Table 4.

Features for fall detection experiments.

No.	Feature	Equation
F1	Mean	$\mu =\frac{1}{N}\sum _{i=1}^N{x}_i$
F2	Standard deviation	$\sigma =\sqrt{\frac{1}{N}\sum _{i=1}^N{\left(x_i-\mu \right)}^2}$
F3	Variance	${\sigma }^2=\frac{1}{N}\sum _{i=1}^N{\left(x_i-\mu \right)}^2$
F4	Standard deviation magnitude	$\left|\sigma \right|=\sqrt{{\sigma }_x^2+{\sigma }_y^2+{\sigma }_z^2}$
F5	Sum vector magnitude	$\left|a\right|=\sqrt{a_x^2+a_y^2+a_z^2}$
F6	Sum vector on horizontal plane	${\left|a\right|}_h=\sqrt{a_x^2+a_z^2}$
F7	Standard deviation of sum vector magnitude	${\sigma }_{\left|a\right|}=\sqrt{\frac{1}{N}\sum _{i=1}^N\left({\left|a\right|}_i-{\mu }_{\left|a\right|}\right)}$
F8	Difference between maximum and minimum values of sum vector magnitude	Δ|a|max−min = max(|a|)−min(|a|)
F9	Root mean square of sum vector magnitude	${\left|a\right|}_{\text{r}ms}=\sqrt{\frac{1}{N}\sum _{i=1}^N{\left|a\right|}_i^2}$
F10	Signal magnitude area	$\mathit{\text{SMA}}=\frac{1}{t}\left(\underset{0}{\overset{t}{\int }}\left|a_x\left(t\right)\right|dt+\underset{0}{\overset{t}{\int }}\left|a_y\left(t\right)\right|dt+\underset{0}{\overset{t}{\int }}\left|a_z\left(t\right)\right|dt\right)$
F11	Activity signal magnitude area	$\text{ASMA}=\frac{1}{t_2-t_1}\left(\underset{t_1}{\overset{t_2}{\int }}\sqrt{a_x^2\left(t\right)+a_y^2\left(t\right)+a_z^2\left(t\right)dt}\right)$
F12	Reference velocity	${\upsilon }_{\mathit{\text{ref}}}=\underset{t_{\mathit{\text{rest}}}}{\overset{t_{\mathit{\text{tilt}}}}{\int }}\left(\left|a\left(t\right)\right|-g\right)dt$
F13	Velocity	υ =∫(|a(t)| − g)dt
F14	Velocity (approximate)	${\upsilon }_2=\sqrt{{\left(\int a_x\left(t\right)dt\right)}^2+{\left(\int a_y\left(t\right)dt\right)}^2+{\left(\int a_z\left(t\right)dt\right)}^2}-\int \mathit{\text{gdt}}$
F15	Vertical acceleration	$a_v=\frac{1}{2g}\left({\left|a\right|}^2-{\left|a\right|}_{\mathit{\text{dynamic}}}^2-g^2\right)$
F16	Maximum vertical acceleration	(av)max=max(az)
F17	Average acceleration change	$\overline{\mathrm{\Delta }\left|a\right|}=\frac{1}{T_n-T_0}\sum _{i=0}^{n-1}\left(\left|a\left(t+1\right)\right|-\left|a\left(t\right)\right|\right)$
F18	Overall acceleration value	aoverall = E[||a|2−E[|a|2]|]
F19	Acceleration amplitude at absolute vertical direction	|av|=|ax sinθz + ay sinθy −az cosθy cosθz|
F20	Angle between device and ground	ρx = sin(ax), ρy = sin(ay), ρz = sin(az)
F21	Angle between device and gravity	${\theta }_x={sin}^{-1}\left(\frac{a_x}{g}\right),{\theta }_y={sin}^{-1}\left(\frac{a_y}{g}\right)$
F22	Angle between z axis and vertical (with respect to the gravity)	$\theta =\text{atan2}\left(\sqrt{{\text{a}}_x^2+{\text{a}}_y^2},{\text{a}}_z\right)$
F23	Tilt angle (with respect to the gravity)	θ=cos−1(aZ)
F24	Inclination angle (with respect to the gravity)	$\theta =co{s}^{-1}\left(\frac{a_z}{g}\right)$
F25	Posture (inclination angle with respect to the gravity, calculated using dot-product method)	$\theta \left(t\right)={cos}^{-1}\left(\frac{{\overrightarrow{g}}_s\left(t\right)\cdot {\overrightarrow{g}}_r}{\left|{\overrightarrow{g}}_s\left(t\right)\right|\cdot \left|{\overrightarrow{g}}_r\right|}\right)\left(\frac{180}{\pi }\right)$
F26	Orientation of person's trunk (with respect to the ground)	$\rho =ta{n}^{-1}\left(\frac{\sqrt{a_x^2+a_y^2}}{a_z}\right)$
