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7/23
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8/23
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9/23
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10/23
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12/23
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5Histogram of Z using unconditional Gaussian for time series generated from HMEs
0 50 100 150 2000.1
0
0.1
0.2Autocorrelation of (Zmean(Z))
0 50 100 150 2000.1
0
0.1
0.2Autocorrelation of (Zmean(Z))
2
0 50 100 150 2000.1
0
0.1
0.2Autocorrelation of (Zmean(Z))
3
0 50 100 150 2000.1
0
0.1
0.2Autocorrelation of (Zmean(Z))
4
-
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13/23
5100 5200 5300 5400 5500 5600 5700 5800 5900 6000
0
0.5
1
Segmentation from GEs
probability
time
5100 5200 5300 5400 5500 5600 5700 5800 5900 6000
2
0
2
Time Series Generated from HMEs
5100 5200 5300 5400 5500 5600 5700 5800 5900 6000
2
0
2
Predictions
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5Histogram of Z using Gated Experts for time serise generated from HMEs
0 50 100 150 2000.1
0
0.1
0.2Autocorrelation of (Zmean(Z))
0 50 100 150 2000.1
0
0.1
0.2Autocorrelation of (Zmean(Z))
2
0 50 100 150 2000.1
0
0.1
0.2Autocorrelation of (Zmean(Z))
3
0 50 100 150 2000.1
0
0.1
0.2Autocorrelation of (Zmean(Z))
4
-
7/31/2019 10.1.1.109.1935
15/23
5100 5200 5300 5400 5500 5600 5700 5800 5900 6000
0
0.5
1
Segmentation from HMEs vs. True
5100 5200 5300 5400 5500 5600 5700 5800 5900 6000
2
0
2
Time Series Generated from HMEs
5100 5200 5300 5400 5500 5600 5700 5800 5900 6000
2
0
2
Predictions
-
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5Histogram of Z using Hidden Markov Experts for time series generated from HMEs
0 50 100 150 2000.1
0
0.1
0.2Autocorrelation of (Zmean(Z))
0 50 100 150 2000.1
0
0.1
0.2Autocorrelation of (Zmean(Z))
2
0 50 100 150 2000.1
0
0.1
0.2Autocorrelation of (Zmean(Z))
3
0 50 100 150 2000.1
0
0.1
0.2Autocorrelation of (Zmean(Z))
4
-
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
2Histogram of Z using unconditional Gaussian for SP500 returns
0 50 100 150 2000.1
0
0.1
0.2Autocorrelation of (Zmean(Z))
0 50 100 150 2000.1
0
0.1
0.2Autocorrelation of (Zmean(Z))
2
0 50 100 150 2000.1
0
0.1
0.2Autocorrelation of (Zmean(Z))
3
0 50 100 150 2000.1
0
0.1
0.2Autocorrelation of (Zmean(Z))
4
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0
0.5
1
Segmentation of GEs on SP500
0
0.5
1
0
0.5
1
May79 Feb82 Nov84 Aug87 May90 Jan93 Oct95
0
0.5
1
5
0
5
Returns of SP500 (from 01/02/77 to 12/31/97)
trunre
turns
2
0
2
Predictions
training set crash test settraining set crash test set
-
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5Histogram of Z using Gated Experts for SP500 returns
0 50 100 150 2000.1
0
0.1
0.2Autocorrelation of (Zmean(Z))
0 50 100 150 2000.1
0
0.1
0.2Autocorrelation of (Zmean(Z))
2
0 50 100 150 2000.1
0
0.1
0.2Autocorrelation of (Zmean(Z))
3
0 50 100 150 2000.1
0
0.1
0.2Autocorrelation of (Zmean(Z))
4
-
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0
0.5
1
Segmentation of HMEs on SP500
0
0.5
1
0
0.5
1
May79 Feb82 Nov84 Aug87 May90 Jan93 Oct95
0
0.5
1
5
0
5
Returns of SP500 (from 01/02/77 to 12/31/97)
True
returns
2
0
2
Predictions
training set crash test set
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
0 50 100 150 2000.1
0
0.1
0.2Autocorrelation of (Zmean(Z))
0 50 100 150 2000.1
0
0.1
0.2Autocorrelation of (Zmean(Z))
2
0 50 100 150 2000.1
0
0.1
0.2Autocorrelation of (Zmean(Z))
3
0 50 100 150 2000.1
0
0.1
0.2Autocorrelation of (Zmean(Z))
4
1.24 1.23 1.22 1.21 1.2 1.19 1.18 1.17 1.16 1.15 1.140
50
100
150
HMEs
GEs
Median of HMEs
Median of GEs
Log likelihood of Unconditional Gaussian
Log Likelihood / N