Intelligent Information Ma nagement, 2011, 3, 65-70
doi:10.4236/iim.2011.33008 Published Online May 2011 (
Copyright © 2011 SciRes. IIM
Cross Correlation of Intra-Day Stock Prices in Comparison
to Random Matrix Theory
Mieko Tanaka-Yamawaki
Department of Information and Knowledge Engineering, Tottori University, Tottori, Japan
Received February 7, 2011; revised March 21, 2011; accepted March 22, 2011
We propose and apply a new algorithm of principal component analysis which is suitable for a large sized,
highly random time series data, such as a set of stock prices in a stock market. This algorithm utilizes the fact
that the major part of the time series is random, and compare the eigenvalue spectrum of cross correlation
matrix of a large set of random time series, to the spectrum derived by the random matrix theory (RMT) at
the limit of large dimension (the number of independent time series) and long enough length of time series.
We test this algorithm on the real tick data of American stocks at different years between 1994 and 2002 and
show that the extracted principal components indeed reflects the change of leading stock sectors during this
Keywords: Principal Component, Random Matrix Theory, Cross Correlation, Eigenvalues, Stock Market
1. Introduction
Many stock market analysts rely on various technical
indicators calculated for individual stocks. However, the
difficulty lies in the fact that the optimal time scale var-
ies by time and they can be calculated only after the price
time series is recorded. Moreover, the average size of the
relevant time scales are not very long, typically shown
by the auto-correlation function of the price time series
vanish after a few ticks. This fact makes the prediction of
future market situation difficult.
Very often, important information to characterize the
market can be obtained by observing the change of active
business sectors, since multiple stocks in the same busi-
ness sectors move coherently, independent of the finan-
cial status or other parameters of individual stocks.
The method was first proposed and applied on stock
prices by Plerou, et al. [1,2] on daily close prices (com-
bined with other data) of American stocks and Laloux, et
al. [3] then applied on daily close prices of Japanese
stocks by Aoyama et al.
This method uses a result of random matrix theory
(RMT) [4-6] on the principal component analysis of the
cross correlation matrix of pairs of independent time
series. Namely, the method attempts to separate the prin-
cipal components from the random noises by identifying
the random part of the eigenvalue spectrum, to the part
that is identical to the theoretically derived formula [7],
out of the eigenvalue distribution of cross-correlation
matrix between pairs of stock time series.
Reference [1] argued that the corresponding eigen-
vector components of the extracted principal components
characterizes the eminent business sectors of the years in
the period of 1991 to 1998. The reason why the entire 7
years was used for one analysis is the restriction of the
methodology requiring N < T, where the length T of each
time series and the number N of independent time series
used for correlation matrix, whose (i, j) element consists
of the inner product between the time series of the i-th
stock and the j-th stock.
In this paper, we show our challenge of applying the
same line of thought as above, on the tick data of the
American market during the period of 1994 to 2002. Due
to the higher frequency of data points, we can compare
the results of different years and describe the historical
change of the market in the scenario of this new method
of principal component analysis based on RMT spectrum.
For the sake of convenience, we name this methodology
as RMT-PCA (RMT-oriented PCA).
The rest of this article is constructed as follows. We
describe the basic method of RMT-PCA in Chapter 2.
Then we show, in Chapter 3, the results of applying the
methodology on 1-hour data extracted from the tick data
of NYSE and compare the results of computation by
showing the eigenvalue spectrum of 3 years, 1994, 1998
and 2002, compared to the theoretical formula [7] of
RMT [6]. We show in this chapter that the truly signifi-
cant eigenvalues are limited to the first few components,
unlike the results of the conventional principal compo-
nent analysis who counts on the first 80 percent of the
accumulated eigenvalues, or other criteria of separating
the signal from the noise. In chapter 4, we examine the
business sectors which eminent components of eigen-
vectors of significant principal components belong to.
We compare the eminent business sectors of 3 different
years, 1994, 1998 and 2002 and show that they indeed
reflect the actual list of active stocks during this period
of time. Chapter 5 is dedicated to the summary and dis-
cussion of this paper.
2. Cross Correlation of P r i ce T im e Se r i es
It is of significant importance to extract sets of correlated
stock prices from a huge complicated network of hun-
dreds and thousands of stocks in a market. In addition to
the correlation between stocks of the same business sec-
tors, there are correlations or anti correlations between
different business sectors.
For the sake of comparison between price time series
of different magnitudes, we often use the profit instead
of the prices [1-5]. The profit is defined as the ratio of
the increment ΔS, the difference between the price at t
and t + Δt, divided by the stock price S(t) itself at time t.
Stt StSt
St St
 
