JP3341801B2 - Correlation dimension analyzer and Lyapunov spectrum analyzer for quasi-stationary time series using adaptive segmentation - 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, in which a correlation dimension and a Lyapunov spectrum can be obtained in a stationary time interval even from quasi-stationary time series data composed of a combination of stationary time intervals. Classification of
Recognition, to a correlation dimension min 析装 location and Lyapunov spectrum analyzer quasi-stationary time series using adaptive segmentation available for identification.

[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. 11 shows a flowchart of a conventional correlation dimension analysis method. (S31) to (S36) in the figure.
Indicates each step.

[0005] In the conventional correlation dimension analysis method, it is assumed that time-series data is generated by an autonomous system equation, and calculation is performed in a stationary time interval.

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, it 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 dimension of the reconstructed attractor is smaller than the dimension 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. 12 shows a flowchart of a conventional Lyapunov spectrum analysis method. Note that (S41) in FIG.
(S46) shows each step.

A conventional Lyapunov spectrum calculation method is as follows.
The calculation was performed assuming that the time-series data was 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 in the time-series equation obtained from observation, that is, the instability of the orbit of the attractor, or the degree of divergence. Well, if this is positive, unstable,
If it is negative, it is stable, and there are Lyapunov exponents for the number of dimensions of a general dynamical system of an ordinary system, and Lyapunov spectrum is a generic term for that. Incidentally, chaos is called a strange attractor and is a set of unstable orbits. So,
The maximum Lyapunov exponent in the Lyapunov spectrum is always a positive value. If the system can be described by a differential equation, chaos has a correlation dimension of 3 or more.

[0018]

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

[0019]

(Equation 6) Tailor the left side more,

[0020]

(Equation 7) Becomes here

[0021]

(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. 13, a certain point X on the orbit
(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

[0023]

(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

[0024]

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

[0025]

[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).

[0026]

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

[0027]

(Equation 13) Than

[0028]

[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.

[0029] To determine all dimensions spectrum Lyapunov exponent, [delta] X (t) and d dimension 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 λ _i (i = 1,2,…, d) is

[0032]

[Equation 17] (S46).

[0033]

The above-described correlation dimension analysis method and Lyapunov spectrum analysis method are means for analyzing a stationary time series obtained from an autonomous system equation. Therefore, the correlation dimension and the Lyapunov spectrum cannot be calculated from the quasi-stationary time series data composed of a combination of stationary time sections.

According to the present invention, the correlation dimension analysis method and the Lyapunov spectrum analysis method for the stationary time series described in the above prior art can be obtained from quasi-stationary time series data composed of a combination of stationary time sections. It is intended to be extended to.

[0035]

[0036]

Correlation dimension analysis device and Lyapunov spectrum analyzer quasi-stationary time series using adaptive segmentation of the present invention SUMMARY OF THE INVENTION, when the spectrum to calculate the spectrum at time t ₀ and time t-series data storing the autocorrelation function calculating circuit for calculating a calculation circuit or the autocorrelation function, when the autocorrelation function distribution in the spectral distribution storage circuit or time t ₀ of the time-series data storing spectral distribution at time t ₀ of the series data difference amount based on the auto-correlation function in the autocorrelation function distribution storage circuit and the time t ₀ of the spectral error amount calculating circuit or time series data to calculate a spectral error amount at time t ₀ and time t of the time series data and the time t for Difference calculation circuit that calculates the difference and determines whether the error or difference exceeds a set threshold A threshold circuit which, by sequence division processing circuit time when the error amount exceeds the threshold, the sequence division processing circuit when cutting out the time interval T ₁ of the to time t ₁ from the time series of the entire used.

[0037]

[Action] In correlation dimension min 析装 location and Lyapunov spectrum component 析装 location according to the present invention, it is possible to obtain the correlation dimension and Lyapunov spectrum even from the quasi-stationary time series data consisting of the constant time interval.

[0038]

DESCRIPTION OF THE PREFERRED EMBODIMENTS One embodiment of the present invention will be described below.

FIG. 1 shows the operation source of the correlation dimension analyzer of the present invention.
3 shows a flowchart for explaining the process . Note that (S1) to (S9) in the figure indicate each step.

First, for the time series data x (t), (S
1), spectrum P _t0 (f) at time t ₀ , time t
Calculating a spectrum P _t (f) in (S2). The following spectral error (SE) is a measure of the difference between the two.
M) is used (S3).

