Patent Publication Number: US-9405795-B2

Title: Stream data processing server and a non-transitory computer-readable storage medium storing a stream data processing program

Description:
BACKGROUND OF THE INVENTION 
     This invention relates to stream data analyzing processing. More particularly, this invention relates to processing of analyzing stream data with the use of an approximate expression. 
     In recent years, along with the increase in the amount of data to be processed, stream data processing systems that allow real-time data compilation and real-time data analysis are attracting attention. Stream data processing systems process stream data, which is a string of time-series data arriving consecutively. 
     Stream data processing systems execute data processing in accordance with a query defined in advance. A query is a scenario indicating data to be processed and the specifics of the processing, and is written in Continuous Query Language (CQL). 
     Stream data keeps arriving consecutively without a break, which makes it necessary to extract data about which computation is performed. Processing of analyzing stream data therefore uses sliding window in order to cut a finite data set out of stream data. 
     There are roughly two types of sliding window, specifically, count-based sliding window for holding n pieces of time-series data that precede a processing target time, and time-based sliding window for holding n hours of time-series data that precede a processing target time. 
     By using sliding window, for example, count-based sliding window, n pieces of input information preceding an arbitrary time can be compiled and analyzed in substantially real time. Stream data processing systems therefore enable one to analyze the state at the current time and deal with a future data change that is predicted. 
     In stream data processing systems, a computer that processes stream data uses sliding window to cut out time-series data, and analyzes the relation between a time and a target value (metrics) with respect to the cut out time-series data. This computer calculates a time-metrics relational expression (approximate expression) as the result of the analysis. A future change in value can thus be predicted. 
     The least square method is known as a method of calculating a relational expression of the relation between a time and a target value. For example, in the case of using count-based sliding window for extracting n pieces of time-series data to approximate the relation between a time x i  and metrics y i  with a linear expression “y=ax+b”, the values of the coefficients a and b are respectively calculated by Expression (1) and Expression (2), where i is a natural number indicating the place in the order of the time-series data. 
     
       
         
           
             
               
                 
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     Expression (1) and Expression (2) are solutions of an equation expressed as Expression (3). 
     
       
         
           
             
               
                 
                   
                     
