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Singular spectrum analysis for a forecasting Singular spectrum analysis for a forecasting
of financial time seriesof financial time series
Financial Academy Financial Academy under the Government of the Russian Federationunder the Government of the Russian Federation
Moscow - 2009Moscow - 2009
SpeakerSpeaker::
Kozlov Alexander A.Kozlov Alexander A.
ReportReport
Content list:Content list:
• Introduction to nonlinear dynamics approach
• Overview of the main methods (including SSA)
• Financial time series analysis and forecasting:
• Schlumberger Limited
• Deutsche Bank
• Honda Motor Co., Ltd.
• Toyota Motor Corp.
• Starbucks
• BP plc.
• Conclusions
• Time series Time series is a series of variable values taken in successive periods of time.• Time series analysisTime series analysis is a part of nonlinear dynamics. • SuppositionSupposition:: market of shares is unstable and chaotic.
• ObjectiveObjective:: analysis and forecasting of stock price time series with nonlinear dynamics methods
• In this report the following questions will be consideredIn this report the following questions will be considered::• Embedding dimension as “space” characteristic
and its estimation
• К2-entropy and Lyapunov exponents as “time” characterics
and their estimation
• SSA forecasting method
IntroductionIntroduction
•The idea of attractor reconstructionThe idea of attractor reconstruction [11][11]::
Satisfactory geometry picture of low-dimensional strange attractor can be obtained if instead of x-variables from dynamic system equations somebody use k-dimensional delay vectors:
},...,,{ )1(1 kiiii xxxz
•Takens theorem [2]Takens theorem [2]::
There is a transformation which can embed to on conditions that .
12 dk
dM kR
It means that:
- k – embedding dimension;
- ),...,( 2,1 kiiii xxxFx
OverviewOverview
__________________________________________________________________________________________________[1] Packard N.H., Crutchfield J.P., Farmer J.D., Shaw R.S.,"Geometry from a time series", Phys.Rev.Lett. 45, p.712,1980.[2] F. Takens, "Dynamical Systems and Turbulence", Lect. Notes in Math, Berlin, Springer. №898, 1981, p. 336.
-4 -3 -2 -1 0 1-8
-7
-6
-5
-4
-3
-2
-1
0
ln C
k
ln r
k
• GrassbergerGrassberger-- Procaccia Procaccia method [3] method [3]::
• Limitation [4]Limitation [4]::
______________________________________________________________________________________[3] P. Grassberger, I. Procaccia, "Characterization of Strange Attractors",Phys.Rev.Lett.,50,346, 1983[4] G.G. Malineckiy, A.B. Potapov, “Actual problems of nonlinear dynamics", М: URSS, 2002
N
jiji zzrH
NrC
1,22
1)(
k
edk
cd
Nd lg2
OverviewOverview
1 2 3 4 5 6 70.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
k
k
de
dc2D
constkKrDwrC qqq ln),(ln
k
•Correlation integral Correlation integral on on r<<1 and k>>1r<<1 and k>>1 [ [44]: ]:
k
k
ed
k
k
ked
k
k
cdk
ed
k
k
2Dcdk
ed
k
1. Find , having curves for each k, starting with k=1;
2. Starting with certain k-number stops growing and stabilizes;
3. This k-number is embedding dimension ;
4. Maximum value of is a so-called correlation dimension (or ) of the attractor.
2Dcdk
ed
k
k
• Having fixed r and investigating dependence С(r*,k) from k (k>>1), somebody can estimate K2-entropy [5].K2-entropy [5].
• K2 defines the time of predictability for the system in “volume” interpretation (growing of the volume in phase space which the system can occupy in the future)
• The time of predictability also can be determined from Lyapunov exponentsLyapunov exponents The maximal one is estimated in Wolf method [6].
________________________________________________________________________________________________________[5] Grassberger, I. Procaccia,"Estimation of the Kolmogorov Entropy from a Chaotic Signal",Phys.Rev.A,vol.28,4,1983,p.2591[6] Wolf A., Swift J.B., Swinney H.L., "Determining Lyapunov exponents from time series", Physica D, 69 (1985), №3, p.285-317.
