Financial Time Series Analysis Based On Normalized Mutual Information Functions

International Journal on Cybernetics & Informatics ( IJCI) Vol.2, No.4, August 2013
DOI: 10.5121/ijci.2013.2401 1
Financial Time Series Analysis Based On
Normalized Mutual Information Functions.
David Blokh
C. D. Technologies Ltd., Israel
david_blokh@012.net.il
ABSTRACT
A method of predictability analysis of future values of financial time series is described. The method is
based on normalized mutual information functions. In the analysis, the use of these functions allowed to
refuse any restrictions on the distributions of the parameters and on the correlations between parameters.
A comparative analysis of the predictability of financial time series of Tel Aviv 25 stock exchange has been
carried out.
KEYWORDS
Nonlinear financial time series, Normalized mutual information function
1. INTRODUCTION
Estimation of the possibility of predicting future values of a time series is an important problem
in the analysis of financial time series [1]. This problem is applicable in econophysics [2] and, in
particular, in the forecast of financial markets [2,3]. In the present paper, the following version of
the problem is examined:
A set of time series is specified. Find time series with the "highest" possibility of predicting future
values in this set.
The principal method of the study of the time series prediction problem is the use of correlation
functions that allow evaluating the possibility of linear prediction of future values of a time series.
Thereby, the gaussianity of distribution and the linearity of correlations are assumed. Rather
often, these conditions do not hold for financial time series [1,4]. In the present paper, functions
of normalized mutual information are used for the analysis of the predictability of financial time
series. It allowed to refuse any restrictions on distributions and correlations.
The possibility of using information functions for the analysis of time series was first noted in [5].
Functions of mutual information were used for the analysis in [1,6,7]. However, mutual
information is an unnormalized value, and it is impossible to compare mutual information
functions of different time series. Therefore, in the present paper, normalized mutual information
is a measure of coupling between two time series. On the one hand, this measure allows the
estimation of nonlinear correlations between time series, and on the other, the comparison of
functions of normalized mutual information of different time series. Normalized mutual
information has been applied with significant results in various fields of medicine [8], in

2
particular in oncology [9,10,11]. The approach was proposed earlier [9], as described in the
monograph [12].
Using the method described in the present paper, a time series with the highest assessment of the
possibility of future values prediction has been chosen from time series with the parameter
"Change" of all companies of Tel Aviv 25 stock exchange. An approach to the clustering of
financial time series similar to that described in the present paper is presented in [13].
Predictability estimation of financial markets also was carried out using statistical methods
[14,15], wavelet analysis [16] and fractals [17].
2. ANALYSIS ALGORITHM
The analysis algorithm consists of four procedures:
- discretization of time series values;
- construction of a normalized mutual information function matrix;
- ranking of columns of the normalized mutual information function matrix;
- application of multiple comparisons method.
2.1 Discretization of Time Series Values
We transform a time series
1( ) ( ( ) ( ) ),...,...,i i i nt t tx x x having continuous values into a
time series
1( ) ( ( ) ( ) ),...,...,i i i nt t ty y y having discrete values. Discretization can be
performed taking into account the properties of the time series [18]. If the time series does not
possess respective properties or we are unaware of them, then we can use formal rules of
discretization [19].
2.2. Construction of normalized mutual information function
Let 1( ) ( ( ) ( ) ),...,...,i i i nt t ty y y represent time series having discrete values and
1( ) ( ( ) ( ) ),...,..., n jji ii t j t ty y y  be a time series ( )i ty with a lag j.
The normalized mutual information of the two time series ( )i ty and ( )i t jy  equals [8,20]
( ) ( ) ( ) ( ) ( ) ( )
( )
( ) ( )
( ; ) ( ) ( ) ( , )
( ) ( )i
i i i i i i
i
i
t t j t t j t t j
j
t j t j
I H H H
H Hy
y y y y y y
c
y
  
 
 
 
where ( )( )i tH y , ( )( )i t jH y  , ( ) ( )( , )i it t jH y y  are entropies of random values ( )i ty ,
( )i t jy  , ( ) ( )i it t jy y  , respectively.

