**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 V tð Þ be represented as a sum

$$\mathbf{V}(\mathbf{t}) = \mathbf{V}\_R(\mathbf{t}) + \mathbf{V}\_S(\mathbf{t}).$$

2.Let *tk* ¼ *k*Δ*t k*ð Þ ¼ 1, … , *N* , *t*<sup>0</sup> ¼ 0, *tN* ¼ *T* points of task *V t*ð Þ by 0, ½ � *T* with a certain step of discreteness Δ*t*; *N* ¼ ½ � *T=*Δ*t* .

We calculate the average value:

$$
\langle V(t) \rangle = \frac{1}{N} \sum\_{k=1}^{N} V(t\_k) \tag{12}
$$

In what follows, we will assume that h i *V t*ð Þ ¼ 0, i.e., signal *V t*ð Þ, is stationary.

3. For stationary signal *V t*ð Þ, the power spectrum *S f* ð Þ (Fourier transform of the autocorrelation function *ψ τ*ð Þ) coincides with *Sc*ð Þ*f* (cosine Fourier transform of *ψ τ*ð Þ).

We set M from condition <sup>4</sup> <sup>3</sup> ≤ *M* ≤ *N* (in practice, take M close to N). We assume that M is an even number. For a time delay of *m<sup>τ</sup>* ¼ 0, 1, … , *M* � 1, we calculate the autocorrelator:

$$\psi(m\_{\mathfrak{r}}) = \frac{1}{N - m\_{\mathfrak{r}}} \sum\_{k=1}^{N-m\_{\mathfrak{r}}} V(k)V(k + m\_{\mathfrak{r}}) \tag{13}$$

4. Let *<sup>f</sup>* <sup>¼</sup> *<sup>q</sup>=<sup>M</sup>* <sup>Δ</sup>*<sup>t</sup>* .

> We calculate the power spectrum *S f* ð Þ¼ *Sc*ð Þ*f Sc*ð Þ¼ *<sup>f</sup>* <sup>1</sup> Δ*t Sc*ð Þ*q*

$$\begin{aligned} S\_{\epsilon}(q) &= \wp(\mathbf{0}) + \wp\left(\frac{\mathcal{M}}{2}\right)(-\mathbf{1})^q + 2\sum\_{m=1}^{\mathcal{M}} \wp(m)\cos\left(\frac{2\pi qm}{\mathcal{M}}\right), \\ \zeta(q = 0, 1, \dots, \mathcal{M} - \mathbf{1}) \end{aligned} \tag{14}$$

9.1. Using the spectrum graph (**Figure 3**), we introduce the constants

*<sup>S</sup>* ð Þ <sup>0</sup> , as well as the threshold value RSS <sup>∗</sup> <sup>¼</sup> <sup>10</sup>10.

*<sup>S</sup>* ð Þ 0 and evaluate the parameters *T*0, *n*0.

, *x* ¼ ln 2*πf*, *a* ¼ *n*0, *b* ¼ *n*<sup>0</sup> ln *T*0,

� � with y*<sup>m</sup>* and x*m*, corresponding to frequencies

*ym* � *ax*^ *<sup>m</sup>* <sup>þ</sup> ^

*<sup>S</sup>* ð Þ 0 and evaluate *T*0.

*y* ¼ *ax* þ *b b*ð Þ ¼ 0 ,

� � � �

1*=n*<sup>0</sup>

.

<sup>0</sup> and evaluate SSð Þ 0 .

*y* ¼ *ax* þ *b b*ð Þ ¼ 0 ,

1 þ ð Þ 2*π*f T0 n0

.

<sup>0</sup> and evaluate n0.

h i � � <sup>2</sup>

*b*

<sup>0</sup> , T<sup>∗</sup>

, *x* ¼ 2*πf*, *a* ¼ *T*0*:*

<sup>0</sup> <sup>¼</sup> <sup>T</sup>^0, where <sup>T</sup>^<sup>0</sup> <sup>¼</sup> ^a.

, a ¼ SSð Þ 0

SSð Þ <sup>0</sup> , where ^

SSð Þ¼ 0 ^a.

,

<sup>0</sup> <sup>¼</sup> <sup>T</sup>^0, where <sup>n</sup>^<sup>0</sup> <sup>¼</sup> ^a, <sup>T</sup>^<sup>0</sup> <sup>¼</sup>

*y* ¼ *ax* þ *b*,

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

and estimate the coefficients a and b using the least squares method (least

b LSM are the estimations of parameters *a* and *b*.

*f* ∗ <sup>0</sup> , *<sup>f</sup>* <sup>∗</sup> , *<sup>f</sup>*

where

*<sup>f</sup> <sup>m</sup>* <sup>¼</sup> *<sup>m</sup>*

*exp* ^ b*=*^a n o.

where

where

**77**

∗ , *S* <sup>∗</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.92264*

*<sup>y</sup>* <sup>¼</sup> ln *Ss*ð Þ <sup>0</sup>

� � � �

We calculate the residual sum of squares

*Ss*ð Þ*<sup>f</sup>* � <sup>1</sup>

RSSð Þ<sup>1</sup> <sup>¼</sup>

If RSSð Þ<sup>1</sup> <sup>&</sup>lt; RSS <sup>∗</sup> , then RSS <sup>∗</sup> <sup>≔</sup> RSSð Þ<sup>1</sup> , <sup>n</sup>^<sup>0</sup> <sup>¼</sup> <sup>n</sup><sup>∗</sup>

