ASYMMETRICAL CORRELATION TEST FOR CONSTRUCTING SUPER BAYESIAN INFLUENCE NETWORKS FOR FINANCIAL INTERMARKET INFLUENCE ANALYSIS

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Nov 7, 2013 (3 years and 7 months ago)

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ASYMMETRICAL CORRELATION TEST

FOR CONSTRUCTING SUPER BAYESIAN
INFLUENCE
NETWORKS

FOR FINANCIAL INTERMARKET INFLUENCE ANALYSIS



Heping Pan

and
Nadejda Soukhoroukova

Intelligent Finance Cluster, Centre for Informatics and Applied Optimization

School of
Information Technology and Mathematical Sciences

University of Ballarat, Mt Helen, Victoria 3353, Australia

h.pan@ballarat.edu.au


ABSTRACT


This
paper presents a computational procedure for asymmetrical correlation test as the first
step for constructing
super Bayesian influence networks with an application in financial
intermarket influence analysis. We start with a data set of multivariate time series without
any prior knowledge about possible influences in the problem domain. With the belief on the
exis
tence of influence patterns, asymmetrical correlation test is developed to detect all
possible asymmetries among all the pairs of random variables. We then use paired t
-
test to
check the statistical significance for each detected asymmetric correlation. Fu
rthermore, we
check the possible existence of ever
-
changing cycles in nonlinear dynamical systems such as
financial markets. This procedure results in a dynamic directed acyclic graph of the random
variables, which provides a graphical basis for a super Ba
yesian influence network. Note that
asymmetrical correlation test can quickly become complicated when the scale space of time
and the expanding set of conditioning variables for each asymmetrical correlation are
introduced. On the other hand, symmetrical o
r undirected correlations should not be ignored
completely, as they may bear additional information for augmenting super Bayesian influence
networks to general super influence networks.




KEYWORDS:
Super Bayesian
Influence
Networks (SB
I
N), Super Influence

Networks
(SIN),

Asymmetrical Correlation Test (ACT), conditional independence test, statistical
significance test, ever
-
changing cycles, dynamic directed acyclic graph,
multivariate time
series prediction, conditional probability distribution, nonlinear d
ynamic system modeling,
exploratory data mining.



1. Introduction


Detecting influence patterns and
modelling

detected influences for a data set of multivariate
time series is a central issue for exploratory data mining and modelling of open nonlinear
dyn
amic systems. Intermarket influences exist in the worldwide financial markets, and
detection and
modelling

of these influences are essential for developing computational
intelligent systems for predicting financial markets of interest. Pan (2004
a
) proposed

an
initial theory of Super Bayesian Influence Networks (SBINs)


nonlinear dynamic Bayesian
networks of probability ensembles of neural networks


as a new form of probabilistic and
connectionist complex system model for multivariate time series predictio
n. In particular,
financial intermarket influence analysis has been used as an underlying application to support
and facilitate the construction and development of the general theory of SBINs. A SBIN is a
dynamic directed acyclic graph for a set of random
variables. The directed links of the graph
correspond to asymmetrical influences between different variables. The conditional
probability distribution for a variable given its parent nodes is provided by a probability
2
/
17

ensemble of neural networks. Thus it i
s not difficult to imagine that SBINs provide one of the
most complicated and most powerful architecture for information fusion and probabilistic and
connectionist
modelling

of multivariate time series. As a follow
-
up of that initial theory paper,
this pap
er provides a computational procedure for initially detecting the influence patterns
that are statistically significant, which corresponds to the first step in constructing a SBIN
from a data set of multivariate time series.


The type of variables we consi
der is continuous random, of which discrete variables in the
discrete Bayesian networks dominant in the literature are a subtype. Due to the general type
of continuous variables, existing approaches for learning the structure of a discrete Bayesian
network

are apparently not applicable, such as conditional independence test based on
contingency tables, because it is not possible to give a general representation of probability
distribution for continuous variables. While multinomial distribution is general e
nough for
discrete variables, Gaussian distribution or Gaussian mixture models are the widely applied
representations

for Gaussian variables
. However, it is known that real
-
world problems often
can not be assumed to follow normal distributions or a mixture

of normals. For example,
relative returns of financial markets exhibit non
-
normal distributions with long and fat tails.
In consideration of this impassable difficulty due to the continuous variable type, we have
developed a technique which we call Asymme
trical Correlation Test (ACT), which is
essentially equivalent to conditional independence test but on continuous variables. The
essence of ACT is to detect asymmetry of influence between a given pair of dynamic random
variables (or systems). Of course, w
e are more interested in asymmetrical influences that are
statistically significant across a sufficient
ly

long period. In nonstationary complex systems
such as financial markets, we are well aware of the possible existence of ever
-
changing
cycles. Therefor
e, we generally expect such cycles
may or may not appear
on the detected
asymmetrical correlations.


