The paper is devoted to statistical methods estimation of probabilistic characteristics of rhythmocardiosignal with increased resolution on the basis of its model in the form of a vector of stationary and stationary related random processes. The hypothesis about the normality of the law of components distribution of the rhythmocardiosignal with increased resolution is confirmed. It was made a decomposition of the statistical estimates of autocorrelation and intercorrelation functions allowed to obtain spectral and inter-spectral power density of vector components, that allowed to reduce the space dimension of diagnostic features in heart rate analysis systems based on rhythmocardiosignals with increased resolution. That allowed to substantiate the vector of diagnostic features in the systems of cardiac rhythm analysis based on the rhythmocardiosignals with increased resolution is substantiated.
Automated heart rhythm analysis systems make it possible to evaluate both the state of the cardiovascular system and the state of the adaptive capacity of the human body as a whole. Most modern heart rate analysis systems are based on the use of stochastic mathematical models of rhythmocardiosignal and methods of its statistical analysis by rhythmocardiogram, which is an ordered set of durations of R-R intervals in a registered electrocardiosignal [1-8].
However, this approach makes it impossible to detect subtle, more detailed features of the heart rhythm, since RR intervals reflect only the change in the duration of the cardiac cycles, and not the whole totality of time intervals between single-phase values of the electrocardiosignal, which makes it impossible to describe the rhythm of hearts in full.
In papers [9,10], in order to provide a more informative description of the heart rhythm, a new approach to its analysis based on rhythmocardiosignal with increased resolution has been substantiated. The classical rhythmocardiogram is embedded in the increased-resolution rhythmocardiogram, which is the basis for increasing the level of informativeness of the heart rate analysis in modern computer systems of functional diagnostics of the human heart condition based on the rhythmocardiogram with increased resolution.
In papers [9, 10], the use of a random variable vector as a mathematical model of rhythmocardiosignal with increased resolution is substantiated. However, this model is a relatively poor mathematical model of rhythmocardiosignal with increased resolution, since it does not allow to study its time dynamics. To take into account the time dynamics of rhythmocardiosignal with increased resolution, it is necessary to use a mathematical apparatus of the theory of random processes, that is, to consider it as a vector of discrete-time random processes.
In this paper, we develop methods for the statistical estimation of the probabilistic characteristics of a rhythmocardiogram with high resolution based on its model in the form of a vector of stationary and stationary related random sequences. 2
F (x1,..., x p , m1,..., m p ) = F (x1,..., x p , m1 + k,..., m p + k ), pTl1...Tlp occurs: One of the simplest stochastic models that take into account the dynamics of rhythmocardiosignal with increased resolution is the vector of ____ ΞL (, m) = Tl (, m), Ω, l = 1, L , m Z stationary and stationary related random processes. In this vector, the index m indicates the cycle number of the electrocardiosignal, and the index l - the reference number of the electrocardiogram within it m -th cycle. The number of samples L per electrocardiosignal cycle determines the resolution of the rhythmcardiosignal, and specifies the number of phases in the cycle of the electrocardiosignal that can be separated by methods of segmentation and detection in solving the problem of automatic formation of the rhythmocardiosignal from the electrocardiosignal.
The defining property of a vector ΞL ( , 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 . Namely, for any distribution function F (x1,..., x p , m1,..., m p ) order p ( p N ) from the family of vector distribution functions ΞL ( , m) of stationary and stationary random sequences such equality related random sequences.
An estimate that converge in the root-mean-square sense to the distribution function F (x1,..., x p , m1,..., m p ) of order p ( p N ) of vector of ΞL ( , m) stationary and stationary related random sequences:
F
pTl1 ...Tlp
(x1,..., x p , m1,..., m p ) = lK.i→.m. 2K1+ 1 k=K−K jp=1 H (x j − Tl j ( , m j + k )),
(
1, x 0, is a Heaviside step function, which is an indicator of a negaFunction H (x) =
0, x 0. tive number.
