**4. Study of nonlinear dynamical systems based on multifractal theory**

Fractal is a kind of geometry structures that have similarity in structure, form or function between the local and the whole. In nature, almost every object is very complex and performs a self-organization phenomenon that is a spatiotemporal structure or state phenomenon by forming spontaneously. From the view of geometry structure, this object has its own self-similarity properties in many parts, called a multifractal system. This structure can often be characterized by a set of coefficients, such as multifractal dimension, wavelet multifractal energy dimension. The multifractal theory reflects the complexity and richness of the nature in essence.

#### **4.1. Self-affine fractal[63-65]**

Computational Intelligence in Electromyography Analysis – 160 A Perspective on Current Applications and Future Challenges

practicability for the small sets of experiment data.

log( / ( )) *<sup>i</sup> <sup>i</sup>* μ *tr* μ

richness of the nature in essence.

dimension *m* of Lorenz system is 2.07, in general, if *d*>*m*, *d* is viable.

*3.7.5. Analysis of the surface EMG signal based on symplectic geometry* 

spectrums of Lorenz chaos time series (see Fig. 25c and Fig. 28a), although the position of noise floor is constantly driven up. However, in the same condition, SVD method cannot give the appropriate embedding dimension (see Fig. 25e and 28b), the results of which are similar to the results of the literature[61]. Besides, no matter the sampling interval is over sampling or under sampling, SG method can always give the appropriate embedding dimension *d* of Lorenz chaos time series (see Fig. 28c and 28d) because the correlation

For the action surface EMG signal (ASEMG) collected from a normal person, SVD method cannot give its appropriate embedding dimension (see Fig. 29a). The method based on correlation theory can do it but costs much time for computation. Here, SG method can fast obtain its embedding dimension. Figure 29b is the symplectic geometry spectrums of action surface EMG signal. The embedding dimension can be chosen as 6, which is the same as that of correlation dimension analysis[3]. This further shows that the SG method has stronger

a. The SVD principal component spectrums b. The symplectic geometry spectrums

**Figure 29.** The analysis of action surface EMG signal, *d*=3, 8, 13, 18, 23, abscissa is *d*, ordinate is

**4. Study of nonlinear dynamical systems based on multifractal theory** 

Fractal is a kind of geometry structures that have similarity in structure, form or function between the local and the whole. In nature, almost every object is very complex and performs a self-organization phenomenon that is a spatiotemporal structure or state phenomenon by forming spontaneously. From the view of geometry structure, this object has its own self-similarity properties in many parts, called a multifractal system. This structure can often be characterized by a set of coefficients, such as multifractal dimension, wavelet multifractal energy dimension. The multifractal theory reflects the complexity and **Definition 1** Let the mapping *<sup>n</sup> <sup>n</sup> S* : *R* → *R* . S is defined by

$$S(\mathbf{x}) = T(\mathbf{x}) + b \tag{69}$$

where *T* is a linear transformation on *Rn*. *b* is a vector in *Rn*. Thus, S is a combination of a translation, rotation, dilation and, perhaps, a reflection, called an affine mapping. Unlike similarities, affine mappings contract with differing ratios in different directions.

**Theorem 1** Consider the iterated function system given by the contractions{ *<sup>m</sup> S* , , *S* <sup>1</sup> }on *<sup>n</sup> D* ⊂ *R* , so that

$$\left| S\_i(\mathbf{x}) - S\_i(\mathbf{y}) \right| \le C\_i \left| \mathbf{x} - \mathbf{y} \right|, \text{ \forall x, y \in D} \tag{70}$$

with ∃ < 1 *Ci* for each *i*. Then there is a unique attractor *F*, i.e. a non-empty compact set such that

$$F = \bigcup\_{i=1}^{n} S\_i(F) \tag{71}$$

Moreover, if we define a transformation S on the class *φ* of non-empty compact sets by

$$S(E) = \bigcup\_{i=1}^{n} S\_i(E) \tag{72}$$

For *E*∈*φ*, and write *Sk* for the *k*th iterate of S (so *S* (*E*) = *E* <sup>0</sup> and ( ) ( ( )) <sup>1</sup> *S E S S E <sup>k</sup> <sup>k</sup>* <sup>−</sup> = for *k* ≥1 ), then

$$F = \bigcap\_{k=1}^{\bullet} S^k(E) \tag{73}$$

for every set *E*∈*φ* such that *Si*(*E*) ⊂ *E* for all *i*.

