JP2007215354A - Method and processing program for estimating electrical load - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a method which finds upper/lower limits for estimating an electrical load and efficiently estimates the electric load. <P>SOLUTION: This program includes steps of: a computing device calculating the mean and the dispersal of estimation distribution by the Gaussian Process developed by an arithmetic unit by means of MAP estimation or Hybrid Monte Carlo method, based on pre-distribution stored in a storage device; and outputting the mean and the dispersal of the calculated estimation distribution. When estimated in this way, a result better than the mean error and the maximum error is achieved for either of the GP (MAP) and the GP (HMC) compared to the MLP and the RBFN of the comparison method. <P>COPYRIGHT: (C)2007,JPO&INPIT

Description

  The present invention relates to a power load prediction method and a power load prediction processing program for obtaining upper and lower limit values and average values of power load prediction by a Gaussian process using hierarchical Bayesian estimation.

  In order to balance supply and demand in an electric power system, the stability and economics of the electric power system are important in a power generation plan. Conventionally, as a method for predicting future events, a method of modeling a prior process that is past actual data and estimating an internal variable of the model is known (Patent Document 1).

JP-A-2005-78519

  In recent years, uncertainties in power systems have increased due to power liberalization and changes in weather conditions. In new power liberalization environments, there is a tendency to avoid excess power generation and maximize profits and minimize risks. As a result, a new power load prediction method that takes uncertainty into account is also eagerly desired in power load prediction that is the basis of a power generation plan. However, in the conventional method, since the upper and lower limits of the power load prediction cannot be obtained systematically, there is a problem that the power load cannot be estimated efficiently.

  This invention is made in view of such a point, and it aims at providing the electric power load prediction method which can obtain | require the upper and lower limits of electric power load prediction, and can estimate electric power load efficiently. .

  A power load prediction method according to the present invention is a power load prediction method for predicting a power load using a computer including a storage device, a calculation device, and an output device, and based on a prior distribution stored in the storage device, The computing device calculates the mean and variance of the predicted distribution by the Gaussian process constructed by the computing device by MAP estimation or hybrid Monte Carlo method, and outputs the calculated mean and variance of the predicted distribution via the output device. And a step of performing.

  The arithmetic unit constructs a Gaussian process by obtaining a MAP estimation value by MAP estimation.

  The arithmetic unit calculates an average and a variance of the predicted distribution by sampling by a hybrid Monte Carlo method.

  A power load prediction processing program according to the present invention is a power load prediction processing program for use in a computer including a storage device, an arithmetic device, and an output device, and predicts a power load, and is stored in the storage device in the computer. Based on the prior distribution, a step in which the arithmetic device calculates an average and variance of the predicted distribution by a Gaussian process constructed by the arithmetic device by MAP estimation or a hybrid Monte Carlo method, and an average and variance of the calculated prediction distribution And a step of outputting via an output device.

  ADVANTAGE OF THE INVENTION According to this invention, the power load prediction method which can obtain | require the upper and lower limits of electric power load prediction, and can estimate electric power load accurately can be provided.

Hereinafter, embodiments of the present invention will be described in detail with reference to the drawings.
In the power load prediction method of this embodiment, the upper and lower limit values and the average value of power load prediction are obtained using a Gaussian process using hierarchical Bayesian estimation. At this time, in this embodiment, the average value and variance of the Gaussian process are obtained by MAP estimation or a hybrid Monte Carlo method described later.

  FIG. 1 is a diagram illustrating a computer system for realizing the power load prediction method according to the present embodiment. This system includes an input device (or receiving device) 1, an arithmetic device 2, a storage device 3, and an output device (or transmitting device) 4. The input device 1 is used when inputting various types of information used for power load prediction such as prior distribution. The arithmetic device 2 constructs a Gaussian process by MAP estimation or a hybrid Monte Carlo method, and performs processing for calculating the mean and variance of the predicted distribution. The output device 4 is used when displaying or transmitting the mean and variance of the predicted distribution.