In particular, if in the formula (
F1T (x) = l.i.m. 1 K H (x − Tl ( , k )), x R, l 1__,_L_, k Z. (
j=1 сsTl1...Tlp (m1,..., mp ) = lK.i→.m. 2K1+1 k=K−KTl1s1 ( , m1 + k )...Tlspp ( , mp + k ), ____ m1,..., m p Z, l1,...,l p {1, L}, k Z .
If s = 2 and p = 2 , then from the formula (4) follows such a convergence in the root-mean-square sense for the covariance function сsTl1Tl2 (m1, m2 ) two stationary and stationary connected random sequences Tl1 ( , m) and Tl2 ( , m) , describing the time distances between single-phase electrocardiosignal samples for l1 -st and l2 -d its phases, such as: If in the formula (5) p = 1 , that is l1 = l2 = ... = l p = l , then we have the convergence of the estimate in the root-mean-square sense to the one-dimensional initial moment function сs (m) s -th order, which is for a stationary random sequence Tl ( , m) is a
Tl constant сsTl = сsTl (m) (the initial moment s -th order), that is:
сsTl = lK.i→.m. 2K1+1 k=K−KTls ( , k ), l 1__,_L_ . (6) If in the formula (6) s = 1 , then we will have the convergence of the estimate in the root-mean-square sense to the initial moment of the first order с1Tl = с1Tl (m) (mathematical expectation) stationary random sequence Tl ( , m) , that is:
с1Tl = lK.i→.m. 2K1+1 k=K−KTl ( , k ), l 1__,_L_ . (7) An estimate that converge in the root-mean-square sense to the mixed initial central p function of order s = s j :
j=1 rsTl1...Tlp (m1,...,mp ) = lK.i→.m. 2K1+1 k =K−K Tl1 (, m1 + k ) − с1Tl1 s1 ... Tlp (, mp + k )− с1 s p ,
Tlp ____ m1,..., mp Z, l1,...,l p {1, L}, k Z .
If s = 2 and p = 2 , then from the formula (8) such a convergence follows in the root-mean-square sense for the correlation function rsTl1Tl2 (m1, m2 ) two stationary and stationary connected random sequences Tl1 ( , m) and Tl2 ( , m) , describing the time intervals between single-phase electrocardiogram samples for l1 -th and l2 -d its phases, that is: 2Tl1Tl2
1 K (m1, m2 ) = lK.i→.m. 2K +1 k =−KTl1 ( , m1 + k )Tl2 ( , m2 + k ),
____ m1, m2 Z, l1,l2 {1, L}, k Z . (5) (8) (9) r 2Tl1Tl2 (m1, m2 ) = lK.i→.m. 2K1+1 k=K−K Tl1 ( , m1 + k )− с1Tl1 Tl2 ( , m2 + k )− с1Tl2 , ____ m1, m2 Z, l1,l2 {1, L}, k Z .
If in the formula (9) s = 2 and p = 1 , that is l1 = l2 = ... = l p = l , then we will have a convergence of the estimate in the root-mean-square sense to the variance r2Tl stationary random sequence Tl ( , m) , that is:
r2Tl = lK.i→.m. 2K1+1 k =K−K(Tl ( , k ) − с1Tl )2 , l 1__,_L_ . (10) The above formulas reflect the convergence in the root-mean-square sense of the statistical estimates to the corresponding probabilistic characteristics of rythmocardisignal with increased resolution, and, therefore, the statistical estimates are consistent.
Since in real computer systems of cardiac rhythm analysis the finite number of cycles of electrocardiosignal is always recorded, this fact should be taken into account also in the statistical estimation of probabilistic characteristics of the rhythmocardiosignal with increased resolution. Namely, the statistical evaluation of the probabilistic characteristics of the rhythmocardiosignal with increased resolution is to obtain the realizations of statistical estimates that can be taken as pproximation to the corresponding probabilistic characteristics of the analyzed rhythmocardiosignal.