12
)1( ~ KT
OverviewOverview
)exp(~),( 222 kKrwrC D
i
1
0
1
01 )(
)(ln
1 n
i i
i
n tL
tL
tt
11
)2( ~ T
SSA forecasting method SSA forecasting method [7][7]::• 1) Construction of the delay matrix
from time series and preliminary changes in it (centering and normalization)
• 2) Finding the components (M) and selection of the most important ones (r) This is equal to search of eigenvectors and и eigenvalues of the matrix .
• 3) Time series reconstruction with r main components and taking average on each diagonal.
• 4) Forecast constraction with «caterpillar» method:Equal to constraction of the new delay vector with one unknowncoordinate.
_____________________________________________________________________________________________________
[20] “The main components of time series: “caterpillar“ method”. Col.articles // ed. D.L. Danilov, А.А. Zhiglyavskiy – St.P.: St.P.
University, 1997. - 308 p.
OverviewOverview
2
1
1
ˆ
ˆ
MN
N
N
N
x
x
x
x
TXX
121
232
1
)1(
MN
MN
NMM
MNM
xxx
xxx
xxx
X
1
2
0
0 M
Λ
ˆ TM r M r X V V X
1ˆ
211 min)ˆˆ(
Nx
NTrMrMN xVVx
)(1
)2(1
)1(1
)(2
)2(2
)1(2
)()2()1(
M
M
MMMM
MM
vvv
vvv
vvv
V
• CCriteriariteria for the selected companies for the selected companies::- long time on the market of shares (NYSE) – more than 10 years;- publicity;- from different sectors;
• Thus the following companiesthe following companies were chosen:
• Schlumberger Limited• Deutsche Bank• Honda Motor Co., Ltd.• Toyota Motor Corp.• Starbucks• BP plc.
• Forecasting parametersForecasting parameters:- delay number - M=20- number of the main components –
• During forecasting llogarithmic profitogarithmic profit is taken in to account:- positive in growth- negative in fall
Analysis and forecastingAnalysis and forecasting
)(
)(ln
1
i
ii tx
txS
edr
1 2 3 4 5 6 7 8 9 10 11
1,0
1,5
2,0
2,5
3,0
k
k
1. Schlumberger Limited1. Schlumberger Limited
•Period from 31.12.1981 to 31.12.2008
•Time series consists of 6814 stock price values (on close).
Analysis and forecastingAnalysis and forecasting
y = -0,1656x - 2,8552
R2 = 1
-5
-4,5
-4
-3,5
-3
-2,5
-2
0 2 4 6 8 10 12
k
ln C
(k)
7ed 12,3cd1656,02 K 04,6)1( T
1597,01 26,6)2( T
1. Schlumberger Limited1. Schlumberger Limited
Analysis and forecastingAnalysis and forecasting
1 2 3 4 5 6 7 8 9 10 11
1,0
1,5
2,0
2,5
3,0
3,5
k
k
2. Deutsche Bank2. Deutsche Bank
•Period from 18.11.1996 to 31.12.2008.
•Time series consists of 3033 stock price values (on close).
Analysis and forecastingAnalysis and forecasting
6ed 26,3cd
y = -0,1208x - 1,4495
R2 = 1
-2,8
-2,6
-2,4
-2,2
-2
-1,8
-1,6
-1,4
-1,2
-1
0 2 4 6 8 10 12
k
ln C
(k)
1208,02 K
58,8)2( T
28,8)1( T
1164,01
2. Deutsche Bank2. Deutsche Bank
Analysis and forecastingAnalysis and forecasting
1 2 3 4 5 6 7 8 9 10 11
1,0
1,5
2,0
2,5
3,0
k
k
3. Honda Motor Co., Ltd.3. Honda Motor Co., Ltd.
•Period from 11.08.1987 to 31.12.2008.
•Time series consists of 5390 stock price values (on close).