<sup>0</sup> , *SS*ð Þ¼ <sup>0</sup> *<sup>S</sup>* <sup>∗</sup>

*<sup>y</sup>* <sup>¼</sup> *Ss*ð Þ <sup>0</sup>

� � � �

P�1 *m*¼0

<sup>0</sup> , n0 <sup>¼</sup> <sup>n</sup><sup>∗</sup>

P�1 *m*¼0

If *RSS*ð Þ<sup>3</sup> <sup>&</sup>lt; *RSS* <sup>∗</sup> , then *RSS* <sup>∗</sup> <sup>≔</sup> *RSS*ð Þ<sup>3</sup> , SSð Þ¼ <sup>0</sup> ^

If RSSð Þ<sup>2</sup> <RSS <sup>∗</sup> , then RSS <sup>∗</sup> ≔ RSSð Þ<sup>2</sup> and T<sup>∗</sup>

*Ss*ð Þ*<sup>f</sup>* � <sup>1</sup>

<sup>y</sup> <sup>¼</sup> Ssð Þ<sup>f</sup> , x <sup>¼</sup> <sup>1</sup>

<sup>S</sup> ð Þ <sup>0</sup> , T0 <sup>¼</sup> <sup>T</sup><sup>∗</sup>

y*<sup>m</sup>* � ^ax*<sup>m</sup>* � �<sup>2</sup>

y*<sup>m</sup>* � ^ax*<sup>m</sup>* � �<sup>2</sup>

*M* X�1 *m*¼0

� � � �

9.2. We set *SS*ð Þ¼ <sup>0</sup> *<sup>S</sup>* <sup>∗</sup>

squares) for sample *ym*, *xm*

where ^a and ^

9.3. We set *<sup>n</sup>*<sup>0</sup> <sup>¼</sup> *<sup>n</sup>*<sup>∗</sup>

We calculate RSSð Þ<sup>2</sup> <sup>¼</sup> *<sup>M</sup>*

We calculate *RSS*ð Þ<sup>3</sup> <sup>¼</sup> *<sup>M</sup>*

9.5. We set SSð Þ¼ <sup>0</sup> <sup>S</sup> <sup>∗</sup>

9.4. We set T0 <sup>¼</sup> <sup>T</sup><sup>∗</sup>

Build a regression

Build a regression

*<sup>M</sup>*�Δ*<sup>t</sup>* ð Þ *<sup>m</sup>* <sup>¼</sup> 0, 1, … , *<sup>M</sup>* � <sup>1</sup> .

Build a regression

**Figure 3.** *Typical curve for function S f* ð Þ*, characterizing a chaotic signal V t*ð Þ *without a resonant component.*

5. We will construct the image *S f* ð Þ (or j j *S f* ð Þ , if in some frequencies *S f* ð Þ<0) in a bilogarithmic scale (**Figure 3**).

From **Figure 3** we find the frequency *f* ¼ *f* <sup>0</sup>, starting from which *S f* ð Þ ceases to stabilize around a certain constant *S*ð Þ 0 .


$$\begin{split} \boldsymbol{\Psi}\_{S,\mathbb{R}}(\boldsymbol{\tau}) &= \frac{1}{N - m\_{\tau}} \sum\_{k=1}^{N-m\_{\tau}} \left[ V\_{S}(k) V\_{S}(k + m\_{\tau}) + V\_{R}(k) V\_{S}(k + m\_{\tau}) + V\_{S}(k) V\_{R}(k + m\_{\tau}) \right], \\ \boldsymbol{\Psi}\_{\tau}(\boldsymbol{m}\_{\tau} = \boldsymbol{0}, 1, \ldots, M - 1) & \tag{15} \end{split} \tag{15}$$

8. We calculate the singular component *SS*ð Þ*f* of spectrum *S f* ð Þ by the formula

$$\begin{aligned} \mathcal{S}\_S(f) &= \frac{1}{\Delta t} \mathcal{S}\_S(q) \\ \mathcal{S}\_S(q) &= \wp\_{S, \mathbb{R}}(0) + \wp\_{S, \mathbb{R}}\left(\frac{M}{2}\right)(-1)^q + 2\sum\_{m=1}^M \wp\_{S, \mathbb{R}}(m)\cos\left(\frac{2\pi qm}{M}\right) \\ (q &= 0, 1, \dots, M - 1) \end{aligned} \tag{16}$$

9. For parameterization *SS*ð Þ*f* , we approximate this function by an interpolation expression:

$$\hat{S}\_{\mathcal{S}}(f) \approx \frac{\mathcal{S}\_{\mathcal{S}}(\mathbf{0})}{\mathbf{1} + \left(2\pi f T\_{\mathbf{0}}\right)^{n\_{\mathcal{0}}}} \tag{17}$$

Parameter *T*<sup>0</sup> by formula (17) will be determined by **Algorithm 1**, assuming that the "experimental" spectrum *SS*ð Þ*f* is calculated by formula (16).

Algorithm 1.

*Flicker-Noise Spectroscopy Method in the Problem of Diagnosing the State of the Cardiovascular… DOI: http://dx.doi.org/10.5772/intechopen.92264*

