JP5175515B2 - Model construction apparatus, model construction method and program - Google Patents

Description

  The present invention relates to a model construction apparatus, a model construction method, and a model construction program suitable for use when, for example, the number of model parameters is large.

  When a certain property of an analysis object can be expressed as a numerical value and the property behaves stochastically, the numerical property is called a random variable. Various stochastic models have been proposed to concisely represent the behavior of one or more random variables using several model parameters θ.

For example, the normal distribution model of the following equation (1) is the probability that the random variable X can take the value x using two model parameters (θ = {μ, σ}), the mean (μ) and standard deviation (σ). Is a probabilistic model that can describe

  There are two types of probability models. One is called a generation model and the other is called a prediction model. The former generation model is a probability model for describing the probability (Pr (X = x)) that X takes a specific value x when there is an attribute variable X that is one or more random variables related to the properties of the object. is there. On the other hand, in the latter prediction model, in addition to the attribute variable X, an objective variable Y that is one or more random variables related to the target property is prepared, and when X takes a specific value x, Y has a specific value y. This is a probability model for describing the probability (Pr (Y = y | X = x)). For example, the one-dimensional normal distribution model and multi-dimensional normal distribution model of Equation (1) are generation models, and linear multiple regression models, generalized linear models, and the like are classified as prediction models.

  The parameters of the probabilistic model can be determined (learned) from learning data collected from a plurality of objects of information obtained by quantifying the property to be modeled. The probabilistic model construction device is a device for determining optimal probabilistic model parameters using read learning data.

  In order to model a complex probabilistic phenomenon using a simple probability distribution such as a normal distribution or a Poisson distribution, a probability model called a mixed model in which two or more simple probability distributions are combined is used. In the mixed model, the behavior of each case may be explained by any one (or combination) of a plurality of simple probability distributions. The model parameters of the mixed model can be determined by EM algorithm or the like using learning data (Non-patent Document 1).

  If the behavior of the random variables related to phenomena on the ground surface such as store sales, land prices, and demographics can be accurately expressed by a probabilistic model, various applications such as store sales, land price prediction, and demographic structural analysis in the target area will be possible. It can be carried out. For that purpose, it is necessary to determine a geospatial model that is a probabilistic model for such a geospatial phenomenon from geospatial data that is learning data including geospatial information.

  In order to model complex geospatial phenomena, it is effective to use a mixed model, but geospatial phenomena may have spatial dependence. For example, when using a mixed model to model a failure variable that is a numerical value of the failure index of equipment at each point in the target area, indicating a complex behavior (such as cannot be expressed with a simple normal distribution) think of. The reason why the failure index distribution does not become a simple normal distribution is that there is a hidden spatial factor called salt damage, and that the probability distribution of the failure index differs between an area close to the sea and an area far from the sea. In such a case, it is necessary to learn the model parameter θ in consideration of the fact that when a certain point is included in the salt damage area, there is a high possibility that the adjacent point is also included in the salt damage area. In order to handle geospatial phenomena with spatial dependence, it is necessary to build a geospatial mixed model that is a mixed model that considers the positional relationship between such points. The EM algorithm cannot perform parameter learning considering spatial dependence.

  As a probabilistic model that takes into account spatial dependence, there is a Markov Random Field (hereinafter referred to as MRF) that is applied to image processing and the like (Non-Patent Document 2). In MRF, the dependence between adjacent points is taken into account by using a spatial dependence parameter λ (hereinafter referred to as MRF parameter). MRF parameters can be determined from image sample data. In general, the learning data for image processing includes information on which pixels and which pixels have different labels, but when building a mixed model using geospatial data, boundary information of such regions is included. This is one of the difficulties in building a geospatial mixed model.

There is a method proposed in Non-Patent Document 3 as a method for constructing a geospatial mixed model in which spatial dependence is considered by MRF. In the method of Non-Patent Document 3, MRF parameter λ is introduced for all combinations of model parameters θ which are n-dimensional continuous value vectors, and θ and λ are estimated by quasi-Newton method and Markov chain Monte Carlo method (hereinafter MCMC method). ing.
Finite Mixture Models, Wiley-Interscience, ISBN: 0471006262 Image Analysis, Random Fields and Markov Chain Monte Carlo Methods: A Mathematical Introduction, Springer, ISBN: 3540442138 Mark S. Kaiser, Noel Cressie, Jaehyung Lee, Spatial Mixture Models based on Exponential Family Conditional Distributions, Statistica Sinica 12, pages 449-474, 2002.

  Although it is possible to build a mixed model by considering geospatial dependence by building a mixed geospatial model using MRF, the existing method uses n × n MRFs for the number n of model parameters θ. The parameter λ was required. In addition, since the exact optimal parameters are obtained by the quasi-Newton method and the MCMC method, there is a problem that a mixed model using a stochastic model with a large number n requires a lot of learning data and calculation time. .

  The present invention has been made to solve the above-described problems, and an object of the present invention is to provide a model construction apparatus, a model construction method, and a model that can efficiently construct a geospatial mixed model even when there are many model parameters. To provide a construction program.

