Open access peer-reviewed chapter

Flicker-Noise Spectroscopy Method in the Problem of Diagnosing the State of the Cardiovascular System

By Abdullayev Namiq Tahir and Ahmadova Khadija Ramiz

Submitted: November 12th 2019Reviewed: March 25th 2020Published: May 11th 2020

DOI: 10.5772/intechopen.92264

Downloaded: 125

Abstract

In the field of research of the cardiovascular system, mainly analysis methods that are strictly mathematically applicable to stationary signals are distinguished; however, nonstationary signals prevail in medical practice, the statistical properties of which vary with time. Often they consist of short-term high-frequency components, followed by long-term low-frequency components. Given this nature of bioelectric potentials, and in particular electrocardiographic signals, the most suitable for their analysis may be the nonlinear dynamics method with the calculation of quantitative characteristics of chaos. This possibility is presented by the flicker-noise spectroscopy method, which takes into account the intermittency effect in a complex dynamic system when sections of chaotic bursts and jumps alternate with relatively long sections of a laminar nature. The analysis of signals of such a dynamic nature is usually based on the use of flicker-noise spectroscopy.

Keywords

  • flicker-noise spectroscopy
  • electrocardiographic signals
  • analysis
  • intermittency
  • diagnostics

1. Introduction

The flicker-noise spectroscopy method is proposed as a general phenomenological (non-model) approach to the analysis of chaotic signals of different nature. The essence of flicker-noise spectroscopy is to give informational significance to the correlation relationships that are realized in sequences of signal irregularities—bursts, jumps, and kinks of derivatives of various orders—as carriers of information about changes occurring at each spatiotemporal level of the hierarchical organization of the dynamic system under study. The autocorrelation function ψτis used as a basic image for extracting information from complex signals in the flicker-noise spectroscopy method [1, 2].

To classify information, this function is not analyzed but some of its transformations (“projections”), such as power spectrum Sf, where fis the signal frequency, and the difference moment (“transition structure function”) Φ2τof the second order. The information extracted from the analysis of dependencies Sfand Φ2τ, built on the basis of time series Vt, has the meaning of correlation times or parameters, characterizing the loss of correlation relationships (“memory”) for the irregularities under consideration such as bursts and jumps.

Moreover, only irregularities of the type of jumps of dynamic variable Vtcontribute to the formation of dependence Φ2τ, and jumps and bursts (outbursts) of chaotic series Vtcontribute to the formation of Sf.

The solution to the problem of predicting the evolution of a complex system and, above all, the search for precursors (precursors) of catastrophic changes in it is associated with the most dramatic changes in dependencies Sfand Φ2τp=23calculated on the basis of high-frequency and low-frequency components Vt.

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2. Splitting an electrocardiographic signal into low-frequency and high-frequency components

The behavior of the electrocardiographic signal, reflecting the functional state of the cardiovascular system, is quite complicated and has the character of randomness.

The most general form of evolution in dynamic variable Vtifor the ith space–time level of the electrocardiographic signal is presented in the form of intermittency, when not all intervals on the time axis are informationally equivalent. Such dynamics of the electrocardiogram (ECG) is characterized (Figure 1) by relatively weak changes in the variable over relatively long time intervals—“laminar phases” with characteristic durations of Tiand sudden interruptions of such evolution by abrupt changes in the value of the dynamic variable in the short intervals of duration τiτi<Ti.

Figure 1.

The dynamics of the electrocardiogram.

Each such abrupt change in the values of a dynamic variable then ends up with values in the subsequent “laminar” section. The magnitude and duration of such jumps, surges, and “laminar” sections are specific for each of the cardiovascular systems, causing a certain contribution to the corresponding power spectrum.

In this case, the studied signal Vtis conveniently represented as the sum of the two terms: the singular term VSt, which is formed only by bursts of the dynamic variable, and the regular term VRt=VtVSt, which is formed after subtracting bursts from the presented signal and determined by the jumps of the dynamic variable and “laminar phases.”

