Fractal variance is used to learn driving behavior for autonomous vehicle.

1. Hypo/ Hyper Variance/ Covariance and its
applications in Autonomous Bus etc
Speaker ： Jun Steed Huang
2020/8/21
For Vision X LLC @ Mountain View, CA94043

2. 1 Background
2 Complex Models
3 Applications
CONTENTS

3. CH
1
Background

4. Executive Summary
With the rapid development of various computing platforms from 5G smart
phones to the 5G edge computing, we can not only calculate the moment of
integer order and the Hurst index with real value, but also the fractional
moment, covariance and Hurst index with complex value.
The real part retains the original physical meaning, while the imaginary part
brings more higher order (up to infinite order in theory!). The information of
moments is helpful to identify the subtle differences between multiple static
(space) or dynamic (time) observations.
It provides insights into the engineering application scenarios of random
variables in various industries: gas explosion (Engineering 工 ), vegetation
coverage (Agriculture 农 ), stock market (Commerce 商 ), automatic driving
(Research 学 ), network abnormality (Army 兵 ), just to name a few.

5. Moment Generating Function 矩量母函数

6. Simple Definition for Hypo/ Gauss/ Hyper Variance
亚方差
方差
超方差

7. Year 1880, Karl Pearson's formula
for two

8. Jun Steed Huang's Fractal formula
today

9. 3P Hypo can be done now!

10. More Party from Hypo to Hyper as well!
We want to analyze the relationship between the four variables of
gas concentration, temperature, pressure and wind rate, let's continue
to define the product moment coefficient of any multiple variables:

11. Compare Pearson with Steed
The new definition, instead of square, uses x.5 power,
which can reflect the asymmetry of fluctuation,
more accurately fit the essential relationship of things,
and can also be extended to multi variable situations.
The disadvantage is that the calculation is relatively complex.
If the variables strictly conform to the Gaussian distribution,
and the fluctuation is statistically symmetric,
or only involves two target variables,
then there is no need to use a new definition.

12. An example of Score
01
02
03
04

13. A picture is worth a thousand words ！
0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7
S t u d e n t s a b o v e A v e r a g e d i s p e r s i o n
- 0 . 4
- 0 . 3
- 0 . 2
- 0 . 1
0
0 . 1
0 . 2
0 . 3
0 . 4
0 . 5
StudentsbelowAveragedispersion
T h e C o m p l e x M o m e n t f r o m 1 . 5 t o 2 . 5 f o r E S L C l a s s S c o r e
S t e e d V a r i a n c e
G a u s s V a r i a n c e
Good Bad

14. In between Momentum and Energy
MOMENERGRY
˯
deng

15. 2013-2017 59 Classes ESL score
0 2 0 0 0 4 0 0 0 6 0 0 0 8 0 0 0 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0
S t u d e n t s a b o v e A v e r a g e d i s p e r s i o n
- 0 .5
0
0 . 5
1
1 . 5
2
2 . 5
3
StudentsbelowAveragedispersion
1 0 4 T h e C o m p l e x M o m e n t f r o m 1 . 5 t o 2 . 5 f o r C l a s s S c o r e

16. Harold Edwin Hurst (1880–1978)
Complex Hurst model
extended traditional Hurst model
by allowing value to be complex
number ， Hurst can be
calculated from IDC,IDI,IDA slope,
i.e. Index of Dispersion for Count,
Interval or Absorption.
The normalization of variance to expectation is called index of dispersion.

17. CH
2
Complex Models

18. Steed's a dozen step away academictree.org ancestors
Laplace is a French analyst, probability theorist and physicist, and an academician of the French Academy of Sciences. In 1812, he
published an important book, Probability Analysis Theory, which summarizes the research of probability theory at that time and
discusses the application of probability in astronomy.
Gauss is a Jewish, famous German mathematician, physicist, astronomer, geodesy, and one of the founders of modern mathematics.
Gauss was considered one of the most important mathematicians in history. He studied the complex plane of integer and enjoyed the
name of the prince of mathematics.
Mandelbrot moved to Paris with his family at an early age. He spent most of his life in the United States and has the triple nationality
of Poland, France and the United States. He studied a wide range from mathematical physics to financial mathematics, but his greatest
achievement was the creation of fractal dimension geometry.
12 Steps 11 Steps 10 Steps

19. Probability Space Fractal Moment
01
02
03
04
•
Fractional center moment:
Given as a discrete random variable,
exist, define order k center moment as
where
 E














 


 i
i
ikik
k E
i
k
EEE )()1()(
0












0
)()1(
i
ikii
EE
i
k

1< k < 3
Gamma Function
(-0.5)! = 1.7725
0.5! = 0.8862
1.5! = 1.3293
2.5! = 3.3234

20. Factorial of Any Number
01
02
03
04
  

 dtetx tx 1
0
  dtetz tz 


 1
0
  0Re z
Gamma function, as an extension of factorial, is a meromorphic function defined in the range of
complex numbers, which is usually written as:
(1)The gamma function on the real number field is defined as:
(2)The gamma function on the complex number is defined as:
among ,This definition can be extended to the gamma function of nonpositive integers
except on the complex plane in the whole complex field by using the analytic extension principle.

