Approaches to Statistical Processing of Rhythmocardiosignal with Increased Resolution Iaroslav Lytvynenko1[0000-0001-7311-4103], Serhii Lupenko1[0000-0002-6559-0721], Vyacheslav Kharchenko2[0000-0001-5352-077X]1, Andrii Horkunenko3[0000-0001-8644-0776], Andrii Zozulya1[0000-0003-1582-3088] 1Ternopil Ivan Puluj National Technical University, Department of Computer Science, Ternopil, Ukraine iaroslav.lytvynenko@gmail.com 2National Aerospace University, Kharkiv, Ukraine v.kharchenko@csn.khai.edu 3I. Horbachevsky Ternopil National Medical University, Department of Medical Physics of Diagnostic and Therapeutic Equipment, Ternopil, Ukraine horkunenkoab@tdmu.edu.ua Abstract. 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 distri- bution of the rhythmocardiosignal with increased resolution is confirmed. It was made a decomposition of the statistical estimates of autocorrelation and inter- correlation 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 in- creased 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. Keywords: methods of statistical estimation, probabilistic characteristics, vec- tor of stationary and stationary-related random sequences, electrocardiogram, rhythmocardiosignal, heart rate. 1 Introduction 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 regis- tered 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 Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). ICST-2020 cardiac cycles, and not the whole totality of time intervals between single-phase val- ues 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 resolu- tion, 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 Methods 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 elec- trocardiosignal, and the index l - the reference number of the electrocardiogram with- in 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 Ξ L ( , 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 pT ...T x1 ,..., x p , m1 ,..., m p order p ( p  N ) from the family of vector distribution l1 lp functions Ξ L ( , m) of stationary and stationary random sequences such equality occurs: ( ) ( ) F pT ...T x1 ,..., x p , m1 ,..., m p = F pT ...T x1 ,..., x p , m1 + k ,..., m p + k , l1 lp l1 lp ____  x1 ,..., x p  R, m1 ,..., m p  Z, l1 ,..., l p  1, L , k  Z . (1)   We present formulas that represent the convergence in the root-mean-square sense of the corresponding statistical estimates to the estimated probabilistic characteristics of ____ the vector of Ξ L ( , m) = {Tl ( , m),    Ω, l = 1, L , m  Z } stationary and stationary related random sequences. An estimate that converge in the root-mean-square sense to the distribution function ( ) F pT ...T x1 ,..., x p , m1 ,..., m p of order p ( p  N ) of vector of Ξ L ( , m) stationary l1 lp and stationary related random sequences: (x1 ,..., x p , m1 ,..., m p ) = lK.i→.m. 2K1+ 1   H (x j − Tl ( , m j + k )), K p F pT ...T j l1 lp k = − K j =1 (2) ____  x1 ,..., x p  R, m1 ,..., m p  Z, l1 ,..., l p  1, L , k  Z.   Function H ( x) = 1, x  0, is a Heaviside step function, which is an indicator of a nega- 0, x  0. tive number. In particular, if in the formula (2) p = 1 , that is l1 = l 2 = ... = l p = l , then we will have one-dimensional F1T (x ) = F1T (x, m ) distribution auto-function of stationary random l l sequence Tl ( , m) , for which from the formulas (2) it follows a convergence in the root-mean-square sense: K ____  F1T (x ) = l.i.m.  H (x − Tl ( , k )), x  R, l  1, L , k  Z. 1 (3) l K → 2 K + 1   k =− K An estimate that converge in the root-mean-square sense to the mixed initial moment p function of order s = s : j =1 j ( ) K → 2K1+ 1 Tls ( , m1 + k ) ...  