The analysis of the electrocardiogram shows that it corresponds to the described dynamics, when bursts in the form of QRS complexes alternate with rather small jumps in the form of Pand Tteeth and extended phases in the form of an isoline.

The information contained in Sfand Φ2τis different, so in order to determine the adequate parameters of the structure under study, it is necessary to analyze the dependencies logSf=Flogfand logΦ2τ=Flogτ.

Let Vtdenotes the dynamic variable, characterizing the ECG signal. We apply the proposed method of splitting the dynamic signal into low-frequency VRtand high-frequency VStcomponents. This method is built by analogy (Figure 2) with the solution of the diffusion equation and is based on the following “relaxation” procedure:

Figure 2.

Typical curve for functionΦ2τ, characterizing a chaotic signalVtwithout a resonant component.

  1. Set the value V1,,VNof signal Vwith a step of discreteness Δt.

  2. Calculate V=1Nk=1nVkand put VRVkV,k=1,,N.

  3. We calculate

    ψmτ=1Nmτk=1NmτVkVk+mτ,mτ=τ/ΔtE1

    where Φ2τ=2ψ0ψτ,σ2=ψ0,τ=mτΔt

    mτ=0,1,,M1,MNE2

  4. We plot Φ2τin bilogarithmic coordinates.

    The asymptotic representation for Φ2τis

    Φ2τ=τ2H1,ifτ<<T12σ2,ifτ>>T1.E3

  5. We take for T1the value τ, at which logΦ2τbegins to stabilize to a constant equal to log2σ2.

  6. Choose a sequence of small τkk=1k0k020,τk<<T1, and construct a regression y=ax+bb=0;y=lnΦ2τ,x=lnτ,a=2H1.

    According to the least squares method (LSM) estimate â, we calculate the estimate H1=â/2.

  7. We calculate

    D=σ2Γ21+H1T1E4

    To calculate Γxfor x=1+H1, put n=103and represent Γxin the form

    Γx=Γx+1x=Γx+2xx+1==Γx+nxx+1x+nt.E5

    The value Γz(we have z=x+n) is calculated by the formula

    Γz=expz12lnzz+12ln2πE6

    with an error of order z1103n103z=x+n.

  8. Denote by ∆t and ∆τ the steps of discreteness in t and τ, and

    ω=DΔτ/Δt2E7

    choose ∆τ so that ω<1/2.

  9. Put MN1, and construct an iterative procedure according to j=0,1,,, according to which the value Vkj+1at the jth step is calculated through the value Vkjaccording to the formula

    Vkj+1=ωVk+1j+ωVk1j+12ωVkjE8

    at j=0we set Vkj=Vk; at k=1and k=M, the values of Vkj+1are calculated by the formulas

    V1j+1=12ωV1j+2ωV2j,VMj+1=12ωVMj+2ωVM1j.E9

    The procedure stops at step j=j0, in which.

    Vkj0+1Vkj0<ε, for Vk=1,,M,k=1,2,,N,

    where εis the given number (e.g., ε=10m+1, where 10mis the error in setting the initial values Vk).

  10. The values Vkj0determine the low-frequency component VRt. Then VtVRt=VStis the high-frequency component of the signal Vt.

The described signal smoothing procedure is focused on minimizing the “high-frequency” information in the “low-frequency” part VRtof the signal and vice versa, minimizing the “low-frequency” information in the “high-frequency” part VStof the signal. This conclusion follows from the diffusion nature of the partial differential equation used

∂V∂τ=2Vt2,E10

represented as a difference equation

Vkj+1VkjΔτ=Vk+1j+Vk1j2VkjΔt2,E11

corresponding to the simplest difference scheme for numerically solving Eq. (10). From (11) we obtain

Vkj+1=Vkj+ΔτΔt2Vk+1j+Vk1j2Vkj.

In notation ω=Δτ/Δt2, the last equation is written in the form (8). From the theory of stability of difference schemes, it is known that this difference scheme will be absolutely stable at ω<1/2.