Here, each term will be briefly described.
The “Bayesian model” is a model for estimating a parameter and data as random variables and estimating a predicted value as a point rather than estimating it as a point. The “Gaussian process (hereinafter referred to as“ GP ”)” is a model that processes a normal distribution as a parameter prior distribution in the Bayesian model. The hierarchical Bayesian estimation model is a Bayesian model that assumes a prior distribution of parameters, that is, considers parameters (hyperparameters) of parameter distribution. Note that hierarchical Bayesian estimation works to alleviate the influence of a specific distribution. “MAP estimation” is an estimation method that determines hyperparameters using gradient information so that the posterior probability is maximized. The “hybrid Monte Carlo method” is a kind of Markov chain Monte Carlo method, and is a method of determining the mean and variance of the posterior distribution by sampling.

  Next, GP will be described in detail. Here, the definition of GP based on the relationship between linear regression in the Bayesian model and a kernel machine that has been actively researched in recent years will be described.

First, as in the case of a neural network, consider using H basis functions Φ _H = {Φ _h (·)}, (h = 1,..., H) and performing regression with the weighted sum. However, it is assumed that H is a very large value. Further, it is assumed that _N explanatory variables x _N = {x _n }, (n = 1,..., N) are given. A matrix R having Φ _h (x _n ) as an element is defined from these basis functions and input variables. The element R _nh of this matrix R is as follows.

Therefore, if the weighted sum of the basis functions is y (x) = {y _n }, (n = 1,..., N), it can be written as the following equation (Equation 2). In Equation 2, u _H = {u _h }, (h = 1,..., H) is a weight corresponding to the basis function.

Further, a prior distribution is set for {u _h }. This prior distribution is assumed to be a normal distribution having an average of “0” independently for each weight and a variance of “σ _u ² ” as shown in the following equation (Equation 3). In Equation 3, “I” means a unit matrix.

  Then, it can be seen from Equation 2 that “y” similarly follows a normal distribution with a mean of zero. If the variance-covariance matrix of “y” is “Q”, the following equation (Equation 4) is obtained.

  Since the variance-covariance matrix “Q” of “y” is expressed by Equation 4, the prior distribution P (y) of “y” can be expressed by the following equation (Expression 5).

Next, the objective variable t _N = {t _n }, (n = 1,..., N) will be considered. Assuming that “y _n ” corresponding to “t _n ” is added with normal noise having an average of 0 and a variance “σ _ν ² ”, the prior distribution of “t” is also represented by the following equation (Equation 6): Normal distribution.

  Here, if the variance-covariance matrix of Expression 6 is expressed as “C”, the variance-covariance matrix “C” is expressed by the following expression (Expression 7).

If the number H of the basis functions is smaller than the number of data N, RR ^T is no longer in the always regular. However, since the noise term “σ _ν ² ” is full rank, “C” is always regular, and numerical stability is obtained. Such an attempt to eliminate the defective setting by adding some condition to the normal setting is called “regularization” and is also a concept behind the derivation of the radial basis function network.

  Now consider “Q” which is a variance-covariance matrix. The (i, j) -th element “Qij” of “Q” is expressed by the following equation (Equation 8).

From Equation 8, it can be seen that what is actually required to calculate “Q” is the inner product calculation between Φ _h (x). Here, it is assumed that this inner product calculation can be expressed by a kernel function such as the following equation (Equation 9).

Of course, Equation 9 does not hold for any function. A necessary and sufficient condition for the expression 9 to be satisfied is that the kernel function satisfies a symmetric half positive definite value. Such a condition is referred to as a “Mercer condition”, and a kernel that satisfies the “Mercer condition” is particularly referred to as a “Mercer kernel”. Further, replacing the inner product of Φ _h (x) with the kernel in this way and avoiding the calculation like Φ _h (x) is called “kernel trick”.

  Since symmetric semidefiniteness is also applicable to ordinary variance-covariance matrices, GP can be regarded as a model in which ordinary linear regression is extended by a kernel. As the “Mercer kernel”, for example, there is a “Gaussian kernel” shown in the following equation (Equation 10). However, in Equation 10, “ω” is the width of the Gaussian kernel.

  From the above, the element of Expression 7 can be expressed as the following expression (Expression 11).

Next, consider the problem of predicting a new target value “t _{N + 1} ” from the obtained “t _N ”. The conditional distribution P (t _{N + 1} | t _N ) of “t _{N + 1} ” is expressed as shown in the following equation (Equation 12) from Bayes' theorem.

