Comparing the performance of FA, DFA and DMA using different synthetic long-range correlated time series Ying-Hui Shao,1, 2 Gao-Feng Gu,1, 2 Zhi-Qiang Jiang,1, 2 Wei-Xing Zhou,1, 2, 3, ∗ and Didier Sornette4, 5, † 1
2
∗
School of Business, East China University of Science and Technology, Shanghai 200237, China Research Center for Econophysics, East China University of Science and Technology, Shanghai 200237, China 3 School of Science, East China University of Science and Technology, Shanghai 200237, China 4 Department of Management, Technology and Economics, ETH Zurich, Zurich, Switzerland 5 Swiss Finance Institute, c/o University of Geneva, Geneva, Switzerland
[email protected]
†
[email protected]
2 0
10
(b-1)
(a-1)
1
(c-1)
(d-1)
10
(e-1)
1
10
0
10
hF i
hF i
10
hF i
hF i
hF i
0 0
10
−1
10
−1
10 0
1
10
10
2
0
10
S
10
(a-2)
0
10
1
10
10
S
2
0
10
10
1
2
0
10
S
0
10
10
10
1
2
0
10
S
10 2
(b-2)
(c-2)
1
10
10
(d-2) 1
0
10
1
2
10
S
(e-2)
1
10
hF i
0
hF i
10
hF i
hF i
hF i
10
10
0
10
0
10
−1
10
1
2
3
10
S
0
10
10
(a)
0
10
1
10
10
2
S
10
10
3
10
0
10
10
−1
−1
0
10
1
10
10
2
3
10
S
0
10
1
10
2
10
3
10
S
0
10
1
(c)
2
10
10
(d)
10
1
10
2
10
S
3
10
(e)
2
0
10
1
10
hF i
0
hF i
hF i
hF i
hF i 0
10
3
10
S
10
1
10
2
10
3
(b) 10
FA DMA,θ=0 DMA,θ=0.5 DFA
1
10
1
10
0
0
10
10
−1
0
1
10
2
10
10
10
S
3
10
4
0
10
10
−1
−1
4
10
1
10
2
10
3
10
S
10
4
10
0
10
10
1
2
10
3
10
4
10
S
0
10
10
1
2
10
3
10
4
10
S
10
0
10
(f-1)
(h-1)
(g-1)
−1
(j-1)
10
(i-1)
−1
10
0
10
10
hF i
−1
10
hF i
hF i
hF i
hF i
−1
−2
10
−2
10
−3
10
−2
10 0
1
10
10
2
0
10
S
1
10 10
(f-2)
0
10
10
S
2
0
10
10
1
2
0
10
S
10
(h-2)
(g-2)
10
1
2
0
10
S
10
(i-2)
0
hF i
hF i
hF i
hF i
hF i
10
−1
10
2
10
S
(j-2)
−1
10 −1
1
10
−1
10
10
−2
10
−2
10
−3
10
−2
10 0
10
10
1
2
3
10
S
0
10
1
10 10
(f)
10
2
3
10
S
0
10
1
10
10
2
3
10
S
0
10
1
10
10
2
3
10
S
0
10
0
(h)
(g)
−1
10
0
10
−1
(i)
−1
10
hF i
hF i
−1
10
2
3
10
S
10
(j)
−2
hF i
hF i
hF i
10 10
1
10
−2
10
−2
10
−3
10
10
−3
10
−4
10
−1
10
0
10
10
1
10
2
10
S
3
4
0
10
10
1
2
10
10
10
S
3
4
0
10
10
1
2
10
3
10
4
10
S
0
10
10
1
2
10
3
10
4
10
S
0
10
10
1
2
10
3
10
4
10
S
10
1
(o-1)
10
0
10
hF i
10
hF i
10
1
(n-1)
0
0
hF i
0
10
10
(m-1)
(l-1) hF i
hF i
(k-1)
0
10
−1
10
−1
10
−1
10 0
1
10
10
2
0
10
S
1
10
10
10
S
2
0
1
10
10
2
0
10
S
1
10
10
2
0
10
S
10
1
(k-2)
10
(l-2)
1
10
10
2
10
S
2
(m-2)
(n-2)
1
10
(o-2)
1
0
0
10
hF i
10
hF i
0
10
hF i
hF i
hF i
10 0
10
−1
10
−1
10
−1
0
10
10 0
10
10
1
2
3
10
S
0
10
1
10 10
(k)
10
2
3
10
S
0
10
1
10
10
2
3
10
S
0
10
1
10
10
2
3
10
S
0
10
1
(l)
1
10
10
(m)
(n)
2
10
0
10
0
10
0
10
1
10
S
2
10
3
4
10
3
10
0
10
1
10
2
10
S
10
3
4
10
(o)
0
10
1
10
2
10
S
3
10
4
10
10
0
10
−1
10
−1
10
−1
10 10
2
10
S
1
hF i
0
hF i
10
hF i
hF i
hF i
0
10
1
10 10
1
10
2
0
10
1
10
2
10
S
3
10
4
10
0
10
1
10
2
10
S
3
10
4
10
FIG. 1. Comparing plots of ⟨F ⟩ against s. The plots labeled (a)-(o) are the same as in the paper where the length of time series is 20000, the plots labeled with (a-1) to (o-1) are the results where the length of time series is 500, and the plots labeled with (a-2) to (o-2) are the results where the length of time series is 2000.
