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Symbolic CTQ-analysis – a new method for studying. . .
Symbolic CTQ-analysis – a new method for studyingof financial indicators
Andrey V. [email protected]
Constructive Cybernetics Research GroupMoscow, Russia, www.rdcn.org
Institute of Control Sciences RASMoscow, Russia
International Conference"Advanced Finance and Stochastics"
June’24 to 28, 2013Steklov Mathematical Institute RAS, Moscow, Russia
𝐶2 A.V. Makarenko, Symbolic CTQ-analysis – a new method for studying. . . AFS’2013 1 / 17
Symbolic CTQ-analysis – a new method for studying. . .
Outline
1 The symbolic CTQ-analysisKey IdeaBasic featuresKey article (in English)
2 T-synchronizationThe general ideaMain characteristics
3 TQ-synchronization of financial indicatorsInitial dataResult of analysis
4 Conclusion
𝐶2 A.V. Makarenko, Symbolic CTQ-analysis – a new method for studying. . . AFS’2013 2 / 17
Symbolic CTQ-analysis – a new method for studying. . .The symbolic CTQ-analysis
Outline section
1 The symbolic CTQ-analysisKey IdeaBasic featuresKey article (in English)
2 T-synchronizationThe general ideaMain characteristics
3 TQ-synchronization of financial indicatorsInitial dataResult of analysis
4 Conclusion
𝐶2 A.V. Makarenko, Symbolic CTQ-analysis – a new method for studying. . . AFS’2013 3 / 17
Symbolic CTQ-analysis – a new method for studying. . .The symbolic CTQ-analysis
Key Idea
Key Idea – shape of trajectories
Denote the sequence (such as the values of some financial indicators):{︁{s𝑘}𝐾𝑘=1, {𝑡𝑘}𝐾𝑘=1
}︁,
s ∈ S ⊂ R𝑁 , 𝑡 ∈ T ⊂ R, 𝑡𝑘+1 > 𝑡𝑘, 𝑘 ∈ K ⊂ N, 𝑛 = 1, 𝑁, 𝑘 = 1, 𝐾.
We introduce the map:
{𝑠(𝑛)𝑘 }𝐾+1
𝑘=0 ⇒ {𝑇𝛼𝜙𝑘 |𝑛}𝐾𝑘=1, 𝑇𝛼𝜙
𝑘 = [𝑇𝛼𝜙𝑘 |1 . . . 𝑇𝛼𝜙
𝑘 |𝑁 ] ,
where 𝑇𝛼𝜙|𝑛 – symbol of T-alphabet:
T𝛼𝜙𝑜 = {T0, T1, T2, T3N, T3P, T4N, T4P, T5N, T5P, T6, T7, T8N, T8P}.
We define symbol of the T-alphabet – select a subsequence {s𝑘−1, s𝑘, s𝑘+1}:T5P
T5N
T6
T4N
T2( )
1n
ks
T3N
T3P
T7
T4P
T1 T8N
T0
T8P( )nks
( )1
nks
( )1
nks
( )nks
( )1
nks
( )1
nks
( )nks ( )
1n
ks
𝐶2 A.V. Makarenko, Symbolic CTQ-analysis – a new method for studying. . . AFS’2013 4 / 17
Symbolic CTQ-analysis – a new method for studying. . .The symbolic CTQ-analysis
Basic features
Symbolic TQ-image
Symbolic TQ-image of sequence {s𝑘}𝐾𝑘=1
Directed graph Γ𝛼𝜑|𝑛 =⟨︀VΓ|𝑛, EΓ|𝑛
⟩︀:
VΓ|𝑛 ⊆ T𝛼𝜙𝑜 – vertex Γ𝛼𝜑|𝑛 and EΓ|𝑛 ⊆ Q𝛼𝜙
𝑜 – edges Γ𝛼𝜑|𝑛.
