JP4887661B2 - Learning device, learning method, and computer program - Google Patents

Description

  The present invention relates to a learning apparatus and a learning method for performing a process for approximating a function for outputting a state value at the next time using time series data including state values from the past to the present as an argument, and a computer program. In particular, the present invention relates to a learning apparatus and a learning method that automatically generate information that cannot be directly known from a given learning sample, and a computer program.

  More particularly, the present invention relates to a learning apparatus and a learning method for performing a process for approximating a function for predictive learning of time series data not following a Markov process and outputting a state value at the next time, and a computer program. In particular, the present invention relates to a learning apparatus and learning method for learning a prediction function of time-series data of a non-Markov process by a method other than a recurrent neural network, and a computer program.

  For example, humans are expected to experience similar events over time, repeatedly observe similar facts, and predict that similar things will occur, and do something Guess that an existing one exists. In this way, learning based on past experiences and acquiring new knowledge and techniques is called “learning”.

  In recent years when information technology (IT) has been developed, research and development for realizing such learning mechanism on a computer system is widely performed. For example, learning is performed using the state value up to the current time as input, and the state value at the next time is estimated or automatically generated based on the learning result. That is, the learning device is equivalent to performing a process of approximating a function that outputs a state value at the next time using time series data including state values from the past to the present as an argument.

  Many learners learn by creating a Markov process model in which the future probability law is determined only from the current state, regardless of the past history. However, there are cases where the learning target, that is, the function to be approximated, does not follow the Markov process (or is a Markov process of second or higher order). For example, even if the current state value is the same value, there are two types of sine waves: when the future tends to increase and when it tends to decrease, the future state is automatically generated only from the current state. I can't do it.

  Such a non-Markov process is thought to be nothing but predictive learning. A typical example of a learning mechanism for predictive learning of a non-Markov process is a recurrent neural network (for example, see Non-Patent Document 1).

  For example, a robot equipped with a recurrent neural network as a learning mechanism moves a movable object in the outside world by a controllable part of the robot itself, and an environment in which the object is placed by a perceptual sensor, and the object Learn the relationship between the movement of each joint part of the robot and the movement of the target object, predict the movement of the target object, and self-learn the motion of moving the target object through novelty-rewarding (For example, see Patent Document 1).

  However, since the recurrent neural network uses an error back propagation method, there is a problem that learning takes time.

  Further, in the recurrent neural network using the error back propagation method, the number of dimensions of the context unit (context information) is much larger than the number originally considered necessary. This can affect the generalization characteristics.

JP 2002-59384 A Elman, J.M. L. Author “Finding structure in time” (Cognitive Science, vol. 14, 1990, pp. 179-211)

  An object of the present invention is to provide an excellent learning device, learning method, and computer program capable of automatically generating information that cannot be directly known from a given learning sample.

  A further object of the present invention is to provide an excellent learning apparatus, learning method, and computer capable of predicting and learning time-series data not following a Markov process and approximating a function that outputs a state value at the next time.・ To provide a program.

  A further object of the present invention is to provide an excellent learning apparatus and learning method, and a computer program capable of learning a prediction function of time-series data of a non-Markov process by a method other than a recurrent neural network. is there.

The present invention has been made in consideration of the above problems, and a first aspect thereof is a time series for predicting a state z _{t + 1} related to a next time t + 1 based on a state z _t related to a certain time t. A learning device for approximating the prediction function F, wherein the state at each time t consists of information x _t and context information c _t to be learned at the time,
The time series information {x _t } of the learning object and the time series {c _t } of the context information at each time t (where t = 1... T) up to the current time T are input as past states {z _t }. Data input means;
Function learning means for learning the time series prediction function F according to a predetermined learning algorithm using the input past state {z _t };
Prediction means for predicting the learning sample {x _t } at each time t until the current time T using the time series prediction function F and the state initial value z ₁ obtained by the learning;
An error calculating means for calculating an error between the learning sample {x _t } at each time t input by the data input means and the predicted value of the learning sample at each time t generated by the prediction means;
Determination means for determining whether learning of the time series prediction function F by the function learning means is completed based on the error;
A learning apparatus comprising:

  The present invention relates to a learning apparatus that learns a time series prediction function F of a non-Markov process. It is common to use a recurrent neural network based on the error back-propagation method to learn the prediction function of non-Markov time series information, but it takes a long time to learn and the number of dimensions of context information is very high. There is a concern that the generalization characteristics may be affected in large numbers. Therefore, in the learning apparatus according to the present invention, a continuous value function approximation method is used as a learning algorithm for learning the time series prediction function F. A representative example of a continuous function approximation method that is guaranteed to converge to a global solution in a short time is Support Vector Regression (hereinafter referred to as SVR).