Tls ( , m p + k ), K сsT ...T m1 ,..., m p = l.i.m. 1 p l1 lp 1 p k =− K ____ m1 ,..., m p  Z, l1 ,..., l p  {1, L }, k  Z . (4) 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 с sT T (m1 , m2 ) two stationary and l1 l2 stationary connected random sequences Tl1 ( , m) and Tl2 ( , m) , describing the time distances between single-phase electrocardiosignal samples for l1 -st and l 2 -d its phases, such as: K с2T T (m1 , m2 ) = l.i.m. Tl1 ( , m1 + k )  Tl2 ( , m2 + k ), 1 l1 l2 K → 2 K + 1 k =− K ____ m1 , m2  Z, l1 , l 2  {1, L }, k  Z . (5) If in the formula (5) p = 1 , that is l1 = l 2 = ... = l p = l , then we have the convergence of the estimate in the root-mean-square sense to the one-dimensional initial moment function с sT (m ) s -th order, which is for a stationary random sequence Tl ( , m) is a l constant с sT = с sT (m ) (the initial moment s -th order), that is: l l K ____   Tl s ( , k ), l  1, L  . 1 сsT = l.i.m. (6) l K → 2 K + 1   k =− K 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 с1T = с1T (m ) (mathe- l l matical expectation) stationary random sequence Tl ( , m) , that is: K ____   Tl ( , k ), l  1, L  . 1 с1T = l.i.m. (7) l K → 2 K + 1   k =− K An estimate that converge in the root-mean-square sense to the mixed initial central p function of order s = s : j =1 j ( ) K → 2K1+ 1  Tl (, m1 + k ) − с1   ...  Tl (, m p + k ) − с1  , K s1 sp rsT ...T m1,..., m p = l.i.m. 1 Tl1 p Tl p l1 lp k =− K ____ m1 ,..., m p  Z, l1 ,..., l p  {1, L }, k  Z . (8) 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 rsT T (m1 , m2 ) two stationary and l1 l2 stationary connected random sequences Tl1 ( , m) and Tl2 ( , m) , describing the time intervals between single-phase electrocardiogram samples for l1 -th and l 2 -d its phases, that is: K r2T T (m1 , m2 ) = l.i.m.   T ( , m + k ) − с    T ( , m + k ) − с , 1 l1 l2 K → 2 K + 1  l1 1 1Tl 1   l2 2 1Tl 2  k =− K ____ m1 , m2  Z, l1 , l 2  {1, L }, k  Z . (9) If in the formula (9) s = 2 and p = 1 , that is l1 = l 2 = ... = l p = l , then we will have a convergence of the estimate in the root-mean-square sense to the variance r2T station- l ary random sequence Tl ( , m) , that is: ( ) K ____  Tl ( , k ) − с1T 2 , l  1, L  . 1 r2T = l.i.m. (10) l K → 2 K + 1 l   k =− K The above formulas reflect the convergence in the root-mean-square sense of the statistical estimates to the corresponding probabilistic characteristics of rythmo- cardisignal with increased resolution, and, therefore, the statistical estimates are con- sistent. 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 char- acteristics of the rhythmocardiosignal with increased resolution is to obtain the reali- zations of statistical estimates that can be taken as pproximation to the correspond- ing probabilistic characteristics of the analyzed rhythmocardiosignal. We write down the expressions to calculate the realizations of the corresponding sta- tistical 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  ____ _____  Ξ L (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 pT ...T x1 ,..., x p , m1 ,..., m p of order p ( p  N ) of vector Ξ L ( , m) sta- l1 lp tionary and stationary related random sequences looks like: (x1 ,..., x p , m1 ,..., m p ) = M − M1 + 1   H (x j − Tl (m j + k )), M − M1 p Fˆ pT ...T j l1 lp 1 k =0 j =1 (11)  _____  ____  x1 ,..., x p  R, m1 ,..., m p  1, M 1 , l1 ,..., l p  1, L .     Where M 1 ( M 1  M ) – maximum value of arguments m1 ,..., m p , which is selected depending 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 = l 2 = ... = l p = l , then we will have an expression to calculate the realization of the statistical estimate Fˆ1T (x ) one- l dimensional auto-distribution function F1T (x ) stationary random sequence Tl ( , m) , l that is: M −M  ( ) 1 ____  Fˆ1T (x ) = H x − Tl (k ) , x  R, l  1, L . 1 (12) l M − M 1 + 1 k =0   An expression to calculate the realization of a statistical estimate of a mixed initial p moment function of order s =  s is given: j =1 j M − M1 l1 lp ( ) M − M1 + 1 Tls (m1 + k ) ...  