**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 ψ τð Þ is 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 *S f* ð Þ, where *f* is the signal frequency, and the difference moment ("transition structure function") <sup>Φ</sup>ð Þ<sup>2</sup> ð Þ<sup>τ</sup> of the second order. The information extracted from the analysis of dependencies *S f* ð Þ and <sup>Φ</sup>ð Þ<sup>2</sup> ð Þ<sup>τ</sup> , built on the basis of time series *V t*ð Þ, 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 *V t*ð Þ contribute to the formation of dependence <sup>Φ</sup>ð Þ<sup>2</sup> ð Þ<sup>τ</sup> , and jumps and bursts (outbursts) of chaotic series *V t*ð Þ contribute to the formation of *S f* ð Þ.

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 *S f* ð Þ and <sup>Φ</sup>ð Þ<sup>2</sup> ð Þ<sup>τ</sup> ð Þ *<sup>p</sup>* <sup>¼</sup> 2, 3, … calculated on the basis of high-frequency and low-frequency components *V t*ð Þ.

## **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 *V t*ð Þ*<sup>i</sup>* for 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 *Ti* and sudden interruptions of such evolution by abrupt changes in the value of the dynamic variable in the short intervals of duration τ*i*ð Þ τ*<sup>i</sup>* <*Ti* .

high-frequency *VS*ð Þ*t* components. This method is built by analogy (**Figure 2**) with the solution of the diffusion equation and is based on the following "relaxation"

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

*Typical curve for function <sup>Φ</sup>*ð Þ *<sup>2</sup>* ð Þ*<sup>τ</sup> , characterizing a chaotic signal V t*ð Þ *without a resonant component.*

*<sup>k</sup>*¼<sup>1</sup>*V k*ð Þ and put *<sup>V</sup>*ð Þ *<sup>R</sup>* <sup>≔</sup> *<sup>V</sup>*ð Þ <sup>k</sup> � h i *<sup>V</sup>* , *<sup>k</sup>* <sup>¼</sup> 1, … , *<sup>N</sup>*.

τ2H1 , *if τ* < <T1 2σ2, *if τ* > >T1*:*

5.We take for T1 the value <sup>τ</sup>, at which log <sup>Φ</sup>ð Þ<sup>2</sup> ð Þ<sup>τ</sup> begins to stabilize to a constant

construct a regression y <sup>¼</sup> ax <sup>þ</sup> b bð Þ <sup>¼</sup> <sup>0</sup> ; y <sup>¼</sup> ln <sup>Φ</sup>ð Þ<sup>2</sup> ð Þ<sup>τ</sup> , x <sup>¼</sup> *ln* <sup>τ</sup>, a <sup>¼</sup> 2H1.

*V*ð Þ *<sup>k</sup> V*ð Þ *<sup>k</sup>*þ*m<sup>τ</sup>* , m<sup>τ</sup> ¼ ½ � τ*=Δ*t (1)

(3)

m<sup>τ</sup> ¼ 0, 1, … , M � 1, M ∈ N (2)

1.Set the value *V*1, … ,*VN* of signal *V* with a step of discreteness Δ*t*.

*N* <sup>X</sup>�*m<sup>τ</sup> k*¼1

1 *N* � *m*<sup>τ</sup>

where <sup>Φ</sup>ð Þ<sup>2</sup> ð Þ¼ <sup>τ</sup> <sup>2</sup>½ � <sup>ψ</sup>ð Þ� <sup>0</sup> ψ τð Þ , *<sup>σ</sup>*<sup>2</sup> <sup>¼</sup> <sup>ψ</sup>ð Þ <sup>0</sup> , τ ¼ m**<sup>τ</sup>** � Δt

<sup>Φ</sup>ð Þ<sup>2</sup> ð Þ¼ <sup>τ</sup>

(

6.Choose a sequence of small f g *τ<sup>k</sup>* ð Þ k ¼ 1, … , k0 k0 ≈ 20 , *τ<sup>k</sup>* < <T1, and

According to the least squares method (LSM) estimate ^a, we calculate the

procedure:

**Figure 2.**

2.Calculate h i *<sup>V</sup>* <sup>¼</sup> <sup>1</sup>

equal to log 2σ<sup>2</sup> ð Þ.

estimate H1 ¼ ^a*=*2.