  The learning device according to the present invention approximates a time series prediction function F for predicting or automatically generating a state related to the next time t + 1 based on a state related to the current time t. Here, context information is used to solve a time series prediction problem related to time series information which is a non-Markov process. Therefore, the state at a certain time includes a learning sample at the time and context information at the same time. If the learning sample is n-dimensional and the context information is m-dimensional, the function F is a time-series prediction function with an (n + m) -dimensional input (n + m) -dimensional output.

In such a case, the object to be learned by the learning algorithm is the time series prediction function F, but since the context information {c _t } is unknown, the context information {c _t } is estimated together with the learning of the function F. There must be. Therefore, in the present invention, the estimation of the context information {c _t } and the learning of the function F are alternately repeated so as to approach the ideal solution.

First, the time series information {x _t } of the learning target and the time series {c _t } of the context information at each time t (where t = 1... T) up to the current time T are set as past states {z _t }. The time series prediction function F is learned from the input past state {z _t } according to the SVR learning algorithm.

Subsequently, in order to evaluate the learning result, the state {z _t } up to the current time T is predicted using the time series prediction function F obtained by learning and the initial value z ₁ of the state. . Then, an error e between the learning sample {x _t } at each time t when data is actually input and the predicted value of the learning sample at each time t predicted using the learned time series prediction function F is calculated, Whether or not learning is completed can be determined based on whether or not the error e falls below a threshold value.

Here, when it is determined that the learning is not finished, the time series prediction function F is relearned after correcting the context information {c _t } at each time t. The context information {c _t } at each time t can be corrected based on the calculated error e. Specifically, the calculated error e can be corrected by changing the context information {c _t } in the direction of the gradient vector obtained as a result of partial differentiation of the calculated error e with the context information {c _t }.

The second aspect of the present invention, a computer processing for when performing approximation of series prediction function F for prediction based on the state z _t for a certain time t the state z _{t + 1} for the next time t + 1 A computer program written in a computer-readable format to be executed on the system, wherein the state at each time t is composed of information _xt and context information _ct to be learned at the time, and is stored in the computer system. In contrast,
The time series information {x _t } of the learning object and the time series {c _t } of the context information at each time t (where t = 1... T) up to the current time T are input as past states {z _t }. Data entry procedure;
A function learning procedure for learning the time series prediction function F according to a learning algorithm based on a continuous value function approximation method using the input past state;
A prediction procedure for predicting a learning sample {x _t } at each time t up to the current time T using the time series prediction function F and the state initial value z ₁ obtained by the learning;
An error calculation procedure for calculating an error between the learning sample {x _t } at each time t input in the data input procedure and the predicted value of the learning sample at each time t generated in the prediction procedure;
A determination procedure for determining whether learning of the time-series prediction function F in the function learning procedure is completed based on the error;
A context correction procedure for correcting the context information {c _t } at each time t based on the error calculated in the error calculation procedure;
When it is determined in the determination procedure that learning has not been completed, the time series prediction function F in the function learning procedure using the state {z _t } at each time t including the context information corrected in the context correction procedure. An iterative learning procedure to re-learn
Is a computer program characterized in that

  The computer program according to the second aspect of the present invention defines a computer program described in a computer-readable format so as to realize predetermined processing on a computer system. In other words, by installing the computer program according to the second aspect of the present invention in the computer system, a cooperative action is exhibited on the computer system, and the learning device according to the first aspect of the present invention. The same effect can be obtained.

  According to the present invention, it is possible to provide an excellent learning device, learning method, and computer program capable of automatically generating information that cannot be directly known from a given learning sample.

  In addition, according to the present invention, it is possible to provide an excellent learning device, learning method, and computer program capable of learning a prediction function of time-series data of non-Markov processes by a method other than a recurrent neural network. Can do.

  Further, according to the present invention, it is possible to provide an excellent learning device, learning method, and computer program that can solve the time-series prediction problem of a non-Markov process using context information.

  In addition, according to the present invention, an excellent learning device that can learn a prediction function of time-series data of a non-Markov process using a continuous value function approximation method that guarantees convergence to a global solution in a short time. And a learning method and a computer program can be provided.