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     SUMMARY OF THE INVENTION 
     In stream data processing systems where time-series data is processed continuously, when a long period of time elapses, in other words, when the value of the time x is large, the value of each term (the respective sums and products thereof) in Expression (1) and Expression (2) is large. Consequently, the computer undergoes a digit overflow in integer representation. Use of a floating point in order to prevent a digit overflow causes digit cancellation in turn. 
     An object of this invention is to provide a stream data processing system capable of calculating an approximate expression at a lower calculation cost while preventing a digit overflow even when the time has a large value. 
     The present invention can be appreciated by the description which follows in conjunction with the following figures, wherein: a stream data processing server for processing stream data, which arrives consecutively from a computer system as a monitoring target, comprising: a processor; a memory coupled to the processor; a storage medium coupled to the processor; and an interface coupled to the processor, for coupling to other devices. The stream data including a plurality of pieces of data to which time stamps are added. The stream data processing server further comprising a stream data processing module for cutting the plurality of pieces of data that are included within a processing range as a target out of the stream data by using a sliding window in accordance with a query registered in advance, and executing analyzing processing for the cut out plurality of pieces of data. The stream data processing module having: an approximate expression calculating module for calculating, by using the cut out plurality of pieces of data, an approximate expression that indicates an association relation between the time stamps and values of the plurality of pieces of data; and an anomaly detecting module for calculating predicted values of the plurality of pieces of data by using the calculated approximate expression, and predicting an anomaly in the computer system based on the calculated predicted values. The approximate expression calculating module being configured to: determine one of the time stamps of the cut out plurality of pieces of data as a time origin; modify the time stamps of the cut out plurality of pieces of data to relative time values in relation to the determined time origin; and use the modified time stamps and the values of the plurality of pieces of data to calculate the approximate expression. 
     According to a mode of this invention, a digit overflow is prevented in the calculation of an approximate expression by modifying the time stamp of data cut out with the use of sliding window as the origin of the time is modified. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The present invention can be appreciated by the description which follows in conjunction with the following figures, wherein: 
         FIG. 1  is a block diagram illustrating a configuration example of a stream data processing system in an embodiment of this invention; 
         FIG. 2  and  FIG. 3  are explanatory diagrams showing an example of a conventional approximation method using the least square method; 
         FIGS. 4A and 4B  are explanatory diagrams showing an example of an approximation method that uses the least square method in the embodiment of this invention; 
         FIG. 5  is an explanatory diagram illustrating a configuration of an approximate expression calculating module according to the embodiment of this invention 
         FIG. 6  is a flow chart illustrating processing that is executed by a state value updating module according to the embodiment of this invention; 
         FIG. 7  takes as an example a case of a count-based window for cutting out seven pieces of data; 
         FIG. 8  is a flow chart illustrating processing that is executed by the state value updating module according to the embodiment of this invention; and 
         FIG. 9  is an explanatory diagram illustrating respective data updating timing of the component modules of the approximate expression calculating module according to the embodiment of this invention. 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
       FIG. 1  is a block diagram illustrating a configuration example of a stream data processing system in an embodiment of this invention. 
     The stream data processing system includes a stream data processing server  100 , a monitoring target system, and a plurality of host computers, here,  130  and  140 . 
     The stream data processing server  100  is coupled via a network  150  to the monitoring target system, which is denoted by  120 , and the plurality of host computers,  130  and  140 . The network  150  can be a WAN, a LAN, or the like. However, this invention is not limited by the format of network connection. 
     The stream data processing server  100  receives stream data transmitted from the monitoring target system  120 , and processes the stream data in accordance with a specified query. The stream data includes a plurality of pieces of data  122  organized in time series. 
     The stream data processing server  100  includes a processor  101 , a memory  102 , a network interface  104 , and storage  150 , which are connected via a bus  103 . 
     The processor  101  executes various types of processing by executing a program that is stored in the memory  102 . 
     The memory  102  stores a program executed by the processor  101  and information necessary to execute the program. Specifically, the memory  102  is provided with a stream data processing module  110 . 
     The stream data processing module  110  processes stream data. At the start of the processing, the stream data processing module  110  reads a query group definition stored in the storage  105 , and configures a query graph based on the read query group definition. The stream data processing module  110  executes the processing in accordance with the query graph. 
     The stream data processing module  110  includes an approximate expression calculating module  112  and an anomaly detecting module  113 . 
     The approximate expression calculating module  112  executes processing of analyzing the plurality of pieces of data  122  cut out by sliding window to calculate an approximate expression. Data cut out by sliding window is hereinafter also referred to as target data, and sliding window is hereinafter referred to as window. 
     The anomaly detecting module  113  uses the calculated approximate expression to detect an anomaly in the monitoring target system  120 , and to predict an anomaly as well. For instance, the anomaly detecting module  113  uses the approximate expression to calculate a predicted value of metrics, and determines whether or not the predicted value is equal to or larger than a given threshold. 
     In the following description of processing, when a sentence has the approximate expression calculating module  112  or the anomaly detecting module  113  as the subject, it means that a program implementing the module  112  or the module  113  is being executed by the processor  101 . 
     The functions of the approximate expression calculating module  112  and the anomaly detecting module  113  may be implemented by hardware. 
     The network interface  104  is an interface for coupling to the network  150 . 
     The storage  105  stores stream data (the data  122 ), a query  132 , and other types of information. Examples of the storage  105  include an HDD, an SSD, and similar storage media. This invention is not limited by the type of storage media. 
     The monitoring target system  120  is a computer system constituted of a plurality of computers (not shown). A system monitor  121  for monitoring data (metrics) to be monitored is executed on the computers (not shown) constituting the monitoring target system  120 . 
     The system monitor  121  collects necessary data from the computers (not shown) constituting the monitoring target system  120 , and generates the data  122  from the collected data. The system monitor  121  transmits the generated data  122  to the stream data processing server  100 . 
     The host computers  130  and  140  are computers used by users of the stream data processing server  100 , and includes a processor (not shown), a memory (not shown), and a network interface (not shown). 
     A program for implementing an anomaly monitoring query operating interface  131  is executed on the host computer  130 . The anomaly monitoring query operating interface  131  is an interface for registering the query  132 , anomaly monitoring query  132  to be exact, and for commanding the execution of the anomaly monitoring query  132 . 
     In a case where the anomaly monitoring query  132  is input, the stream data processing server  100  analyzes the anomaly monitoring query  132  to configure a query graph for executing stream data processing. The stream data processing server  100  processes stream data in accordance with the query graph. 
     An anomaly monitoring process  141  is executed on the host computer  140 . The anomaly monitoring process  141  is a process for displaying a processing result to the user based on a result  142 , which is transmitted from the stream data processing server  100 , in order to notify of an error or the like. 
     Alternatively, the provision of the anomaly monitoring query operating interface  131  and the execution of the anomaly monitoring process  141  may be handled by a single computer. 
     A conventional approximation method using the least square method is described first. 
       FIG. 2  and  FIG. 3  are explanatory diagrams showing an example of the conventional approximation method using the least square method. An approximation method of a linear expression is described below. 
     In  FIGS. 2 and 3 , a horizontal axis x represents time and a vertical axis y represents metrics. Examples of the metrics include the utilization ratio of a processor to be monitored and the utilization ratio of the network bandwidth. A time means a time stamp added to data. 
     In  FIG. 2 , in a case where the least square method is applied to the seven pieces of target data cut out by a count-based window  200 , a linear expression such as a line  201  is calculated as the approximate expression. In  FIG. 3 , in a case where the least square method is applied to the seven pieces of target data cut out by a count-based window  300 , a linear expression such as a line  301  is calculated as the approximate expression. 
     The coefficients a and b can be calculated by Expression (1) and Expression (2). As can be seen in Expression (1) and Expression (2), calculating the coefficients a and b involves obtaining the values of Expression (4) to Expression (7). 
     