T7
T6
T5PT3N T5NT3P
T0
T1 T2
T4P T4NT8N T8P The main formalisms:
∙ CTQ-symmetryof trajectories;
∙ TQ-bifurcations;∙ TQ-complexity;∙ T-synchronization;∙ Q-control.
𝐶2 A.V. Makarenko, Symbolic CTQ-analysis – a new method for studying. . . AFS’2013 5 / 17
Symbolic CTQ-analysis – a new method for studying. . .The symbolic CTQ-analysis
Key article (in English)
Key article (in English)
A.V.M., Structure of Synchronized Chaos Studied by SymbolicAnalysis in Velocity–Curvature Space, Technical Physics Letters, 38:2(2012), 155—159, arXiv: 1203.4214.
A.V.M., Multidimensional Dynamic Processes Studied by SymbolicAnalysis in Velocity–Curvature Space, Computational Mathematics andMathematical Physics, 52:7 (2012), 1017—1028.
A.V.M., Measure of Synchronism of Multidimensional ChaoticSequences Based on Their Symbolic Representation in a T-Alphabet,Technical Physics Letters, 38:9 (2012), 804–808, arXiv: 1212.2724.
𝐶2 A.V. Makarenko, Symbolic CTQ-analysis – a new method for studying. . . AFS’2013 6 / 17
Symbolic CTQ-analysis – a new method for studying. . .T-synchronization
Outline section
1 The symbolic CTQ-analysisKey IdeaBasic featuresKey article (in English)
2 T-synchronizationThe general ideaMain characteristics
3 TQ-synchronization of financial indicatorsInitial dataResult of analysis
4 Conclusion
𝐶2 A.V. Makarenko, Symbolic CTQ-analysis – a new method for studying. . . AFS’2013 7 / 17
Symbolic CTQ-analysis – a new method for studying. . .T-synchronization
The general idea
The general idea of T-sync
DefinitionThe components of the sequence {s𝑘}𝐾𝑘=1 are T-synchronized by a count 𝑘,if the corresponding sequence {𝑇𝛼𝜙
𝑘 }𝐾𝑘=1 following equalityholds 𝐽𝛼𝜙
𝑠𝑦𝑚 [𝑇𝛼𝜙𝑘 ] = 1, where:
𝐽𝛼𝜙𝑠𝑦𝑚 [𝑇𝛼𝜙
𝑘 ] =
{︂1 𝑇𝛼𝜙
𝑘 |1 = . . . = 𝑇𝛼𝜙𝑘 |𝑛 = . . . = 𝑇𝛼𝜙
𝑘 |𝑁 ,0 otherwise.
Anti-synchronization: 𝑠(𝑛)𝑘 → −1 · 𝑠(𝑛)
𝑘 .
+1 T0 T1 T2 T3N T3P T4N T4P T5N T5P T6 T7 T8N T8P
−1 T0 T2 T1 T5P T5N T4P T4N T3P T3N T7 T6 T8N T8P
Lag-synchronization:{︁𝑇𝛼𝜙𝑘 |1 → 𝑇𝛼𝜙
𝑘+ℎ1|1, . . . , 𝑇𝛼𝜙
𝑘 |𝑁 → 𝑇𝛼𝜙𝑘+ℎ𝑁
|𝑁}︁.
𝐶2 A.V. Makarenko, Symbolic CTQ-analysis – a new method for studying. . . AFS’2013 8 / 17
Symbolic CTQ-analysis – a new method for studying. . .T-synchronization
Main characteristics
Integral factor and Time structure
Particular integral factor:
𝛿𝛼𝜙𝑚,h =
1
𝐾* + 1 − 𝑘*
𝐾*∑︁𝑘=𝑘*
𝐽[︀𝑇𝛼𝜙𝑘 | {𝑚, h}
]︀,
where: 𝑘* = 1 + max (ℎ1, . . . , ℎ𝑁 ), 𝐾* = 𝐾 + min (ℎ1, . . . , ℎ𝑁 ).
