Abstract:
A current first state, of a first temporal sequence of respective first states of a dynamically modifiable system, is determined. The first current state of the system is determined by combining a first system-inherent information flow comprising past system information of the system with a second system-inherent information flow comprising future system information in the first current state. The first current state is then determined from the combination.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS  
         [0001]    This application is based on and hereby claims priority to PCT Application No. PCT/DE02/03494 filed on Sep. 17, 2002 and German Application No. 101 46 222.0 filed on Sep. 19, 2001, the contents of which are hereby incorporated by reference.  
         BACKGROUND OF THE INVENTION  
         [0002]    The invention relates to determining a current first state of a first temporal sequence of respective first states of a dynamically modifiable system.  
           [0003]    It is known from S. Hayken, Neural Networks: A Comprehensive Foundation, McMillan College Publishing Company, Second Edition, ISBN 0-13-273350-1, pp. 732-789, 1999 (“the Hayken reference”) how to employ an arrangement for imaging temporally modifiable state descriptions to describe a dynamic system. The arrangement is implemented by interconnected computing elements used to effect the imaging.  
           [0004]    A dynamic system or, as the case may be, dynamic process is in general customarily described by a state transition description which is not visible to an observer of the dynamic process and by an output equation describing observable variables of the technical dynamic process.  
           [0005]    A relevant structure of a dynamic system of this type is shown in FIG. 2 a.    
           [0006]    The dynamic system  200  is subject to the influence of an external input variable u of pre-definable dimension, with an input variable u t  at a time t being designated u t :  
           [0007]    u t             
           [0008]    where 1 designates a natural number.  
           [0009]    The input variable u t  at a time t causes a modification to the dynamic process running in the dynamic system  200 .  
           [0010]    An internal state s t  (s t           ) of pre-definable dimension m at a time t is unobservable for an observer of the dynamic system  200 .  
           [0011]    A state transition of the internal state s t  of the dynamic process is caused as a function of the internal state s t  and of the input variable u t , and the state of the dynamic process changes to a follow-on state s t+1  at a following time t+1.  
           [0012]    The following applies here:  
           s t+1 =f(s t ,u t ),  (1) 
           [0013]    where f(.) designates a general imaging rule.  
           [0014]    An output variable y t  at a time t observable by an observer of the dynamic system  200  depends on the input variable u t  and on the internal state s t .  
           [0015]    The output variable y t  (y t  z, 3  ) is of a pre-definable dimension n.  
           [0016]    The dependency of output variable y t  on the input variable u t  and on the internal state s t  of the dynamic process is determined by the following general rule:  
           y t =g(s t ,u t ),  (2) 
           [0017]    where g(.) designates a general imaging rule.  
           [0018]    A system of interconnected computing elements in the form of a neural network of interconnected neurons is employed in the Hayken reference to describe the dynamic system  200 . The connections between the neurons of the neural network are weighted. The weights of the neural network are combined in a parameter vector v.  
           [0019]    Thus an internal state of a dynamic system which is subject to a dynamic process depends, according to the following rule, on the input variable u t  and the internal state of the preceding time s t , and on the parameter vector v:  
           s t+1 =NN(v,s t ,u t ),  (3) 
           [0020]    where NN(.) designates an imaging rule determined by the neural network.  
           [0021]    The arrangement known from the Hayken reference and referred to as a Time Delay Recurrent Neural Network (TDRNN) is trained in a training phase in such a way that a target variable y d   t  is in each case determined on a real dynamic system for an input variable u t . The tuple (input variable, determined target variable) is referred to as a training datum. A plurality of such training data form a training data record.  
           [0022]    Temporally succeeding tuples (u t−4 , y t−4   d ), (u t−3 , y t−3   d ), (u t−2 , y t−2   d ) of times (t−4, t−3, t−3, . . . ) of the training data record in each case have a pre-defined time step.  
           [0023]    The TDRNN is trained by the training data record. An overview of various training methods can also be found in the Hayken reference.  
           [0024]    It must be emphasized at this point that it is only possible to discern the output variable y t  at a time t of the dynamic system  200 : the “internal” system state s t  is unobservable.  
           [0025]    The following cost function E is customarily minimized in the training phase:  
               E   =         1   T            ∑     t   =   1     T            (       y   t     -     y   t   d       )     2         -&gt;     min     f   ,   g           ,           (   4   )                               
 
           [0026]    Where T designates the number of times taken into consideration.  
           [0027]    An overview of fundamentals of neural networks and the possible applications of neural networks in the area of the economy is furthermore given in H. Rehkugler and H. G. Zimmermann, Neuronale Netze in der Ökonomie, Grundlagen und finanzwirtschaftliche Anwendungen, published by Franz Vahlen, Munich, ISBN 3-8006-1871-0, pp. 3-90, 1994 (“the Rehkugler et al. reference”).  
           [0028]    The known arrangements and methods particularly have the disadvantage that a dynamic system or, as the case may be, process requiring to be described can only be described by them with insufficient accuracy. This is because the imaging employed in the case of the arrangements and methods is only able to simulate the state transition description of the dynamic process with insufficient accuracy.  
