singular spectrum analysis for a forecasting of financial time series financial academy under the...

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).


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


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.



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


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.



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.


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.



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.


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.



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


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.



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.


•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


)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!

singular spectrum analysis for a forecasting of financial time series financial academy under the...

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time seriestime series

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nonlinear dynamics methods

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