Full integral factor:
𝛿𝛼𝜙 = max𝑚
maxh
𝛿𝛼𝜙𝑚,h, 0 6 𝛿𝛼𝜙 6 1,
DefinitionSynchronized domain 𝑆𝐷 – a collection of samples of thesequence {𝑇𝛼𝜙
𝑘 }𝐾𝑘=1, for which we have the condition:
𝑆𝐷𝑟 :{︀𝐽𝛼𝜙𝑠𝑦𝑚
[︀𝑇𝛼𝜙𝑘′
]︀= 1, 𝐽𝛼𝜙
𝑠𝑦𝑚
[︀𝑇𝛼𝜙
𝑘−
]︀= 0 ∨ 𝑘− = 0,
𝐽𝛼𝜙𝑠𝑦𝑚
[︀𝑇𝛼𝜙
𝑘+
]︀= 0 ∨ 𝑘+ = 𝐾 + 1
}︀,
where 𝑘′ = 𝑏𝑆𝐷𝑟 , 𝑏𝑆𝐷
𝑟 + 𝐿𝑆𝐷𝑟 , 𝑘− = 𝑏𝑆𝐷
𝑟 − 1, 𝑘+ = 𝑏𝑆𝐷𝑟 + 𝐿𝑆𝐷
𝑟 + 1, 𝑟 –domain number, 𝑟 = 1, 𝑅𝑆𝐷, and besides 𝑅𝑆𝐷 6 (𝐾 + 1) div 2.
𝐶2 A.V. Makarenko, Symbolic CTQ-analysis – a new method for studying. . . AFS’2013 9 / 17
Symbolic CTQ-analysis – a new method for studying. . .T-synchronization
Main characteristics
Analytical characteristics of Time structure
Spectral density synchronous domains 𝑆𝐷:
𝐻𝑆𝐷[︁𝐿𝑆𝐷
]︁=
𝑅𝑆𝐷∑︁𝑟=1
𝛿[𝐿𝑆𝐷𝑟 , 𝐿𝑆𝐷],
Conditional entropy of the structure of synchronous domains, for 𝛿𝛼𝜙 > 0:
𝐸𝑆𝐷𝑐𝑛𝑑 = −
𝐾∑︁𝑖=1
𝑃𝑆𝐷 [𝑖] ln𝑃𝑆𝐷 [𝑖], 𝑃𝑆𝐷[︁𝐿𝑆𝐷
]︁=
𝐻𝑆𝐷[︀𝐿𝑆𝐷
]︀𝐾∑︀𝑖=1
𝐻𝑆𝐷 [𝑖]
.
Relative conditional entropy structure of synchronous domains:
∆𝐸 =𝐸𝑆𝐷
𝑐𝑛𝑑
�̂�𝑆𝐷𝑐𝑛𝑑
, �̂�𝑆𝐷𝑐𝑛𝑑 = ln𝑊, 𝑊 =
⌊︂√17 + 8 𝛿𝛼𝜙 𝐾 − 3
2
⌋︂.
Map of synchronization:
𝑀𝑆𝐷𝑘 =
{︃𝐿𝑆𝐷
𝑟 𝑏𝑆𝐷𝑟 6 𝑘 6 𝑏𝑆𝐷
𝑟 + 𝐿𝑆𝐷𝑟 ,
0 otherwise.
𝐶2 A.V. Makarenko, Symbolic CTQ-analysis – a new method for studying. . . AFS’2013 10 / 17
Symbolic CTQ-analysis – a new method for studying. . .TQ-synchronization of financial indicators
Outline section
1 The symbolic CTQ-analysisKey IdeaBasic featuresKey article (in English)
2 T-synchronizationThe general ideaMain characteristics
3 TQ-synchronization of financial indicatorsInitial dataResult of analysis
4 Conclusion
𝐶2 A.V. Makarenko, Symbolic CTQ-analysis – a new method for studying. . . AFS’2013 11 / 17
Symbolic CTQ-analysis – a new method for studying. . .TQ-synchronization of financial indicators
Initial data
The rates of world currencies on Ruble
2000 2005 2010
20
30
40
50
USD EUR JPH CHF GBP
Year
Rub RU
The period: from 01.01.1999 to 31.03.2013. The sample size: 3 545 counts.