We write down the expressions to calculate the realizations of the corresponding statistical estimates of the probabilistic characteristics of the vector of ____ ΞL ( , m) = Tl ( , m), Ω, l = 1, L , m Z stationary and stationary related ran dom sequences when some long realization is given ____ _____ ΞL (m) = Tl (m), l = 1, L , m = 1, M , where M - the number of registered complete cycles from which the rhythmocardiosignal with increased resolution is formed. An expression for calculating the realization of a statistical estimate of a distribution function F (x1,..., x p , m1,..., m p ) of order p ( p N ) of vector ΞL ( , m) stadepending on the number of averages in the realization of statistics to provide the required level of accuracy and assurance of statistical estimation.
In particular, if in the formula (11) p = 1 , that is l1 = l2 = ... = l p = l , then we will have an expression to calculate the realization of the statistical estimate Fˆ1Tl (x) onedimensional auto-distribution function F1Tl (x) stationary random sequence Tl ( , m) , that is:
Fˆ1Tl (x) =
1 M −M1H (x − Tl (k )), x R, l 1__,_L_.
M − M1 +1 k =0 (12) An expression to calculate the realization of a statistical estimate of a mixed initial p moment function of order s = s j is given:
j=1 сˆsTl1...Tlp (m1,..., m p ) =
1 M −MT1 s1 (m1 + k )...Tlspp (m p + k ), M − M1 +1 k =0 l1
____ m1,..., m p {1, M1}, l1,...,l p {1, L} . (13) If s = 2 and p = 2 , then from formula (13) follows the expression to calculate the realization of the statistical estimate сˆsTl1Tl2 (m1, m2 ) of the covariance function с (m1, m2 ) two stationary and stationary-related random sequences Tl1 ( , m) and sTl1Tl2 Tl ( , m) , that describe the time distances between single-phase samples of electro2 cardiosignal for l1 -st and l2 -d its phases, in particular: сˆ2Tl1Tl2 (m1, m2 ) = M − M11 +1 Mk−=M0T1l1 (m1 + k )Tl2 (m2 + k ), m1, m2 1_,_M___1 , l1, l2 1__,_L_ . (14)
If in the formula (13) p = 1 , that is l1 = l2 = ... = l p = l , then we get an expression to сsTl calculate the realization of the statistical estimate сˆsTl of stationary random sequence Tl ( , m) , in particular: the initial moment s -th order сˆsTl = M − M11 +1 Mk−=M0T1ls (k ), l 1__,_L_ . (15) If in the formula (15) s = 1 , then we get an expression to calculate the realization of the statistical estimate сˆ1Tl of the initial moment of the first order с1Tl (mathematical expectation) stationary random sequence Tl ( , m) , that is:
сˆ1Tl = M − M11 +1 Mk−=M0T1l (k ), l 1__,_L_ . (16) An expression to calculate the realization of a statistical estimate of a mixed initial p central function of order s = s j is given:
j=1 rˆsTl1...Tlp (m1,...,mp ) = M − M11 + 1 M −kM=01 +1Tl1 (m1 + k ) − сˆ1Tl1 s1 ... Tl p (mp + k )− сˆ1Tlp s p , _____ ____ m1,..., m p {1, M1}, l1,...,l p {1, L} . (17) If s = 2 and p = 2 , then from the formula (17) follows the expression to calculate the realization of the statistical estimation of the correlation function rsTl1Tl2 (m1, m2 ) of rˆ2Tl1Tl2 (u) = rˆ2
M − M11 +1 Mk−=M01 Tl1 (k ) − сˆ1Tl1 Tl2 (k + u) − сˆ1Tl2 ,
__________ _____ ___ u = 0, M1 −1, m1, m2 {1, M1}, l1, l2 {1, L} . late the realization of the variance estimate r2Tl If in the formula (19) u = 0 , а l1 = l2 = l , then we will have an expression to calcustationary random sequence Tl ( , m) , that is: rˆ2Tl =
M1−1 kM=1 (Tl ( , k )− с1Tl )2 , l 1__,_L_ .