         SUMMARY OF THE INVENTION  
         [0029]    One possible object underlying the invention is accordingly to disclose a method and an arrangement for the computer-assisted imaging of temporally modifiable state descriptions enabling a state transition description of a dynamic system to be described with improved accuracy, with the disclosed arrangement and method not exhibiting the disadvantages of the known arrangements and methods. Theby  
           [0030]    In the method for determining a current first state of a first temporal sequence of respective first states of a dynamically modifiable system the following procedural steps are carried out in a first state space:  
           [0031]    a second temporal sequence of respective second states of the system is determined in a second state space, the second temporal sequence having at least one current second state and one older second state temporally preceding the current second state,  
           [0032]    a third temporal sequence of respective third states of the system is determined in the second state space, the third temporal sequence having at least one future third state and one younger third state temporally succeeding the future third state,  
           [0033]    the current first state is determined by a first transformation of the current second state from the second state space to the first state space and of a second transformation of the future third state from the second state space to the future state space.  
           [0034]    The arrangement for determining a current first state of a first temporal sequence of respective first states of a dynamically modifiable system in a first state space has interlinked computing elements, with the computing elements in each case representing a state of the system and with the links in each case representing a transformation between two states of the system, wherein  
           [0035]    first computing elements are set up in such a way that it is possible to determine a second temporal sequence of respective second states of the system in a second state space, the second temporal sequence having at least one current second state and one older second state temporally preceding the current second state,  
           [0036]    second computing elements are set up in such a way that it is possible to determine a third temporal sequence of respective third states of the system in the second state space, the third temporal sequence having at least one future third state and one younger third state temporally succeeding the future third state,  
           [0037]    a third computing element is set up in such a way that it is possible to determine the current first state by a first transformation of the current second state from the second state space to the first state space and of a second transformation of the future third state from the second state space to the future state space.  
           [0038]    The system is especially suitable for carrying out the methods or one of their developments explained below.  
           [0039]    Viewed in clear terms, the first current state of the system is determined by combining a first system-inherent information flow comprising past system information of the system with a second system-inherent information flow comprising future system information in the first current state and then determining the first current state from the combination.  
           [0040]    The further developments described below relate both to the methods and to the arrangement.  
           [0041]    The method and apparatus described below can be implemented both in software and in hardware form, for example using special electrical circuitry.  
           [0042]    The method and apparatusdescribed below, can also be implemented by a computer-readable storage medium on which is stored a computer program which carries out the described method.  
           [0043]    The method and apparatus, or a development thereof described below, can also be implemented by a computer program product having a storage medium on which is stored a computer program which carries out the described method.  
           [0044]    The inventors propose that two, temporally succeeding second states of the second temporal sequence are in each case coupled to each other by a third transformation.  
           [0045]    This coupling by the third transformation can be embodied in such a way that a temporally younger second state is determined from a temporally older second state.  
           [0046]    In an embodiment, two temporally succeeding third states of the third sequence can furthermore in each case be coupled to each other by a fourth transformation.  
           [0047]    This coupling by the fourth transformation can be embodied in such a way that a temporally older third state is determined from a temporally younger third state.  
           [0048]    The inventors propose that a younger second state of the second temporal sequence temporally succeeding the current second state is determined  
           [0049]    by the third transformation of the current second state and  
           [0050]    by a fifth transformation of the current state from the first state space to the second state space.  
           [0051]    It is also possible to determine a current third state of the third temporal sequence temporally preceding the future third state  
           [0052]    by the fourth transformation of the future third state, and  
           [0053]    by a sixth transformation of the current state from the first state space to the second state space.  
           [0054]    The accuracy of the description of a state transition description of a dynamic system can be improved by determining any error there may be between the determined first current state and a pre-specified current first state. Error determining of this type is referred to as “error correction”.  
           [0055]    The description of a state transition description can be improved if external state information of the system is in each case added to the second states of the second temporal sequence and/or to the third states of the third temporal sequence.  
           [0056]    Moreover, a state of the system can be described by a vector of pre-definable dimension.  
           [0057]    The method may be used in order to determine a dynamic characteristic of the dynamically modifiable system, with the first temporal sequence of the respective first states describing the dynamic characteristic.  
           [0058]    An instance of a dynamic characteristic of this type is that of an electrocardiogram, with the respective first temporal sequence of the respective first states being signals of an electrocardiogram.  
           [0059]    The dynamic characteristic can also be that of an economic system, with the first temporal sequence of the respective first states in this case being economic, macroeconomic, or microeconomic states described by a corresponding economic variable.  
           [0060]    The method may make it possible to determine the dynamic characteristic of a chemical reactor, with the first temporal sequence of the respective first states being described by chemical state variables of the chemical reactor.  
           [0061]    A further embodiment is used in order to predict a state of the dynamically modifiable system with, in this case, the determined first current state being used as the predicted state.  
           [0062]    A development provides for fourth computing elements which are in each case linked to a first computing element and/or to a second computing element and which are set up in such a way as to enable a fourth state of a fourth temporal sequence of respective fourth states of the system to be routed to, in each case, one of the fourth computing elements, with each fourth state containing external state information of the system.  
           [0063]    A further embodiment furthermore provides for embodying at least one part of the computing elements as artificial neurons and/or at least one part of the links between the computing elements on a variable basis.  
           [0064]    It is further possible to provide a measuring system for recording physical signals with which states of the dynamically modifiable system are described.  
           [0065]    Developments can also be used for processing speech.  
           [0066]    In a development of this type it is possible, for example, for  
           [0067]    the external state information to be a first item of speech information of a word and/or syllable and/or phoneme being spoken, and for  
           [0068]    the current first state to comprise a second item of speech information of the word and/or syllable and/or phoneme being spoken.  
