EP1148978A1

EP1148978A1 - Method and device for determining a regularity

Info

Publication number: EP1148978A1
Application number: EP99958075A
Authority: EP
Inventors: Friedhelm R. Drepper
Original assignee: Individual
Current assignee: Individual
Priority date: 1998-11-21
Filing date: 1999-11-19
Publication date: 2001-10-31
Also published as: US6651025B1; AU1554400A; DE19853765C1; DE19956053A1; WO2000030815A1

Abstract

The invention relates to a device and/or a method for determining the phenomenological regularity of one or more cyclically fluctuating technical or scientific quantities, as well as their mutual influence on each other. According to the invention regularity is determined as an autoregressive model of successive phase angles which are each defined as a homogeneous function from the zero degree of at least two successive values of an initial quantity. By skilled selection of the homogeneous function and a matching implicit estimation formulation the reconstruction of the phase dynamic acquires a capacity for retransformation into the initial quantities and therefore, potentially, the characteristic of an immersion. The periodicity of the phase angles permits approximation of the behaviour rule by a finite Fourier series. The explicit form of the motion equation can be used for prediction and/or control as well as to identify and characterize the mutual phase synchronization of several processes.

Description

Method and device for determining regularity

Description

The invention relates to a method for determining the regularity of a cyclically fluctuating scientific or technical variable, and a device for carrying out the method.

The scientific or technical size is referred to below as (technical) SIZE-A. A fluctuating variable is a variable that oscillates over time. A cycle is the time span between three consecutive averages of the technical SIZE-A. An average - passage of value is the point in time at which the technical SIZE-A is equal to the average technical SIZE-A.

Examples of cyclically fluctuating physical or scientific parameters are e.g. the amount of fuel injected into an internal combustion engine, the resulting pressure in the cylinder of the engine or the resulting torque, the breathing activity or the heartbeat length of a person.

With the regularity of several variables, phase-synchronous fluctuations are of particular interest:

From the publication “Hirsch J.A, and B. Bishop. Respiratory sinus arrhythmia in humans: how breathing pattern modulates heart rate, Am. J. Physiol. 241: H620-H629, 1981 "it is known that the breathing activity and the heartbeat length of a stress-free, healthy person run" synchronously ".

According to the publication "Engel P., Hildebrandt G. and Scholz H.-G. 269, (1967) Measuring the phase coupling between heartbeat and breathing in humans, Plügers Arch. Ges. Phy- This means that heartbeats prefer to fall in certain breathing phases.

Both types of cardio-respiratory synchronization are seen as an expression of a healthy vegetative enervation of the heart. Weaknesses in synchronization in a physically and psychologically stress-free state are, depending on the form, either associated with autonomic neuropathies (e.g. diabetes mellitus) or with an increased risk of heart attack.

The German patent application with the official file number 197 18 806.0 describes a device and a method for determining the regularity of heartbeat speeds. The device has a clock for setting an inhalation and / or exhalation frequency and a detector. An EKG measuring device is used as a detector for measuring the heart rate. Furthermore, the device contains a memory in which reference data are stored and an evaluation means for generating a value which depends on the measured heartbeat rate and on the stored reference data.

With the device, the regularity of the heartbeat speeds of a stress-free and healthy person is determined. The measure for regularity is the determined flat pattern, which is obtained as follows.

While the subject breathes in the specified rhythm, his EKG is recorded. The length of the subject's heartbeat is determined from the device. High-pass filtering with a cutoff frequency that corresponds to the respiratory rate is carried out. The individual heartbeat lengths are converted into a two-dimensional representation. The heartbeat length of a heartbeat is plotted against the heartbeat length of the preceding heartbeat. An ellipse typically results from this plot. The coordinates from this plot are then transformed into polar coordinates (angles and radii). The mean heartbeat length serves as the center of the polar coordinate system. An angle is assigned to each heartbeat length.

The time sequence of the angles is then converted again into a two-dimensional representation. For this purpose, the angle assigned to a heartbeat is plotted against the angle of the previous heartbeat. This application typically results in a pattern that resembles a two-step staircase.

A deviation from this typical pattern in a subject allows conclusions to be drawn about physical or psychological changes.

Research has shown that this method is less suitable for the elderly. In particular, this method requires rhythmic breathing. It takes time for the subject to breathe stress-free at the specified rate. The process cannot therefore be carried out very quickly.

The object of the invention is to create a simple, fast method for determining the regularity of a physical variable, including the possibility of determining the regularity of influencing this variable by a further physical variable. The object of the invention is also to provide a device for carrying out the method.

The object is achieved by a method with the features of the main claim and by a device with the features of the auxiliary claim. Advantageous embodiments result from the related claims.

According to the method, values Ai (in particular in succession), in particular more than three values Ai, A _j , ... the SIZE-A determined per middle cycle. The indices i, j are used for consecutive numbering of the determined values.

At least two determined values Ai, A ₂ , ... are mapped to a size by a homogeneous function, which will be called ANGLE1 in the following. The values do not necessarily have to follow one another directly. The mapping rule is given by A— »W.

The homogeneous function W (A _i; A ₂ , ...) fulfills the equation

W (.. A _x, A _2,.) (A.. Aι _be arbitrarily *, a * _be arbitrarily A _2,.) = W for each _be a iiebig / wherein a iiebig _be any positive number.