Such a relaxation procedure realizes the maximum rate of entropy generation and uses the relationship of entropy and Fisher information, which is a quantitative measure of the heterogeneity of the distribution density of the analyzed data array.

3. Parameterization of the singular component of the ECG signal

The procedure for parameterizing the singular part of the signal consists of the following sequence of steps [3]:

  1. Let Vtbe represented as a sum

    Vt=VRt+VSt.

  • Let tk=kΔtk=1N,t0=0,tN=Tpoints of task Vtby 0Twith a certain step of discreteness Δt;N=T/Δt.

    We calculate the average value:

    Vt=1Nk=1NVtkE12

    In what follows, we will assume that Vt=0, i.e., signal Vt, is stationary.

  • For stationary signal Vt, the power spectrum Sf(Fourier transform of the autocorrelation function ψτ) coincides with Scf(cosine Fourier transform of ψτ).

    We set M from condition 43MN(in practice, take M close to N). We assume that M is an even number. For a time delay of mτ=0,1,,M1, we calculate the autocorrelator:

    ψmτ=1Nmτk=1NmτVkVk+mτE13

  • Let f=q/MΔt.

    We calculate the power spectrum Sf=Scf

    Scf=1ΔtScqScq=ψ0+ψM21q+2m=1M21ψmcos2πqmM,q=01M1E14

  • We will construct the image Sf(or Sf, if in some frequencies Sf<0) in a bilogarithmic scale (Figure 3).

    From Figure 3 we find the frequency f=f0, starting from which Sfceases to stabilize around a certain constant S0.

  • Let ff¯be the frequency interval in the region of the graph Sf(or Sf), preceding the first strong peak of the power spectrum Sf, corresponding to “irregularity-burst.” We assume that Sfincreases at fff¯and Ss0—a certain number from interval SfSf¯.

  • We calculate the autocorrelator ψS,Rτaccording to the formula.

    ψS,Rτ=1Nmτk=1NmτVSkVSk+mτ+VRkVSk+mτ+VSkVRk+mτ,mτ=01M1E15

  • We calculate the singular component SSfof spectrum Sfby the formula

    SSf=1ΔtSSqSSq=ψS,R0+ψS,RM21q+2m=1M21ψS,Rmcos2πqmMq=01M1E16

  • For parameterization SSf, we approximate this function by an interpolation expression:

    ŜSfSS01+2πfT0n0E17

    Parameter T0 by formula (17) will be determined by Algorithm 1, assuming that the “experimental” spectrum SSfis calculated by formula (16).

  • Figure 3.

    Typical curve for functionSf, characterizing a chaotic signalVtwithout a resonant component.

    Algorithm 1.

    9.1. Using the spectrum graph (Figure 3), we introduce the constants f0,f¯,f¯,SS0, as well as the threshold value RSS=1010.

    9.2. We set SS0=SS0and evaluate the parameters T0, n0.

    Build a regression

    y=ax+b,

    where

    y=lnSs0Ssf1,x=ln2πf,a=n0,b=n0lnT0,

    and estimate the coefficients aand busing the least squares method (least squares) for sample ymxmwith ymand xm, corresponding to frequencies fm=mMΔtm=01M1.

    We calculate the residual sum of squares

    RSS1=m=0M1ymâxm+b̂2,

    where âand b̂LSM are the estimations of parameters aand b.

    If RSS1<RSS, then RSSRSS1, n̂0=n0, T0=T̂0, where n̂0=â, T̂0=expb̂/â.

    9.3. We set n0=n0, SS0=SS0and evaluate T0.

    Build a regression

    y=ax+bb=0,

    where

    y=Ss0Ssf11/n0,x=2πf,a=T0.

    We calculate RSS2=m=0M1ymâxm2.

    If RSS2<RSS, then RSSRSS2and T0=T̂0, where T̂0=â.

    9.4. We set T0=T0, n0=n0and evaluate SS0.

    Build a regression

    y=ax+bb=0,

    where

    y=Ssf,x=11+2πfT0n0,a=SS0

    We calculate RSS3=m=0M1ymâxm2.