The model construction apparatus as one aspect of the present invention is:
Geospatial data storage means for storing geospatial data having a plurality of cases including at least one variable representing the property of the evaluation object by numerical value and position data indicating a position in geospatial space;
Parameter storage means for storing parameter information representing parameters of a plurality of probability models obtained by modeling the probability distribution of the variables;
Application model information storage means for storing application model information representing the probability model to be applied for each position in the geographic space;
A probability model to be applied to each position in the geospace, and a probability model applied to one or more neighboring positions included in a predefined neighborhood range for each position in the geospace. Model dependence calculation means for calculating model dependence information in which a dependence between the two probability models is numerically expressed for each set of the same or different two probability models based on a relationship;
Model dependence information storage means for storing the model dependence information calculated by the model dependence calculation means;
The probability model to be applied for each position in the geospatial is selected from the plurality of probability models so that the likelihood of the geospatial data with respect to the set of the parameter information and the model dependency information is high. A probability model selection means for updating the applied model information to indicate the probability model selected for each position;
Based on the updated application model information, each of the plurality of groups is configured to divide the geospatial data into a plurality of groups to which the same probability model is applied, and to maximize a predetermined model criterion. Parameter learning means for learning parameters of the probability model corresponding to and updating the parameter information to indicate the learned parameters of each probability model;
Is provided.

A model construction method as one aspect of the present invention includes:
Geospatial data storage means for storing geospatial data having a plurality of cases including at least one variable representing the property of the evaluation object by numerical value and position data indicating a position in geospatial space;
Parameter storage means for storing parameter information representing parameters of a plurality of probability models obtained by modeling the probability distribution of the variables;
Application model information storage means for storing application model information representing the probability model to be applied for each position in the geographic space;
Preparation steps, and
A probability model to be applied to each position in the geospace, and a probability model applied to one or more neighboring positions included in a predefined neighborhood range for each position in the geospace. A model dependency information calculating step for calculating model dependency information that represents numerically the dependency between the two probability models for each set of the same or different two probability models based on the relationship;
Storing the model dependency information in a model dependency information storage means;
The probability model to be applied for each position in the geospatial is selected from the plurality of probability models so that the likelihood of the geospatial data with respect to the set of the parameter information and the model dependency information is high. A probability model selection step of updating the applied model information to indicate the probability model selected for each position;
Based on the updated application model information, each of the plurality of groups is configured to divide the geospatial data into a plurality of groups to which the same probability model is applied, and to maximize a predetermined model criterion. Learning a parameter of the probability model corresponding to and updating the parameter information to indicate the learned parameter of each probability model; and
Is provided.

The model construction program as one aspect of the present invention is:
Accessing geospatial data storage means for storing geospatial data having a plurality of cases including at least one variable representing the property to be evaluated numerically and position data indicating a position in geospatial;
Accessing parameter storage means for storing parameter information representing a parameter of each of a plurality of probability models modeling the probability distribution of the variable;
Accessing application model information storage means for storing application model information representing the probability model to be applied for each position in the geographic space;
A probability model to be applied to each position in the geospace, and a probability model applied to one or more neighboring positions included in a predefined neighborhood range for each position in the geospace. A model dependency calculating step for calculating model dependency information that represents numerically the dependency between the two probability models for each set of the same or different two probability models based on the relationship;
Storing the model dependency information in a model dependency information storage means;
The probability model to be applied for each position in the geospatial is selected from the plurality of probability models so that the likelihood of the geospatial data with respect to the set of the parameter information and the model dependency information is high. A probability model selection step of updating the applied model information to indicate the probability model selected for each position;
Based on the updated application model information, each of the plurality of groups is configured to divide the geospatial data into a plurality of groups to which the same probability model is applied, and to maximize a predetermined model criterion. Learning a parameter of the probability model corresponding to and updating the parameter information to indicate the learned parameter of each probability model; and
Is provided.

  According to the present invention, a mixed model can be efficiently constructed even when there are many model parameters.

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

  FIG. 2 is a configuration diagram showing an embodiment of a mixed geospatial model construction apparatus according to the present invention. As shown in FIG. 2, the geospatial mixed model construction apparatus according to the present invention includes a geospatial data storage unit (geospatial data storage unit) 201, a mixed model learning unit (parameter learning unit) 202, and a mixed model parameter storage. Means (parameter storage means) 203, point state maximum likelihood estimation means (probability model selection means) 204, point state storage means (applied model information storage means) 205, point state MRF parameter estimation means (model dependence calculation means) 206, A point state MRF parameter storage means (model dependence information storage means) 207 is provided.

  Each means can be realized, for example, as a program module. In this case, the function of each means can be realized by executing a program including each program module in the computer system shown in FIG. This computer system includes a CPU 102 for executing program instructions, a main storage device 103 such as a memory, an external storage device 104 such as a hard disk, a magnetic disk device or a magneto-optical disk device, an input device 105 for inputting data by a user, A display device 106 for displaying data and a bus 101 for connecting them to each other are provided.