This quantity does not depend on the unit, or the size,
of the prices which make us possible to deal with many
time series of different magnitude. More convenient
quantity, however, is the log-profit defined by the dif-
ference between log-prices.
 
log logrtSt tSt
Since it can also be written as
 
log Stt
rt St
and the numerator in the log can be written as S(t) +
 
log 1St St
rt St St
 
It is essentially the same as the profit r(t) defined on
Equation (1). The definition in Equation (2) has an ad-
The correlation Ci,j between two stocks, i and j, can be
written as the inner product of the two log-profit time
rt and
 
iji j
t (5)
We normalize each time series in order to have the
zero averages and the unit variances as follows.
1, ,
rt r
Here the suffix i indicates the time series on the i-th
member of the total N stocks.
The correlations defined in Equation (5) makes a
square matrix whose off-diagonal elements are in general
smaller than one.
CiNj N (7)
and its diagonal elements are all equal to one due to
CiN (8)
Moreover, it is symmetric
ij ji
CCi NjN  (9)
As is well known, a real symmetric matrix C can be
diagonalized by the similarity transformation 1
by an orthogonal matrix V satisfying t
VV, each
column of which consists of the eigenvectors of C.
such that
1, ,
Cv vN (11)
where the coefficient k
is the k-th eigenvalue.
Equation (11) can also be written explicitly by using
the components as follows.
,, ,
ij kjkki
Cv v
The eigenvectors in Equation (10) form an ortho-
normal set. Namely, each eigenvector is normalized
to the unit length
kk kn
vv (13)
and the vectors of different suffices k and l are orthogo-
nal to each other.
kl knln
vv (14)
Copyright © 2011 SciRes. IIM
Equivalently, it can also be written as follows by using
Kronecker’s delta.
,kl kl
vv (15)
The right hand side of Equation (15) is zero(one) for
. The numerical solution of the eigenvalue
problem of a real symmetric matrix can easily be ob-
tained by repeating Jacobi rotations until all the off-di-
agonal elements become close enough to zero.
3. RMT-Oriented Principal Component
Analysis (RMT-PCA)
The diagonalization process of the correlation matrix C
by repeating the Jacobi rotation is equivalent to convert
the set of the normalized set of time series in Equation (6)
into the set of eigenvectors.
 
txtV (16)
It can be written explicitly using the components as
 
ytvx t
The eigenvalues can be interpreted as the variance of
the new variable discovered by means of rotation toward
components having large variances among N independ-
ent variables. Namely,
 
11 1
,, ,
il lim m
tl m
il imlm
vx tvxt
vv C
 
 
since the average i
of yi over t is always zero based
on Equation (6) and Equation (17). For the sake of sim-
plicity, we name the eigenvalues in descending order,
The theoretical base underlying the principal compo-
nent analysis is the expectation of distinguished magni-
tudes of the principal components compared to the other
components in the N dimensional space. We illustrate in
Figure 1 the case of 2 dimensional data (x, y) rotated to a
new axis z = ax + by and w perpendicular to z, in which z
being the principal component and this set of data can be
described as 1 dimensional information along this prin-
cipal axis.
If the magnitude of the largest eigenvalue 1
of C is
significantly large compared to the second largest eigen-
value, then the data are scattered mainly along this prin-
cipal axis, corresponding to the direction of the eigen-
vector 1 of the largest eigenvalue. This is the first
principal component. Likewise, the second principal
component can be identified to the eigenvector 2
v of
the second largest eigenvalue perpendicular to 1. Ac-
cordingly, the 3rd and the 4th principal components can be
identified as long as the components toward these direc-
tions have significant magnitude. The question is how
many principal components are to be identified out of N
possible axes.
One criterion is to pick up the eigenvalues larger than
1. The reason behind this scenario is the conservation of
trace, the sum of the diagonal elements of the matrix,
under the similarity transformation. Due to Equation (8),
we obtain
which means there exists m such that 1
for m
and 1
for m. This criterion is too loose to
use for the case of the stock market having N > 400.
There are several hundred eigenvalues that are larger
than 1, and many of the corresponding eigenvector
components are literally random and do not carry useful
Another criterion is to rely on the accumulated contri-
bution. It is recommended by some references to regard
the top 80 percent of the accumulated contribution are to
be regarded as the meaningful principal components.
This criterion is too loose for the stock market of N >
400, for m easily exceeds a few hundred.
A new criterion proposed in [1-4] and examined re-
cently in many real stock data is to compare the result to
the formula derived in the random matrix theory [6].
According to the random matrix theory (RMT, hereaf-
ter), the eigenvalue distribution spectrum of C made of
random time series is given by the following formula [7]
 