9.1. Using the spectrum graph (**Figure 3**), we introduce the constants

*f* ∗ <sup>0</sup> , *<sup>f</sup>* <sup>∗</sup> , *<sup>f</sup>* ∗ , *S* <sup>∗</sup> *<sup>S</sup>* ð Þ <sup>0</sup> , as well as the threshold value RSS <sup>∗</sup> <sup>¼</sup> <sup>10</sup>10. 9.2. We set *SS*ð Þ¼ <sup>0</sup> *<sup>S</sup>* <sup>∗</sup> *<sup>S</sup>* ð Þ 0 and evaluate the parameters *T*0, *n*0. Build a regression

$$
\mathbf{y} = a\mathbf{x} + b,
$$

where

$$\mathcal{Y} = \ln \left| \frac{\mathbb{S}\_{\mathfrak{s}}(\mathbf{0})}{\mathbb{S}\_{\mathfrak{s}}(f)} - \mathbf{1} \right|, \mathfrak{x} = \ln 2\mathfrak{x}f, a = n\_0, b = n\_0 \ln T\_0, \mathfrak{x}$$

and estimate the coefficients a and b using the least squares method (least squares) for sample *ym*, *xm* � � with y*<sup>m</sup>* and x*m*, corresponding to frequencies *<sup>f</sup> <sup>m</sup>* <sup>¼</sup> *<sup>m</sup> <sup>M</sup>*�Δ*<sup>t</sup>* ð Þ *<sup>m</sup>* <sup>¼</sup> 0, 1, … , *<sup>M</sup>* � <sup>1</sup> .

We calculate the residual sum of squares

$$\text{RSS}^{(1)} = \sum\_{m=0}^{M-1} \left[ \mathcal{y}\_m - \left( \hat{a} \mathbf{x}\_m + \hat{b} \right) \right]^2,$$

where ^a and ^ b LSM are the estimations of parameters *a* and *b*.

If RSSð Þ<sup>1</sup> <sup>&</sup>lt; RSS <sup>∗</sup> , then RSS <sup>∗</sup> <sup>≔</sup> RSSð Þ<sup>1</sup> , <sup>n</sup>^<sup>0</sup> <sup>¼</sup> <sup>n</sup><sup>∗</sup> <sup>0</sup> , T<sup>∗</sup> <sup>0</sup> <sup>¼</sup> <sup>T</sup>^0, where <sup>n</sup>^<sup>0</sup> <sup>¼</sup> ^a, <sup>T</sup>^<sup>0</sup> <sup>¼</sup> n o.

*exp* ^ b*=*^a

> 9.3. We set *<sup>n</sup>*<sup>0</sup> <sup>¼</sup> *<sup>n</sup>*<sup>∗</sup> <sup>0</sup> , *SS*ð Þ¼ <sup>0</sup> *<sup>S</sup>* <sup>∗</sup> *<sup>S</sup>* ð Þ 0 and evaluate *T*0. Build a regression

$$y = a\mathbf{x} + b(b=0),$$

where

$$\mathcal{Y} = \left| \frac{\mathbb{S}\_{\mathfrak{s}}(\mathbf{0})}{\mathbb{S}\_{\mathfrak{s}}(f)} - \mathbf{1} \right|^{1/n\_0}, \mathfrak{x} = 2\mathfrak{x}f, a = T\_0.$$

We calculate RSSð Þ<sup>2</sup> <sup>¼</sup> *<sup>M</sup>* P�1 *m*¼0 y*<sup>m</sup>* � ^ax*<sup>m</sup>* � �<sup>2</sup> . If RSSð Þ<sup>2</sup> <RSS <sup>∗</sup> , then RSS <sup>∗</sup> ≔ RSSð Þ<sup>2</sup> and T<sup>∗</sup> <sup>0</sup> <sup>¼</sup> <sup>T</sup>^0, where <sup>T</sup>^<sup>0</sup> <sup>¼</sup> ^a. 9.4. We set T0 <sup>¼</sup> <sup>T</sup><sup>∗</sup> <sup>0</sup> , n0 <sup>¼</sup> <sup>n</sup><sup>∗</sup> <sup>0</sup> and evaluate SSð Þ 0 . Build a regression

$$y = a\mathbf{x} + b(b=0),$$

where

$$\mathbf{y} = \mathbf{S}\_{\mathbf{s}}(\mathbf{f}), \mathbf{x} = \frac{1}{1 + (2\pi \mathbf{f} \, \mathbf{T}\_0) \, \mathbf{n}\_0}, \quad \mathbf{a} = \mathbf{S}\_{\mathbf{S}}(\mathbf{0}).$$

We calculate *RSS*ð Þ<sup>3</sup> <sup>¼</sup> *<sup>M</sup>* P�1 *m*¼0 y*<sup>m</sup>* � ^ax*<sup>m</sup>* � �<sup>2</sup> . If *RSS*ð Þ<sup>3</sup> <sup>&</sup>lt; *RSS* <sup>∗</sup> , then *RSS* <sup>∗</sup> <sup>≔</sup> *RSS*ð Þ<sup>3</sup> , SSð Þ¼ <sup>0</sup> ^ SSð Þ <sup>0</sup> , where ^ SSð Þ¼ 0 ^a. 9.5. We set SSð Þ¼ <sup>0</sup> <sup>S</sup> <sup>∗</sup> <sup>S</sup> ð Þ <sup>0</sup> , T0 <sup>¼</sup> <sup>T</sup><sup>∗</sup> <sup>0</sup> and evaluate n0.

Build a regression

$$\mathbf{y} = a\mathbf{x} + b(b=0),$$

where

$$\mathbf{y} = \ln \left| \frac{\mathbf{S}\_{\mathbf{s}}(\mathbf{0})}{\mathbf{S}\_{\mathbf{s}}(\mathbf{f})} - \mathbf{1} \right|, \mathbf{x} = \ln \left( 2\pi \mathbf{f} \mathbf{T}\_{\mathbf{0}} \right), \mathbf{a} = \mathbf{n} \mathbf{o}.$$

We calculate *RSS*ð Þ <sup>4</sup> <sup>¼</sup> *<sup>M</sup>* P�1 *m*¼0 y*<sup>m</sup>* � ^ax*<sup>m</sup>* � �<sup>2</sup> .

