Page 199 - Special Topic Session (STS) - Volume 1

STS425 Arifah B. et al.
2.2 Long memory estimation by using Detrended Fluctuation
Analysis
The detrended fluctuation analysis (DFA) is developed by Peng et al.
(1994) to investigate the correlation of long range power law of DNA
nucleotides. The DFA analysis can estimates the scaling H of the series in
nonstationary case and it can eliminate the spurious detection of long-
range dependence. Additionaly, DFA implements the dispersion
measurements that take the squared fluctuations around the time series
trend. Conducting the DFA on the sub period avoids the effect of
nonstationaries . However, DFA is still can be used to examine the long
and short range correlation in both stationary and non-stationary series.
The first step of DFA is to integrate the time series ( )y k with N

samples in purpose of analysation and this series will be divided into n
non-overlapping segments. Next, for each segment, the local trend y n ( )k
will be calculated by using the least-square regression. The ( )y k series is
detrended by subtracting the y ( )k in each segment. Finally, the root-
n
mean-square fluctuation can be calculated as follows:

1 N
2
F ( )n   [ ( )y k  y ] (9)
N k 1  n ( )k
The pervious process will be repeated at a range of different window
sizes n for the whole signal. To test the self similarity or fractal properties,
the log-log graph of n against F(n) is created. If the plot is linear, then
the power law scaling is exist. Then, the slop  of the plot line will be
employed to characterize the fluctuations of the series as follows:

( )F n  Cn (10)
log ( )F n   logn logC (11)

where C is constant and  is the correlation value and represents the
Hurst exponent H estimation. If 0< <1 then it will have the properties
of fractional Brownian motion. The  values that can be used to explain
the series of self-correlations are summarised by Chen et al. (2017).
Bardet and Kammoun (2008) showed the DFA asymptotic properties
for the fractional Gaussian noise. These properties can be used for long
range dependent processes in stationary case. The asymptotic behaviour
of the DFA for the FGN can be written as:
F ( )n  c ( , ). ,H n H (12)

where c is a positive function. Bardet and Kammoun (2008) also
investigated the convergence of long range dependence parameter
estimator. They found it has a reasonable convergence rate in the semi-

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