in the limit of
,, consNTQTN t.
Figure 1. A set of four 2-dimensional data points are char-
acterized as a set of 1-dimensional data along z axis.
Copyright © 2011 SciRes. IIM
where T is the length of the time series and N is the total
number of independent time series (i.e. the number of
stocks considered). This means that the eigenvalues of
correlation matrix C between N normalized time series of
length T distribute in the following range.
 1
are the upper bound and the lower bound of Equation
(20). The proposed criterion in our RMT-PCM is to use
the components whose eigenvalues, or the variance, are
larger than the upper bound given by RMT.
4. Cross Correlation of Intra-Day Stock
In this chapter we report the result of applying the
method of RMT_PCM on intra-day stock prices. The
data sets we used are the tick-wise trade data (NYSE-
TAQ) for the years of 1994 - 2002. We used price data
for each year to be one set. In this paper we mention our
result on 1994, 1998 and 2002.
One problem in tickdata is the lack of regularity in the
traded times. We have extracted N stocks out of all the
tick prices of American stocks each year that have at
least one transaction in the 1-hour block including every
hour of the days between 10 am to 3 pm. More precisely,
among the actual transactions executed between 9:30 to
10:30, the price closest to 10 o’clock is taken as the price
at 10am. This provides us a set of price data of N sym-
bols of stocks with length T, for each year. For 1994,
1998 and 2002, the number of stock symbols N as 419,
490, and 569, respectively. The data length T was 1512,
six (per day) times 252, the number of working days of
the stock market in the above three years. This method
can be called as “block-tick” method in comparison to
“before-tick” method to be mentioned in Chapter 6. The
stock prices thus obtained becomes a rectangular matrix
of where represents the stock symbol
and represents the executed time of the
S1, ,i
1, ,k
The i-th row of this price matrix corresponds to the
price time series of the i-th stock symbol, and the k-th
column corresponds to the prices of N stocks at the time
We summarize the algorithm that we used for extract-
ing significant principal components from 1 hour price
matrix in Table 1.
Table 1. The algorithm to extract the significant principal
components (RMT-PCA) to be applied on tick-wise stock
Algorithm of RMT-PCA:
1) Select N stock symbols for which the traded price exist for all
1, ,t
. (6 times a day, at every hour from 10 am to 3 pm, on
every working day of the year).
2) Compute log-return r(t) for all the stocks. Normalize the time
series to have mean = 0, variance = 0, for each stock symbol,
1, ,
3) Compute the cross correlation matrix C and obtain eigenval-
ues and eigenvectors
4) Select discrete eigenvalues larger than
in Equation (22),
the upper limit of the RMT spectrum, Equation (20), and the con-
tinuum spectrum extending the region larger than
Following the procedure described so far, we obtain
the distribution of eigenvalues shown in Figure 2 for the
1-hour stock prices for N = 419 and T = 1512 in 1994.
The histogram shows the eigenvalues (except the
largest λ1 = 46.3), λ2 = 5.3, λ3 = 5.1, λ4 = 3.9, λ5 = 3.5, λ6
= 3.4, λ7 = 3.1, λ8 = 2.9, λ9 = 2.8, λ10 = 2.7, λ11 = 2.6, λ12 =
2.6, λ13 = 2.6, λ14 = 2.5, λ15 = 2.4, λ16 = 2.4, λ17 = 2.4 and
the bulk distribution of eigenvalues under the theoretical
maximum, λ+ = 2.3. These are compared with the RMT
curve of Equation (20) for Q = 1512/419 =3.6.
Corresponding result of 1998 data gives the eigen-
value distribution shown in Figure 3 for N=490 and T =
1512. In 1998, for N = 490, T = 1512, there are 24 ei-
genvalues: λ1 = 81.12, λ2 = 10.4 λ3 = 6.9, λ4 = 5.7, λ5 =
4.8, λ6 = 3.9, λ7 = 3.5, λ8 = 3.5, λ9 = 3.4, λ10 = 3.2, λ11
= 3.1, λ12 = 3.1, λ13 = 3.0, λ14 = 2.9, λ15 = 2.9, λ16 =
2.8, λ17 = 2.8, λ18 = 2.8, λ19 = 2.7, λ20 = 2.7, λ21 = 2.6,
λ22 = 2.6, λ23 = 2.5, λ24 = 2.5 and the bulk distribution