If *RSS*ð Þ <sup>4</sup> < *RSS* <sup>∗</sup> , then n<sup>∗</sup> <sup>0</sup> ¼ n^0, where n^<sup>0</sup> ¼ ^a.

As a result of **Algorithm 1**, we obtain the three parameters *SS*ð Þ¼ <sup>0</sup> *<sup>S</sup>* <sup>∗</sup> *<sup>S</sup>* ð Þ 0 , *<sup>n</sup>*<sup>0</sup> <sup>¼</sup> *<sup>n</sup>*<sup>∗</sup> <sup>0</sup> , *<sup>T</sup>*<sup>0</sup> <sup>¼</sup> *<sup>T</sup>*<sup>∗</sup> <sup>0</sup> , characterizing the interpolation expression (17) for the singular component of the spectrum *SS*ð Þ*f* .

**Figure 5.**

**Figure 6.**

**Figure 7.**

**79**

*Graphs of the singular component of this signal.*

*DOI: http://dx.doi.org/10.5772/intechopen.92264*

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

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

*Graphs of the spectral power of the ECG signal for the "atrial arrhythmia."*

### **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 4–6**, 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.

In **Figures 7–9**, as an example, similar relationships for an ECG signal with a range of "atrial arrhythmia" are presented.

**Figure 4.** *Graphs of the spectral power of the ECG signal for the norm.*

*Flicker-Noise Spectroscopy Method in the Problem of Diagnosing the State of the Cardiovascular… DOI: http://dx.doi.org/10.5772/intechopen.92264*

**Figure 5.** *Graphs of the singular component of this signal.*

#### **Figure 6.**

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

#### *Data Acquisition - Recent Advances and Applications in Biomedical Engineering*

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.,

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

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

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

Ψð Þ¼ *τ* h i *V t*ð Þ,*V t*ð Þ þ *τ* , (18)

*:*

Ψð Þ*τ* cos 2ð Þ *πf τ dτ* (19)

(20)

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

using artificial neural networks).

*DOI: http://dx.doi.org/10.5772/intechopen.92264*

and, therefore, to cosine conversion

and second-order difference moment

spectral regions of dependence *S f* ð Þ.

*Sc*ð Þ¼ *f*

ð *T=*2

�*T=*2

D

ponent of the signal irregularities in the spectral dependence *S f* ð Þ.

below in the form of the following sequence of operations:

<sup>Φ</sup>ð Þ<sup>2</sup> ð Þ¼ *<sup>τ</sup> V t*ðÞ� *V t*<sup>ð</sup> <sup>þ</sup> *<sup>τ</sup>*<sup>j</sup>

Here *V t*ð Þ is a stationary signal ð Þ h i *V t*ð Þ ¼ 0 , and h i� is a symbol of the average

The developed method of signal parametrization is based on the fact that the introduced "irregularities-bursts" and "irregularities-jumps" contribute to various

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") com-

Based on the remaining (after subtracting the "burst" contribution) spectral dependence, we can now determine the structure function <sup>Φ</sup>ð Þ<sup>2</sup> ð Þ*<sup>τ</sup>* , 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

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))

<sup>2</sup> � � <sup>E</sup>

different, ECG signals.

function [4].

value.

**81**

**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**).

The high specificity of *S f* ð Þ patterns obtained in the study of the cardiovascular system in the norm with the indicated pathologies can be used to diagnose diseases. Dependence *S f* ð Þ built on the basis of different ECGs and the corresponding informative parameters obtained by them differ from each other, which gives


**Table 1.**

*Informative parameters of the singular component of ECG signals.*

*Flicker-Noise Spectroscopy Method in the Problem of Diagnosing the State of the Cardiovascular… DOI: http://dx.doi.org/10.5772/intechopen.92264*

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].

$$
\Psi(\tau) = \langle V(t), V(t+\tau) \rangle,\tag{18}
$$

and, therefore, to cosine conversion

$$\mathcal{S}\_{\mathfrak{c}}(f) = \int\_{-T/2}^{T/2} \Psi(\mathfrak{r}) \cos \left( 2\mathfrak{r}f \mathfrak{r} \right) d\mathfrak{r} \tag{19}$$

and second-order difference moment

$$\Phi^{(2)}(\mathbf{r}) = \left\langle \left| \mathbf{V}(\mathbf{t}) - \mathbf{V}(\mathbf{t} + \mathbf{r}) \right|^2 \right\rangle. \tag{20}$$

Here *V t*ð Þ is a stationary signal ð Þ h i *V t*ð Þ ¼ 0 , and h i� 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 *S f* ð Þ.

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 *S f* ð Þ.

Based on the remaining (after subtracting the "burst" contribution) spectral dependence, we can now determine the structure function <sup>Φ</sup>ð Þ<sup>2</sup> ð Þ*<sup>τ</sup>* , 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))

$$\mathbf{S\_{rR}(f) = S(f) - S\_s(f)} \tag{21}$$

• We set *RSS* <sup>∗</sup> <sup>¼</sup> <sup>10</sup>10, T1 <sup>¼</sup> <sup>T</sup><sup>∗</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.92264*

• A preliminary estimate *T*<sup>∗</sup>

The value <sup>Φ</sup>ð Þ<sup>2</sup> ð Þ*<sup>τ</sup>* is taken as *<sup>T</sup>*<sup>∗</sup>

*<sup>y</sup>* <sup>¼</sup> ln <sup>Φ</sup>ð Þ<sup>2</sup> ð Þ� *<sup>τ</sup>* <sup>Φ</sup>ð Þ<sup>2</sup>

LSM-estimates *a*^ and ^

We calculate

Build a regression

where

maximum value <sup>Φ</sup>ð Þ<sup>2</sup> ð Þ*<sup>τ</sup>* <sup>≈</sup> <sup>2</sup>*σ*2.