If an IFS consists of affine contractions { *<sup>m</sup> S* , , *S* <sup>1</sup> } on *Rn*, the attractor *F* guaranteed by Theorem 1 is termed a self-affine set.

Since self-affine time series have a power-law dependence of the power-spectral density function on frequency, .self-affine time series exhibit long-range persistence. For a practical data, one can use the relationship of power spectrum and frequency to determine if the data has the self-affine fractal characteristic.

#### **4.2. Spectrum analysis[65, 66]**

Let a time series be *x*(*t*) , *t* ∈[0, *T*]. Its spectrum is given by

$$X(f,T) = \int\_0^T \mathbf{x}(t)e^{2\mathbf{a}\dagger t}dt\tag{74}$$

where *f* is frequency. The power spectrum of *f* is defined by

$$S(f) = \frac{1}{T} \left| X(f, T) \right|^2 \tag{75}$$

If the power spectrum obeys a power law

$$S(f) = K \cdot f^{-\beta} \tag{76}$$

Nonlinear Analysis of Surface EMG Signals 163

(80)

=

1

≠

1

 

*l*

lim

*q*

 

*<sup>l</sup> <sup>q</sup>*

<sup>−</sup> <sup>=</sup> <sup>=</sup>

*l C l D <sup>N</sup> <sup>l</sup>*

ln( ) ln( ( )) lim ( )

→

*<sup>l</sup> <sup>q</sup>*

surface EMG signals during the arm movements.

**Table 4.** The self-affine *D* of surface EMG signals during movements

dimension of the signal.

EMG signals.

0

→

0

μ

lim 1 1

= →

1 0

The above *Dq* is the multifractal dimension method based on Grassberger and Procaccia. The generalized correlation integral *C* (*l*) *<sup>q</sup>* which can be obtained from an experimental time series yields in a plot ln *C* (*l*) *<sup>q</sup>* vs ln*l* straight lines with slopes *Dq*. For *q*=0, the *D*0 is called the topological dimension, fractal dimension or capacity dimension. For *q*=1, the *D*1 is called the information dimension. For *q*=2, the *D*2 is called correlation dimension. The function *Dq* is monotonically decreasing with *q* and gives information about the inhomogeneity of the attractor. For simple fractals, called monofractals, such as a homogeneous attractor, the multifractal dimension *Dq* is constant. In the general case of multifractal objects, the values of *Dq* monotonically decrease as *q* increase[67]. The shape of the *Dq* can be considered a criterion confirming that the object is a nonuniform fractal. Furthermore, it can be determined if the object is a nonlinear, complex structure by using the multifractal

*N l i*

( ) 1

=

( )ln ( )

*C C*

ln( ) ln [ ( )]

μ

μ

*C*

*<sup>q</sup> <sup>l</sup>*

*<sup>q</sup> <sup>l</sup>*

*q i*

ln( )

*<sup>i</sup> <sup>i</sup> <sup>i</sup>*

**4.4. Analysis of surface EMG signal based on multifractal dimension** 

The surface EMG signal is a complicated physiological signal. Its distribution is clearly uneven (see Figure 30). When the surface EMG signal is studied by using the fractal method, one should first determine if the surface EMG signal is fractal. Then, its corresponding fractal dimension *D* can be estimated by Eq.(77) under a certain resolution. Figure 30 shows the self-affine fractal analysis of the surface EMG signals from Channel 1 during finger flexion, finger tension, forearm pronation and forearm supination (the results of Channel 2 are similar to those of Channel 1). It can be seen that the surface EMG signals have selfaffine fractal characteristics. The results explain the physiological mechanism of the surface

