3.7 Analysis in long time data
3.7.1 Calculation with overlapping
in the log-log scale power spectra whether it is in the market modes or in the other modes.
Such characteristics provide evidences as the presence of intermittent fluctuations.
3.7 Analysis in long time data
3.7.1 Calculation with overlapping
To confirm the phenomenon we have observed, we use the whole year data to check it again. First, we check the eigenvalue of the cross correlation matrix. Here is the eigen-value of the cross correlation matrix from 1996 whole year data.
0 10 20 30 40 50 60
Data from 1996 with overlap
Raw data
Figure 3.37: The eigenvalue of correlation matrix from raw data in different τ (SP500 1996 full year)
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Data from 1996 with overlap
with HF1MA
Figure 3.38: The eigenvalue of correlation matrix from data with HF1MA in different τ . (SP500 1996 full year)
From figure 3.37 and figure 3.38 we can find out that the biggest eigenvalue of the data with HF1MA is always larger than the raw data for different τ . The result is same as the calculation we have done before. The reason is that taking the average of the shifting windows (HF1MA) would strength the connection between each stock. The reason that the correlation matrix does not have the discrete largest eigenvalue is probably caused by the length of the data and causing the mean of the correlation matrix elements close to zero. It will be more clearly verified by the following figure
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國立 政 治 大 學
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N a tio na
l C h engchi U ni ve rs it y
-0.5 0 0.5 1
Value of element 0
2 4 6 8 10 12
τ=612 Sec.
τ=3600 Sec.
τ=18000 Sec.
τ=36000 Sec.
τ=54000 Sec.
τ=72000 Sec.
Distribution of Correlation matrix elements for data with overlap
Data from S&P 500 in 1996
Figure 3.39: Distribution of correlation’s elements, calculated with overlap. The whole year’s length of data is 162500, due to our calculation method, the length of data in dif-ferent τ will be difdif-ferent. The length of the data for 612 Sec. is 162583. For 3600 Sec. is 162400. For 18000 Sec. is 162000. For 36000 Sec. is 161500. For 54000 Sec. is 161000.
For 72000 Sec. is 160500. (1996 SP 500 full year)
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國立 政 治 大 學
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N a tio na
l C h engchi U ni ve rs it y
-0.6 -0.4 -0.2 0 0.2 0.4 0.6
Value of element 0
10 20 30 40 50
τ=612 Sec.
τ=3600 Sec.
τ=18000 Sec.
τ=36000 Sec.
τ=54000 Sec.
τ=72000 Sec.
Distribution of Correlation elements for random number with overlap
Data from random number,length 162500
Figure 3.40: Distribution of correlation’s elements, calculated with overlap. The length of data is 162500, due to our calculation method, the length of data in different τ will be different. The length of the data for 612 Sec. is 162583. For 3600 Sec. is 162400. For 18000 Sec. is 162000. For 36000 Sec. is 161500. For 54000 Sec. is 161000. For 72000 Sec. is 160500. (Random case)
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國立 政 治 大 學
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N a tio na
l C h engchi U ni ve rs it y
0 0.02 0.04 0.06 0.08 0.1
Variance 0
500 1000 1500 2000
τ
Variance of correlation matrix element vs Tau
1996 S&P 500 , with overlapping
Figure 3.41: The variance of correlation matrix element versus the τ , calculated with overlap. The length of the data for 612 Sec. is 12983. For 3600 Sec. is 12900. For 18000 Sec. is 12500. For 36000 Sec. is 12000. For 54000 Sec. is 11500. For 72000 Sec. is 11000. (1996 SP 500 full year)
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國立 政 治 大 學
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N a tio na
l C h engchi U ni ve rs it y
0 0.02 0.04 0.06 0.08 0.1
Variance 0
10 20 30 40 50 60
λM
Correlation matrix’s largest eigenvalue vs variance
with overlap , 1996 S&P 500
Figure 3.42: The variance of correlation matrix element, versus the largest eigenvalue of correlation matrix, calculated with overlap. The length of the data for 612 Sec. is 12983.
For 3600 Sec. is 12900. For 18000 Sec. is 12500. For 36000 Sec. is 12000. For 54000 Sec. is 11500. For 72000 Sec. is 11000. (1996 SP 500 full year)
We can find some difference between the whole year data and monthly data in the distribution of correlation’s elements in figure 3.39 and figure 3.5. In figure 3.39 and figure 3.41, we can observed that as the τ grows, the variance of the distribution will grow too. The feature of figure. 3.39 is similar to figture. 3.40, but for figure 3.39 it’s mean value is not 0, it shift right from the zero line. The reason that variation grows with τ and λM is due to the random feature. In random case, while the τ grows the variation will grows, too. The correlation matrix element can be describe by
Cij = ¯C + ηijσ2+ b(τ ) (3.23)
Where µ represent the means, η is a small constant, σ2represent the variance. b(τ ) is some
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國立 政 治 大 學
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N a tio na
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factor related to τ .