       
         
           
             
               
                 
                   
                     
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     Values expressed by Expression (4) to Expression (7) are each hereinafter referred to as value S(t) when there is no need to distinguish one from another. 
     However, the calculation method described above has the following problems. 
     A first problem is that, when an absolute value X of the time is significantly greater than a time difference w (w&lt;&lt;X) as shown in  FIG. 3 , a difference in time value between pieces of data within the window ends up being a relatively meaningless value. The values of the coefficients a and b therefore cannot be calculated correctly. 
     This corresponds to the case where an increase in the value of the time x raises the number of digits of the values of Expression (4) to Expression (7) and consequently causes a digit overflow. 
     Thus in this invention, the time origin is moved with sliding the window. Specifically, the approximate expression calculating module  112  first determines the time stamp of one piece of data  122  out of the cut out pieces of data  122  as the origin. The approximate expression calculating module  112  next modifies the time stamps of the cut out pieces of data  122  to relative times in relation to the modified origin. Expression (4) to Expression (7), too, are changed through the time stamp modifying processing described above. 
     In the case where a time stamp x t  of an arbitrary piece of data is determined as the time origin, for example, Expression (4) to Expression (7) are transformed into Expression (8) to Expression (11). In short, the first problem is solved by using relative times. 
     
       
         
           
             
               
                 
                   
                     
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     When Expression (8) to Expression (11) are applied, Expression (1) and Expression (2) can be expressed as Expression (12) and Expression (13). 
     