At least two determined values A _{1 + k} , A ₂₊ , ... are mapped to a variable ANGLE2 (A—> W) by the homogeneous function W (Aι, A ₂ , ...). Two values Aι _{+ m} , A _{2 + m} , ... are advantageously mapped to a variable ANGLE3 (A- »W) by the homogeneous function W (Aχ, A ₂ , ...). k, m are positive integers that are greater than or equal to 1. Furthermore, m is greater than k. Preferably m = 2k.

A is a set of SIZE-A values. W is a value from the set WINKEL1, WINKEL2, ..., which is also called angle W below. An archetype of (AW) comprises as many elements of A as are necessary to form a value "ANGLE1, ANGLE2, ...". If an archetype of (A—> W) becomes zero, the associated angle W is not defined In order to achieve good results, k should be chosen to be relatively small, in particular smaller than the largest index in the original image of WINKEL1. Successive angles thus have overlapping original images. Due to the overlap of the original images, successive angles cannot be transformed back independently of one another it turns out that in a sequence of angles there are prohibited combinations of angles for which the reverse transformation of this combination of angles is either not possible or results in an A in which at least one complete archetype of (A—> W) becomes zero and / or at least one element of A becomes singular. All other combinations in a sequence of angles are allowed combinations of angles. An element of A becomes singular if the amount of this element approaches infinity (diverges).

The equation (implicit rule of conduct)

MT (G [ANGLE (n + 1), ANGLE (n), ...]) = F [ANGLE (n), ANGLE (n-1), ...; pl, p2, ...] set up. The function MT (x) is an arbitrary, preferably non-falling monotonous function of an argument x. The function F can depend on one or more arguments "WINKEL1, WINKEL2, ....".

The function G [ANGLE (n + 1), ANGLE (n), ...] has at least 2 arguments and is chosen so that G has singularities where prohibited combinations of ANGLE (n + 1), ANGLE (n), ... are available. A function y (xl, x2, ...) has a singularity if there is a combination of xl, x2, ... where the amount of y becomes infinite, [n represents a running index. For example, is ANGLE (n + 1) with n = 2 equal to ANGLE3. ]

The parameters pl, p2, ... of the function F [ANGLE (n), ANGLE (n-1), ...; pl, p2, ...] are adapted to the determined ANGLE1, ANGLE2, ANGLE3, ... This is done for example by regression or interpolation. Linear regression makes the adjustment particularly easy.

MT (G [ANGLE (n + 1), ANGLE (n), ...]) =

F [ANGLE (s), ANGLE (n-1), ...; pl, p2, ...] is used to reproduce the regularity.

Since the regularity was derived from a scientific or technical SIZE-A, it is about WO 00/30815 _ g _ PCT / EP99 / 08877

also by a scientific or technical quantity. The regularity of a cyclically fluctuating technical SIZE-A is used, for example, to regulate a technical process (such as the injection of fuel into an engine) so that a predetermined normal state is reached. The regularity of a technical SIZE-A can be used by detecting deviations from a typical regularity and thus disturbances in a technical or scientific process.

There are also technical or scientific processes, such as the beating of a heart, in which the type of disturbance can be determined by detecting deviations from the regularity in the normal state. A diagnosis is always possible if the discrepancy is typical for one or more types of faults. In order to be able to diagnose a fault 1, for example, it is first determined how the regularity deviates from the normal state when the fault 1 is present. The result serves as a reference pattern 1. Future deviations from the regularity of the normal state are compared with reference pattern 1. If these deviations agree sufficiently with the reference pattern 1, this is at least an indication of the presence of the disturbance 1.

If a fluctuating variable is measured as a function of time, the resulting measurement signal (measured value) contains interference components. Interference components are e.g. high-frequency noise, low-frequency temperature-dependent shift phenomena or low-frequency fluctuations in physiological processes in organisms.

A technical SIZE-A is determined from a measured value (measuring signal) in particular by removing the interference from the measured value. This is done in particular by high and / or low pass filtering. In an advantageous embodiment of the claimed method, the equation

MT (G [ANGLE (n + 1), ANGLE (n), ...]) = F [ANGLE (n), ANGLE (n-1), ...; pl, p2, ...] after

ANGLE (n + 1) = H (ANGLE (n), ANGLE (n-1), ...; pl, p2, ...) solved and the solved equation used as a regularity. This is an immediate reproduction of the regularity. The implicit representation of regularity is converted into an explicit one. The explicit code of conduct is better suited to carry out reconstructions or controls of technical or scientific processes.

In a further embodiment of the claimed method, all functions of angles, in particular the functions F [ANGLE (n), ANGLE (n-1), ...; pl, p2, ...] and G [ANGLE (n + 1), ANGLE (n), ...] are selected so that they are for each of the angular arguments ANGLE (n + 1), ANGLE (n), ANGLE ( n-1), ... have a period. With a finite range of values for the angle arguments, the period length P is equal to this range of values. The value range is the difference between the largest and the smallest angle. With an unlimited range of values of the angle arguments there is a minimum period length P. All whole multiples of P then also represent periods. In this case, each angle W can be assigned a reduced angle W ¹ within a reduced value range of length P, for which all functions have the same Accepting the value as for W. ^λ results from the modulo (rest) function, for example as W = mod (W, P). Without limiting the generality, a finite range of W can also be assumed in this case. A function f has thus a period if f (largest angle) = f (smallest angle).

In a further embodiment of the claimed method, the homogeneous function W (AI, A2, ...) is selected so that the reverse transformation of a sequence of angles has prohibited combinations of angles.

In a further embodiment of the claimed method, the homogeneous function W (Aι, A ₂ , ...) is selected so that the back transformation of a sequence of angles has at least one central permitted combination of angles that lies at the intersection of manifolds on which forbidden combinations of angles.