_____ ____ m1, m2 {1, M1}, l1,l2 {1, L} .
Since for stationary and stationary random sequences, correlation functions are functions of only one integer argument u , which is equal to u = m1 − m2 , then their statistical estimates also depend on only one argument u . In this case, assuming the ergodicity of the stationary components of the vector ΞL ( , m) , then the formula (18) will look like this: two stationary and stationary related random sequences Tl1 ( , m) and Tl2 ( , m) , that describe the time distances between single-phase samples of electrocardiosignal for l1 -st and l2 -d its phases, that is: rˆ2Tl1Tl2 (m1, m2 ) =
M − M11 +1 Mk−=M01 Tl1 (m1 + k ) − сˆ1Tl1 Tl2 (m2 + k ) − сˆ1Tl2 , 3
The most comprehensive information on the probabilistic characteristics of a increased-resolution rhythmocardiogram is contained in the distribution function family F (x1,..., x p , m1,..., m p ), p N, l1,...,l p 1__,_L_ of vector ΞL ( , m) stationary pTl1...Tlp and stationary-related random sequences, and all the other probabilistic characteristics (mixed, central, initial moment functions of different orders) are derived from this family. However, due to the high computational complexity of the methods of statistical estimation of multidimensional vector distribution functions ΞL ( , m) , it is necessary to study increased-resolution rhymocardialsignals to substantiate their types of distribution, in particular, to test the statistical hypothesis for the normality (Gaussian) of the stationary components of a vector, which, if confirmed, will allow us to apply the model of the studied rhythmocardiogram within the framework of spectralcorrelation theory, in particular instead of a tedious, computationally complex estimation of distribution functions, to apply simpler computational procedures for estimating spectral-correlation characteristics of rhythmocardiosignals with increased resolution. (18) (19) (20) Let's test the hypothesis for normality of the distribution law of components of a vector ΞL ( , m) . To do this, we apply the Pearson's agreement criterion ( 2 -test), that allow to establish consistency (or inconsistency) of empirical and theoretical distributions of vector components ΞL ( , m) . Empirical distribution of vector components ____ ______ ΞL ( , m) = Tl ( , m), Ω, l = 1, L , m = 1, M , estimated by making a histogram.
That is, the interval at which all the values of the realization fall into Tl ( m) l -th ___ component Tl ( , m) divided into I subintervals {(Sil , Sil+1), i = 1, I} with durations {li = Sil+1 − Sil , i = 1_,_I_} and for each interval Sil the number is calculated hil (empirical frequency), which is equal to the ratio of the number of realization values Tl ( m) l -th component Tl ( , m) , falling into the interval li , to their total number M , that is:
nil , i = _1_,_I_, l = 1_,_L_ .
hil = li M ____ The set of pairs {(li , hil ),i = 1, I } for realization Tl ( m) l -th component Tl ( , m) can be presented either as a table or graphically as a histogram, for example in Figure 2 shows the results of such calculations for the components of the vector. In the test 2 - square as a measure of the empirical frequency deviation hil from the corresponding theoretical probability pil the value is used 2 hil − pil 2 = I M .
i=1 pil (21) (22) Value 2 in expression (22) is a random variable, the distribution of which at M → , tend to 2 - distribution Pq ( x) , which depends on the parameter q , which is called the number of freedom degrees equal to:
q = I − s −1, (23) where s - the number of theoretical distribution parameters, against which the hypothesis about the consistency of empirical and theoretical distributions is tested. In the case of the normal distribution of the stationary components of the vector ΞL ( , m) s = 2 .
