JP3189197B2 - Correlation dimension analyzer and Lyapunov spectrum analyzer - Google Patents

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to signal analysis, and more particularly, to approximating a certain time interval as an autonomous system equation even if time-series data is deterministically generated by a non-autonomous system equation. The correlation dimension and Lyapunov spectrum can be obtained in the time window section,
Recognition, in which according to correlation dimensions min 析装 location and Riapuno <br/> full spectrum component 析装 location available to identify.

[0002]

2. Description of the Related Art Time series analysis using a nonlinear dynamical system is performed by first assuming that time series data obtained from observation is obtained from an equation (differential equation) including the time series variables and hidden variables. Calculate quantities that characterize nonlinear systems, such as and Lyapunov exponents.

[0003] The correlation dimension is a dimension calculated when an analysis is performed to determine how many equations can be described in the above equation. For example, if the correlation dimension is 4.5, a 5-dimensional equation ( (5 variables).

FIG. 6 shows a flowchart of a conventional correlation dimension analysis method. (S31) to (S37) in the figure indicate each step.

[0005] In the conventional correlation dimension analysis method, calculation is performed on the assumption that time-series data is generated by an autonomous system equation.

First, it is assumed that the time-series data x (t) follows a d-dimensional dynamical system (S31), and the attractor is reconstructed by an embedding operation known per se (S32).
A set of data of the total number N is created from the time-series data x (t) to the d-dimensional reconstructed state space as follows. If the time delay is τ, the set of data is

[0007]

(Equation 1) This means that N d-dimensional points X (t) have been obtained.

A point on the reconstructed attractor is X
Assuming (i), the correlation integral for calculating the correlation dimension is expressed by the following equation (S33).

[0009]

(Equation 2) Here, the above equation is a Heaviside function represented by I (t) = 1 (t ≧ 0) and 0 (t <0). This correlation integral is scaled in the appropriate region of r as follows:

[0010]

(Equation 3) Taking the log of both sides of this equation,

[0011]

(Equation 4) Becomes By plotting logr on the horizontal axis and log C ^d (r) on the vertical axis, a correlation index ν (d) is obtained as the slope of this graph (S34).

Generally, since the dimension of the original attractor in the time series is unknown, the embedding dimension d is also unknown.

Therefore, the correlation index ν (d) is calculated while increasing the value of the embedding dimension d (S35, S37). If the dimensions of the reconstructed attractor are smaller than the dimensions of the original attractor, ν (d) is approximately equal to d, and as d increases, ν (d) saturates and asymptotically approaches the correlation dimension D _c ( S36).

FIG. 7 shows a flowchart of a conventional Lyapunov spectrum analysis method. (S41) to (S4) in FIG.
6) shows each step.

The conventional Lyapunov spectrum analysis method is as follows.
The calculation has been performed on the assumption that the time-series data x (t) is generated by an autonomous system equation.

First, it is assumed that the time-series data x (t) follows a d-dimensional dynamic system (S41), and the attractor is reconstructed by an embedding operation (S42). This operation is the same as the attractor reconstruction in the state space by the embedding operation in the correlation dimension analysis method described above.

The Lyapunov exponent is a quantity that indicates the degree of instability of the orbit in the state space of the time-series equation obtained from observation, that is, the orbit instability or divergence of the attractor. If this is positive, it is unstable; if it is negative, it is stable. Usually, there are Lyapunov exponents for the number of dimensions of the system, and Lyapunov spectrum is a generic term for these.
Incidentally, chaos is called a strange attractor and is a set of unstable orbits. Therefore, the largest Lyapunov exponent in the Lyapunov spectrum is always positive. If the system can be described by a differential equation, chaos has a correlation dimension of 3 or more.

Now, a general dynamical system

[0019]

(Equation 5) And Assuming that a small change in X (t) is δX (t),

[0020]

(Equation 6) Tailor the left side more,

[0021]

(Equation 7) Becomes here,

[0022]

(Equation 8) Is the Jacobian matrix at t of F. This DF (X
(t)) is estimated from N d-dimensional points obtained by the embedding operation.

