Table 6 Parameters used for the extraction of features

MAV—mean absolute value [38, 58, 59]

\(\mathrm{MAV}=\frac{1}{N}\sum_{i=1}^{N}\left|{x}_{i}\right|\) (2)

MAVFD—mean absolute value of the first difference [38, 39, 60]

\(\mathrm{MAVFD}=\frac{1}{N-1}\sum_{i=1}^{N-1}\left|{x}_{i+1}-{x}_{i}\right|\) (3)

MAVSD—mean absolute value of the second difference [38, 60]

\(\mathrm{MAVSD}=\frac{1}{N-2}\sum_{i=1}^{N-2}\left|{x}_{i+2}-{x}_{i}\right|\) (4)

RMS—root mean square [38, 58, 61, 62]

\(\mathrm{RMS}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}{\left({x}_{i}\right)}^{2}}\) (5)

Peak—maximum value of the vector, considering only positive values from the window [38]

\(\mathrm{Peak}=\mathrm{max}{\left\{{x}_{i}\right\}}_{i=1}^{N}\) (6)

Frequency—(Zero Crossing, FMean, FPeak, F50, F80, Power3.5–7.5)

Zero Crossing (ZC) [38, 58, 59, 61]

Given two consecutive samples x_i and x_i+1, the counting of zero crossings, ZC, is increased if:

\(\left\{{x}_{i}>0 \mathrm{and} {x}_{i+1}<0\right\} or \left\{{x}_{i}<0 \mathrm{and} {x}_{i+1}>0\right\}\) (7)

FMean—mean frequency [38, 40, 41, 58, 62]

\(\mathrm{FMean}= \frac{\sum_{i=1}^{N}{(P}_{n}\left(i\right)*{f}_{n}\left(i\right)) }{\sum_{i=1}^{N}{P}_{n}\left(i\right)}\) (8)

where \({P}_{n}\) is the power spectrum; \({f}_{n}\) is the vector frequency of \({P}_{n}.\)

FPeak—frequency at which maximum power occurs [40, 41, 63]

\(FPeak=fn \mathrm{where} \left\{\mathrm{max}{\left\{{Pn}_{i}\right\}}_{i=1}^{N}\}\right.\) (9)

F50—median frequency [38, 40, 41, 58, 62, 63]

\(F50=\sum_{i=1}^{F50}{P}_{n}\left(i\right)= \sum_{F50}^{N}{P}_{n}\left(i\right)=\frac{1}{2}\sum_{i=1}^{N}Pn(i)\) (10)

F80—total power frequency of Pn below 80% [41, 64]

\(F80=frequency \mathrm{where} \left\{\sum_{i=1}^{F80}{P}_{n}\left(i\right)= 0.8*\sum_{i=1}^{N}Pn(i)\right\}\) (11)

Power3.5–7.5—Power in frequency band 3.5–7.5 Hz [42]

\(\mathrm{Power}3.5\_7.5=\sum_{{f}_{n}=3.5}^{{f}_{n}=7.5}Pn(i)\) (12)

Statistic- (VAR, RANGE, INTQ, SKEW, KURTOSIS)

VAR—variance [38, 58, 61]

\(\mathrm{VAR}={\sigma }^{2}=\sum_{i=1}^{N}{\left({x}_{i}-\overline{x }\right)}^{2}\) (13)

where \(\overline{x }\)—mean of the signal and \(\sigma \)—standard deviation

RANGE—amplitude range [26, 38]

\(\mathrm{RANGE}=\mathrm{max}{\left\{{x}_{i}\right\}}_{i=1}^{N}-\mathrm{min}{\left\{{x}_{i}\right\}}_{i=1}^{N}\) (14)

INTQ—interquartile range [38, 65, 66]

\(\mathrm{INTQ}=Q3-Q1\) (15)

where \({Q}_{3}\) is the third quartile and \({Q}_{1}\) is the first quartile

SKEWNESS—asymmetry [39, 64, 67]

\(\mathrm{SKEWNESS}=\frac{\frac{1}{n}\sum_{i=1}^{N}{\left({x}_{i}-\overline{x }\right)}^{3}}{{\sigma }^{3}}\) (16)

KURTOSIS—flattening [39, 64, 67]

\(\mathrm{KURTOSIS}=\frac{\frac{1}{n}\sum_{i=1}^{N}{\left({x}_{i}-\overline{x }\right)}^{4}}{{\sigma }^{4}}\) (17)

Entropy—(approximate entropy and fuzzy entropy)

ApEn—approximate entropy [38, 61, 64, 68, 69]

Entropy is an analysis tool used with goal of quantifying the regularity of a signal, returning a value between 0 and 2, where 0 indicates signal predictability based on previous values and 2 indicates signal unpredictability [70]

Given a time series composed of N samples {x(1), x(2), …, x(N)} and m a sequence of vectors starting from x(1) until x(N-m + 1), defined by

\(x\left(i\right)=\left\{x\left(i\right), x\left(i+1\right), \dots , x\left(i+m-1\right)\right\}, i=1, \dots , N-m+1\)

The distance between two vectors x(i) and x(j), is defined as being the maximum distance between such elements—d[x(i), x(j)].

For each value of i smaller than N-m + 1, calculate \({C}_{i}^{m}\), defined as

\(number of j such as:(d\left[x\left(i\right), x(j)\le r)/(N-m+1)\right]\)

Following this, calculate \({C}^{m}\left(r\right)\) given by

\({C}^{m}\left(r\right)={\left(N-m+1\right)}^{-1}\sum_{i=1}^{N-m+1}ln{C}_{i}^{m}(r)\).

The approximate entropy is given by (18)

\(ApEn\left(m,r,N\right)={C}^{m}\left(r\right)-{C}^{m+1}(r)\) (18)

where m—window length; r—tolerance and ln is the natural logarithm

FuzzyEn—fuzzy entropy [38, 61, 71]

Given a time series composed of N samples {x(1), x(2), …, x(N)} and m a sequence of vectors starting from x(1) until x(N-m + 1), calculate the degree of similarity between the vectors x(i) and x(j) defined by the fuzzy function

\({d}_{[x\left(i\right),x\left(j\right)]}^{m}=\mu ({d}_{ij}^{m},r)\)

where \({d}_{ij}^{m}\) is the largest difference between the elements of vectors x(i) and x(j).

For each vector x(i) calculate the mean of all degrees of similarity with its neighbors (j ≠ i)

For each value of i smaller or equal to N-m + 1, calculate \({P}_{i}^{m}\)(r), given by

\({P}_{i}^{m}\left(r\right)={\left(N-m+1\right)}^{-1}\sum_{j=1}^{N-m}{d}_{[x\left(i\right),x\left(j\right)]}^{m}\)

\({P}^{m}\left(r\right)={\left(N-m\right)}^{-1}\sum_{i=1}^{N-m}{P}_{i}^{m}\left(r\right)\)

\({P}^{m+1}\left(r\right)={\left(N-m\right)}^{-1}\sum_{i=1}^{N-m}{P}_{i}^{m+1}\left(r\right)\)

Fuzzy entropy is given by (19)

\(FuzzyEn\left(m,r,N\right)=\mathit{ln}{P}^{m}(r)-\mathit{ln}{P}^{m+1}(r)\) (19)