In a further embodiment of the claimed method, the function G is selected such that G [ANGLE (n + 1), ANGLE (n), ...] has at least one central singularity for which there is an arbitrarily small (convex) neighborhood of There are combinations of angles in which the function G assumes any value, and the central singularity is also at a point which has the property of a central permitted combination of the previous subclaim. (The neighborhood of a central singularity can, for example, be imagined as two 180 ° spiral staircases with infinitely small steps, which become increasingly steeper at the two ends.) In the place of a central singularity, the function G degenerates into a quantitative function.

In a further embodiment of the claimed method, the function F is chosen so that F has no singularities. This method step further improves the functioning of the method.

In one embodiment of the claimed method, W (A _i; A ₂ , ...) = arctan2 (Lι, L ₂ ) is selected as a homogeneous function. Li and L ₂ are linear combinations according to L _x = (ciAi + c ₂ A ₂ + ...), L ₂ = (diAi + d ₂ A ₂ + ...). This specification of the homogeneous function W represents an example of suitable homogeneous functions.

The following applies: arctan2 (x, y) = arctan (x / y) if x> 0, arctan2 (x, y) = arcta [(x / y) + π] if x <0.

In a further embodiment of the claimed method, the function G becomes the function

G [ANGLE (n + 1), ANGLE (n)] = tan (ANGLE (n + 1) + &) * y (ANGLE (n)) used. This is an example of a suitable function G. This configuration of the function G helps to ensure that singularities occur at the required points. The function H can be specified explicitly in this embodiment if the preferably non-falling monotonous function MT (x) can be analytically inverted.

In a further improved embodiment of the claimed method, F [WINKEL (n), WINKEL (n-1), ...; pl, p2, ...] = p ₀ + ∑ {pι * sin (f * i * ANGLE (n)) + ppi * cos (f * i * ANGLE (n))} with f = 2π / (value range from W) selected. Σ is the sum of i = l to m. If the value range of W is unlimited, the reduced value range is to be used.

In a further embodiment of the method, F [ANGLE (n), ANGLE (n-1), ...; pl, p2, ...] = Σ {pai * sin [f * i * ANGLE (n) + αi]} + Σ {pbj * sin [f * j * ANGLE (n-l) + ßj]} +

Σ {pc _k * sin <f * k * [ANGLE (n) - ANGLE (n-1) + χ _k ]>} + Σ {pd _m * sin <f * m * [ANGLE (n) + ANGLE (n -1) + δ _m ]>}, where i = 0, ..., m _a , j = 0, ..., m _b , k = 0, ..., m _C m = 0, ... , m ^ and f> 0. It is summed up via the respective index (Σ). The respective index is i, j, k or m. In a further embodiment of the method, additional method steps are carried out for at least one further scientific or technical SIZE-B. The homogeneous function for mapping the SIZE-B is not necessarily identical to the one mentioned above and therefore not the range of values of the angle W defined thereby. Finally MT (G [WINKELA (n + 1), WINKELA (n), .. .]) = F [WINKELA (n), WINKELB (n), ...; pl, p2, ...] used to reproduce the regularity of SIZE-A including the influence of SIZE-B. The corresponding equation for the SIZE-A values has therefore been expanded to include the argument regarding the SIZE-B values. To differentiate, the value ANGLE is also identified by an A or a B. WINKELA (n) is assigned to SIZE-A and was previously referred to as WINKELA (n). WINKELB (n) is assigned to SIZE-B.

The extension now enables not only a statement about the regularity of a scientific or technical quantity, but also a statement about the regularity of several scientific or technical quantities - including their mutual influence (correlation). This is the area of phenomenological reconstruction methods of multivariate processes. Different approaches are already used here, e.g. polynomial approaches or so-called radial basis functions. The method presented here is particularly suitable for the reconstruction of (possibly only temporarily) synchronous or phase-synchronous dynamics. One of the advantages of the method is that qualitative and quantitative properties of the phase synchronization can be read directly from the (estimated) equation of motion or based on the reconstruction of the phase angle.

Since, according to the method, the mutual influence can be determined quantitatively as a correlation or phase synchronization, it is mδg- WO 00/30815 _ JJ _ PCT / EP99 / 08877

lent to indicate the regularity of another correlating SIZE-A by observing the regularity of a SIZE-B. For example, SIZE-B is a person's breathing and SIZE-A is their heartbeat. SIZE-B, in another example, may be injecting fuel into an engine and SIZE-A may be torque.

In a further embodiment of the method

MT (G [WINKELA (n + 1), WINKELA (n), ...]) = pO +

Σ {pai * sin [f * i * WINKELA (n) + αi]} +

Σ {pbj * sin [f * j * WINKELB (n) + ßj]} +

Σ {pek * sin <f * k * [WINKELA (n) - WINKELB (n) + χk]>} +

Σ {pdm * sin <f * m * [WINKELA (n) + WINKELB (n) + δm]>}, where i = 0, ..., ma, j = 0, ..., mb, k = 0 , ..., mc and f> 0 is used to represent the regularity of SIZE-A. The

Regularity can include the influence of SIZE-B.

If a function is specified with only one argument, for example f (x), functions that depend on further arguments according to f (x, y) may also be suitable.

A device for performing the method has a measuring probe for determining values Ai of SIZE-A. A computer is connected to the measuring probe in such a way that the determined values are automatically fed into the computer. A program is installed on the computer. The program carries out the procedural steps.

