**4.4.3 Effects of aging on dynamic stability**

Equation (15) calculates the sum of single feature value by age.

Subject's walking is one kind of periodic movements and the same events will happen during different walking cycles, so the similarity of the data between adjacent cycles to

This paper used dynamic time warping (DTW) to calculate this similarity, which is a method for flexible pattern-matching scheme. It translates, compress and expands a pair of patterns so similar features within the two patterns are matched (Li, 2003). Fig. 28 and

( ) ( 1) <sup>1</sup>

(a) The source data of left toe at superior

(b) Compare between two adjacent cycles

According to FL model, it's easy to get corresponding position, velocity and acceleration motion data of 19 points. Therefore, 19×3=57 features are extracted to describe human

Fig. 28. Similarity between two adjacent cycles by DTW

Equation (15) calculates the sum of single feature value by age.

**4.4.2 Extracting of dynamic stability features** 

**4.4.3 Effects of aging on dynamic stability** 

dynamic stability.

 

1

(14)

(, )

*j j*

1

*DTW D S n n*

where *D* is the number of walking cycles, *D(Si)* is the average of similarity at feature *Si*.

*n*

*j i*

1

**4.4.1 Definition of dynamic stability** 

assess subject's walking stability.

Equation (14) show details.

$$S(A\_{k'}, S\_i) = \frac{1}{m \times l\_k} \sum\_{j=1}^{m} D\_j(S\_i) \tag{15}$$

where *m* is the number of subjects in the same age class, *lk* is the average leg length of subjects in the same age class, *Dj(Si)* is similarity value of the selected feature *i* by Equation (14), is a single stability value of the same feature *i* in the same age class *Ak*, and *k* is the number of age class.

According to Equation (14), similarity value of all selected features in the same age group was calculated by age.

$$F(A\_k) = \sum\_{i=1}^{p} S(A\_{k'} S\_i) \tag{16}$$

where *p* is the number of features, in this case, *p* is equal to 32, and *k* is equal to 5. Fig. 29 shows trend on the change of dynamic stability.

Fig. 29. Statistic on stability of all dynamic features

This figure tells us that if all dynamic features are employed to do statistics on such kind of stability, the dynamic stability is decreasing with ageing except the twenties. It seems that all of those features are desirable.

#### **4.5 Feature selection on dynamic stability**

The previous section did statistic on all 57 dynamic stability. To reduce the computational complexity, and more importantly, to get more important and contributing features, this paper did feature selection on those 57 dynamic features. On the one hand, it could simplify the method of data acquisition; On the other hand, it is more persuasive by analyzing those selected features.

This section ties to find the best features reflecting the relationship between age and walking stability among those 57 ones, which include 257-1 different kinds of feature combination. A classic method of feature selection, which is the cooperation of adaptive genetic algorithm and support vector machine, was used to do it.

#### **4.5.1 Improved crossover operation**

In order to avoid that better solutions with high fitness disappear in a standard genetic algorithm although the algorithm is accommodated again and again. This paper proposed a formulation to adjust crossover probability (*pc*) between average fitness and maximum fitness, as shown in Equation (17).

Human Walking Analysis, Evaluation and Classification Based on Motion Capture System 385

1

*e*

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

, max

where *M* is the confusion matrix, *b* is a coefficient that balances the total correctness and the

classes, *ei* is the sum of non-diagonal entries of *i*-th row of *M*, *m* is the sum of all entries of *M*,

After 18 generations of GA, 32 walking stability features were selected with classification

To compare number of markers between before and after feature selection, those 32 features are marked with 14 red markers, as shown in Fig. 30. It means that there is (30-14)/30=53.3%

,  *w n*

*<sup>c</sup> <sup>c</sup> <sup>b</sup> otherwise*

is the importance weight between classes, *n* is the number of

*i* .

is the average value of *e wi i <sup>c</sup>*

0 1

0, 0

0

*n w n <sup>i</sup> <sup>i</sup>* 

reduction on markers, which could simplify equipment to a large extent.

Fig. 30. Red markers corresponding to selected features

counted together.