       
         
           
             
               
                 
                   
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       FIGS. 4A and 4B  are explanatory diagrams showing an example of an approximation method that uses the least square method in the embodiment of this invention. 
     The approximate expression calculating module  112  sets the time stamp of latest data  411  within a window  410  as the origin. 
     In the example of  FIGS. 4A and 4B , a time origin O m  before the window  410  is slid is the time stamp X t−1  of data  413 . With the sliding of the window  410 , the origin O m  is moved to a time stamp X t  of data  423 . Data  424  is deleted as the window is slid. 
     How the origin is determined in this invention is not limited to the one described above, and any time stamp within the window  410  can be set as the origin. 
     Expression (4) to Expression (7) in this case are expressed as Expression (8) to Expression (11). A digit overflow of Expression (4) to Expression (7) due to an increase in the absolute value of the time can be avoided as a result. The first problem can thus be solved by moving the time origin. 
     However, there is a second problem in that an increase in window size raises the cost of calculating Expression (8) to Expression (11). 
     Thus in this invention, an incremental calculation method is used, that uses a value S(t) calculated before the window is slid to obtain a value S(t) calculated next. Specifically, Expression (8) to Expression (11) are transformed into Expression (14) to Expression (17).
 
 S   x ( t )= S   x ( t −1)+ d   t   −z   t   (14)
 
 S   x2 ( t )= S   x2 ( t −1)+2 d   t   S   x ( t −1)+ nd   t   2   −z   t   2   (15)
 
 S   yx ( t )= S   yx ( t −1(+ d   t   S   y ( t −1)− z   t   y   t−n   (16)
 
 S   y ( t )= S   y ( t −1)+ y   t   −y   t−n   (17)
 
     Symbols d t  and z t  are values indicating time differences, and are defined by Expression (18) and Expression (19).
 
 d   t   =x   t−1   −x   t   (18)
 
 z   t   =x   t−n   −x   t   (19)
 