The procedural steps relate to a method for determining the phenomenological regularity of one or more cyclically fluctuating scientific or technical parameters, including their mutual influence. The invention determines the regularity as an autoregressive model of successive phase angles, each as a homogeneous function of zero degree at least two successive values of an output variable are defined. By skillful selection of the homogeneous function and a suitable implicit estimation approach, the reconstruction of the phase dynamics acquires the property of being transformable back to the output quantities and thus potentially the property of an embedding. The periodicity of the phase angle enables the behavioral rule to be approximated by a finite Fourier series. The explicit form of the equation of motion can be used to predict and / or control, as well as to identify and characterize the mutual phase synchronization of several processes.

The invention is explained in more detail below with the aid of two examples.

EXAMPLE 1

The first example reconstructs a strongly asymmetrical sine function using a time series which, according to the formula A _t = 4/9 SIN (2PI * t / Tau) * (3-SIN (2PI * t / Tau)) with t = 0, 1, 2, ..., 17 was generated and is specified as column SIZE-A in Table 1. The (incommensurable) period length Tau was chosen as 7.777777.

Table 1 t SIZE-A angle WA reconstructor-constructor

W (A _{1 #} A ₂ , A ₃ ) = arctan2 (A ₁ + 2A ₂ + A ₃ , root (3) Ai-root (3) A ₃ ) is selected as a homogeneous function, with the linear combinations Lj Ai, A ₂ , A ₃ ) = Ai + 2A ₂ + A ₃ and L ₂ (Aι, A ₂ , A ₃ ) = root (3) A _X - root (3) A ₃ .

The following angles - large WA are determined as a result of the illustration (A — W): WAi = W (Aι, A ₂ , A ₃ ) = arctan2 (A _! + 2A ₂ + A ₃ , root (3) A _λ - root (3) A ₃ ), WA ₂ = W (A ₂ , A ₃ , A ₄ ) = arctan2 (A ₂ + 2A ₃ + A ₄ , root (3) A ₂ - root (3) A ₄ ), .. .

The numerical values of all angle sizes can be found in Table 1 in the Angle WA column. The feasibility of the inverse transformation of (A-W) plays a special role in determining the function G. As an intermediate step of the back transformation, (A—> W) can be given the following implicit form, ι (A _n , A _{n +} ι, A _{n + 2} ) = R _n * cos (WA _n ) (la)

L ₂ (A _n , A _{n +} ι, A _{n + 2} ) = R _n * sin (WA _n ) (lb) for n = 1, 2, ..., 15 and with the radii R _n = root (Li ² (A _n , A _{n +} ι, A _{n + 2} ) + L ₂ ² (A _n , A _{n + 1 /} A _{n + 2} )).

Eliminating A _n from the two equations for WA _n and A _{n + 3} from the two equations for WA _{n +} ι results in the following solvability conditions, R _n * [root (3) cos (WA _n ) -sin (WA _n ) ] =

Rn ₊ i * [root (3) cos (WA _{n + 1} ) + sin (WA _{n + 1} )], which can also be written as

R _n * sin (WA _n + 2π / 3) = R _{n + 1} * cos (WA _{n +} ι- π / 6). (2)

Since the radii R _n cannot become negative, the angular functions on both sides of the last equation have to perform common sign changes, that is to say they become zero together. If the angle function becomes zero on only one side, this results in a prohibited angle combination for which the system of equations (la) and (lb) does not conform to A _1; A ₂ , ..., Aι ₇ can be resolved. The proposed combinations lie on straight lines whose intersections represent central permitted combinations. The following function G becomes singular or zero for prohibited angle combinations and has central singularities at the central permitted combinations of angles, G [WA (n + l), WA (n)] = tan [WA (n + l) -π / 6] * sin [WA (n) +2 π / 3] with θ = -π / 6, Y (WA) = sin [WA + 2π / 3] and MT = 1. The following function values GA are determined in Table 2:

GAi = G (WA ₂ , WA _X ) = tan (WA ₂ -π / 6) sin (WAι + 2 π / 3), GA ₂ = G (WA ₃ , WA ₂ ) = tan (WA ₃ -π / 6 ) sin (WA ₂ +2 π / 3), .... F [ANGLE (s), ANGLE (n-1), ...; pl, p2, ...] = p ₀ + pl * sin (ANGLE (n)) + ppl * cos (ANGLE (n)) + p2 * sin (2 * ANGLE (n)) + pp2 * cos (2 * ANGLE (s) chosen. The following angular function values, sinA, cosA, sin2A, cos2A in Table 2, are determined in detail: sinAi = sin (WAι), sinA ₂ = sin (WA ₂ ), ..., cosAχ = cos (WAi), cosA ₂ = cos (WA ₂ ), ..., sin2Aι = sin (2 * WAι), sin2A ₂ = sin (2 * WA ₂ ), ..., cos2Aι = cos (2 * WAι), cos2A ₂ = cos (2 * WA ₂ ), ....