ACT can quickly become complicated or too computationally expensive when the scale space
of time and the
expanding
set of conditioning variables for a give
n pair of variable are
introduced into the Multiresolution Conditional Asymmetrical Correlation Test

(MCACT)
.


Throughout the description of this computational procedure, we present an application on a
data set of 16 financial markets which are known
a pr
iori

to be of the same type and
interrelated. Due to the commercial nature of this application, we shall not disclose more
information about these 16 markets. But in fact, when we started dealing with this data set,
we had no prior knowledge about possible

influences among these markets.


For more information on the background of Bayesian networks, influence networks, and
structural learning, see references such as [Shachter and Kenley 1989; Spirtes et al 1992;
Jordan 1998; Pan and Liu 2000;
Pan
2002a, 200
2b; Pearl 2002]. For backgrounds on
financial market modelling, see [Engle and Granger 1987; Alexander 2001; Pan from 2002
on].


2. Super Bayesian Influence Networks

(SBINs)

Re
-
Introduced


Following an initial theory of Super Bayesian Influence Networks (S
BINs) proposed by Pan
(2004
a
), a SBIN for a set of dynamic random variables

X
)
(
t
=
))
(
,
),
(
),
(
(
2
1
t
X
t
X
t
X
n










(1)

is defined as

SBIN
(
X
(
t
)) =
(
V
,
L
,
P
)









(2)

where t is the time,
V
,
L
,
P

are respectively the set of nodes, set
of directed links and set of
conditional probability distributions of the
SBIN
, defined as

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/
17


V

=
{
X
(
t+1
),

F
X
(
t
)},
F
X
(
t
) = {
)
(
t
FX
,


X
X

}






(
3
)


L

= {
))
1
(
)
(
(

t
X
t
FX

,
))
1
(
)
(
(

t
X
t
Y

,


X
X

and



X
Y
}



(
4
)


P

= {
|
)
1
(
(

t
X
P
))
(
),
(
t
t
FX
X


,


X
X

}







(5)

where
)
(
t
FX

represents a feature vector extracted from the univariate time series of
X
,
which is most closely related to the most recent data,


X

represents the set of parents of
X
.


Definition 1:

For a pair of dynamic random variables
)
(
t
X
and
)
(
t
Y

(as
shown in (4)
)
,
Y
is
called a
parent

of
X

if
)
(
t
Y
influences
)
1
(

t
X
, but
)
(
t
X
has no influence on
)
1
(

t
Y
. We
denote such an
asymmetrical influen
ce

by
))
1
(
)
(
(

t
X
t
Y


or simply,
)
|
(
Y
X

or written as
)
(
X
Y


. For a variable
X
, the set of all its parents is denoted by


X
.


The conditional probability distribution
|
)
1
(
(

t
X
P
))
(
),
(
t
t
FX
X


is prov
ided

by



|
)
1
(
(

t
X
P
))
(
),
(
t
t
FX
X



=
PENN
(X(t+1);
))
(
),
(
t
t
FX
X







(6)

where
PENN

is a probability ensemble of neural networks which takes the feature vector
)
(
t
FX
and the parents
)
(
t
X


as inputs and generates a probability distribution over the
domain of
X(t+1)
.


How to define and extract a feature vector and how to construct
a
PENN

are beyond the
scope of this paper.

For more information on

these aspects, see [Pan et al 2003, 2004; Pan
2004b].


Definition 2:

For a set of dynamic random variables
X
)
(
t

as detailed in (1), we call the
complete set
S

of asymmetrical influences among all the variables the
basic structure

of a
SBIN for
X
)
(
t
, and
S
can be written as

S

=
),
{(
X
Y




X
X

and


Y
X

}







(7)

Apparently,
S

is a subset of
L

defined by (4).


