JPH0628335A - Chaos calculator - Google Patents

Abstract

(57) [Summary] [Purpose] The measurement data (time series) is arithmetically processed to determine the chaos contained in the data, and the order contained in the seemingly chaotic measurement data is inferred. [Configuration] A function to generate a chaotic attractor in a four-dimensional number space from the input discrete data by the embedding method of turkens, and a coordinate transformation of the chaotic attractor in the four-dimensional number space is sequentially performed and projected to a two-dimensional number space. When calculating the Lyapunov exponent from the chaotic attractor in the four-dimensional number space with a calculation function, allow a certain margin for determining the orthogonality of adjacent vectors, and also calculate the distance between vectors in a specific range The computer has been given a computing function to do so.

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a computer capable of performing arithmetic processing on time-series measurement data that looks seemingly disorderly and determining chaos included in the data.

[0002]

2. Description of the Related Art Conventionally, chaos of continuous (infinite) time series data generated by a mathematical model has been determined.

[0003]

However, since the time-series data that we can actually measure and use for analysis is discrete and finite, it is necessary to perform arithmetic processing on this to determine its chaos. No valid calculator was provided. Therefore, for example, even if time-series data of biometric information is measured, it cannot be calculated whether or not the biometric information is chaotic.

[0004]

The present invention is a chaotic calculator including a calculating means for calculating a chaotic attractor and / or a Lyapunov exponent from input time-series discrete data, wherein the calculating means is the input data. Has a function of generating a chaotic attractor by expanding it into a four-dimensional number space according to the procedure of the embedding method of Turkens, and a function of projecting a chaotic attractor in a four-dimensional number space into a three-dimensional number space by coordinate transformation. It is intended to provide a characteristic chaos calculator.

First, chaos will be explained. In the present invention, chaos means an integrated body which can include the above-mentioned cosmos, unlike chaos and anti-order which are used in comparison with cosmos which is an ordered integrated body. Is a well-defined mathematical and physical concept that has rules and laws, and even though the law itself is causal, the future prediction of the outcome is probable by probability. It is a phenomenon that becomes uncertain.

That is, although it is deterministic, in reality, a slight error is a phenomenon that is amplified and becomes unpredictable under the influence of nonlinearity, and randomness generated deterministically. It is chaos.

[0007] Therefore, chaos shows that there is a fundamental limit to the predictability, and in the sense that a considerable phenomenon, which was conventionally considered only stochastically, can derive an ordered structure. It is predictable.

The topology that characterizes the long-term behavior of chaos is called a chaos attractor, which is a mathematical structure in which the behavior of a system that generates chaos converges. The Lyapunov exponent is a numerical value indicating the degree of sharp dependence on the initial value, which is one of the characteristics of chaos.
When the first Lyapunov exponent, which is the largest of the Lyapunov exponents (n Lyapunov exponents are obtained in an n-dimensional number space), has a positive value, the original time series data is chaos.

In the present invention, when the measured time-series data is digital, it is as it is, and when it is analog, it is converted into a digital signal to generate time-series discrete data, and then a four-dimensional number space is created by a technique of embedding a turkens. , Chaotic attractor by CR, by projecting this into 3D number space and then into 2D number space.
It can be displayed on a display means such as T.

Further, in the present invention, when computing the Lyapunov exponent from the chaotic attractor in the four-dimensional number space, a certain margin is provided for the determination of the orthogonality of the adjacent vectors, and the vectors between the vectors within the specific range are given. By calculating the distance, the calculation of the Lyapunov exponent converges efficiently even with finite data.

[0011]

Embodiments of the present invention will be described with reference to the accompanying drawings, taking as an example the case where time series data of a pulse wave waveform is input. The same applies to the processing of other time series data. Further, in this embodiment, the pulse wave measuring device capable of outputting the time-series discrete data is connected to the computer to input the time-series discrete data. It is also possible to substitute for another possible measuring device, or to input and execute time-series discrete data which has already been measured and accumulated in a magnetic storage medium or the like.

FIG. 1 shows the whole apparatus (A) in the case of inputting time-series discrete data from a pulse wave waveform measuring apparatus,
The apparatus is a pulse wave waveform measuring apparatus (1) capable of outputting time-series discrete data, a small computer (2) as a computing means and a storage means, a CRT display (3) as a display means, and a printer (4). And a power supply (5) for supplying power to these.

The feature of the present invention resides in the small computer (2). First, the overall processing procedure of the small computer (2) will be described with reference to FIG.