3 1.6
1.4 1
1.2
(a-1)
−0.2 0 10
10
1
10
S
0.6 0.2
0.6
(b-1)
(c-1) 1
10
−0.2 0 10
2
10
S
1.2
0.8 0.4
0.2
−0.2 0 10
2
hHout i
0.2
hHout i
hHout i
hHout i
1 0.6
hHout i
1
1
10
2
S
0.4
0 0 10
10
(d-1) 1
10
0 0 10
2
10
S
1
10
2
10
S
0.6 0.2
1.2
hHout i
0.2
hHout i
hHout i
0.6
1.2
1 0.6
−0.2 0 10
1
10
10
S
2
0.2
10
1
2
S
0
3
10
10
10
10
1
2
S
10
(e-2)
0
0 0 10
3
10
0.8 0.4
(d-2)
(c-2)
−0.2 0 10
3
10
0.8 0.4
(b-2)
(a-2)
hHout i
1
1
10
2
10
S
10
3
0
1
10
10
2
3
10
S
10
1.6
hHout i
1
0.2
0.6 0.2
1.2
0.6
(b) 1
2
10
3
10
4
10
S
−0.2 0 10
10
0.8 0.4
0.2
(a) −0.2 0 10
1.2
1
hHout i
DFA DMA,θ=0 DMA,θ=0.5 FA
0.6
hHout i
1
hHout i
hHout i
(e-1)
1.6
1
hHout i
0.8
0.4
1
2
10
10
10
S
3
4
10
(e)
(d)
(c) −0.2 0 10
0.8
1
2
10
3
10
4
10
S
0 0 10
10
1
2
10
3
10
10
S
10
0 0 10
4
1
2
10
3
10
4
10
S
10
1.4
1.2
0.2
−0.2 10
0
10
1
10
S
(h-1)
(g-1) 1
10
S
1.2
hHout i
hHout i
hHout i
0.8 0.4 0
(f-2) 1
10
10
S
2
3
(g-2)
10
10
0
2
S
0.8 0.4
1
10
10
S
2
3
0
10
10
10
1
2
3
10
S
10
S
1.4
1
1
0.6
(i-2) 0
10
0 0 10
2
10
1.4
0.2
(h-2)
0
0
1
10
10
1.2
0.2 −0.2 0 10
1
10
0.8 0.4
(i-1)
0
−0.2 0 10
2
10
1 0.6
0.4
0.2
−0.2 0 10
2
0.6
1.2
0.8
hHout i
(f-1)
hHout i
0.6
hHout i
0.2
hHout i
0.6
1
hHout i
1
hHout i
hHout i
1
10
2
10
S
10
3
1
10
2
10
S
0.6
(j-2)
0.2 1
10
(j-1)
0
1
10
1.2
1.2
1.4
1.4
0.8
0.8
1
1
10
2
3
10
S
10
0.2
0
−0.2 0 10
0
(g)
(f) 1
2
10
3
10
S
−0.4 0 10
4
10
10
0.4
1
2
10
10
S
3
−0.4 0 10
4
10
0.6 0.2
0.2
1
2
10
3
10
S
−0.2 0 10
4
10
10
(j) 1
2
10
3
10
S
−0.2 0 10
4
10
10
1.4
0.6 0.2
(k-1)
−0.2 10
0
10
1
10
S
2
0.6 0.2
−0.2 (l-1) 0 10
1
10
−0.2 0 10
2
10
S
(m-1) 1
10
0.4
3
4
10
S
10
S
1.2 0.8 0.4
(n-1) 0
2
1
10
10
1.2
10
0 0 10
2
10
S
(o-1) 1
10
2
10
S
1.4 1.4
0.4 0
(k-2)
−0.2 0 10
1
10
10
S
2
3
10
10
0.6 0.2
(l-2)
0.6
1
10
10
S
2
−0.2 0 10
3
10
1
10
2
S
0
3
10
1
10
10
10
2
10
S
10
3
1
10
2
10
S
3
10
4
10
10
(l) 0
1
2
10
S
3
10
4
10
0.6 0.2
3
10
−0.2 0 10
1
10
2
10
S
3
10
4
10
1 0.6 0.2
(o)
(n)
(m)
10
hHout i
hHout i
0.6 0.2
−0.2 0 10
2
10
S
1.4 1
0.4 0
(k)
1
10
1.4
0.8
hHout i
hHout i
0.2
(o-2) 0
10
1 0.6
0.6 0.2
(n-2)
(m-2)
0
1
0.2
1.2
1
1
hHout i
0.2
hHout i
hHout i
0.6
hHout i
1
0.8
hHout i
0.8
0
1
hHout i
2
10
1.6
hHout i
0.2
hHout i
0.6
1
10
1.2
1
hHout i
1
hHout i
hHout i
1
0.6
(i)
(h)
10
hHout i
0.4
hHout i
0.6
hHout i
hHout i
hHout i
1
−0.2 0 10
1
10
2
10
S
3
10
4
10
−0.2 0 10
1
10
2
10
S
3
10
4
10
FIG. 2. Comparing local slopes of the fluctuation functions. The plots labeled (a)-(o) are the same as in the paper where the length of time series is 20000, the plots labeled with (a-1) to (o-1) are the results where the length of time series is 500, and the plots labeled with (a-2) to (o-2) are the results where the length of time series is 2000.
