JP2008234094A - Prediction model selection method, prediction method, parameter estimation device, and program thereof - Google Patents

Abstract

A model is selected by calculating an AIC with respect to a common regression equation, selecting an estimated value of the original difference equation, and calculating a fitness with the data, and selecting a model at the stage of regression analysis. .
An actual data input / storage unit for inputting actual data stored in an AIC calculation unit and a parameter estimation / estimated value calculation unit; a model determination command unit for commanding determination of a selected model; A model information input unit that inputs model information to the AIC calculation unit and the parameter estimation / estimation value calculation unit, and AIC calculation is performed by the determination command from the model determination command unit, and the parameter estimation / estimation value calculation unit The AIC calculation unit that outputs the most suitable model for the model, and the model information input unit receive information on how to convert the parameters of the regression analysis into the parameters of each model. A parameter estimation / estimated value calculation unit for calculating the estimated value and a result output unit are provided.
[Selection] Figure 1

Description

  The present invention is a phenomenon of monotonically increasing that can be approximated by an exponential or S-shaped curve, and finally becoming a certain constant value (saturation value). The present invention relates to a prediction method, a parameter estimation device, and a program thereof.

  Computer virus infection number prediction model that predicts the number of computer virus infections, software reliability growth model that predicts software reliability or the number of bugs, population prediction model that predicts population changes, organisms that predict changes in the number of still life individuals The population prediction model and the spread prediction model for predicting the spread of new products, technologies, and services are often described by an exponential curve or an S-shaped curve. A plurality of models have been proposed, and when data is actually given, which model is most suitable for the data is often determined by an empirical rule, and there are few criteria for model selection.

  As a standard for model selection, in the software reliability growth model, Shigeru Yamada: Software Reliability Model-Criteria and Application, Nikka Giren) pp. 125-129 (see Non-Patent Document 1). These are for evaluating the quality of a plurality of models after the curve finally reaches a constant value (saturation value), or for selecting a model. Therefore, when data is actually given and prediction is necessary, it is not sufficient as an index for determining which one of a plurality of models is preferable.

  As an index for selecting a model from actual data, Japanese Patent Application Laid-Open No. 2003-186701 (see Patent Document 1) or D.S. Saoh and S.M. Yamada: Discrete Equations and Software Reliability growth models, Proceedings of 12th International Symposium on Reliable Empowering, and IE Hereinafter, this method will be described. If the event of interest is a software reliability growth model, it is the cumulative bug count, but for each phenomenon listed at the beginning, it should be read as the event of interest. In the following, the software reliability growth model will be described as an example.

In this model selection method, the actual data is an index for determining which of the logistic curve model showing the S-shaped curve and the Gompertz curve model is applied. The index is shown below, and a model having a smaller value of C _L (n) and C _G (n) is a suitable model.

ここで、Ｄ_ｉは第ｉ期における実データの値、

Here, _Di is the actual data value in the i period,

はそれぞれロジスティック曲線モデル、ゴンペルツ曲線モデルによって得られた第ｉ期における推定値である。この推定値は、厳密解を持つ差分方程式を用いて回帰分析によって与えられる。まず、ロジスティック曲線モデルの回帰式は以下の通りである。
Ｙ_ｎ＝Ａ＋ＢＬ_ｎ＋１ （３）
ここで、

Are estimated values in the i-th period obtained by the logistic curve model and the Gompertz curve model, respectively. This estimate is given by regression analysis using a difference equation with an exact solution. First, the regression equation of the logistic curve model is as follows.
Y _n = A + BL _{n + 1} (3)
here,

式（３）を用いて回帰分析を行い、得られたＡ，Ｂの推定値から元の差分方程式のパラメータであるｋ，α，ｍを求め、厳密解

Regression analysis is performed using equation (3), and k, α, m that are parameters of the original difference equation are obtained from the obtained estimated values of A and B, and an exact solution is obtained.

により推定値を求め、Ｃ_Ｌ（ｎ）を計算する。
一方、ゴンペルツ曲線モデルの回帰式は、

To obtain an estimated value and calculate C _L (n).
On the other hand, the regression equation of the Gompertz curve model is

で与えられる。ここで、

Given in. here,

である。式（８）を用いて回帰分析を行い、得られたＡ，Ｂの推定値から元の差分方程式のパラメータであるｋ，ａ，ｂを求め、厳密解

It is. Regression analysis is performed using equation (8), and k, a, b, which are parameters of the original difference equation, are obtained from the obtained estimated values of A, B, and an exact solution is obtained.