    If RSS3<RSS, then RSSRSS3, SS0=ŜS0, where ŜS0=â.

    9.5. We set SS0=SS0, T0=T0and evaluate n0.

    Build a regression

    y=ax+bb=0,

    where

    y=lnSs0Ssf1,x=ln2πfT0,a=n0.

    We calculate RSS4=m=0M1ymâxm2.

    If RSS4<RSS, then n0=n̂0, where n̂0=â.

    As a result of Algorithm 1, we obtain the three parameters SS0=SS0, n0=n0, T0=T0, characterizing the interpolation expression (17) for the singular component of the spectrum SSf.

    4. Informative diagnostic parameters of the singular component of the ECG signal

    During a computational experiment, electrocardiographic signals with a normal state of the cardiovascular system and pathological signals (“tachycardia,” “arrhythmia,” and “atrial fibrillation”) were analyzed. We used data from the public site www.PhysioNet.org for the II standard lead. The ECG removal parameters (type of lead, sampling frequency, time, number of samples, and signal amplitude) are included in the sample. The sampling rate for various samples varies from 125 to 1000 Hz. The values of the presented samples, taking into account the sign discharge, correspond to the use of a 12-bit ADC.

    In Figures 46, the graphs of the spectral power of the ECG signal for the norm, the singular component of this signal, and the estimation of the singular component of this signal are presented.

    Figure 4.

    Graphs of the spectral power of the ECG signal for the norm.

    Figure 5.

    Graphs of the singular component of this signal.

    Figure 6.

    Graphs of the estimation of the singular component of this signal.

    In Figures 79, as an example, similar relationships for an ECG signal with a range of “atrial arrhythmia” are presented.

    Figure 7.

    Graphs of the spectral power of the ECG signal for the “atrial arrhythmia.”

    Figure 8.

    Graphs of the singular component of the “atrial arrhythmia.”

    Figure 9.

    Graphs of the estimation of the singular component of the “atrial arrhythmia.”

    For all the considered states of the cardiovascular system, the same dependencies were obtained, and based on the obtained dependencies, informative parameters of the singular component of the ECG signals were calculated (Table 1).

    ECG signaln0T0Ss(0)
    Norm0.34140.0042437.8090
    Atrial fibrillation0.38360.0036334.3640
    Ventricular tachycardia0.41230.0032197.3580
    Atrial arrhythmia0.40130.005943.7105

    Table 1.

    Informative parameters of the singular component of ECG signals.

    The high specificity of Sfpatterns obtained in the study of the cardiovascular system in the norm with the indicated pathologies can be used to diagnose diseases. Dependence Sfbuilt on the basis of different ECGs and the corresponding informative parameters obtained by them differ from each other, which gives reason to consider these dependencies as patterns characterizing the condition of the patient under study. The obtained informative parameters can be considered as distinguishing features for the differential diagnosis of cardiovascular diseases (e.g., using artificial neural networks).

    This approach shows the possibility of flicker-noise spectroscopy as a method that allows you to establish significant differences in the original, visually not very different, ECG signals.

    5. Parameterization of the regular part of the ECG signal and determination of informative diagnostic parameters

    The determination of the parameters of a chaotic signal given on a limited interval T is set on the basis of the flicker-noise spectroscopy method, taking into account the contributions of the “resonant components” to the autocorrelation function [4].

    Ψτ=VtVt+τ,E18

    and, therefore, to cosine conversion

    Scf=T/2T/2Ψτcos2πfτE19

    and second-order difference moment

    Φ2τ=VtV(t+τ2.E20

    Here Vtis a stationary signal Vt=0, and is a symbol of the average value.

    The developed method of signal parametrization is based on the fact that the introduced “irregularities-bursts” and “irregularities-jumps” contribute to various spectral regions of dependence Sf.

    In fact, the first step in the parameterization of irregularities was to isolate the “burst” (singular), most “high-frequency” (the so-called “flicker-noise tail”) component of the signal irregularities in the spectral dependence Sf.