**4.5.5 Effects of aging on dynamic stability after feature selection** 

Fig. 30 told us that the selected markers are almost symmetrical. To increase the symmetry, other three features are added, such as LHIP in position, RELB in velocity, RSHO and LELB

The difference between this method and previous one is the number of dynamic features. Therefore, the same statistic method will be applied on this one. Another, the similarity calculated by DTW is used to assess the stability of specific dynamic feature. Because DTW doesn't care about the data unite, all features from position, velocity and acceleration will be

in acceleration. Table 3 shows these fours features with gray background in details.

max

balance achievement, *<sup>w</sup>*

namely total number of patterns, 1

**4.5.4 Selected stable features** 

correctness from 89.4% to 94.7%.

*U e w e w*

$$p\_c^i = \begin{cases} p\_c^{i-1} \* \frac{f'-f\_{avg}}{f\_{\text{max}}-f\_{avg}}, f' \ge f\_{avg} \\ p\_c^{i-1} \* \frac{f\_{avg}-f'}{f\_{\text{max}}-f\_{avg}}, f' \prec f\_{avg} \end{cases} \tag{17}$$

where *fmax* is the maximum fitness of current population, *favg* is the average fitness of current population, *f'* is the larger fitness between two individuals in crossover operation.

#### **4.5.2 Improved mutation operation**

Just as crossover operation, there are the same problems with mutation operation. If probability of mutation (*pm*) is undersize, new individual can't be generated easily, inversely, genetic algorithm will be a pure searching process.

To solve this problem, this paper improved it as shown in Equation (18).

$$p\_m^i = \begin{cases} p\_m^{i-1} \ast \frac{f - f\_{avg}}{f\_{\text{max}} - f\_{avg}}, f \ge f\_{avg} \\ p\_m^{i-1} \ast \frac{f\_{avg} - f}{f\_{\text{max}} - f\_{avg}}, f \prec f\_{avg} \end{cases} \tag{18}$$

where *f* is the fitness of individual going to mutate. All other parameters have the same meaning as Equation (17).

#### **4.5.3 Improved support vector machine (SVM)**

According to information of age classification, SVM was used to separate datasets and assess fitness of specific feature combination during feature selection. This paper improves SVM in two parts: classification balancing and evaluation.

A conventional SVM is to build a decision function *fc(x)* for each class *C*. Then use Equation (19) as the predicted class label.

$$d(\mathbf{x}) = \arg\max(f\_c(\mathbf{x})) \tag{19}$$

However, this equation may fail to work in some skewed inseparable distribution. Therefore, it's improved as Equation (20), which suggests a function *pc(f)* to balance values of *fc(x)* in Equation (19).

$$d(\mathbf{x}) = \arg\max(p\_c(f\_c(\mathbf{x}))) \tag{20}$$

Another problem is about evaluation. Generally, correctness is calculated by *correctNumber/totalNumber*, but it does not fit well to skewed distributions. It's improved as Equation (21) described.

$$F(M, b, \vec{w}) = 1 - (1 - b) \times \sum\_{i=1}^{n} \frac{e\_i}{m} - \mathsf{U} \tag{21}$$

$$LI = \begin{cases} 0, & \sum\_{i=1}^{n} e\_i = 0\\ & \max\_{0 \le i \le n} \frac{e\_i w\_i}{c\_i} \sum\_{i=1}^{n} \left| \frac{e\_i w\_i}{c\_i} - \xi \right|, & \\ b \times \frac{\max \max i}{0 \le i \le n} \times \frac{\sum\_{i=1}^{n} |e\_i|}{n\zeta}, & otherwise \end{cases}$$

where *M* is the confusion matrix, *b* is a coefficient that balances the total correctness and the balance achievement, *<sup>w</sup>* is the importance weight between classes, *n* is the number of classes, *ei* is the sum of non-diagonal entries of *i*-th row of *M*, *m* is the sum of all entries of *M*,

namely total number of patterns, 1 *n w n <sup>i</sup> <sup>i</sup>* , is the average value of *e wi i <sup>c</sup> i* .