           [0069]    It can also be a provision of a development of this type for  
           [0070]    the first item of speech information to include a classification of the word and/or syllable and/or phoneme being spoken and/or an item of pause information about the word and/or syllable and/or phoneme being spoken, and/or  
           [0071]    for the second item of speech information to include an item of articulation information about the word and/or syllable and/or phoneme being spoken and/or an item of sound length information about the word and/or syllable and/or phoneme being spoken.  
           [0072]    It is further possible in a development of this type for  
           [0073]    the first item of speech information to include an item of phonetic and/or structural information about the word and/or syllable and/or phoneme being spoken, and/or  
           [0074]    for the second item of speech information to include an item of frequency information about the word and/or syllable and/or phoneme being spoken and/or a duration of sound length of the word and/or syllable and/or phoneme being spoken. 
       
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0075]    These and other objects and advantages of the present invention will become more apparent and more readily appreciated from the following description of the preferred embodiments, taken in conjunction with the accompanying drawings of which:.  
         [0076]    [0076]FIG. 1 is a sketch of an arrangement according to a first exemplary embodiment (KRKNN);  
         [0077]    [0077]FIGS. 2 a  and  2   b  are a first sketch of a general description of a dynamic system and a second sketch of a description of a dynamic system which is based on a “causal retro-causal” relationship;  
         [0078]    [0078]FIG. 3 shows an arrangement according to a second exemplary embodiment (KRKFKNN);  
         [0079]    [0079]FIG. 4 is a sketch of a chemical reactor by which variables are measured which are further processed using the arrangement according to the first exemplary embodiment;  
         [0080]    [0080]FIG. 5 is a sketch of an arrangement of a TDRNN, the arrangement being developed over time with a finite number of states;  
         [0081]    [0081]FIG. 6 is a sketch of a traffic control system modeled using the arrangement within the framework of a second exemplary embodiment;  
         [0082]    [0082]FIG. 7 is a sketch of an alternative arrangement according to a first exemplary embodiment (KRKNN with released connections);  
         [0083]    [0083]FIG. 8 is a sketch of an alternative arrangement according to a second exemplary embodiment (KRKFKNN with released connections);  
         [0084]    [0084]FIG. 9 is a sketch of an alternative arrangement according to a first exemplary embodiment (KRKNN);  
         [0085]    [0085]FIG. 10 is a sketch of a speech processing process using an arrangement according to a first exemplary embodiment (KRKNN);  
         [0086]    [0086]FIG. 11 is a sketch of a speech processing process using an arrangement according to a second exemplary embodiment (KRKFKNN). 
     
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT  
       [0087]    Reference will now be made in detail to the preferred embodiments of the present invention, examples of which are illustrated in the accompanying drawings, wherein like reference numerals refer to like elements throughout.  
         [0088]    First Exemplary embodiment: Chemical reactor  
         [0089]    [0089]FIG. 4 shows a chemical reactor  400  filled with a chemical substance  401 . Chemical reactor  400  includes a mixer  402  by which chemical substance  401  is mixed. Other chemical substances  403  flowing into chemical reactor  400  react during a pre-definable period of time in chemical reactor  400  with chemical substance  401  already contained in chemical reactor  400 . A substance  404  flowing out of chemical reactor  400  is let off from chemical reactor  400  via an output.  
         [0090]    Mixer  402  is connected via a lead to a control unit  405  by which a mixing frequency of mixer  402  can be set via a control signal  406 .  
         [0091]    Provision is furthermore made for a measuring device  407  by which concentrations of chemical substances contained in chemical substance  401  are measured.  
         [0092]    Measuring signals  408  are routed to a computer  409 , digitized in the computer via an input/output interface  410  and an analog/digital converter  411 , and stored in a memory  412 . A processor  413  is connected, as is memory  412 , to analog/ digital converter  411  via a bus  414 . Computer  409  is furthermore connected via input/output interface  410  to control unit  405  of mixer  402 , and computer  409  thus controls the mixing frequency of mixer  402 .  
         [0093]    Computer  409  is furthermore connected via input/output interface  410  to a keyboard  415 , a computer mouse  416 , and a monitor screen  417 .  
         [0094]    Chemical reactor  400  is thus subject as a dynamic technical system  250  to a dynamic process.  
         [0095]    Chemical reactor  400  is described by a state description. An input variable u t  of the state description in this case comprises details of the temperature prevailing in chemical reactor  400 , the pressure prevailing in chemical reactor  400 , and the mixing frequency set at time t. Input variable u t  is thus a three-dimensional vector.  
         [0096]    The aim of modeling, described below, of chemical reactor  400  is to determine the dynamic development of substance concentrations in order to enable efficient production of a pre-definable target substance flowing out as substance  404 .  
         [0097]    This is done using the arrangement which is described below and shown in FIG. 1.  
         [0098]    The dynamic process underlying the described reactor  400  and having what is termed a “causal retro-causal” relationship is described by a state transition description which is not visible to an observer of the dynamic process and by an output equation describing observable variables of the technical dynamic process.  
         [0099]    A structure of the type of a dynamic system having a “causal retro-causal” relationship is shown in FIG. 2 b.    