    Based on the remaining (after subtracting the “burst” contribution) spectral dependence, we can now determine the structure function Φ2τ, which contains the contributions from the “jump” and “resonance” components that slowly change against its background. The next steps are to parameterize the “higher frequency” (of those remaining) “hopping” (regular) component using the least squares method.

    It must be borne in mind that when solving the signal parametrization problem under consideration, problems arise due to the limited averaging interval T. For this reason, in particular, it is the “experimental” dependence V(t) constructed on the basis of observed signal S(f) that may turn out to be negative in some frequency intervals. Therefore along with four in such cases, S(f) is introduced into consideration.

    The procedure for parameterizing the regular part of the signal is presented below in the form of the following sequence of operations:

    1. From the extreme spectrum S(f), we subtract the singular component Ss(f) calculated by the interpolation formula (we denote the result by SrR(f))

      SrRf=SfSsfE21

    The resulting difference characterizes the contribution of the “resonant” components Srfand the “irregularities-jumps” SRfto the general dependence Sf. If it turns out that SrRf<0in some frequency intervals, we assume SrRfSrRf.

  • Take the inverse cosine Fourier transform of SrRf

    ψrRτ=20fmaxSfcos2πfτdf,ττ=T/4E22
    fmax=14Δt,τ=kΔτk=1k0,Δτ=T/4k0,k0=500.

    Put a=0,b=fmax,h=fmax/n,n=100, SrRfcos2πfτ=gfτand apply the trapezoid formula:

    abgfτdf=hgaτ2+ga+hτ+ga+2hτ++gbhτ+gbτ2

  • We calculate

    ΦrR2τ=2ψrR0ψrRτ,τ=kΔτk=1k0
    .

  • Put Φ˜r2τ=Φ˜rR2τ.

  • We denote

    Φ˜2τ=Φr2τ+ΦR2τ.E23

    where ΦR2τis given by the interpolation formula:

    ΦR2τ=2σ121Γ2H1+1τT12H1,τ<<T1,2σ121Γ1H1τT1H11expτT12E24

  • Compare the experimental structural function Φ2τ, determined by the formula

    Φ2τ=2ψ0ψτ,E25

    where

    ψmτ=1Nmτk=1NmτVkVk+mτmτ=τ/ΔtE26

    with function Φ2τdetermined by formula (20) using the least squares method.

    • We set RSS=1010, T1=T1

    • A preliminary estimate T1of parameter T1can be obtained using the asymptotic representation of structure function Φ2τ(Figure 10).

    1. The value Φ2τis taken as T1for small delays, at which Φ2τ2σ2takes the maximum value Φ2τ2σ2.

    2. We estimate parameters σ1,H1at τ<<T1.

    3. Build a regression

    Figure 10.

    Graph of functionΦ2τin bilogarithmic coordinates.

    y=ax+b,

    1. where

    y=lnΦ2τΦr2,x=lnττ1,a=2H1,b=2lnσ1Γ2H1+1.

    1. LSM-estimates âand b̂are obtained on the basis of sequence τk,k=1k1close to τ=0, using representation (21) for Φ̂2τ.

    2. We calculate

    RSS1=k=1k1ykâxk+b2,

    • where ykand xkcorrespond to delays τk.

    • If RSS1RSS, then go to Section 6.5.

    • Otherwise, set RSSRSS1, σ1=σ̂1, and H1=Ĥ1, where Ĥ1=â2and σ̂1=Γ2Ĥ1+1expb̂/2.

    • Given σ1=σ1, we estimate H1,T1at τ<<T1

    • Build a regression

    y=ax+bb0,

    • where a=2H1, b=lnΓ2H1+12H1lnT1, y=lnΦ2τΦr22σ12, x=lnτ.

    • LSM grades âand b̂are obtained by sequence τk,k=1k1.

    • We calculate

    RSS2=k=1k1ykâxk+b̂.

    • If RSS2RSS, then go to Section 6.5.

    • Otherwise, set H1=Ĥ1, T1=T̂1, where Ĥ1=â2, T̂1=Γ2H1+112H1expb2H1.