With the notion of SBIN re
-
introd
uced above following [Pan 2004] and with these definitions,
we now can state the objective of this work reporte
d

here in exact mathematical terms, that is
to detect or discover the basic structure
S
including all asymmetrical influences among a
given set o
f dynamic random variables
X
)
(
t

from a given data set of historical time series of
X
)
(
t
.


Note that for a standard (non
-
dynamic) Bayesian network, the basic structure
S

is the
complete structure

L
. But a SBIN include
s univariate influences
)),
1
(
)
(
{(

t
X
t
FX



X
X
}
on top of the basic structure
S
.


3. Conditional Independence Test Revisited


Two random variables
X
and
Y
are said to b
e independent if


)
(
)
(
)
,
(
Y
P
X
P
Y
X
P











(8)

where P denotes probability. The independence is equivalent to the zero mutual information
)
,
(
Y
X
I
, which is defined as














)
(
)
(
)
,
(
log
)
(
)
(
)
,
(
log
)
,
(
)
,
(
)
,
(
Y
p
X
p
Y
X
p
E
y
p
x
p
y
x
p
y
x
p
Y
X
I
y
x
Y
y
X
x




(9)

where
y
x
,
den
ote the point value of
Y
X
,
respectively, and
)
,
(
y
x
p
denotes the joint
probability of
y
x
,
, E denotes expectation.

4
/
17


X
and
Y
are said to be conditionally independ
ent given a third variable
Z
if


)
|
(
)
|
(
)
|
,
(
Z
Y
P
Z
X
P
Z
Y
X
P








(10)

This conditional independence is equivalent to the zero conditional mutual
information

)
|
,
(
Z
Y
X
I
, which is defined as









)
|
(
)
|
(
)
|
,
(
log
)
|
,
(
)
,
,
(
Z
Y
p
Z
X
p
Z
Y
X
p
E
Z
Y
X
I
z
y
x






(11)

Conditional independence and conditional mutual information can be generalized to a set of
conditioning variables.


For discrete Bayesian networks, joint or conditional probabilities involved in the formulas
(8)
-
(11) can be calculated from contingency tabl
es (counting through all the possible discrete
states for each discrete variable) [Spirtes et al 1992, Pan 2002a, 2002b]. The conditional
independence can be tested using the conditional deviance. The exact distribution of the
deviance
is
, in general, intr
actable, and we have to rely on asymptotic and approximate
results. Asymptotically, the deviance approximates to the
2

-
distribution with the same
degrees of freedom.


However, for continuous variables, there is no general representatio
n for joint or conditional
probabilities. Under the assumption of Gaussian and conditional Gaussian distributions, the
mutual information
)
,
(
Y
X
I

is reduced to correlation, and the conditional mutual information
)
|
,
(
Z
Y
X
I
is re
duced to conditional correlation (also called partial correlation in statistics
textbooks). In general, correlation (including conditional correlation) provides a practical
approximate estimate of mutual information (including conditional mutual informatio
n) even
we do not know prior distribution of variables. In reality, any nonlinear correlation must lead
to correlation when the time span is minimized, so the standard correlation to a nonlinear
correlation is like a tangent to a curve.


4. Asymmetrical Co
rrelation Test


Let us first define a specific covariance
)
,
,
,
,
(
l
t
t
Y
X
Cov
y
x
and a specific standard deviation
)
,
,
(
l
t
X
Std
x


))
,
(
)
(
))(
,
(
)
(
(
1
1
)
,
,
,
,
(
1
l
t
Y
t
t
v
Y
l
t
X
v
X
l
l
t
t
Y
X
Cov
y
y
x
l
t
t
v
x
y
x
x
x












(12)









1
2
))
,
(
)
(
(
1
1
)
,
,
(
l
t
t
v
x
x
x
x
l
t
X
v
X
n
l
t
X
Std






(13)

both involve the interval average
)
,
(
0
l
t
X






1
0
0
0
)
(
1
)
,
(
l
t
t
v
v
X
l
l
t
X









(14)

For two dynamic random variables
)
(
t
X

and
)
(
t
Y
, the standard correlation
)
,
),
(
),
(
(
0
l
t
t
Y
t
X
C
is defined as


)
,
,
(
)
,
,
(
)
,
,
,
,
(
1
1
)
,
),
(
),
(
(
0
0
0
0
0
l
t
Y
Std
l
t
X
Std
l
t
t
Y
X
Cov
n
l
t
t
Y
t
X
C







(15)

However, the standard correlat
ion
)
,
),
(
),
(
(
0
l
t
t
Y
t
X
C
does not have much prediction value.