7.We calculate

**73**

3.We calculate

*N* P*<sup>n</sup>*

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

ψð Þ¼ *m*<sup>τ</sup>

4.We plot <sup>Φ</sup>ð Þ<sup>2</sup> ð Þ<sup>τ</sup> in bilogarithmic coordinates. The asymptotic representation for <sup>Φ</sup>ð Þ<sup>2</sup> ð Þ<sup>τ</sup> is

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 *V t*ð Þ is conveniently represented as the sum of the two terms: the singular term *VS*ð Þ*t* , which is formed only by bursts of the dynamic variable, and the regular term *VR*ðÞ¼ *t V t*ðÞ� *VS*ð Þ*t* , 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 *P* and *T* teeth and extended phases in the form of an isoline.

The information contained in *S f* ð Þ and <sup>Φ</sup>ð Þ<sup>2</sup> ð Þ<sup>τ</sup> is different, so in order to determine the adequate parameters of the structure under study, it is necessary to analyze the dependencies log *S f* ð Þ¼ *<sup>F</sup>*ð Þ log *<sup>f</sup>* and log <sup>Φ</sup>ð Þ<sup>2</sup> ð Þ¼ <sup>τ</sup> *<sup>F</sup>*ð Þ log <sup>τ</sup> .

Let *V t*ð Þ denotes the dynamic variable, characterizing the ECG signal. We apply the proposed method of splitting the dynamic signal into low-frequency *VR*ð Þ*t* and

**Figure 1.** *The dynamics of the electrocardiogram.*

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

**Figure 2.** *Typical curve for function <sup>Φ</sup>*ð Þ *<sup>2</sup>* ð Þ*<sup>τ</sup> , characterizing a chaotic signal V t*ð Þ *without a resonant component.*

high-frequency *VS*ð Þ*t* components. This method is built by analogy (**Figure 2**) with the solution of the diffusion equation and is based on the following "relaxation" procedure:

1.Set the value *V*1, … ,*VN* of signal *V* with a step of discreteness Δ*t*.


$$\Psi(m\_{\pi}) = \frac{1}{N - m\_{\pi}} \sum\_{k=1}^{N-m\_{\pi}} V\_{(k)} V\_{(k + m\_{\pi})}, m\_{\pi} = [\pi/\Delta t] \tag{1}$$

where <sup>Φ</sup>ð Þ<sup>2</sup> ð Þ¼ <sup>τ</sup> <sup>2</sup>½ � <sup>ψ</sup>ð Þ� <sup>0</sup> ψ τð Þ , *<sup>σ</sup>*<sup>2</sup> <sup>¼</sup> <sup>ψ</sup>ð Þ <sup>0</sup> , τ ¼ m**<sup>τ</sup>** � Δt

$$\mathbf{m}\_{\mathbf{t}} = \mathbf{0}, \mathbf{1}, \dots, \mathbf{M} - \mathbf{1}, \mathbf{M} \in \mathbf{N} \tag{2}$$

4.We plot <sup>Φ</sup>ð Þ<sup>2</sup> ð Þ<sup>τ</sup> in bilogarithmic coordinates.

The asymptotic representation for <sup>Φ</sup>ð Þ<sup>2</sup> ð Þ<sup>τ</sup> is

$$\Phi^{(2)}(\tau) = \begin{cases} \tau^{2\mathcal{H}\_1}, & \text{if } \tau < \tau \mathcal{T}\_1 \\ 2\sigma^2, & \text{if } \tau > \mathcal{T}\_1. \end{cases} \tag{3}$$


According to the least squares method (LSM) estimate ^a, we calculate the estimate H1 ¼ ^a*=*2.

7.We calculate

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

$$\mathbf{D} = \frac{\sigma^2}{\Gamma^2 \left(\mathbf{1} + \mathbf{H}\_1^\*\right) \cdot \mathbf{T}\_1^\*} \tag{4}$$

Vjþ<sup>1</sup> <sup>k</sup> � <sup>V</sup><sup>j</sup> k Δτ <sup>¼</sup> Vj

<sup>k</sup> þ

Vjþ<sup>1</sup> <sup>k</sup> <sup>¼</sup> Vj

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

Eq. (10). From (11) we obtain

In notation *<sup>ω</sup>* <sup>¼</sup> Δτ*=*ð Þ <sup>Δ</sup><sup>t</sup> <sup>2</sup>

following sequence of steps [3]:

*ψ τ*ð Þ).