In view of self-affine fractal characteristics, only one single fractal dimension is not easy to characterize the dynamics of surface EMG signals for different actions (see Table 4). There is little difference for the self-affine fractal dimensions of the four actions, where each type of action signals was chosen 100 sets of the data. The data length is 1000 points. In other words, it is difficult to identify the surface EMG signals of the different actions by using a single fractal dimension. The multifractal dimension values should be used to describe the action

Finger flexion Finger tension Forearm pronation Forearm supination

Channel 1 -0.2402±0.0725 -0.2571±0.0947 -0.0280±0.3250 0.0692±0.1418 Channel 2 0.0738±0.5734 -0.3199±0.2842 -0.0901±0.2591 -0.2343±0.2134

for large *f*, the time series *x*(*t*) has the self-affine fractal characteristic. The self-affine fractal dimension *D* is given

$$D = (\mathbb{S} - \beta) / \mathbb{Z} \tag{77}$$

*S*(*f*) is plotted as a function of *f* with log-log scaling. β is the negative of the slope of the bestfit straight line in the range of large *f*. Note that the value of β is a measure of the strength of persistence in a time series. β>1 reflects strong persistence and nonstationary. 1>β>0 describes weak persistence and stationary. β=0 shows uncorrelated stationary. β<0 indicates antipersistence and stationary. In all cases, however, a self-affine time series with a non-zero β has long-range (as well as short-range) persistence and anti-persistence. For small β, the correlations with large lag are small but are non-zero. This can be contrasted with time series that are not self-affine; these may have only short-range persistence (either strong or weak).

Although the self-affine mapping are varied in a continuous way, the dimension of the selfaffine set need not change continuously. Unfortunately, the self-affine fractal situation is much more complicated. It is quite difficult to obtain a general formula for the dimension of self-affine sets. It is not enough that only one fractal dimension is used to describe the selfaffine fractal time series. The multifractal dimensions have been proposed to describe this kind of the time series[67-72].

#### **4.3. Multifractal dimension**

For a measured time series of a multifractal system, its trajectory in phase space is often attracted to a bounded fractal object called strange attractor for which a whole set of dimension *Dq* has been introduced which generalize the concept of the Hausdorff dimension. Let *X*1, ..., *Xn* be a point of the attractor in the phase space. The probability that the trajectory point is found within a ball of radius *l* around one of the inhomogeneously distributed points of the trajectory is denoted by

$$\mu(C\_i) = \frac{1}{n} \sum\_{i=1}^{n} \theta(l - \left\|{X\_i - X\_j}\right\|) \tag{78}$$

where θ( ) *<sup>X</sup>* is the Heaviside step function. If *<sup>X</sup>* <sup>≥</sup> <sup>0</sup> , θ() 1 *X* = ; otherwise, θ() 0 *X* = .

The *q*-order correlation integral is defined by

$$C\_q(l) = \left(\frac{1}{n} \sum\_{j=1}^{n} (\mu(C\_i))^q\right)^{\bigvee\_{q=1}^{q}}\tag{79}$$

The multifractal dimension *Dq* can be computed by the following equation:

$$D\_q = \lim\_{l \to 0} \frac{\ln(C\_q(l))}{\ln(l)} = \begin{cases} \frac{1}{q-1} \lim\_{l \to 0} \frac{\ln \sum\_{i=1}^{N(l)} [\mu(C\_i)]^q}{\ln(l)} & q \neq 1 \\\frac{\sum\_{i=1}^{N(l)} \mu(C\_i) \ln \mu(C\_i)}{\ln(l)} & q = 1 \end{cases} \tag{80}$$

The above *Dq* is the multifractal dimension method based on Grassberger and Procaccia. The generalized correlation integral *C* (*l*) *<sup>q</sup>* which can be obtained from an experimental time series yields in a plot ln *C* (*l*) *<sup>q</sup>* vs ln*l* straight lines with slopes *Dq*. For *q*=0, the *D*0 is called the topological dimension, fractal dimension or capacity dimension. For *q*=1, the *D*1 is called the information dimension. For *q*=2, the *D*2 is called correlation dimension. The function *Dq* is monotonically decreasing with *q* and gives information about the inhomogeneity of the attractor. For simple fractals, called monofractals, such as a homogeneous attractor, the multifractal dimension *Dq* is constant. In the general case of multifractal objects, the values of *Dq* monotonically decrease as *q* increase[67]. The shape of the *Dq* can be considered a criterion confirming that the object is a nonuniform fractal. Furthermore, it can be determined if the object is a nonlinear, complex structure by using the multifractal dimension of the signal.