5
/
17

In order to detect predictive influences, we now define a new notion, Asymmetrical
Correlation
)
,
,
,
(
0
l
t
Y
X
AC
, as


)
,
,
(
)
,
1
,
(
)
,
,
1
,
,
(
1
1
)
,
),
(
),
1
(
(
)
,
,
,
(
0
0
0
0
0
0
l
t
Y
Std
l
t
X
Std
l
t
t
Y
X
Cov
n
l
t
t
Y
t
X
C
l
t
Y
X
AC








(16)

This definition of Asymmetrical Corr
elation is also called Interval Asymmetrical Correlation
as it is defined over a period of
]
2
,
1
[
0
0



l
t
t

for
)
(
t
X

and a period of
]
1
,
[
0
0


l
t
t

for
)
(
t
Y

and it uses a single average
)
,
1
(
0
l
t
X


of
)
(
t
X

and
)
,
(
0
l
t
Y
of
)
(
t
Y
, being constant
over
their respective periods.



For time series, it may be more practical to use an adaptive moving average
)
,
(
~
l
t
X

such as
the simple

moving average defined as







t
l
t
v
v
X
l
l
t
X
1
)
(
1
)
,
(
~









(17)

Accordingly, we can also define moving covariance
)
,
,
,
(
l
t
Y
X
MCov

and moving standard
deviation
)
,
,
(
l
t
X
MStd

by

))
,
(
~
)
(
))(
,
(
~
)
(
(
1
1
)
,
,
,
,
(
1
l
v
Y
v
Y
l
v
X
v
X
l
l
t
Y
X
MCov
t
l
t
v















(18)









t
l
t
v
x
l
v
X
v
X
n
l
t
X
MStd
1
2
))
,
(
~
)
(
(
1
1
)
,
,
(





(
19)

Then we can define Moving Correlation
)
,
,
,
,
(
l
t
Y
X
MC

by




























t
l
t
v
l
v
Y
MStd
l
v
Y
v
Y
l
v
X
MStd
l
v
X
v
X
l
l
t
Y
X
MC
1
)
,
,
(
)
,
(
~
)
(
)
,
,
(
)
,
(
~
)
(
1
1
)
,
,
,
,
(






(20)

With these definitions, we can define Moving Asymmetrical Correlation
)
,
,
,
(
l
t
Y
X
MAC
by


)
,
1
,
,
,
(
)
,
,
,
(
l
t
Y
X
MC
l
t
Y
X
MAC








(21)


The (Interval)Asymmetrica
l Correlation
)
,
,
,
(
0
l
t
Y
X
AC
defined by (16) and the Moving
Asymmetrical Correlation
)
,
,
,
(
l
t
Y
X
MAC
defined by (21) provide two types of
Asymmetrical Correlation for detecting asymmetrical influences between variables.


Theoretically, the A
symmetrical Correlation and Moving Asymmetrical Correlation can be
generalized to Conditional Asymmetrical Correlation (CAC) and Conditional Moving
Asymmetrical Correlation (CMAC). However, in order to define these notions, we need to
define Conditional Co
rrelation
))
(
|
,
),
(
),
(
(
0
t
Z
l
t
t
Y
t
X
C
, also called partial correlation,

))
,
),
(
),
(
(
1
))(
,
),
(
),
(
(
1
(
)
,
),
(
),
(
(
)
,
),
(
),
(
(
)
,
),
(
),
(
(
))
(
|
,
),
(
),
(
(
0
2
0
2
0
0
0
0
l
t
t
Z
t
Y
C
l
t
t
Z
t
X
C
l
t
t
Z
t
Y
C
l
t
t
Z
t
X
C
l
t
t
Y
t
X
C
t
Z
l
t
t
Y
t
X
C






(22)

Based on this definition of
))
(
|
,
),
(
),
(
(
0
t
Z
l
t
t
Y
t
X
C
, CAC and CMAC can be defined in a
way similar to the reasoning from (16)
-
(21).


The conditioning variable
)
(
t
Z
in (22) can also be generalized to a set of conditioning
variables.


In the following discussions, we may use
)
,
(
Y
X
AC
for general Asymmetrical Correlation
from
X

to
Y
, while
it may actually refer to Interval Asymmetrical Correlation
)
,
,
,
(
0
l
t
Y
X
AC
or Moving Asymmetrical Correlation
)
,
,
,
(
l
t
Y
X
MAC
.