Tls (m p + k ), сˆsT ...T m1 ,..., m p = 1 1 p p 1 k =0 ____ m1 ,..., m p {1, M 1}, 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 сˆsT T (m1 , m2 ) of the covariance function l1 l2 с sT T (m1 , m2 ) two stationary and stationary-related random sequences Tl1 ( , m) and l1 l2 Tl2 ( , m) , that describe the time distances between single-phase samples of electro- cardiosignal for l1 -st and l 2 -d its phases, in particular: M −M  1 сˆ2T T (m1 , m2 ) = Tl1 (m1 + k )  Tl2 (m2 + k ), 1 l1 l2 M − M 1 + 1 k = 0    _____  ____  m1 , m2  1, M 1 , l1 , l 2  1, L  . (14)     If in the formula (13) p = 1 , that is l1 = l 2 = ... = l p = l , then we get an expression to calculate the realization of the statistical estimate с̂ sT the initial moment s -th order l с sT of stationary random sequence Tl ( , m) , in particular: l M −M ____   1 Tls (k ), l  1, L  . 1 сˆsT = (15) l M − M 1 + 1 k =0   If in the formula (15) s = 1 , then we get an expression to calculate the realization of the statistical estimate сˆ1T of the initial moment of the first order с1T (mathematical l l expectation) stationary random sequence Tl ( , m) , that is: M −M ____   1 Tl  (k ), l  1, L  . 1 сˆ1T = (16) l M − M 1 + 1 k =0    An expression to calculate the realization of a statistical estimate of a mixed initial p central function of order s =  s is given: j =1 j M − M 1 +1 ( ) M −M  T (m + k ) − сˆ   ...   T (m + k ) − сˆ  , sp +1   s1 1 rˆsT ...T m1,..., m p =  l 1  l1 lp l 1 1  1  p  Tl1 p Tl p 1 k =0 _____ ____ m1 ,..., m p  {1, M 1}, 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 rsT T (m1 , m2 ) of l1 l2 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 l 2 -d its phases, that is: M −M  1 rˆ2T T (m1 , m2 ) =  T (m + k ) − сˆ    T (m + k ) − сˆ , 1 M − M 1 + 1 k =0  1 1   2  l1 l2 l 1 1Tl l2 2 1Tl _____ ____ m1 , m2  {1, M 1}, l1 , l 2  {1, L } . (18) Since for stationary and stationary random sequences, correlation functions are func- tions of only one integer argument u , which is equal to u = m1 − m2 , then their statis- tical estimates also depend on only one argument u . In this case, assuming the er- godicity of the stationary components of the vector Ξ L ( , m) , then the formula (18) will look like this: M −M  1 rˆ2T T (u ) = rˆ2T T (m1 − m2 ) =  T (k ) − сˆ    T (k + u ) − сˆ , 1 M − M 1 + 1 k =0  1 1   2  l1 l2 l1 l2 l 1Tl l2 1Tl __________ _____ ___ u = 0, M 1 − 1, m1 , m2  {1, M 1}, l1 , l 2  {1, L} . (19) If in the formula (19) u = 0 , а l1 = l 2 = l , then we will have an expression to calcu- late the realization of the variance estimate r2T stationary random sequence l Tl ( , m) , that is: ( ) M ____  Tl ( , k ) − с1T 2 , l  1, L  . 1 (20) rˆ2T = l M − 1 k =1 l   3 Normality hypothesis test of the vector components The most comprehensive information on the probabilistic characteristics of a in- creased-resolution rhythmocardiogram is contained in the distribution function family  ____  ( ) FpTl ...Tl x1 ,..., x p , m1 ,..., m p , p  N, l1 ,..., l p  1, L  of vector Ξ L ( , m) stationary  1 p   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 statisti- cal estimation of multidimensional vector distribution functions Ξ L ( , m) , it is nec- essary 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 spectral- correlation theory, in particular instead of a tedious, computationally complex estima- tion of distribution functions, to apply simpler computational procedures for estimat- ing spectral-correlation characteristics of rhythmocardiosignals with increased resolu- tion. Let's test the hypothesis for normality of the distribution law of components of a vec- tor Ξ L ( , m) . To do this, we apply the Pearson's agreement criterion (  2 -test), that allow to establish consistency (or inconsistency) of empirical and theoretical distribu- tions of vector components Ξ L ( , m) . Empirical distribution of vector components  ____ ______  Ξ L ( , 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 S il the number is calculated hil (empir- ical 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 num- ber M , that is: nil ____ ___ . (21) hil = , i = 1, I , l = 1, L i  M l ____ 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 p il the value is used 2  hil   − pil  I M  2 = i =1  l pi  . (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 hy- pothesis about the consistency of empirical and theoretical distributions is tested. In the case of the normal distribution of the stationary components of the vector Ξ L ( , 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 quan- tile  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 The results of statistical analysis 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 auto- matic formation of rhythmocardiogram with high accuracy, the realization received ____ ______ Ξ3 ( m) = {Tl ( m), l = 1,3 , m = 1,245} of tricomponent vector ____ ______ Ξ3 ( , m) = {Tl ( , m),    Ω, l = 1,3 , m = 1,245} of stationary and stationary- related random sequences. T ( , m) The first component 1 of this vector is a random stationary sequence, which describes the duration P - intervals in the electrocardiosignal for all of its 245 recorded cycles. T ( m) The plot of realization 1 of this component is shown in Figure 1,а. Second T2 ( , m) component of this vector is a random stationary sequence, which describes R the duration - intervals in the electrocardiosignal. T ( m) The plot of realization 2 of the second component is shown in the figure 1,b. T3 ( , m) The third component 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. T1  (m) 86 19 31 T2  (m) T3  (m) 84 18 30 82 29 80 17 78 28 16 27 76 74 15 26 72 25 70 14 24 68 13 23 66 12 22 64 21 62 11 m 60 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 0 20 40 60 80 100 120 140 160 180 200 a) b) c) Fig. 1. Plot of realization: a) T1 (m) the first component T1 ( , m) , which describes the duration P - intervals in the electrocardiosignal; b) T2 (m) the second compo- nent 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. Table 1. The results of application Station- Quantile value The value of re- ary com-  q2 at  = 0.05 alization a random Hypothesis testing ponent result and q = 7 variable  2 number 1 14,067 1,26  2   q2 (confirmed) 2 14,067 1,02  2   q2 (confirmed) 3 14,067 0,49  2   q2 (confirmed) Thus, based on the results of normality hypothesis testing of the distribution of the stationary components of a random vector Ξ L ( , m) by Pearson's criterion, it is found that these results do not contradict the hypothesis for normality of its distribu- tion. Normality of the vector Ξ L ( , m) is the basis for the substantiation of diagnos- tic 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 Ξ L ( , m) of stationary and stationary-related random sequences, it is sufficient to statistically evaluate only the vector ____ С1L = {с1T , l = 1, L } his mathematical expectations according to formula (16) and the l ___ matrix of correlation functions R T = [r2T T (u ), l1 , l2 = 1, L] according to formula (19). l1 l2 5 Choice substantiation of diagnostic features in cardiac rhythm analysis systems by rhythmocardiosignals with increased resolution An important step in the development of information systems of cardiac rhythm anal- ysis is a substantiated choice of diagnostic features set, which will be used for auto- mated procedure for diagnostic decision making. There are mainly two requirements for this set of diagnostic features. The first requirement is the informativeness re- quirement of many diagnostic features, and the second - the requirement of minimali- ty 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 fea- tures. Such informativeness of diagnostic features is determined by two of their char- acteristics, 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 fea- tures. 