Research on the Accurate Measurement Method of Athletes’ Physical Consumption Using Intelligent Wearable Devices in Table Tennis Training

Linear regression modeling:

Regression algorithms are relative to categorical algorithms and are related to the type of value of the target variable Y that you want to predict. If the target variable Y is a categorical variable, such as predicting the gender of the user, predicting the color of the flowers, etc., a classification algorithm is needed to fit the data and make a prediction; if Y is a continuous variable, such as predicting the revenue of the product, predicting the performance of the sales, etc., a regression model is needed [25]. Linear regression is a simple, versatile and easy to understand model. The equation is obtained by using Y as the dependent variable and X as the independent variable: (1)Y=α+βX$$Y = \alpha + \beta X$$

When parameters α and β are given, a straight line is displayed within the coordinate plot. When only one value of X is used to predict the value of Y, it is called a one-way regression. Univariate linear regression is a calculation to find the most appropriate a straight line to fit the data. Suppose there is a scatter plot based on a set of data, one-way linear regression is to find a straight line that fits the data points in the scatter plot as closely as possible, and the difference between the calculated theoretical data and the actual value is the error, which is denoted by μ in statistics. Thus the equation is obtained: Y = α + βX + μ. The point of linear regression analysis is to find the best straight line that fits the actual data. Here it is necessary to introduce the concept of residuals, the residuals are the difference between the true value and the predicted value, which can be understood as the distance or gap between the two, as shown in the following equation: (2)e=y−y^$$e = y - \hat y$$

For the dependent variable x₁, there is a corresponding independent variable y₁, and the prediction of the resulting y^i$${\hat y_i}$$, after calculating the value of ei=yi−y^i$${e_i} = {y_i} - {\hat y_i}$$, and then square it, and then calculate each point in the data, and finally add up all the ei2$$e_i^2$$ can be calculated to fit a straight line and the actual value of the error between. The so-called best-fitting straight line is the smaller the value of the calculated sum of squares of the residuals, i.e. the smaller the error the better the fit. In order to calculate the minimum sum of squares of the residuals, you can make Q=∑i=1n[yi−(α+βxi)]2$$Q = \sum\limits_{i = 1}^n {{{\left[ {{y_i} - \left( {\alpha + \beta {x_i}} \right)} \right]}^2}}$$, the use of calculus for the principle of the extreme value of the α and β partial derivatives, and make its first-order derivatives equal to 0 to solve α and β, that is, the least squares estimation method: (3)∂Q∂α=2Σ(yi−α−βxi)(−1)=0$$\frac{{\partial Q}}{{\partial \alpha }} = 2\Sigma \left( {{y_i} - \alpha - \beta {x_i}} \right)( - 1) = 0$$ (4)∂Q∂β=2Σ(yi−α−βxi)(−xi)=0$$\frac{{\partial Q}}{{\partial \beta }} = 2\Sigma \left( {{y_i} - \alpha - \beta {x_i}} \right)\left( { - {x_i}} \right) = 0$$

When there are two or more independent variables, a computational analysis of multiple linear regression is required. Multiple linear regression analysis addresses the following problems: (1) Determining whether there is a correlation between several specific variables. (2) Predicting or controlling the value of another variable based on the value of one or more variables, with the possibility of knowing its accuracy. (3) Factor analysis, which factors are significant and which are minor with respect to the common effects among several independent variables of a variable.

In multiple linear regression, a dependent variable begins to be influenced by more than one independent variable, so its equation takes the form: (5)y=β0+β1x1+⋯βnxn+ε$$y = {\beta _0} + {\beta _1}{x_1} + \cdots {\beta _n}{x_n} + \varepsilon$$ (6)E(y)=β0+β1x1+⋯βnxn$$E(y) = {\beta _0} + {\beta _1}{x_1} + \cdots {\beta _n}{x_n}$$

Since an observation in multiple linear regression is no longer a scalar but a vector, the observations of the independent variables become (1,x11,⋯,x1n)$$\left( {1,{x_{11}}, \cdots ,{x_{1n}}} \right)$$, (1,x21,…,x2p)$$\left( {1,{x_{21}}, \ldots ,{x_{2p}}} \right)$$, while the observations of the corresponding dependent variables remain unchanged, so that each row of these observations is superimposed to become a vector or matrix. (7)y=[ y1 y2 ⋮ yn]X=[ 1 x11 … x1p 1 x21 x2p ⋮ ⋮ 1 xn1 ⋯ xnp]$$y = \left[ {\begin{array}{*{20}{c}} {{y_1}} \\ {{y_2}} \\ \vdots \\ {{y_n}} \end{array}} \right]X = \left[ {\begin{array}{*{20}{c}} 1&{{x_{11}}}& \ldots &{{x_{1p}}} \\ 1&{{x_{21}}}&{}&{{x_{2p}}} \\ \vdots &{}&{}& \vdots \\ 1&{{x_{n1}}}& \cdots &{{x_{np}}} \end{array}} \right]$$ (8)∈=[ ∈1 ∈2 ⋮ ∈n]β=[ β0 β1 ⋮ βn]$$ \in = \left[ {\begin{array}{*{20}{c}} {{ \in _1}} \\ {{ \in _2}} \\ \vdots \\ {{ \in _n}} \end{array}} \right]\beta = \left[ {\begin{array}{*{20}{c}} {{\beta _0}} \\ {{\beta _1}} \\ \vdots \\ {{\beta _n}} \end{array}} \right]$$

After determining the multiple regression equation, the coefficient of determination r² and the standard error of estimate can be used to determine the goodness of fit of the equation. The coefficient of determination is used to explain how much of the change in Y is explained by the change in the selected independent variable, and if the value of r² is too large, the presence of multicollinearity between the variables needs to be considered; the standard error of estimation is used to explain the average deviation between the actual value of Y and the estimated value of Y_i when the independent variable is given.

Multiple linear regression equations and univariate linear models are tested in the same way by using t to test each partial regression coefficient and F to test the significance of the entire multiple regression model. The significance test of the equation is used to test whether the equation is valid or not, if the test shows that it is not significant, then the equation is not valid. F test: used for all the independent variables (x1,⋯,xn)$$\left( {{x_1}, \cdots ,{x_n}} \right)$$ in the whole for the y linear significance, F test need to look at the Significant F value. P-value and Significant F value is generally required to be less than 0.05, and the smaller the result the more significant.

After determining the multiple regression equation, it is also necessary to calculate the variance inflation factor (VIF) of each independent variable to determine whether there is a correlation between the independent variables. If there is a high degree of correlation or complete correlation between the independent variables, then there is multicollinearity among the independent variables, and the effects of each variable on the dependent variable cannot be accurately distinguished. Based on the variance inflation factor, it is possible to determine whether there is multicollinearity between each independent variable. It is determined whether the VIF of each independent variable is greater than 5, if so, this variable needs to be eliminated and if the VIF is less than 5, then it can be assumed that the multiple regression equation that has been determined does not suffer from severe multicollinearity.

Residual analysis is used to evaluate the fit of the linear regression model to the actual data. [26]. When testing a multiple linear regression model, the residual plots of the simple linear regression equation between each independent and dependent variable can be analyzed separately. Such a multiple linear regression model can be considered valid if the scatter in the residual plots is shown to be relatively random and when it is not obvious that there is a certain pattern in the scatter.

In this study, different hitting styles were tested at low ball speed, high ball speed and simulated free singles situations. Table 1 shows the predicted MET values for GT3X and the measured MET values for K4b². The average K4b² measured MET values for all phases of combined and free singles were 9.2 and 9.4, which were not significantly different from each other, while there was a significant difference between the predicted MET values of the GT3X for different parts of the body. The predicted values of the waist, thigh and ankle joints in both modes were significantly lower than the predicted values of the wrist and handle sites, which were smaller than the measured values; while the predicted values of the wrist and handle were significantly larger than the measured values.

The degree of error in the MET prediction of the ankle was significantly lower than that of the other sites, while the degree of error in the clapper handle and wrist (overestimation) was significantly greater than that of the waist, thigh, and ankle (underestimation); the variability between the predicted values of the waist, thigh, and ankle was small, whereas the variability between the wrist and the clapper handle was large, with the clapper handle predicted values being significantly greater than those predicted for the wrist in terms of the degree of overestimation.

The results of the paired t-tests are shown in Table 2, with highly significant differences between MET predictions and K4b2 measurements for different sites of GT3X in all phases and free singles mode.

The statistical analysis of the predicted EE values and the measured EE values of K4b² is shown in Table 3, the average measured EE values of K4b² in free singles and in all stages of synthesis were 10.7 kccals/min and 10.5 kccals/min, which were not significantly different from each other, while there was a significant difference between the predicted values of EE in different parts of the GT3X. The predicted values of waist, thigh and ankle joints in both modes were significantly lower than the predicted values of wrist and handle sites and smaller than the measured values; while the predicted values of wrist and handle sites were significantly larger than the measured values.

The degree of error in EE prediction was significantly lower for the ankle than for the other sites, while the degree of error was significantly greater for the clapper handle and wrist (overestimation) than for the waist, thigh, and ankle (underestimation); the variability between the predicted values for the waist, thigh, and ankle was small, while the variability between the wrist and the clapper handle was large, with the clapper handle predicted values being significantly more overestimated than the wrist predicted values.

The results of the paired t-tests are shown in Table 4, with highly significant differences between the predicted values of EE and the K4b² measurements for the different wearing parts of the GT3X in all phases and in free singles mode.

The consistency of the predicted values of EE and MET for GT3X at different wearing sites was further compared with the measured values of K4b², with K4b² and GT3X and the mean as the horizontal coordinate and the difference between K4b² and GT3X as the vertical coordinate, where the scatter falls closer to the 0-difference-mean line in the confidence interval, the better the consistency is.

Figures 1 and 2 show the results of the free singles EE, MET analysis, and Figures 3 and 4 show the results of the combined exercise EE, MET analysis.

Figure 1.

Free single-play EE parts of the consistency scatter diagram

Figure 2.

Free single-play MET parts of the consistency scatter diagram

Figure 3.

Each stage of the EE various bits of the consistency scatter diagram

Figure 4.

Each stage of the MET various bits of the consistency scatter diagram

The results show that, both in free singles and integrated sports mode, the scatters of EE and MET predicted values of GT3X in waist, ankle and thigh areas fall significantly in the region above the 0 value within the 95% confidence interval, which indicates that the consistency with K4b² is low, and that there is a significant underestimation of all the three; the scatters of the predicted values of EE and MET of GT3X in racket handle and wrist areas fall significantly in the region below the 0 value within the 95% confidence interval, and are significantly higher than those of the K4b². The scatter of EE and MET predicted values of GT3X in the handle and wrist area fell in the region below the value of 0 within the 95% confidence interval, and was obviously farther away than the upper region, indicating that it was in lower consistency with K4b², and there was obvious overestimation in both, and the degree of overestimation of the predicted values of the handle was even higher; meanwhile, the degree of error of the predicted values of the handle and the wrist was obviously greater than that of the predicted values of the waist, ankle, and thigh areas.