3.7 Analysis in long time data
3.7.2 Calculation without overlapping
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3.7.2 Calculation without overlapping
In order to understand the bulk structure inside the spectra, we adjust our calculation method from with overlap to without overlap. And do the same calculation specially in power spectra for the data without overlap. Due to the length of the data, we only can show the spectra in small τ . First, we check the distribution of correlation’s elements. For the data calculate without overlapping in figure 3.52 and figure 3.54, we can observe same feature like with overlapping that as the τ grows, the variance grows too, and the mean value shifts to right as the length decrease.
-0.2 0 0.2 0.4 0.6
Value of element 0
2 4 6 8
10 τ=360 Sec.
τ=612 Sec.
τ=1080 Sec.
τ=1440 Sec.
τ=1800 Sec.
Distribution of Correlation elements for data without overlap
Data from S&P 500 in 1996
Figure 3.52: Distribution of correlation’s elements, calculated without overlap. The whole year’s length of data is 162500, but in the calculation of no overlapping, the length of the data is quite different. The length of the data for 360 Sec is 14772. For 612 Sec is 9027. For 1080 Sec is 5241. For 1440 Sec is 3963. For 1800 Sec is 3186. (1996 SP 500 full year)
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0.004 0.0045 0.005 0.0055 0.006 0.0065 0.007
Variance 20
30 40 50 60
λM
Correlation matrix’s largest eigenvalue vs variance
without overlap , 1996 S&P 500
Figure 3.53: The variance of correlation matrix element, versus the largest eigenvalue of correlation matrix, calculated without overlap. The whole year’s length of data is 162500, but in the calculation of no overlapping, the length of the data is quite different. The length of the data for 360 Sec is 14772. For 612 Sec is 9027. For 1080 Sec is 5241. For 1440 Sec is 3963. For 1800 Sec is 3186. (1996 SP 500 full year)
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10 20 30 40 50 60 70
λsymmetric 20
30 40 50 60
λM
The largest eigenvalue from real data and symmetric matrix
Real market data without overlap, 1996
Figure 3.54: The largest eigenvalue compute from real market data and random symmet-ric matrix.The real market data is calculated without overlapping. (1996 SP 500 full year) The red line represent the slope =1 which means the largest eigenvalue compute from ran-dom symmetric matrix equal to the largest eigenvalue compute from the real market data.
the length of the data is quite different. The length of the data for 360 Sec is 14772. For 612 Sec is 9027. For 1080 Sec is 5241. For 1440 Sec is 3963. For 1800 Sec is 3186.
For figure 3.55, we compare the calculating with or without overlapping , the black and red line has the same τ but different length of the data. The length of the data in-fluence the mean value of the distribution of correlation’s elements. For the calculating with overlapping shows the symmetric feature (black and red line), but that feature can not been seen in the calculating without overlapping (green line).
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12 with overlap, data length = 162483
with overlap, data length =12983 without overlap, data length= 9027
Distribution of Correlation elements for data with or without overlap
Data in τ=612 Sec. from S&P 500 in 1996
Figure 3.55: Distribution of correlation’s elements, calculate with or without overlapping in same τ = 612 Second. The length of the black line is 162483, and the red line is 12983.
The black and red line are calculating with overlapping. The green line is calculating without overlapping and it’s length of data is 9027. (1996 SP 500 full year)
Second, we check the temporal part again.
0 1 2 3 4 5 6 7
Discrete Fourier Transformation of Karhunen-Loeve expansion time part
Raw data without overlap , Mode 1 τ= 612 Sec
Figure 3.56: The Fourier transform of Karhunan-Loeve expansion temporal part mode 1(with the smallest eigenvalue) for the raw data in τ = 612Sec .(SP 500 1996 full year )
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Discrete Fourier Transformation of Karhunen-Loeve expansion time part
Data without overlap in HF1MA, Mode 1 τ= 612 Sec
Figure 3.57: The Fourier transform of Karhunan-Loeve expansion temporal part mode 1(with the smallest eigenvalue) for the data with HF1MA in τ = 612Sec .(SP 500 1996 full year )
Discrete Fourier Transformation of Karhunen-Loeve expansion time part
Raw data without overlap , Mode 345 τ= 612 Sec
Figure 3.58: The Fourier transform of Karhunan-Loeve expansion temporal part mode 345(with the largest eigenvalue) for the raw data in τ = 612Sec .(SP 500 1996 full year )
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0 1 2 3 4 5 6 7
ω 0
0.01 0.02 0.03 0.04 0.05 0.06
S(ω)
Discrete Fourier Transformation of Karhunen-Loeve expansion time part
Data without overlap in HF1MA, Mode 345 τ= 612 Sec
Figure 3.59: The Fourier transform of Karhunan-Loeve expansion temporal part mode 345(with the largest eigenvalue) for the data with HF1MA in τ = 612Sec .(SP 500 1996 full year )
We can easily find out that the bulk structure inside the spectra is no longer exist in figure 3.56 and figure 3.58. However, the bulk structure is still exist in figure 3.57 and figure 3.59. By comparing with the calculation we have done before, the bulk structures may caused by the overlap.