  According to the learning method of the present invention, learning can be completed faster than the recurrent neural network using the error back-propagation method, and the learning can be converged with less dimension number m of context information. Can do.

  Other objects, features, and advantages of the present invention will become apparent from more detailed description based on embodiments of the present invention described later and the accompanying drawings.

  Hereinafter, embodiments of the present invention will be described in detail with reference to the drawings.

  In the present invention, in order to learn the prediction function of the time series information of the non-Markov process, a continuous value function approximation method is used instead of the recurrent neural network using the error back propagation method. SVR can be given as a representative example of a continuous value function approximation method that guarantees convergence to a global solution in a short time.

The SVR estimates an n-dimensional input one-dimensional output real value function f: R ⁿ → R in the form shown in the following equation (1).

Here, z∈R ⁿ input vectors, s _i ∈R ⁿ is support vector, K (.) Is a kernel function, b is a scalar value called bias term.

According to the learning algorithm of SVR, when T learning samples {(z _k , y _k ) | k = 1... T} and a kernel function K (.) Are given, It is possible to uniquely determine s _j , θ _j , and b that can explain them well.

Here, a prediction problem of multidimensional time series information is considered. When a time series {x _t εR ⁿ | t = 1... T} is given as a learning sample, an n-dimensional input n is obtained by setting the learning sample x as an n-dimensional vector as shown in the following equation (2). The dimension output time series prediction function F: R ⁿ → R can be configured by SVR.

  However, although the function F can predict time series information of a Markov process, there is a problem that time series information of a non-Markov process (or a second or higher order Markov process) cannot be predicted.

  There are two common methods for solving the time series prediction problem of non-Markov processes. One is a method using time delay input, and the other is a method using context information. In the present embodiment, the latter context information is used. In the following, the mechanism for introducing context information is explained in detail.

Series prediction function F when it is defined by the above formula (2) as input learning samples x _t n-dimensional at the current time t, the next time t + n-dimensional in one learning sample x _t that is predicted based on this _{Has +1} as output. Here to introduce contextual information, define a made of n-dimensional training samples {x _t} and m-dimensional context information _{{c t} (n + m} ) dimensional states {z _t}, expanded learning sample To do. Then, the function F shall be possible to predict the state z _{t + 1} for the next time t + 1 based on the state z _t for the current time t. Therefore, the function F to be learned by the SVR learning algorithm is expressed as the following equation (3).

In such a case, the object to be learned by the learning algorithm is the time series prediction function F, but since the context information {c _t } is unknown, the context information {c _t } is estimated together with the learning of the function F. There must be. Therefore, in the present embodiment, the estimation of the context information {c _t } and the learning of the function F are alternately repeated so as to approach the ideal solution. The algorithm for function learning and context information correction in this case is as follows.

(1) Input time series information to be learned and time series of context information.
Here, the time-series information {x _t } to be learned is n-dimensional and is a non-Markov process that cannot be predicted from only the previous learning sample. Further, since the context information {c _t } is unknown, a randomly generated value is used as the initial value. As will be described later, the context information is repeatedly corrected until the learning is completed. Here, the i-th generated context information is represented as {c ⁽ⁱ⁾ _t }. The context information {c _t } has m dimensions (where m is arbitrary).

(2) Predict a (n + m) -dimensional state {z ⁽ⁱ⁾ _t } composed of time series information {x _t } to be learned and time series {c ⁽ⁱ⁾ _t } of context information according to an algorithm based on SVR. A time series prediction function F is learned.

(3) Predict the past state {z _t } using the learned function F: R ^{n + m} → R ^{n + m} and the initial value z ⁽ⁱ⁾ ₁ of the state.

(4) An error between the predicted time series information of the learning target and the time series information of the learning target actually given is calculated. For example, the square error e is obtained, and if e is equal to or less than the threshold value, it is determined that the learning is finished.

(5) When the error e between the predicted time series information of the learning target and the actually given time series information does not fall within the threshold, the context information is corrected and the learning of the function F is performed again. . The context information is corrected using the calculated error e.

Then, until the predicted error of the time-series information to be learned falls below the threshold value, the correction of the context information {c _t } and the learning of the function F are alternately repeated as i ← i + 1.

In the above equation (8), in order to estimate the context information {c ⁽ⁱ⁾ _t }, the error e in the direction of the gradient vector obtained as a result of partial differentiation with the i-th estimated value {c ⁽ⁱ⁾ _t } c ⁽ⁱ⁾ _t } is changed. This is the same as the steepest descent approach. The gradient vector in the i-th iteration is expressed as the following equation (9).