#### **4.5.4 Selected stable features**

384 Health Management – Different Approaches and Solutions

1 ' max

\* ,

*f f <sup>p</sup> f f f f*

> 1 ' max

*<sup>p</sup> f f f f*

\* ,

where *fmax* is the maximum fitness of current population, *favg* is the average fitness of current

If probability of mutation (*pm*) is undersize, new individual can't be generated easily,

max

*i avg*

\* ,

*f f <sup>p</sup> f f f f*

*m avg*

*m avg avg*

<sup>c</sup> *dx f x* ( ) arg max( ( )) (19)

<sup>c</sup> ( ) arg max( ( ( ))) *<sup>c</sup> dx p f x* (20)

*<sup>m</sup>* (21)

\* ,

*<sup>p</sup> f f f f*

max

where *f* is the fitness of individual going to mutate. All other parameters have the same

According to information of age classification, SVM was used to separate datasets and assess fitness of specific feature combination during feature selection. This paper improves

A conventional SVM is to build a decision function *fc(x)* for each class *C*. Then use Equation

However, this equation may fail to work in some skewed inseparable distribution. Therefore, it's improved as Equation (20), which suggests a function *pc(f)* to balance values

Another problem is about evaluation. Generally, correctness is calculated by *correctNumber/totalNumber*, but it does not fit well to skewed distributions. It's improved as

*<sup>e</sup> FMbw b U*

( , , ) 1 (1 )

1

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

'

(17)

(18)

*c avg*

*c avg avg*

'

*i avg*

*avg i*

*<sup>p</sup> f f*

population, *f'* is the larger fitness between two individuals in crossover operation.

Just as crossover operation, there are the same problems with mutation operation.

1

*avg i*

*i avg*

1

*<sup>p</sup> f f*

*i avg*

 

*c*

inversely, genetic algorithm will be a pure searching process.

*m*

**4.5.3 Improved support vector machine (SVM)** 

SVM in two parts: classification balancing and evaluation.

To solve this problem, this paper improved it as shown in Equation (18).

**4.5.2 Improved mutation operation** 

meaning as Equation (17).

(19) as the predicted class label.

of *fc(x)* in Equation (19).

Equation (21) described.

After 18 generations of GA, 32 walking stability features were selected with classification correctness from 89.4% to 94.7%.

To compare number of markers between before and after feature selection, those 32 features are marked with 14 red markers, as shown in Fig. 30. It means that there is (30-14)/30=53.3% reduction on markers, which could simplify equipment to a large extent.

Fig. 30. Red markers corresponding to selected features

#### **4.5.5 Effects of aging on dynamic stability after feature selection**

Fig. 30 told us that the selected markers are almost symmetrical. To increase the symmetry, other three features are added, such as LHIP in position, RELB in velocity, RSHO and LELB in acceleration. Table 3 shows these fours features with gray background in details.

The difference between this method and previous one is the number of dynamic features. Therefore, the same statistic method will be applied on this one. Another, the similarity calculated by DTW is used to assess the stability of specific dynamic feature. Because DTW doesn't care about the data unite, all features from position, velocity and acceleration will be counted together.

Human Walking Analysis, Evaluation and Classification Based on Motion Capture System 387

As a response to the assumption that elder the person is, less stable his gait is. This method assesses walking stability by searching the best contributing features and doing statistics on them. The result shows that walking stability truly becomes worse as ageing except the

Gait symmetry analysis is a part of normal gait analysis. Our research is based on the Fourteen-Linkage Walking Model of human. The detail of this model can be seen in chapter 2. We all know, gait symmetry reflects the general characteristics of human walk gait, and it is an important indicator to assess the function of the human walk. Especially in the human aging process the recession of brain and central nervous system and physiological function

Here, we mainly aim to investigate the footprint properties of the left and right foot. See

**Right step length**

**Left toe out**

Walking base

will affect the lower limb gait of left or right side, and lead gait mutation.

Fig.32, the symmetric properties were step length and toe out angle of two feet.

Stride length

**Right toe out** 

Fig. 31. Statistic on stability of selected dynamic features

group of twenties.

**5.1 Footprint symmetry** 

**5. Walking symmetry analysis** 

**Left step length**

Fig. 32. Features of footprint for symmetry analysis


Table 3. Selected features in three types

Fig. 31. Statistic on stability of selected dynamic features

As a response to the assumption that elder the person is, less stable his gait is. This method assesses walking stability by searching the best contributing features and doing statistics on them. The result shows that walking stability truly becomes worse as ageing except the group of twenties.