We estimate parameters *σ*1, *H*<sup>1</sup> at *τ* < <*T*1.

*r* n o, *<sup>x</sup>* <sup>¼</sup> ln *<sup>τ</sup>*

close to *<sup>τ</sup>* <sup>¼</sup> 0, using representation (21) for <sup>Φ</sup>^ ð Þ<sup>2</sup> ð Þ*<sup>τ</sup>* .

RSSð Þ<sup>1</sup> <sup>¼</sup> <sup>X</sup>

• where y*<sup>k</sup>* and x*<sup>k</sup>* correspond to delays *τk*.

• If *RSS*ð Þ<sup>1</sup> ≥*RSS* <sup>∗</sup> , then go to Section 6.5.

*b=*2 n o.

• Otherwise, set *RSS* <sup>∗</sup> ≔ *RSS*ð Þ<sup>1</sup> , *σ* <sup>∗</sup>

*Graph of function* <sup>Φ</sup>ð Þ <sup>2</sup> ð Þ*<sup>τ</sup> in bilogarithmic coordinates.*

*<sup>σ</sup>*^<sup>1</sup> <sup>¼</sup> <sup>Γ</sup><sup>2</sup> *<sup>H</sup>*^ <sup>1</sup> <sup>þ</sup> <sup>1</sup> � � exp ^

**Figure 10.**

**83**

*k*1

*k*¼1

1

asymptotic representation of structure function <sup>Φ</sup>ð Þ<sup>2</sup> ð Þ*<sup>τ</sup>* (**Figure 10**).

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

*y* ¼ *ax* þ *b*,

*τ*1

<sup>1</sup> of parameter *T*<sup>1</sup> can be obtained using the

� �, *<sup>a</sup>* <sup>¼</sup> <sup>2</sup>*H*1, *<sup>b</sup>* <sup>¼</sup> 2 ln *<sup>σ</sup>*<sup>1</sup>

*b* are obtained on the basis of sequence f g *τ<sup>k</sup>* ,ð Þ *k* ¼ 1, … , *k*<sup>1</sup>

,

<sup>1</sup> <sup>¼</sup> *<sup>H</sup>*^ 1, where <sup>H</sup>^ <sup>1</sup> <sup>¼</sup> *<sup>a</sup>*^

<sup>2</sup> and

<sup>y</sup>*<sup>k</sup>* � ð Þ ^ax*<sup>k</sup>* <sup>þ</sup> <sup>b</sup> � �<sup>2</sup>

<sup>1</sup> <sup>¼</sup> *<sup>σ</sup>*^1, and H<sup>∗</sup>

Γ2

ð Þ *<sup>H</sup>*<sup>1</sup> <sup>þ</sup> <sup>1</sup> *:*

<sup>1</sup> for small delays, at which <sup>Φ</sup>ð Þ<sup>2</sup> ð Þ*<sup>τ</sup>* <sup>≈</sup>2*σ*<sup>2</sup> takes the

The resulting difference characterizes the contribution of the "resonant" components S*r*ð Þf and the "irregularities-jumps" *SR*ð Þ*f* to the general dependence *S f* ð Þ. If it turns out that *SrR*ð Þ*f* <0 in some frequency intervals, we assume S*<sup>r</sup>*Rð Þf ≔ j j S*<sup>r</sup>*Rð Þf .

2. Take the inverse cosine Fourier transform of SrRð Þf

$$\boldsymbol{\mu}\_{r\mathbf{R}}(\tau) = 2 \int\_0^{\mathbf{f}\_{\text{max}}} \mathbf{S}(\mathbf{f}) \cos \left(2\pi \mathbf{f} \tau \right) \mathbf{d}\mathbf{f}, \left(\tau \le \tau^\* \right) = \mathbf{T}/4 \mathbf{} \tag{22}$$

$$\mathbf{f}\_{\text{max}} = \frac{\mathbf{1}}{4\Delta \mathbf{t}}, \tau = k \cdot \Delta \tau (k = 1, \dots, k\_0), \Delta \tau = \frac{T/4}{k\_0}, k\_0 = 500.$$

Put *a* ¼ 0, *b* ¼ *f* max, *h* ¼ *f* max*=n*, *n* ¼ 100, *SrR*ð Þ� *f* cos 2ð Þ¼ *πf τ g f* ð Þ , *τ* and apply the trapezoid formula:

$$\int\_{a}^{b} \mathbf{g}(f, \tau)d\mathbf{f} = h\left(\frac{\mathbf{g}(a, \tau)}{2} + \mathbf{g}(a+h, \tau) + \mathbf{g}(a+2h, \tau) + \dots + \mathbf{g}(b-h, \tau) + \frac{\mathbf{g}(b, \tau)}{2}\right)$$

3. We calculate

$$\Phi\_{r\mathbb{R}}^{(2)}(\tau) = \mathfrak{Z}[\boldsymbol{\psi}\_{r\mathbb{R}}(\mathbf{0}) - \boldsymbol{\psi}\_{r\mathbb{R}}(\tau)], \tau = k \cdot \Delta\tau(k = 1, \dots, k\_0)$$