#### **4.4. Analysis of surface EMG signal based on multifractal dimension**

Computational Intelligence in Electromyography Analysis – 162 A Perspective on Current Applications and Future Challenges

*S*(*f*) is plotted as a function of *f* with log-log scaling.

describes weak persistence and stationary.

fit straight line in the range of large *f*. Note that the value of

β

− β

β

β

>1 reflects strong persistence and nonstationary. 1>

=0 shows uncorrelated stationary.

for large *f*, the time series *x*(*t*) has the self-affine fractal characteristic. The self-affine fractal

β

*D* = (5 −

β

has long-range (as well as short-range) persistence and anti-persistence. For small

antipersistence and stationary. In all cases, however, a self-affine time series with a non-zero

correlations with large lag are small but are non-zero. This can be contrasted with time series that are not self-affine; these may have only short-range persistence (either strong or

Although the self-affine mapping are varied in a continuous way, the dimension of the selfaffine set need not change continuously. Unfortunately, the self-affine fractal situation is much more complicated. It is quite difficult to obtain a general formula for the dimension of self-affine sets. It is not enough that only one fractal dimension is used to describe the selfaffine fractal time series. The multifractal dimensions have been proposed to describe this

For a measured time series of a multifractal system, its trajectory in phase space is often attracted to a bounded fractal object called strange attractor for which a whole set of dimension *Dq* has been introduced which generalize the concept of the Hausdorff dimension. Let *X*1, ..., *Xn* be a point of the attractor in the phase space. The probability that the trajectory point is found within a ball of radius *l* around one of the inhomogeneously

> ( ) <sup>1</sup> ( ) 1 = − − = *n <sup>i</sup> <sup>i</sup> <sup>X</sup> <sup>i</sup> <sup>X</sup> <sup>j</sup> <sup>l</sup> <sup>n</sup>*

> > θ

*q q*

( 1) <sup>1</sup>

<sup>−</sup> *<sup>n</sup>*

() 1 *X* = ; otherwise,

θ

= =

*<sup>q</sup> Ci <sup>n</sup> <sup>C</sup> <sup>l</sup>* <sup>1</sup>

The multifractal dimension *Dq* can be computed by the following equation:

*j*

( ( )) ) <sup>1</sup> ( ) ( μ

*S* ( *f* ) = *K* ⋅ *f* (76)

)/ 2 (77)

is the negative of the slope of the best-

is a measure of the strength of

(78)

(79)

() 0 *X* = .

θ

β

β>0

<0 indicates

β, the

If the power spectrum obeys a power law

dimension *D* is given

β

weak).

where

θ

persistence in a time series.

kind of the time series[67-72].

**4.3. Multifractal dimension** 

distributed points of the trajectory is denoted by

The *q*-order correlation integral is defined by

μ*C*

( ) *<sup>X</sup>* is the Heaviside step function. If *<sup>X</sup>* <sup>≥</sup> <sup>0</sup> ,

The surface EMG signal is a complicated physiological signal. Its distribution is clearly uneven (see Figure 30). When the surface EMG signal is studied by using the fractal method, one should first determine if the surface EMG signal is fractal. Then, its corresponding fractal dimension *D* can be estimated by Eq.(77) under a certain resolution. Figure 30 shows the self-affine fractal analysis of the surface EMG signals from Channel 1 during finger flexion, finger tension, forearm pronation and forearm supination (the results of Channel 2 are similar to those of Channel 1). It can be seen that the surface EMG signals have selfaffine fractal characteristics. The results explain the physiological mechanism of the surface EMG signals.

In view of self-affine fractal characteristics, only one single fractal dimension is not easy to characterize the dynamics of surface EMG signals for different actions (see Table 4). There is little difference for the self-affine fractal dimensions of the four actions, where each type of action signals was chosen 100 sets of the data. The data length is 1000 points. In other words, it is difficult to identify the surface EMG signals of the different actions by using a single fractal dimension. The multifractal dimension values should be used to describe the action surface EMG signals during the arm movements.


