JP2005036289A - Model parameter decision method and program therefor, and model prediction method and program therefor - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a model parameter decision method capable of obtaining the parameter value of a model formula highly precisely by a multiple regression calculating method or the like by the replenishment of other appropriate data even when past actual data belonging to the same group as the model object is less, and to provide a modrl prediction method capable of predicting the correct model by using the model formula. <P>SOLUTION: The model parameter decision method comprises following steps. Regarding the kind of steel in the model objective, the actual data of M group (N>M>0), in the past actual data of the kind of steel belonging to the same group as the objective kind of steel, are selected (N: necessary number of the data) (S1). The neighborhood (index of the neighborhood) between the past actual data regarding to the kind of steel in the other group and values in the actual data in the same group as the M group, is calculated (S2). Value in (N-M) group extracted from the actual data in the other group having little index of the neighborhood, in order, and this value is added to the actual data in the M group (S3) and the value of the model parameter is decided by using the data in the N group with the multiple regression calculation etc. (S4). Further, the predicted value is corrected with a deviation between the predicted value in the model formula and the observed value of the model objective. <P>COPYRIGHT: (C)2005,JPO&NCIPI

Description

  The present invention provides a model parameter determination method for accurately determining a value of a model parameter in a model equation of a predetermined model object based on past performance data, and a model object using the model equation thus determined. The present invention relates to a model prediction method for predicting observation values. In particular, model parameters suitable for determination of model parameters for predicting observation values such as carbon concentration, temperature, phosphorus concentration, etc. in converters and application to model prediction The present invention relates to a determination method and a model prediction method.

The main role of the converter is to remove the carbon content in the hot metal by blowing and to keep the carbon content and temperature after treatment as the target. This converter is charged with the molten iron discharged from the blast furnace and subjected to treatments such as desiliconization, dephosphorization, and desulfurization.
FIG. 5 is a schematic diagram showing a schematic configuration of the converter.
In the converter 1, oxygen is blown from the lance 2 to the hot metal at a speed exceeding Mach, and the oxygen is combined with the carbon content in the hot metal (decarburization) to generate a gas mainly composed of CO, which is exhaust gas. And discharged outside the furnace. Here, since decarburization (2C + O ₂ → 2CO) is an exothermic reaction, the temperature rises to a very high temperature while the hot metal changes into molten steel by blowing.
During this blowing process, it is difficult to continuously measure the carbon concentration and temperature because of the high temperature and adverse environment. For this reason, it is common to insert a sublance 3 into the converter 1 in the latter half of the blowing process and to measure (detect) the carbon concentration and temperature (observed values) of the molten steel. Based on the measured values (observed values), the time series changes in carbon concentration and molten steel temperature are predicted in real time using a predetermined prediction model, and both the carbon concentration and molten steel temperature reach the target value (target range). Stop blowing. When the molten steel temperature is higher than the target, the molten steel is cooled by introducing steel ore into the converter through the auxiliary material charging hopper 4.

Conventionally, a converter prediction model is created by first creating a model expression based on a predetermined theoretical expression, and the value of the parameter of the model expression (hereinafter referred to as model parameter) is used as the past actual data (actual operation of the converter). Often based on data). Specifically, the model parameters are determined using a multiple regression calculation method, a quadratic programming method, or the like so that the predicted value based on the model formula approximates the value of the past actual data.
Here, in order to determine model parameters with high accuracy in a model object with large disturbances and uncertain factors such as converters, the hot metal and molten steel blowing conditions (components, temperature, properties, etc.) to be performed as much as possible are as much as possible. Requires a lot of past performance data in common.
On the other hand, in converters that carry out high-mix low-volume production, there are various blowing conditions, so there are not many past data on the same blowing conditions as the blowing conditions that will be used in the future. Many.
Therefore, in the past, past performance data was grouped for each of those with relatively similar blowing conditions (substantially the same type), and past data (subjects of the same type) belonging to the same group as the blowing conditions to be performed from now on. Actual data was used to determine model parameters.
Patent Document 1 discloses a batch that uses a time series model formula that considers not only the current operation factor but also the past operation error and the past operation factor in the model formula for predicting the error in the refining calculation. A refining control method for a type melting refining furnace (converter, etc.) has been proposed.
In Patent Document 2, a certain number (N) of data near the current charge input value (neighbor data) is extracted from past performance data, and the outputs of the N neighboring data are averaged. A method has been proposed in which the average value is used as the current charge output estimation value, and the standard deviation of the outputs of the N neighboring data is used as the output estimation error.
Japanese Patent Laid-Open No. 2-190413 JP 2002-236119 A

However, for special materials with a low frequency of operation and new steel grades, the accumulation of past performance data is small, so the number of past performance data necessary for multiple regression calculations does not exist and model parameters can be obtained. There is a problem that the prediction accuracy of the model cannot be improved or the prediction accuracy of the model is remarkably deteriorated. In this case, it was necessary to cope with the deterioration of accuracy by using model parameters of other groups, or by manually correcting the model parameters with experience and trial and error.
For special materials with low operation frequency, it is necessary to go back to the past in order to collect the required accumulated data for the required number. In this case, there was a problem that a highly accurate model could not be obtained because the influence of changes over time of the converter was greatly reflected between past performance data.
On the other hand, the technique disclosed in Patent Document 1 relates to the improvement of the structure of a model formula, and does not particularly characterize a model parameter determination method performed using a statistical analysis method such as a multiple regression calculation method. Therefore, the above-mentioned problem cannot be solved.
Further, the technique disclosed in Patent Document 2 does not have to determine the structure of the model formula, but has a problem that it cannot be used when optimizing the operation input because there is no model formula. . Of course, it is possible to use a powerful technique that simulates the predicted output value for all patterns of operation input, but such is not practical.
In addition, if a model formula (theoretical formula is better), a certain degree of accuracy can be obtained by extrapolation / interpolation calculation even in areas where there is no past actual data, or areas where abnormal data due to measurement errors, etc. are partly included. However, the technique disclosed in Patent Document 2 that does not have a model equation has a problem that prediction accuracy is not compensated at all in a region where there is no past performance data, and it is easily affected by abnormal data. .
Accordingly, the present invention has been made in view of the above circumstances, and the object of the present invention is to provide other appropriate data even when there is little past performance data of an object that is the same as the model object or belongs to the same group. Model parameter determination method that can obtain the parameter value of the model formula with high accuracy by calculation such as multiple regression calculation method and quadratic programming method, etc. The object is to provide a model prediction method for accurately predicting an observation value of a model object using an equation.

In order to achieve the above object, the present invention provides a model parameter determination method for determining a value of a model parameter in a model formula of a predetermined model object based on past performance data, and a predetermined parameter used for determining the value of the model parameter. When past performance data of a target of substantially the same type as the predetermined model object is given as a part of a usage result data group that is a required number of the past performance data, the past of the target of the substantially same type is given. Approximate degree calculation step for obtaining closeness of values between past record data and past past record data of a target other than the target of substantially the same type, and the use record based on the closeness of the value obtained by the approximate degree calculation step A deficient data determining step for determining a deficiency of the data group from the past actual data of the target other than the target of the substantially same type, and the past actual data of the substantially same type A parameter value determining step for determining a value of the model parameter based on the use result data group to which the data determined in the insufficient data determining step is added. is there.
As a result, even if there is a shortage of past performance data for targets of the same type (same group) as the model target, appropriate data (result data for model targets) from the performance data of targets of other types (other groups) Data close to) is replenished.

Here, as the deficient data determination step, for example, among the actual result data of objects other than the target of the substantially same type, the actual usage data is sequentially from the closest value obtained by the approximation calculating step. One that determines the deficiency of the group is conceivable.
In addition, a tolerance value of the proximity of the value is set in advance, and until there is a shortage of actual data of the target other than the target of the substantially same type having the proximity of the value that falls within the range of the tolerance value, It is also possible to calculate the closeness of the value.

Further, as the model formula, for example, an expression expressed by a primary line system combination of the model parameters including a plurality of parameters can be considered.
In addition, the model equation includes the converter static model, the converter dynamic model, the main raw material compounding amount calculation model in the converter, the auxiliary material input amount calculation model in the converter, the alloy iron addition amount calculation model in the converter, Alternatively, a desulfurization (S) model, a dephosphorization (P) model, a desiliconization (Si) model, a temperature model, or the like in hot metal pretreatment that is a pre-process of the converter may be considered.

The present invention may also be understood as a model prediction method for predicting a predetermined observation value of a model object using a model whose model parameters are determined by the model parameter determination method.
In this case, in a model prediction method for predicting a predetermined observation value of a model object using a model expression of the predetermined model object, the predicted value based on the model expression determined by the model parameter determination method described above and the predetermined value A model prediction method characterized by having a predicted value correction step of correcting the predicted value based on the model formula based on the deviation from the observed value of the model object can be considered.

Further, the present invention may be a model parameter determination program for causing a computer to execute each step in the model parameter determination method.
Similarly, it may be understood as a model prediction program for causing a computer to execute each step in the model prediction method.

According to the present invention, as described above, even if there is a shortage of past performance data of a target of substantially the same type as the model target, appropriate data (model target (Data close to the actual data) is replenished, so that the value of the model parameter can be obtained with high accuracy by calculation such as multiple regression calculation method or quadratic programming method.
If the predicted value based on the model formula is corrected based on the deviation between the predicted value based on the model formula whose parameter has been determined by the model parameter determination method according to the present invention and the observed value of the model object, an offset error will occur in the predicted value of the model. Even if it occurs, the error can be compensated, and more accurate model prediction is possible.

Hereinafter, embodiments of the present invention will be described with reference to the accompanying drawings so that the present invention can be understood. The following embodiment is an example embodying the present invention, and does not limit the technical scope of the present invention.
FIG. 1 is a flowchart showing the procedure of the model parameter determination method according to the embodiment of the present invention, FIG. 2 is a diagram schematically showing the Euclidean norm in the n-dimensional space of past performance data, and FIG. FIG. 4 is a scatter diagram showing an example of past performance data used in the model parameter determination method according to the embodiment of the invention, and FIG. 4 uses the model parameter determination method according to the embodiment of the present invention and the conventional model parameter determination method. FIG. 5 is a schematic diagram showing a schematic configuration of a converter that is a model object to which the present invention can be applied.

Hereinafter, the procedure of the model parameter determination method according to the embodiment of the present invention will be described using the flowchart of FIG. 1 assuming that the model object is a converter.
Here, before entering the procedure of FIG. 1, it is assumed that a model formula Ym of a converter blowing process for a steel type A to be produced is given, and a plurality of models included in the model formula Ym are given. Assume that N sets of past performance data (usage performance data group) are required when parameters are obtained by multiple regression calculation based on past performance data. The necessary number N is determined empirically from the model structure, the number of parameters, data variations, and the like. Possible model formulas include a converter static model, a converter dynamic model, a main raw material compounding amount calculation model in a converter, a secondary material input amount calculation model in a converter, or an alloy iron addition amount calculation model in a converter. .
(Step 1)
First, M sets (N>M> 0) of performance data are selected from the past performance data of the target steel type A or a steel type of the same type (which belongs to the same group) that approximates the target steel type A (S1).
In this step, as a rule, past performance data for steel type A (same type) is selected, but if past performance data for steel type A does not exist or is extremely small, the condition of steel type A (steel material) Past performance data on steel grades (steel grades belonging to the same group) that can be said to be of the same type within a predetermined condition range with the composition, blowing temperature, properties, etc. .
Even if there are N or more past performance data for steel type A, past performance data for a predetermined period or more, such as measurement errors, which are considered to have a large effect of changes over time of the converter, and results with obvious abnormalities such as measurement errors. It may be possible not to select data.
Further, when there are sufficient (N or more) recent actual data with no abnormality for steel type A, the present invention is not applicable.
Hereinafter, the M pieces of past result data are referred to as the same group result data (the past result data of a target of substantially the same type as the model target).
Here, each performance data consists of a set of n-dimensional (n) data corresponding to n explanatory variables, and the same group performance data (each set) is _represented by p _k = (p _1k , p _2k ,... , P _nk ), (k = 1, 2,... M). k represents the number (1st to Mth) of the same group performance data.

この（１）式における係数β（β＞０）を２とすると，前記重み付き距離ｒ_ikは，ユークリッドノルムとなる。
図２は，過去の実績データのｎ次元空間におけるユークリッドノルムを模式的に表した図である。
さらに，前記近傍指数Ｊ_iは，次の（２）式により計算する。

ここで，係数αは，（０＜α≦２）とすることが望ましい。
このような近傍指数Ｊ_iの計算を，予め与えられた前記他グループ実績データ全てについて計算する。前記他グループ実績データとしては，前記同一グループ実績データを除く他の全ての実績データを与えることや，計算の効率化のため，鋼種Ａと条件が比較的近似する鋼種についての実績データのみを与えること，さらには，モデル対象（転炉等）の経時変化を考慮して最近の所定期間内の実績データのみを与えること等が考えられる。
また，モデルの説明変数（実績データ）は，標準偏差が１となるように，或いはさらに平均値が０となるように正規化しておくことが望ましい。 (Step 2)
Next, it is an index that represents the closeness of values between past performance data (hereinafter referred to as other group performance data) and other steel types other than steel type A or other steel types belonging to the same group as steel type A. A certain neighborhood index is calculated (S2).
Here, the neighborhood index is an index obtained by calculating a weighted distance in the n-order space for each of the same group performance data (for M pieces), and adding these together.
Specifically, the i-th (the number of the plurality of other group actual data) is the other group actual data x _i = (x _1i , x _2i ,..., X _ni ), and for each axis in the nth-order space. When the weight (weight of each explanatory variable) is w _j (j = 1, 2,..., N), the weighted distance r _ik is calculated by the following equation (1).

When the coefficient β (β> 0) in the equation (1) is 2, the weighted distance r _ik is the Euclidean norm.
FIG. 2 is a diagram schematically showing the Euclidean norm in the n-dimensional space of past performance data.
Further, the neighborhood index J _i is calculated by the following equation (2).

Here, the coefficient α is preferably (0 <α ≦ 2).
Such calculation of the neighborhood index J _i is performed for all the other group performance data given in advance. As the other group performance data, all other performance data except the same group performance data is given, and only the performance data for steel types whose conditions are relatively similar to steel grade A is given for efficient calculation. In addition, it is conceivable that only actual data within the latest predetermined period is given in consideration of changes over time of the model object (converter, etc.).
In addition, it is desirable to normalize the explanatory variables (actual data) of the model so that the standard deviation becomes 1 or the average value becomes 0.

In the example described above, the required number N of actual data is set in advance, but the present invention is not limited to this. For example, the above steps are repeated while adjusting the required number N, and the neighborhood coefficient J _i becomes smaller. Thus, the optimum N may be found.
Similarly, the above steps may be repeated while adjusting the weight coefficient w _j in order to find the optimum combination of weights.

(Step 3)
Next, among the other group performance data for which the neighborhood index J _i has been obtained, (NM) pairs are extracted in ascending order of the neighborhood index J _i , and this is divided into M pieces of the same group performance data. In addition to the data, the data used for the calculation of model parameter values (used data group).
(Step 4)
Finally, using the N sets of usage data groups, multiple regression calculations, least-squares method (linear programming) with linear constraints, principal component regression (PCR), PLS (Partial Least Squares) The model parameter value is calculated (determined) by, for example. It is known that principal component regression and PLS are effective when a strong correlation (multicollinearity) exists between input variables (explanatory variables).
In addition to determining the shortage of the usage record data group in order from the one having the smallest neighborhood index J _i (the one having a close value), an allowable value for the neighborhood index J _i is set in advance, It is also conceivable to perform calculation of the neighborhood index J _i until (NM) sets of performance data having the neighborhood index J _i falling within the allowable value range are obtained.

Next, using Fig. 3 and Fig. 4, the model parameter determination method according to the procedure described above is based on actual converter operation data (past performance data) in the converter temperature prediction model (converter dynamic model). The results applied to the model parameter determination will be described.
FIG. 3 is a scatter diagram showing an example of past performance data used to determine model parameters (the amount of oxygen blown from the lance 2, the amount of molten iron (charge amount) in the converter, the measured value by the sub lance 3, etc.). Yes, the horizontal axis and the vertical axis respectively represent predetermined data item 1 (explanatory variable 1) and data item 2 (explanatory variable 2) in the converter operation data. The value of each data item is normalized to mean value 0 and standard deviation 1.
In FIG. 3, there are four (= M) data for regression of the target steel type (A) corresponding to the same group actual result data, which are indicated by black circles “●” in the figure.
In addition, the regression data of other steel types corresponding to the other group performance data extracted (selected) by the model parameter determination method (according to the above-described procedure) according to the present invention is 96 pieces (= N−M). Medium and dark dots. That is, the actual data number N required for model parameter determination (such as multiple regression calculation) is N = 100.
In the figure, 20 data indicated by white circles “◯” are verification data of the target steel type (A) (model target), and the target steel type (A) used for verification of the determined model parameters. It is actual data of.
Other data are actual data of other steel types other than those extracted for regression.
As shown in FIG. 3, it can be seen that, by the model parameter determination method according to the present invention, other steel type data close to (near) the regression data of the target steel type is extracted for regression.

FIG. 4 shows a model parameter determination method (procedure of FIG. 1) according to an embodiment of the present invention and a conventional model parameter determination method based on actual converter operation data (actual data, FIG. 3 shows an example thereof). It is a graph which compares the model prediction precision at the time of using.
Specifically, it represents the magnitude of the evaluation index (vertical axis) of the prediction accuracy of the converter model when the number of M is changed to 4, 8, 12, and 16 (horizontal axis). As the evaluation index, a standardized standard deviation of the difference (error) between the model predicted value and the value of the verification data is used. The predicted value of this model is obtained by multiple regression calculation using 100 (= N) regression data selected (extracted) by the conventional method and the method of the present invention described above, and the model parameter is applied. This is a predicted value obtained by calculating a predetermined observation value of the converter using the converted converter model. The horizontal axis is normalized as M / N (N = 100).
Here, as a conventional method, a multiple regression calculation is performed using only the regression data of the given M steel types (with (N−M) data deficient), and model parameters are calculated. Asked.
In order to eliminate the bias in the verification results, the method of selecting the M target steel types and the method of selecting the 20 target steel types were random, and the above operation was repeated 10,000 times.
In FIG. 4, the graph representing the prediction accuracy according to the conventional method in the case of M = 4 (M / N = 0.04) is interrupted, but its value is 12.24.
FIG. 4 shows that the model parameter determination method according to the present invention can dramatically improve the model accuracy compared to the conventional method. In particular, in the conventional method, when the data for regression of the target steel type is small (for example, when M = 4 or 8), the prediction accuracy deteriorates sharply. It can be seen that the prediction accuracy is stable even with a small amount of data, and that the model accuracy can be dramatically improved in a region where the number of data of the target steel type is small.
This is because, by supplementing data similar to the target steel type from other steel types, the accuracy of each data for regression is inferior, but using many of them reduces the influence of disturbances and uncertain factors. .
As a result, it is possible to perform decarburization control and temperature control of converters with high accuracy even in special steel types and new steel types that are infrequently operated, enabling high-yield production and high-quality steel production. Become.
In addition, since a model is used, it is possible to predict with high accuracy by extrapolation / interpolation even in a region where past performance data does not exist.

By the way, a model whose parameters are determined by the model parameter determination method according to the present invention may have an offset error (average deviation). That is, regarding the error (deviation) between the predicted value by the model and the actually detected observation value, the standard deviation of the error is small (= the model accuracy is good), but the average value of the error may be large. In such a case, the offset error may be compensated as follows.
In order to simplify the explanation, the model formula Ym (prediction formula for temperature, etc.) of the target steel type A is defined as the following formula (3).
Ym = ΘTX (3)
here,
Θ = [θ0 θ1 θ2... ΘN] T: Parameter (coefficient) vector X = [1 x1 x2... XN] T: A measured value or a value calculated from a measured value or the like. This expression (3) is an expression expressed by a linear system combination of a model parameter Θ composed of a plurality of parameters (θ0, θ1,...).
Further, when the correction term Δ is added additively to the model equation (3), the corrected model equation Yp becomes the following equation (4).
Yp = ΘTX + Δ (4)
Alternatively, it is conceivable to provide a correction term in a multiplicative manner as in the following equation (4 ′).
Yp = ΘTXΔ (4 ′)
In general, if the error is almost constant regardless of the predicted value of model equation (3), the model equation may be corrected additively as in equation (4). If the error tends to increase with increasing, the model equation may be corrected in a multiplicative manner as in equation (4 ′).

Here, consider a case where the model equation is corrected additively as in equation (4). The correction term at a certain time point k is Δ (k), the value of the model expression is Ym (k), and the observed value detected from the model object is Yr (k). In this case, it is conceivable to learn the correction term by updating Δ as needed as in the following equation (5).
Δ (k + 1) = γ {Yr (k) −Ym (k)} + (1−γ) Δ (k), 0 <γ ≦ 1 (5)
Although exponential smoothing is used here, it may be possible to update Δ with a moving average of past finite sections.
As described above, if the predicted value based on the model formula obtained by the model parameter determination method according to the present invention is corrected based on the deviation (error) from the observed value of the model target, Offset error can be compensated, and more accurate model prediction is possible.

  INDUSTRIAL APPLICABILITY The present invention can be used in various fields involving modeling, such as parameter determination and model prediction of a model to be controlled such as a converter, and model parameter determination and model prediction in gene analysis.

The flowchart showing the procedure of the model parameter determination method which concerns on embodiment of this invention. The figure which represented typically the Euclidean norm in the n-dimensional space of past performance data. The scatter diagram showing an example of the past performance data used with the model parameter determination method which concerns on embodiment of this invention. The graph which compares the model prediction precision at the time of using the model parameter determination method which concerns on embodiment of this invention, and the conventional model parameter determination method. The schematic diagram showing the schematic structure of the converter which is a model object which can apply this invention.

Explanation of symbols

DESCRIPTION OF SYMBOLS 1 ... Converter 2 ... Lance 3 ... Sub lance 4 ... Sub raw material input hopper S1, S2, ... Process procedure (step)

Claims (7)

In a model parameter determination method for determining a value of a model parameter in a model equation of a predetermined model object based on past performance data,
When past performance data of a target that is substantially the same as the predetermined model object is given as a part of a use performance data group that is a predetermined required number of the past performance data used to determine the value of the model parameter ,
A degree of approximation calculation step for obtaining a closeness of a value between past performance data of the substantially identical target and past performance data of a target other than the substantially identical target;
A deficient data determination step of determining a deficiency in the use result data group based on the proximity of the value obtained in the approximation degree calculation step from past result data of a target other than the substantially same target;
A parameter value determining step for determining a value of the model parameter based on the use record data group obtained by adding the data determined in the shortage data determination step to the past record data of the substantially same target;
A model parameter determination method characterized by comprising: The deficient data determination step determines deficiencies in the usage result data group in order from the closest to the value obtained by the approximation degree calculation step among the actual result data other than the substantially identical target. The model parameter determination method according to claim 1. The model parameter determination method according to claim 1 or 2, wherein the model formula is represented by a linear system combination of the model parameters including a plurality of parameters. The model formula is a static model of the converter, a dynamic model of the converter, a main raw material compounding amount calculation model in the converter, a secondary material input amount calculation model in the converter, an alloy iron addition amount calculation model in the converter, or a hot metal reserve The model parameter determination method according to claim 1, wherein the model parameter determination method is any one of a desulfurization model, a dephosphorization model, a desiliconization model, and a temperature model in processing. In a model prediction method for predicting a predetermined observation value of a model object by a model formula of the predetermined model object,
A predicted value based on the model formula based on a deviation between the predicted value based on the model formula determined by the model parameter determination method according to any one of claims 1 to 4 and the observed value of the predetermined model target. A model prediction method comprising: a predicted value correction step for correcting. A model parameter determination program for causing a computer to execute a process of determining a value of a model parameter in a model equation of a predetermined model object based on past performance data,
When past performance data of a target that is substantially the same as the predetermined model object is given as a part of a use performance data group that is a predetermined required number of the past performance data used to determine the value of the model parameter ,
Approximation degree calculation processing for determining the closeness of values between past performance data of the substantially identical target and past performance data of a target other than the substantially identical target;
A deficient data determination process for determining a deficiency of the use result data group based on the proximity of the value obtained by the approximation degree calculating step from past result data of a target other than the substantially same target;
A parameter value determination process for determining a value of the model parameter based on the use result data group obtained by adding the data determined in the insufficient data determination step to the past result data of the substantially identical object;
Model parameter determination program for causing a computer to execute. In a model prediction program for causing a computer to execute a process of predicting a predetermined observation value of a model object by a model formula of the predetermined model object,
The predicted value based on the model formula is corrected based on the deviation between the predicted value based on the model formula for which the model parameter has been determined by executing the model parameter determination program according to claim 6 and the observed value of the predetermined model target. A model prediction program for causing a computer to execute processing.

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