     In the case where old data is not deleted with the sliding of the window, z t  in Expression (12) is “0”. 
     In this embodiment, the value S(t) is treated as a state value. This eliminates the need to execute SUM calculation of Expression (8) to Expression (11) in order to calculate the value S(t), and solves the second problem. 
     When carrying out the incremental calculation method, a third problem arises in that the updated value S(t) needs to be treated as an input again. In other words, the method necessitates recursive computation processing in which the updated value is used as a new input to calculate the next updated value. However, performing recursive computation with simple loop processing means that the calculation processing never converges due to concurrent input and output, which makes carrying out recursive computation processing difficult. 
     Thus in this invention, a time difference is generated between data inputting processing and data updating processing. A feature of this invention is that component modules of the approximate expression calculating module  112  are configured so that a time difference is generated between data inputting processing and data updating processing. 
       FIG. 5  is an explanatory diagram illustrating the configuration of the approximate expression calculating module  112  according to the embodiment of this invention. 
     The approximate expression calculating module  112  includes a data inputting module  501 , an initial state generating module  502 , a state value storing module  503 , a window data storing module  504 , a data storing module  505 , a state value updating module  506 , a coefficient calculating module  507 , and a data outputting module  508 . 
     The data inputting module  501  receives an input of target data. Target data is input in a time stamp-metrics format. In other words, target data in an (x, y) format is input. 
     In this invention where incremental processing is executed, target data is input to the data inputting module  501  in time series one piece at a time. 
     The initial state generating module  502  generates an initial value S(0) of the state value and outputs the generated initial value S(0), in a case where target data is input to the approximate expression calculating module  112  for the first time. Specifically, the initial values of Expression (8) to Expression (11) are “0”. 
     The state value storing module  503  stores the state value S(t). In a case where target data is input for the first time, the state value storing module  503  stores the initial value S(0). 
     The window data storing module  504  stores target data cut out by the window. The window data storing module  504  also executes data updating processing as the window is slid. Specifically, the following processing is executed. 
     The window data storing module  504  determines whether or not there is data to be deleted from a window as the window is slid. In other words, the window data storing module  504  determines whether or not there is target data that moves out of a window with the sliding of the window. 
     In the case where there is data to be deleted from the window, the window data storing module  504  outputs the data to the state value updating module  506 . Data to be deleted as a window is slid is hereinafter also referred to as expired data. 
     In the case of count-based sliding window, the timing of inputting target data and the timing of inputting expired data are the same. In the case of time-based sliding window, on the other hand, the timing of inputting target data and the timing of inputting expired data differ from each other. Processing of the window data storing module  504  therefore varies depending on whether count-based sliding window or time-based sliding window is used. The concrete specifics of the processing are described later with reference to  FIGS. 6 and 8 . 
     The data storing module  505  stores target data that is used to calculate the state value S(t). The target data stored in the data storing module  505  is one that immediately precedes target data received by the data inputting module  501  in time-series order. For example, in a case where the data inputting module  501  receives target data (x 6 , y 6 ), the data storing module  505  stores target data (x 5 , y 5 ). 
     The state value updating module  506  uses Expression (14) to Expression (17) to calculate the state values S(t), in a case where values are input from the respective component modules. 
     Updating processing differs for count-based sliding window and time-based sliding window. This is because the timing of inputting target data and the timing of inputting expired data differ from each other. Details of the processing executed by the state value updating module  506  are described later with reference to  FIGS. 6 and 8 . 
     The coefficient calculating module  507  substitutes the state values S(t) in Expression (12) and Expression (13) to calculate the coefficients a and b, in a case where the state values S(t) calculated by the state value updating module  506  are input. 
     The data outputting module  508  generates an approximate expression based on the calculated coefficients a and b, and outputs the generated approximate expression to the anomaly detecting module  113 . 
     The state value storing module  503 , the window data storing module  504 , and the data storing module  505  have a function of storing data in the storage area of the memory  102 . 
     The data flow of the approximate expression calculating module  112  is described below. 
     When target data is input, the data inputting module  501  outputs the target data to the initial state generating module  502 , the window data storing module  504 , the data storing module  505 , and the state value updating module  506 . 
     The data output to the data storing module  505  has a small delay caused by the data inputting module  501 . 
     In the case where target data is input for the first time, the initial state generating module  502  generates the initial state value S(0), and outputs the generated initial state value S(0) to the state value storing module  503 . 
     The data storing module  505  outputs currently stored target data to the state value updating module  506  and, after the short time elapses, stores new target data. This allows the data storing module  505  to hold target data that immediately precedes input target data in time-series order in a case where processing is executed for the input target data. 
     The state value storing module  503  outputs the currently stored state values S(t) to the state value updating module  506 . 
     The state value updating module  506  substitutes values input from the data inputting module  501 , the state value storing module  503 , the window data storing module  504 , and the data storing module  505  in Expression (14) to Expression (17) to calculate state values S(t+1), which indicate a state after the window is slid. The state value updating module  506  outputs the calculated state values S(t+1) to the state value storing module  503  and the coefficient calculating module  507 . 
     The updated values S (t+1) output to the state value storing module  503  have a small delay caused by the state value updating module  506 . In other words, the state value updating module  503  is updated with the state values S(t+1) after the short time elapses. 
     This is because, if the updated values S(t+1) are input to the state value storing module  503  without a delay, new updated values S(t+1) are calculated with the updated values S(t+1) as an input, which gives rise to a problem in that the computation processing never converges. 
     This invention therefore involves causing a small delay in order to maintain consistency between input and output. 
     By providing a small delay in a case where data is input to the state value storing module  503  and the data storing module  505  as described above, recursive processing can be carried out. 
     The data flow of the approximate expression calculating module  112  is accomplished by a recursive query described in Japanese Patent Application Laid-open No. 2010-108152. 