[0041]

(Equation 18) Based on the above-mentioned spectral error (SEM), time t
₀ Compared to the spectrum of, when the spectral error amount above a certain threshold is detected from the time-series data at time t (S4), as shown in FIG. 5, steady time so far as t ₁ Adaptive segmentation is performed as a time section T ₁ (= t ₁ −t ₀ ) (S5). afterwards,
By implantation method for the time interval T ₁ reconstructs the attractor (S6), calculate the correlation dimension. This method performs the same operation time series on data as the conventional method. As a result, the correlation dimension in the steady interval T ₁ has been determined (S7) ~ (S9). Next, the spectrum at the time t _{1 at} which the segmentation was performed and the spectrum at the next time t are compared, and the above-described segmentation operation is repeated, thereby similarly obtaining the correlation dimension and the Lyapunov spectrum. By repeating these operations, the correlation dimension and the Lyapunov spectrum are calculated from the quasi-stationary time series data composed of a combination of stationary time sections.

Further, a method based on an autocorrelation function having temporal symmetry can be used as a measure of the segmentation. The degree of difference DIFF of the autocorrelation function between the two reference sections and the test section is defined as a linear sum of the degree of difference ADIFF of the amplitude of the autocorrelation function and the degree of difference FDIFF of the frequency distribution.

[0043]

[Equation 19] ATHR and FTHR are threshold values for the degree of difference between amplitude and frequency, respectively. Since each term is normalized,
Segmentation is performed when DIFF exceeds one.

ADIFF is the value P of the time lag 0 of the autocorrelation function between the two reference sections and the test section as shown in FIG.
_a, with P _b

[0045]

(Equation 20) Is defined as

FDIFF considers the two reference sections normalized as shown in FIG. 6 (b) and the first positive value part of the autocorrelation function of the test section, and calculates the area by the areas B and C in FIG. 6 (b).

[0047]

(Equation 21) Is defined as

FIG. 2 shows the operation of the correlation dimension analyzer of the present invention.
3 shows another flowchart for explaining the principle . FIG.
The difference from the above lies in steps (S2 ') and (S3').
That is, in the embodiment shown in FIG. 1, step (S2),
In (S3), the spectrum and the spectrum error amount are calculated, but the embodiment in the figure is different in that the autocorrelation and the difference amount are calculated. However, in that it is determined whether cut from a time series of the entire time interval T ₁ of the until time t ₁ that exceeds, the operation principle is the same.

FIG. 3 is a flowchart for explaining the operation principle of the Lyapunov spectrum analyzer of the present invention. Note that (S11) to (S20) in the figure indicate each step. Steps (S11) to (S16) correspond to steps (S1) to (S6) of the embodiment of FIG. 1, and steps (S17) to (S20) correspond to step (S43) of FIG.
(S46), the description of the operation is omitted.

FIG. 4 shows another flowchart for explaining the operation principle of the Lyapunov spectrum analyzer of the present invention. The difference from FIG. 3 is that steps (S12 ′) and (S12
13 '), which is exactly the same as that described in the difference between FIG. 1 and FIG.

FIG. 7 is a block diagram showing the configuration of an embodiment of the correlation dimension analyzer of the present invention.

In this figure, 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 on the measured time-series data.

Reference numeral 3A denotes a spectrum calculation circuit for calculating the spectrum of the time-series data at time t ₀ and time t. 4A is a circuit for storing the spectral distribution in the spectrum distribution storage circuit, when the series data time t ₀ and time t. 5A is a spectrum error amount calculation circuit,
This is a circuit for calculating the amount of spectral error at time t ₀ and time t of time-series data. Reference numeral 6 denotes a threshold value circuit which determines whether or not the error amount exceeds a set threshold value. 7 is a time series division processing circuit, when the threshold is exceeded, a circuit for cutting out from the time series of the entire time interval T ₁ of the until time t ₁ beyond. 8 is an embedding operation circuit,
A circuit for reconstructing the attractor using cut time interval T _1. 9 is a correlation dimension calculating circuit, a circuit for calculating the correlation dimension from reconstructed state space vector of the time interval T _1.

The operation is as described with reference to the flowchart of FIG.

FIG. 8 is a block diagram showing the configuration of another embodiment of the correlation dimension analyzer of the present invention. In this figure, reference numeral 3B denotes an autocorrelation function calculation circuit which calculates an autocorrelation function at time t ₀ and time t of time-series data. Reference numeral 4B denotes an autocorrelation function distribution storage circuit which stores an autocorrelation function distribution at time t ₀ of time-series data. 5B is a difference calculation circuit, which is a time t ₀ of the time series data.
And a circuit for calculating a difference amount based on the autocorrelation function at time t. Others are the same as the embodiment of FIG.

