A novel fractional dimension derivation along with the exact algorithm involving the fractional norm definition and the fractional center moment extension, which ends up a complex Hurst parameter is presented. It reveals the underline stock marketing uncertainty drivers from both real and imaginary angles. We take the e-commence cluster of online related companies as an example.

1. Complex Hurst and Fractional Analysis
for Stock Market Fluctuation
Presenter: Yufan Hu
Authors: Qing Zou, Yufan Hu, Jun Steed Huang
ICIMSA2015 May 26-28 Tokyo Japan

2. About the Authors
• Qing Zou is a student with Suqian College,
who wrote 2 books on number theory.
• Yufan Hu is a master major math with
Loughborough University.
• Jun Steed Huang is a distinguished
professor.

3. Traditional Hurst Parameter
• Denoted by H, in honor of Harold
Edwin Hurst and Ludwig Otto Hölder
• Meaning: Index of long-range
dependence
• 0 1H≤ <

4. Meaning of Hurst Parameter
• : a time series with long-
term switching between high and low
values in adjacent pairs, e.g. High—Low
or Low-High
• : a time series with long-
term positive autocorrelation, e.g.
High—High or Low-Low
• H=0.5 : a completely uncorrelated
memory less series, e.g. Poisson point
process
0 0 .5H≤ <
0 .5 1H< <

5. Defining Fractional Norm
• Fractional Norm:
, 1, 1,p
x l p r∀ ∈ > ≥
2 2
1
1 1 11 21 2 2
1
2 2
1
1 1
2
sup : ,( 1)
inf : ,(0 1)
yn n
i ir
rn i ir y
rr
i
yn ni
i ir
y i i
r
x y x
x x
x y x
−
− = =≤−
=
−
≤ = =
    
 ∈ >  
      
= =  
      
∈ < <   
    
∑ ∑
∑
∑ ∑
ℚ
ℚ

6. Defining Fractional Moment
• Fractional center moment:
Given as a discrete random variable,
exist, define order k center moment as
where
ξ ξE








−





=−= ∑
∞
=
− i
i
ikik
k E
i
k
EEE )()1()(
0
ξξξξµ
∑
∞
=
−
−





=
0
)()1(
i
ikii
EE
i
k
ξξ
1< k < 3

7. Defining Complex Hurst
• Complex Hurst Parameter CH
IDC: Index of Dispersion for Count
: mean of
m: the last point in the sample
CH =
1+log
IDC(m)
IDC(1)
⋅log
Level(1)
Level(m)
2
IDC =
E(ξ − Eξ)k
µ
µ ξ

8. Level of Grouping
•Time span between two successive checks
1: 1 day 2: 2 days
5: 1 week 10: 2 weeks
20: 1 month 65: 3 months
130: 6 months 260: 1 year

9. Complex Hurst
•
• Real part: same as traditional H
• Imaginary part: Uncertainty of real part
−0.5 < Re(CH),Im(CH) <1.5

10. Forward, Backward & Irreversibility
(p1, p2,…, p700)
• Forward Hurst FH: (p1, p2,…, p700)
• Backward Hurst BH: (p700, p699,…, p1)
• Irreversibility Hurst IH:
earliest latest
2
FH BH
IH
−
=
IH ∈ (−0.5,1.5)

11. Irreversibility
The real part of IH is used to judge if
the stock market will repeat itself in
future, the lower the value, the better
chance it will repeat, the high the
value, unlikely it will.

12. Example of Real Part
• MATLAB code:
http://www.mathworks.com/matlabcentral/fileexchange/49803-
calculate-complex-hurst/content/complexHurstV1.m

13. Example of Real & Imaginary Value
Company Re(FH)
F=2.5
Im(FH)
F=2.5
Re(BH)
F=1.5
Im(BH)
F=1.5
Re(IH)
ΔF=0.5
Im(IH)
ΔF=0.5
eBay 1.2529 -0.0477 0.0645 0.0196 0.5942 -0.0337
Microsoft 0.9756 0.0308 0.7929 -0.0103 0.0914 0.0206
Baidu 0.7681 0.0285 0.8119 -0.0088 -0.0219 0.0186
Vodafone 1.1049 0.0213 0.7720 -0.0070 0.1665 0.0142
Nokia 0.9958 0.0151 0.7891 -0.0059 0.1034 0.0105
Apple 1.0746 0.0137 0.7870 -0.0042 0.1438 0.0090
Samsung 1.2304 -0.0108 0.7128 0.0022 0.2588 -0.0065
Amazon 1.2035 0.0052 0.7332 -0.0015 0.2352 0.0034

14. Conclusion
• We have defined Complex Hurst and
related
• FH, BH, IH for stock market quote
estimation
• Tested their meaning of real &
imaginary parts

15. Thank you!