により推定値を求め、Ｃ_Ｇ（ｎ）を計算する。

To obtain an estimated value and calculate C _G (n).

JP 2003-186701 A Shigeru Yamada: Software Reliability Model-Standards and Applications, Nikka Giren's pp. 125-129 D. Saoh and S.M. Yamada: Discrete Equations and Software Reliability growth models, Proceedings of 12th International Symposium on Reliability, 17th International Recombination Empowering, (IEEE

As described above, in the conventional technique, the model is determined by comparing the sum of the relative square errors between the actual data and the estimated value. When the determination is performed based on such an error, the number of parameters used in the models to be compared needs to match. This is because as the number of parameters increases, the conformity increases accordingly. For this reason, the conventional technique has a problem in that it cannot be compared between models having different numbers of parameters.
In addition, in the prior art, in order to compare the actual data and the estimated value, the parameter obtained by the regression equation is converted to the parameter of the original difference equation, and the estimated value is not substituted until the converted parameter is substituted into the exact solution. And the estimated value must be compared with the actual data. For this reason, in order to select a model, there is also a problem that it has to be converted into an estimated value for each model only for the model selection.

(the purpose)
It is an object of the present invention to select a model by calculating an AIC for a common regression equation, to compare models with different models, and to estimate the original difference equation. It is an object of the present invention to provide a prediction model selection method, a prediction method, a parameter estimation device, and a program thereof that enable selection of a model at the stage of regression analysis without calculating the degree of fit with data.

In the prediction model selection method of the present invention, when rewriting a differential equation to a difference equation, by performing a difference method that is not a normal forward difference or a center difference, a method that satisfies the above-described regression equation properties is derived, In order not to consider the discretization error from the differential equation, the difference equation derives a difference equation that has an exact solution and uses this difference equation to convert it into a regression equation for parameter estimation. Enables model selection based on AIC (Akaike Information Criterion).
Note that AIC is used to compare models with different numbers of parameters, but the regression equation must include all models in order to calculate AIC. In other words, by setting any one of the regression equation parameters to 0, the model must be another model.

  According to the present invention, a model can be selected by calculating an AIC with respect to a common regression equation, and models can be compared even with models having different numbers of parameters. Furthermore, it is possible to select a model at the stage of regression analysis without obtaining the estimated value of the original difference equation and calculating the degree of fitness with the data.

Hereinafter, embodiments of the present invention will be described in detail with reference to mathematical expressions and drawings.
First, a common regression equation obtained from a plurality of models will be described.
Here, one common regression equation for estimating parameters of the exponential model and the S-shaped model (logistic curve model, Riccati equation model) is shown. One of the common parameters for estimating the parameters of exponential model and sigmoid model (logistic curve model, Riccati equation model) as the cumulative number of bugs discovered up to the nth period (for example, one day or one week) The regression equation is

と表される。ここで、

It is expressed. here,

である。この回帰式を用いて回帰分析、ＡＩＣの計算によるモデル選択を行う。
式（１３）は、

It is. Using this regression formula, model selection is performed by regression analysis and AIC calculation.
Equation (13) is

のとき指数形モデルの回帰式

Regression formula for exponential model

となる。回帰分析によりＡ，Ｂが推定できれば、式（１７）（１８）をａ_ｅ，ｂ_ｅそれぞれについて解き、

It becomes. If A and B can be estimated by regression analysis, equations (17) and (18) are solved for a _e and b _e, respectively.

また、δ＝１とおくことで、ａ_ｅ，ｂ_ｅの推定値を求めることができる。注目している事象の飽和値はａ_ｅの推定値によって与えられる。指数形モデルの差分方程式とその厳密解は、

Further, by placing a [delta] = _1, it is possible to obtain an estimate of a e, _{b e.} Saturation value of the target to which the event is given by the estimated value of a _e. The difference equation of the exponential model and its exact solution are

であり、時間間隔δ→０で微分方程式とその厳密解

The differential equation and its exact solution at time interval δ → 0

に一致する。
推定値に必要なｍ_ｅはデータ数をＮ、既に得られているデータＭ_ｎ，ｎ＝１，２，・・・，Ｎとして、

Matches.
M _e necessary for the estimated value is N, and the data M _n , n = 1, 2,.

