From the plots below we can easily separate into two part by significantly difference.
For figure 3.12 and figure 3.11, we can tell that mode 345 (with the biggest eigenvalue) it has smaller frequency than other mode, no matter it is raw data or data with HF1MA.
And the different between the data with HF1MA and the raw data for mode 345 (with the biggest eigenvalue) is that the HF1MA has more smooth curve, which is cause by the average of shifting windows, than the raw data. As the τ grows, the frequency of the bj(t) decrease. It suits for both raw data or data with HF1MA.
For figure 3.11, figure3.12, shows that they have much higher frequency than market mode. However, it’s hard to tell the different between them. As the result, we are going to analyze them by the power spectra S(ω) under Fourier transform of the time-wise part bk(t) for the 345 modes in the next section.
3.5 Fourier transform of the time-wise part from the Karhunan-Loeve expansion
In order to find out what’s in the figure 3.11 and figure 3.12, we are going to use the Fourier transformation to expand the Karhunan-Loeve expansion temporal part bj(t) by the following equation
And we sum the real part and imagine part as power spectra from Fourier transform by
S(ω) =√
Im2+ Re2 (3.22)
First, we compare the Fourier transform of Karhunan-Loeve expansion temporal part
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for the raw data with the different τ in the mode 1(with the smallest eigenvalue) .
0 1 2 3 4 5 6
Discrete Fourier Transformation of Karhunen-Loeve expansion time part
Raw data, Mode 1 τ= 3600 Sec
Figure 3.13: The Fourier transform of Karhunan-Loeve expansion temporal part mode 1(with the smallest eigenvalue) for the raw data with τ = 3600Sec. The length of data for 3600 second is 11600.(SP500 1996 Jan)
0 1 2 3 4 5 6
Discrete Fourier Transformation of Karhunen-Loeve expansion time part
Raw data, Mode 1 τ= 36000 Sec
Figure 3.14: The Fourier transform of Karhunan-Loeve expansion temporal part mode 1(with the smallest eigenvalue) for the raw data with τ = 36000Sec. The length of data for 36000 second is 10700.(SP500 1996 Jan)
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0 1 2 3 4 5 6
ω 0
0.02 0.04 0.06 0.08
S(ω)
Discrete Fourier Transformation of Karhunen-Loeve expansion time part
Raw data, Mode 1 τ= 72000 Sec
Figure 3.15: The Fourier transform of Karhunan-Loeve expansion temporal part mode 1(with the smallest eigenvalue) for the raw data with τ = 72000Sec. The length of data for 72000 second is 9700.(SP500 1996 Jan)
We can actually count the pack number while the τ is small, the pack number equals to the τ /36. As the τ goes up, the pack number and the frequency also goes up. And we found out that there are some bulk structure inside the spectra, it will discuss in the next section using power spectra in log-log scale to analyze.
Second, we compare the Fourier transform of Karhunan-Loeve expansion temporal part for the data with HF1MA in different τ and in mode 1 (with the smallest eigenvalue).
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0 1 2 3 4 5 6
ω 0
0.02 0.04 0.06 0.08 0.1
S(ω)
Discrete Fourier Transformation of Karhunen-Loeve expansion time part
Data with HF1MA, Mode 1 τ= 3600 Sec
Figure 3.16: The Fourier transform of Karhunan-Loeve expansion temporal part mode 1(with the smallest eigenvalue) for the data with HF1MA with τ = 3600Sec .The length of data for 3600 second is 10950. (SP500 1996 Jan)
0 1 2 3 4 5 6
ω 0
0.05 0.1 0.15 0.2
S(ω)
Discrete Fourier Transformation of Karhunen-Loeve expansion time part
Data with HF1MA, Mode 1 τ= 36000 Sec
Figure 3.17: The Fourier transform of Karhunan-Loeve expansion temporal part mode 1(with the smallest eigenvalue) for the data with HF1MA with τ = 36000Sec. The length of data for 36000 second is 10050.(SP500 1996 Jan)
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0 1 2 3 4 5 6
ω 0
0.05 0.1 0.15 0.2
S(ω)
Discrete Fourier Transformation of Karhunen-Loeve expansion time part
Data with HF1MA, Mode 1 τ= 72000 Sec
Figure 3.18: The Fourier transform of Karhunan-Loeve expansion temporal part mode 1(with the smallest eigenvalue) for the data with HF1MA with τ = 72000Sec. The length of data for 72000 second is 9050.(SP500 1996 Jan)
For the data with HF1MA, the difference between different τ is just same as the result of the Fourier transform of Karhunan-Loeve expansion temporal part with raw data, as the τ goes up, the pack number and the frequency also goes up. But they show different fea-tures between data with HF1MA and raw data. While the data with HF1MA, the spectra will concentrate to the both end sides, it’s because that taking the average of the shifting windows (HF1MA) would eliminate the noise from the time series .
We are wondering that, what will be in the market mode, so we are going to do the same comparison as below, with the market mode (with the biggest eigenvalue) .