4. Put <sup>Φ</sup><sup>~</sup> ð Þ<sup>2</sup> *<sup>r</sup>* ð Þ¼ *<sup>τ</sup>* <sup>Φ</sup><sup>~</sup> ð Þ<sup>2</sup> *rR* ð Þ*τ* . 5. We denote

$$
\tilde{\Phi}^{(2)}(\mathfrak{r}) = \Phi\_r^{(2)}(\mathfrak{r}) + \Phi\_R^{(2)}(\mathfrak{r}).\tag{23}
$$

where Φð Þ<sup>2</sup> *<sup>R</sup>* ð Þ*τ* is given by the interpolation formula:

$$\Phi\_{R}^{(2)}(\tau) = \begin{cases} 2\sigma\_{1}^{2} \cdot \frac{1}{\Gamma^{2}(H\_{1}+1)} \left(\frac{\tau}{T\_{1}}\right)^{2H\_{1}}, \ \tau < \tau T\_{1}, \\\\ 2\sigma\_{1}^{2} \left[1 - \Gamma^{-1}(H\_{1}) \left(\frac{\tau}{T\_{1}}\right)^{H\_{1}-1} \exp\left(-\frac{\tau}{T\_{1}}\right)\right]^{2} \end{cases} \tag{24}$$

6. Compare the experimental structural function <sup>Φ</sup>ð Þ<sup>2</sup> ð Þ*<sup>τ</sup>* , determined by the formula

$$\Phi^{(2)}(\tau) = \mathcal{Z}[\psi(\mathbf{0}) - \psi(\tau)],\tag{25}$$

where

$$\begin{aligned} \Psi(m\_{\tau}) &= \frac{1}{N - m\_{\tau}} \sum\_{k=1}^{N-m\_{\tau}} V\_{(k)} V\_{(k+m\_{\tau})} \\ m\_{\tau} &= [\tau/\Delta t] \end{aligned} \tag{26}$$

with function <sup>Φ</sup>ð Þ<sup>2</sup> ð Þ*<sup>τ</sup>* determined by formula (20) using the least squares method.

*Flicker-Noise Spectroscopy Method in the Problem of Diagnosing the State of the Cardiovascular… DOI: http://dx.doi.org/10.5772/intechopen.92264*


The value <sup>Φ</sup>ð Þ<sup>2</sup> ð Þ*<sup>τ</sup>* is taken as *<sup>T</sup>*<sup>∗</sup> <sup>1</sup> for small delays, at which <sup>Φ</sup>ð Þ<sup>2</sup> ð Þ*<sup>τ</sup>* <sup>≈</sup>2*σ*<sup>2</sup> takes the maximum value <sup>Φ</sup>ð Þ<sup>2</sup> ð Þ*<sup>τ</sup>* <sup>≈</sup> <sup>2</sup>*σ*2. We estimate parameters *σ*1, *H*<sup>1</sup> at *τ* < <*T*1.

Build a regression

$$
\mathcal{Y} = a\mathfrak{x} + b,
$$

where

$$y = \ln\left\{\Phi^{(2)}(\tau) - \Phi\_r^{(2)}\right\}, \mathbf{x} = \ln\left\{\frac{\tau}{\tau\_1}\right\}, a = 2H\_1, b = 2\ln\frac{\sigma\_1}{\Gamma^2(H\_1 + 1)}.$$

LSM-estimates *a*^ and ^ *b* are obtained on the basis of sequence f g *τ<sup>k</sup>* ,ð Þ *k* ¼ 1, … , *k*<sup>1</sup> close to *<sup>τ</sup>* <sup>¼</sup> 0, using representation (21) for <sup>Φ</sup>^ ð Þ<sup>2</sup> ð Þ*<sup>τ</sup>* .

We calculate

$$\mathbf{RSS}^{(1)} = \sum\_{k=1}^{k\_1} \left[ \mathbf{y}\_k - (\hat{\mathbf{a}} \mathbf{x}\_k + \mathbf{b}) \right]^2 \mathbf{s}$$


**Figure 10.** *Graph of function* <sup>Φ</sup>ð Þ <sup>2</sup> ð Þ*<sup>τ</sup> in bilogarithmic coordinates.*


$$y = a\mathbf{x} + b(b \neq \mathbf{0}),$$

• where *<sup>y</sup>* <sup>¼</sup> <sup>Φ</sup>ð Þ<sup>2</sup> ð Þ� *<sup>τ</sup>* <sup>Φ</sup>ð Þ<sup>2</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.92264*

*τ<sup>k</sup>* ¼ *T* � *k k*ð Þ ¼ 1, … , *k*<sup>1</sup> .

*a*^*=*2 p .

• If *RSS*ð Þ <sup>4</sup> > > *RSS* <sup>∗</sup> , then go to Section 6.5.

• Otherwise, we set *RSS* <sup>∗</sup> <sup>¼</sup> *RSS*ð Þ <sup>4</sup> , *<sup>σ</sup>* <sup>∗</sup>

<sup>1</sup> , and*T*<sup>1</sup> <sup>¼</sup> *<sup>T</sup>* <sup>∗</sup>

*k*1 *k*¼1

<sup>1</sup> , H<sup>∗</sup> <sup>1</sup> , T<sup>∗</sup> 1 .

• We calculate *<sup>σ</sup>*^<sup>1</sup> <sup>¼</sup> ffiffiffiffiffiffiffi

• We calculate *RSS*ð Þ <sup>4</sup> <sup>¼</sup> <sup>P</sup>

• 6.5. Suppose *RSS* <sup>∗</sup> , *σ* <sup>∗</sup>

**of the ECG signal**

*<sup>H</sup>*<sup>1</sup> <sup>¼</sup> *<sup>H</sup>*<sup>∗</sup>

**Figure 11.**

**85**

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

*<sup>r</sup>* , *<sup>x</sup>* <sup>¼</sup> <sup>1</sup> � <sup>Γ</sup>�<sup>1</sup>

y*<sup>k</sup>* � ^ax*<sup>k</sup>* � �<sup>2</sup> ð Þ *<sup>H</sup>*<sup>1</sup> *<sup>τ</sup> T*1 � �*<sup>H</sup>*1�<sup>1</sup>

• We calculate the least squares method (LSM) estimation by sequence f g *τ<sup>k</sup>* ,

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

.