**Table 4.** The self-affine *D* of surface EMG signals during movements

**Figure 30.** The analysis study of self-affine of surface EMG signal: Curve is power spectrum of surface EMG signal, line is straight line fit related part of curve; Abscissa is lg(*f*), ordinate is lg(Psd)

Here, we use the above multifractal dimension theory to analyze the action surface EMG signals. For the surface EMG signals of the four actions in channel 1, the multifractal analysis results are shown in Figure 31. The results of channel 2 are omitted since they are similar to those of channel 1. In the figure, the *Dq*-*q* curves are calculated under q =8, 7, 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, -7, -8. It can be seen that the *Dq*-*q* curves have a certain range. The results indicate that the surface EMG signals are non-uniform fractal structure signals. These are consistent with the results of the above self-affine fractal analysis. The parameter values with *q* can be used to classify the data. In theory, it will be more reasonable that multifractal dimensions are used to describe the surface EMG signals. However, the actual calculation process of the multifractal dimensions is very time-consuming. For the surface EMG signals, it is extremely difficult to meet the requirements of real-time classification.

Nonlinear Analysis of Surface EMG Signals 165

d. The multi-fractal dimensions of curves in Fig. c; abscissa is *q*, ordinate is *Dq*

f. The multi-fractal dimensions of curves in Fig. e; abscissa is *q*, ordinate is *Dq*

> h. The multi-fractal dimensions of curves in Fig. e; abscissa is *q*, ordinate is *Dq*

c. The multi-fractal curves of EMG signal during finger tension; abscissa is ln(*l*), ordinate is ln(C*q*(*l*))

e. The multi-fractal curves of EMG signal during forearm pronation; abscissa is ln(*l*), ordinate is ln(C*q*(*l*))

g. The multi-fractal curves of EMG signal during forearm pronation; abscissa is ln(*l*), ordinate is ln(C*q*(*l*))

**5. Conclusion and future research** 

method, and so on.

**Figure 31.** The multi-fractal analysis of surface EMG signals during movements

In order to investigate whether the essence of the surface EMG signal is stochastic or deterministic nonlinear (even chaotic), some emerging nonlinear time series analysis approaches are discussed in this chapter. These techniques are based on detecting and describing determinisitic structure in the signal, such as surrogate data method, VWK model method, chaotic analysis method, symplectic geometry method, fractal analysis

b. The multi-fractal dimensions of curves in Fig. a; abscissa is *q*, ordinate is *Dq*

**Figure 31.** The multi-fractal analysis of surface EMG signals during movements

#### **5. Conclusion and future research**

Computational Intelligence in Electromyography Analysis – 164 A Perspective on Current Applications and Future Challenges

> a. The multi-fractal curves of EMG signal during finger flexion; abscissa is ln(*l*), ordinate is ln(C*q*(*l*))

a. finger flexion b. finger tension

c. forearm pronation d. forearm supination

**Figure 30.** The analysis study of self-affine of surface EMG signal: Curve is power spectrum of surface

Here, we use the above multifractal dimension theory to analyze the action surface EMG signals. For the surface EMG signals of the four actions in channel 1, the multifractal analysis results are shown in Figure 31. The results of channel 2 are omitted since they are similar to those of channel 1. In the figure, the *Dq*-*q* curves are calculated under q =8, 7, 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, -7, -8. It can be seen that the *Dq*-*q* curves have a certain range. The results indicate that the surface EMG signals are non-uniform fractal structure signals. These are consistent with the results of the above self-affine fractal analysis. The parameter values with *q* can be used to classify the data. In theory, it will be more reasonable that multifractal dimensions are used to describe the surface EMG signals. However, the actual calculation process of the multifractal dimensions is very time-consuming. For the surface EMG signals, it is extremely difficult to meet the requirements of real-time classification.