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Discrete Fourier Transformation of Karhunen-Loeve expansion time part
Raw data, Mode 345 τ= 3600 Sec
Figure 3.19: The Fourier transform of Karhunan-Loeve expansion temporal part mode 345(with the biggest eigenvalue) for the raw data with τ = 3600Sec. The length of data for 3600 second is 11600.(SP500 1996 Jan)
0 1 2 3 4 5 6
Discrete Fourier Transformation of Karhunen-Loeve expansion time part
Raw data, Mode 345 τ= 36000 Sec
Figure 3.20: The Fourier transform of Karhunan-Loeve expansion temporal part mode 345(with the biggest eigenvalue) for the raw data with τ = 36000Sec. The length of data for 36000 second is 10700.(SP500 1996 Jan)
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Discrete Fourier Transformation of Karhunen-Loeve expansion time part
Raw data, Mode 345 τ= 72000 Sec
Figure 3.21: The Fourier transform of Karhunan-Loeve expansion temporal part mode 345(with the biggest eigenvalue) for the raw data with τ = 72000Sec. The length of data for 72000 second is 9700.(SP500 1996 Jan)
0 1 2 3 4 5 6
Discrete Fourier Transformation of Karhunen-Loeve expansion time part
Data with HF1MA, Mode 345 τ= 3600 Sec
Figure 3.22: The Fourier transform of Karhunan-Loeve expansion temporal part mode 345(with the biggest eigenvalue) for the data with HF1MA with τ = 3600Sec. The length of data for 3600 second is 10950.(SP500 1996 Jan)
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Discrete Fourier Transformation of Karhunen-Loeve expansion time part
Data with HF1MA, Mode 345 τ= 36000 Sec
Figure 3.23: The Fourier transform of Karhunan-Loeve expansion temporal part mode 345(with the biggest eigenvalue) for the data with HF1MA with τ = 36000Sec.The length of data for 36000 second is 10050.(SP500 1996 Jan)
0 1 2 3 4 5 6
Discrete Fourier Transformation of Karhunen-Loeve expansion time part
Data with HF1MA, Mode 345 τ= 72000 Sec
Figure 3.24: The Fourier transform of Karhunan-Loeve expansion temporal part mode 345(with the biggest eigenvalue) for the data with HF1MA with τ = 72000Sec. The length of data for 72000 second is 9050.(SP500 1996 Jan)
For market mode (with the biggest eigenvalue), we observed that not only the spectra will concentrate to the both end sides, it almost become a spectra with ultimate peak in both end sides and almost zero for other place.
In comparison with market mode and other mode, the market mode has the same
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nomenon with the other mode while the τ is small, but for larger τ the temporal part of the Karhunan-Loeve expansion from market mode become a different spectra that has a ultimate peak in both end and almost zero for other place. To understand the phenomenon observed from power spectra, we make a log-log scale in power spectra in the next sec-tion.
3.6 Power spectra from log-log scale discrete Fourier trans-formation
In last section, we observed the bulk structure in small mode, and ultimate peak in both end in market mode, so we try to dig out the information in log-log scale.
0.001 0.01 0.1 1
ω 0.0001
S(ω)
Discrete Fourier Transformation of Karhunen-Loeve expansion time part
Raw data, Mode 345 τ= 72000 Sec
Figure 3.25: log-log scale plot of the Fourier transform of Karhunan-Loeve expansion temporal part mode 345(with the biggest eigenvalue) for the raw data with τ = 72000Sec.
The length of data for 72000 second is 9700. (SP500 1996 Jan)
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0.001 0.01 0.1 1
ω 0.0001
S(ω)
Discrete Fourier Transformation of Karhunen-Loeve expansion time part
Data with HF1MA, Mode 345 τ= 72000 Sec
Figure 3.26: log-log scale plot of the Fourier transform of Karhunan-Loeve expansion temporal part mode 345(with the biggest eigenvalue) for the data with HF1MA with τ = 72000Sec. The length of data for 72000 second is 9050.(SP500 1996 Jan)
0.001 0.01 0.1 1
ω 0.0001
S(ω)
Discrete Fourier Transformation of Karhunen-Loeve expansion time part
Raw data, Mode 345 τ= 36000 Sec
Figure 3.27: log-log scale plot of the Fourier transform of Karhunan-Loeve expansion temporal part mode 345(with the biggest eigenvalue) for the raw data with τ = 36000Sec.
The length of data for 36000 second is 10700.(SP500 1996 Jan)
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0.001 0.01 0.1 1
ω 0.0001
S(ω)
Discrete Fourier Transformation of Karhunen-Loeve expansion time part
Data with HF1MA, Mode 345 τ= 36000 Sec
Figure 3.28: log-log scale plot of the Fourier transform of Karhunan-Loeve expansion temporal part mode 345(with the biggest eigenvalue) for the data with HF1MA with τ = 36000Sec. The length of data for 36000 second is 10050.(SP500 1996 Jan)
The power spectra S(ω) of the market modes(with the biggest eigenvalue) in log-log scale for the largest τ considered in this study (τ = 72000 s). With or without the manip-ulation of taking moving averages(HF1MA) are shown in figure. 3.26 and figure. 3.25.