         [0100]    Dynamic system  250  is subject to the influence of an external input variable u of pre-definable dimension, with an input variable u t  at a time t being designated u t :  
         [0101]    u t             
         [0102]    where 1 designates a natural number.  
         [0103]    The input variable u t  at a time t causes a modification to the dynamic process running in the dynamic system  250 .  
         [0104]    An internal state of system  250  at a time t, which is a state that is unobservable for an observer of system  250 , in this case comprises a first internal partial state s t  and a second internal partial state r t .  
         [0105]    A state transition of the first internal partial state s t−1  of the dynamic process to a follow-on state s t  is caused as a function of the first internal partial state s t−1  at an earlier time t−1 and of the input variable u t .  
         [0106]    The following applies here:  
         s t =f1(s t−1 ,u t ),  (5) 
         [0107]    where f1(.) designates a general imaging rule.  
         [0108]    Viewed in clear terms, the first internal partial state s t  is influenced by an earlier first internal partial state s t−1  and by input variable u t . A relationship of this type is usually referred to as “causality”.  
         [0109]    A state transition of the first internal state r t+1  of the dynamic process to a follow-on state r t  is caused as a function of the second internal partial state r t+1  at a succeeding time t+1 and of input variable u t .  
         [0110]    The following applies here:  
         r t =f2(r t+1 ,u t ),  (6) 
         [0111]    where f2(.) designates a general imaging rule.  
         [0112]    Viewed in clear terms, in this case the second internal partial state r t  is influenced by a later second internal partial state r t+1 , generally, therefore, by an expectation about a later state of dynamic system  250 , and by input variable u t . A relationship of this type is referred to as “retro-causality”.  
         [0113]    An output variable y t  at a time t, which is a variable that is observable for an observer of dynamic system  250 , therefore depends on the input variable u t , the first internal partial state s t , and the second internal partial state r t+1 .  
         [0114]    Output variable y t  (y t            ) is of a pre-definable dimension n.  
         [0115]    The dependency of output variable y t  on input variable u t , the first internal partial state s t , and the second internal partial state r t+1  of the dynamic process is stated by the following general rule:  
         y t =g(s t ,r t+1 ),  (7) 
         [0116]    where g(.) designates a general imaging rule.  
         [0117]    An arrangement of interconnected computing elements in the form of a neural network of interconnected neurons is employed to describe dynamic system  250  and its states. The network is shown in FIG. 1 and is referred to as a “causal retro-causal” neural network (KRKNN).  
         [0118]    The connections between the neurons of the neural network are weighted. The weights of the neural network are combined in a parameter vector v.  
         [0119]    In the neural network, the first internal partial state s t  and the second internal partial state r t  depend, according to the following rules, on input variable u t , the first internal partial state s t−1 , the second internal partial state r t+1 , and parameter vectors v s , v t , v y :  
         s t =NN(v s ,s t−1 ,u t ),  (8) 
         r t =NN(v r ,r t+1 ,u t ),  (9) 
         y t =NN(v y ,s t ,r t ),  (10) 
         [0120]    where NN(.) designates a general imaging rule specified by the neural network.  
         [0121]    KRKNN  100  according to FIG. 1 is a neural network developed across four times t−1, t, t+1, and t+2.  
         [0122]    Essential features of a neural network developed across a finite number of times are described in the Hayken reference.  
         [0123]    To make it easier to understand the principles underlying the KRKNN, FIG. 5 shows the known TDRNN as a neural network  500  developed across a finite number of times.  
         [0124]    Neural network  500  shown in FIG. 5 has an input layer  501  with three partial input layers  502 ,  503 , and  504  each containing a pre-definable number of input computing elements to which at a pre-definable time t input variables u t , which is to say temporal sequence values described below, can be applied.  
         [0125]    Input computing elements, which is to say input neurons, are connected via variable connections to neurons of a pre-definable number of concealed layers  505 .  
         [0126]    Neurons of a first concealed layer  506  are herein connected to neurons of the first partial input layer  502 . Neurons of a second concealed layer  507  are furthermore connected to neurons of the second input layer  503 . Neurons of a third concealed layer  508  are connected to neurons of the third partial input layer  504 .  
         [0127]    The connections between the first partial input layer  502  and the first concealed layer  506 , the second partial input layer  503  and the second concealed layer  507 , and the third partial input layer  504  and the third concealed layer  508  are the same in each case. The weights of all connections are in each case contained in a first connection matrix B.  
         [0128]    Neurons of a fourth concealed layer  509  are connected by their inputs to outputs of neurons of the first concealed layer  506  according to a structure provided by a second connection matrix A 2 . Outputs of the neurons of the fourth concealed layer  509  are furthermore connected to inputs of neurons of the second concealed layer  507  according to a structure provided by a third connection matrix A 1 .  
         [0129]    Neurons of a fifth concealed layer  510  are furthermore connected by their inputs to outputs of neurons of the second concealed layer  507  according to a structure provided by the third connection matrix A 2 . Outputs of the neurons of the fifth concealed layer  510  are connected to inputs of neurons of the third concealed layer  508  according to a structure provided by the third connection matrix A 1 .  
         [0130]    This type of connection structure applies in equivalent terms to inputs of a sixth concealed layer  511  which, according to a structure provided by the second connection matrix A 2 , are connected to outputs of the neurons of the third concealed layer  508  and, according to a structure provided by the third connection matrix A1, are connected to neurons of a seventh concealed layer  512 .  