    • Given σ1=σ1, H1=H1, we estimate T1at τ<<T1.

    • Build a regression

    y=ax+bb=0,

    1. where

    y=lnΦ2τΦr2τ2σ12/Γ2H1+1,x=τ,a=1T1.

    • In LSM, a score of âwill be obtained by sequence τk,k=1k1k1<<k0.

    • We calculate T1=1/â

    • We calculate RSS3=k=1k1ykâxk2.

    • If RSS3RSS, then go to Section 6.5.

    • Otherwise, we set RSS=RSS3, T1=T̂1.

    • Given H1=H1, T1=T1, we estimate σ1at τ>>T1.

    • Build a regression

    y=ax+bb=0,

    • where y=Φ2τΦr2, x=1Γ1H1τT1H11expτT12, a=2σ1.

    • We calculate the least squares method (LSM) estimation by sequence τk,τk=Tkk=1k1.

    • We calculate σ̂1=â/2.

    • We calculate RSS4=k=1k1ykâxk2.

    • If RSS4>>RSS, then go to Section 6.5.

    • Otherwise, we set RSS=RSS4, σ1=σ̂1.

    • 6.5. Suppose RSS, σ1, H1, T1.

    • As a result of the proposed algorithm, we obtain the three parameters σ1=σ1, H1=H1, and T1=T1, characterizing the interpolation expression (21) for ΦR2τ.

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    6. Informative diagnostic parameters of the regular component of the ECG signal

    Using the above algorithm, we obtained the graphs of functions Φ2τin bilogarithmic coordinates for the normal state of the cardiovascular system and a number of “catastrophic” arrhythmias (ventricular tachycardia, atrial fibrillation, atrial arrhythmia). An example is given of such a dependence for the state of the cardiovascular system—“ventricular tachycardia” (Figure 11) and atrial arrhythmia (Figure 12). When conducting a computational experiment, we used the experimental data from the publicly available website www.PhysioNet.org.

    Figure 11.

    Dependence logΦ(2)(τ) for ventricular tachycardia.

    Figure 12.

    Dependence logΦ(2)(τ) for atrial rhythm.

    For the considered conditions of the cardiovascular system, on the basis of the obtained dependencies, the informative parameters of the regular component of the ECG signals were calculated (Table 2).

    No.ECG signalσ1H1T1
    1Norm0.5511.13315.080
    2Atrial fibrillation0.43511.3880.0640
    3Ventricular tachycardia0.5110.9130.6840
    4Atrial arrhythmia0.20811.29811.560

    Table 2.

    Informative parameters of the regular component.

    Thus, for the considered functional conditions of the cardiovascular system, three informative parameters n0, T0, Ss(0) for the singular component of the ECG signal and three informative diagnostic parameters σ1, H1, T1 for the regular component of the ECG signal were obtained by flicker-noise spectroscopy.

    A complex of six diagnostic parameters can be used to diagnose catastrophic conditions of the cardiovascular system (e.g., using an artificial neural network, where these parameters are considered as input data).

    7. Fluctuation dynamics of electrocardiograms and the choice of sampling frequency of the studied signals

    In the general case, when analyzing a complex chaotic signal measured at a certain sampling frequency fd, a set of the indicated parameters is determined that characterizes the correlation interconnections in the sequences of irregularities-jumps and irregularities-bursts characteristic of a given signal determined with a sampling frequency of fd. Thus, one of the main factors allowing to realize the allocation of the contribution of irregularities to the analyzed real signals is the variation of the used frequencies fd. If the analyzed time series is obtained at a sufficiently high sampling frequency fd, then the analysis of dependencies Φ2τand S(f), calculated on the basis of time series obtained from the initial time series with a decreasing sampling frequency, allows us to estimate the measure of “stability” of parameters σ1, T1 and H1 (for Φ2τ) and the measure of variability of parameters Ss0, T0 and n0 (for S(f)).

