Steed variance is an extension of Pearson's.

1. The definition of Steed Variance/
Covariance and their applications
Speaker：Jun S. Huang
February 6，2020
UCSD ITA Workshop

2. 1 Background
2 Complex Models
3 Applications
CONTENTS

3. CH
1
Background

4. Executive Summary
The average or expectation of an engineering observation random quantity is the first
order origin moment. The deviation of random variable from the average is called
error.
The quantity used to measure the degree of correlation between the errors of two
variables is called covariance. Variance is a special case of covariance, when two
variables are the same. The variance is the second-order central moment. The
normalization of covariance to standard deviation is called Pearson correlation
coefficient.
The normalization of variance to expectation is called index of dispersion, and the
derivative of the index of dispersion to the observation time interval of the
accumulation span plus one then divided by two is Hurst index. The Hurst value
reflects the long memory fluctuation.
By turning integer order moment to fractional order counterpart, we can have more
fun, and it is called Steed version of everything!

5. Executive Summary Continue
With the rapid development of various computing platforms from mobile phones to
the edge computing, we can not only calculate the moment of integer order and the
Hurst index with real value, but also calculate 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 such as gas explosion (Engineering), vegetation coverage
(Agriculture), stock market (Commerce), autonomous driving (Research), network
abnormality (Military).

6. Simple Definition for Hypo/ Gauss/ Hyper Variance

7. Year 1880, Pearson's formula at London College

8. Steed's formula today at Suqian College

9. Three Hypo Cov can be done now!

10. More 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

13. A picture is worth a thousand words！
Good Bad

14. In between Momentum and Energy
MOMENERGRY
˯
deng

15. 2013-2017 59 Classes ESL score

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.

17. NDVI
1、Normalized Difference Vegetation Index
The NDVI is a simple graphical indicator that can be used to analyze remote sensing
measurements, from a space and assess whether the target contains live green vegetation:
NDVI=(NIR-R)/(NIR+R）
where R and NIR stand for the spectral reflectance measurements acquired in the red
(visible) and near-infrared regions, respectively.
2、NDVI value
If there is more reflected radiation in near-infrared wavelengths than in visible wavelengths,
then the vegetation in that pixel is likely to be dense and may contain forest. NDVI is directly
related to the photosynthetic capacity and hence energy absorption of plant canopies. The
index goes from -1 to 1, even in populated urban areas the value of normal NDVI is positive.
Negative values are likely to be disturbed in the atmosphere like snow or rain.
3、NDVI Data
NDVI NASA MODIS Terra 500M，MOD09GA WGS84（http://www.gscloud.cn/）

18. Typical NDVI Daily Average for Regions
ArcGIS/ENVI

19. CH
2Complex Models

20. A Few 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.

21. Probability Space Fractal Moment
• 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

22. Factorial of any number
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.
 

 dtetx tx1
0
 dtetz tz 


 1
0
  0Re z

23. 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

24. Complex Hurst Process Step
F < 2 F > 2
1.5000 2.5
1.3571 3.5
1.2778 4.5
1.2273 5.5
1.1923 6.5
1.1667 7.5
Steps:
 Divide China into east and west
 Divide 10 years into first 5 years and second 5 years
 Calculate hypo/ hyper variances respectively
 Arithmetically or geometrically etc weighted the slopes
 Average variances to obtain complex slope
 Convert the pair of slope into Hurst

25. CH
3Applications

26. 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

27. Brian Storm Optimization by Yuhui Shi in 2010

28. BSO（20 by 30 rounds）
0
5
10
15
20
0
10
20
30
0
5
10
15
20
25
30
brain storm round
Predicting Gas Emission with BSO
complex fractional round
gasfitnessvalue
0 5 10 15 20 25
-1
0
1
2
3
4
5
6
24 hours
gasemission
Predicting Gas Emission with CFD
data
trend
The preliminary analysis of the data shows that the gas outburst has certain
periodicity related to the gravitational wave.

29. 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

30. East
Space
Analysis
West
Comparing the
eastern and western
regions, we can see that
the larger the
population in the west is,
the greater the degree
of vegetation
destruction. With the
growth of population,
this kind of destruction
is becoming more and
more serious. On the
contrary, the
destruction of
vegetation in the west is
relatively slow and less
than that in the East.

31. Zou Qing, of the IOWA University，et al
Proof the Complex Model, and his Langlands Book.
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
In 1796, when Gauss was 19 years old, he discovered
the ruler drawing method of regular 17 sides, which
solved the difficult problem which has been outstanding
for more than 2000 years since Euclidean time. In the
same year, the law of quadratic reciprocity was
published and proved. That is, the supreme program of
the mathematical kingdom: the embryonic form of the
Langlands program!

32. Stock market analysis by Yufan Hu, Carleton U.
Ebay and Samsung

33. Complex Hursts

34. Shenzhen Autonomous Bus learning Good driving habits
https://www.researchgate.net/publication/332369723_Method_and_System_Based_on_Reinforcement_Learning_jiyuqianghuaxuexixingdefangfajixitong

35. Sask U,Botao Zhu's thesis with Blue attack Red net flow

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Thank you for your time!
JunHuang@Cunet.Carleton.Ca