Therefore, we compare the largest eigenvalue from the real market data and the largest eigenvalue compute from random symmetric matrix. We can compute the largest eigen-value from random symmetric matrix as
λM = (N− 1) ¯C + 1 + σ2/ ¯C
(3.24)
where n represent the rank of the matrix, ¯C represent the means and σ2 represent the variance. [12]
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國立 政 治 大 學
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N a tio na
l C h engchi U ni ve rs it y
5 10 15
λsymmetric 10
20 30 40 50
λM
The largest eigenvalue from real data and symmetric matrix
Real market data with overlap, 1996
Figure 3.43: The largest eigenvalue compute from real market data and random sym-metric matrix.The real market data is calculated with overlapping. (1996 SP 500) The red line represent the slope =1 which means the largest eigenvalue compute from random symmetric matrix equal to the largest eigenvalue compute from the real market data.
From figure 3.42 we can know that the variance relates to the largest eigenvalue, but the largest eigenvalue compute from random symmetric matrix does not equal to the the largest eigenvalue compute from the real market data, moreover Eq.3.24 do not evaluated when ¯C ≈ 0. It might be that the random symmetric matrix does not capture the Wishart matrix’s structure. There may be some factor relate to τ behind it.
Second, we check the stock part (eigenvector) from 1996 whole year data. We show the modes with the top four eigenvalue and the mode with the smallest eigenvalue in dif-ferent τ .
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Eigenvector for typical mode,Raw, τ =3600 sec
λM =12.01
Figure 3.44: The eigenvector of correlation matrix from raw data in typical modes. The length of data for 3600 second is 11600. (SP500 1996 full year)
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Eigenvector for typical mode,HF1MA, τ=3600 sec
λM =25.32
Figure 3.45: The eigenvector of correlation matrix from data with HF1MA in typical modes. The length of data for 3600 second is 10950. (SP500 1996 full year)
From figure 3.44 and figure 3.45, the stock part still remain the same features in market mode, but some points in market mode cross the zero line. It is a little different from calculation in 1996 Jan. .
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Eigenvector for typical mode,Raw, τ=72000 sec
λM =54.87
Figure 3.46: The eigenvector of correlation matrix from raw data in typical modes. (SP500 1996)
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Eigenvector for typical mode,HF1MA, τ =72000 sec
λM =64.79
Figure 3.47: The eigenvector of correlation matrix from data with HF1MA in typical modes. (SP500 1996 full year)
From figure 3.46 and figure 3.47 shows the same result in small τ . The reason that the stock part in market mode is not all positive as the result we have done before is that market mode is close to or even merged into the continuous band, in the cases of smaller tau.
Third, we check the temporal part from 1996 full year data. We show the modes with the top four eigenvalue and the mode with the smallest eigenvalue in different τ .
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國立 政 治 大 學
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N a tio na
l C h engchi U ni ve rs it y
0.01 0.1 1
0.0001 1
DFT for typical mode,Raw, τ =3600 sec
λM =12.01
0.01 0.1 1
0.0001
λ4=8.79
0.01 0.1 1
0.0001 1
λ2=11.39
0.01 0.1 1
ω
0.0001
λmin=0.13
0.01 0.1 1
0.0001 1
S (ω)
λ3=9.74
Figure 3.48: The Karhunan-Loeve expansion temporal part from raw data in typical modes. (SP500 1996 full year)
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國立 政 治 大 學
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N a tio na
l C h engchi U ni ve rs it y
0.01 0.1 1
0.0001
DFT for typical mode,HF1MA, τ=3600 sec
λM =25.32
0.01 0.1 1
1e-08 0.0001 1
λ4=19.61
0.01 0.1 1
0.0001 0.001 0.01
λ2=22.64
0.001 0.01 0.1 1
ω
0.0001
λmin=1.45 E-004
0.01 0.1 1
0.0001 0.001 0.01
S (ω)
λ3=21.11
Figure 3.49: The Karhunan-Loeve expansion temporal part from data with HF1MA in typical modes. (SP500 1996 full year)
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國立 政 治 大 學
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N a tio na
l C h engchi U ni ve rs it y
0.001 0.01 0.1 1
1e-08 0.0001
DFT for typical mode,Raw, τ =72000 sec
λM =54.87
0.01 0.1 1
1e-08 0.0001
λ4=39.17
0.001 0.01 0.1 1
1e-08 0.0001
λ2=45.10
0.001 0.01 0.1 1
ω
0.0001 1
λmin=7.64 E-003
0.001 0.01 0.1 1
1e-08 0.0001
S (ω)
λ3=41.64
Figure 3.50: The Karhunan-Loeve expansion temporal part from raw data in typical modes. (SP500 1996 full year)
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DFT for typical mode,HF1MA, τ =72000 sec
λM =64.79
Figure 3.51: The Karhunan-Loeve expansion temporal part from data with HF1MA in typical modes. (SP500 1996 full year)
For the figure from 3.48 to 3.51, we have the same result that we have seen : the market mode leads the long term tendency, and other modes leads the variation in short time and have the straight line with a slope equal to −1 in the log-log plot. And the smaller scale variations in S(ω) are in fact well-organized when we examine the spectra for smaller τ . For figure 3.48, the power spectra with the smallest eigenvalue doesn’t toward down in low frequency regime, but the others has the smallest eigenvalue still retained some features in market modes, like they have straight line with a slope equal to −1 within the bulk of major continuous band, except the presence of a peak at low frequency regime.
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國立 政 治 大 學