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 di- mension of diagnostic features space, which, as a consequence, provides the mini- mum computational complexity of algorithms for diagnostic decision making. Let’s substantiate the diagnostic features set to evaluate the state of regulatory mech- anisms 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 mathe- l matical expectations calculated according to expression (16) and an estimates matrix ___ 2T T (u ), l1 , l 2 = 1, L] , which were calculated according ˆ = [rˆ of correlation functions R T l1 l2 to the formula (19). One of the obvious ways to reduce the number of diagnostic fea- tures of the rhythmocardiosignal is to take into account the fact of symmetry ___ ( rˆ2T T (u ) = rˆ2T T (u ), l1, l2 = 1, L ) estimates matrix of correlation functions l1 l2 l2 l1 ___ 2T T (u ), l1 , l 2 = 1, L] ˆ = [rˆ R , indicating that it is sufficient to evaluate only those ele- T l1 l2 ments of the matrix R̂T , what lie on its diagonal and above the diagonal, that is, such ___ ____ an ordered set Rˆ T = [rˆ2T T (u ), l1 = 1, L, l 2 = l1 , L] . On the diagonal of this matrix, when l1 l2 l1 = l2 , autocorrelation functions estimates are placed, and the elements of the matrix R̂T , which are placed above its diagonal, that is, when l1  l2 , are estimates of cross- correlation functions. ___ 2T T (u ), l1 , l 2 = 1, L] without losing the informativeness, ˆ = [rˆ Therefore, the matrix R T l1 l2 ___ ____ we can replace with the triangular matrix Rˆ T = [rˆ2T T (u ), l1 = 1, L, l 2 = l1 , L] . l1 l2 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 ___ ____ 2T T (u ), l1 = 1, L, l 2 = l1 , L] , in particular, by using a discrete Fourier transform ˆ = [rˆ R T l1 l2 of autocorrelation estimates and cross-correlation functions from this matrix. That is, ___ ____ 2T T (u ), l1 = 1, L, l 2 = l1 , L] correlation functions ˆ = [rˆ instead of a triangular matrix R T l1 l2 ___ ____ can be used by a triangular matrix Sˆ T = [ Sˆ2T T ( ), l1 = 1, L, l 2 = l1 , L] , elements of l1 l2 which are Fourier-images of the corresponding estimates of the correlation functions from the matrix R̂T . That is, Fourier-images from the matrix Ŝ T are calculated as follows: M 1 −1 − j 2u  __________ ___ ____ Sˆ2T T ( ) = rˆ2T T (u )  e M1 ,  = 0, M1 − 1, l1 = 1, L, l2 = l1, L, j = − 1 . (24) l1 l2 l1 l2 u =0 Based on the Bessel inequality, as diagnostic features we will not choose the whole  __________  set Sˆ2T T ( ),  = 0, M 1 − 1  samples of functions Sˆ 2T T ( ) , but only a subset of  12  l l l1 l2  __________  their first M 2 ( M 2  M 1 ) samples Sˆ 2T T ( ),  = 0, M 2 − 1 , which contribute to  12  l l the full energy of evaluation rˆ2T T (u ) correlation function is not less than 95%. l1 l2 Here is an example of the statistical evaluation of vector elements ____ С1L = {с1T , l = 1, L } mathematical expectations, elements of the matrix of correlation l  ___  functions RT = r2T T (u ), l1, l2 = 1, L  and matrix elements of the Fourier-images   l1 l2  ___ ____  Sˆ T =  Sˆ2T T ( ), l1 = 1, L, l2 = l1, L by one realization  12  l l ____ ______ Ξ3 ( m) = {Tl ( m), l = 1,3 , m = 1,245} tricomponent vector ____ ______ Ξ3 ( , m) = {Tl ( , m),    Ω, l = 1,3 , m = 1,245 } stationary and stationary-related random sequences. Figure 3 shows the plot of realization rˆ2T T (u ) statistical estimation of autocorrela- 11 tion function r2T T (u ) ( l1 = l2 = 1 ) of three vector components Ξ3 ( , m) . 