6
/
17


For any given pair of variables
X

and
Y
, the Asymmetrical Correlation

Test (ACT) is to
check whether


)
,
(
)
,
(
X
Y
AC
Y
X
AC









(23)

consistently, or


)
,
(
)
,
(
Y
X
AC
X
Y
AC









(24)

consistently, over the whole historical time period of a given data set. If inequality (23) holds
consistently, we say
X

influences
Y
, and express this relation as


Y
X

, also written as
)
|
(
X
Y







(25)

Else if inequality (24) holds consistently, we write

X
Y

, also written as
)
|
(
Y
X







(26)


However, consistency has to be tested through a series of time samples in terms of statistical
significance.



5. Statistical Significance Test of Asymmetrical Correlations


Let
T
denote the whole historical
time period of the data set,


]
,
,
2
,
1
[
L
T












(27)

For example, in financial markets, a time series of (27) may be daily or minutely prices,
depending on the nature of the data set.


Let all the time intervals for testing asymmetrical corr
elations have the equal length
L
l

,
and two consecutive time intervals have a time shift
L
h

. Let
m
denote the total number of
time intervals covering the whole time period, apparently,









h
L
m










(28)

where



denotes the integer part of a positive real number.


For any given pair of variables
X

and
Y
, a series of Asymmetrical Correlation pairs can be
calculat
ed, so a series of differences will be

),
,
,
,
(
)
,
,
,
(
)
,
,
(
{
0
0
0
l
t
X
Y
AC
l
t
Y
X
AC
t
Y
X
d



for
}
)
1
(
1
,
,
2
1
,
1
,
1
0
h
m
h
h
t







(29)

similarly for using
)
,
,
,
(
l
t
Y
X
MAC

instead of
)
,
,
,
(
0
l
t
Y
X
AC
.


Let
)
,
(
Y
X
d
and
)
,
(
Y
X


be the sample mean and p
opulation mean for
)}
,
,
(
{
0
t
Y
X
d
defined
by (29),
the

hypotheses to be tested are



Null hypothesis
:
0
H

0
)
,
(

Y
X








(30)



First alternative hypothesis
:
1
H
0
)
,
(

Y
X







(31)



Seco
nd alternative hypothesis
:
2
H
0
)
,
(

Y
X







(32)


We can use t
-
statistic, denoted by

(
in order to differentiate from the time
t
)
,


m
Y
X
Std
Y
X
d
)
,
(
)
,
(











(33)

where
)
,
(
Y
X
Std
is the standard deviation of
)}
,
,
(
{
0
t
Y
X
d
of (29).


7
/
17

Setting the significance level

, for example
%
5


usually, it corresponds to a critical
value


of t
-
statistics

with degrees of
freedom
.
1

m

The m
-
paired t
-
test can be concluded
with one of the three outcomes



If




, accept
:
0
H

there is no directed influence between
X

and
Y
;


(34)



Else if




, accept
:
1
H

Y
X

, or write
)
|
(
X
Y





(35)



Else if





, accept
:
2
H
X
Y

, or write,
)
|
(
Y
X





(36)


In fact, there is a fourth possible outcome: both
)
,
(
Y
X
AC
and
)
,
(
X
Y
AC
are positive and
significant, or both are negative and significant. This is to say there is a significant undirected
or mutual depe
ndence or influence between
X

and
Y
. This type of influences are not
considered in SBIN models, but they are included in general Super Influence Networks (SINs)
[Pan 2004a].


6. An Application of Multivariable Fi
nancial Time Series


We have applied the approach of Asymmetrical Correlation Test as described above to an
application of multivariate financial time series. In this application, there are 16 interrelated
markets of the same type but with different term s
tructures, denoted as


X
)
(
t
=
))
(
,
),
(
),
(
(
16
2
1
t
X
t
X
t
X

,
16

n






(37)

The data set contains 16 time series of daily close prices for a common period of about two
years. Here we let the time interval have a length of
22

l
trading
days, corresponding
approximately to one month. The
re are totally
26

m

time intervals, all non
-
overlapping, so
.
22

h


Fig.1 and Fig.2 show the first 8 markets
))
(
,
),
(
),
(
(
8
2
1
t
X
t
X
t
X

the se
cond 8 markets
))
(
,
),
(
),
(
(
16
10
9
t
X
t
X
t
X

respectively.