によって求める。求められたｍ_ｅ，ａ_ｅ，ｂ_ｅを式（２２）に代入することにより、ｎ期目の推定値を求めることができる。Ｍｎがある値になるのに、今後どれだけの時間がかかるかといった予測が得られる。
次に、式（１３）において、

Ask for. The obtained _{m e,} a e, a _{b e} by substituting the equation (22), it is possible to obtain the estimate of n period th. A prediction can be obtained as to how long it will take for Mn to reach a certain value.
Next, in equation (13):

とすると、ロジスティック曲線モデルの回帰式

Then, regression formula of logistic curve model

となる。回帰分析によりＢ，Ｃが推定できれば、式（２８）（２９）をｋ_ｌ，α_ｌそれぞれについて解き、

It becomes. If B and C can be estimated by regression analysis, equations (28) and (29) are solved for k _l and α _l respectively.

また、δ＝１とおくことで、ｋ_ｌ，α_ｌの推定値を求めることができる。注目している事象の飽和値ｋ_ｌの推定値が求められる。
ロジスティック曲線モデルの差分方程式とその厳密解は、

Further, by setting δ = 1, estimated values of k _l and α _l can be obtained. The estimated value of the saturation value k _l of an event of interest is required.
The difference equation of the logistic curve model and its exact solution are

であり、時間間隔→０で微分方程式と厳密解

And the differential equation and exact solution at time interval → 0

に一致する。推定値に必要なｍ_ｌはデータ数をＮ、既に得られているデータＭ_ｎ，ｎ＝１，２，・・・，Ｎとして、

Matches. The required _ml for the estimated value is N, and the data M _n , n = 1, 2,.

によって求める。求められたｍ_ｌ，ｋ_ｌ，α_ｌを式（３３）に代入することにより、ｎ期目の推定値を求めることができる。これにより、Ｍ_ｎがある値になるのに今後どれだけの時間がかかるかといった予測が得られる。
次に、式（１３）において、

Ask for. By substituting the determined m _l , k _l , and α _l into the equation (33), the estimated value for the n-th period can be determined. This provides a prediction of how long it will take for M _n to reach a certain value.
Next, in equation (13):

のとき、リカッチ方程式モデルとなる。このリカッチ方程式モデルは、分野により名称が異なる。例えば、ソフトウェア信頼度成長モデルでは、習熟Ｓ字形モデルと呼ばれ、新製品、技術、およびサービスの普及を予測する普及予測モデルでは、Ｂａｓｓモデルとも呼ばれる。共通することは、モデルを記述する微分方程式がリカッチ方程式であることである。分野によりパラメータが異なることがあるが、適当な変換により別の分野でのモデルのパラメータに書き換えることができる。

Then the Riccati equation model. This Riccati equation model has different names depending on the field. For example, the software reliability growth model is called a proficiency S-shaped model, and the popularization prediction model for predicting the diffusion of new products, technologies, and services is also called a Bass model. What is common is that the differential equation describing the model is the Riccati equation. The parameters may vary depending on the field, but can be rewritten to model parameters in another field by appropriate conversion.

となる。
回帰分析に母Ａ，Ｂ，Ｃが推定できれば、式（４０）（４１）（４２）をＫ_ｒ，ｋ_ｒ，α_ｒそれぞれについて解き、

It becomes.
If the mothers A, B, and C can be estimated in the regression analysis, the equations (40), (41), and (42) are solved for K _r , k _r , and α _r, respectively.

また、δ＝１とおくことで、Ｋ_ｒ，ｋ_ｒ，α_ｒの推定値を求めることができる。注目している事象の飽和値ｋ_ｒ＋Ｋ_ｒがこれにより得られる。
式（１３）のパラメータとリカッチ方程式モデルの差分方程式とその厳密解は

Further, by setting δ = 1, estimated values of K _r , k _r , and α _r can be obtained. The saturation value k _r + K _{r of the} event of interest is thereby obtained.
The parameters of equation (13), the difference equation of the Riccati equation model and its exact solution are

に一致する。推定値に必要なｍｒはデータ数をＮ、既に得られているデータＭｎ，ｎ＝１，２，・・・，Ｎとして

Matches. The mr required for the estimated value is N, and the data Mn, n = 1, 2,.