When the small computer (2) is started (100), the initial setting (101) is first performed, and the line-of-sight direction of a four-dimensional number space, which will be described later, is set (102) and various calculation processing of pulse wave waveform data is performed. Pre-calculate and store constants frequently used in (103) and open CRT display (3) (104)
And display the menu (105).

In the menu, as shown in FIG. 3, a power-on sequence (107) for receiving a power-on response from the pulse wave measuring device (1) and setting the entire device (A) in a usable state, Clear the chaotic attractor display window (108), zoom in to increase the display size of the chaotic attractor (109), zoom out to reduce the display size of the chaotic attractor (110), and make the input data fit within the window. Pulse wave to display (1
11), save input data (112), read saved input data and display its chaotic attractor (116), 4
Rotation of a chaotic attractor by changing the line-of-sight direction of the dimension space (113) (Note that 3 patterns of angles that make it easy to understand the structure of the input data found in the trials so far are registered so that these can be specified easily. The next data (114) drawn from the input data as a chaotic attractor on the window, and END (11) which terminates the operation of the entire device (A).
5) etc. Next, the calculation for displaying the chaotic attractor will be described by taking the pulse wave waveform data as an example.

FIG. 4 shows an outline of the above calculation. The pulse wave waveform data is decomposed into 12 bits at a sampling frequency of 200 Hz (120), and the time-series discrete data is embedded in a four-dimensional number space to generate chaos. Create an attractor (12
1).

Next, the chaotic attractor of the four-dimensional number space is projected on the three-dimensional number space (122), and the chaotic attractor projected on the three-dimensional number space is projected on the two-dimensional number space and output to the screen ( 123).

During the above calculation, a menu is called to rotate the chaotic attractor so that the chaotic attractor can be viewed from an arbitrary direction, and the chaotic attractor can be enlarged or reduced and discrete data of time series can be displayed. It is possible to save, read the saved data, and display the chaotic attractor calculated from it.

In order to increase the calculation speed, the time-series discrete data is of integer type, and constants frequently used in the above calculation are calculated and stored in advance. In addition,
In the above calculation, embedding the time-series discrete data in the four-dimensional number space by the method of embedding the turkens means that the numerical value at a certain time point of the time-series discrete data input to the small computer (2) one after another is first. Is set to the numerical value X of the axis, and from this point (for example, if the constant interval τ = 10), the tenth numerical value is the second axial numerical value Y, and the twentieth numerical value is the third axial numerical value Z, 30 The four-dimensional vector is formed by these numerical values with the numerical value W of the fourth axis as the numerical value W of the fourth axis.
The chaotic attractor of the time-series discrete data is formed in the four-dimensional number space by the plurality of vectors thus created by the numerical values X, Y, Z, W.

Then, in order to match the unit vector n1 = (n1, n2, n3, n4) in the viewing direction of the four-dimensional number space with the fourth axis, the following matrix calculation is performed, and each vector (X, Y,
Z, W) is converted into (X ', Y', Z ', W').

[0021]

[Equation 1]

Since the attractor formed in the four-dimensional number space cannot be displayed as a figure on the CRT display (3), the following matrix calculation is performed and the coordinate X "of the projection point in the three-dimensional number space is calculated. , Y ", Z" are calculated.

[0023]

[Equation 2]

Then, the coordinates X "', Y"' obtained by projecting the coordinates of the three-dimensional number space onto the two-dimensional number space are calculated by the following equation.

X "'= Y" cosβ-X "cosα Y"' = Z "-X"'tan β where α and β are the three-dimensional number space x and y axes are the two-dimensional number space x and y, respectively. The angle with the axis.

A CRT display (3) using the coordinates X "', Y"' in the two-dimensional number space thus obtained as display means
, Or output to the printer (4). If chaos is inherent in the input time-series discrete data, a chaotic attractor with a unique shape is output for each and it can be determined that it is chaos. Incidentally, the pulse wave data is chaos and shows a unique chaos attractor as described later.

Next, the Lyapunov exponent will be described. Chaos has a property of strongly depending on an initial value. This property is called initial value dependence, and the numerical value indicating the degree of initial value dependence by an index is called the Lyapunov exponent. Then, the Lyapunov exponent can be calculated by calculating how far the distance between two adjacent trajectories drawn by the chaotic attractor is when the unit time elapses.