The initial data are taken from the official web-site Central Bank of Russia:http://www.cbr.ru/eng/currency_base/dynamics.aspx
Remark: Data extraction and processing was carried out in the programWolfram Mathematica 9.
𝐶2 A.V. Makarenko, Symbolic CTQ-analysis – a new method for studying. . . AFS’2013 12 / 17
Symbolic CTQ-analysis – a new method for studying. . .TQ-synchronization of financial indicators
Result of analysis
Integral factor of TQ-synchronization
The studied combinations:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
USDEURJPHCHFGBP
v
– direct components – anti- components: ( ) ( )1n nk ks s
Integral factor of TQ-synchronization:
0 5 10 15 20 25
-1
1
10
-210
-310
v
– real data [ http://www.cbr.ru/eng/ ]
– synthetic data (homogeneous Markov chain,
Monte Carlo simulation, 1000 trials,
The ranges of values by in probability (empirical) 0.999)
𝐶2 A.V. Makarenko, Symbolic CTQ-analysis – a new method for studying. . . AFS’2013 13 / 17
Symbolic CTQ-analysis – a new method for studying. . .TQ-synchronization of financial indicators
Result of analysis
Time structure of TQ-synchronization
The studied combinations:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
USDEURJPHCHFGBP
v
– direct components – anti- components: ( ) ( )1n nk ks s
Entropy and Spectral density of synchronous domains:1
5
10
15
20
26
1 10 20 30 390 5 10 15 20 250.0
0.2
0.4
0.6
0.8
countv
v
SDL
SDE
SDH
0 20 400
𝐶2 A.V. Makarenko, Symbolic CTQ-analysis – a new method for studying. . . AFS’2013 14 / 17
Symbolic CTQ-analysis – a new method for studying. . .TQ-synchronization of financial indicators
Result of analysis
Time structure of TQ-synchronization
Map of synchronization (samples for 3 combinations):
2000 2005 2010
0
10
20
30
40
Year
Count 5v EUR – CHF
2000 2005 2010
0
2
4
6
8
10
12Count 12v EUR – CHF – GBP
2000 2005 2010
0
2
4
6
8
10
12
Count 19v USD – EUR – CHF
𝐶2 A.V. Makarenko, Symbolic CTQ-analysis – a new method for studying. . . AFS’2013 15 / 17
Symbolic CTQ-analysis – a new method for studying. . .Conclusion
Outline section
1 The symbolic CTQ-analysisKey IdeaBasic featuresKey article (in English)
2 T-synchronizationThe general ideaMain characteristics
3 TQ-synchronization of financial indicatorsInitial dataResult of analysis
4 Conclusion
𝐶2 A.V. Makarenko, Symbolic CTQ-analysis – a new method for studying. . . AFS’2013 16 / 17
Symbolic CTQ-analysis – a new method for studying. . .Conclusion
Summary
∙ The CTQ-analysis method uses the term shape of the trajectory in thespace S × K.
∙ The strongest plus CTQ-analysis methods – focus onmultidimensionality and nonstationarity studied processes and systems.
∙ For some currencies detected nonrandom long periods of simultaneouschanges in their exchange rates.
∙ It is further planned multiscale CTQ-analysis of these financialindicators for investigation of their temporal structure in order tostudy the mechanism and causes of synchronicity.
Thank you for your attention!
𝐶2 A.V. Makarenko, Symbolic CTQ-analysis – a new method for studying. . . AFS’2013 17 / 17