<sup>1</sup> ¼ *σ*^1.

• As a result of the proposed algorithm, we obtain the three parameters *<sup>σ</sup>*<sup>1</sup> <sup>¼</sup> *<sup>σ</sup>* <sup>∗</sup>

Using the above algorithm, we obtained the graphs of functions <sup>Φ</sup>ð Þ<sup>2</sup> ð Þ*<sup>τ</sup>* in 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

**6. Informative diagnostic parameters of the regular component**

� � n o

<sup>1</sup> , characterizing the interpolation expression (21) for <sup>Φ</sup>ð Þ<sup>2</sup>

exp � *<sup>τ</sup> T*1 2

, *a* ¼ 2*σ*1.

1 ,

*<sup>R</sup>* ð Þ*τ* .


$$\text{RSS}^{(2)} = \sum\_{k=1}^{k\_1} \left[ \mathbf{y}\_k - \left( \hat{\mathbf{a}} \mathbf{x}\_k + \hat{\mathbf{b}} \right) \right].$$


$$\text{where}$$
 
$$\text{where}$$

$$\mathcal{Y} = \ln \left\{ \frac{\Phi^{(2)}(\mathfrak{r}) - \Phi\_r^{(2)}(\mathfrak{r})}{2\sigma\_1^2/\Gamma^2(H\_1+1)} \right\}, \mathfrak{x} = \mathfrak{r}, a = \frac{1}{T\_1}.$$


$$\mathbf{y} = a\mathbf{x} + b(b=\mathbf{0}),$$

*Flicker-Noise Spectroscopy Method in the Problem of Diagnosing the State of the Cardiovascular… DOI: http://dx.doi.org/10.5772/intechopen.92264*


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

Using the above algorithm, we obtained the graphs of functions <sup>Φ</sup>ð Þ<sup>2</sup> ð Þ*<sup>τ</sup>* in 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 11.** *Dependence logΦ(2)(τ) for ventricular tachycardia.*

#### *Data Acquisition - Recent Advances and Applications in Biomedical Engineering*

#### **Figure 12.** *Dependence logΦ(2)(τ) for atrial rhythm.*


#### **Table 2.**

*Informative parameters of the regular component.*

(**Figure 12**). When conducting a computational experiment, we used the experimental data from the publicly available website www.PhysioNet.org.

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**).

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 <sup>Φ</sup>ð Þ<sup>2</sup> ð Þ*<sup>τ</sup>* and *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 <sup>σ</sup>1,*T*<sup>1</sup> and *<sup>H</sup>*<sup>1</sup> (for <sup>Φ</sup>ð Þ<sup>2</sup> ð Þ*<sup>τ</sup>* ) and the measure of variability of

**No. ECG signal Singular component Regular component**

*DOI: http://dx.doi.org/10.5772/intechopen.92264*

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

1 Norm 437.80 0.0042 0.3414 0.55 11.133 15.080 2 Ventricular tachycardia 197.358 0.0032 0.4123 0.51 10.913 0.6840 3 Atrial fibrillation 334.364 0.0036 0.3836 0.435 11.388 0.0640 4 Atrial Arrhythmia 43.7105 0.0059 0.4013 0.208 11.298 11.560

*fd***, Hs** *N* **Singular component Regular component <sup>4</sup>***S*ð Þ **<sup>0</sup>**

500 29.859 437.80 0.0042 0.3414 0.55 11.133 15.080 0.05 250 14.930 403.72 0.0028 0.4187 0.5044 10.845 1.38 0.08

*fd***, Hs** *N* **Singular component Regular component <sup>4</sup>***S*ð Þ **<sup>0</sup>**

500 29.859 197.358 0.0032 0.4123 0.5180 10.913 0.6840 0.026 250 14.930 175.80 0.0034 0.3446 0.517 15.200 0.340 0.10

*Ss***(0)** *T***<sup>0</sup>** *n σ***<sup>1</sup>** *H***<sup>1</sup>** *T***<sup>1</sup>**

*Ss***(0)** *T***<sup>0</sup>** *n***<sup>0</sup>** *σ***<sup>1</sup>** *H***<sup>1</sup>** *T***<sup>1</sup>**

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

*Ss***(0)** *T***<sup>0</sup>** *n***<sup>0</sup>** *σ***<sup>1</sup>** *H***<sup>1</sup>** *T***<sup>1</sup>**

*N*

*N*

The high specificity of dependencies <sup>Φ</sup>ð Þ<sup>2</sup> ð Þ*<sup>τ</sup>* and *<sup>S</sup>*(*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 *fd* times. 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

parameters *Ss*ð Þ 0 ,*T*<sup>0</sup> and *n*<sup>0</sup> (for *S*(*f*)).

shown in **Table 3**.