Fig.1 The daily close charts of the first 8 markets
))
(
,
),
(
),
(
(
8
2
1
t
X
t
X
t
X




8
/
17



Fig.
2

The daily close charts of the second 8 markets
))
(
,
),
(
),
(
(
16
10
9
t
X
t
X
t
X



For 16 variables, there are about
120

pairs of variables to be tested for asymmetrical
correlations. The level of significance
%
5


,

t
he degrees of freedom
=25, and therefore
t
he critical value for t
-
statistics
.
708
.
1




On this level of significance,
62 asym
metrical
influences have been detected, for which
0
H
has been rejected.
Among them,
49 influences

have
1
H

accepted, with
,
617
.
14
047
.
2



13 influences have
2
H

accepted, with
.
063
.
2
382
.
5






From the

results we
can conclude

that
1
H

has been accepted more
often and on a higher significance level than
2
H
. This reflects an inherent structure among
the 16 markets.


The 62 detected significant in
fluences can be clustered into 4 groups

(quarters)

according to
different levels of significance:


Quarter I

contains 15 influences

{
),
12
|
4
(

(5|12),

),
12
|
6
(

),
12
|
7
(

),
12
|
8
(

),
12
|
11
(

(4|13),
),
13
|
5
(

),
13
|
6
(

),
13
|
7
(

)
13
|
8
(
, (11|13), (8|15), (7|16), (8|16)
}
.

This is the most significant
group.
Fig.3

shows
asymmetrical
correlation

coefficients

for
)
13
|
7
(
over all the time
intervals (also called subperiods) as an example from this quarter.


Quarter II
contains 15 influences

{
),
12
|
3
(

),
12
|
10
(

),
13
|
3
(

),
13
|
10
(

),
14
|
6
(

),
14
|
7
(

),
14
|
8
(

(10|14), (11|14),
),
15
|
6
(

),
15
|
7
(

(5|16),
),
16
|
6
(

(11|16),
),
16
|
15
(
}. This is the
second most significant group. Fig
.

4 shows asymmetrical correlation coe
fficients for
)
14
|
6
(
as an example from this quarter.


Quarter III

contains
1
6

influences {
(14|12), (2|13), (14|13), (15|13),
),
14
|
3
(

),
14
|
4
(

),
14
|
5
(

(3|15),

),
15
|
4
(

),
15
|
5
(

),
15
|
10
(

),
15
|
11
(

(3|16),
),
16
|
4
(
),
16
|
10
(

)
16
|
14
(
}
. Fig.
5

shows
)
15
|
10
(
as an example from this
quarter
.


Quarter IV

contains

16
influences {
),
7
|
8
(

),
2
|
10
(

),
3
|
10
(

),
4
|
10
(
),
5
|
10
(
),
6
|
10
(

),
7
|
10
(

),
8
|
10
(

),
9
|
10
(

(10|11), (2 |12),
)
12
|
15
(
, (2|14),

(2|15), (2|16), (13|16)
}. Fig. 6
shows
)
3
|
10
(
as an example from this
quarter
.

9
/
17




Fig. 3 Asymmetrical Correlation
)
13
|
7
(

from
Quarter I





Fig. 4 Asymmetrical Correlation
)
14
|
6
(

from
Quarter II





F
ig. 5 Asymmetrical Correlation
)
15
|
10
(

from
Quarter III

10
/
17




Fig. 6 Asymmetrical Correlation
)
3
|
10
(

from
Quarter IV




Fig. 7 The significant directed influences
: monthly based asymmetrical correlation model


11
/
17

7. Dynamic

Directed Acyclic Graphs

(t
-
statistics based models)


Putting all the detected, statistically significant directed influences together, we will obtain a
dynamic directed acyclic graph, which is the basic structure for a SBIN. Fig.
7

shows the
significant
influences from
all the quarters

together.


Recall that in the proposed model the correlation is computed over 26 non
-
overlapping time
intervals. This model can be called a
monthly based asymmetrical correlation model
.
However, it might be interesting t
o use some other models. In our experiments we also
studied a
weekly based asymmetrical correlation model

and two models, based on Moving
Asymmetrical Correlation (monthly and weekly). The parameters
l

and
h
of t
he
corresponding models are established as follows:

monthly based asymmetrical correlation model:
22

l
,
22

h
;

weekly based asymmetrical correlation model:
5

l
,
5

h
;

monthly based

moving asymmetrical correlation model:
22

l
,
1

h
;

weekly based moving asymmetrical correlation model:
5

l
,
1

h
.