> b. The multi-fractal dimensions of curves in Fig. a; abscissa is *q*, ordinate is *Dq*

EMG signal, line is straight line fit related part of curve; Abscissa is lg(*f*), ordinate is lg(Psd)

In order to investigate whether the essence of the surface EMG signal is stochastic or deterministic nonlinear (even chaotic), some emerging nonlinear time series analysis approaches are discussed in this chapter. These techniques are based on detecting and describing determinisitic structure in the signal, such as surrogate data method, VWK model method, chaotic analysis method, symplectic geometry method, fractal analysis method, and so on.

The surrogate data method and VWK model mehtod are used to detect the surface EMG signal for arm movement and muscle fatigue. The results show that the surface EMG signal has deterministic nonlinear components. Moreover, our algorithm of surrogate data based on the null hypothesis 3 is proved that can completely satisfy the requirment of the null hypothesis 3. The VWK method with surrogate data can illuminate that not only the action but also fatigue surface EMG signals contain nonlinear dynamic properties.

Nonlinear Analysis of Surface EMG Signals 167

(82)

(85)

(86)

*SMS* ~ *M* <sup>−</sup><sup>1</sup> ∴ (83)

(84)

<sup>∴</sup> According Definition 2,

<sup>−</sup><sup>1</sup> ∴ *SMS* is also a Hamilton matrix.

**Theorem 2 proof:** 

**Theorem 5 proof:** 

**Theorem 6 proof:** 

 

matrix. According Definition 1 and 2, there is

− \* 0 0 *A*

 

=

= −

So Hamilton matrix *M* keeps unchanged at symplectic similar transform.

Let *S*1, *S*2,…, *S*n as symplectic matrix, respectively. According Definition 1, there are

− −∗

− − − −

*n*

 

 

0

*P J P*

0

\*

*P P*

 

0

*P*

1 1 1

=

*<sup>n</sup> <sup>n</sup> JS J S*

( )

*J S S S J*

1 2

( )−∗

*S S S S S S*

\*

*H JH*

= = =

So the product of sympletcic matrixes is also a symplectic matrix.

−∗ −∗ −∗

*n*

*n*

In order to prove that *H* matrix is symplectic matrix, we only need to prove *H JH* = *J* \* .

 − =

0

\*

*P P*

0

\*

 =

0

*P*

2 1 1

1

−

1

*JS J JS J J J JS J*

*n*

*JS J S JS J S*

1 2

1 1

− −∗ − −∗

2

1

= =

= −

*<sup>A</sup>* is a Hamilton matrix.

Let *S* as a symplectic transform matrix, *M* as a Hamilton matrix. Then −<sup>1</sup> *S* is also symplectic

1 1 1 1 11

− − − − −−

( )

*J SMS J JSJ JMJ JS J S MS*

( )

−∗ ∗ ∗

1

<sup>∗</sup> <sup>−</sup>

( )

*SMS*

Chaotic analysis techniques are reviewed and applied to investigate the surface EMG signals. The results show that the surface EMG signals have high-dimension chaotic dynamics by using correlation dimension and largest Lyapunov expoent techniques. For the estimation of embedding dimension, symplectic principal component analysis method is introduced and discussed. In comparison with correlation dimension algorithm and SVD analysis, symplectic geometry analysis is both very simple and reliable. The results show that symplectic geometry method is useful for determining of the system attractor from the experiment data.

The fractal theory is applied to study the fractal feature of the action surface EMG signal collected from forearm of normal person. The results show that he action surface EMG signal possesses the self-affine fractal characteristic. So, it is difficult to describe the surface EMG signals by using a single fractal dimension. The multifractal dimensions are used to analyze the action surface EMG signals during the arm movements. The results indicate that the surface EMG signals are non-uniform fractal structure signals. The multifractal dimension values can be used to identify the surface EMG signals for different movements.

For the nonlinear characteristics of the surface EMG signals, chaos and fractal theories will play the leading role in the nonlinear study of the surface EMG signals. The related methods need to be further researched and developed although these techniques have been applied to analyze the surface EMG signals. It provides a new way for the study of the quantitative analysis of physiology and pathology, sports medicine, clinical medical diagnostics and bionics of robot limb motion.