によって求める。求められたｍ_ｒ，Ｋ_ｒ，ｋ_ｒ，α_ｒを式（４７）に代入することにより、ｎ期目の推定値を求めることができる。これにより、Ｍ_ｎがある値になるのに、今後どれだけの時間がかかるかといった予測が得られる。

Ask for. By substituting the determined m _r , K _r , k _r , and α _r into the equation (47), it is possible to determine the estimated value for the nth period. As a result, it is possible to predict how long it will take for M _n to reach a certain value.

FIG. 1 is a block diagram of a model selection / prediction apparatus according to an embodiment of the present invention.
FIG. 2 is a flowchart showing the operation procedure of the model selection / prediction method.
The number of occurrences of an event of interest (a number of found bugs in the case of a software reliability growth model) is input to the actual data input / storage unit 11 for each predetermined period such as one day or one week. The actual data input unit stores the data value input for each period and the accumulated data value up to each period (step 101). In response to a model determination request (step 102) from the model determination command unit 12, the AIC calculation unit (model determination unit) 14 calculates the AIC using the data stored in the actual data input / storage unit 11. The AIC calculation unit (model determination unit) 14 receives model information input from the model information input unit 13, that is, which model uses all the parameters A, B, and C of the equation (13), or Information about whether the parameter is 0 is stored.

  In the case of a multiple regression analysis such as Equation (13), the AIC can be calculated as follows.

ここでＮはその時点でのデータの数、ｐはパラメータ数、Ｓｅは残差平方和で、

Where N is the number of data at that time, p is the number of parameters, Se is the residual sum of squares,

と表される。ここで

It is expressed. here

は回帰分析により推定されたパラメータの値である。
指数形モデル、ロジスティック曲線モデル、リカッチ方程式モデルのそれぞれについてＡＩＣをＡＩＣ計算部（モデル判定部）１４で求める（ステップ１０３，１０４，１０５）。
指数形モデルのときは、
ｐ＝２ （５３）

Is the value of the parameter estimated by regression analysis.
For each of the exponential model, logistic curve model, and Riccati equation model, the AIC is calculated by the AIC calculation unit (model determination unit) 14 (steps 103, 104, and 105).
For the exponential model,
p = 2 (53)

ロジスティック曲線モデルのときは、

For a logistic curve model,

リカッチ方程式モデルのときは、

For the Riccati equation model,

である。
それぞれ求められたＡＩＣの値が最も小さいモデルが最も実データに適合するモデルである。ＡＩＣ計算部（モデル判定部）１４は、パラメータ推定・推定値算出部１５に対して最も適合するモデルを出力する（ステップ１０８）。
パラメータ推定・推定算出部１５は、最も適合するモデルを受け取り、そのモデルのパラメータおよび各時間周期ごとの推定値を計算する。パラメータ推定・推定値算出部１５は、モデル情報入力部１３から回帰分析のパラメータＡ，Ｂ，Ｃから各モデルのパラメータにどのように変換するのか、という情報を得て記憶している。
パラメータ推定・推定算出部１５は、最も適合するモデル名、最も適合するモデルのパラメータ、推定値を結果出力部１６で出力する（ステップ１０７，１０８，１０９）。

It is.
The model having the smallest AIC value obtained is the model most suitable for the actual data. The AIC calculation unit (model determination unit) 14 outputs the model most suitable for the parameter estimation / estimated value calculation unit 15 (step 108).
The parameter estimation / estimation calculation unit 15 receives the most suitable model, and calculates a parameter of the model and an estimated value for each time period. The parameter estimation / estimated value calculation unit 15 obtains and stores information on how to convert the parameters A, B, and C of the regression analysis from the model information input unit 13 to the parameters of each model.
The parameter estimation / estimation calculation unit 15 outputs the most suitable model name, the most suitable model parameter, and the estimated value in the result output unit 16 (steps 107, 108, and 109).

  The procedure of the model selection / prediction method in FIG. 2 is programmed, stored in a recording medium such as a CD-ROM, mounted on the computer in the model selection / prediction apparatus of the present invention, and the program is installed in this computer. , The present invention can be easily realized. Moreover, the program can be generalized by downloading the program to another computer via a network.

1 is a block diagram of a model selection / prediction device according to an embodiment of the present invention. It is a flowchart which shows the procedure of the model selection / prediction method which concerns on embodiment of this invention.