**87**

**Table 3.**

**Table 4.** *Norm.*

**Table 5.**

*Ventricular tachycardia.*

Thus, for the considered functional conditions of the cardiovascular system, three informative parameters *n*0,*T*0, *Ss*(0) for the singular component of the ECG signal and three informative diagnostic parameters *σ*1, *H*1,*T*<sup>1</sup> 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 irregularitiesjumps 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

*Flicker-Noise Spectroscopy Method in the Problem of Diagnosing the State of the Cardiovascular… DOI: http://dx.doi.org/10.5772/intechopen.92264*


**Table 3.**

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


**Table 4.** *Norm.*


#### **Table 5.**

*Ventricular tachycardia.*

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 <sup>Φ</sup>ð Þ<sup>2</sup> ð Þ*<sup>τ</sup>* and *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 <sup>σ</sup>1,*T*<sup>1</sup> and *<sup>H</sup>*<sup>1</sup> (for <sup>Φ</sup>ð Þ<sup>2</sup> ð Þ*<sup>τ</sup>* ) and the measure of variability of parameters *Ss*ð Þ 0 ,*T*<sup>0</sup> and *n*<sup>0</sup> (for *S*(*f*)).

The high specificity of dependencies <sup>Φ</sup>ð Þ<sup>2</sup> ð Þ*<sup>τ</sup>* and *<sup>S</sup>*(*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 *fd* times.

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**.

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.

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

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

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

For the computational experiment, a perceptron three-layer network with direct

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**). 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

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 *fd* of 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 *fd* can be used as an additional input parameter of an artificial neural network for recognition

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

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

**9. The choice of artificial neural network and its characteristics**

signal.

considered as input data.

connections was chosen (**Figure 13**).

*DOI: http://dx.doi.org/10.5772/intechopen.92264*

degree of training was about 0.01.

pathologies without retraining the entire system.

of the state of the cardiovascular system.

*The structural diagram of the proposed artificial neural network.*

**Figure 13.**

**89**

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 <sup>Φ</sup>ð Þ<sup>2</sup> ð Þ<sup>τ</sup> also occur at small *<sup>τ</sup>*, which are caused by the contribution of local changes in the values of the "laminar" signal sections. Therefore, with an increase of *fd*, parameters *T*<sup>0</sup> and *n*0, characterizing the highfrequency region of dependence *S*(*f*), and parameters *H*<sup>1</sup> and *T*1, characterizing the dependence of <sup>Φ</sup>ð Þ<sup>2</sup> ð Þ<sup>τ</sup> for small *<sup>τ</sup>*, change. The value of parameter *<sup>σ</sup>*<sup>1</sup> and the nature of spectral dependence *S*(*f*) change to a much lesser extent. Small variations in the standard deviation parameter *<sup>σ</sup>*<sup>1</sup> indicate a smaller dependence of function <sup>Φ</sup>ð Þ<sup>2</sup> ð Þ<sup>τ</sup> on *fd*. At the same time, the signal analysis in flicker-noise spectroscopy reveals the dynamics of changes in parameters *H*<sup>1</sup> and *T*<sup>1</sup> at small *τ*, as well as parameters *T*<sup>0</sup> and*n*0, characterizing dependence *S*(*f*) in the high-frequency region. Since dependence *S*(*f*) is determined by the number of *M* terms 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* ¼ 4*=N* when 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 <sup>Φ</sup>ð Þ<sup>2</sup> ð Þ<sup>τ</sup> and *<sup>S</sup>*(*f*), calculated on the basis of time series with decreasing sampling frequency, allows you to evaluate the measure of "stability" of parameters *σ*1,*T*1, and *<sup>H</sup>*1, determined on the basis of <sup>Φ</sup>ð Þ<sup>2</sup> ð Þ<sup>τ</sup> , and the measure of variability of the parameters *Ss*(0),*T*0, and *n*0, 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: *T*0, determining some characteristic time within which the measured dynamic variable is interconnected *V t*ð Þ*<sup>i</sup>* ; *n*0, dimensionless parameter that effectively determines how this relationship is lost as frequencies decrease to 1*=*2*πT*0; and *s*ð Þ 0 , 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 <sup>Φ</sup>ð Þ<sup>2</sup> ð Þ*<sup>τ</sup>* with parameters *<sup>T</sup>*1, *<sup>τ</sup>*1, and *<sup>H</sup>*1. In this case, parameter *<sup>T</sup>*<sup>1</sup> determines the characteristic time at which the values of the dynamic variables *V t*ð Þ*<sup>i</sup>* do not correlate. To obtain reliable values of variance *<sup>σ</sup>*<sup>2</sup> 1, it is necessary to calculate it at time intervals exceeding *T*1. In this case, parameter *H*<sup>1</sup> shows by what law the relationship between the quantities *V t*ð Þ*<sup>i</sup>* measured at different time instants is lost—the Hurst exponent.

*Flicker-Noise Spectroscopy Method in the Problem of Diagnosing the State of the Cardiovascular… DOI: http://dx.doi.org/10.5772/intechopen.92264*

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**).

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**).

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 *fd* of 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 *fd* can 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

**Figure 13.** *The structural diagram of the proposed artificial neural network.*

**References**

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[1] Timashev SF. Flicker-Noise Spectroscopy: Information in Chaotic Signals. Moscow: FIZMATLIT; 2007.

*DOI: http://dx.doi.org/10.5772/intechopen.92264*

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

[2] Timashev SF, Polyakov YS. Review of flicker noise spectroscopy in electrochemistry. Fluctuation and Noise Letters. 2007;**7**(2):R15-R47

[3] Abdullaev NT, Dyshin OA, Gasankulieva MM. Flicker noise spectroscopy of electrocardiographic signals. Biomedical Engineering. 2016;

[4] Abdullaev NT, Dyshin OA, Gasankulieva MM. Parameterization of the regular component of the ECG signal for diagnosis of the critical states of the cardiovascular system. Biomedical Engineering. 2016;**50**(3):

[5] Abdullaev NT, Gasankulieva MM, Dzhabieva ID. Application of neural network technology in flicker-noise spectroscopy of electrocardiograms. Information Technologies. 2018;**6**:

#### **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).*

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.