As shown in FIG. 8, a certain point X on the trajectory
(T) is selected, and a point X (k (i)), (i = 1, 2,..., M) on the same orbit that enters a hypersphere of radius ε centered on that point is denoted by M
Pick out pieces. Then, the displacement vector of each point in the hypersphere of radius ε viewed from X (t) is

[0024]

(Equation 9) Becomes When time passes by m, X (t) becomes X (t +
m), X (k (i)) becomes X (k (i) + m). Therefore, the displacement vector z (i) at time t + m is

[0025]

(Equation 10) Becomes If ε is small, linear approximation

[0026]

[Equation 11] (S43). The A _t is approximately equal to DF (X (t)), consider a method of determining the A _t. Where at least 2
Request A _t to the average distance S minimized by multiplication (S44).

[0027]

(Equation 12) (K, l) of A _t the component a (k, l) and to the

[0028]

(Equation 13) Than

[0029]

[Equation 14] Where y (i, k) and z (i, k) are vectors y
(I) represents the k-th component of z (i). If there is no degeneracy in M ≧ n A _t the can be determined uniquely. To find all dimensional spectra of the Lyapunov exponent, δX (t)
Are the d-dimensional orthogonal basis vectors u ₁ (t), u ₂ (t), u _i (t),
..., u _d (t) and see the change. First,

[0030]

(Equation 15) To obtain e _i (t + 1), and then construct a new orthogonal basis vector by the Gram-Schmidt method (S45).

[0031]

(Equation 16) <> Is a dot product. The above conversion is repeated using u _i (t + 1) as an orthogonal basis vector. At this time, the Lyapunov spectrum Jλ _i (i = 1,2,…, d) is

[0032]

[Equation 17] (S46).

[0033]

In the above-described conventional correlation dimension analysis method and Lyapunov spectrum analysis method, the observed time-series data is only for the autonomous system equations.

The present invention relates to a method for analyzing the correlation dimension and Lyapunov spectrum for time series data obtained from the autonomous system equation described in the above-mentioned prior art. It is intended to be extended.

[0035]

[0036]

[0037]

A correlation dimension analyzer according to the present invention comprises a time series measurement circuit for measuring time series data, an A / D conversion circuit for A / D converting an output signal of the time series measurement circuit. A time-series window division processing circuit for cutting out a certain time section of the output of the A / D conversion circuit from the entire time-series data, an embedding operation circuit for reconstructing an attractor for the window time, and the reconstructed operation circuit. A correlation dimension calculation circuit for obtaining a correlation dimension from the attractor,
Further, the time-series window division processing circuit has a function of gradually shifting the time window by time and obtaining a correlation dimension at that time.

[0038]

The Lyapunov spectrum analyzer is
A time-series measurement circuit for measuring time-series data, an A / D conversion circuit for A / D-converting an output signal of the time-series measurement circuit, and a time section having an output of the A / D conversion circuit as a whole time series A time-series window division processing circuit that cuts out data, an embedding operation circuit that reconstructs an attractor with respect to the window time, and a Lyapunov spectrum calculation circuit that obtains a Lyapunov spectrum from the reconstructed attractor, further comprising: The window division processing circuit has a function of gradually shifting the time window by time and obtaining a Lyapunov spectrum at that time.

[0040]

[Action] In correlation dimension min 析装 location and Lyapunov spectrum component 析装 location according to the present invention, by approximating the autonomous equations only time a non-autonomous system equations, the time-series data generated from a non-autonomous system equations On the other hand, the correlation dimension and the Lyapunov spectrum are obtained, and by repeating the operation, the correlation dimension and the Lyapunov spectrum at that time are obtained.

[0041]

Embodiments of the present invention will be described below.

FIG. 1 shows the operation of the correlation dimension analyzer of the present invention.
FIG. 2 shows a flowchart of an example of a correlation dimension analysis method for explaining the principle, and FIG. 2 shows a Lyapunov spectrum for explaining the operation principle of the Lyapunov spectrum analyzer of the present invention.
4 shows a flowchart of an example of a vector analysis method. In addition,
(S1) to (S6) and (S11) to (S17) in the figure indicate respective steps.

