Lecture 5 – part 1
Detection of self-similar and quasi-periodic processes in complex systems: How quantitatively to describe their behavior?
Professor Raoul R. Nigmatullin
Radio-electronics and Informative-Measurements
Technics Department, (RE&IMT department)
Kazan National-Research Technical University (KNRTU-KAI),
Karl Marx str.10, 420011, Kazan, Tatarstan, Russia
e-mail: renigmat@gmail.com
Plan of the lecture 5 (part 1 and part 2)
1. Properties of the Self-Similar (SS)- systems. How to see their SS (fractal) properties and solve inverse problem: to restore the desired scaling equation?
2. Examples: Economic data, Meteo-data
3.Perspectives of further research.
4.Properties of the QP-systems. How to see the QP properties and describe them in terms of the generalized Prony’s spectrum (GPS) ?
5.How to find the period T?
6.Examples: Acoustic data recorded from test hole (TGT)
7.Perspectives of further research.
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How to extract the logic and compress the text material? The secrets of the ex-excellent student.
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Mnemonic symbols that help to
order and see the logic of material. You can use it for writing any paper, lecture and etc.
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The basic message for the audience
The Self-Similar scenario.
How to prove the presence of SS property based on a set of data points? How to restore the proper scaling equation?
Algorithm of recognition?
Comments
How to justify SS 
principle on real
data?
What is QP process? Periodic function x Non-Periodic envelope/component
The Quasi-Periodic scenario.
How to prove the presence of the QP? How to restore the corresponding functional equation?
Memory between measurements? The meaning of the Prony’s decomposition?
How to prove the existence of memory between measurements?
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Introductory part
Conventional laws: |
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models and simple |
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Complex systems: |
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Principles?! |
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Another space? |
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Range( ) |
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H |
, 0 H 1. |
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Stdv( ) |
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Env( ) a1 |
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Here the current time [ j (j=1,2,…,N)] is associated with the length of the discrete random sequence having N discrete points.
Uncontroll- able factors are weak
Uncontroll- able factors are strong
Envelope of
the SRA
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Introductory part
Fractal system, SS process? How to prove it?
It is necessary to find the fractal dimension (D), to determine the limits of applicability of the scaling properties and to prove the evidence/absence of log-periodic oscillations [9] that accompany any scaling process in time or space.
How to find the convincing arguments for the skeptical scholar if a scientist has only a set of numerical data characterizing the response of a complex system and nothing else?
In this presentation continuing these ideas mentioned above we want to prove that based on the justified (for many random sequences) SS principle it is possible to prove that many random data have self-similar structure and, thereby, to solve the inverse problem.
It implies to reveal the desired fitting function that enables to fit many self-similar and random functions satisfying to the SS principle.
The fitting parameters of this function can be used for comparing two or more random data series with each other. (the idea of reduction of initial set of random parameters)
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Some important details of the problem
S(z ) 1S(z) s0
Solution of the
Scaling Equation
Here S(z) defines a physical value that depends on the argument z which can be associated with any current variable as time, frequency, coordinate and etc. In general, it can accept real or complex value. The parameters and 1 denote the scaling
factors. The constant s0 represents a possible shift.
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ln( 1) |
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s0 |
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1 cos(bnt) |
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S(z) A0 z |
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Pr(ln z), |
ln |
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C(t) |
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, C(bt) aC(t), a ? |
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1 1 |
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Pr(ln
Pr(ln
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2 k |
z) A0 Ack cos |
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z ln ) Pr(ln z).
ln z As sin 2 k ln k
ln z , ln
Log-periodic function
The problem can be formulated as follows: It is necessary to justify the self-similar structure of random data characterizing the behavior of the complex system under analysis.
Then it is necessary to develop a procedure for the fitting of the data to the function S(z), to find the desired fitting parameters and to restore the functional equation for S(z).
Random sequence (RS) |
S(z) |
If Random Sequence |
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is fractal! |
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The description of the algorithm
Step 1. Reduction an interval to three incident (invariant) points. Does it prove SS – principle of the curve?
Let us choose some interval [x0, xk-1] containing a set of k data points {(x0, y0), … , (xk-1, yk-1)}.
One can reduce this information into three incident points if the first point is associated with the mean value of the amplitudes and the other two points are associated to their maximal and minimal values, correspondingly. So, this selection represents the simplest reduction of the given set of k random points to three characteristic points p1=mean{y0, … , yk-1},
p2=max{y0, … , yk-1}, p3=min{y0, … , yk-1}. We suppose that for any finite set of k data points
these three incident points exist. In general, these chosen sets can contain different number |
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of points. |
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Pmn |
Fig.1. Initial data that show the distribution |
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of prices of gold (for one Troy Ounce in |
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KiloUSD |
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kiloUSD). These data are taken from the site |
1.5 |
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http://www.indexmundi.com/commodities/. |
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Ounce in |
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Troy |
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September |
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September 1982 |
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2012 year |
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for |
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mean Price |
0.5 |
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0.0 |
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-50 |
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1(0.9.1982) < months <360(0.9.2012) |
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Compression program (reduction to 3 IP)
The reduction procedure for a single Sequence having N data points
The reduction procedure for the matrix having M columns
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