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

**75**

<sup>Δ</sup>*<sup>t</sup>* .

Δ*t Sc*ð Þ*q*

*Sc*ð Þ¼ *q ψ*ð Þþ 0 *ψ*

ð Þ *q* ¼ 0, 1, … , *M* � 1

*Sc*ð Þ¼ *<sup>f</sup>* <sup>1</sup>

1. Let V tð Þ be represented as a sum

We calculate the average value:

We set M from condition <sup>4</sup>

we calculate the autocorrelator:

*ψ*ð Þ¼ *m<sup>τ</sup>*

We calculate the power spectrum *S f* ð Þ¼ *Sc*ð Þ*f*

*M* 2 � �

certain step of discreteness Δ*t*; *N* ¼ ½ � *T=*Δ*t* .

will be absolutely stable at *ω* <1*=*2.

<sup>k</sup>þ<sup>1</sup> <sup>þ</sup> Vj

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

<sup>k</sup>þ<sup>1</sup> <sup>þ</sup> <sup>V</sup><sup>j</sup>

� �*:*

corresponding to the simplest difference scheme for numerically solving

theory of stability of difference schemes, it is known that this difference scheme

**3. Parameterization of the singular component of the ECG signal**

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.

The procedure for parameterizing the singular part of the signal consists of the

V tðÞ¼ V*R*ð Þþ t V*S*ð Þt *:*

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

*N* X *N*

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

assume that M is an even number. For a time delay of *m<sup>τ</sup>* ¼ 0, 1, … , *M* � 1,

*N* <sup>X</sup>�*m<sup>τ</sup> k*¼1

ð Þ �<sup>1</sup> *<sup>q</sup>* <sup>þ</sup> <sup>2</sup> <sup>X</sup>

*M* <sup>2</sup> � <sup>1</sup>

*m*¼1

*ψ*ð Þ *m* cos

*k*¼1

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

h i *V t*ð Þ <sup>¼</sup> <sup>1</sup>

1 *N* � *m<sup>τ</sup>*

Δτ ð Þ <sup>Δ</sup><sup>t</sup> <sup>2</sup> Vj <sup>k</sup>‐<sup>1</sup> � 2Vj k

> <sup>k</sup>‐<sup>1</sup> � 2Vj k

, the last equation is written in the form (8). From the

ð Þ <sup>Δ</sup><sup>t</sup> <sup>2</sup> , (11)

*V t*ð Þ*<sup>k</sup>* (12)

*V k*ð Þ*V k*ð Þ þ *m<sup>τ</sup>* (13)

2*πqm M* � �, (14)

<sup>3</sup> ≤ *M* ≤ *N* (in practice, take M close to N). We

To calculate <sup>Γ</sup>ð Þ <sup>x</sup> for x <sup>¼</sup> <sup>1</sup> <sup>þ</sup> H1, put n <sup>¼</sup> <sup>10</sup>3and represent <sup>Γ</sup>ð Þ <sup>x</sup> in the form

$$\Gamma(\mathbf{x}) = \frac{\Gamma(\mathbf{x} + \mathbf{1})}{\mathbf{x}} = \frac{\Gamma(\mathbf{x} + \mathbf{2})}{\mathbf{x}(\mathbf{x} + \mathbf{1})} = \dots = \frac{\Gamma(\mathbf{x} + \mathbf{n})}{\mathbf{x}(\mathbf{x} + \mathbf{1}) \cdot (\mathbf{x} + \mathbf{n}\mathbf{t})}.\tag{5}$$

The value Γð Þz (we have z ¼ x þ n) is calculated by the formula

$$\Gamma(\mathbf{z}) = \exp\left\{ \left( \mathbf{z} - \frac{\mathbf{1}}{2} \right) \ln z - \mathbf{z} + \frac{\mathbf{1}}{2} \ln 2\pi \right\} \tag{6}$$

with an error of order z�<sup>1</sup> <sup>≈</sup>10�<sup>3</sup> <sup>n</sup>≈10<sup>3</sup> <sup>z</sup> <sup>¼</sup> <sup>x</sup> <sup>þ</sup> n.