    The high specificity of dependencies Φ2τand S(f) obtained by analyzing the state of complex systems can be used to diagnose diseases, as well as a combination of these parameters for their classification. We analyzed the four types of ECG signals—normal and cardiac “catastrophic” arrhythmias that directly threatened the patient’s life, ventricular tachycardia, atrial fibrillation, and atrial arrhythmia. To identify the characteristics of the analyzed signals, it is necessary to evaluate the entire set of digitized data of V(t) electrocardiograms for the indicated states of the cardiovascular system. When conducting the computational experiment, the experimental data from the public site www.PhysioNet.org were used.

    The signals were taken from the II standard lead for ∼60 s with a sampling frequency of fd = 500 Hs and containing N = 29,859 values. Thus, a time series of ECG signals was obtained at a sufficiently high sampling frequency of fd, since it can be used to obtain a set of new time series at sampling frequencies of less than fdtimes.

    The results of the corresponding analysis for the indicated functional conditions of the cardiovascular system at a sampling frequency of ECG signals fd = 500 Hs are shown in Table 3.

    No.ECG signalSingular componentRegular component
    Ss(0)T0n0σ1H1T1
    1Norm437.800.00420.34140.5511.13315.080
    2Ventricular tachycardia197.3580.00320.41230.5110.9130.6840
    3Atrial fibrillation334.3640.00360.38360.43511.3880.0640
    4Atrial Arrhythmia43.71050.00590.40130.20811.29811.560

    Table 3.

    Informative diagnostic parameters for various functional conditions of the cardiovascular system.

    We will carry out a comparative analysis of informative parameters for the two states of the cardiovascular system: normal (Table 4) and ventricular tachycardia (Table 5) for sampling frequencies fd = 500 Hs and fd = 250 Hs.

    fd, HsNSingular componentRegular component4S0N
    Ss(0)T0n0σ1H1T1
    50029.859437.800.00420.34140.5511.13315.0800.05
    25014.930403.720.00280.41870.504410.8451.380.08

    Table 4.

    Norm.

    fd, HsNSingular componentRegular component4S0N
    Ss(0)T0nσ1H1T1
    50029.859197.3580.00320.41230.518010.9130.68400.026
    25014.930175.800.00340.34460.51715.2000.3400.10

    Table 5.

    Ventricular tachycardia.

    From the obtained tables, it follows that with increasing sampling frequency fd, the high-frequency contribution to the power spectrum S(f) increases due to the inclusion of “bursts” in the analyzed signal corresponding to the increased frequency fd. In this case, changes in dependence Φ2τalso occur at small τ, which are caused by the contribution of local changes in the values of the “laminar” signal sections. Therefore, with an increase of fd, parameters T0 and n0, characterizing the high-frequency region of dependence S(f), and parameters H1 and T1, characterizing the dependence of Φ2τfor small τ, change. The value of parameter σ1 and the nature of spectral dependence S(f) change to a much lesser extent. Small variations in the standard deviation parameter σ1 indicate a smaller dependence of function Φ2τon fd. At the same time, the signal analysis in flicker-noise spectroscopy reveals the dynamics of changes in parameters H1 and T1 at small τ, as well as parameters T0 andn0, characterizing dependence S(f) in the high-frequency region. Since dependence S(f) is determined by the number of Mterms in a discrete expression for S(f), it is convenient to use normalized expressions obtained by multiplying S(f) by a factor of 1/M=4/Nwhen changing the sampling frequencies. With this normalization, functional differences in dependence S(f), due to the use of signals measured at different sampling frequencies, are detected more explicitly.

    Thus, when analyzing a complex chaotic signal during flicker-noise spectroscopy, a set of parameters is determined that characterize the correlation relationships in the sequences of irregularity-jumps and irregularity-bursts characteristic of this signal, determined with a sampling frequency of fd. The analysis of dependencies Φ2τand S(f), calculated on the basis of time series with decreasing sampling frequency, allows you to evaluate the measure of “stability” of parameters σ1, T1, and H1, determined on the basis of Φ2τ, and the measure of variability of the parameters Ss(0), T0, and n0, concerning dependence S(f).