     The micro time only needs to be a shorter length of time than the time precision of the time stamp. For instance, when the time stamp has a time precision of 1 millisecond, a delay of 1 microsecond or 1 nanosecond is sufficient. 
     The processing flows of the respective component modules are described next. Count-based sliding window and time-based sliding window have different processing procedures, which are described separately. The processing procedure for count-based sliding window is described first. 
     In the case of count-based sliding window, the timing at which target data is input and the timing at which data turns into expired data are the same. However, no data turns into expired data, in a case where the count of pieces of target data cut out by a count-based window is equal to or less than a data count set to the count-based window. 
     The state value updating module  506  therefore needs to vary the calculation expression for the case where expired data is input and for the case where expired data is not input. 
       FIG. 6  is a flow chart illustrating processing that is executed by the state value updating module  506  according to the embodiment of this invention. 
     In a case where target data is input from the data inputting module  501 , the state value updating module  506  first calculates the time difference d t  (Step S 601 ). 
     The state value updating module  506  next determines whether or not there is expired data (Step S 602 ). 
     Specifically, the state value updating module  506  determines whether or not expired data has been input from the window data storing module  504 . In the case where expired data has been input, it is determined that there is expired data. 
     Determining that there is expired data, the state value updating module  506  calculates the time difference z t  (Step S 603 ). 
     In the case where it is determined that there is no expired data, the state value updating module  506  sets the time difference z t  to “0” (Step S 605 ). 
     The state value updating module  506  substitutes values input from the respective component modules in Expression (14) to Expression (17) to calculate the respective state values (Step S 604 ), and ends the processing. 
     The timing of updating data in the approximate expression calculating module  112  is described next. 
       FIG. 7  is an explanatory diagram illustrating respective data updating timing of the component modules of the approximate expression calculating module  112  according to the embodiment of this invention. 
       FIG. 7  takes as an example a case of a count-based window for cutting out seven pieces of data. 
     The data inputting module  501  receives an input of target data (x 6 , y 6 ). 
     At this point, target data (x 5 , y 5 ), which immediately precedes the input data in time-series order, has been stored in the data storing module  505 . The data storing module  505  outputs the target data (x 5 , y 5 ) to the state value updating module  506  and, after a short time elapses since the input of target data (x 6 , y 6 ), updates the target data (x 5 , y 5 ) with the target data (x 6 , y 6 ). 
     When the target data (x 6 , y 6 ) is input, the state value updating module  506  uses the target data (x 5 , y 5 ), the state values S(5), and the target data (x 6 , y 6 ) to calculate the state values S(6). The state value updating module  506  outputs the state values S(6) to the state value storing module  503 . 
     At the time the target data (x 6 , y 6 ) is input, the state values S(5) calculated from data that immediately precedes the input data in time-series order have been stored in the state value storing module  503 . The state value storing module  503  is updated to the state values S(6) after a short time elapses since the input of the state values S(6) from the state value updating module  506 . 
     The same updating processing is executed in a case where other pieces of target data are input. 
     The window data storing module  504  outputs expired data to the state value updating module  506 . In the example of  FIG. 7  where a count-based window for cutting out seven pieces of data is used, data turns into expired data for the first time in a case where data (x 8 , y 8 ) is input. The expired data, (x 1 , y 1 ), has been input prior to the data (x 8 , y 8 ) by seven pieces of data. The state value updating module  506  therefore uses mathematical expressions where the time difference z t  is “0” to calculate the state values S(t) until the data (x 8 , y 8 ) is input. 
     The processing procedure for time-based sliding window is described next. 
     In the case of time-based sliding window, the timing of inputting target data and the timing of inputting expired data differ from each other. The state value updating module  506  therefore needs to vary the calculation expression for updating processing depending on the type of input data. 
       FIG. 8  is a flow chart illustrating processing that is executed by the state value updating module  506  according to the embodiment of this invention. 
     The state value updating module  506  determines whether or not the input data is expired data (Step S 801 ). The determining in this step is the same as in Step S 602 , and a description thereof is therefore omitted here. 
     In a case where the input data is determined as expired data, the state value updating module  506  calculates the time difference z t  (Step S 802 ), and sets the time difference d t  to “0” (Step S 803 ). 
     This is because, with no new data input, there is no need to move the time origin. 
     The state value updating module  506  substitutes the respective values in Expression (14) to Expression (17) to calculate the state values S(t) (Step S 804 ), and ends the processing. 
     In a case where it is determined that the input data is not expired data, in other words, in a case where the input data is target data, the state value updating module  506  calculates the time difference d t  (Step S 805 ), and sets the time difference z t  to “0” (Step S 806 ). 
     The state value updating module  506  substitutes the respective values in Expression (14) to Expression (17) to calculate the state values S(t) (Step S 803 ), and ends the processing. 
     The timing of updating data in the approximate expression calculating module  112  is described next. 
       FIG. 9  is an explanatory diagram illustrating respective data updating timing of the component modules of the approximate expression calculating module  112  according to the embodiment of this invention. 
     The description given here takes as an example a case where the size of a time-based window is “T”. 
     In time-based sliding window, the timing of inputting target data and the timing of inputting expired data differ from each other. However, the updating method of the data inputting module  501 , the state value storing module  503 , and the data storing module  505  is the same as in count-based sliding window, and a description thereof is therefore omitted here. 
     In the example of  FIG. 9 , no data turns into expired data at the time the target data (x 5 , y 5 ) is input. The state value updating module  506  at this point executes Steps S 805 , S 806 , and S 804  to calculate the state values S(6). 
     In a case where the time x 7  equals x 1 +T, in other words, in a case where a time T which is the window size elapses since the input of data (x 1 , y 1 ), the data (x 1 , y 1 ) turns into expired data (x 1 , y 1 ), and the expired data (x 1 , y 1 ) is input to the state value updating module  506 . The state value updating module  506  at this point executes Steps S 802 , S 803 , and S 804  to calculate the state values S(7). 
     Modification Example 
     This invention is not limited to linear approximate expressions, and is also applicable to cases of approximation to a high-order polynomial. 
     For example, in the case of a quadratic approximate expression “y=ax 2 +bx+c”, coefficients a, b, and c can be obtained by solving a simultaneous equation expressed as Expression (20). 
     