Table 2

Table 2 is used as input for standard software for multiple linear regression, with the column GA representing the dependent variable and the remaining columns the independent variables. The result is:

Regression statistics Multiple correlation coefficient 0 9992 coefficient of determination 0 9984 adjusted coefficient of determination 0 9978 standard error 0 0366 observations 16

ANOVA degrees of freedom

(df)

Regression 4

Residue 11

Total 15 Coefficients of intersection 0.1931

X variable 1 1.1509

X variable 2 0.1533

X variable 3 0.1873

X variable 4 -0.1967

Po = 0.1931 pl = 1.1509, ppl = 0.1533, p2 = 0.1873, pp2 = -0.1967

tan [ANGLE (n + 1) -π / 6] * sin [ANGLE (n) +2 π / 3] = p ₀ + pl * sin (ANGLE (n)) + ppl * cos (ANGLE (n)) + p2 * sin (2 * ANGLE (n)) + pp2 * cos (2 * ANGLE (n)) represents the implicit form of regularity. The explicit form of the recursion rule,

ANGLE (n + 1) = arctan2 {sin [ANGLE (n) +2 π / 3], p ₀ + pl * sin (ANGLE (n)) + ppl * cos (ANGLE (n)) + p2 * sin (2 * ANGLE (n)) + pp2 * cos (2 * ANGLE (n))} +

π / 6, leads with angle (1) = WAi to the reconstructed angles W_r of column 4 in table 1. Since all angles W_r are specified uniformly in the interval -π <= W_r _n <= π, W_r is reduced to the basic interval W_r _n = mod (ANGLE (n) + π, 2π) - π.

Two further iterations are carried out for the reconstruction of the output SIZE-A, A_r, (column 5 in Table 1). Reshaping the solvability condition (2) leads to the recursion formula

R_r _{n + 1} = R_r _n * sin (W_r _n + 2pi / 3) / cos (W_r _{n +} ι- π / 6) and elimination of A _{n + 2} from equations (la) and (lb) leads to A r _{n +} ι = R r _n * cos (W r _n -π / 6) / root (6) - A r _n. R_rι = root (Lχ ² (A _!, A ₂ , A ₃ ) + L ₂ ² (Ai, A _{2 /} A ₃ )) and A_r _x = Ai are used as starting values. The last two columns of Table 1 contain improved reconstructions of WA and SIZE-A angles, which were calculated based on 40 SIZE-A values and using an order 6 Fourier sum. The example suggests that qualitatively correct reconstructions of the dynamics are possible even with relatively short time series.

In addition to the angle definition used in the first example, which can be understood as a strongly localized Hilbert phase, the above steps can also be used for Lι (A _n , A _{n +} ι, A _{n + 2} ) = A _n - 2A _{n +} ι + A _{n + 2} can be carried out analogously. In this case, since both linear combinations, Li and L ₂ , have the property of a high-pass filter, this high-pass phase is particularly suitable when there is unsteadiness.

EXAMPLE 2

The second example determines the regularity of medical data. The respiratory flow and the heartbeat length RR are measured. The measured values are freed from interference by high-pass filtering. The result is SIZE-A and SIZE-B. The SIZE-A column in the following table thus results from high-pass filtering from the raw RR data and the SIZE-B column results from high-pass filtering from the uncalibrated raw respiratory flow data. Table 3

Time respiratory flow SIZE-B RR SIZE-A WB WA

11. .147 2906. .20 1. .668 1. .061 0, .033 3, .096 5. .906

12, .164 2944. .19 38. .799 1. .017 -0. .013 1. .528 2, .591

13, .204 2892. .00 -15. .083 1, .040 0, .008 5, .912 1, .364

14, .276 2879. .24 -29, .338 1, .072 0. .038 4, .238 0, .327

15, .324 2932. .32 23, .347 1, .048 0, .013 2, .469 5, .001

16, .316 2932. .03 22, .368 0, .992 -0, .044 0, .764 3, .260

17, .347 2894. .98 -16, .303 1, .031 -0, .005 5, .653 1. .749

18, .412 2887. .01 -24, .875 1, .065 0, .029 4 .132 0, .490

19, .463 2931. .08 20, .359 1, .051 0, .015 2 .456 5, .083

20, .458 2939. .35 29, .296 0, .995 -0, .040 0 .963 3, .338

21, .484 2892. .61 -17, .782 1, .026 -0, .008 5 .738 1, .799

22, .551 2886. .12 -23, .717 1, .067 0, .034 4 .069 0, .450

23, .599 2928, .62 20, .297 1, .048 0, .017 2 .434 5, .095

24, .588 2921, .85 13, .560 0, .989-0, .041 0 .589 2, .932

25, .626 2890, .38 -20, .037 1, .038 0, .009 5 .307 1, .374

26,699 2884, .54 -28, .664 1, .073 0, .044 4 .102 6, .010

27, .716 2935. .13 19. .181 1. .017 -0, .012 2, .552 4. .493

28, .691 2937. .16 17., 782 0. .975 -0. .056 0, .748 3. .536

29, .700 2906. .90 -15. .853 1. .009 -0, .023 5, .555 2. .541

30, .750 2918. .10 -5. .934 1. .050 0, .016 3, .500 5. .893

31, .780 2960. .35 37, .137 1, .030 -0. .007 1, .729 4. .524

W (Aχ, A) = arctan2 (Ai, A ₂ ) is selected as a homogeneous function, with the linear combinations L _x = Ai and L ₂ = A ₂ and analogously W (B _!, B ₂ ) = arctan2 (B _!, B ₂ ).

The following angle values WA and WB are determined in detail:

WAi = W (Aι, A ₂ ) = arctan2 (Aι, A ₂ ), WA ₂ = W (A ₂ , A ₃ ) = arctan2 (A ₂ , A ₃ ), ... and Bi = W (Bι, B ) = arctan2 (Bi, B ₂ ), WB2 = W (B2, B3) = arctan2 (B2, B3), ...

The numerical values of all angle sizes can be found in table 3 in the columns WA and WB.

Function G becomes function

G [ANGLE _A (n + l), ANGLE _A (n)] = tan (ANGLE _A (n + 1)) * sin (ANGLE _A (n) selected (θ = 0, MT = 1).