         [0131]    Neurons of an eighth concealed layer  513  are in turn connected according to a structure provided by the first connection matrix A 2  to neurons of the seventh concealed layer  512  and, via connections according to the third connection matrix A 1 , to neurons of a ninth concealed layer  514 . The information contained in the indices in the respective layers in each case indicates the time t, t−1, t−2, t+1, t+2 to which in each case the signals which can be tapped at or, as the case may be, routed to the outputs of the respective layer relate (u t , u t−1 , u t−2 ).  
         [0132]    An output layer  520  has three partial output layers, a first partial output layer  521 , a second partial output layer  522 , and a third partial output layer  523 . Neurons of the first partial output layer  521  are connected according to a structure provided by an output connection matrix C to neurons of the third concealed layer  508 . Neurons of the second partial output layer are likewise connected according to a structure provided by the output connection matrix C to neurons of the eighth concealed layer  512 . Neurons of the third partial output layer  523  are connected according to output connection matrix C to neurons of the ninth concealed layer  514 . The output variables for in each case a time t, t+1, t+2 (y t , Y t+1 , y t+2 ) can be tapped at the neurons of partial output layers  521 ,  522 , and  523 .  
         [0133]    Proceeding from this what is termed the ‘shared weights’ principle, which is to say the principle that equivalent connection matrices in a neural network have the same weights at a respective time, the arrangement shown in FIG. 1 will be explained below in the formed condition.  
         [0134]    The sketches described below are each to be understood such that each layer or, as the case may be, each partial layer has a pre-definable number of neurons, which is to say computing elements.  
         [0135]    Partial layers of a layer each represent a system state of the dynamic system described by the arrangement. Partial layers of a concealed layer accordingly each represent an “internal” system state.  
         [0136]    The respective connection matrices are of any dimension and each contain the weight values applying to the relevant connections between the neurons of the respective layers.  
         [0137]    The connections are directional and identified in FIG. 1 by arrows. An arrow direction indicates a “computing direction”, in particular an imaging direction or a transformation direction.  
         [0138]    The arrangement shown in FIG. 1 has an input layer  100  with four partial input layers  101 ,  102 ,  103 , and  104 , with the possibility of routing in each case temporal sequence values u t−1 , u t , u t+1 , u t+2  at in each case a time t−1, t, t+1 or, as the case may be, t+2 to each the partial input layer  101 ,  102 ,  103 ,  104 .  
         [0139]    The partial input layers  101 ,  102 ,  103 ,  104  of input layer  100  are in each case connected via connections according to a first connection matrix A to neurons of a first concealed layer  110  with in each case four partial layers  111 ,  112 ,  113 , and  114  of the first concealed layer  110 .  
         [0140]    The partial input layers  101 ,  102 ,  103 ,  104  of input layer  100  are additionally in each case connected via connections according to a second connection matrix B to neurons of a second concealed layer  120  with in each case four partial layers  121 ,  122 ,  123 , and  124  of the second concealed layer  120 .  
         [0141]    The neurons of the first concealed layer  110  are in each case connected according to a structure provided by a third connection matrix C to neurons of an output layer  140 , which in its turn has four partial output layers  141 ,  142 ,  143 , and  144 .  
         [0142]    The neurons of the output layer  140  are in each case connected according to a structure provided by a fourth connection matrix D to the neurons of the second concealed layer  120 .  
         [0143]    The neurons of output layer  140  are also in each case connected according to a structure provided by an eighth connection matrix G to the neurons of the first concealed layer  110 .  
         [0144]    The neurons of the second concealed layer  120  are furthermore in each case connected according to a structure provided by a seventh connection matrix H to the neurons of the output layer  140 .  
         [0145]    Moreover, partial layer  111  of the first concealed layer  110  is connected via a connection according to a fifth connection matrix E to the neurons of partial layer  112  of the first concealed layer  110 .  
         [0146]    All other partial layers  112 ,  113 , and  114  of the first concealed layer  110  also have corresponding connections.  
         [0147]    Viewed in clear terms, this means all partial layers  111 ,  112 ,  113 , and  114  of the first concealed partial layer  110  are interconnected according to their temporal sequence t−1, t, t+1, and t+2.  
         [0148]    Partial layers  121 ,  122 ,  123 , and  124  of the second concealed layer  120  are at this particular time interconnected in opposite directions.  
         [0149]    In this case, partial layer  124  of the second concealed layer  120  is connected via a connection according to a sixth connection matrix F to the neurons of partial layer  123  of the second concealed layer  120 .  
         [0150]    All other partial layers  123 ,  122 , and  121  of the second concealed layer  120  also have corresponding connections.  
         [0151]    Viewed in clear terms, all partial layers  121 ,  122 ,  123 , and  124  of the second concealed partial layer  120  are in this case interconnected counter to their temporal sequence, therefore t+2, t+1, t, and t−1.  
         [0152]    According to the connections described, an “internal” system state s t , s t+1  or, as the case may be, S t+2  of partial layer  112 ,  113  or, as the case may be,  114  of the first concealed layer is formed in each case from the associated input state u t , u t+1  or, as the case may be, u t+2 , from the temporally preceding output state y t−1 , y t  or, as the case may be, y t , and from the temporally preceding “internal” system state s t−1 , s t  or, as the case may be, s t .  