Fig. 8 The significant directed influences
: weekly based a
symmetrical correlation model

12
/
17

In the case of the monthly based asymmetrical correlation model 62 asymmetrical influences
have been detected (the level of significance
%
5


). Fig. 7 shows the structure for a SBIN
for this model. In the ca
se of the weekly based asymmetrical correlation model for the same
level of significance 59 pairs have been detected (comparing to the previous model, one pair
appeared and four pairs disappeared). The distribution of the influences inside the quarters
fo
r these two models is quite similar (Fig. 8). For the moving asymmetrical correlation
models we observe that the same level of significance has been reached a larger number of
asymmetric influences: 107 influences in the case of the monthly based asymmetr
ical
correlation model and 75 influences in the case of the weekly based asymmetrical correlation
model . It was found that if we take 62 the most significant influences detected in the last
two models and distribute them into the quarters similar to how

it has been done for the first
two models. Fig. 9 shows the structure for a SBIN for the monthly based moving
asymmetrical correlation model, Fig. 10 shows the structure for a SBIN for the weekly based
moving asymmetrical correlation model.






Fig. 9
The significant directed influences
: monthly based moving asymmetrical correlation
model


13
/
17





Fig. 1
0

The significant directed influences
: weekly based moving asymmetrical correlation
model



The experiments show that the obtained results are quite sen
sitive to the choice of the
parameters
l

and
h

if the goal is to remove non
-
significant influences (under a certain level).
However, if the goal is to remove a certain amount of non
-
significant influences (for
e
xample, half of the possible influences according to their t
-
values) the results are not very
sensitive to the choice of the parameters. Therefore, in this paper we present the results
obtained for the monthly based asymmetrical correlation model.


8. Dyna
mic Directed Acyclic Graphs

(correlation and partial correlation based models)


In the previous section we tested the significance of the influences according to the
corresponding t
-
values. In our study we also develop some other models that are based on
t
esting of absolute values of asymmetric correlations and asymmetric partial correlations. We
study two models: Dominant asymmetric correlation model and Minimal asymmetric
correlation model.

14
/
17


The models have been constructed as follows:

1.

Detecting of sign
ificant influences by means of t
-
test (in our experiments we use the
monthly asymmetric correlation model, where 62 influences have been detected as
significant).

2.

Arranging of the influences into the quarter according to their dominant or minimal
asymmetr
ic correlations.


The dominant asymmetric correlation for a chosen pair
)
,
(
Y
X

is calculated as
|}
)
,
(
|
|,
)
,
(
max{|
)
,
(
X
Y
AC
Y
X
AC
Y
X
DAC

.

If
|
)
,
(
|
)
,
(
Y
X
AC
Y
X
DAC

, then the influence
Y
X

is significant, otherwise the
influence
X
Y

is significant.



Fig. 1
1

The significant directed influences
: Dominant asymmetrical correlation model


Suppose that on the first stage it was found that a market
X

is influenced by markets
k
Y
Y
Y
,
,
,
2
1

,
then the minimal asymmetric correlations are calculated as

)}},
|
,
(
{
min
),
,
(
min{
)
,
(
,
,
1
i
j
j
i
k
i
j
j
Y
X
Y
CAC
X
Y
AC
X
Y
MINAC





,
,
1
k
j



15
/
17

where
)
|
,
(
i
j
Y
X
Y
CAC

is the conditional asymmetric correlation which can be obtained
from (22).




Fig. 1
2

The significant directed
influences
: Minimal asymmetric correlation model


Fig. 11 shows the structure for a SBIN for the Dominant asymmetrical correlation model and
Fig. 12 shows the structure for a SBIN for the Minimal asymmetrical correlation model. The
distribution of the inf
luences into the quarters in the case of these two models are quite
similar to each other and to the distribution of influences in the case of the t
-
statistics based
models (Fig. 7).
It confirms that the obtained significant influences are indeed very
str
ong

.