Application of 2 - test implies that some level of significance is given previously (for example, = 0.01; = 0.05 ), which makes it possible to calculate the quantile q2 of distribution 2 for a given and q . If the value 2 , calculated by the formula (22), more than q2 , then it is considered, that a theoretical distribution (for example, normal) is in poor agreement with the results of observations at a given level of significance . Conversely, if the value is calculated 2 less than q2 , then it is considered, that the theoretical and empirical distributions is in good agreement. 4
In order to obtain the probable result of verification for the normality of the law of distribution of the rhythmocardialsignal with the increased resolution, the realization of the electrocardiosignal in the second lead, which contained 245 cardiac cycles and was generated by the work of the heart of the patient with a conditional norm, was processed. From the registered electrocardiogram according to the method of automatic formation of rhythmocardiogram with high accuracy, the realization received ____ ______ Ξ3 ( m) = {Tl ( m), l = 1,3 , m = 1,245} of tricomponent vector of stationary and
stationary____ ______ Ξ3 ( , m) = {Tl ( , m), Ω, l = 1,3 , m = 1,245} related random sequences.
The first component T1 ( , m) of this vector is a random stationary sequence, which describes the duration P - intervals in the electrocardiosignal for all of its 245 recorded cycles.
T1 (m) The plot of realization
of this component is shown in Figure 1,а. Second
The plot of realization component T2 ( , m) of this vector is a random stationary sequence, which describes the duration R - intervals in the electrocardiosignal.
T2 (m)
of the second component is shown in the figure 1,b.
The third component T3 ( , m) of this vector is a random stationary sequence, which describes the duration T - intervals in the electrocardiosignal. Plot of realization T3 (m)
the third component is shown in the figure 1,c. 19 T1 (m) 18 17 16 15 14 13 12 11 10 the duration P - intervals in the electrocardiosignal; b) T2 (m) the second component T2 ( , m) , which describes the duration R - intervals in the electrocardiosignal; c) T3 (m) third component T3 ( , m) , which describes the duration T - intervals in the electrocardiosignal
a) b) c) Fig. 2. Histogram for: a) the first component T1 ( , m) , which describes the duration P - intervals in the electrocardiosignal; b) the second component T2 ( , m) , which describes the duration R - intervals in the electrocardiosignal; c) third components T3 ( , m) , which describes the duration T - intervals in the electrocardiosignal The number of freedom degrees was chosen equal q = 7 , level of signification = 0.05 , і, respectively quantile 2 - distribution with q freedom degrees 02.95,7 = 14.07 . Figures 2, a-c show histograms for realizations T1 (m) , T2 (m) and T3 (m) the corresponding three stationary components of the vector Ξ3 ( , m) . Table 1 shows the results of application 2 - test for checking the normality of the law of distribution of three stationary components of a vector Ξ3 ( , m) , that set the rhythmcardiosignal with increased resolution.
Stationary component number 1 2 3 Thus, based on the results of normality hypothesis testing of the distribution of the stationary components of a random vector ΞL ( , m) by Pearson's criterion, it is found that these results do not contradict the hypothesis for normality of its distribution. Normality of the vector ΞL ( , m) is the basis for the substantiation of diagnostic features in systems of cardiac rhythm analysis according to rhythmocardiogram with increased resolution within the spectral-correlation theory, which significantly reduces the computational complexity of such analysis. In this case, to evaluate the probabilistic structure of the vector ΞL ( , m) of stationary and stationary-related random sequences, it is sufficient to statistically evaluate only the vector ____ С1L = {с1Tl , l = 1, L} his mathematical expectations according to formula (16) and the matrix of correlation functions RT = [r2
Tl1Tl2
___ (u), l1, l2 = 1, L] according to formula (19).
An important step in the development of information systems of cardiac rhythm analysis is a substantiated choice of diagnostic features set, which will be used for automated procedure for diagnostic decision making. There are mainly two requirements for this set of diagnostic features. The first requirement is the informativeness requirement of many diagnostic features, and the second - the requirement of minimality of their number.