       
         
           
             
               
                 
                   
                     
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     In the case of an m-th order approximate expression, solutions of a simultaneous equation having the sum of powers of x and the sum of products of a power of x and y as coefficients are generally obtained. Here, a state value Sxm(t) which is the sum of m-th powers of x at the time x t  is expressed by Expression (21). A state value Sxm(t−1) which is the sum of m-th powers of x at the time x t−1  is then expressed by Expression (22). 
     
       
         
           
             
               
                 
                   
                     
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     It is understood that the state value Sxm(t) can be calculated by incremental computation from the immediately preceding state value Sxm(t−1) as in the case of linear approximate expressions by transforming Expression (21) into Expression (23). 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
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     Similarly, an approximate expression can be calculated for the sum of products of a power of x and y with the use of Expression (24) and Expression (25). In other words, the approximate expression calculating module  112  can calculate coefficients of an m-th order polynomial by a recursive and incremental calculation method with the configuration of  FIG. 5 . 
     
       
         
           
             
               
                 
                   
                     
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                   25 
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     In the expressions given above, m≧1 is satisfied, as well as Sx1(t)=Sx(t), Syx1(t)=Syx(t), Sx0(t)=n, and Syx0(t)=Sy(t). 
     According to a mode of this invention, a digit overflow can be prevented by moving the time origin in concert with the sliding of the window. In addition, recursive and incremental computation processing can be carried out by causing a small delay, in a case where target data and a state value are updated. This allows the approximate expression calculating module  112  to calculate an approximate expression at a reduced calculation cost. 
     The embodiment of this invention has now been described. However, the embodiment is merely an exemplification given to describe this invention, and the range of application of this invention is not limited to the exemplary mode alone. Any combination of the embodiments described above can also constitute an embodiment of this invention.