Analysis and forecastingAnalysis and forecasting
7ed 80,2cd
y = -0,1495x - 3,1127
R2 = 0,9997
-5
-4,5
-4
-3,5
-3
-2,5
-2
0 2 4 6 8 10 12
k
ln C
(k)
1495,02 K 69,6)1( T
1442,01 94,6)2( T
3. Honda Motor Co., Ltd.3. Honda Motor Co., Ltd.
Analysis and forecastingAnalysis and forecasting
4. Toyota Motor Corp.4. Toyota Motor Corp.
1 2 3 4 5 6 7 8 9 10 11
1,0
1,5
2,0
2,5
k
k
•Period from 13.04.1993 to 31.12.2008.
•Time series consists of 3956 stock price values (on close).
Analysis and forecastingAnalysis and forecasting
6ed 64,2cd
y = -0,1175x - 1,8895
R2 = 0,9999
-3,5
-3
-2,5
-2
-1,5
-1
-0,5
0
0 2 4 6 8 10 12
k
ln C
(k)
1175,02 K 51,8)1( T1175,02 K
1222,01 18,8)2( T
4. Toyota Motor Corp.4. Toyota Motor Corp.
Analysis and forecastingAnalysis and forecasting
1 2 3 4 5 6 7 8 9 10 110,5
1,0
1,5
2,0
2,5
k
k
5. Starbucks5. Starbucks
•Period from 26.06.1992 to 31.12.2008.
•Time series consists of 4161 stock price values (on close).
Analysis and forecastingAnalysis and forecasting
7ed 34,2cd
y = -0,1389x - 3,9295
R2 = 0,9998
-5,5
-5
-4,5
-4
-3,5
-3
0 2 4 6 8 10 12
k
ln C
(k)
1389,02 K 20,7)1( T
1342,01 45,7)2( T
5. Starbucks5. Starbucks
Analysis and forecastingAnalysis and forecasting
1 2 3 4 5 6 7 8 9 10 110,5
1,0
1,5
2,0
2,5
k
k
6. BP plc.6. BP plc.
•Period from 03.01.1977 to 31.12.2008.
•Time series consists of 8076 stock price values (on close).
Analysis and forecastingAnalysis and forecasting
6ed 45,2cd
y = -0,1403x - 3,0905
R2 = 1
-5
-4,5
-4
-3,5
-3
-2,5
0 2 4 6 8 10 12
k
ln C
(k)
13,7)1( T1403,02 K
1369,01 31,7)2( T
6. BP plc.6. BP plc.
Analysis and forecastingAnalysis and forecasting
•Final results of analysis are in the table:
2K
КомпанияКомпания
Schlumberger Limited 7 3,12 0,1656 0,1597 6,04 6,26
Deutsche Bank 6 3,26 0,1208 0,1164 8,28 8,58
Honda Motor Co. Ltd. 7 2,8 0,1495 0,1442 6,69 6,94
Toyota Motor Corp. 6 2,64 0,1175 0,1222 8,51 8,18
Starbucks 7 2,34 0,1389 0,1342 7,20 7,45
BP plc. 6 2,45 0,1403 0,1369 7,12 7,31
ed cd
Analysis and forecastingAnalysis and forecasting
)1(T2K 1 )2(T
КомпанияКомпания 2007 , % 2008 , %
Schlumberger Limited 75 75
Deutsche Bank 81 69
Honda Motor Co. Ltd. 75 58
Toyota Motor Corp. 75 81
Starbucks 86 57
BP plc. 71 71
•Percentage of coincidence between logarithmic profit signs of forecast and real time series
• Nonlinear dynamics methods applied to stock price time series led to a “space” and “time” analysis of the trading system. Thus we determined number of the main components (=embedding dimension) and time of predictability (according to K2-entropy and Lyapunov exponents) for each company.
• Obtained results have both fundamental and applied sense for economics.
• Complex analysis permitted to make a forecast on the basis of SSA method (“caterpillar”). Forecasted values and logarithmic profit fits the real ones quite well.
• Thus SSA forecasting method can be a useful instrument in quantitative analysis of any risks connected with financial time series.
ConclusionsConclusions
Thank You for attention!Thank You for attention!