3
The normalized mutual information ( )i jc is then calculated as a function of the lag j.
Properties of the Normalized Mutual Information Function ( )i jc [8,21]:
1) 0  ( )i jc  1;
2) ( )i jc = 0 if and only if ( )tiy and ( )t jiy  are mutually independent;
3) ( )i jc = 1 if and only if there exists a functional relationship between ( )tiy
and ( )t jiy  .
Consider a set of time series
1 1 1 1 1( ) ( ( ) ( ) ), , ( ) ( ( ) ( ) ),... ,...{ ,..., ... ,..., }m m m nnX t t t t t tx x x x x x 
having continuous values. Performing discretization, we obtain a set of time series
1 1 1 1 1( ) ( ( ) ( ) ) ( ) ( ( ) ( ) ), , ,...{ ,..., ,... ,..., }... m m m nnY t t t t t ty y y y y y 
We compute the normalized mutual information functions ( )i jc for the time series
iy
1 i m  ,1 j k  and obtain m×k matrix of function values ( )[ ]i jc .
2.3. Ranking of Columns of Normalized Mutual Information Function Matrix
Each row of ( )[ ]i jc matrix is a normalized mutual information function of time series, and each
column contains the values of normalized mutual information functions corresponding to the
same lag. For each column of ( )[ ]i jc matrix, we rank its entries and assign the rank 1 to the
smallest entry of the column. We obtain m×k matrix of ranks ( )[ ]i jr , with each column of the
matrix containing ranks from 1 to m.
We estimate the predictability of the i-th time series as compared to other time series by
the sum of all the entries of i-th row of the matrix ( )[ ]i jr .
Such estimation allows us to use multiple comparisons of rank statistics for the comparison of
time series predictability.

4
2.3. Application of Multiple Comparisons Method
We compare rank sums using the Newman-Keuls test [22]. The application of this test allows us
to obtain an estimation adequate to the problem content. This test is successfully used for the
analysis of biomedical data [22,23,24] and clustering financial time series [13].
3. PREDICTABILITY ESTIMATION OF TEL AVIV 25 STOCK
EXCHANGE COMPANIES
We perform the predictability estimation of Tel-Aviv 25 stock exchange companies. Obviously,
similar estimates can be also performed for other markets. The importance of such estimation is
noted in [2]. The complexity of this problem consists in the fact that the financial time series
contain nonlinearities [4].
Let ( )ji tp constitute the Adjusted Closing Price of i-th company on the
jt day, and ( )ji tx –
the parameter “Change” equal to
1
1
( ) ( )
( )
( )
j j
j
j
i i
i
i
t t
t
t
p p
x p




.
We consider the changes in the parameter “Change” for all Tel-Aviv 25 stock exchange
companies [25] in the period from June 8, 2010 till December 14, 2011 (374 days). Thus, we are
dealing with a set of time series 251( ),..., ( ){ }t tx x , and the value of each series is the
parameter “Change”. Thus, we have 25 time series, and each series contains 374 elements.
We carry out, step by step, the four procedures mentioned in the previous section.
1. The discretization rule is as follows [19]:
Having performed the discretization, we obtain a set of time series 251( ),..., ( ){ }t ty y , whose
elements assume the values of –2, –1, 0, 1, 2. The discretization is such that the quantities of
elements in each category are “approximately equal”.
2. Then we compute the matrix of normalized mutual information functions – Table 1.

5
3. We rank entries of each column of the normalized mutual information function matrix –
Table 2.
Let us consider Table 2 as the Friedman statistical model [26], and examine the row effect of this
table.
Hypotheses
H0: There is no row effect (“null hypothesis”).
H1: The null hypothesis is invalid. Critical range. The sample is “large”, therefore, the critical
range is the upper 1%-range of 2
24 distribution.
Let us calculate the 2
 -criterion [26]. This gives us 2
 = 56.05.
The critical range is 2
24 > 42.98. Since 56.05 > 42.98, the null hypothesis with respect to Table
2 is rejected. Thus, according to the Friedman test, the row effect has been found. Hence, there is
a difference between the rows under consideration.
4. For multiple comparisons, we use the Newman-Keuls test [22].
1
12.08j j
R R 
 
, where Rj and Rj+1 are elements of the column “Sum of ranks” in the j-th
and (j+1)-th rows of Table 3, respectively. By multiple comparisons, we construct the
predictability estimation shown in Table 3.
The obtained predictability estimation possesses the following properties: for two neighboring
sets of Table 3, the smallest element of one set and the greatest element of another located nearby
are significantly different ( T =0.01); elements belonging to the same set do not differ from
each other ( T =0.01). The differences between Set 1 (STRAUSS GROUP) and all the elements
(companies of the stock exchange) are statistically significant ( T =0.01). Thus, the time series
of the parameter “Change” of STRAUSS GROUP company is the most predictable in comparison
with time series of other Tel Aviv 25 Stock Exchange companies.

6
Table 1. Matrix of normalized mutual information functions.
Table 2. Ranked entries of each column of the normalized mutual information function matrix.

7
Table 3. Predictability estimation of Tel Aviv 25 Stock Exchange Companies
4. CONCLUSION
The analysis of financial time series based on the functions of normalized mutual information
allows predictability assessment without the assumptions of gaussianity and linearity. The use of
these functions also allows us to compare the predictabilities of different time series. At present,
there are no such advanced prediction methods for the functions of normalized mutual
information as for the analysis based on correlation functions. The development of prediction
methods for time series analysis based on the functions of normalized mutual information should
become an object of further research.
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