of eigenvalues under the theoretical maximum, λ+ = 2.46.
These are compared with the RMT curve of Equation (20)
for Q = 1512/490 = 3.09.
Similarly, we obtain N=569 and T=1512 for 2002 data
as shown in Figure 4, there are 19 eigenvalues, λ1 =
166.6, λ2 = 20.6, λ3 = 11.3, λ4 = 8.6, λ5 = 7.7, λ6 = 6.5, λ7
= 5.8, λ8 = 5.3, λ9 = 4.1, λ10 =4.0, λ11 = 3.8, λ12 = 3.5, λ13
= 3.4, λ14 = 3.3, λ15 = 3.0, λ16 = 3.0, λ17 = 2.9, λ18 = 2.8 =
3.0, λ19 = 2.6, and the bulk distribution under the theo-
retical maximum, λ+ = 2.61. These are compared with the
RMT curve of Equation (12) for Q = 1512/569 = 2.66.
However, a detailed analysis of the eigenvector com-
ponents tells us that the random components do not nec-
essarily reside below the upper limit of RMT, λ+, but
percolates beyond the RMT limit if the sequence is not
perfectly random. Thus it is more reasonable to assume
that the border between the signal and the noise is
somewhat larger than λ+. This interpretation also explains
the fact that the eigenvalue spectra always spreads beyond
Copyright © 2011 SciRes. IIM
Figure 2. Distribution of eigenvalues of correlation matrix
of N = 419 stocks for T = 1512 data in 1994 compared to the
corresponding RMT in Equation (20) for Q = T/N = 3.6.
Figure 3. Distribution of eigenvalues of correlation matrix
of N = 490 stocks for T = 1512 data in 1998 compared to the
corresponding RMT in Equation (20) for Q = T/N = 3.09.
Figure 4. Distribution of eigenvalues of correlation matrix
of N = 569 stocks for T = 1512 data in 2002 compared to the
corresponding RMT in Equation (20) for Q = T/N = 2.66.
λ+. It seems there is no more mathematical reason to de-
cide the border between signal and noise.
However, we have some insight upon this point based
on our computer simulations. Namely, the extended con-
tinuum over λ+ occurs even when the time series are ran-
dom, as long as the return, r(t), instead of the original
data, S(t), is used in Equations (1)-(4) in the process of
computing the correlation matrix in Equation (5). This
causes a characteristic deviation from the randomness
and the continuum spectrum extending over λ+. We shall
discuss on this point in later publication and limit our-
selves to make a statement in the step 4 of the RMT-PCA
algorithm in Table 1.
We return to data analysis in order to obtain further
insight for extracting principal components of stock cor-
5. Eigenvectors as the Principal Components
The eigenvector v1 corresponding to the largest eigen-
value is the 1st principal component. For 1-hour data of
1994 where we have N = 419 and T = 1512 (Figure 2),
the major components of U1 are giant companies such as
GM, Chrysler, JP Morgan, Merrill Lynch, and DOW
Chemical. The 2nd principal component v2 consists of
mining companies, while the 3rd principal component v3
consists of semiconductor manufacturers, including Intel.
The 4th principal component v4 consists of computer and
semiconductor manufacturers, including IBM, and the 5th
component v5 consists of oil companies. The 6th and later
components do not have distinct features compared to
the first 5 and can be regarded as random [8].
For 1-hour data of 1998 where we have N = 490 and T
= 1512 (Figure 3), the major components of v1 are made
of banks and financial services. The 2nd principal com-
ponent v2 consists of 10 electric companies, while v3 con-
sists of banks and financial services, and U4 consists of
semiconductor manufacturers. The 6th and later compo-
nents do not have distinct features compared to the first 5
components and regarded as random.
For 1-hour data of 2002 where we have N = 569 and T
= 1512 (Figure 4), the major components of v1 are
strongly dominated by banks and financial services,
while v2 are strongly dominated by electric power sup-
plying companies, which were not particularly visible in
1994 and 1998.
The above observation summarized in Table 2 indi-
cates that Appliances/Car and IT dominated the indus-
trial sector in 1994, which have moved toward the