The first requirement regarding the informative nature of the diagnostic features is the ability to distinguish between different state of the system under study on these features. Such informativeness of diagnostic features is determined by two of their characteristics, that is, sensitivity of diagnostic features to change of a state of regulatory mechanisms of cardiovascular system and organism as a whole, and also insensitivity to various non-informative noise factors (interferences) which are always present in a rhythmocardiosignal. One of the possible quantitative informativeness indicators of diagnostic features is the ratio of the average distance between the diagnostic classes (training sets) and the average diameter of the corresponding classes that corresponds to different states of the cardiovascular system in the metric space of diagnostic features. If this ratio is significant, then the components of the diagnostic feature vector are considered informative.
The minimum number requirement of diagnostic features provides the minimum dimension of diagnostic features space, which, as a consequence, provides the minimum computational complexity of algorithms for diagnostic decision making. Let’s substantiate the diagnostic features set to evaluate the state of regulatory mechanisms of the cardiovascular system and the organism as a whole, that is, such sets of diagnostic features, which, on the one hand, are informative, and on the other - have a minimum number. First, let's focus on the procedure for providing the minimum number of diagnostic features by rhythmocardiosignals with increased-resolution. Since the hypothesis for normality of distribution of the rhythmocardiosignal with increased resolution was previously confirmed, as described above, the initial set of ____ diagnostic features is a numeric vector Сˆ 1L = {сˆ1T , l = 1, L} point estimates of mathel matical expectations calculated according to expression (16) and an estimates matrix ___ of correlation functions Rˆ T = [rˆ2 (u), l1, l2 = 1, L] , which were calculated according
Tl1Tl2 to the formula (19). One of the obvious ways to reduce the number of diagnostic feaRˆ T = [rˆ2Tl1Tl2 (u), l1, l2 = 1_,_L_ ] , indicating that it is sufficient to evaluate only those elements of the matrix Rˆ T , what lie on its diagonal and above the diagonal, that is, such ___ ____ an ordered set Rˆ T = [rˆ2Tl1Tl2 (u), l1 = 1, L, l2 = l1, L] . On the diagonal of this matrix, when l1 = l2 , autocorrelation functions estimates are placed, and the elements of the matrix Rˆ T , which are placed above its diagonal, that is, when l1 l2 , are estimates of crosscorrelation functions.
Therefore, the matrix Rˆ T = [rˆ2
Tl1Tl2
___ (u), l1, l2 = 1, L] without losing the informativeness, ___ ____ we can replace with the triangular matrix Rˆ T = [rˆ2Tl1Tl2 (u), l1 = 1, L, l2 = l1, L] . Another way to reduce the number of diagnostic features in cardiac rhythm analysis information systems on the basis of rhythmocardiosignal with increased resolution is to use spectral decompositions of the triangular matrix elements themselves ___ ____ Rˆ T = [rˆ2 (u), l1 = 1, L, l2 = l1, L] , in particular, by using a discrete Fourier transform
Tl1Tl2 of autocorrelation estimates and cross-correlation functions from this matrix. That is, ___ ____ instead of a triangular matrix Rˆ T = [rˆ2 (u), l1 = 1, L, l2 = l1, L] correlation functions
Tl1Tl2 can be used by a triangular matrix Sˆ T = [Sˆ2 Tl1Tl2 which are Fourier-images of the corresponding estimates of the correlation functions from the matrix Rˆ T . That is, Fourier-images from the matrix Sˆ T are calculated as ___ ____ ( ), l1 = 1, L, l2 = l1, L] , elements of follows: the full energy of evaluation rˆ2Tl1Tl2 Here is an of the
− j2u
M1 −1 __________ ___ ____
Sˆ2Tl1Tl2 ( ) = u=0 rˆ2Tl1Tl2 (u) e M1 , = 0, M1 − 1, l1 = 1, L, l2 = l1, L, j = − 1 . (24) Based on the Bessel inequality, as diagnostic features we will not choose the whole set Sˆ2Tl1Tl2 ( ), = _0_,_M___1_−__1_ samples of functions Sˆ2Tl1Tl2 ( ) , but only a subset of their first M 2 ( M 2 M 1 ) samples Sˆ2Tl1Tl2 ( ), = 0__,_M___2__−_1_ , which contribute to (u ) correlation function is not less than 95%.