         [0153]    Furthermore, according to the connections described, an “internal” system state r t−1 , r t  or, as the case may be, r t+1  of partial layer  121 ,  122  or, as the case may be,  123  of the second concealed layer  120  is formed in each case from the associated output state y t−1 , y t  or, as the case may be, y t+1 , from the associated input state u t−1 , u t  or, as the case may be, u t+1 , and from the temporally succeeding “internal” system state r t , r t+1  or, as the case may be, r t+2 .  
         [0154]    In partial output layers  141 ,  142 ,  143 , and  144  of output layer  140  a state is in each case formed from the associated “internal” system state s t−1 , s t , s t+1  or, as the case may be, S t+2  of a partial layer  111 ,  112 ,  113  or, as the case may be,  114  of the first concealed layer  110 , and from the temporally preceding “internal” system state r t , r t+1 , r t+2  or, as the case may be, r t+3  (not shown) of a partial layer  122 ,  123  or, as the case may be,  124  of the second concealed layer  120 .  
         [0155]    At an output of first partial output layer  141  of output layer  140  it is therefore possible to tap a signal which is dependent on the “internal” system states (s t ,r t ).  
         [0156]    The same applies analogously to partial output layers  142 ,  143 , and  144 .  
         [0157]    The following cost function E is minimized during the training phase of the KRKNN:  
               E   =         1   T            ∑     t   =   1     T            (       y   t     -     y   t   d       )     2         -&gt;     min     f   ,   g           ,           (   11   )                               
 
         [0158]    Where T designates the number of times taken into consideration.  
         [0159]    A back-propagation method is employed as the training method. The training data record is obtained in the following manner from chemical reactor  400 .  
         [0160]    Concentrations at defined input variables are measured with measuring device  407  and routed to computer  409 , where they are digitized and grouped in a memory into temporal sequence values x t  together with the relevant input variables corresponding to the measured variables.  
         [0161]    The weight values of the respective connection matrices are matched during training. In clear terms, matching takes place in such a way that the KRKNN describes the dynamic system simulated by it, in this case the chemical reactor, as accurately as possible.  
         [0162]    The arrangement from FIG. 1 is trained using the training data record and the cost function E.  
         [0163]    The arrangement from FIG. 1 trained according to the above described training method is used to control and monitor chemical reactor  400 . For this, a predicated output variable y t+1  is determined from input variables u t−1 , u t . The output variable is then routed as a control variable, where applicable after any editing that may be required, to control unit  405  for controlling mixer  402  and to control equipment  430  for controlling the feed flow (see also FIG. 4).  
         [0164]    Second exemplary embodiment: Predicting a rental price  
         [0165]    [0165]FIG. 3 shows a development of the KRKNN which is shown in FIG. 1 and described within the framework of the above embodiments.  
         [0166]    The developed KRKNN shown in FIG. 3, what is termed a causal retro-causal error-correcting neural network (KRKFKNN), is used for predicting a rental price.  
         [0167]    The input variable u t  in this case comprises details of a rental price, an offer of accommodation space, an inflation figure, and an unemployment rate, the details which relate to a residential area under examination being determined in each case at the end of the calendar year (December values). The input variable is thus a four-dimensional vector. A temporal sequence of the input variables including a plurality of temporally succeeding vectors has time steps of in each case one year.  
         [0168]    The aim of modeling the establishment of a rental price, as described below, is to predict a future rental price.  
         [0169]    The arrangement described below and shown in FIG. 3 is used to provide a description of the dynamic process of establishing a rental price.  
         [0170]    Components from FIG. 1 are given the same reference numerals where the embodiment is the same.  
         [0171]    The KRKFKNN additionally has a second input layer  150  with four partial input layers  151 ,  152 ,  153 , and  154 , with the possibility of routing in each case temporal sequence values y t−1   d , y t   d , y t+1   d , y t+2   d  at in each case a time t−1, t, t+1 or, as the case may be, t+2 to each the partial input layer  151 ,  152 ,  153 ,  154 . The temporal sequence values y t−1   d , y t   d , y t+1   d , y t+2   d  are herein output values measured on the dynamic system.  
         [0172]    Partial input layers  151 ,  152 ,  153 ,  154  of input layer  150  are in each case connected via connections according to a ninth connection matrix, which is a negative identity matrix, to neurons of output layer  140 .  
         [0173]    A differential state (y t−1 -y t−1   d ), (y t -y t   d ), (y t+1 -y t+1   d ), and (y t+2 -y t+2   d ) is thus formed in each case in partial output layers  141 ,  142 ,  143 , and  144  of the output layer.  
         [0174]    The procedure for training the above described arrangement corresponds to that for training the arrangement according to the first exemplary embodiment.  
         [0175]    Third exemplary embodiment: Traffic modeling and tailback predicting  
         [0176]    A third exemplary embodiment described below describes an instance of traffic modeling and is used to predict tailbacks.  
         [0177]    The arrangement according to the first exemplary embodiment is used in the third exemplary embodiment (see also FIG. 1).  
         [0178]    The third exemplary embodiment differs, however, in each case from the first exemplary embodiment and the second exemplary embodiment in that the variable t originally employed as a time variable is in this case employed as a location variable t.  
         [0179]    An original description of a state at time t thus in the third exemplary embodiment describes a state at a first location t. The same applies in each case analogously to a state description at a time t−1 or, as the case may be, t+1 or, as the case may be, t+2.  
