Mathematical function data model analysis and synthesis system based on short-term human movement

The FDA method is a statistical method that has emerged in recent years. It is a development and extension of traditional statistical analysis methods. FDA is often used for economic data, meteorological data etc., but the application to sports data series is rare [3]. The changes of speed, acceleration and other variables in the process of human movement are continuous, while the corresponding data collected by sensors are all discrete. Because of this, the FDA's research on exercise data is reasonable. The general process of the FDA is as follows. For an observation data series, y = (y₁,y₂, …, y_n)^T, we build 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 function x (t) of the observation data sequence at t_i, and ∈_i represents the noise. To estimate the value of x (t_i) in formula (1), we use the method of basis function expansion to select a set of basis functions Φ(t) = (φ₁(t),φ₂(t), …, φ_K (t)). At the same time, the discrete data is converted into a linear combination of basis functions, (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 [4]. The most direct method to solve the coefficient vector is the least-squares method, and the solution is as follows to minimise the residual sum of squares (3) SMSSE(y|c)=∑i=1n[yi−∑k=1Kckϕk(ti)]2 SMSSE(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. (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)

Different sample function matrices are obtained according to different types of behaviour data. However, the morphological curve characteristics of the corresponding functions in different sample function matrices are different [8]. This difference in morphology is the basis for the classification of sample sets, and this difference can be reflected in one movement cycle as shown in Figure 2. Therefore, we can distinguish between different behaviours by extracting only one period of data.

Fig. 2

We select one element f (t) from the sample function matrix of Eq. (11) and find its characteristic points. This paper selects the extreme point set f (t) as the feature point set of the function and then selects the point in the feature point set that makes f (t) to obtain the minimum value the starting point of the motion cycle [9]. Finally, each element function in the sample function matrix uses this starting point as the starting point of the motion cycle.

We choose the minimum extreme point as the starting point of the cycle and achieve data alignment. The minimum extreme point of the acceleration function in the vertical direction of the foot is the point where the acceleration value is the smallest during the movement. This time point often corresponds to when the person's foot is raised to the highest point. All the periodic behaviours based on footsteps include this movement process [10]. Therefore, we use this point as the starting point of the movement, which can align all sample functions of the same action here and also align periodic functions of different behaviours at this starting point.

This article uses Windows7 64-bit operating system and Matlab software for experiments. The data set we used comes from the wearable sensor behaviour recognition database (WARD) provided by the University of Berkeley in the United States. This data set binds sensors to 5 parts of the body namely left wrist, right wrist, waist, left ankle and right ankle. Each sensor consists of a three-axis accelerometer and a two-axis gyroscope. The sampling frequency is 30 Hz. The data set includes 13 types of daily behaviours. When collecting data, 20 collectors are required to do each action 5 times. Therefore, a total of 1,300 samples are included in the data set. The 1000 samples of 10 types of dynamic behaviours selected are shown in Table 1.

WARD uses 5 sensors to collect data and each sensor has 5 measurement units. Therefore, the values of N and M are 5. So the sample function matrix is stated as follows: (12) S=[f11(t)f21(t)f31(t)f41(t)f51(t)f12(t)f22(t)f32(t)f42(t)f52(t)f13(t)f23(t)f33(t)f43(t)f53(t)f14(t)f24(t)f34(t)f44(t)f54(t)f15(t)f25(t)f35(t)f45(t)f55(t)] S = \left[ {\matrix{ {f_1^1(t)} & {f_2^1(t)} & {f_3^1(t)} & {f_4^1(t)} & {f_5^1(t)} \cr {f_1^2(t)} & {f_2^2(t)} & {f_3^2(t)} & {f_4^2(t)} & {f_5^2(t)} \cr {f_1^3(t)} & {f_2^3(t)} & {f_3^3(t)} & {f_4^3(t)} & {f_5^3(t)} \cr {f_1^4(t)} & {f_2^4(t)} & {f_3^4(t)} & {f_4^4(t)} & {f_5^4(t)} \cr {f_1^5(t)} & {f_2^5(t)} & {f_3^5(t)} & {f_4^5(t)} & {f_5^5(t)} \cr } } \right] When sampling frequency F = 30 Hz, 45 points in the sensor sequence have been obtained with good experimental results. Note that1.5 s is close to the time to complete an action. In our experiment, the time for an ordinary person to complete an exercise cycle is T_data = 1.5 s. In this paper, 45 points are used as the data length of a period. From Eq. (9), the length of the feature vector is N × M × [l/d] = 1125.

To calibrate the starting point of the cycle, it is necessary to select one of the element functions in the sample function matrix of Eq. (10) as the cycle's starting point. We use this period starting point 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 of the calibration of this function is when a person's foot is raised to the highest point during exercise [11]. Therefore, we use 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 to be more accurate.

Figure 3 shows a sample of the collector walking normally. We extract the acceleration data of the sensor at the left ankle in the vertical direction and periodically extract the results. From Figure 3, it can be seen that the algorithm used by us achieves a more accurate function of the motion data – ground cycle extraction.

Fig. 3

In the simulation experiment, the K-CV method is adopted to select test samples and training samples. In the experiment, the data set is divided into K = 20 groups. We conducted a total of 20 experiments and they show that the algorithm used by us has a recognition rate of 97.5% for periodic human behaviour based on the WARD dataset. The specific results are shown in Table 2.

Table 3 and Figure 4 show the simulation results of different algorithms in the same environment using the WARD data set in Table 1. Table 3 and Figure 4 show that the recognition rate of this algorithm is higher than other algorithms [12]. The choice of the classifier is significant for behaviour classification. Table 4 results from a classification using different classifiers after extracting feature vectors according to algorithm used by us. It can be seen from Table 4 that the feature vector extracted by the algorithm can achieve a higher recognition rate.

Fig. 4