The threshold circuit 6 is a circuit for determining whether or not the difference exceeds a set threshold.

The operation is as described with reference to the flowchart of FIG.

FIG. 9 is a block diagram showing the configuration of one embodiment of the Lyapunov spectrum analyzer of the present invention. In the figure, reference numeral 10 denotes Lyapunov spectrum calculation circuit, a circuit for calculating the Lyapunov spectrum than reconstructed state space vector of the time interval T _1, the other is the same as FIG.

The operation is as described with reference to the flowchart of FIG.

FIG. 10 is a block diagram showing the configuration of another embodiment of the Lyapunov spectrum analyzer of the present invention.
In the figure, 3B, 4B, and 5B are the same as those shown in FIG. 8, and the others are the same as FIG.

The operation is as described with reference to the flowchart of FIG.

[0062]

Effect of the Invention] As described above in detail, correlation dimension min 析装 location and Lyapunov spectrum component 析装 location of the quasi-stationary time series using adaptive segmentation of the invention, the first time in the time-series data And the amount of difference from the autocorrelation function at the first time or
When a spectral error amount or a difference amount of the autocorrelation function exceeding a certain threshold value is detected from the time-series data, the time up to that point is adaptively segmented as a stationary time interval, and the time interval is embedded. Reconstruct the attractor by the method, calculate the correlation dimension or Lyapunov spectrum, repeat the above segmentation operation, find the correlation dimension or Lyapunov spectrum in that time section, and obtain a quasi-stationary time series consisting of a combination of stationary time sections The correlation dimension or Lyapunov spectrum from the data can be determined.

[Brief description of the drawings]

Explaining the operation principle of the correlation dimension analysis device of the present invention; FIG
It is a flowchart for the.

Explaining the operation principle of the correlation dimension analysis device of the present invention; FIG
9 is a flowchart showing another example for the operation.

[3] Operation of the Lyapunov spectrum analyzer of the present invention
It is a flowchart for explaining a principle .

FIG. 4 shows the operation of the Lyapunov spectrum analyzer of the present invention.
9 is another flowchart for explaining the principle .

FIG. 5 is an explanatory diagram of adaptive segmentation based on a spectral error amount according to the present invention.

FIG. 6 is a diagram showing an autocorrelation function between a reference section and a test section, and a normalized autocorrelation function between the reference section and the test section.

FIG. 7 is a block diagram showing a configuration of an embodiment of a correlation dimension analyzer of the present invention.

FIG. 8 is a lock diagram showing the configuration of another embodiment of the correlation dimension analyzer of the present invention.

FIG. 9 is a block diagram showing a configuration of one embodiment of a Lyapunov spectrum analyzer of the present invention.

FIG. 10 is a block diagram showing a configuration of one embodiment of a Lyapunov spectrum analyzer of the present invention.

FIG. 11 is a flowchart for explaining a conventional correlation dimension analysis method.

FIG. 12 is a flowchart for explaining a conventional Lyapunov spectrum analysis method.

FIG. 13 shows a relationship z between the displacement vectors y (i) and z (i).
(I) = Aty (i).

[Explanation of symbols]

 Reference Signs List 1 time series measurement circuit 2 A / D conversion circuit 3A spectrum calculation circuit 3B autocorrelation function calculation circuit 4A spectrum distribution storage circuit 4B autocorrelation function distribution storage circuit 5A spectrum error amount calculation circuit 5B difference amount calculation circuit 6 threshold circuit 7 Time series division processing circuit 8 Embedding operation circuit 9 Correlation dimension calculation circuit 10 Lyapunov spectrum calculation circuit

──────────────────────────────────────────────────続 き Continued on the front page (72) Inventor Akira Hiraiwa 1-1-6 Uchisaiwai-cho, Chiyoda-ku, Tokyo Nippon Telegraph and Telephone Corporation (72) Inventor Noboru 1-6-6 Uchisaiwai-cho, Chiyoda-ku, Tokyo Japan Inside Telegraph and Telephone Corporation (72) Inventor Masatoshi Horiguchi 1-1-6 Uchisaiwaicho, Chiyoda-ku, Tokyo Nippon Telegraph and Telephone Corporation (56) References JP-A-8-278958 (JP, A) JP-A-7- 55226 (JP, A) Kazuyuki Aihara, Chaos Seminar, Kaibundo, Japan, April 15, 1994, p. 100-105, p. 150-154 Tatsuya Yamaguchi, Toshinori Nagano, Masahiro Nakagawa, Study on Fractal Properties of Speech (Fractal Dimension of Vowels), IEICE Technical Report [Speech], Japan, January 19, 1993, Vol. 92, No. 410, SP92-130, p. 57-64 Tatsuya Yamaguchi, Masahiro Nakagawa, Fractal Characteristics of Speech and Its Evaluation Method, IEICE Technical Report [Speech], Japan, October 14, 1993, Vol. 93, no. 266, SP93-74, p. 79-86 Sabanal Salvador, Masahiro Nakagawa, Research on Speech Recognition Based on Fractal Properties, IEICE Technical Report [Medical Electronics and Biotechnology], Japan, May 26, 1995, Vol. 95, No. 84, MBE95-6, p. 37-44 (58) Field surveyed (Int. Cl. ⁷ , DB name) G01L 11/00 G01L 15/02 G01L 15/08 H03H 17/02 JICST file (JOIS)

