JP2001522094A - Method and apparatus for classifying a first time series and at least one second time series - Google Patents

Abstract

(57) [Summary] Examine the time series for their statistical dependencies and train the neural networks so that each one models the conditional probability of one time series. Using a neural network, a proxy time series is obtained, and a measure for the statistical dependence of the sample value of each time series and the sample value of the proxy time series is obtained. These measures are compared with each other to see if the dynamics of the underlying individual time series are represented by Markov processes of order (n ₁ ,..., _{N N} ).

Description

DETAILED DESCRIPTION OF THE INVENTION

The invention relates to the classification of a first time series and at least one second time series.

In various applications, it is advantageous to analyze dynamic systems for statistical dependencies and to pre-characterize the measured signal.

In this case, the predetermined measurement signal x has an arbitrary number of sample values _xt , which are sampled with a step width w (see FIG. 2). This corresponds to _obtaining a statistical linear or non-linear dependence between a plurality of sample values xt. Depending on a predetermined number of past sample values v whose statistical dependencies are being analyzed, the information obtained by the analysis is used to prescribe a predetermined number of future values z.

[0004] The concept of information flow is known from the literatures [1] and [3] in order to analyze the statistical dependence on the one-dimensional case, ie on one time series {x _t }. In this type of approach, the loss of information between a past value and a future value in a configurable number of steps is determined. Depending on the characteristics of the information flow, it is possible to conclude the statistical dependence of the sample values and future values.

A method for obtaining a proxy time series with respect to a set time series, that is, a method of obtaining a predetermined number of sample values of a signal and a basis regarding the proxy is described in Documents [2] and [6].

A proxy for a predetermined time series is understood to be one time series having the same predetermined statistical characteristics as the set time series.

[0007] Various statistical estimators, such as neural networks, kernel estimators (Ke
The prospects for rnel-Schaetzer) are described in Refs. [4] and [8].

The cumulant is specifically understood to be a coefficient of a Taylor expansion of a log function of a Fourier transform of a probability density. The basis for cumulants is described in Ref. [5]. The development of the characteristic function by cumulant is also described in reference [5].

Reference [6] describes learning of a neural network by the maximum likelihood method.

A Markov process of order (order) m is understood to be a time-discrete random process. Here, future values depend only on values that occurred in the past in m steps.

The rank of a time series (order) is understood to be the order of the sample values in one time series in detail according to the magnitude of the sample values.

A method of classifying a time series is well known from the literature [9], and here, a predetermined number of substitutes are obtained for the time series. For the time series and the surrogate, a non-linear correlation between the time series value and the surrogate value is determined by a cumulant-based approach. The time series is classified based on the non-linear correlation.

Another method for classifying time series is well known from document [10]. In this method, a dynamic system is modeled according to its probability distribution. In the neural network, learning is performed by the maximum likelihood method according to the probability of a non-linear Markov process of order m.

An object of the present invention is to provide a method and an apparatus for classifying a plurality of time series in terms of the statistical dependence of sample values.

This object is achieved by a method according to claim 1 and an apparatus according to claim 11.

In this method, a non-linear Markov process for a first time series is modeled by a first statistical estimator. The non-linear Markov process for the second time series is modeled by a second statistical estimator. For the first time series, at least one surrogate time series is formed using a first statistical estimator. For the second time series, at least one surrogate time series is formed using a second statistical estimator. A first measure for the statistical dependence between the sample values in the first time series and the sample values in the second time series is formed for a predetermined number of future sample values.
Furthermore, a second measure for the mutual statistical dependence between the values in the surrogate time series is formed for a predetermined number of future sample values. A difference measure is formed from the first measure and the second measure. The classification includes: a) assigning the first time series and the second time series to the first group if the difference measure is less than a predetermined threshold; If not less than the threshold, the first time series and the second
Are assigned to the second group.

The apparatus has a processor unit, which is configured to model a non-linear Markov process for a first time series with a first statistical estimator. The nonlinear Markov process for the second time series is the second
Is modeled by a statistical estimator. For the first time series, at least one surrogate time series is formed using a first statistical estimator. For the second time series, at least one surrogate time series is formed using a second statistical estimator. A first measure for the statistical dependence of the sample values in the first time series and the sample values in the second time series is formed for a predetermined number of future sample values.
Furthermore, a second measure for the mutual statistical dependence between the values of the surrogate time series is formed for a certain number of sample values that are present in the future. A difference measure is formed from the first measure and the second measure. The classification includes: a) assigning the first time series and the second time series to the first group if the difference measure is less than a predetermined threshold; If not less than the threshold, the first time series and the second
Are assigned to the second group.