Next, a theoretical calculation method of the Lyapunov exponent will be described in order to explain the procedure for calculating the Lyapunov exponent employed in the present invention. As shown in FIG. 9, the reference trajectory is x (t) t = {0,1,2… nτ}, and the point ahead of τ time from x (0) is x (τ), x (0) Points close to
y (τ), y (0) exists on the orbit where y (t) and y (0) are advanced τ time, and y (τ) is the distance between y (0) and x (0). When d (0), the distance d (τ) between x (τ) and y (τ) can be expressed by the following equation.

[0029]

[Equation 3]

This means that the distance between the orbits that were initially close to each other exponentially expands with time, and the numerical value λ indicating the expansion ratio of this distance is an index of Lyapunov. The Lyapunov exponent λ can be calculated by the following equation.

[0031]

[Equation 4]

Next, a specific method of calculating the Lyapunov exponent based on the above theory will be described.

Reference trajectory x (t) t = {0,1,2 ... n
If τ} is present, let y ₀ (0) be the point that is orthogonal to the vector x (0) x (1) and is separated by a certain unit distance, and y ₀ (t) be the trajectory where y ₀ (0) exists. .

The distance between x (0) and y ₀ (0) is d ₀ (0), and the point x
The point after τ time of (0) is x (τ), and the point after τ time of y0 (0) is y.
The distance between ₀ (τ), x (τ) and y (τ) is d (τ).

Next, a point in the same direction as the vector x (τ) y ₀ (τ) and separated by a certain unit distance is y ₁ (0), and an orbit where y ₁ (0) exists is y ₁ (t) , The distance between x (τ) and y ₁ (0) is d ₁ (0)
, X (tau) point a x after between _{τ (2τ), y 1 (} 0) of the point after tau time y ₁ (τ), the distance between x (2.tau) and y ₁ (τ) d ₁ (τ)
And

Next, points in the same direction as the vector x (2τ) y ₁ (τ) and separated by a certain unit distance are y ₂ (0), x (2τ)
And the distance between y ₂ (0) and d ₂ (0), and the point after τ time of x (2τ)
The point after τ time of x (3τ), y ₂ (0) is y ₂ (τ), x (3τ)
And the distance between y ₂ (τ) and d ₂ (τ). By repeating this operation, d ₀ (τ) / d ₀ (0) in the first τ time
Times the next τ time, d ₁ (τ) / d ₁ (0) times, then the next τ
In time, the distance between the x (t) orbit and the adjacent orbit increases by d ₂ (τ) / d ₂ (0) times, and after n τ time, d _n-1 (τ) / d
It turns out that it becomes _n-1 (0).

The Lyapunov exponent λ is the average of the expansion ratios of the distances between the orbits per unit time, and can be calculated by the following equation.

[0038]

[Equation 5]

The above is a theoretical calculation method of the Lyapunov exponent λ, but it is difficult to calculate the Lyapunov exponent of time-series discrete data actually obtained by this calculation method. The reason is that the data used as a model for the theoretical Lyapunov exponent calculation method generates chaos data by an appropriately defined formula (mathematical model), and therefore continuous infinite data can be taken. Therefore, in contrast to the conditions described below, the actually obtained time-series discrete data does not always have a data point at a desired position in the number space, and the number of data is finite.

It is necessary that the vector A '(unit vector) at a point A on the four-dimensional vector orbit be present at the orthogonal position of the orbit, but this is not always the case in time-series discrete data that can be actually measured. Not exclusively.

The next vector B "at a certain point B
It is necessary that the (unit vector) exists at a position orthogonal to the trajectory at the point B, but this is not always found in reality.

Although it is necessary that the angle formed by the vectors B'and B "is small, it is not always small in reality.

Then, the closest vector at each point is searched. For example, when searching the vector B ″, if the search is performed simply on the condition that it is close to B ′, the vector at the point next to the point B is taken. there is a possibility.

Since the number of data is finite, if the point A is near the end of the time-series discrete data group, the next data point (point A plus τ) cannot be taken.

Since the time-series discrete data actually obtained is a finite number, it is possible that different data points have the same value.

Therefore, in the present invention, the conditions for adopting the data points used in the calculation of the Lyapunov exponent are set as follows.

That is, in a four-dimensional number space, a vertex is placed on the above-mentioned orbit, and a cone having a small apex angle with the vector B'as the center line is set, and the vector within this cone is adopted. did.

By the above, it is possible to avoid the possibility of taking the vector of the adjacent point unless the vector B ″ is very small and is not substantially in the same direction as the trajectory direction.

Further, the angle θ with respect to the trajectory of each vector
Is the vector of each data is (X, Y, Z, W), and the vector of the orbit direction is (X ', Y', Z ', W'),

[0050]

[Equation 6]

It is possible to check the orthogonality between the vector and the orbit by using the above equation.

