**2. Estimation of the asymmetry of electromyographic signals using the statistical distribution of asymmetry coefficients**

Functional asymmetry is an integral feature of the human brain, which is manifested in various forms of human behavior and motor activity [8].

There are different approaches to assessing the asymmetry of different human organs. The asymmetry coefficient (KAs) was calculated as the ratio of the difference in EMG values of symmetrical muscles on the side of its minimum (X1) and maximum (X2) decrease to a larger value [5]:

$$\mathbf{K\_{As}} = \frac{(X\_1 - X\_2) \ast \mathbf{100\%}}{X\_1}.\tag{1}$$

The method of a normalized quantitative indicator, the asymmetry coefficient, was used to assess the EEG asymmetry [9, 10]. What is universal for its calculation is the formula

$$(\mathbf{A} - \mathbf{B})/(\mathbf{A} + \mathbf{B})\mathbf{x}\mathbf{100}\%,\tag{2}$$

where A is the numerical characteristic of the EEG of the left hemisphere, and B is the right one.

#### **2.1 Method**

The skewness coefficient is a widely used parameter in statistical analysis that characterizes the skewness of a statistical distribution.

The central moment of distribution can be calculated by the formula [11]:

$$m\_k^{(0)} = E(\xi - m\_i)^k = \begin{cases} \int (\varkappa - m\_i)^k f(\varkappa) d\varkappa, \text{if } \xi \text{ uninterupted;}\\ \sum\_i (\varkappa\_i^0 - m\_i)^k p\_i, \text{ if } discrete. \end{cases} \tag{3}$$

Here k is order; ξ is a discrete random variable with possible values *xi* and probabilities of their realization *pi* , (*i* ¼ **1**, **2**, **3**, … *:*).

By formula (1), it is easy to understand that if the density *f* <sup>ξ</sup>ð Þ *x* (or the sequence of probabilities *<sup>P</sup>* <sup>ξ</sup> <sup>¼</sup> *<sup>x</sup>***<sup>0</sup>** *i* � � is symmetric with respect to the mean value *<sup>m</sup>***<sup>1</sup>** <sup>¼</sup> *<sup>E</sup>*<sup>ξ</sup> (i.e. *f m*ð Þ� **<sup>1</sup>** � *<sup>x</sup> f m*ð Þ **<sup>1</sup>** <sup>þ</sup> *<sup>x</sup>* ), then all odd central moments (if they exist) *<sup>x</sup>*ð Þ **<sup>0</sup> <sup>2</sup>***k*þ**<sup>1</sup>** are equal to zero. Therefore, any odd, non-zero torque can be considered as a

*Methods and Tools for Assessing Muscle Asymmetry in the Analysis of Electromyographic… DOI: http://dx.doi.org/10.5772/intechopen.103061*

characteristic of the asymmetry of the corresponding distribution. The simplest of these characteristics is *m*ð Þ **<sup>0</sup> <sup>3</sup>** and is taken as the basis for calculating the so-called asymmetry coefficient *γ***1**—a quantitative characteristic of the degree of skewness of the distribution [10]:

$$\gamma\_1 = \frac{\frac{1}{N} \sum\_{i=1}^{N} (\mathbf{S}\_i - \mathbf{m}\_x)^3}{\sigma^3} \tag{4}$$

where *mx* <sup>¼</sup> **<sup>1</sup>** *N* P*<sup>N</sup> <sup>i</sup>*¼**<sup>1</sup>***Si* – sample mean, *<sup>σ</sup>***<sup>2</sup>** <sup>¼</sup> **<sup>1</sup>** ð Þ *N*�**1** P*<sup>N</sup> <sup>i</sup>*¼**<sup>1</sup>**ð Þ *Si* � *mx* **<sup>2</sup>** - sample variance, *Si*, **1** : *N* - time series.

All symmetric distributions will have zero skewness. Probability distributions with the "long part" of the density curve located to the right of the top are characterized by positive asymmetry, and distributions with the "long part" of the density curve located to the left of its top are negatively skewed [10].

The range of variability of the asymmetry coefficient is determined from �3 to +3.

At the same time, it is generally accepted that asymmetry above 0.5 (regardless of sign) is significant and less than 0.25 is insignificant. With a symmetric distribution, the kurtosis coefficient Ek = 0. If Ek <0, then the distribution has a flat-topped character, and if Ek > 0, then it is peaked. We determined the fluctuation of qualitative features of the variational series by the total variance, based on the theorem of addition of the variance of the share of a feature.

Thus, the variation statistics for assessing the contractile properties of muscles is based on the asymmetry coefficient. By the degree of deviation of the asymmetry coefficient from the median, as a rule, one can judge the value of the Gaussian distribution density. The closer the skewness indicator is to the median, the higher the Gaussian distribution density. The deviation of the asymmetry from the median is determined by the standard deviation from the mean. For a normal distribution, 95% of the values are within two standard deviations of the mean and 68% are within one standard deviation [11].

#### **2.2 Experimental part**

For the experiment, 6 muscles of the lower extremities of a patient with a diagnosis of asymmetry were selected. Measuring signals were received from 12 leads, in pairs from the right and left parts of the limbs (**Table 1**).

To estimate the asymmetry coefficient, formula (4) was used and the skewness function was implemented by Matlab [12]. The experimental results obtained are given in **Table 2**.

As already noted, asymmetry above 0.5 (regardless of sign) is significant and less than 0.25 is insignificant. According to the results of the experiment, we can say that on the left side of the limbs for the muscles a-gm-lp, a-qfm-rf, there is a great tendency to asymmetry, and for a-bfm, both the left and right parts, there is a tendency to asymmetry. For the muscles a-qfm-vm, a-gm-mp, a slight asymmetry can be observed on the left side, and a tendency towards asymmetry for the right side.

To visualize the results, you can present them in the form of a histogram (**Figure 1**).

To assess the method used, the results obtained are compared with the values obtained by the Myograph, which are given in **Figure 2**.

Based on the results in **Figure 2**, we can say that the patient has left-sided asymmetry in the a-qfm-vl muscles, since the highest voltage is observed on the


#### **Table 1.**

*Measuring muscle signals used in the experiment.*


#### **Table 2.**

*The value of the asymmetry coefficient depending on the location of the muscles.*

**Figure 1.**

*Diagram of the asymmetry coefficient of paired muscles.*

right side (from the bottom of the fourth drain). It is also not very convenient to compare this value with the value of the paired muscles (second line from the top).

The experimental data help not only to conveniently compare the results, but also to give a predictive conclusion about the possibility of the expected muscle pathology. The value between the interval 0.25 < x < 0.5 can be used as the bias towards asymmetry.

It is also possible to use the difference in the asymmetry coefficients for convenient observation of the results (**Figure 3**).

The results in **Figure 3** show the positions of the asymmetry during the measurement and may allow immediate observation.

*Methods and Tools for Assessing Muscle Asymmetry in the Analysis of Electromyographic… DOI: http://dx.doi.org/10.5772/intechopen.103061*

#### **Figure 2.**

*Measurement results with myograph.*

#### **Figure 3.**

*The difference in the coefficient of asymmetry of paired muscles.*