Claims (4)

(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.
/ D conversion circuit, a spectrum calculation circuit for obtaining a spectrum of the time-series data at time t ₀ and time t, a spectrum distribution storage circuit for obtaining a spectrum distribution of the time-series data at time t ₀ , and a time of the time-series data A spectrum error amount calculation circuit for calculating a spectrum error amount at t ₀ and time t, a threshold circuit for determining whether or not the error amount exceeds a set threshold value, If the threshold is exceeded, exceeds a sequence division processing circuit when cutting out the time interval T ₁ of the to time t ₁ from the time series of whole, embedded operating circuit to reconstruct the attractor for the time interval T ₁ If, suitable for comprising the correlation dimension is calculated, it repeats the above segmentation operation from time t _1, the calculated Mel correlation dimension calculating circuit correlation dimension at each time period Correlation dimension analysis apparatus of quasi-stationary time series using a specific segmentation. 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.
/ D conversion circuit, an autocorrelation function calculation circuit for obtaining an autocorrelation function at time t ₀ and time t of time series data, and an autocorrelation function distribution storage for obtaining an autocorrelation function distribution at time t ₀ of the time series data Circuit and time t of time series data
A difference calculation circuit for calculating a difference based on the autocorrelation function between ₀ and time t, a threshold circuit for determining whether the error exceeds a set threshold, and determination of the threshold circuit time t ₁ but if it exceeded the threshold, which exceeded
A time series division processing circuit for extracting the time section T ₁ from the entire time series, an embedding operation circuit for reconstructing an attractor for the time section T ₁ , a correlation dimension is calculated, and the above segmentation operation is performed. repeated from time t _1, the correlation dimension analysis apparatus quasi-stationary time series using adaptive segmentation is characterized in that a determined Mel correlation dimension calculating circuit correlation dimension at each time interval. 3. 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.
/ D conversion circuit, a spectrum calculation circuit for obtaining a spectrum of the time-series data at time t ₀ and time t, a spectrum distribution storage circuit for obtaining a spectrum distribution of the time-series data at time t ₀ , and a time of the time-series data A spectrum error amount calculation circuit for calculating a spectrum error amount at t ₀ and time t, a threshold circuit for determining whether or not the error amount exceeds a set threshold value, If the threshold is exceeded, exceeds a sequence division processing circuit when cutting out the time interval T ₁ of the to time t ₁ from the time series of whole, embedded operating circuit to reconstruct the attractor for the time interval T ₁ When, the spectral distribution is calculated and repeats the above-described segmentation operation from time t _1, the Lyapunov spectrum calculated Mel rear <br/> flop Nof spectrum calculation for each time period A Lyapunov spectrum analyzer for quasi-stationary time series using adaptive segmentation characterized by comprising a circuit. 4. 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.
/ D conversion circuit, an autocorrelation function calculation circuit for obtaining an autocorrelation function at time t ₀ and time t of time series data, and an autocorrelation function distribution storage for obtaining an autocorrelation function distribution at time t ₀ of the time series data Circuit and time t of time series data
A difference calculation circuit for calculating a difference based on the autocorrelation function between ₀ and time t, a threshold circuit for determining whether the error exceeds a set threshold, and determination of the threshold circuit time t ₁ but if it exceeded the threshold, which exceeded
A time series division processing circuit for extracting the time section T ₁ from the entire time series, an embedding operation circuit for reconstructing the attractor for the time section T ₁ , a spectrum distribution is calculated, and the above segmentation operation is performed. repeated from time t _1, asked Me the Lyapunov spectra at each time interval
Lyapunov spectrum calculation circuit and Lyapunov spectrum analysis apparatus quasi-stationary time series using adaptive segmentation that comprising the that.