F27	Device orientation change	$\theta =co{s}^{-1}\left(\frac{a_x{\mu }_x+a_y{\mu }_y+a_z{\mu }_z}{\sqrt{{\mu }_x^2+{\mu }_y^2+{\mu }_z^2}.\sqrt{a_x^2+a_y^2+a_z^2}}\right)\left(\frac{180}{\pi }\right)$
F28	Orientation change	θ = ā(tb) ā(ta)
F29	Orientation angle (with respect to the gravity)	$\theta =co{s}^{-1}\left(\frac{a_z}{\sqrt{a_x^2+a_y^2+a_z^2}}\right)$
F30	Ratio between two consecutive angles	${\theta }_{\mathit{\text{ratio}}}=\frac{\theta \left(t_i\right)}{\theta \left(t_{i+1}\right)}$
F31	Difference between two consecutive angles	Δθ=θ(ti+1) − θ(ti)
F32	Sagittal angle (with respect to the gravity)	${\theta }_s=-ta{n}^{-1}\left(\frac{a_y}{a_z}\right)\left(\frac{180}{\pi }\right)$
F33	Lateral angle (with respect to the gravity)	${\theta }_l=ta{n}^{-1}\left(\frac{a_x}{\sqrt{1-a_x^2}}\right)\left(\frac{180}{\pi }\right)$
F34	Horizontal angle from x-axis in xy-plane	${\theta }_h=ta{n}^{-1}\left(\frac{a_x}{a_y}\right)$
F35	Vertical angle from x-axis	${\theta }_v=si{n}^{-1}\left(\frac{\sqrt{{a_x}^2+{a_y}^2}}{\left|a\right|}\right)=co{s}^{-1}\left(\frac{a_z}{\left|a\right|}\right)$
F36	Jerk (rate of acceleration change)	$\frac{\mathrm{\Delta }a}{\mathrm{\Delta }t}=\frac{a_x\left(t_i\right)-a_x\left(t_{i-1}\right)}{0.001}$
F37	Trunk angle	${\theta }_{\mathit{\text{pitch}}}=\underset{t=-1.2s}{\overset{t=0.5s}{\int }}{\omega }_{\mathit{\text{pitch}}}\left(t\right)dt,{\theta }_{\mathit{\text{roll}}}=\underset{t=-1.2s}{\overset{t=0.5s}{\int }}{\omega }_{\mathit{\text{roll}}}\left(t\right)dt$
F38	Trunk angular acceleration	${\alpha }_{\mathit{\text{pitch}}}=\frac{d}{dt}{\left\{{\omega }_{\mathit{\text{pitch}}}\right\}}_{-0.5s}^{0.5s},{\alpha }_{\mathit{\text{roll}}}=\frac{d}{dt}{\left\{{\omega }_{\mathit{\text{roll}}}\right\}}_{-0.5s}^{0.5s}$
F39	Resultant angular acceleration	${\alpha }_r=\sqrt{{\alpha }_{\mathit{\text{pitch}}}^2+{\alpha }_{\mathit{\text{roll}}}^2}$
F40	Resultant angular velocity	${\omega }_{\text{r}}=\sqrt{{\omega }_{\mathit{\text{pitch}}}^2+{\omega }_{\mathit{\text{roll}}}^2}$
F41	Resultant change in trunk angle	${\theta }_{\text{r}}=\sqrt{{\theta }_{\mathit{\text{pitch}}}^2+{\theta }_{\mathit{\text{roll}}}^2}$
F42	Differential pressure	$\mathrm{\Delta }{p}_i=\frac{t}{2}\left[\left(\sum _{k=i}^{k=i+(2/t)}{p}_k-\sum _{k=i-(t/2)}^{k=i}{p}_k\right)\right]$
F43	Multiple regression equation	Y = −0.139 + 0.0195X1 + 0.0163X2
F44	Maximum acceleration derivative	N/A
F45	Maximum peak-to-peak acceleration amplitude	N/A
F46	Maximum peak-to-peak acceleration derivative	N/A
F47	Timestamp of falling body to be at rest	N/A
F48	Timestamp of body's initial contact to ground	N/A
F49	Time difference between when inclination angle exceed a critical angle and inclination velocity has local maximum above a threshold	N/A
F50	Variation of |a| around 1 g	N/A

Notes: N = number of data samples, x = observation, i = index of data sample, g = 9.81 ms⁻², a_x, a_y, a_z, are acceleration values along the x- (sideward), y-(forward), and z- (upward) axes, respectively, a_¯ = average acceleration vector, t_a = time before fall, t_b = time after fall, t_tilt = time when body tilts, t_rest = initial time when body is at rest, g⃗_s = gravity vector estimated with respect to the body segment, g⃗_r = the reference gravitational vector, X₁ = the absolute peak value in the movement direction, X₂ = the absolute peak value in the horizontal direction.