The following function values GA are determined in detail: GAi = G (WA ₂ , WA _X ) = tan (WA ₂ ) sin (WAι), GA ₂ = G (WA ₃ , WA ₂ ) = tan (WA ₃ ) sin (WA ₂ ), the numerical values of all sizes are shown in Table 4 in the

Find column GA.

With m _a = 1, m _b = l, m _c = 1 and πi _d = 0 and f = l, F [ANGLE _A (n), ANGLE _A (n-1), ..., ANGLE _B (n) , ANGLE _B (n-1), ...; pl, p2, ...] = Po + pa * sin [ANGLE _A (n) + αj _. ] + pb * sin [ANGLE _B (n) + ß] + pc * sin [ANGLE _A (n) - ANGLE _B (n) + χ _k ] is selected. According to the addition theorem of the sine function, this approach is equivalent to

F [ANGLE _A (n), ANGLE _A (n-1), ..., ANGLE _B (n), ANGLE _B (n-1), ...; pl, p2, ...] = po + pa * sin [ANGLE _A (n)] + ppa * cos [f * ANGLE _A (n)] + pb * sin [ANGLE _B (n)] + ppb * cos [ f * ANGLE _B (n)] + pc * sin [ANGLE _A (n) - ANGLE _B (n) _k ] + ppc * cos [ANGLE _A (n) - ANGLE _B (n) _k ]

The following angular function values sinA, cosA, sinB, cosB, sinAB, cosAB are determined in detail: sinAi = sin (WAι), sinA ₂ = sin (WA ₂ ), ... and cosAi = cos (WAi), cosA ₂ = cos (WA ₂ ), ... and sinB _! = sin (WBχ), sinB ₂ = sin (WB ₂ ), ... and cosBi = cos (WBι), cosB ₂ = cos (WB ₂ ), ... and sinABi = sin (WABχ), sinAB ₂ = sin (WAB ₂ ), ... and cosABi = cos (WABi), cosAB ₂ = cos (WAB ₂ ), ... and the numerical values of all angular function values are in table 4 in the columns sinA, cosA, sinB, cosB, sinAB to find cosAB. Table 4

GA sinB cosB sinAB cosAB sinA cosA

0. .226 0, .046 -0. .999 -0, .326 -0, .945 -0, .368 0, .930

2 .497 0 .999 0 .043 -0. .874 0. .486 0, .524 -0, .852

0, .332 -0, .362 0. .932 -0, .987 -0, .163 0, .979 0, .205

-1. .082 -0, .889 -0, .457 -0, .695 -0, .719 0, .322 0, .947

-0. .114 0. .623 -0. .782 -0. .573 -0, .820 -0, .959 0, .285

0, .656 0, .692 0, .722 -0, .602 -0, .799 -0, .118 -0, .993

0, .525 -0, .589 0, .808 -0, .691 -0, .723 0, .984 -0, .177

-1. .214 -0, .836 -0, .548 -0, .480 -0, .877 0, .471 0, .882

-0, .186 0, .633 -0, .774 -0, .492 -0, .870 -0 .932 0 .362

0, .843 0, .821 0, .571 -0, .694 -0, .720 -0, .196 -0, .981

0. .471 -0, .519 0, .855 -0, .715 -0, .699 0, .974 -0, .226

-1. .083 -0, .800 -0, .600 -0 .459 -0 .888 0 .435 0 .900

0, .198 0, .650 -0, .760 -0, .462 -0, .887 -0, .928 0, .373

1, .047 0, .556 0, .832 -0, .717 -0, .698 0, .208 -0, .978

-0 .275 -0 .828 0 .560 -0 .711 -0 .703 0 .981 0 .195

-1, .211 -0, .820 -0, .573 -0, .944 -0, .330 -0, .270 0, .963

-0, .407 0, .556 -0, .831 -0, .932 -0, .362 -0, .976 -0, .218

0, .264 0, .680 0, .733 -0, .346 -0, .938 -0, .384 -0, .923

-0, .232 -0 .665 0 .746 0 .127 -0 .992 0 .565 -0, .825

-1, .992 -0, .351 -0 .937 -0 .680 -0 .733 -0, .380 0, .925

0, .513 0, .987 -0, .158 -0, .340 -0, .940 -0, .982 -0, .187

The values from Table 4 are used as input for standard software for multiple linear regression, with the column GA representing the dependent variable and the remaining columns the independent variables. The result is:

Regression statistics

Multiple correlation coefficient. 0.94298

Coefficient of determination 0.88922

Adjusted coefficient of determination 0.84174

Standard error 0.38874

Observations 21

Degrees of freedom

Regression 6

Residue 14

Total 20

Coefficients

Intersection 0.79554

X variable 1 1.65197

X variable 2 0.21054

X variable 3 0.71818 X variable 4 0.61474

X variable 5 1.07526

X variable 6 0.22753

Po = 0.796 pa = 1.075, ppa = 0.228 pab = 0.718, ppab = 0.615 pb = 1.652, ppb = 0.211.

tan (ANGLE _A (n + l)) * sin (ANGLE _A (n)) = p ₀ + pa * sin [ANGLE _A (n)] + ppa * cos [f * ANGLE _A (n)] + pb * sin [ANGLE _B (n)] + ppb * cos [f * ANGLE _B (n)] + pab * sin [ANGLE _A (n) ANGLE _B (n) _k ] + ppab * cos [ANGLE _A (n) ANGLE _B (n) _k ] represents the implicit form of regularity. Regularity can also be given in an explicit form: ANGLE _A (n + l) = (arctan2 (sin (ANGLE _A (n), p ₀ + pa * sin [ ANGLE _A (n)] + ppa * cos [f * ANGLE _A (n)] + pb * sin [ANGLE _B (n)] + ppb * cos [f * ANGLE _B (n)] + pab * sin [ANGLE _A (n) - ANGLE _B (n) _k ] + ppab * cos [ANGLE _A (n) - ANGLE _B (n) _k ])

The figure shows a typical result, which, however, was obtained with other data.

