Table 4.

Feature extraction from time-series data of multiple sensors

Features Name	Description	Equations
Mean	Mean value calculation for accelerometer and gyroscope	$$\mu =\frac{1}{N}\sum _{i=1}^{N}S{x}_i$$ (2)
Standard deviation	Finds the sensor’s data spread around the mean.	$$\sigma =\sqrt{\frac{1}{N}\sum _{i=0}^{N-1}{(S{x}_i-\mu )}^2}$$ (3)
Skewness	The measure for the degree of symmetry in the variable distribution.	$$S{k}_x=\frac{{\sum }_{i=1}^N{(S{x}_i-\mu )}^3}{{\left({\sigma }_x\right)}^3}$$ (4)
Kurtosis	The measure of tailedness in the variable distribution.	$$K{t}_x=\frac{{\sum }_{i=1}^N{(S{x}_i-\mu )}^4}{{\left(N\sigma \right)}^4}$$ (5)
Maximum value	Calculates the maximum value of the accelerometer (x,y,z).	$$Ac{c}_{max}=maxS{x}_i$$ (6)
Minimum value	Shows the minimum value of the accelerometer (x,y,z).	$$Ac{c}_{min}=minS{x}_i$$ (7)
Entropy	Essential for differentiating between activities.	$$Entropy=-\frac{1}{N}\sum _{i=0}^{N-1}p{x}_{i}logp{x}_i$$ (8)
Cosine Similarity	To distinguish between activities that fluctuate along an axis.	$$cos\theta =\frac{S_x.\ast S_y}{\left|\left|S_x\right|\right|\left|\left|S_y\right|\right|}$$ (9)
Root mean square	Calculates the angular movement along the x, y, and z axes, accordingly.	$$RM{S}_x=\sqrt{\frac{1}{N}\sum _{i=1}^{N}G{x}_i}$$ (10) Where $G{x}_i$ is sample of x-axis gyroscope.
The absolute time difference between peaks	Computed by taking the absolute difference between the maximum and minimum peak times.	$$ATD=\left|t_{maxpeak}-t_{minpeak}\right|$$ (11)
Frequency domain features	To find frequency domain features of acceleration data based on fast Fourier transform (FFT).	$$H\left(k\right)=\sum _{n=0}^{N-1}x\left(n\right)e^{-j2\pi \left(\frac{kn}{N}\right)}$$ (12)
Quartile Range	To find the middle number between the minimum and the median of the sample data.	$$Q_1=l+\frac{h}{f}(\frac{n}{4}-C)$$ (13)