         [0180]    From the analog transfer of the time variability to a location variability, it furthermore ensues that locations t−1, t, t+1, and t+2 are arranged consecutively along a travel route in a pre-defined direction of travel.  
         [0181]    [0181]FIG. 6 shows a road  600  along which automobiles  601 ,  602 ,  603 ,  604 ,  605 , and  606  are driving.  
         [0182]    Conductor loops  610 ,  611  integrated in road  600  register electrical signals in a known manner and route the electrical signals  615 ,  616  to a computer  620  via an input/output interface  621 . The electrical signals are digitized into a temporal sequence in an analog/digital converted  622 , which is linked to input/output interface  621 , and stored in a memory  623  connected via a bus  624  to analog/digital converter  622  and to a processor  625 . Control signals  951  from which a pre-defined specified speed  652  can be set in a traffic control system  650 , or other details of traffic regulations which are displayed to drivers of the vehicles  601 ,  602 ,  603 ,  604 ,  605 , and  606  via the traffic control system  650 , are routed to the traffic control system  650  via input/output interface  621 .  
         [0183]    The following local state variables are used in this case for traffic modeling:  
         [0184]    Speed of traffic flow v,  
         [0185]    Vehicle density p  
       (     ρ   =     number                 of                 vehicles                 per                 kilometer                   Fz   km         )                         
 
         [0186]    Traffic flow q  
         (       q   =     number                 of                 vehicles                 per                 hour                   Fz   h         ,     (     q   =     v   *   p       )       )     ,                         
 
         [0187]    (q=v*p)), and  
         [0188]    Speed restrictions  952  displayed at one time in each case by traffic control system  950 .  
         [0189]    The local state variables are measured as described above using conductor loops  610 ,  611 .  
         [0190]    These variables (v(t), p(t), q(t)) thus represent a state of the technical “traffic” system at a specific time t. From the variables an assessment r(t) takes place of, in each case, a current state in terms of traffic flow and homogeneity, for instance. The assessment can be on either a quantitative or a qualitative basis.  
         [0191]    The traffic dynamic is modeled in two phases within the framework of this exemplary embodiment:  
         [0192]    From prediction variables determined during the application phase, control signals  651  are formed by which it is indicated which speed restriction should be selected for a future time period (t+1).  
         [0193]    Alternatives to the exemplary embodiments  
         [0194]    Some alternatives to the above described exemplary embodiments are presented below.  
         [0195]    Alternative areas of application:  
         [0196]    The arrangement described in the first exemplary embodiment can also be used to determine a dynamic characteristic of an electrocardiogram (ECG). This will facilitate the early detection of indicators pointing to an increased risk of heart attack. A temporal sequence comprising ECG values measured on a patient is used as an input variable.  
         [0197]    In a further alternative to the first exemplary embodiment an arrangement according to the first exemplary embodiment is used for an instance of traffic modeling according to the third exemplary embodiment.  
         [0198]    The variable t originally used (in the first exemplary embodiment) as a time variable is in this case used, as described within the framework of the third exemplary embodiment, as a location variable t.  
         [0199]    The explanations about this given for the third exemplary embodiment apply analogously.  
         [0200]    In a third alternative to the first exemplary embodiment the arrangement according to the first exemplary embodiment is used within the framework of speech processing (FIG. 10). Basic principles of speech processing of this type are known from J. Hirschberg, Pitch accent in context: predicting intonational prominence from text, Artificial Intelligence 63, pp. 305-340, Elsevier, 1993 (“the Hirschberg reference”).  
         [0201]    The arrangement (KRKNN)  1000  is employed in this case in order to determine the articulation in a sentence  1010  being articulated.  
         [0202]    For this, the sentence  1010  being articulated is broken down into its component words  1011  and these are each classified  1012  (part-of-speech tagging). The classifications  1012  are each coded  1013 . Each code  1013  is extended to include pause information  1014  (phrase-break information) in each case indicating whether a pause is made after the respective word when sentence  1010  being articulated is spoken.  
         [0203]    This type of coding of a sentence being articulated is known from the Hirschberg reference and K. Ross et al., Prediction of abstract prosodic labels for speech synthesis, Computer Speech and Language, 10, pp. 155-185,1996 (“the Ross et al. reference”).  
         [0204]    A temporal sequence  1016  is formed from the extended codes  1015  of the sentence in such a way that a temporal sequence of states of the temporal sequence corresponds to the succession of words in the sentence  1010  being articulated. The temporal sequence  1016  is applied to arrangement  1000 .  
         [0205]    For each word  1011  the arrangement then determines articulation information  1020  (HA: main stress or, as the case may be, strongly articulated; NA: secondary stress or, as the case may be, weakly articulated; KA: no stress or, as the case may be, not articulated) indicating whether the relevant word is spoken with an articulation.  
         [0206]    The explanations about this given for the first exemplary embodiment apply analogously.  
         [0207]    The arrangement described in the second exemplary embodiment can also be used in an alternative embodiment to predict a macroeconomic dynamic characteristic, for example the course of an exchange rate, or other economic parameters including, for instance, those of a stock exchange quotation. With a prediction of this type an input variable is formed from temporal sequences of relevant macroeconomic or, as the case may be, economic parameters such as interest rates, currencies or inflation rates.  