9. Ever
-
Changing Cycles, Scale Space of Time and Conditional Asymmetrical


Correlation Test


Ever
-
changing cycles are the most elusive dynamics in financial markets. We may see some
shadows of ever
-
changing cycles in the examples shown in Fi
g.
3,
4,

5,

6. But we may not
16
/
17

see obvious patterns from there. The scale
space

of time is also worth further investigation. It
is known that financial markets and many complex systems exhibit fractal nature in their
behaviours
. Multiresolution asymmetrical

correlation test has the potential to capture fractal
influences among different markets. Typically the influences detected from the asymmetrical
correlation test may need to go through further conditional asymmetrical correlation tests, so
that spurious
or redundant influences may be removed. This will render the final SBIN
models more efficient and more reliable.


10. Conclusions


This paper
has
described a computational procedure of Asymmetrical Correlation Test (ACT)
for detecting directed influences w
ithin multivariate time series. This test is used as the first
step for constructing a Super Bayesian Influence Network (SBIN), by generating a basic
structure of dynamic directed acyclic graph. A real
-
world application on 16 financial markets
has shown th
at the ACT can effectively detect most of the significant directed influences
from all the possible combinations of these markets. However, much work remains to be
done for further research on ever
-
changing cycles, scale space of time and conditional
asymm
etrical correlations.


R
eferences




Alexander C. (2001):
Market Models
. Wiley.



Engle R.F. and Granger C.W.J. (1987): Co
-
integration and error correction: Representation,
estimation, and testing.
Econometrica
55:251
-
276.



Jordan M.I. (1998):
Learning in Graph
ical Models.

MIT Press.



Pan H.P.

(2004a): An Initial Theory of Super Bayesian Influence Networks


nonlinear
dynamic Bayesian networks of probability ensembles of neural networks. Submitted to
International Conference on Computational Intelligence for Mod
elling, Control and
Automation (CIMCA’2004b), July 2004, Gold Coast, Australia.



Pan H.P. (2004
b
):
A Swingtum Theory of Intelligent Finance integrating technical,
quantitative and fundamental analysis of the financial markets.
Accepted for oral
presentation

by
the
24
th

International Symposium on Forecasting
, Sydney, July 2004.



Pan H.P., Ghosh R., and Yearwood J. (2004): An ensemble of overfitting neural networks
for generating probability distribution of predictions. Accepted for oral presentation, ibid.




P
an H.P., Tilakaratne C., Yearwood J
.

(2003): Predicting the Australian Stock Market
Index Using Neural Networks Exploiting Dynamical Swings and Intermarket Influences.
Lecture Notes in Artificial Intelligence


Proc. 16
th

Australian Joint Conference on
Art
ificial Intelligence
, Springer Verlag. A revised version accepted by
the
Journal of
Research and Practice in Information Technology
(Official Journal of Australian Computer
Society).



Pan H.P. (2003a): A Unified Science of Intelligent Finance


Part 1.
Jour
nal of ATAA
(Australian Technical Analysts Association)
,
2003(July/August):11
-
21
.



Pan H.P. (2003b): A Unified Science of Intelligent Finance


Part 2.
Journal of ATAA,
2003(September/October)
:14
-
19.



Pan H.P. (2003c): A joint review of technical and quant
itative analysis of financial markets
towards a unified science of intelligent finance.
Proc. 2003 Hawaii International
Conference on Statistics and Related Fields
, June 5
-
9, Hawaii, USA.



Pan H.P. (2003d): Swingtum


A computational theory of fractal dynam
ic swings and
physical cycles of stock market in a quantum price
-
time space. Ibid.

17
/
17



Pan

H.P. and Liu L. (2000)
: Fuzzy Bayesian networks
-

A general formalism for
representation, inference and learning with hybrid Bayesian networks.
International
Journal of
Pattern Recognition and Artificial Intelligence
, 14(7): 941
-
962.



Pan H.P. (2002a): Learning Bayesian Networks I


A Theory.
Proceedings of 5
th

International Conference on Information Fusion
, USA, July.



Pan H.P. (2002b): Learning Bayesian Networks II


A

Computational Algorithm. Ibid.



Pearl J. (2000):
Causality: Models, Reasoning, and Inference
. Cambridge University Press.



Shachter R.D. and Kenley C.R. (1989): Gaussian Influence Diagrams.
Management
Science

35(5):527
-
550.



Spirtes P., Glymour C. and Schei
nes R. (1992):
Causation, Prediciton, and Search
. Lecture
Notes in Statistics 81. Springer.