dominance of Finance, Food, and Electric Power Supply
in 2002.
The advantage in our analysis over that of reference [1]
is the use of tick-wise time series. Every year, we have
large enough length of T = 1512 for all the stocks we
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Table 2. Business sectors of top 10 components of 5 princi-
pal components in 1994, 1998 and 2002.
vk 1994 1998 2002
Finance (4), IT (2),
Appliances/Car (3) Finance (8) Finance (9)
v2 Mining (7), Finance (2) Electric (10) Food (6)
v3 IT (10) Finance (3) Electric (10)
v4 IT (7), Drug (2) IT (10)
Food (4),
Finance (2),
Electric (4)
v5 Oil (9) Mining (6) Electric (9)
Table 3. Business sectors of top 10 components of 5 princi-
pal components for the combined data of 1990-1996 recon-
structed from the data given in Table 1 of reference [1]).
vk 1990 - 1996
v1 (Diverse business sectors)
v2 Semiconductor (6),Computer (4)
v3 Gold (8), Building material (1), Semiconductor/Memory chips (1)
v4 Gold (7), Bank (2), Wireless communications (1)
v5 (Diverse business sectors)
have used, by taking 6 points per day by selecting the
trades nearest to the center of the 6 blocks. This made us
possible to analyze each year’s data separately and com-
pare different years in the flow of history as in Table 2.
This was not possible by using only the daily-close price
data as in reference [1]. In order to show this fact, we
refer the Table 1 of reference [1] by reconstructing its
content of the first five eigenvectors into Table 3.
6. Summary and Future Perspectives
In this paper, we propose a new algorithm, RMT-PCA
(RMT-oriented PCA) and examined its validity and ef-
fectiveness by using the real stock data of 1-hour price
time series extracted from the tick-wise stock data of
NYSE-TAQ database of 1994, 1998, and 2002. We have
shown that this method provides us a handy tool to
compute the principal components v1 - v5 in a reasonably
simple procedure.
We have also tested the method by using two different
machine-generated random numbers and have shown
that those random numbers work well for a wide range of
parameters, N and Q, only if we shuffle to randomize the
machine-generated random numbers.
The use of 1-hour price time series made us possible
to compare the results of different years, since there are
approximately T = 1500 data points per each year for the
number of stocks about N = 500. Since the Q parameter
(T to N ratio) around 3 or larger is the safe area for the
usage of Equation (20), we can use 30 minutes, or even
15 minutes data as long as the number of stocks traded at
every these time interval are kept large enough to guar-
antee Q > 3. Another possibility is to consider the ‘be-
fore-tick’ data, in which we take the past traded price
instead of sticking to the actual traded price during the
specified time interval. The advantage of “before-tick”
method is twofold. One is the possibility of choosing
shorter time interval for one data, such as a quarterly data
or a monthly data by yielding larger N and T. Another is
the simplicity in dealing with the raw tick-data without
taking the time consuming process of extracting actual
trades within each block of time interval, as in the
“block-tick” method taken in this paper.
7. References
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Amaral and H. E. Stanley, “Random Matrix Approach to
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Amaral and H. E. Stanley, “Universal and Nonuniversal
Properties of Cross Correlations in Financial Time Se-
ries,” Physical Review Letters, Vol. 83, 1999, pp.
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[5] R. N. Mantegna and H. E. Stanley, “An Introduction to
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[6] M. L. Mehta, “Random Matrices,” 3rd Edition, Academic
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[9] M. Tanaka-Yamawaki, “Applying Random Matrix The-
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