4
−0.6 0
0.2
0.4
0.6
Hin
0.8
1
0
Hout − Hin
0.4
0.6
Hin
0.8
1
0
−0.4
(a-2)
0
0.2
0.4
0.6
Hin
0.8
1
0.4
0.6
Hin
0.8
1
0
−0.4
(b-2)
0
0.2
0.4
0.6
Hin
0.8
−0.6 0
1
Hout − Hin
0
0.2
0.4
0.6
Hin
0.8
0.2
0.4
0.6
Hin
0.8
−0.4
(d-2)
0
0.2
0.4
0.6
Hin
0.8
−0.6 0
1
0.2
−0.2
−0.2
Hin
0.8
1
0.2
0.4
0.6
Hin
Hout − Hin
0.2 0
−0.2
0.2
0.4
0.6
Hin
0.8
1
Hin
0.6 0.4
0.2
Hout − Hin
0
−0.2
0.2
0.4
0.6
Hin
0.8
1
0.2
0.6
Hin
0.8
1
0.4
0
0.2
0.4
0.6
Hin
0.8
1
0.6
Hin
0.2 0
−0.6 (h-1) 0 0.2
0.4
0.6
Hin
0.8
1
0.4
0.2
0.2
0
−0.6 0
0.2
0.4
0.6
Hin
0.8
1
−0.6 (i-2) 0 0.2
0.2
0.2
0.2
0.2
Hout − Hin
0.4
−0.2
0
0.4
0.6
Hin
0.8
1
−0.4 0
(h) 0.2
0.4
0.6
Hin
0.8
1
−0.4 0
DFA
DMA,θ=0
DMA,θ=0.5
FA
−0.4 0
1
0.2
0.4
0.6
Hin
0.8
1
0.4
0.6
0.8
1
0.4
0.6
0.8
1
0.4
0.6
0.8
1
Hin
Hin
0
−0.2
−0.2
(g)
1
−0.4
(h-2)
0.4
−0.4 (f) 0 0.2
0.8
0
0.4
−0.2
0.6
Hin
−0.2
0.4
0
0.8
0.4 0.2 0 −0.2 −0.4 −0.6 (i-1) −0.8 0 0.2
0.4
−0.4
(g-2)
0.2
(e) 0.2
−0.2
−0.4 0.4
1
−0.4
(g-1)
−0.2
−0.4
0.8
−0.2
0.4
0.2
−0.6 (f-2) 0 0.2
0.6
0.4
−0.6 0
0.4
Hout − Hin
0.4
0.6
−0.4
−0.6 (f-1)
Hout − Hin
0.2
−0.2
−0.4 0
1
Hout − Hin
Hout − Hin
0.4
0.8
0.4
(e-2)
−0.2
Hout − Hin
0.6
−0.4 0
1
0
(d)
Hout − Hin
0.4
−0.2
Hout − Hin
0.2
0
(c) −0.4 0
0.8
Hout − Hin
(b) −0.4 0
Hout − Hin
(a) −0.4 0
Hout − Hin
0.4
0.2
Hout − Hin
0.4
0.2
Hout − Hin
0.4
0.2
−0.2
0.6
Hin
0
0.4
0
0.4
−0.2
0.2
0
0.2
0.2
0.4
0
(e-1)
0.4
0
1
0
−0.6 0
1
0.2
−0.4
0.2
−0.4
−0.2
(c-2)
0.4
−0.2
0.4
0.2
−0.2
−0.2
0
−0.4 (d-1)
0.4
0.2
Hout − Hin
Hout − Hin
0
−0.2
Hout − Hin
0.2
0.2
−0.2
−0.6 (c-1) 0 0.2
0.4
0.2
0
−0.4
−0.4 (b-1)
(a-1)
0.2
−0.2
−0.2
0.4
−0.4
0
Hout − Hin
−0.4
0.2
0.6 0.4
Hout − Hin
−0.2
0.4
Hout − Hin
0
0.6
0.4
Hout − Hin
0.2
0.6
Hout − Hin
0.4
Hout − Hin
Hout − Hin
0.6
0.2
0.4
0.6
Hin
0.8
1
−0.4 (i) 0 0.2
Hin
FIG. 3. Comparing impacts of the scaling range on the Hurst index estimates. The plots labeled (a)-(o) are the same as in the paper where the length of time series is 20000, the plots labeled with (a-1) to (o-1) are the results where the length of time series is 500, and the plots labeled with (a-2) to (o-2) are the results where the length of time series is 2000.
