xp , m1 ,...,mp = F x1 ,..., xp , m1 + k,...,mp + k , x1 ,...,xp R, m1 ,...,mp Z, l1 ,...,lp _1_,L__, k Z . (1) x1 ,..., xp , m1 ,...,m in the case, when l1 = l2 = ... = l p = l is a p distribution function
FpTl x1 ,..., xp , m1 ,...,mp l - stationary components Tl (ω' , m) of vector ΞL(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 F1Tl stationary random sequence Tl (ω' , m) . x, m auto-function distribution of In the case where equality l1 = l2 = ... = l p = l is not executed then the distribution function x1 ,..., xp , m1 ,...,m is a p -dimensional compatible distribution function for several (at p least two) stationary components of a vector ΞL(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 ΞL(ω' , m) of stationary and stationary related random sequences most fully describes its probabilistic structure, however, methods for statistically estimating the distribution function F x1 ,..., xp , m1 ,...,mp have too high computational pTl ...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 ΞL(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 ).
So, if there is a mixed initial momentary function с sTl ...Tl 1 p
p m1 ,...,mp order s = s j vector`s j=1 ΞL(ω' , m) stationary and stationary related random sequences, then it has equality: m1 ,...,mp = M Tls1 ω' , m1 ... Tls p ω' , mp = с 1 p sTl ...Tl 1
p m1 ,...,mp Z, l1 ,...,lp _1_,L__, k Z .
m1 + k,...,mp + k , If there is a mixed central momentary function r sTl ...Tl 1 p
p m1 ,...,mp order s = s j vector`s j=1 ΞL(ω' , m) stationary and stationary related random sequences, then it has equality: r sTl ...Tl 1 p
s1 m1 ,...,mp = M Tl1 ω' , m1 с1Tl1 ... Tlp ω' , mp с1 sp = Tlp
= rsTl ...Tl 1 p
m1 + k,...,mp + k , m1 ,...,mp Z, l1 ,...,lp _1_,L__, k Z . Where с1 Tl1 ,...,с1 is the set of first-order initial moments (mathematical expectations) of
Tl p stationary random sequences from the set T (ω' , m),...,Tl (ω' , m).
l1 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 ΞL(ω' , m) stationary and stationary related random sequences are presented as a matrix of covariance functions: (2) (3) с2T2T1 с 2TpT1
2Tl1Tl2 which can be presented more compactly as:
СT = с2 Tl1Tl2 m1 ,m2 , l1 ,l2 = _1_,L_ , where each of its elements is a covariance function сsTl1Tl2 m1 ,m2 , which is given as: m1 ,m2 = M Tl ω' , m1 Tl ω' , m2 , m1 ,m2Z, l1 , l2 _1_,L_.
1 2
Vector`s components ΞL(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 = m m . Therefore, the covariance matrix of this random vector can be represented as 1 2 follows: СT = с2Tl1Tl2 u, l1 ,l2 = _1_,L_ ,
where each of its elements is a covariance function с2Tl1Tl2 u , which is equal to: с 2Tl1Tl2 u = с2Tl1Tl2
m1 m2 , u, m1 ,m2Z, l1 , l2 _1_,L_.
Provided that l1 = l2 = l , the covariance function сsTlTl u is an auto-covariance function l stationary components Tl (ω' , m) of vector ΞL(ω' , m) , which describes the time distances between single-phase intervals of electrocardiogram for l -phase. If l1 l2 , that means the covariance function с
u is the mutual covariance function for two stationary components of a vector ΞL(ω' , m) , 2Tl1Tl2 they describe the time distances between single-phase intervals of electrocardiogram l1 and l2 phase.
Mixed central second-order momentary functions for a vector ΞL(ω' , m) stationary and stationary related random sequences are presented as a matrix of correlation functions:
RT = r2T1T1 m1 ,m2 r2T1T2 m1 ,m2 r2T1Tp m1 ,m2 r2T2T1 m1 ,m2 r2T2T2 m1 ,m2 r2T2Tp m1 ,m2 , (9) r2TpT1 m1 ,m2 r2TpT2 m1 ,m2 r2TpTp m1 ,m2 which can be presented more compactly as:
RT = r2 Tl1Tl2
m1 ,m2 , l1 ,l2 = _1_,L_ , where each of its elements is a correlation function rsTl1Tl2 m1 ,m2 , which is given as: 2Tl1Tl2
m1 ,m2 = M Tl1 ω' , m1 с1T Tl2 ω' , m2 с1 l1 , T l2 m1 ,m2 Z, l1 , l2 _1_,L_.
The components of the vector ΞL(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 = m m . This correlation matrix of this random vector can be represented as: 1 2
RT = r2 u, l1 ,l2 = _1_,L_ ,
Tl1Tl2 where each of its elements is a correlation function r2Tl1Tl2 u , which is equal to: r 2Tl1Tl2 u = r2Tl1Tl2 m m2 , u, m1 ,m2Z, l1 , l2 _1_,L_.
1 Provided that l1 = l2 = l , correlation function r 2TlTl components Tl (ω' , m) of vector ΞL (ω' , m), which describes the time distances between single-phase intervals of electrocardiogram for l -phase. If l1 l2 , then the correlation function r mutual correlation function for two stationary components of a vector ΞL(ω' , m) , description of time distances between single-phase intervals of electrocardiogram l1 and l2 -phase.
Figures 1-4 show the results of statistical processing of the high resolution rhythmocardio signal, by statistical evaluation of its corresponding probability characteristics. 2Tl1Tl2
u is a u is auto-correlation function l -stationary (11) (12) (13)
(m) realizations of the first component T1(ω' , m) and second component T2(ω' , m)of the vector rhythmocardiogram, that describes duration: а) P -intervals of electrocardio signal; b) R - intervals of electrocardio signal component T2(ω' , m), of the vector rhythmocardiogram describing the duration accordingly: а) P intervals of electrocardio signal; b) R - intervals of electrocardio signal (m) realizations of the first component T1(ω' , m) and second 2T1T1 u
u statistical estimates of autocorrelation functions r 2T1T1
u ( l1 = l2 = 1) first component T1(ω' , m) and second component T2(ω' , m), what describing the duration accordingly: а) P - intervals of electrocardio signal; b) R - intervals of electrocardio signal.
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.