11 Table 2 presents the statistical evaluation results of the mathematical expectations of the stationary components of the vector Ξ3 ( , m) . rˆ1 (u ) rˆ (u ) rˆ (u ) 3T1T1 5 2T1T1 15 T1T1 4 2 10 3 1 2 5 1 0 -0 -0 -1 -1 -5 -2 -2 u -3 u -10 u -4 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 a) b) c) Fig. 3. Plot of realization: a) rˆ2T T (u ) statistical estimation of autocorrelation function 11 r2T T (u ) ( l1 = l 2 = 1 ), the first component T1 ( , m) , which describes the duration P - 11 intervals in the electrocardiosignal; b) r2T T (u ) statistical estimation of autocorrela- 2 2 tion function r2T T (u ) ( l1 = l 2 = 2 ) the second component T2 ( , m) , which describes 33 the duration R - intervals in the electrocardiosignal; c) rˆ2T T (u ) statistical estimation 33 of autocorrelation function r2T T (u ) ( l1 = l2 = 3 ) third component T3 ( , m) , which 33 describes the duration T - intervals in the electrocardiosignal Table 2. The statistical evaluation results Stationary component Statistical estimation realization value of math- number ematical expectation 1 с1T = 14,88 1 2 с1T = 25,02 2 3 с1T = 73,82 3 Figure 4 shows the graphs of realization of statistical estimates of the cross- correlation functions of the vector components Ξ3 ( , m) . 1.5 rˆ3T T (u ) 2 rˆ3T T (u ) 3 rˆ3T T (u ) 11 11 11 1.0 2 0.5 1 1 0.0 -0 0 -0.5 -1 -1.0 -1 -2 -1.5 u u -3 u -2 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 a) b) c) Fig. 4. Plot of realization: a) rˆ2T T (u ) statistical estimation of the cross-correlation 12 function r2T T (u ) ( l1 = 1, l 2 = 2 ) first T1 ( , m) and second T2 ( , m) vector compo- 12 nents Ξ3 ( , m) ; b) rˆ2T T (u ) statistical estimation of the cross-correlation function 13 r2T T (u ) ( l1 = 1, l 2 = 3 ) first T1 ( , m) and third T3 ( , m) vector components 13 Ξ3 ( , m) ; c) rˆ2T T (u ) statistical estimation of the cross-correlation function r2T T (u ) 13 13 ( l1 = 1, l 2 = 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 Sˆ ( ) statistical estimation of cross-spectral power density S ( ) 2T1T2 2T1T2 ( l1 = 1, l 2 = 2 ) first T1 ( , m) and second T2 ( , m) vector components Ξ3 ( , m) . a) b) c) Fig. 5. Plot of realization: a) S 2T T ( ) statistical estimation of power spectral density ˆ 12 S 2T T ( ) ( l1 = l 2 = 1 ) the first component T1 ( , m) , which describes the duration P - 12 intervals in the electrocardiosignal; b) Sˆ 2T T ( ) statistical estimation of power spec- 2 2 tral density S 2T T ( ) ( l1 = l 2 = 2 ) the second component T2 ( , m) , which describes 2 2 the duration R - intervals in the electrocardiosignal; c) Sˆ 2T T ( ) statistical estimation 33 of power spectral density S 2T T ( ) ( l1 = l 2 = 3 ) third component T3 ( , m) , which 33 describes the duration T - intervals in the electrocardiosignal a) b) c) Fig. 6. Plot of realization: a) S 2T T ( ) statistical estimation of cross-spectral power ˆ 12 density S 2T T ( ) ( l1 = 1, l 2 = 2 ) first T1 ( , m) and second T2 ( , m) vector compo- 12 nents Ξ3 ( , m) ; b) Sˆ 2T T ( ) statistical estimation of cross-spectral power density 13 S 2T T ( ) ( l1 = 1, l2 = 3 ) first T1 ( , m) and third T3 ( , m) vector components 13 Ξ3 ( , m) ; c) Sˆ 2T T ( ) statistical estimation of cross-spectral power density S 2T T ( ) 2 3 2 3 ( l1 = 2, l2 = 3 ) second T2 ( , m) and third T3 ( , m) vector components Ξ3 ( , m) . 6 Conclusions The methods of statistical estimation of probabilistic characteristics of rhythmocardi- osignal with increased resolution on the basis of model in the form of a vector of sta- tionary and stationary related random sequences are developed in the paper. Conduct- ed statistical experiments confirmed the hypothesis for the normality of the law of distribution of components of the vector rhythmocardiosignal. 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