According to the present invention, it is possible to obtain a statistical dependency between multi-dimensional time series, that is, a statistical dependency between a plurality of time series.

[0019] Further embodiments of the invention are set out in the dependent claims.

Another embodiment advantageously uses each one non-linear neural network as a statistical estimator. This is because neural networks are very well suited for estimating probability density.

The invention can be used in various fields of application. Therefore, electronic brain imaging EEG
Alternatively, the statistical dependence between a plurality of measurement signals of the electronic electrocardiography EKG can be determined.

The invention can also very advantageously be used for financial market analysis. In this case, the signal characteristics describe, for example, the characteristics of a stock or foreign exchange market.

Embodiments of the present invention are shown in the drawings and will be described in detail below.

FIG. 1 is a schematic diagram illustrating the invention in the individual elements of the invention.

FIG. 2 is a diagram showing the course of a measurement signal f which is converted into a time series {x _t } by sampling with a step width w.

FIG. 3 is a schematic diagram illustrating a computer implementing the present invention.

The first time series {x _t } and the second time series {y _t } each have a predetermined number of sample values x _t and y _t of the signal, in particular of the electrical signal.

These signals are measured by a measuring instrument MG (see FIG. 3),
Supplied to The computer implements the method described below,
This result is supplied to the means for subsequent processing WV.

[0029] Each time series {x _t}, with respect to {y _t}, neural networks NN _x, order n _x using NN _y, nonlinear Markov process of n _y are modeled.

The information flow of the first time series {x _t } or the second time series {y _t } is approximated by a nonlinear Markov process of order m.

For an arbitrary number N of time series that can be considered in this way, the time series is given by: ｛x _{t ｋ k} , k = 1,..., N (1) Each time series ｛ For x _t ｝ _k , a measure of the statistical dependence of the value ｛x _{t + r} ｝ _k (the value in the r-step future) is formed. In this case, n _j past sample values (j = 1,.
., N) are taken into account.

The characteristics of this system are represented by the following differences between the probability density functions P ():

[0033]

(Equation 3)

If the statistical dependence disappears within the future r step, equation (2) becomes

[0035]

(Equation 4)

[0036]

[0037] By applying the maximum likelihood method, the neural network NN _x, NN _y is trained to approximate the nonlinear Markov process of order n _x, n _y. According to the maximum likelihood method, a learning rule that maximizes the product of probabilities is followed. Therefore, each neural network NN _x , NN _y is a conditional probability

[0038]

(Equation 5)

The estimation of is performed.

To perform the analysis, relatively high order cumulants are used instead of probability densities. By transforming equation (3) into Fourier space, the characteristic function

[0041]

[Outside 1]

Is formed. These characteristic functions indicate the Fourier transform of the probability density of the equation (3).

Equation (3) is

[0044]

(Equation 6)

Where, where

[0046]

[Equation 7]:

[0047]

(Equation 8)

And

[0049]

(Equation 9)

The following equation holds:

[0051]

(Equation 10)

In this case, Φ ^k (.) Represents the Fourier transform of the probability density,

[0053]

[Outside 2]

The function in Fourier space by

[0055]

[Equation 11]

Represents a variable of

[0057]

(Equation 12)

Is as follows.

When these characteristic functions are expanded in the cumulant described in [5], the following equation is obtained;

[0060]

(Equation 13)

[0061]

[Outside 3]

Each cumulant is indicated by.

Substituting this extension into equation (5) yields the following condition:

[0064]

[Equation 14]

However, the restriction here

[0066]

(Equation 15)

Is attached.

This condition applies if all kinds of statistical dependencies within the r steps in the future become zero.

In the case of statistical independence between sample values, equation (14) simplifies to:

[0070]

(Equation 16)

The sample value of each time series

[0072]

[Outside 4]

A measure for the statistical dependence between is formed.

The statistical dependence of the sample values of the time series {x _t } _{k in} the r-step future is given by:

[0075]

[Equation 17]

The statistical dependence of the sample values of the different time series {x _t } _k is described by the following rules: The scale m _k (r) is the dynamic system for which the time series {x _t } _k is determined, respectively. represents characterization based on the information flow cumulants, when the measure m _k (r) is in each sequence {x _t} values by taking into account the n _j number of values in the past of _j {x t _{+ r}} Quantify the dependence of _k .

If the further measure m _kk (r) takes the value 0, the sample values of the respective time series are statistically independent of one another. Increasing the positive value of each further measure m _kk (r) indicates an increasing statistical dependence on the sample values of each time series {x _t } _k .