  A method for obtaining the gradient vector will be described below. However, the subscript (i) is omitted assuming the i-th time thereafter (the variable i is reused for another purpose).

  Consider the case where the kernel function K (.) To be used is a Gaussian RBF function. The definition is as in the following formula (10).

First, consider the gradient of the context information c _T-1 at time t = T−1. The forward propagation from c _T-1 to the error function e _T is expressed by the following equation (11).

In the above equation, M _k is the number of support vectors in the k-th function f _k . Further, for convenience, the following equation (12) is set.

As described above, the error e _T components of c _T-1 | when partially differentiated by _{{c T-1 i = 1} ... m}, the following equation (13).

  For reference, each partial differentiation is as shown in the following equation (14).

From the above, the change amount Δc _{T−1, i} of c _T−1 = {c _{T−1, j} | i = 1... M} is expressed by the following equation (15). Here, α is an arbitrary learning coefficient.

So far, the change of the context information c _T-1 that directly affects the error e _T at the last time has been considered. Similarly, in {c _t | t = 1... T−2}, the error function e may be partially differentiated by c _t . However, the context information c _t at time t rather than give only impact to the error e _{t + 1} of time t + 1, it affects from t + 1 to all of the future of over the T. Therefore, the change amount Δct _{, i} is as shown in the following equation (16).

  Moreover, the following is mentioned as an example of a change of the algorithm which changes context information.

(1) An expression for _obtaining c ^{(i + 1)} _t in the algorithm is represented by the following expression (17).

(2) By applying a low-pass filter in the time direction to the context information {c _t }, it is possible to extract context information having a larger time scale than the learning sample {x _t }.

(3) By changing the pass frequency of the low-pass filter for each variable and allowing them to coexist, phenomena of different time scales can be intentionally separated.

  FIG. 1 shows a functional configuration of a learning device 1 according to an embodiment of the present invention. The learning apparatus 1 shown in FIG. 1 includes an input unit 11, an initialization unit 12, a function approximation unit 13, a prediction unit 14, an error calculation unit 15, a determination unit 16, and a context correction unit 17. . Although the learning device 1 may be designed as a dedicated hardware device, it may be configured in such a manner that a computer program for realizing each functional module is started on a general computer system.

The input unit 11 inputs n-dimensional time series information {x _t } to be learned. The learning target is time-series information of a non-Markov process that cannot be predicted from only the previous learning sample. In addition, the initialization unit 12 randomly generates an initial value of the m-dimensional context information {c _t }.

The n-dimensional learning data and m-dimensional context data input from the input unit 11 are input to the function approximating unit 13 as an (n + m) -dimensional state {z _t }. The function approximating unit 13 performs learning, that is, function approximation, of the time series prediction function F for predicting the state z _{t + 1} at the next time t + 1 from the state z _{t at} a certain time t by the learning algorithm of SVR.

The prediction unit 14 tries to predict the state {z _t } at each time t using the prediction function F approximated by the function and the initial value z ⁽ⁱ⁾ ₁ of the state.

  The error calculation unit 15 calculates an error e between the learning data predicted by the prediction unit 14 and the learning data actually input from the input unit 11.

  The determination unit 16 compares the error e calculated by the error calculation unit 15 with a threshold value, and determines that learning has been completed if the error e is equal to or less than the threshold value. And the prediction function F at the time of completion | finish determination is output as a learning result by the learning apparatus 1. FIG.

When the error e between the predicted time series information of the learning target and the actually given time series information does not fall within the threshold, the context correction unit 17 corrects the context information {c _t }, The prediction function F is learned again. The context correction unit 17 corrects the context information using the error e calculated by the error calculation unit 15. Specifically, varying the contextual information {c _t} the calculated error e in the direction of the contextual information {c _t} in partial differential obtained as a result of the gradient vector corrected (above). The correction of the context information {c _t } and the learning of the function F are alternately repeated until the predicted error in the time-series information to be learned falls below the threshold.

  FIG. 2 shows a processing procedure for learning the time series prediction function F while the learning device 1 corrects the context information in the form of a flowchart. Hereinafter, the processing procedure will be described with reference to FIG.

  First, learning data is input from the input unit 11, and an initial value of context data is generated by the initialization unit 12 (step S1).

  Next, the function approximating unit 13 generates the prediction function F, that is, approximates the function with reference to the learning data and the context data (step S2).