First, for the time series data x (t), a time section in which the non-autonomous system equation can be approximated to the autonomous system equation is cut out from the entire time series. FIG. 3 shows the entire time series data x
FIG. 11 is an explanatory diagram of a window process for cutting out a certain time section from (t). The horizontal axis represents time t, the vertical axis represents amplitude D, T represents the window width (time), and the windows are represented by W ₁ , W ₂ ,.
The time series data at that time is captured. Since the window width is T in this manner, for example, x (1)
Is calculated in accordance with the conventional correlation dimension and Lyapunov spectrum analysis method described with reference to FIGS. At this time, the embedding dimension d is set to a value larger than the estimated dimension of the system.
This prevents the correlation dimension and Lyapunov spectrum obtained from the window width T from converging in the assumed dimension d. This operation is repeated for the time-series data x (t) in the order of t = 1, 2,..., N, and the correlation dimension and Lyapunov spectrum at that time are calculated. That is, as compared with the conventional correlation dimension analysis method shown in FIG. 6 and the Lyapunov spectrum analysis method shown in FIG.
2) and (S12) are basically different in that they include the processing steps.

The steps (S3) to (S6) in FIG.
Correspond to steps (S32) to (S36) in FIG. 6, and steps (S13) to (S17) in FIG. 2 correspond to steps (S42) to (S46) in FIG. Omitted.

FIG. 4 is a block diagram showing the configuration of one embodiment of the correlation dimension analyzer of the present invention, and FIG. 5 is a block diagram showing the configuration of one embodiment of the Lyapunov spectrum analyzer of the present invention.

In FIG. 4, reference numeral 1 denotes a time series measurement circuit which measures time series data. Reference numeral 2 denotes an A / D conversion circuit which performs A / D conversion of the measured time series. 3
Is a time series window division processing circuit, which is a circuit for cutting out a time section in which a non-autonomous system equation can be approximated to an autonomous system equation from the entire time series data, and cuts out the window width T described with reference to FIG.

Reference numeral 4 denotes an embedding operation circuit which reconstructs an attractor from the sampled window division time series,
This is a circuit that forms a reconstructed state space. Reference numeral 5 denotes a correlation dimension calculation circuit which calculates a correlation dimension from a reconstructed state space vector from the time series window-divided by the window division processing circuit 3. The operation is, as shown in the flowchart of FIG. 1, the time series x (t) measured by the time series measurement circuit 1 in step (S1),
In 2), window processing for cutting out a certain time section from the entire time series x (t) is performed, and the subsequent steps (S3) to
(S6) is substantially the same as steps (S32) to (S36) in FIG.

Next, the Lyapunov spectrum analyzer of FIG. 5 will be described. In FIG. 5, reference numerals 1 to 4 are the same as those of the correlation dimension analyzer of FIG.

Reference numeral 6 denotes a Lyapunov spectrum calculation circuit for calculating a Lyapunov spectrum from a reconstructed state space vector from time-series data divided into windows. This operation is performed as shown in the flowchart of FIG. 2. However, compared to the conventional Lyapunov spectrum analysis method of FIG. 7, in the present invention, a certain time is obtained from the entire time-series data x (t) in step (S12). A fundamental difference is that window processing for cutting out a section is performed.

[0050]

Effect of the Invention] above, correlation dimension analysis device that written to the present invention, when a series measuring circuit when measuring series data, the output signal of the time-series measurement circuit A / D converter to A / D conversion Circuit, a time-series window division processing circuit that cuts out a certain time section of the output of the A / D conversion circuit from the entire time-series data, an embedding operation circuit that reconstructs an attractor for the window time, and this reconstruction A correlation dimension calculation circuit for obtaining a correlation dimension from the attractor obtained, and the time-series window division processing circuit has a function of gradually shifting the time window by time and obtaining a correlation dimension at that time. since the time that the non-autonomous equations only to approximate the autonomous equations, it can also be obtained correlation dimension with respect to time-series data generated from a non-autonomous system equations, the operation It can be obtained correlation dimension at that time by repeating.