[0078]

(Equation 18)

Therefore, for modeling a Markov process of order (n ₁ ,..., _{N N} ), the probability density

[0080]

[Equation 19]

To approximate, a two-layer feedforward neural network is trained.

According to the following rules, the neural network for each time series {x _t } _k is trained as follows. That is, the neural network is trained to perform an estimation of the probability density of each of the Markov processes of order (n ₁ , ..., n _N ):

[0083]

(Equation 20)

Where H is the number of Gaussian functions, l is the number of hidden neurons in each neural network, and

[0085]

(Equation 21)

Represents parameters of each neuron network. L (d) is defined according to equation (19). Therefore, the conditional probability density is described by the weighted sum of the normal distribution, and its weight

[0087]

[Outside 5]

Average value

[0089]

[Outside 6]

And dispersion

[0091]

[Outside 7]

Are given as starting quantities for three different neural networks. These include the first s component of the vector v ^{k, r} as an s-dimensional input quantity as a multilayer perceptron (see equations (23) to (25)). Training is performed according to the maximum likelihood method described in [7].

N neural networks are trained and N conditional probability densities for N-dimensional Markov processes of order (n ₁ ,..., _{N N} ) are estimated.

[0094] After the completion of the training, the neural network, starting from the value of the first n _k of the time series {x _{t} k,} respectively, when using the new Markov process of the original time series {x _{t} k} A sequence can be generated.

The first value of s

[0096]

(Equation 22)

Is supplied to the neural network. The neural network mimics the conditional probability density each time. According to the so-called Monte Carlo method, new values according to the respective probability density

[0098]

[Outside 8]

Is formed and these values are set to a new time series

[0100]

[Outside 9]

Used as the new value of.

A new set of s of input values (s-Tupel)

[0103]

(Equation 23)

Is again supplied to the respective neural networks. This technique of feedback combining the newly generated values is repeated an arbitrary number of times. New time series

[0105]

[Outside 10]

Has the same distribution as the original time series {x _t } _k, and thus the surrogate time series

[0107]

[Outside 11]

Is processed to form Thus, by repeating this iteration, any number of surrogate time series of M can be determined in advance.

[0109]

[Outside 12]

Is generated (step 101).

Time series {x _t } _k and proxy time series

[0112]

[Outside 13]

Is subjected to the statistical test described in [6] (step 10).
2).

The null hypothesis defines that each dynamic system can be described sufficiently accurately by an s-dimensional Markov process of order (n ₁ ,..., _{N N} ).

For testing the null hypothesis, the so-called Student's t-test is used. In the Student's t-test, one surrogate measure for each surrogate time series

[0116]

[Outside 14]

Is obtained. Surrogate scale

[0118]

[Outside 15]

Is calculated in the same way as the calculation of the measure m _k (r) for the statistical dependence on the time series {x _t } _k .

The dependence of ^r is caused by the last component of the vector v ^{k, r} . That is, the statistical dependence between the last _nk value of each time series {x _t } _k and the value present at r steps in the future is measured.

Surrogate scale

[0122]

[Outside 16]

From the following equation:

[0124]

(Equation 24)

The surrogate average value according to

[0126]

[Outside 17]

And Surrogate Standard Deviation

[0128]

[Outside 18]

Is obtained.

The Student's t-test is performed according to the following equation:

[0131]

(Equation 25)

By forming the test value t _k (r), a value is determined which compares the proxy time series with the scale m _k (r) of the original time series {x _t } _k . This value is also called significance t _k (r). For example 1 ≦ r ≦ 10 If the holds, if the significance t _k for all r (r) is less than 1833 (M = 10), the null hypothesis is accepted.

For simplicity, the null hypothesis is that the time series {x _t } is the last n ₁ values of the time series {x _t } _{1, ... And} the time series {x _t } _N Are described by a multidimensional Markov process in which the last n _N values of are considered.

If the null hypothesis is accepted, the time series {x _t ｝ _k that is sought as a first group of time series that can be described by a Markov process of each order (n ₁ ,..., _{N N} ) Are classified (step 103) and the method ends.

If the null hypothesis is rejected, the order of the Markov process (n ₁ ,..., _{N N} ) is increased and the method is repeated starting with training of the neural network (step 104).

When there is no detailed knowledge about the dynamic system sought, it is started with the following start values: n ₁ = 1, n _j = 0 as j = 2 _,.

The number of time series to be searched is arbitrary. In the case of two time series, only two neural networks are required.

