Moments / Long Range Correlation Measures

 

This group of measures does not represent a distinct class but it is rather a summary of descriptive statistics (simple moment measures of the time series), measures of moments of the first and second differences of the time series (the Hjorth parameters) and measures of long range correlation (Hurst exponent and Detrended Fluctuation analysis).

The Hjorth parameters of mobility and complexity are simple measures of signal complexity based on the second moment of the signal and its first and second difference. These measures have been used in the analysis EEG and they are ‘... clinically useful tools for the quantitative description of an EEG’ as quoted from the original paper in

Hjorth B (1970), EEG Analysis Based on Time Domain Properties, Electroencephalography and Clinical Neurophysiology, Vol 29, 306-310.

On the other hand, the Hurst exponent and Detrended Fluctuation Analysis (DFA) have been used as measures of long range correlation in the analysis of non-stationary time series, mostly in economics and geophysics. For the Hurst exponent, see the books

Mandelbrot B. (1997), Fractals and Scaling in Finance, Springer Verlag.

Edgar E.P. (1996), Chaos and Order in the Capital Markets: A New View of Cycles, Prices, and Market Volatility, Second Edition, John Wiley and Sons, Inc.

DFA has been popular in many applications since it was first introduced in

Peng C.-K., Buldyrev S.V., Havlin S., Simons M., Stanley H.E., Goldberger A.L. (1994), Mosaic Organization of DNA Nucleotides. Physical Review E, Vol 49, pp 1685-1689.

Both Hurst exponent and DFA estimate the scaling of the energy of the time series (either in terms of variance or power spectrum) at different time scales.

 

Descriptive Statistics

The following self-explained simple measures of the time series are included in this group.

Time Series mean (MeanTimSer)

Time Series median (MedianTimS)
Time Series variance (VarianceTS)
Time Series Standard Deviation (StdDeviaTS)
Time Series intequartile range (IntQuarRangTS)
Time Series skewness (SkewnessTS)
Time Series kurtosis (KurtosisTS)

 

Hjorth parameters

The following two Hjorth parameters are implemented:

 

Hjorth mobility (HjorthMobi)

It is the square root of the ratio of the second moment of the first difference of the time series to the second moment of the time series.

 

Hjorth Complexity (HjorthComp)

It is the square root of the difference of two ratios: a) the second moment of the second difference of the time series to the second moment of the first difference of the time series and b) the second moment of the first difference of the time series to the second moment of the time series.

 

Long Range Correlation

The following two measures of long range correlation are implemented:

 

Hurst exponent (HurstExpon)

The Hurst exponent is calculated using the rescaled range (R/S) technique.

 

Detrended Fluctuation Analysis (DetFluctAna)

The simple form of linear DFA is used where first degree polynomials are fitted to the time series segments to derive the residuals.

 

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