    8. The use of neural network technology in flicker-noise spectroscopy of an electrocardiogram

    Based on a computational experiment, dependencies were obtained for the normal state of the cardiovascular system and a number of “catastrophic” arrhythmias (ventricular tachycardia, atrial fibrillation, atrial arrhythmia). We used the experimental data from the public website www.PhysioNet.org.

    As a result of analyzing the power spectrum S(f), informative parameters were obtained for the singular component of the ECG signal: T0, determining some characteristic time within which the measured dynamic variable is interconnected Vti; n0, dimensionless parameter that effectively determines how this relationship is lost as frequencies decrease to 1/2πT0; and s0, contribution to the power spectrum S(f), determined by the most high-frequency singular component [5].

    The parameterization of the regular component of the ECG signal is carried out using expression Φ2τwith parameters T1, τ1, and H1. In this case, parameter T1determines the characteristic time at which the values of the dynamic variables Vtido not correlate. To obtain reliable values of variance σ12, it is necessary to calculate it at time intervals exceeding T1. In this case, parameter H1shows by what law the relationship between the quantities Vtimeasured at different time instants is lost—the Hurst exponent.

    Thus, when analyzing a complex chaotic signal, which is an ECG signal, we consider a set of six parameters, characterizing the correlation relationships in the sequences of irregularities—“jumps” and irregularities—“bursts” inherent in this signal.

    9. The choice of artificial neural network and its characteristics

    The obtained values of the parameters of the singular and regular component of the ECG signals can be used for differential diagnosis of the functional state of the cardiovascular system using artificial neural networks, where these parameters are considered as input data.

    For the computational experiment, a perceptron three-layer network with direct connections was chosen (Figure 13).

    Figure 13.

    The structural diagram of the proposed artificial neural network.

    To train the neural network, the backpropagation algorithm was used. The training time was about 240 s, the maximum network error was about 0.05, and the degree of training was about 0.01.

    To recognize the pathologies of the cardiovascular system, a modular version of the structure of the construction of neural network blocks can be used (Figure 14).

    Figure 14.

    A modular version of the construction of a neural network for recognition of pathologies (the number of input parameters, the number of neurons in the intermediate layer, the number of pathologies analyzed).

    The structure includes several parallel neural network modules, built on the basis of the structure of a multilayer perceptron. The advantage of this structure is the concentration of resources of each module on the recognition of only one pathology, which helps to reduce the likelihood of an error in the wrong conclusion for the whole system. In addition, the functionality of an artificial neural network is expanded by increasing the number of neural network modules to recognize new pathologies without retraining the entire system.

    The main factor that allows one to distinguish the contribution of irregularities to the analyzed electrocardiographic signals is the variation of the used sampling frequencies fdof the real signal. An analysis of the dependencies of the power spectrum and the second-order difference moment calculated on the basis of time series with a varying sampling frequency makes it possible to evaluate the measure of “stability” for the regular component and between the “variability” of its informative parameters for the singular component. In this case, parameter fdcan be used as an additional input parameter of an artificial neural network for recognition of the state of the cardiovascular system.

    The presentation of electrocardiographic signals in the form of successive irregularities allows the use of flicker-noise spectroscopy in the analysis of such signals. The chaotic signal represented by the time series during flicker-noise spectroscopy allows one to parameterize these signals and determine informative diagnostic parameters, characterizing the functional state of the cardiovascular system. The set of informative parameters, as well as the sampling frequency of the signal, which determines the dynamics of changes in these parameters, allows the classification of heart diseases using a neural network.

    © 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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    Abdullayev Namiq Tahir and Ahmadova Khadija Ramiz (May 11th 2020). Flicker-Noise Spectroscopy Method in the Problem of Diagnosing the State of the Cardiovascular System, Data Acquisition - Recent Advances and Applications in Biomedical Engineering, Bartłomiej Płaczek, IntechOpen, DOI: 10.5772/intechopen.92264. Available from:

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