Further, by setting the upper and lower limits of the absolute value of each vector and adopting the vector within this range, the vector B ″ is very small and is substantially the same as the orbit direction. However, it is possible to avoid the possibility of taking the vectors of adjacent points, and it is possible to set the value in which the best convergence is obtained by observing the chaotic attractor.

The elongation rate of each vector thus obtained is converted into a logarithm with a base of 2, and the arithmetic mean value thereof is calculated as the first value.
Lyapunov exponent λ ₁ . In order to execute the above calculation, the program (50) shown in FIGS. 6 and 7 is stored in the small computer (2).

That is, when the calculation of the first Lyapunov exponent λ ₁ is started (51), first, a point A, which is a reference for judging a data adoption condition, is set near the start end of the vector orbit ( 52). This point A is postponed as the calculation progresses.

Next, it is judged whether or not there is a margin to take the next point B (point after τ) (53), and if there is a margin (53Y).
This point B is adopted as the next point (54), temporary data is searched for at this point B (55), if found (55Y), temporary data is searched for at the next point (56), and if not found Search from the beginning of the discrete data in the series (57). Next, if the data found in this way meets the above-mentioned data adoption conditions (5
8Y), adopt this data point (59), and if not (58N), update the vector size range in the data point adoption condition (6
0), if it does not deviate from the upper limit of this range (61N), it returns to step (55), and if it deviates (61Y), the reference point A is moved to the next point B (62), and step (55) Return to 53).

With respect to the data points thus adopted, it is judged whether or not there is a margin to develop the above-mentioned vector A'to the vector B '(63). If there is a margin (63Y), the vector B'of the vector B'is determined. Check orthogonality with orbit (64).

If there is no room in step (63) (64
N) and step (64) if not orthogonal (64N),
Return to step (55).

Then, it is judged whether or not a retry described later is being performed (65), and if the retry is being performed (65Y), the angle formed with the previous vector is calculated (66), and this angle is large. Case (6
6L) returns to step (55), and if it is smaller (66S) and if the retry is not being performed in step (65) (65N), the vector B'is determined as a vector evolved from the vector A '(67). Then, the point A and the vector A'are stored for the retry (68). At the end of the calculation, if there is no room to take the next point in step (53) (53
N), end the operation (E).

Next, the provisional vector B "at the point B is searched (70). If this vector B" does not meet the data point adoption condition (71N), the vector size range of the above condition is updated. (72), but still exceeding the upper limit (73
Y), set the retry flag (74), step (55)
Return to.

If the upper limit is not exceeded in step (73) (73N), the process returns to step (70). And step
If there is a vector B "that meets the data adoption conditions in (71)
(71Y), it is judged whether or not this vector B "is orthogonal to the trajectory (75). If it is not orthogonal (75N), step (7)
Returning to (0), if they are orthogonal (75Y), the angle formed by each vector B ', B "at the point B is calculated (76), and whether or not this angle is sufficiently small, that is, within the above-mentioned cone It is judged whether or not there is B "in (77), and if the angle is not small (7
7N) returns to step (70), and if smaller (77Y), the elongation rate from vector A ′ to vector B ′ due to the movement from point A to point B is converted to a logarithm with base 2. (7
8) Then, this number is arithmetically averaged to obtain the first Lyapunov exponent λ ₁ (79).

Then, it is judged whether or not the point B currently calculated is the end of the time-series discrete data (80), and if it is not the end (80N), the above point B is the reference point for the next calculation. (Corresponding to point A above) (81), converting the vector B ″ into a unit vector and substituting it into the reference vector for the next calculation (corresponding to vector B ′ above) (82), for retrying The point B and the vector B'are stored (83), and the point next to the point B is calculated (8
4), calculate a vector evolved from the calculated reference vector of the next point (85), update the coordinates for orthogonal check of the vector used in the next calculation (86), and return to step (70), Repeat the calculation of.

When it is determined in step (80) that point B is the end of the time-series discrete data (80Y), the operation is terminated (E).

That is, in steps (52) to (68), a reference point that meets the data point adoption condition is searched mainly by referring to the vector B'at the next point, and steps (70) to (77).
In this case, the developed vector that matches the same condition is searched, and as described above, by setting the adoption condition of the data point to be adopted and searching the data point that meets this condition, the unqualified data Prevent points from being included in the calculation, and if no eligible data point is found, instead of abandoning the data point search at that point, widen the range of vector size of the recruitment conditions. By searching and using the next best data point,
The first Lyapunov exponent λ ₁ can be calculated with high accuracy.