The figure shows the cardiac angular velocity profile (WA _{n + 1} -WA _n ), which shows the change in the angle of the heartbeat length modulation as a function of the instantaneous values of the cardiac angle WA _n and the respiratory angle WB _n .

This profile describes a hand-shaking type of synchronization. Non-negative values of the cardiac angular velocity for a certain range of respiratory angles WB _n means that the dynamics of the cardiac angle come to a standstill or even reverse until the respiratory angle WB _n comes into the complementary range. (Both angular velocities are negative on average and angular are given in units of π.)

Root (pab ² + ppab ² ) can be interpreted as a measure of the strength of the cross interaction from size-B to size-A.

Another example is the time course of the injection of fuel into an engine and its torque. The regularity can be determined in the same way in such a case.

Deviations from the determined image signal faults. Typical deviations can signal the type of fault.

Claims

claims

1. Procedure for determining the regularity of a cyclically fluctuating scientific or technical SIZE-A with the steps:

• values Ai of SIZE-A are determined,

• a homogeneous function W (Aχ, A ₂ , ...) is chosen which (a _be iiebi * Aχ, a _be ii _e big * A ₂ , ...) = W (Ai, A ₂ , ... ) for all i ^ _bel _eb ig satisfied, where a _se ii _eb i _g is an arbitrary positive number,

• at least two determined values A _1; A ₂ , ... are mapped to the value ANGLE1 (A → W) by the homogeneous function W (A _{1 #} A ₂ , ...),

At least two determined values A _{1 + k} , A _{2 + k} , ... with k> 1 are mapped to a variable ANGLE2 by the homogeneous function W (Aχ, A ₂ , ...),

According to MT (G [ANGLE (n + 1), ANGLE (n), ...]) = F [ANGLE (n),

ANGLE (n-1), ...; pl, p2, ...] an equation is established,

Where G [ANGLE (n + 1), ANGLE (n), ...] has singularities where there are forbidden combinations of ANGLE (n + 1), ANGLE (n), ...,

• whereby a combination of angles is prohibited if the reverse transformation of this combination of angles is either not possible or results in an A in which at least one complete original image of (A—> W) becomes zero and / or at least one value of A becomes singular ,

• the parameters pl, p2, ... of the function F [ANGLE (n), ANGLE (n-1), ...; pl, p2, ...] are adapted to the values ANGLE1, ANGLE2, ANGLE3, ..., - MT (G [ANGLE (n + 1), ANGLE (n), ...]) = F [ANGLE ( n), ANGLE (n-1), ...; pl, p2, ...] is used to reproduce the regularity.

2. The method according to the preceding claim, in which more than three values Ai, A _j , ... the SIZE-A are determined per average cycle.

3. The method according to any one of the preceding claims, in which at least two determined values A _{1 + m} , A _{2 + m} , ... with m> k by the homogeneous function W (A _X , A ₂ , ...) on a Size ANGLE3 are shown.

4. The method according to any one of the preceding claims, wherein the equation MT (G [ANGLE (n + 1), ANGLE (n), ...]) = F [ANGLE (n), ANGLE (n-1),. ..; pl, p2, ...] according to ANGLE (n + 1) = H (ANGLE (n), ANGLE (n-1), ...; pl, p2, ...) solved and the solved equation used as a regularity becomes.

5. The method according to any one of the preceding claims, wherein the function

F [ANGLE (s), ANGLE (n-1), ...; pl, p2, ...] for each of the arguments ANGLE (n), ANGLE (n-1), ... has a period and the function

G [ANGLE (n + 1), ANGLE (n), ...] has a period for each of the arguments ANGLE (n + 1), ANGLE (n), ...

6. The method according to any one of the preceding claims, wherein the homogeneous function W (Aχ, A ₂ , ...) is selected so that the reverse transformation of a sequence of angles has prohibited combinations of angles.

7. The method according to any one of the preceding claims, in which the homogeneous function W (A _X , A ₂ , ...) is selected so that the back transformation of a sequence of angles has at least one central allowed combination of angles which in the intersection of There are manifolds on which there are prohibited combinations of angles.

8. The method according to any one of the preceding claims, wherein the function G [ANGLE (n + 1), ANGLE (n), ...] is selected so that G has at least one central singularity for which there is an arbitrarily small neighborhood of angle combinations in which the function G takes any value, and the central singularity is also in a position which has the property of a central permitted combination of angles of the previous subclaim.

9. The method according to any one of the preceding claims, wherein the function F is chosen so that F has no singularities.

10. The method according to any one of the preceding claims, in which W (Aι, A ₂ , ...) = arctan2 (L _l7 L ₂ ) is selected as a homogeneous function, wherein Li = (cχAι + c ₂ A ₂ + ... ) and where L ₂ = (diAi + d ₂ A ₂ + ...) are linear combinations.

11. The method according to any one of the preceding claims, in which as function G the function G [ANGLE (n + 1), ANGLE (n)] = tan (ANGLE (n + 1) + θ) * y (ANGLE (n)) is used.