The following publications have been cited within the framework of this document:

[0139]

[Table 1]

[Brief description of the drawings]

FIG. 1 is a schematic diagram showing the present invention in individual elements.

FIG. 2 is a schematic diagram showing the course of a measurement signal obtained by converting a step width w in a time series by leading sampling.

FIG. 3 is a schematic diagram showing a computer for implementing the present invention.

[Procedure amendment]

[Submission date] June 14, 2000 (2000.6.14)

[Procedure amendment 1]

[Document name to be amended] Statement

[Correction target item name] Claims

[Correction method] Change

[Correction contents]

[Claims]

1. A method of classifying a plurality of time-series with a sample value of a physical signal pre-determined number, respectively, the computer is provided, at the computer, against a plurality of time-series by statistical estimator each model the nonlinear Markov process, to form a respective least <br/> Kutomo one surrogate time series by using the respective statistical estimator to said plurality of time-series Te, samples of the plurality of time series the first measure of the statistical dependence between the reciprocal values to form the sample values of a predetermined number of future a second measure of the statistical dependence between the reciprocal of the value of the proxy time series, in the future formed for a predetermined number of sample values positioned, the first to form a measure and difference scales from a second measure of, if the difference measure is less than a predetermined threshold, said plurality of time-series A method for classifying a plurality of time series, characterized in that the classification is performed by assigning the plurality of time series to a first group and otherwise assigning the plurality of time series to a second group.

Wherein the plurality of time series, forming a plurality of surrogate time series using the respective statistical estimators The method of claim 1, wherein.

3. The method according to claim 1, wherein a neural network is used as the statistical estimator.

4. The method according to claim 1, wherein a plurality of time series are processed so as to form a deformed Gaussian time series by obtaining a normally distributed random number and selecting the random numbers according to the rank of the time series. The method according to any one of claims 3 to 7.

5. The first scale is:

(Equation 1) The method according to any one of claims 1 to 4.

The method of if 6. plurality of time series is assigned to the second group, iteratively repeated until the plurality of time series is assigned to the first group, Markov process in each iteration 6. The method according to any one of claims 1 to 5 , wherein the order is increased.

7. Each of a plurality of time series representing the market course of the financial markets, any one process of claim 1 6.

8. Each of a plurality of time series representing the curve of the EEG, any one process of claim 1 6.

9. Each of a plurality of time series representing the curve of the electrocardiogram, any one process of claim 1 6.

10. A device for classifying a plurality of time-series with a sample value of a physical signal number which is determined in advance, respectively, the apparatus has a processor, such that the processor can be carried out the following steps is configured, i.e., each nonlinear Markov process for the plurality of time series by statistical estimator is modeled, respectively little by using the respective statistical estimator to said plurality of time-series < At least one surrogate time series is formed, and a first measure of the statistical dependence between the sample values of the plurality of time series is formed for a predetermined number of future sample values; the second measure of the statistical dependence between the reciprocal values of the time series is formed for a predetermined number of sample values located in the future, whether the first measure and the second measure of Differences scale is formed, if the difference measure is less than a predetermined threshold value, said plurality of time series is assigned to the first group, said plurality of time-series otherwise is assigned to the second group An apparatus for classifying a plurality of time series, wherein classification is performed by:

11. profile processor unit is configured as a plurality of surrogate time series using the respective integrated <br/> meter estimation device for a plurality of time series is formed, claims An apparatus according to claim 10 .

12. profile processor unit, the neural network is configured to be utilized as a statistical estimator, according to claim 10 or 11 Claims.

13. The processor unit is configured as follows, Sunawa Chi, Gaussian number turbulent sought is random numbers normally distributed is thus deformed to be sorted according to the rank of a time series of respectively a plurality of time series is configured to be respectively processed, any one process of claim 11 14 as time series is formed of.

14. The first measure is:

(Equation 2) Apparatus according to any one of claims 10 to 13 .

15. profile processor unit, in the case where a plurality of time series is assigned to the second group the methods, the plurality of time series is repeated iteratively until assigned to a first group, 15. The method according to claim 10 , wherein the order of the Markov process increases with each iteration.

16. profile processor unit, a plurality of time-series by being configured to market the course of the financial markets are expressed Apparatus according to any one of claims 10 15.

17. profile processor unit, a plurality of time-series by being configured to curve of the respective EEG is represented device according to any one of claims 10 15.

18. profile processor unit is configured to curve of the electrocardiogram <br/> views, respectively is represented by a plurality of time series, apparatus of any one of claims 10 15 .