The numerator on the right side of Equation 12 has a normal distribution by assumption. Also, since “t _N ” has already been obtained, the denominator is a constant. Therefore, the distribution P (t _{N + 1} | t _N ) can be expressed as the following formula (formula 13). In Equation 13, “R” is a variance-covariance matrix from “t ₁ ” to “t _{N + 1} ”.

Here, “R” and “R ⁻¹ ” are divided as shown in the following formula (Formula 14).

The official common dividing inverse matrix, each element of "R ^-1" by the following equation (equation 15) can be represented by "C ^-1" represents "R ^-1" using "C ^-1" be able to.

From Expression 14 and Expression 15, the average “μ _{n + 1} ” and the variance “σ ² _{n + 1} ” in the distribution of Expression 13 are as shown in the following Expression (Expression 16).

  As described above, GP is defined based on the relationship between linear regression in the Bayesian model and a kernel machine that has been actively researched in recent years.

  As described above, if the format of the covariance function is determined, the predicted distribution in GP can be easily obtained by matrix / vector operations. Therefore, the covariance function plays an important role in the prediction in GP. However, the parameter (for example, “w” in Equation 10) in the covariance function cannot be determined as it is. In GP, the parameter of the covariance function is further regarded as a random variable, and a parameter is determined by setting a prior distribution and obtaining a posterior distribution from Bayes' theorem.

  Since all of the above models are discussions after the parameters of the covariance function are determined, these depend on the probability distribution of the parameters of the covariance function. In this way, parameters set higher than normal parameters are called “hyper parameters”, and capturing them in the Bayesian framework is called “hierarchical Bayes model”.

Next, a method for determining and predicting hyperparameters will be described.
Here, a calculation method of the predicted distribution when the prior distribution P (θ) is set to the hyperparameter θ = {θ _k }, (k = 1,..., K) will be described. In this case, the predicted distribution is represented by the following equation (Equation 17) from Bayes' theorem. In Expression 17, “D” is given data.

  Further, for P (θ | D), the following equation (Equation 18) is established from Bayes' theorem.

  Here, P (θ | D) is a quantity (hyperparameter marginal likelihood) that represents how likely the data can be expressed by the hyperparameter, and is sometimes referred to as “ABIC” or “evidence”. The logarithm of this term is expressed by the following equation (Equation 19). In Expression 19, “det” is a determinant of a matrix.

  As can be seen from Equation 19, the integration of Equation 17 is very complex and cannot be done analytically. Therefore, approximate calculation is required. Hereinafter, two methods for performing approximation will be described.

"MAP (Maximum A Posteriori) estimation"
First, MAP estimation will be described. MAP estimation is a technique for performing point estimation on a value that maximizes the posterior probability. That is, assuming that the estimated MAP value is “θ _MAP ”, an approximation is performed in which P (t _{N + 1} | x _{N + 1} , D) is P (t _{N + 1} | x _{N + 1} , D, θ _MAP ). Since “θ” is fixed, the predicted distribution is a normal distribution. In order to do this, it is sufficient to maximize Equation 18 using the logarithm of Equation 19 and the hyperparameter prior distribution. The partial derivative of Equation 19 is given by the following equation (Equation 20). In Equation 20, Tr is a matrix trace.

  Therefore, if the partial derivative of lnP (θ) can be obtained, it is possible to maximize Equation 18 using an optimization method such as a conjugate gradient method.

"Sampling with Hybrid Monte Carlo"
Next, sampling by hybrid Monte Carlo will be described. As another method for approximating the predicted distribution of Expression 17, there is a method of calculating an expected value or the like by sampling using a Markov chain Monte Carlo method. If a sample can be efficiently taken from a region having a high probability, an approximate value with sufficiently high accuracy can be obtained even if the number of samples is small.

  However, this region is a very small region in the high-dimensional region, and it is difficult to take a sample from this region in a simple random walk. In this example, the derivative can be calculated as shown in Equation 19. In this case, it is effective to use a hybrid Monte Carlo (hereinafter referred to as “HMC”) which is a kind of Markov chain Monte Carlo method. "HMC" is a hybrid method that combines Metropolis standards and Hamiltonian dynamical system simulation. In general, in the Monte Carlo method, a sample from a posterior distribution is obtained, and then approximation of Expression 17 is performed using the following expression (Expression 21).

In Equation 21, “S” is the number of samples used for prediction, and “θ ^(S) ” is the sth θ sample.

In “HMC”, gradient information is used to determine a direction to be followed in order to find a region having a high probability. On the other hand, moment variables p = {p _k }, (k = 1,..., K) are introduced to determine the state of the motion to be searched. From the moment variable, the kinetic energy ki (p) is determined by the following equation (Equation 22). However, in Expression 22, “m = {m _k }, (k = 1,..., K)” means a virtual mass.

The probability distribution P (p) of the kinetic energy can be modeled by a Gibbs distribution as shown in the following equation (Equation 23). In Expression 23, “Z _k ” is a normalization constant.

  In addition, “m” in Expression 22 may be all set to “1” for convenience, and when set to “1”, the probability distribution of Expression 23 is a standard normal distribution. On the other hand, “θ” is made to correspond to the position variable, the potential energy is E (θ) = − lnP (θ | D), and the probability distribution is exp (−E (θ)) proportional to P (θ | D). And Then, the Hamiltonian that is the total energy can be determined from the potential energy and the kinetic energy as shown in the following equation (Equation 24).

  Based on this Hamiltonian, a dynamical system simulation using the virtual time “τ” is performed. This system is governed by the differential equation:

  However, this differential equation is generally a complex equation and cannot be solved directly. Usually, a discrete approximation method called leapfrog of the following equation (Equation 26) is used. In Expression 26, “ε” is a step size.

This is repeated L times to create a proposed state (θ ^* , p ^* ) after time ετ from the current state (θ, p). The acceptance / rejection of this state is determined according to the Metropolis standard. The Metropolis standard is a method of accepting the proposed state with the probability of the following equation (Equation 27).

  Furthermore, this time, the persistence parameter α (0 ≦ α ≦ 1) by Horowits is introduced in order to converge more quickly with the target distribution. In a normal HMC, a new moment variable is sampled every time L iterations are performed, but when the parameter α is introduced, the moment is updated by the following equation (Equation 28). In Expression 28, “ν” is a random variable obtained from N (0, m).

  By introducing this parameter α, the search direction to the next state in the θ space becomes similar, thus contributing to avoiding a random walk. From the obtained samples, the mean and variance of the posterior distribution of the estimated average estimated value are estimated by the following equations (Equation 29, Equation 30).

  Since it is considered that the initial sampling algorithm of the sth prediction variance has not converged to the target distribution, usually some percentage of the initial sample is discarded without being used for prediction. This is called burn-in.

Hereinafter, a processing method for power load prediction of this example will be described.
In this example, a short-term power load prediction is performed using GP, and a power load prediction method capable of expressing load uncertainty with error bars is provided. Since the power load prediction problem includes uncertain elements such as seasonal variations, a certain amount of error always occurs. It is certainly important to suppress the error, but it is also important to predict the range of observations that can be taken to express uncertainty and grasp the trend of the observations. In addition, such information can be useful information for the operator. For model construction of GP, MAP estimation and HMC are used. The flow of each method is shown below.

"MAP estimation procedure"
FIG. 2 is a flowchart illustrating an example of power load prediction processing based on MAP estimation. In the power load prediction process, the computing device 2 first determines the hyper parameter prior distribution format, its parameters, and the covariance function format (step S11). Next, the arithmetic unit 2 maximizes Expression 18 starting from an appropriate initial value to obtain θ _MAP (step S12). Furthermore, the arithmetic unit 2 obtains the average and variance of the predicted distribution using Equation 16 using θ _MAP (step S13).

"HMC procedure"
FIG. 3 is a flowchart illustrating an example of power load prediction processing by the HMC. In the power load prediction process, the computing device 2 first determines the hyper parameter prior distribution format, its parameters, and the covariance function format (step S21a). The computing device 2 determines the number of HMC iterations, leapfrog iterations, step size, persistence, and burn-in ratio (step S21b). The initial value of θ is determined by an appropriate method such as a random number. In addition, the arithmetic unit 2 generates an initial value of the moment variable p with a standard normal random number (step S22). Then, the arithmetic device 2 executes the following (steps S23 to S28) until the number of HMC iterations determined in step S21 is satisfied.

In steps S23 to S28, the arithmetic unit 2 first sets the current state as (θ, p), performs leapfrog with a step size “ε” L times, and creates a proposed state (θ ^* , p ^* ) (step S23). S23). Next, the arithmetic unit 2 accepts the proposed state if − (H (θ ^* , p ^* ) − H (θ, p)) <0, and (θ, p) ← (θ ^* , p ^* ) and (Step S24).

The arithmetic unit 2 accepts the proposed state if rnd () <esp [− (H (θ ^* , p ^* ) − H (θ, p))], and (θ, p) ← (θ ^* ) ^. , P ^* ) (step S25). Otherwise, the arithmetic unit 2 rejects the proposed state and sets (θ, p) ← (θ, −p) (step S26). Note that rnd () is a uniform random number in the interval [0, 1].

  Next, the computing device 2 updates the moment according to Equation 28 (step S27). Then, if the burn-in is not performed, the arithmetic unit 2 employs θ as a sample (step S28).

  If the number of HMC iterations determined in step S21 is satisfied (step S29), the arithmetic unit 2 evaluates the prediction average and the prediction variance of the posterior distribution using the equations 29 and 30 (step S30).

  Hereinafter, simulation results of power price prediction in which short-term power load prediction is performed using the GP according to the present embodiment and load uncertainty is expressed by error bars will be described.

"Simulation conditions"
First, simulation conditions will be described. In this simulation, the maximum power load of weekday data from the summer of 1999 to 2002 (June to September) was used as sample data, and the data of 2002 was used as test data. However, special day data such as Obon is excluded from the data. FIG. 4 is a diagram showing input variables to be used. Note that the days of the week are converted into bits as shown in FIG. Therefore, the input dimension is 11 dimensions.

Moreover, the function of following Formula (Formula 31) was used as a covariance function used for GP. In Equation 31, “G” is an input dimension, lnv, lnw, and lna are hyperparameters, and “δ _ij ” is a Kronecker delta.

  The covariance function of Equation 31 is a symmetric semi-definite value when “v”, “w”, and “a” are positive. Therefore, the positive values of “v”, “w”, and “a” are designed to be naturally satisfied by using lnv, lnw, and lna as hyperparameters.

  Also, a weight “w” is provided for each input dimension independently. By doing so, “w” of a variable that contributes greatly to prediction takes a large value, while “w” of a variable that does not contribute much to prediction takes a small value. That is, the prediction can be made flexible. In addition, since the day of the week has been converted into bits as described above, it can be expected that knowledge of which day of the week contributed to the prediction will be obtained.

  The prior distribution of “w” is an independent gamma distribution, and the prior distributions of “v” and “a” are independent normal distributions. The gamma distribution is expressed by Expression 32, and the normal distribution is expressed by Expression 33. “Α” is a shape parameter, “β” is a scale parameter, “μ” is an average, “σ” is a standard deviation, and “Γ ()” is a gamma function.

  FIG. 6 is an explanatory diagram showing a parameter relating to the prior distribution of the hyperparameters of the GP. FIG. 7 is an explanatory diagram showing parameters in the GP using HMC (hereinafter referred to as GP (HMC)). The conjugate gradient method was used for optimization of GP using MAP estimation (hereinafter referred to as GP (MAP)). When calculating the error, the average value of the obtained distribution was used for calculating the error in GP (MAP) · GP (HMC).

  On the other hand, as a comparative method, a multilayer perceptron (hereinafter referred to as MLP) having one intermediate layer and a radial basis function network (hereinafter referred to as RBFN) were used. The MLP learning algorithm uses the back propagation method, and the RBFN learning algorithm first gives the initial values of the center and width of the basis function as K-means, and alternately performs the update and calculation of the connection weight. The conjugate gradient method was used to update the center and width, and the least square method was used to calculate the joint weight. Further, both MLP and RBFN were regularized (penalty coefficient: λ) by adding the sum of squares of the connection weights as a penalty to the evaluation function. The parameters of these comparison methods are shown in FIG. However, the input / output variables were standardized and used with an average of 0 and a variance of 1 in all methods.

"simulation result"
A simulation result based on the above simulation conditions will be described. First, FIG. 9 shows the average error and the maximum error of each method. According to FIG. 9, GP (MAP) left results exceeding 0.4063% and 0.3512% in average error and 1.27996% and 0.9535% in maximum error as compared with MLP and RBFN as comparative methods.

  Similarly, GP (HMC) showed an average error of 0.4086% and 0.3535%, and a maximum error of 1.3038% and 1.2437%. Therefore, it can be seen that GP has a higher prediction capability in short-term power load prediction than MLP and RBFN. Further, when paying attention to the GP, both the average error and the maximum error were slightly, but the GP (HMC) exceeded the GP (MAP).

  In addition, FIG. 10 and FIG. 11 show the prediction results of all days in GP (MAP) and GP (HMC) and error bars (3σ) that are indicators of uncertainty, respectively. As shown in these figures, the feature of this example is that uncertainty can be expressed using error bars.

  As described in the error report, the difference between the two methods was also slight in the daily forecast results. Looking at the error bars, it can be seen that the observed values are within ± 3σ for both methods on all days. Actually, the ratio of the observed values within ± nσ was 73.42% within ± σ, 93.67% within ± 2σ, and 100.00% within ± 3σ. From these results, it can be seen that the GP has a high ability not only for the prediction result of the next day maximum power but also for the calculation of its uncertainty.

  Finally, the average value of the hyperparameter “w” in GP (MAP) is shown in FIG. From the figure, it can be seen that in GP (MAP), the maximum temperature, average temperature, discomfort index, and maximum power on the day contributed to the prediction. On the other hand, paying attention to the bit days of the week, the value of the hyper parameter “w” was small on every day, but Monday and Friday showed higher values than other days. Therefore, it can be said that the beginning of the week and the weekend have more important information than other days of the week.

  From the above, it was shown that GP is superior not only in the prediction capability as compared with the conventional method, but also in that additional information can be obtained regarding error bars that are the upper and lower limit values of the prediction value.

  As described above, in the present embodiment, since the short-term power load prediction is performed by applying the GP, the following effects can be obtained. That is, since the weight is determined for each dimension in the covariance function by the GP and determined from the Bayes' theorem, it is possible to perform more flexible prediction than in the past. Further, since the power load prediction is performed by the short-term power load prediction by GP, both the average error and the maximum error can be reduced as a result of comparison with the actual data, and the prediction with high accuracy can be performed. Furthermore, since prediction is performed using prediction distribution instead of point estimation, error bars can be calculated, and uncertainty in short-term power load prediction can be expressed.

It is a block diagram of the electric power load prediction system concerning one embodiment of the present invention. It is a flowchart which shows the example of the electric power load prediction process by MAP estimation. It is a flowchart which shows the example of the electric power load prediction process by HMC. It is explanatory drawing which shows the example of an input variable. It is explanatory drawing which shows the example of the information bit-ized about the day of the week. It is explanatory drawing which shows the parameter regarding the prior distribution of the hyper parameter of GP. It is explanatory drawing which shows the parameter in GP (HMC). It is explanatory drawing which shows the example of the parameter of a comparison method. It is explanatory drawing which shows the average error and maximum error of each method. It is explanatory drawing which shows the prediction result and error bar in GP (MAP). It is explanatory drawing which shows the prediction result and error bar in GP (HMC). It is explanatory drawing which shows the average value of the hyper parameter in GP (MAP).

Explanation of symbols

DESCRIPTION OF SYMBOLS 1 ... Input device 2 ... Arithmetic device 3 ... Memory | storage device 4 ... Output device

Claims (4)

A power load prediction method for predicting a power load using a computer having a storage device, a computing device, and an output device,
Based on the prior distribution stored in the storage device, the arithmetic device calculates the mean and variance of the predicted distribution by a Gaussian process constructed by the arithmetic device by MAP estimation or hybrid Monte Carlo method;
A power load prediction method comprising: outputting the average and variance of the calculated prediction distribution via the output device. The power load prediction method according to claim 1, wherein the arithmetic device constructs a Gaussian process by obtaining a MAP estimated value by MAP estimation. The power load prediction method according to claim 1, wherein the arithmetic device calculates an average and a variance of a prediction distribution by sampling by a hybrid Monte Carlo method. A power load prediction processing program for predicting a power load used in a computer having a storage device, a computing device, and an output device,
In the computer,
Based on the prior distribution stored in the storage device, the arithmetic device calculates the mean and variance of the predicted distribution by a Gaussian process constructed by the arithmetic device by MAP estimation or hybrid Monte Carlo method;
A power load prediction processing program for executing the step of outputting the average and variance of the calculated prediction distribution via the output device.

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