12. The method according to any one of the preceding claims, in which F [ANGLE (n), ANGLE (n-1), ...; pl, p2, ...] = p ₀ + Σ {pi * sin (f * i * ANGLE (n)) + pp _± * cos (f * i * ANGLE (n))} where the sum is over i (i = l, ..., m), m = whole, positive number and f = 2π / (value range of W).

13. The method according to any one of the preceding claims, in which F [ANGLE (n), ANGLE (n-1), ...; pl, p2, ...] =

Σ {pa _± * sin [f * i * ANGLE (n) + oti]} +

Σ {pbj * sin [f * j * ANGLE (n-l) + ßj]} +

Σ {pc _k * sin <f * k * [ANGLE (n) - ANGLE (n-1) + χ _k ]>} +

Σ {pd _m * sin <f * m * [ANGLE (n) + ANGLE (n-1) + δ _m ]>} +

..., where i = 0, ..., m _a , j = 0, ..., m _b , k = 0, ..., m _c and m = 0,

..., ma and f> 0, oti, ß, χ _k , δ _{m are} arbitrary real numbers.

14. The method according to any one of the preceding claims, in which the following steps are carried out for a scientific or technical SIZE-B:

• values Bi of SIZE-B are determined,

• a homogeneous function W (Bχ, B ₂ , ...) is chosen, the W (a _be iebig * Bι, a _belie big * B ₂ , ...) = W (B _1; B ₂ , ... abeiiebig met) for all, where a iiebig _be any positive number,

At least two determined values Bi, B ₂ , ... are mapped to the value ANGLE _B 1 (B → W) by the homogeneous function W (Bι, B ₂ , ...),

• At least two determined values B _{1 + k} , B ₂₊ / • • • With k> 1, the homogeneous function W (Bχ, B ₂ , ...) maps to a variable ANGLE _B 2,

According to MT (G [ANGLE _A (n + 1), ANGLE _A (n), ...]) = F [ANGLE _A (n), ANGLE _B (n), ...; pl, p2, ...] an equation is established,

the parameters pl, p2, ... of the function F [ANGLE _A (n), ANGLE _B (n), ...; pl, p2, ...] are sent to the Values ANGLE _A 1, ANGLE _A 2, ANGLE _A 3, ..., ANGLE _B 1, ANGLE _B 2, ANGLE _B 3, ... adjusted,

MT (G [ANGLE _A (n + l), ANGLE _A (n), ...]) = F [ANGLE _A (n), ANGLE _B (n), ...; pl, p2, ...] is used to reflect the regularity of SIZE-A, including the influence of SIZE-B.

15. The method according to the preceding claim, in which the equation MT (G [ANGLE _A (n + 1), ANGLE _A (n), ...]) = F [ANGLE _A (n), ANGLE _B (n), ...; pl, p2, ...] according to ANGLE _A (n + l) = H [ANGLE _A (n), ANGLE _B (n), ...; pl, p2, ...] solved and the solved equation is used to represent the regularity of SIZE-A including the influence of SIZE-B.

16. The method according to one of the preceding claims, in which G [ANGLE _A (n + 1), ANGLE _A (n), ...] = p ₀ +

Σ {pai * sin [f * i * ANGLE _A (n) + a]} +

Σ {pbj * sin [f * j * ANGLE _B (n) + ßj]} +

Σ {pc _k * sin <f * k * [ANGLE _A (n) - ANGLE _B (n) + χ _k ]>} +

Σ {pd _m * sin <f * m * [ANGLE _A (n) + ANGLE _B (n) + δ _m ]>} +

..., where i = 0, ..., ma, j = 0, ..., mb, k = 0, ..., mc, m = 0,

..., md and f> 0, to reflect the regularity of SIZE-A including the influence of SIZE-

B is used.

17. The method according to any one of the preceding claims, in which the regularity and / or correlation is reproduced graphically on a screen or by a drawing created by means of a printer. WO 00/30815 _ ₂ g. PCT / EP99 / 08877

18. Device for performing the method according to one of claims 1 to 17 with

A measuring probe for determining values Ai of SIZE-A,

With a computer which is connected to the measuring probe in such a way that the determined values are fed into the computer,

With a program installed in the computer with which the following steps can be carried out:

A homogeneous function W (Aχ, A ₂ , ...) is formed, the W (a _be ιiebig * Aχ, a _be iiebi _g * A ₂ , ...) = W (Ai, A ₂ , ... eiiebig met) for all, where a iiebig _be any positive number,

At least two determined values A _x , A ₂ , ... are mapped to the value ANGLE1 by the homogeneous function W (A, A ₂ , ...),

At least two determined values A _{1 + k} , A _{+ k} , ... with k> 1 are mapped to a variable ANGLE2 by the homogeneous function W (Aχ, A, ...),

• according to MT (G [ANGLE (n + 1), ANGLE (n), ...]) =

F [ANGLE (s), ANGLE (n-1), ...; pl, p2, ...] becomes one

Established equation,

• whereby a combination of angles is prohibited if the reverse transformation of this combination of angles is either not possible or results in an A in which at least one complete original image of (A—> W) becomes zero and / or at least one value of A becomes singular , the parameters pl, p2, ... of the function

F [ANGLE (s), ANGLE (n-1), ...; pl, p2, ...] are adapted to the values ANGLE1, ANGLE2, ANGLE3, ..., - MT (G [ANGLE (n + 1), ANGLE (n), ...]) = F [ANGLE ( n), ANGLE (n-1), ...; pl, p2, ...] is used to reproduce the regularity.