To be more accurate, we check the typical modes in log-log scale in power spectra.
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0.001 0.01 0.1 1
0.0001
DFT without overlap for typical mode,Raw, τ =612 sec
λM =36.41
0.001 0.01 0.1 1
0.0001
λ4=1.98
0.001 0.01 0.1 1
0.0001
λ2=3.71
0.001 0.01 0.1 1
ω
0.0001
λmin=0.80
0.001 0.01 0.1 1
0.0001
S (ω)
λ3=2.46
Figure 3.60: The Karhunan-Loeve expansion temporal part from raw data without overlap in typical modes.
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DFT without overlap for typical mode,HF1MA, τ =612 sec
λM =71.34
Figure 3.61: The Karhunan-Loeve expansion temporal part from data without overlap in HF1MA in typical modes.
The feature that the spectra has slope equals−1 is no longer been seen in figure 3.60, but can still been seen in figure 3.61. The feature that peak present in market mode at low frequency regime is also gone in both data with HF1MA and the raw data.
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Chapter 4 Discussion
This study is working on the time characteristic in stock-stock cross correlation. Data source is from SP500. We choose 345 stocks with the higher trading frequencies. We an-alyze different modes in K-L expansion. Each mode has stock part and temporal part. We found that the time characteristic in market mode and those in other mode are different. In market mode, it shows the low frequency variation feature, no matter in raw data or in data with HF1MA, we can clearly tell the differences between market mode and mode 1 (with the smallest eigenvalue). The market mode leads the long term tendency, and other modes leads the variation in short time. In power spectra from Fourier transform of temporal part, we found some specific properties. As τ grows, the pack number goes up too. And we found that there are some bulk structure inside the spectra. In log-log scale , we observed some feature. For market mode, we observed that there is a low frequency peak. Both raw data and data with HF1MA have this feature in different τ , but only in market mode.
We can see well recognized regular packs, going along with the exponent−1 feature in the log-log scale power spectra whether it is in the market modes or in the other modes.
Such characteristics provide evidences of the presence of intermittent fluctuations. Those calculation are under log return with overlapping time interval, however. Some features change and some remain for the log returns without overlapping time interval. In power spectra with Fourier transform of temporal part, the regular packs can not be seen for the raw data but still be seen for the data with HF1MA. In log-log scale, the low frequency peak does not exist either in raw data or in data with HF1MA, and well recognized
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ular packs, going along with the exponent−1 feature is also gone. The bulk structure is still seen in the data with HF1MA, but not seen in the raw data. By comparing the log return with or without overlapping time intervals, the result suggest that the low frequency peak is caused by the calculation with overlapping. The bulk structure is also caused by overlapping, and the HF1MA also can cause the bulk structure. In the distribution of cor-relation matrix elements, we can tell the difference between calculations with or without overlapping time intervals. For those with overlapping, the distribution of correlation ma-trix elements show the symmetric features and the mean value approach to 0. As for those without overlapping, it does not show the symmetric feature and the mean value shifts to right(positive) as the sequence length decreases.
The features we have observed can provide a way to distinguish between two different markets. By such mappings in two different market.Their features can help us to find the similarity and difference between them. Some features we have observed in power spectra from Fourier transform of temporal part, maybe can be explained by the coupled random walk model proposed in 2004 [6] . The well recognized regular pikes, going along with the exponent−1 is similar to the 1/f chaotic system, and may find some links between them.
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