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Table 8 Summary of the invariant domain

From: Review and classification of variability analysis techniques with clinical applications

Domain Assumptions

Features

Feature Assumptions

Transformation used

References

The information is held in those properties of the time series that are invariant-i.e. not supposed to change over either time or space.

Allan scaling exponent

The time series is modeled as a point process, and the ratio between the second order moment of the difference between the number of events of two successive windows and the mean number of events has scale-invariant properties

Point process

[49, 89, 90]

 

Detrended fluctuation analysis

The standard deviation of the detrended cumulative time series has scale-invariant properties

 

[4, 63, 92–94]

 

Diffusion Entropy

The time series is modeled as a family of diffusion processes, which Shannon entropies has scale-invariant properties

 

[54, 95–97]

 

Embedding scaling exponent

The variance of the attractor at different embedding dimensions has scale-invariant properties

Phase space representation

[98]

 

Fano scaling exponent

The time series is modeled as a point process, and the variance of the number of events divided by the mean number of events has scale-invariant properties

Point process

[49, 89, 90]

 

Higuchi's algorithm

The length of the time series at different windows has scale-invariant properties

 

[54, 101–104]

 

Index of variability

The time series is modeled as a point process, and the variance of the number of events has scale-invariant properties

Point process

[2]

 

Multifractal exponents

Multiple scaling exponents characterize the time series

Wavelet transform

[54, 105, 106]

 

Power spectrum scaling exponent

Stationarity, the power spectrum follows a 1/fb like behaviour

Power spectrum

[92]

 

Probability distribution scaling exponent

The distribution of the data has scale--invariant properties

Bin transformation

[63, 107]

 

Rescaled detrended range analysis

The range (difference between maximum and minimum value) of a time series has scale-invariant properties

 

[92]

 

Scaled windowed variance

The standard deviation of the detrended time series has scale-invariant properties

 

[92]

 

Correlation dimension

The time series is extracted from a dynamical system, and the number of points in the phase space that are closer than a certain threshold has scale-invariant properties

Phase space representation

[40, 91]

 

Finite growth rates

The time series is extracted from a dynamical system, which is described by its dependence on the initial conditions (how two points that are close in space and time separate after a certain amount of time)-the ratio between the final and the initial time is an invariant of the system

Phase space representation

[86]

 

Kolmogorov-Sinai entropy

The time series is extracted from a dynamical system, and it is possible to predict which part of the phase space the dynamics will visit at a time t+1, given the trajectories up to time t

Phase space representation

[40, 91]

 

Largest Lyapunov exponent

The time series is extracted from a dynamical system, which is described by its dependence on the initial conditions (how two points that are close in space and time separate after a certain amount of time)-the distance grows on average exponentially in time, and the exponent is an invariant of the system

Phase space representation

[40, 41, 99, 100]