  Next, the prediction unit 14 tries to predict the learning data according to the generated prediction function F using the initial value of the learning data (step S3).

  Next, the error calculation unit 15 calculates a difference between the learning data predicted using the prediction function F and the learning data actually input from the input unit 11, and calculates a prediction error (step S4).

  The error calculation result is input to the determination unit 16. The determination unit 16 determines whether the approximation of the prediction function generated by the function approximation unit 13 is sufficient based on the calculated error (step S5).

  If the determination unit 16 makes an end determination, the learning ends.

On the other hand, when the end determination is not issued, the context correction unit 17 corrects the context data according to the error (step S6). And it returns to step S1 and learning of the prediction function F is performed again. The correction of the context information {c _t } and the learning of the function F are repeated alternately until the predicted error in the time-series information to be learned falls below the threshold value.

  Finally, an experimental example in which the learning mechanism according to the present embodiment is applied to sine wave time series prediction will be described.

FIG. 3 shows learning samples {x _t εR | t = 1... 60} at this time. The horizontal axis is time t, and the vertical axis is the value x _t. The illustrated learning sample is composed of three sine waves.

FIG. 4 shows an initial state before learning. In the upper part of the figure, learning samples {x _t εR | t = 1... 60} composed of three sine waves similar to those shown in FIG. 3 and initial context information randomly generated at each time t are shown. The value {c ⁽¹⁾ _t ∈ R | t = 1... 60} is shown. In the lower part of the figure, the time series value of the predicted learning sample and the predicted context information are shown.

In the first estimation of context information {c _t } and learning of the function F, since the context information {c ⁽¹⁾ _t } is random, as shown in the figure, the predicted value of the predicted learning data is actually It is different from the input learning data. The purpose of alternately estimating the context information {c _t } and learning the function F is to make the two time series values the same.

FIG. 5 shows how learning of the prediction function F converges by repeatedly performing estimation of context information {c _t } and learning of the function F alternately. As shown in the figure, the context information {c ⁽ⁱ⁾ _t } has a waveform with the same period shifted from the learning sample {x _t } by a half phase.

The second row from the top in the figure shows that the predicted value of the learning sample has the same waveform as the actual learning sample {x _t }. Moreover, it can be seen that the sine wave is continuously predicted even after the time t = 61 when it is not learned.

  Further, the third row from the top in the figure shows the result of predicting the function F while adding noise to the predicted value of the learning sample. It can be seen that the same waveform as the learning sample is drawn in the same manner as when no noise is added.

  Further, in the fourth row from the top in the figure, a learning curve is shown in which the horizontal axis represents the number of steps and the vertical axis represents RMSE (Root Mean Squared Error). In the figure, it can be seen that learning has converged after 10 times.

  As described above, the learning method according to the present invention learns a prediction function of time-series data of non-Markov processes using context information according to a continuous value function approximation method in which convergence to a global solution is guaranteed in a short time. can do. At that time, it is possible to complete the learning faster than the recurrent neural network using the error back propagation method, and to sufficiently converge the learning with less dimension number m of context information. I want you to understand.

  The present invention has been described in detail above with reference to specific embodiments. However, it is obvious that those skilled in the art can make modifications and substitutions of the embodiment without departing from the gist of the present invention.

  In the present specification, the description has been made mainly on the embodiment in which the prediction function F is learned mainly according to the SVR, but the gist of the present invention is not limited to this. For example, the present invention can be similarly applied to a learning algorithm based on a continuous value function approximation method other than SVR and a learning device to which another learning algorithm is applied.

  In short, the present invention has been disclosed in the form of exemplification, and the description of the present specification should not be interpreted in a limited manner. In order to determine the gist of the present invention, the claims should be taken into consideration.

FIG. 1 is a diagram showing a functional configuration of a learning device 1 according to an embodiment of the present invention. FIG. 2 is a flowchart showing a processing procedure for the learning device 1 to learn the time series prediction function F while correcting the context information. FIG. 3 is a diagram for explaining an experimental example in which the learning mechanism according to the present invention is applied to time series prediction of a sine wave. FIG. 4 is a diagram for explaining an experimental example in which the learning mechanism according to the present invention is applied to time series prediction of a sine wave. FIG. 5 is a diagram for explaining an experimental example in which the learning mechanism according to the present invention is applied to sine wave time series prediction.

