Table 6 Features extracted from the signals

Root mean square (RMS)

[35,36,37,38]

\(RMS= \sqrt{\frac{1}{N}\sum_{n=1}^{N}{x(n)}^{2}}\)

where \(N\) is the number of elements of \(X\) (\(X=\{{x}_{1},{x}_{2},\dots ,{x}_{n}\}); x\left(n\right)\) is the \(n\)th element

Peak

[35]

Maximum value of the signal

Mean absolute value (MAV)

[35, 36, 39]

The patterns are organized into windows and the average value of each window is used as the value of the feature

\(MAV=\frac{1}{S}{\sum }_{m=1}^{S}\left|\left.{X}_{m}\right|\right.\)

S is the number of samples per window; \({X}_{m}\) is the m-th sample of the window

Mean absolute value of the first difference (MAVFD)

[35, 40, 41]

\(MAVFD=\frac{1}{N-1}\sum_{n=1}^{N-1}\left|x\left(n+1)-\right.\right.\left.x \left(n\right)\right|\)

Mean absolute value of the second difference (MAVSD)

[35, 40]

\(MAVSD=\frac{1}{N-2}\sum_{n=1}^{N-2}\left|x\left(n+2)-\right.\right.\left.x \left(n\right)\right|\)

Mean frequency (FMEAN)

[35, 36, 38, 42, 43]

\(FMEAN=\frac{\sum_{n=1}^{N}{(P}_{n}\left(n\right)*{f}_{n}(n))}{\sum_{n=1}^{N}{P}_{n}\left(n\right)}\)

where \({P}_{n}\) is the Power spectrum; \({f}_{n}\) is the vector frequency of \({P}_{n}\); \(N\) is the number of samples

Zero crossing (ZC)

[35,36,37, 39]

Computes how many times the signal crosses zero

Peak frequency (FPEAK)

[42,43,44]

FPEAK is a frequency at which the maximum power occurs

FPEAK = maximum(\({P}_{n}\))

Median frequency (F50)

[35, 36, 38, 42,43,44]

\(\sum_{n=1}^{F50}{P}_{n}\left(n\right)=\sum_{F50}^{N}{P}_{n}\left(n\right)=\frac{1}{2}*\sum_{n=1}^{N}{P}_{n}\left(n\right)\)

Frequency for which 80% of the total power of P_n is below this value (F80)

[43, 45]

\(\sum_{n=1}^{F80}{P}_{n}\left(n\right)=0.8*\sum_{n=1}^{N}{P}_{n}\left(n\right)\)

Power in frequency band 3.5–7.5 Hz (Power3.5_7.5)

[46]

\(Power3.5\_7.5=\sum_{{f}_{n}=3.5}^{{f}_{n=7.5}}{P}_{n}\left(n\right)\)

Approximate entropy (ApEn)

[35, 37, 45, 47, 48]

According to [48]:

• For a time series of sample N {u(1), u(2), u(3)…u(N)} given m, forms sequences of vectors x(1) through x(N-M + 1), defined by x(i) = {u(i), u (i + 1),…, u (i + m—1)}, i = 1,…, N—m + 1;

• Compute the distance between the vectors x(i) and x(j) defined as the maximum difference between each element of the vectors (d[x(i), x(j)]);

• For each i ≤ N-m + 1, compute \({C}_{i}^{m}(r)\), which is defined as: (\(number of j such as d[x\left(i\right), x\left(j\right)]\le r)/(N-m+1)\);

• Define: \({C}^{m}\left(r\right)={\left(N-m+1\right)}^{-1}{\sum }_{i=1}^{N-m+1}ln{C}_{i}^{m}\left(r\right)\);

• The approximated entropy is defined by:

\(ApEn\left(m,r,N\right)={C}^{m}\left(r\right)-{C}^{m+1}\left(r\right)\)

where m is the length of the comparative window; r is the tolerance; ln is the natural logarithm

Fuzzy entropy (FuzzyEn)

[35, 37, 49]

According to [50, 51]:

• For a time series of sample N {u(1), u(2), u(3)…u(N)} given m, forms sequences of vectors x(1) through x(N-M + 1), defined by x(i) = {u(i), u (i + 1),…, u (i + m—1)}, i = 1,…, N—m + 1;

• Compute the similarity degree between the vectors x(i) and x(j) defined by a fuzzy function: \({d}_{[x\left(i\right),x\left(j\right)]}^{m}=\mu ({d}_{ij}^{m},r\)), where \({d}_{ij}^{m}\) is the maximum difference between each element of the vectors;

• For each vector x(i) average all the similarity degree of its neighboring vectors (i ≠ j);

• For each i ≤ N-m + 1, compute \({P}_{i}^{m}(r)\), which is defined as: \({P}_{i}^{m}\left(r\right)={\left(N-m+1\right)}^{-1}\sum_{j=1,j\ne i}^{N-m}{d}_{[x\left(i\right),x\left(j\right)]}^{m}\);

• Define: \({P}^{m}\left(r\right)={\left(N-m\right)}^{-1}{\sum }_{i=1}^{N-m}{P}_{i}^{m}\left(r\right)\) and \({P}^{m+1}\left(r\right)={\left(N-m\right)}^{-1}{\sum }_{i=1}^{N-m}{P}_{i}^{m+1}\left(r\right)\);

• The fuzzy entropy is defined by:

\(FuzzyEn\left(m,r,N\right)={lnP}^{m}\left(r\right)-{lnP}^{m+1}\left(r\right)\)

Variance (VAR)

[35,36,37]

\({\mathrm{VAR}=\sigma }^{2}=\frac{1}{N-1}{\sum }_{n=1}^{N}\left(x\left(n\right)-\overline{x }\right){ }^{2}\)

\(\overline{x }\)–average of the samples

Range (RANGE)

[35, 52]

Difference between the maximum and minimum value of the signal

Range interquartile (IntlA)

[35, 53, 54]

\(\mathrm{IntIA}= {Q}_{3}-{Q}_{1}\)

\({Q}_{3}\)—third quartile;

\({Q}_{1}\)—first quartile

Skewness (Skewness)

[41, 45, 55]

\(\mathrm{Skewness}=\frac{\frac{1}{n}\sum_{n=1}^{N}{(x\left(n\right)-\overline{x })}^{3}}{{\sigma }^{3}}\)

Kurtosis (Kurtosis)

[41, 45, 55]

\(\mathrm{Kurtosis}=\frac{\frac{1}{n}\sum_{n=1}^{N}{(x\left(n\right)-\overline{x })}^{4}}{{\sigma }^{4}}\)