21. Approximation
1 2 3 4
1 、 K=1.5 ： hypo variance
2 、 K=2 ： standard variance
3 、 K=2.5 ： hyper variance
Complex Hurst ：
CH =
1+ log
IDC(m)
IDC(1)
×log
Level(1)
Level(m)
2
where ：
IDC =
E(x - Ex)k
m
is the last pointm
is meanm x
Matlab code: ：
for s = 1:(m-1)
slopeTmp = slopeTmp+((m-s)/(m*(m-1)/2))*((log(I(s+1))-log(I(s)))/(log(L(s+1))-log(L(s))));
end

22. CH
3
Applications

23. Shenzhen Autonomous Bus Chief Scientist Action List
1. Wireless system around 90 meters, when people is detected, reduce 1/8 of speed;
2. Infrared laser radar detection system around 60 meters, when detected, reduce 1/4 of speed;
3. 77G millimeter wave radar detection system around 30 meters, detected, reduce 1/2 of speed;
4. Visible light (infrared at night) sight system around 10 meters, tapping on the brakes to inform;
5. Green light (purple structured light at night) sight system around 5 meters, hit the break as usual;
6. Ultrasonic wave detection system around 2 meters, hit the break as hard as it could;
7. Touch detection system (reverse airbag is set) around half meters.

24. Autonomous Bus learning Good Driving habits
https://www.researchgate.net/publication/332369723_Method_and_System_Based_on_Reinforcement_Learning_jiyuqianghuaxuexixingdefangfajixitong
https://www.mathworks.com/matlabcentral/profile/authors/620531
2

25. A Coal Mine Article Written by a
Graduate at the University of New York
https://www.researchgate.net/publication/309666155_Analysis_of_Risk_Management_for_the_Coal_Mine_Operations

26. Brain Storm Optimization （ 20 by 30 rounds ）
0
5
1 0
1 5
2 0
0
1 0
2 0
3 0
0
5
1 0
1 5
2 0
2 5
3 0
b r a in s t o r m r o u n d
P r e d i c t i n g G a s E m i s s io n w i t h B S O
c o m p le x fr a c t io n a l r o u n d
gasfitnessvalue
0 5 1 0 1 5 2 0 2 5
- 1
0
1
2
3
4
5
6
2 4 h o u r s
gasemission
P r e d ic t in g G a s E m is s io n w it h C F D
d a t a
t r e n d
The preliminary analysis of the data shows that the gas outburst has certain
periodicity related to the gravitational wave.

27. Typical Normalized Difference Vegetation Index
Daily Average for Regions
ArcGIS/ENVI
https://www.researchgate.net/publication/328717334_Complex_Hurst_Model_for_NDVI

28. 100
101
102
103
From Day to Year
10-3
10-2
10-1
100
101
ModulusofNDVIDispersionIndex
Modulus for NDVI Complex Hurst Diagram
F=1.5 1st
F=2.0 1st
F=2.5 1st
F=1.5 2nd
F=2.0 2nd
F=2.5 2nd
0 1 2 3 4 5 6
From Day to Year
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
AngleofNDVIDispersionIndex
Angle for NDVI Complex Hurst Diagram
F=1.5 1st
F=2.0 1st
F=2.5 1st
F=1.5 2nd
F=2.0 2nd
F=2.5 2nd
1 、 Whole NDVI map
1st is the first 5 years ， 2nd is the second 5 years, the traditional slope overlaps
closely, complex slopes are different.
NDVI Time Analysis

29. Zou Qing, of the Iowa University ， et al
Proof the Complex Model, and his Learning paper.
If you want to know the original proof, you can refer to ：
Definition of Complex Hurst and Fractional Analysis for
Stock Market Fluctuation:
https://link.springer.com/book/10.1007/978-3-662-47200-2
https://cmsworkshops.com/ICASSP2020/Papers/ViewPaper.asp?PaperNum=5529
IEEE ICASSP2020
Paper Title SAMPLING OF SURFACES AND LEARNING FUNCTIONS IN HIGH
DIMENSIONS
Authors Qing Zou, Mathews Jacob, University of Iowa, United States
Session COLL-L1: Session 3H: Processing of Highly complex, Heterogeneous,
High-dimension data

30. Stock Market Analysis by Yufan Hu, Carleton U.
Ebay and Samsung

31. Papers with Blue attack Red Network Flow

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