──────────────────────────────────────────────────の Continued on the front page (72) Inventor Rosalia Siripo Florence Via O Vecchi 61 F-term (reference) 5B056 BB61 BB64

Claims (20)

[Claims] 1. A method for classifying a first time series and a second time series each having a predetermined number of sample values, comprising: providing a computer, wherein the computer uses a first statistical estimator. Model a nonlinear Markov process for a first time series, model a nonlinear Markov process for a second time series with a second statistical estimator, and use a first statistical estimator for the first time series Forming at least one surrogate time series, using the second statistical estimator for the second time series to form at least one surrogate time series, Forming a first measure of the statistical dependence on the sample values of the time series of 2 for a predetermined number of future sample values; The second measure is
Forming a difference measure from the first measure and the second measure for a predetermined number of sample values located in the future; if the difference measure is less than a predetermined threshold, the first time series and the second Time series
The first time series and the second time series are characterized in that the first time series and the second time series are classified by assigning the first time series and the second time series to the second group. How to classify. 2. The method according to claim 1, wherein a plurality of second time series are taken into account. 3. The method of claim 1, further comprising: forming a plurality of proxy time series for the first time series and the second time series using the first statistical estimator and the second statistical estimator, respectively. 1
Or the method of 2. 4. The method according to claim 1, wherein a neural network is used as the statistical estimator. 5. A modified Gaussian time series instead of the first time series and / or the second time series and / or at least one further time series,
The method according to any one of claims 1 to 4, wherein the time series is formed by obtaining a normally distributed random number and selecting the random number according to the rank of the time series. 6. The first measure is: A method according to any one of claims 1 to 5. 7. The method according to claim 1, wherein the first time series and the second time series are assigned to the first group when the first time series and the second time series are assigned to the second group. A method according to any of the preceding claims, wherein iterative repetition is performed until each iteration increases the order of the Markov process. 8. The method according to claim 1, wherein the first time series and the second time series represent a market progress of a financial market. 9. The method according to claim 1, wherein the electroencephalographic profile is represented by a first time series and a second time series. 10. The method according to claim 1, wherein the course of the electrocardiogram is represented by a first time series and a second time series. 11. An apparatus for classifying a first time series having a predetermined number of sample values and at least one second time series, each comprising a processor unit, wherein the processor unit comprises: A first statistical estimator models a non-linear Markov process for a first time series, a second statistical estimator models a non-linear Markov process for a second time series, At least one surrogate time series is formed by using one statistical estimator, and at least one surrogate time series is formed by using a second statistical estimator for the second time series; A first measure of the statistical dependence of the sample values of the time series of the second time series and the sample values of the second time series is formed for a predetermined number of future sample values; The second measure of the statistical dependencies between mutual values at representative time series,
A difference measure is formed from a predetermined number of sample values located in the future, and a difference measure is formed from the first measure and the second measure. Is the first time series
And the first and second time series are otherwise assigned to the second group.
An apparatus for classifying a first time series and at least one second time series, wherein the first time series and at least one second time series are classified. 12. The apparatus according to claim 11, wherein said processor unit is configured such that another time series is taken into account. 13. The processor unit is configured to form a plurality of surrogate time series using a statistical estimator for the first time series and the second time series, respectively. 13. The device according to 11 or 12. 14. The apparatus according to claim 11, wherein the processor unit is configured such that a neural network is used as a statistical estimator. 15. The processor unit, wherein a modified Gaussian time series is used instead of the first time series and / or the second time series and / or at least one further time series. 15. The method according to any of claims 11 to 14, wherein the sequence is formed such that normally distributed random numbers are determined and the random numbers are sorted according to the rank of the time series. 16. The first measure is: Apparatus according to any one of claims 11 to 15. 17. The method as claimed in claim 17, wherein the first time series and the second time series are assigned to a second group when the first time series and the second time series are assigned to a second group.
17. The method according to any one of claims 11 to 16, wherein the time series of is repeated iteratively until it is assigned to a first group, with each iteration increasing the order of the Markov process. 18. The processor unit according to claim 11, wherein the processor unit is configured such that a first time series and a second time series represent a market progress of a financial market.
An apparatus according to any one of claims 7 to 10. 19. The processor unit according to claim 11, wherein the processor unit is configured such that a course characteristic of the electroencephalogram is represented by a first time series and a second time series.
An apparatus according to any one of the preceding claims. 20. The processor unit according to claim 11, wherein the processor unit is configured so that a course characteristic of the electrocardiogram is represented by a first time series and a second time series.
An apparatus according to any one of the preceding claims.