[0051] Further, the present invention Brighter Apu Nof spectrum analyzer written, when a series measuring circuit when measuring series data, A to an output signal of the time-series measurement circuit for A / D conversion
A / D conversion circuit, a time-series window division processing circuit for cutting out a time section having an output of the A / D conversion circuit from the entire time-series data, an embedding operation circuit for reconstructing an attractor for the window time, A Lyapunov spectrum calculation circuit for obtaining a Lyapunov spectrum from the reconstructed attractor, and the time series window division processing circuit performs the time window by gradually shifting the time window, and obtains a Lyapunov spectrum at that time. By approximating the non- autonomous system equation with the autonomous system equation for a certain time, the Lyapunov spectrum can be obtained even for the time-series data generated from the non-autonomous system equation. There is an advantage that a Lyapunov spectrum at can be obtained.

[Brief description of the drawings]

FIG. 1 illustrates the principle of operation of a correlation dimension analyzer of the present invention.
Is a flowchart showing an example of a correlation dimension analysis method for performing the above.

FIG. 2 shows the operation of the Lyapunov spectrum analyzer of the present invention.
One of the Lyanov spectrum analysis methods to explain the principle
It is a flowchart which shows an example .

FIG. 3 is an explanatory diagram of window processing for cutting out a certain time section from the entire time-series data x (t) according to the present invention.

FIG. 4 is a block diagram showing a configuration of an embodiment of a correlation dimension analyzer of the present invention.

5 is a block diagram showing the configuration of an embodiment of the Lyapunov spectrum analyzer of the present invention.

FIG. 6 is a flowchart illustrating an example of a conventional correlation dimension analysis method.

FIG. 7 is a flowchart illustrating an example of a conventional Lyapunov spectrum analysis method.

FIG. 8 shows a relation z between the displacement vectors y (i) and z (i).
It is a figure which shows (i) = Aty (i).

[Explanation of symbols]

 Reference Signs List 1 time series measurement circuit 2 A / D conversion circuit 3 time series window division processing circuit 4 embedding operation circuit 5 correlation dimension calculation circuit 6 Lyapunov spectrum calculation circuit

──────────────────────────────────────────────────続 き Continuing on the front page (72) Inventor Noboru Sonehara 1-1-6 Uchisaiwaicho, Chiyoda-ku, Tokyo Nippon Telegraph and Telephone Corporation (72) Inventor Masatoshi Horiguchi 1-16-1 Uchisaiwaicho, Chiyoda-ku, Tokyo Japan (56) References JP-A-7-64966 (JP, A) JP-A-6-146857 (JP, A) Technical report of the Institute of Electronics, Information and Communication Engineers vol. 95 no. 47 1995 pp41-48 (58) Fields investigated (Int. Cl. ⁷ , DB name) G06F 17/10 G06F 19/00 500 EPAT (QUESTEL) INSPEC (DIALOG) JICST file (JOIS) WPI (DIALOG)

Claims (2)

(57) [Claims] 1. A time series measurement circuit for measuring time series data, and an A / D converter for A / D converting an output signal of the time series measurement circuit.
A / D conversion circuit, a time-series window division processing circuit for cutting out a time section having an output of the A / D conversion circuit from the entire time-series data, an embedding operation circuit for reconstructing an attractor for the window time, A correlation dimension calculation circuit for obtaining a correlation dimension from the reconstructed attractor, and the time series window division processing circuit performs a function of obtaining the correlation dimension at that time by gradually shifting the time window by time. A correlation dimension analyzer, comprising: 2. A time series measurement circuit for measuring time series data, and an A / D converter for A / D converting an output signal of the time series measurement circuit.
A / D conversion circuit, a time-series window division processing circuit for cutting out a time section having an output of the A / D conversion circuit from the entire time-series data, an embedding operation circuit for reconstructing an attractor for the window time, A Lyapunov spectrum calculation circuit for obtaining a Lyapunov spectrum from the reconstructed attractor, and the time-series window division processing circuit performs the time window by gradually shifting the time window, and has a function of obtaining a Lyapunov spectrum at that time. A Lyapunov spectrum analyzer, comprising: