Analysis and synthesis of function data of human movement

The FDA is a statistical method that has emerged in recent years. It is a development and extension of traditional statistical analysis methods. The FDA's application of exercise data research is reasonable. For an observation data sequence y = (y₁,y₂, …, y_n)^T, it is possible to establish a model (1) yi=x(ti)+∈i,i=1,2,⋯,n {y_i} = x({t_i}) + { \in _i},i = 1,2, \cdots ,n Among them, x(t_i) represents the value of the observation data sequence's function and ∈_i, represents the noise. To estimate the value of x(t_i) in Eq. (1), the basis function expansion method is used, i.e. a set of basis functions Φ(t) = (φ₁(t),φ₂(t), …, φ_K(t)) is selected to transform the discrete data into a linear combination of basis functions, namely (2) x(ti)=∑k=1Kckϕk(ti),i=1,2,⋯,n x({t_i}) = \sum\limits_{k = 1}^K {c_k}{\phi _k}({t_i}),i = 1,2, \cdots ,n After the basis function is determined, a set of values of the coefficient vector c = (c₁,c₂, …,c_K)^T uniquely determines a function. The most direct method to solve the coefficient vector is the least square method to minimise the residual sum of squares. (3) SMSS(y|c)=∑i=1n[yi−∑k=1Kckϕk(ti)]2 SMSS(y|c) = \sum\limits_{i = 1}^n {\left[ {{y_i} - \sum\limits_{k = 1}^K {c_k}{\phi _k}({t_i})} \right]^2} For periodic observation data, the Fourier series is often used as the basis function, namely (4) ϕ1(t)=1,ϕ2(t)=sin(ωt),ϕ3(t)=cos(ωt),⋯,ϕ2k(t)=sin(kωt),ϕ2k+1(t)=cos(kωt) {\phi _1}(t) = 1,{\phi _2}(t) = \sin (\omega t),{\phi _3}(t) = \cos (\omega t), \cdots ,{\phi _{2k}}(t) = \sin (k\omega t),{\phi _{2k + 1}}(t) = \cos (k\omega t) Therefore, the Fourier expansion of the function x(t) of the observation data sequence is (5) x(t)=c1+c2sin(ωt)+c3cos(ωt)+⋯+c2ksin(kωt)+c2k+1cos(kωt) x(t) = {c_1} + {c_2}\sin (\omega t) + {c_3}\cos (\omega t) + \cdots + {c_{2k}}\sin (k\omega t) + {c_{2k + 1}}\cos (k\omega t) Among them is c_i(i = 1,2, …,2k + 1).

SVM is a machine learning method. Its essence is a linear classifier that maximises the interval in the feature space. The learning strategy is to find a hyperplane in the feature space that maximises the sample interval, i.e. to solve a convex quadratic programming problem in the process [3]. The crucial part of SVM is to solve the following optimisation problems (6) min12‖ω‖2+C∑i=1nξis.t.y(ω.xi+b)≥1−ξi,ξi≥0i=1,2,⋯,n \matrix{ {\min {1 \over 2}{{\left\| \omega \right\|}^2} + C\sum\limits_{i = 1}^n {\xi _i}} \cr {s.t.y(\omega .{x_i} + b) \ge 1 - {\xi _i},{\xi _i} \ge 0} \cr {i = 1,2, \cdots ,n} \cr } Among them, ξ_i is called the slack variable, and C is the penalty parameter. The kernel function is one of the critical factors of SVM. This article mainly adopts the SVM model with the radial basis function as the kernel function, and the form is as follows: (7) K(x,x1)=exp[−‖x−x1‖22σ2] K(x,{x_1}) = \exp \left[ { - {{{{\left\| {x - {x_1}} \right\|}^2}} \over {2{\sigma ^2}}}} \right] where σ is the core radius.

According to different types of behavioural data, different sample function matrices are obtained. The morphological curve characteristics of the corresponding functions in different sample function matrices are different. This difference in morphology is the basis for the classification of the sample set, and this difference can be reflected in one exercise cycle, as shown in Figure 1. Therefore, only one cycle of data can be extracted to distinguish different behaviours.

Fig. 1

Selecting the starting point of the period through the element function of the matrix in Eq. (13) is essential for extracting period data. The method adopted in this paper is to calibrate the starting point of a period through the information of the derivative function of the fitting function (as shown in Figure 2) and then find a motion cycle. From the sample function matrix of Eq. (13), we select one of the element functions, abbreviated as f (t), and find its characteristic point. It is reasonable to use the characteristic point as the starting point of the cycle. This paper selects f (the set of extreme points of t); that is, all the zero points in the function image in Figure 2 are used as the function's feature point set. Then the point in the feature point set that makes f (t) obtain the minimum is selected as the starting point of the motion cycle. Finally, each element function in the sample function matrix uses this starting point as the starting point of the motion cycle.

Fig. 2

We choose the minor extreme point as the starting point of the cycle and achieve data alignment. From the perspective of the function, since there are multiple extreme points in the same function, the minor extreme point is selected to unify the standard so that the primary function represents the different functions that are aligned at the same starting point, thereby facilitating comparison of the data of different shapes and changing trends in a period. In a practical sense, the minimum extreme point of the vertical acceleration function of the foot is the minimum acceleration value during the movement. This point in time often corresponds to when a person's foot is raised to the highest point, and the periodic behaviour based on foot movement includes this movement process [5]. Therefore, we use this point as the starting point of the movement. All sample functions of the same action are aligned here, and periodic functions of different behaviours can be aligned at this starting point.

After the starting point of the period is calibrated, a period of data can be extracted according to the fitting function. In this paper, the fitting function is resampled to obtain approximately one period of motion data to express the curve characteristics of the function in one period. The sampling frequency, the time for an ordinary person to complete a motion cycle and the sampling interval of resampling directly determine the length of the resampling period sequence [6]. Assuming that the sampling frequency of the sensor is F, the time for an ordinary person to complete a motion cycle is T_data (estimable), and each element function of the sample function matrix S is discretised. In other words, the function of each measurement unit of the sensor determines the period of resampling, which starts at the starting point t^*; and the periodic data of the kth measurement unit can be expressed as (14) υk=(ak(t*),ak(t*+d),⋯,ak(t*+l−1)) {\upsilon _k} = ({a_k}({t^*}),{a_k}({t^*} + d), \cdots ,{a_k}({t^*} + l - 1)) Among them k = 1,2, …,M,d is the sampling interval, l represents the period length (the number of resampling points) when the sampling interval d = 1, and l is determined by the sampling frequency F and the motion cycle time T_data, i.e. l = F × T_data. The feature vector is extracted for each behaviour sample. First, according to Eq. (14), a vector is constructed in the form of Eq. (15) for each sensor, as follows: (15) Vj=(υ1,υ2,⋯,υM),j=1,2,⋯,N {V^j} = ({\upsilon _1},{\upsilon _2}, \cdots ,{\upsilon _M}),j = 1,2, \cdots ,N N represents the total number of sensors, and M represents the total number of measurement units for each sensor. Then the V^j of all sensors can be arranged into the following row vector: (16) V=(V1,V2,⋯,VN) V = ({V^1},{V^2}, \cdots ,{V^N}) In Eq. (16), we arrange the periodic data sequence of all measurement units of all sensors in a sample into an N × M × [l/d] dimensional vector, which is used as the feature vector of the sample for subsequent classification and recognition.

This paper uses the SVM method with the radial basis function as the kernel function for classification. For the radial basis kernel function, the parameter C in Eq. (6) and the parameter σ in Eq. (7) are essential for the SVM. The influence is more significant, but the value of the parameter is often judged based on experience. This paper uses the K-fold cross-validation (K-CV) method to select the optimal parameters. This method can effectively avoid the occurrence of over-learning and under-learning and thus obtain an ideal classification model.

This article uses Windows 7 64-bit operating system and Matlab software to conduct experiments. The data set was used from the wearable sensor behaviour recognition database (WARD) provided by the University of Berkeley in the United States. Sensors are bound to five parts of the wrist, waist, left ankle, and right ankle [7]. Each sensor is composed of a three-axis accelerometer and a two-axis gyroscope, i.e. composed of five measurement units. The sensor sampling frequency is 30 Hz. The data constituting the set encompasses 13 types of daily behaviours. When collecting data, 20 collectors (13 men and 7 women) must do each action five times. Therefore, the data set includes a total of 1,300 samples. This article selects 10 types of dynamic behaviours, with 1000 samples; see Table 1.

WARD uses five sensors to collect data. Each sensor has five measurement units (a three-axis accelerometer and a two-axis gyroscope can measure acceleration in three directions and angular velocity in two directions). Therefore, Eq. (13) gives the values of both N and M as 5.

When the sampling frequency F = 30 Hz, good experimental results have been obtained by taking 45 points in the sensor sequence, and the time of 1.5 s is close to the time to complete a behavioural action. In this article, the time for an ordinary person to complete a motion cycle is T_data = 1.5 s. In Eq. (14), F takes the value 30, d takes the value 1 and l = F × T_data = 45. In summary, the value of 45 points in this paper is used as the data length of a period.

Therefore, the length of the eigenvector in Eq. (16) is N ×M ×[l/d] = 1125. To calibrate the starting point of the period, it is necessary to select one of the element functions in the sample function matrix of Eq. (16) as the starting point of the period, and the starting point of the period as the starting point for the resampling of each element function in the sample function matrix. Since the acceleration function of the human body in the vertical direction can better reflect the difference of different behaviours and steps, the starting point calibrated by this function is the person in the movement process, at the time when the foot is raised to the highest point [8]. Figure 3 is a comparison between the left foot data and the waist data. From Figure 3, it can be seen that the acceleration data change in the vertical direction of the footsteps is more evident than the data change in the vertical direction of the waist. Therefore, this paper uses the vertical acceleration function of the left ankle sensor in the WARD data set as the function of the starting point of the calibration cycle, which is more accurate.

Fig. 3

Figure 4 shows the result of selecting a sample of the collector walking normally, extracting the acceleration data in the vertical direction of the sensor at the left ankle (that is, the data of a single measurement unit), and performing periodic extraction. Figures 4(a) and 4(b) show that the algorithm in this paper achieves a relatively accurate periodic extraction of the motion data function.

Fig. 4

Figure 5 shows all samples of normal walking behaviour and upstairs behaviour. The acceleration data in the vertical direction of the left foot sensor is periodically extracted [9]. It can be seen from Figure 5 that the sample data are aligned at the uniform starting point of the cycle, the data length is approximately one cycle and the periodic data of different behaviours have morphological differences.

Fig. 5

In the simulation experiment, we adopted the K-CV method to select test samples and training samples. During the experiment, we divided the data set into K = 20 groups. Each time, 1 group was selected as the test sample, and the remaining 19 groups were used as the training sample, giving rise to a total of 20 experiments. Experiments show that the algorithm in this paper has a recognition rate of 97.5% for periodic human behaviours based on the WARD data set. The specific results are shown in Table 2.

Table 3 and Figure 6 show the results of simulation experiments with different algorithms in the same environment using the WARD data set in Table 3. As shown in Table 3 and Figure 6, in terms of algorithm recognition rate, the recognition rate of the algorithm in this paper is high. It is similar to traditional algorithms [10]. Therefore, it cannot be compared with traditional classification methods in terms of algorithm performance.

Fig. 6

The choice of the classifier is significant for behaviour classification. Table 4 is the result of classification using different classifiers (SVM, DT, naive Bayes, K-nearest neighbour) after extracting the feature vector according to the algorithm of this paper. From Table 4, It can be seen that based on the feature vectors extracted by the algorithm in this paper, the SVM method can achieve a higher recognition rate.