Next, the calculation of the second Lyapunov exponent λ ₂ will be described. The calculation of the second Lyapunov exponent λ ₂ is basically the same as the calculation of the first Lyapunov exponent λ ₁ described above. At the point A1, the vectors A2 and A3 adapted to the data point adoption condition are taken, and the four-dimensional number space is calculated. On the triangle A1-A2-
A3 is formed, and then the vectors A2 and A3 are
Vectors B2 and B3 that have evolved to 1 form a triangle B1-B
2-B3 is formed, and then the newly formed eligible vectors B'and B "at the point B1 form a triangle B1-B'-B",
When the angle formed by the triangles B1-B2-B3 and the triangles B1-B'-B "is sufficiently small, the triangles A1-A2-A3 to the triangles B1-
The second Lyapunov exponent λ ₂ is calculated by calculating the area growth rate from B2 to B3, converting this to a logarithm with base 2, and arithmetically averaging this numerical value.

The angle formed by each triangle is the vector B
The angle formed by the above triangle is defined by the angle formed by the combined vector of 2 and B3 and the combined vector of B'and B ". The chaotic attractor and the Lyapunov exponent drawn by the above method are used for the pulse wave waveform. 8 illustrates an example of the case of discrete data of a series.
FIG. 9 shows a state in which the subject is reading (magazine).

First and second Lyapunov exponents (λ ₁ , λ ₂ )
Are shown at the bottom of the drawing.

FIG. 10 shows another subject (Mr. K, woman, health).
11 is reading (math text), Figure 12 is reading (manga book), and Figure 13 is chaotic attractor and Lyapunov exponent when looking at vaguely beautiful pictures. Is.

FIG. 14 shows chaos attractor and Lyapunov index in a relaxed state before treatment of Mr. S who has a history of neurosis, during treatment, and in FIG. 16 after recovery by treatment.

Since a chaotic attractor as described above is drawn from a time series discrete data that is not chaotic and a positive value cannot be obtained for the first Lyapunov exponent, the drawing method of the chaotic attractor and the Lyapunov exponent in this embodiment are not possible. It has been proved that the calculation method of is suitable as a chaos determination method.

[0070]

[Brief description of drawings]

FIG. 1 is an explanatory diagram showing a configuration of a device.

FIG. 2 is an explanatory diagram showing an overall processing procedure of a small computer.

FIG. 3 is an explanatory diagram of a menu.

FIG. 4 is an explanatory diagram of a calculation procedure for displaying a chaotic attractor.

FIG. 5 is an explanatory diagram showing a relationship between orbits and points in Lyapunov index calculation.

FIG. 6 is a flowchart of a calculation process for obtaining a Lyapunov number.

FIG. 7 is a flowchart of a calculation process for obtaining a Lyapunov number.

FIG. 8 is a diagram showing a chaotic attractor of a pulse wave.

FIG. 9 is a diagram showing a chaotic attractor of a pulse wave.

FIG. 10 is a diagram showing a chaotic attractor of a pulse wave.

FIG. 11 is a diagram showing a chaotic attractor of a pulse wave.

FIG. 12 is a diagram showing a chaotic attractor of a pulse wave.

FIG. 13 is a diagram showing a chaotic attractor of a pulse wave.

FIG. 14 is a diagram showing a chaotic attractor of a pulse wave.

FIG. 15 is a diagram showing a chaotic attractor of a pulse wave.

FIG. 16 is a diagram showing a chaotic attractor of a pulse wave.

Claims (3)

[Claims] 1. A chaotic calculator including a computing means for computing a chaotic attractor and / or a Lyapunov exponent from input time-series discrete data, wherein the computing means follows the procedure of a method for embedding the input data into a takens. A chaotic computer having a function of generating a chaotic attractor by expanding in a four-dimensional number space and a function of projecting a chaotic attractor in a four-dimensional number space into a three-dimensional number space by coordinate transformation. 2. The calculation means has a function of projecting a chaotic attractor in the three-dimensional number space onto a two-dimensional number space by coordinate transformation.
The chaos calculator described in the item. 3. The calculation means adopts a vector that falls within a certain cone at a certain point on the trajectory of the proximity vector in the four-dimensional number space, and adopts a vector whose absolute value is within a predetermined range. The chaos computer according to claim 1 or 2, further comprising a function of calculating a Lyapunov exponent.