example statistical evaluation of vector elements ____ С1L = {с1 , l = 1, L} mathematical expectations, elements of the matrix of correlation Tl
functions RT = r2 Tl1Tl2
___ (u), l1,l2 = 1, L and matrix elements of the Fourier-images
(u ) ( l1 = l2 = 1 ), the first component T1( , m) , which describes the duration P 2T1T1 intervals in the electrocardiosignal; b) r2T2T2 (u ) statistical estimation of autocorrela(u ) ( l1 = l2 = 2 ) the second component T2 ( , m) , which describes tion function r2T3T3 the duration R - intervals in the electrocardiosignal; c) rˆ2T3T3 (u ) statistical estimation of autocorrelation function r2T3T3 describes the duration T - intervals in the electrocardiosignal
(u ) ( l1 = l2 = 3 ) third component T3 ( , m) , which Stationary component number 1 (u ) statistical estimation of the cross-correlation function
T3 ( , m) vector components Ξ3 ( , m) ; c) rˆ2T1T3 (u ) statistical estimation of the cross-correlation function r2T1T3 (u ) ( l1 = 1, l2 = 3 ) first T1( , m) and third T3 ( , m) vector components Ξ3 ( , m) Figure 5 shows graphs of realization of statistical estimates of the cross-correlation functions of the vector components Ξ3 ( , m) Figure 6 shows realization plot ˆ S ( ) statistical estimation of cross-spectral power density 2T1T2 2T1T2 ( l1 = 1, l2 = 2 ) first T1( , m) and second T2 ( , m) vector components Ξ3 ( , m) . S ( ) 2T1T2
a) b) c) Fig. 5. Plot of realization: a) Sˆ2T1T2 ( ) statistical estimation of power spectral density S ( ) ( l1 = l2 = 1) the first component T1( , m) , which describes the duration P tral density S2T2T2 intervals in the electrocardiosignal; b) Sˆ2T2T2 ( ) statistical estimation of power spec( ) ( l1 = l2 = 2 ) the second component T2 ( , m) , which describes of power spectral density S2T3T3 describes the duration T - intervals in the electrocardiosignal the duration R - intervals in the electrocardiosignal; c) Sˆ2T3T3 ( ) statistical estimation ( ) ( l1 = l2 = 3 ) third component T3 ( , m) , which density S
2T1T2 nents Ξ3 ( , m) ; b) Sˆ
2T1T3 a) b)
c) 2T1T2 Fig. 6. Plot of realization: a) Sˆ
( ) statistical estimation of cross-spectral power ( ) ( l1 = 1, l2 = 2 ) first T1 ( , m) and second T2 ( , m) vector compo( ) statistical estimation of cross-spectral power density S 2T1T3
( ) ( l1 = 1, l2 = 3 ) first T1 ( , m) and third T3 ( , m) vector components Ξ3 ( , m) ; c) Sˆ 2T2T3 ( ) statistical estimation of cross-spectral power density S 2T2T3 ( ) ( l1 = 2, l2 = 3 ) second T2 ( , m) and third T3 ( , m) vector components Ξ3 ( , m) .
The methods of statistical estimation of probabilistic characteristics of rhythmocardiosignal with increased resolution on the basis of model in the form of a vector of stationary and stationary related random sequences are developed in the paper. Conducted statistical experiments confirmed the hypothesis for the normality of the law of distribution of components of the vector rhythmocardiosignal. The conducted decomposition of statistical estimates of autocorrelation and сross-correlation functions made it possible to obtain spectral and сross-spectral power densities of the vector components, which allowed to reduce the dimension space of diagnostic features in systems of analysis of cardiac rhythm according to the rhythmocardiosignals with increased resolution.
The developed statistical methods can be used in developing of specialized software in automated cardio-diagnostic complexes, in particular, rhythm analysis subsystems.