Explanation of symbols

11 actual data input / storage unit 12 model determination command unit 13 model information input unit 14 AIC calculation unit (model determination unit)
15 Parameter estimation / estimated value calculation unit 16 Result output unit

Claims (8)

A predictive model selection method that selects a model that best describes a phenomenon that monotonously increases and finally becomes a certain value, and predicts an event that occurs in the phenomenon,
By inputting data from the actual data input / storage unit to multiple models approximating the phenomenon input from the model information input unit, the differential equations describing the multiple models are converted into differential equations. Converted,
The AIC calculation unit connected to the model information input unit and the actual data input / storage unit uses a common regression equation for estimating a parameter from the plurality of difference equations, Calculate the AIC, select the model with the lowest AIC value,
The parameter estimation / estimated value calculation unit connected to the AIC calculation unit obtains the parameter of the difference equation that is the original model from the regression equation parameters according to the model selected by the AIC calculation unit. A prediction model selection method characterized by estimating an event that occurs and predicting when the event reaches a certain value. In the prediction model selection method according to claim 1,
A predictive model selection method comprising at least an exponential curve and an S-shaped curve as the model. In the prediction model selection method according to claim 1 or 2,
A prediction model selection method comprising: selecting a model in which a differential equation converted from a differential equation representing the model has an exact solution. In the prediction model selection method in any one of Claim 1 to 3,
The model describing the phenomenon includes a computer virus infection number prediction model that predicts the number of computer virus infections, a software reliability growth model that predicts software reliability or the number of bugs, a population prediction model that predicts population changes, A prediction model selection method characterized by being at least one of an organism population prediction model for predicting a change in the number of individuals and a diffusion prediction model for predicting the spread of new products, technologies, and services. Select the model that best describes the phenomenon that monotonically increases and eventually reaches a certain value, and predicts how long it will take for the exact solution of the model's difference equation to reach a certain value A method,
By inputting data from the actual data input / storage unit to multiple models approximating the phenomenon input from the model information input unit, the differential equations describing the multiple models are converted into differential equations. Converted,
In the AIC calculation unit connected to the model information input unit and the actual data input / storage unit, a common regression equation for estimating parameters from the plurality of difference equations is used. Calculate AIC, select model,
The parameter estimation / estimated value calculation unit connected to the AIC calculation unit obtains a regression equation parameter according to the model selected by the AIC calculation unit, and at the time when an event occurring in the phenomenon becomes a certain value. A prediction method characterized by performing prediction. A parameter estimation device that selects a model that best describes a phenomenon that monotonously increases and finally becomes a certain value, and estimates parameters of the selected model.
An actual data input / storage unit for inputting actual data stored in the AIC calculation unit and the parameter estimation / estimation value calculation unit;
A model determination command unit for commanding determination of the selected model;
A model information input unit for inputting model information to the AIC calculation unit and the parameter estimation / estimated value calculation unit;
In response to a determination command from the model determination command unit, actual data from the actual data input / storage unit and model information from the model information input unit are received, AIC calculation is performed, and a parameter estimation / estimated value calculation unit An AIC calculation unit that outputs a model most suitable for
The model information input unit receives information on how to convert the parameters of the regression analysis into the parameters of each model, receives the most suitable model from the AIC calculation unit, receives the model parameters and each time period A parameter estimation / estimated value calculation unit for calculating an estimated value;
A model parameter estimation apparatus comprising: a model name of the most suitable model output from the parameter estimation / estimation value calculation unit; a parameter of the most suitable model; and a result output unit that outputs an estimation value. A program for selecting a prediction model that selects a model that best describes a phenomenon that monotonously increases and finally becomes a certain value, and predicts an event that occurs in the phenomenon,
A procedure for receiving a plurality of models approximating a phenomenon from the model information input unit to a computer of the actual data input / storing unit and the AIC calculation unit, a procedure for receiving data from the actual data input / storing unit by the AIC calculation unit, Using a procedure for converting each differential equation describing the model into a respective difference equation and a common regression equation for estimating parameters from the plurality of difference equations, an AIC is calculated for the regression equation A program for selecting a prediction model for executing a procedure and a procedure for selecting a model having the smallest AIC value. A program for selecting a prediction model that selects a model that best describes a phenomenon that monotonously increases and finally becomes a certain value, and predicts an event that occurs in the phenomenon,
A procedure for receiving data of the model selected by the AIC calculation unit in the computer of the parameter estimation / estimation value calculation unit, and obtaining the parameter of the difference equation that is the original model from the parameters of the regression equation according to the selected model, A prediction model selection program for executing a procedure for estimating an event occurring in the phenomenon and a procedure for predicting a point in time when the event becomes a certain value.

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