The spectra share the same feature that they are straight line with a slope equal to−1 in the log-log plot. And the different between with or without the manipulation of taking mov-ing averages(HF1MA) is that data with HF1MA has less noise than raw data. The noise in data with HF1MA has been eliminated by taking the average of the shifting windows.
And for the raw data, the noise is still standing there. With or without the manipulation of taking moving averages(HF1MA) share the same feature, but the raw data has larger variation at high frequency regime. Same result is also seen in figure. 3.28 and figure.
3.27 .
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0.001 0.01 0.1 1
ω 0.0001
S(ω)
Discrete Fourier Transformation of Karhunen-Loeve expansion time part
Data with HF1MA, Mode 1 τ= 72000 Sec
Figure 3.29: log-log scale plot of the Fourier transform of Karhunan-Loeve expansion temporal part mode 1(with the smallest eigenvalue) for the data with HF1MA with τ = 72000Sec. The length of data for 72000 second is 9050.(SP500 1996 Jan)
0.001 0.01 0.1 1
ω 0.0001
S(ω)
Discrete Fourier Transformation of Karhunen-Loeve expansion time part
Raw data, Mode 1 τ= 72000 Sec
Figure 3.30: log-log scale plot of the Fourier transform of Karhunan-Loeve expansion temporal part mode 1(with the smallest eigenvalue) for the raw data with τ = 72000Sec.
The length of data for 72000 second is 9700.(SP500 1996 Jan)
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0.001 0.01 0.1 1
ω 0.0001
S(ω)
Discrete Fourier Transformation of Karhunen-Loeve expansion time part
Data with HF1MA, Mode 1 τ= 36000 Sec
Figure 3.31: log-log scale plot of the Fourier transform of Karhunan-Loeve expansion temporal part mode 1(with the smallest eigenvalue) for the data with HF1MA with τ = 36000Sec. The length of data for 36000 second is 10050.(SP500 1996 Jan)
0.001 0.01 0.1 1
ω 0.0001
S(ω)
Discrete Fourier Transformation of Karhunen-Loeve expansion time part
Raw data, Mode 1 τ= 36000 Sec
Figure 3.32: log-log scale plot of the Fourier transform of Karhunan-Loeve expansion temporal part mode 1(with the smallest eigenvalue) for the raw data with τ = 36000Sec.
The length of data for 36000 second is 10700.(SP500 1996 Jan)
As for the power spectra S(ω) of the first modes (with the smallest eigenvalue) in log-log plot, 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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0.001 0.01 0.1 1
ω 0.0001
S(ω)
Discrete Fourier Transformation of Karhunen-Loeve expansion time part
Raw data, Mode 345 τ= 3600 Sec
Figure 3.33: log-log scale plot of the Fourier transform of Karhunan-Loeve expansion temporal part mode 345(with the biggest eigenvalue) for the raw data with τ = 3600Sec.
The length of data for 3600 second is 11600.(SP500 1996 Jan)
0.001 0.01 0.1 1
ω 0.0001
S(ω)
Discrete Fourier Transformation of Karhunen-Loeve expansion time part
Data with HF1MA, Mode 345 τ= 3600 Sec
Figure 3.34: log-log scale plot of the Fourier transform of Karhunan-Loeve expansion temporal part mode 345(with the biggest eigenvalue) for the data with HF1MA with τ = 3600Sec. The length of data for 3600 second is 10950. (SP500 1996 Jan)
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0.001 0.01 0.1 1
ω 0.0001
S(ω)
Discrete Fourier Transformation of Karhunen-Loeve expansion time part
Raw data, Mode 1 τ= 3600 Sec
Figure 3.35: log-log scale plot of the Fourier transform of Karhunan-Loeve expansion temporal part mode 1(with the smallest eigenvalue) for the raw data with τ = 3600Sec.
The length of data for 3600 second is 11600.(SP500 1996 Jan)
0.01 0.1 1
ω 0.0001
b j(ω)
Discrete Fourier Transformation of Karhunen-Loeve expansion time part
Data with HF1MA, Mode 1 τ= 3600 Sec
Figure 3.36: log-log scale plot of the Fourier transform of Karhunan-Loeve expansion temporal part mode 1(with the smallest eigenvalue) for the data with HF1MA with τ = 3600Sec. The length of data for 3600 second is 10950.(SP500 1996 Jan)
We found that smaller scale variations in S(ω) are in fact well-organized when we examine the spectra for smaller τ . Figures 3.33 and figures. 3.34 include the spectra for the market mode. And typical continuous band modes in figures.3.35 and 3.36.
We can see well recognized regular packs, going along with the exponent −1 feature
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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.