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

$$
\alpha = \mathbf{D} \cdot \Delta \mathbf{r} / (\Delta \mathbf{t})^2 \tag{7}
$$

choose Δτ so that ω< 1*=*2.

9. Put M ≔ N � 1, and construct an iterative procedure according to j ¼ 0, 1, … , , according to which the value Vjþ<sup>1</sup> <sup>k</sup> at the jth step is calculated through the value Vj <sup>k</sup> according to the formula

$$\mathbf{V}\_{\mathbf{k}}^{\dagger+1} = \boldsymbol{\alpha} \mathbf{V}\_{\mathbf{k}+1}^{\dagger} + \boldsymbol{\alpha} \mathbf{V}\_{\mathbf{k}\cdot 1}^{\dagger} + (\mathbf{1} - \mathbf{2}\boldsymbol{\alpha}) \mathbf{V}\_{\mathbf{k}}^{\dagger} \tag{8}$$

at j <sup>¼</sup> 0 we set Vj <sup>k</sup> <sup>¼</sup> <sup>V</sup>ð Þ <sup>k</sup> ; at k <sup>¼</sup> 1 and k <sup>¼</sup> M, the values of Vjþ<sup>1</sup> <sup>k</sup> are calculated by the formulas

$$\mathbf{V\_1^{j+1}} = (\mathbf{1} - 2a)\mathbf{V\_1^j} + 2a\mathbf{V\_2^j}, \quad \mathbf{V\_M^{j+1}} = (\mathbf{1} - 2a)\mathbf{V\_M^j} + 2a\mathbf{V\_{M-1}^j}.\tag{9}$$

The procedure stops at step j ¼ j <sup>0</sup>, in which.

$$\left| \left| \mathbf{V}\_{\mathbf{k}}^{\flat\_0+1} - \mathbf{V}\_{\mathbf{k}}^{\flat\_0} \right| < \varepsilon \text{, for } \mathcal{V}\_{\mathbf{k}} = \mathbf{1}, \dots, \mathcal{M}, \mathbf{k} = \mathbf{1}, 2, \dots, \mathcal{N}, \varepsilon$$

where <sup>ε</sup> is the given number (e.g., <sup>ε</sup> <sup>¼</sup> <sup>10</sup>‐mþ<sup>1</sup> , where 10‐mis the error in setting the initial values Vk).

10. The values V<sup>j</sup> 0 <sup>k</sup> determine the low-frequency component V*R*ð Þt . Then V tð Þ� V*R*ðÞ¼ t V*S*ð Þt is the high-frequency component of the signal V tð Þ.

The described signal smoothing procedure is focused on minimizing the "highfrequency" information in the "low-frequency" part V*R*ð Þt of the signal and vice versa, minimizing the "low-frequency" information in the "high-frequency" part V*S*ð Þt of the signal. This conclusion follows from the diffusion nature of the partial differential equation used

$$\frac{\partial \mathbf{V}}{\partial \mathbf{\tau}} = \frac{\partial^2 \mathbf{V}}{\partial \mathbf{t}^2},\tag{10}$$

represented as a difference equation

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

$$\frac{\mathbf{V\_{k}^{j+1}} - \mathbf{V\_{k}^{j}}}{\Delta \pi} = \frac{\mathbf{V\_{k+1}^{j}} + \mathbf{V\_{k\cdot 1}^{j}} - 2\mathbf{V\_{k}^{j}}}{\left(\Delta \mathbf{t}\right)^{2}},\tag{11}$$

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

$$\mathbf{V}\_{\mathbf{k}}^{j+1} = \mathbf{V}\_{\mathbf{k}}^{j} + \frac{\Delta \mathbf{r}}{\left(\Delta \mathbf{t}\right)^{2}} \left(\mathbf{V}\_{\mathbf{k}+1}^{j} + \mathbf{V}\_{\mathbf{k}\cdot 1}^{j} - 2\mathbf{V}\_{\mathbf{k}}^{j}\right) \dots$$

In notation *<sup>ω</sup>* <sup>¼</sup> Δτ*=*ð Þ <sup>Δ</sup><sup>t</sup> <sup>2</sup> , 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.