Explanation of symbols

DESCRIPTION OF SYMBOLS 1 ... Learning apparatus 11 ... Input part 12 ... Initialization part 13 ... Function approximation part 14 ... Prediction part 15 ... Error calculation part 16 ... Determination part 17 ... Context correction part

Claims (5)

A learning apparatus for performing an approximation of the series prediction function F when for predicting the state z _{t + 1} for the next time t + 1 based on the state z _t for a certain time t, the state z _t at each time t is the time It consists of learning target information x _t and context information c _{t at t} ,
As the past state {z _t }, the learning target time-series information {x _t } at each time t (provided that t = 1... T) up to the current time T is input, and until the current time T is reached. a data input means to generate a time series {c _t} contextual information randomly or inputting the time series {c _t} randomly contextual information generated,
Function learning means for learning the time series prediction function F according to a predetermined learning algorithm using the input past state {z _t };
Prediction means for predicting the learning sample {x _t } at each time t until the current time T using the time series prediction function F and the state initial value z ₁ obtained by the learning;
An error calculating means for calculating an error between the learning sample {x _t } at each time t input by the data input means and the predicted value of the learning sample at each time t generated by the prediction means;
Context correcting means for changing the context information {c _t } in the direction of the gradient vector obtained as a result of partial differentiation of the error e calculated by the error calculating means with the context information {c _t } ;
Determination means for determining whether learning of the time series prediction function F by the function learning means is completed based on the error;
Equipped with,
When the determination means determines that the learning has not ended , the time learning function {z _t } including the context information corrected by the context correction means is given to the function learning means to obtain a time series prediction function. Re-learn F,
A learning apparatus characterized by that. The function learning means learns the time series prediction function F according to a learning algorithm based on a continuous value function approximation method.
The learning apparatus according to claim 1. The function learning means learns a time series prediction function F according to a learning algorithm based on Support Vector Regression.
The learning apparatus according to claim 2 , wherein: State z for a certain time t _t The state z for the next time t + 1 based on _{t + 1} Is a learning method for approximating the time series prediction function F for predicting the state z at each time t _t Is information x to be learned at the time t _t And context information c _t Consists of
Past state {z _t }, The time series information {x of the learning target at each time t (where t = 1... T) up to the current time T. _t } And context information time series up to the current time T {c _t } At random, or a time series {c of randomly generated context information _t }, A data input step for inputting
The input past state {z _t }, A function learning step of learning the time series prediction function F according to a learning algorithm based on a continuous value function approximation method,
Time series prediction function F and state initial value z obtained by learning ₁ Using the learning sample {x at each time t until the current time T _t } For predicting,
Learning sample {x at each time t input in the data input step _t } And an error calculation step of calculating an error between the prediction value of the learning sample at each time t generated in the prediction step;
The error e calculated in the error calculation step is used as context information {c _t } In the direction of the gradient vector obtained as a result of partial differentiation with {} _t } To modify the context and modify the context,
A determination step of determining whether learning of the time series prediction function F in the function learning step is completed based on the error;
Have
When it is determined in the determination step that the learning has not been completed, the state {z at each time t including the context information corrected in the context correction step _t } Is used to learn the time series prediction function F in the function learning step again.
A learning method characterized by that. State z for a certain time t _t The state z for the next time t + 1 based on _{t + 1} Is a computer program written in a computer-readable format so as to execute on a computer system a process for approximating a time series prediction function F for predicting a state z at each time t. _t Is information x to be learned at the time t _t And context information c _t For the computer system
Past state {z _t }, The time series information {x of the learning target at each time t (where t = 1... T) up to the current time T. _t } And context information time series up to the current time T {c _t } At random, or a time series {c of randomly generated context information _t } The data input procedure to input},
A function learning procedure for learning the time series prediction function F according to a learning algorithm based on a continuous value function approximation method using the input past state;
Time series prediction function F and state initial value z obtained by learning ₁ Using the learning sample {x at each time t until the current time T _t } Prediction procedure for predicting
Learning sample {x at each time t input in the data input procedure _t } And an error calculation procedure for calculating an error between the prediction value of the learning sample at each time t generated in the prediction procedure;
The error e calculated by the error calculation procedure is used as context information {c _t } In the direction of the gradient vector obtained as a result of partial differentiation with {} _t } To modify the context and modify the context,
A determination procedure for determining whether learning of the time-series prediction function F in the function learning procedure is completed based on the error;
When it is determined in the determination procedure that learning has not been completed, the state {z at each time t including the context information corrected in the context correction procedure {z _t } To repeat the learning of the time series prediction function F in the function learning procedure,
A computer program for executing

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