         [0208]    In a further alternative to the second exemplary embodiment the arrangement according to the second exemplary embodiment is employed within the framework of speech processing (FIG. 11). Basic principles of speech processing of this type are known from R. Haury et al., Optimisation of a Neural Network for Pitch Contour Generation, ICASSP, Seattle, 1998 (“the Haury et al. reference”), C. Traber, FO generation with a database of natural FO patterns and with a neural network, G. Bailly and C. Benoit eds., Talking Machines: Theories, Models and Applications, Elsevier, 1992 (“the Traber reference”), E. Heuft et al., Parametric Description of FO-Contours in a Prosodic Database, Proc. ICPHS, Vol. 2, pp. 378-381, 1995 (“the Heuft et al. reference”), and C. Erdem, Topologieoptimierung eines Neuronalen Netzes zur Generierung von FO-Verlaeufen durch Integration unterschiedlicher Codierungen, Tagungsband ESSV, Cottbus, 2000 (“the Erdem reference”).  
         [0209]    In this case, namely of syllable-based speech processing, arrangement (KRKFKNN)  1100  is employed for modeling the frequency contour of a syllable of a word in a sentence.  
         [0210]    Modeling of this type is also known from the Haury et al. reference, the Traber reference, the Heuft et al. reference, and the Erdem reference.  
         [0211]    For this, the sentence  1110  being modeled is broken down into syllables  1111 . For each syllable a state vector  1112  is determined which describes the syllable phonetically and structurally.  
         [0212]    A state vector  1112  of this type contains timing information  1113 , phonetic information  1114 , syntax information  1115 , and stress information  1116 .  
         [0213]    A state vector  1112  of this type is described in the Ross et al. reference.  
         [0214]    A temporal sequence  1117  is formed from state vectors  1112  of syllables  1111  of the sentence being modeled in such a way that a temporal sequence of states of the temporal sequence  1117  corresponds to the succession of syllables  1111  in the sentence  1110  being modeled. The temporal sequence  1117  is applied to arrangement  1100 .  
         [0215]    Arrangement  1100  then determines for each syllable  1111  a parameter vector  1122  with parameters  1120 , fomaxpos, fomaxalpha, lp, rp, describing frequency contour  1121  of the respective syllable  1111 .  
         [0216]    Parameters  1120  of this type and the description of a frequency contour  1121  by the parameters  1120  are known from the Haury et al. reference, the Traber reference, the Heuft et al. reference, and the Erdem reference.  
         [0217]    The explanations about this given for the second exemplary embodiment apply analogously.  
         [0218]    Structural Alternatives  
         [0219]    [0219]FIG. 7 shows a structural alternative to the arrangement from FIG. 1 according to the first exemplary embodiment.  
         [0220]    Components from FIG. 1 are given the same reference numerals in FIG. 7 where the embodiment is the same.  
         [0221]    In contrast to the arrangement shown in FIG. 1, in the alternative arrangement according to FIG. 7 connections  701 ,  702 ,  703 ,  704 ,  705 ,  706 ,  707 ,  708 ,  709 ,  710 , and  711  have been released or, as the case may be, interrupted.  
         [0222]    The alternative arrangement, namely a KRKNN with released connections, can be used in both a training phase an application phase.  
         [0223]    The alternative arrangement is trained and applied in a manner analogous to that described for the first exemplary embodiment described.  
         [0224]    [0224]FIG. 8 shows a structural alternative to the arrangement from FIG. 3 according to the second exemplary embodiment.  
         [0225]    Components from FIG. 3 are given the same reference numerals in FIG. 8 where the embodiment is the same.  
         [0226]    In contrast to the arrangement shown in FIG. 3, in the alternative arrangement according to FIG. 8 connections  801 ,  802 ,  803 ,  804 ,  805 ,  806 ,  807 ,  808 ,  809 ,  810 ,  811 ,  812 , and  813  have been released or, as the case may be, interrupted.  
         [0227]    The alternative arrangement, namely a KRKFKNN with released connections, can be used in both a training phase an application phase.  
         [0228]    The alternative arrangement is trained and applied in a manner analogous to that described for the second exemplary embodiment described.  
         [0229]    It must be noted that it is possible to use the KRKNN with released connections only in the training phase and the KRKNN (without the released connections according to the first exemplary embodiment) in the application phase.  
         [0230]    It is also possible to use the KRKNN with released connections only in the application phase and the KRKNN (without the released connections according to the first exemplary embodiment) in the training phase.  
         [0231]    The same applies analogously to the KRKFKNN and the KRKFKNN with released connections.  
         [0232]    A further structural alternative to the arrangement according to the first exemplary embodiment is shown in FIG. 9.  
         [0233]    The arrangement according to FIG. 9 is a KRKNN with fixed-point recurrence.  
         [0234]    Components from FIG. 1 are given the same reference numerals in FIG. 8 where the embodiment is the same.  
         [0235]    In contrast to the arrangement shown in FIG. 1, in the alternative arrangement according to FIG. 9 additional connections  901 ,  902 ,  903 , and  904  are closed.  
         [0236]    Additional connections  901 ,  902 ,  903 , and  904  each have a connection matrix GT with weights.  
         [0237]    The alternative arrangement can be used in both a training phase and an application phase.  
         [0238]    The alternative arrangement is trained and applied in a manner analogous to that described for the first exemplary embodiment.  
         [0239]    The invention has been described in detail with particular reference to preferred embodiments thereof and examples, but it will be understood that variations and modifications can be effected within the spirit and scope of the invention.