  In FIG. 2, the geospatial data storage unit 201 includes target variable data that is a random variable obtained by quantifying various properties related to a geospatial phenomenon to be modeled, and position data that represents a geographical position. Geospatial data including is stored. A plurality of cases are stored in the geospatial data.

  If you want to build a generation model, the target variable data contains one or more attribute variables X. If you want to build a prediction model, the target variable data contains one or more attribute variables X and one or more target variables Y. Must be included.

  As explained in the background art section, when there is an attribute variable X that is one or more random variables related to the target property, the generation model has a probability that X takes a specific value x (Pr (X = x)) Is a stochastic model for describing On the other hand, in addition to the attribute variable X, the prediction model prepares an objective variable Y that is one or more random variables related to the target property, and the probability that Y takes a specific value y when X takes a specific value x This is a probabilistic model for describing (Pr (Y = y | X = x)). For example, a one-dimensional normal distribution model or a multidimensional normal distribution model is a generation model, and a linear multiple regression model, a generalized linear model, or the like is classified as a prediction model. Here, the set of the attribute variable X and the objective variable Y in the prediction model corresponds to, for example, L variables, and the prediction model (probability model) is given by LS (S is an integer of 1 or more) variables. It can be said that the probability distribution of the remaining S variables is modeled.

  The position data is a collection of information on the positions of the respective cases of the target variable data. For example, when the space is divided into a grid, an index of the grid position is included. In addition, the latitude and longitude of the object, polygon information, and the like may be included.

  FIG. 4 (A) shows an example of geospatial data, Pos attribute (position attribute) which is position data called a grid position index, an X attribute representing a failure occurrence index (for example, the number of failure occurrences) of equipment in each lattice, and The ID attribute corresponding to the case number is included. FIG. 4B shows an ID arranged in a grid according to Pos (position), and FIG. 4C shows an X arranged in a grid according to Pos (position). In this example, there are 5 × 5 = 25 lattices.

  In the mixed model parameter storage means 203, the parameters {θ1,..., Θk} of each probability model are stored in the mixed model parameter θ according to a predetermined number k of probability models constituting the mixed model (k is an integer of 2 or more). Is remembered as

FIG. 5 shows an example of a mixed model parameter for k = 3. The model parameter θ includes model parameters {θa, θb, θc} of three probability models labeled {a, b, c}. The parameters of each probability model are composed of the average μ and standard deviation σ of the one-dimensional normal distribution of the following equation (1), and the number of model parameters is n = 2. Therefore, in this example, the total number of mixed model parameters is n × k = 6.

  In the point state storage means 205, discrete value information for identifying which of the probability models included in the mixed model is used at each point (Pos) is stored as a point state S (applied model information). FIG. 6B is an example of a point state when the geospatial data of FIG. 4 and the mixed model parameters of FIG. 5 are used, and one of {a, b, c} for each of 25 grids. The identification label is given.

ここで、N(Si)はSiの近傍である。また、λはN(Si)(Siの近傍)のラベル値がSiのラベル値に与える影響をモデル化するためのパラメータであり、λ=0の場合には近傍のラベル値によるSiのラベル値への影響はないとみなすことができる。 The spot state MRF parameter storage means 207 stores the Markov random field spatial dependence parameter λ (model dependence information) regarding the spot state S. When the spatial dependency of the point state S is modeled by a Markov random field, for example, the probability that a certain point Si becomes the label a can be expressed as in Expression (2) (Non-patent Document 2).

Here, N (Si) is in the vicinity of Si. Λ is a parameter for modeling the effect of the label value of N (Si) (in the vicinity of Si) on the label value of Si. When λ = 0, the label value of Si is determined by the label value in the vicinity. It can be assumed that there is no impact on

  In the following description, a group of positions adjacent to each other vertically and horizontally in a two-dimensional space (plane) is defined as “neighbor” (neighbor range). However, the definition of the neighborhood may be different depending on the purpose of model construction. For example, it can be considered that only the position on the left side when viewed from a certain position is set as the vicinity, and the positions on the upper side, the right side, and the upper side are not included in the vicinity. In addition, there may be a case where two positions ahead, down, left, and right as viewed from a certain position are defined as neighborhoods. In addition, when the spatial data has three-dimensional position information, the neighborhood may be defined in the three-dimensional space.

  FIG. 10B shows an example of the point state MRF parameter when the second-order neighborhood is adopted. Here, (a, a) = 1 represents a dependency relationship that the probability that the point Si is a becomes higher when the label a exists in the vicinity, and (a, c) =-1 indicates that the point Si is a (A, b) = 1 indicates that the probability that the point Si is a has no effect even if there is a label b in the vicinity ((a, b) = 1) Non-patent document 2).

  The mixed model learning unit 202 learns the mixed model parameter using the geospatial data stored in the geospatial data storage unit 201 and the point state stored in the point state storage unit 205, and uses the learned mixed model parameter as a mixed model. Store in the parameter storage means 203.

  FIG. 3 is a block diagram showing an embodiment of the mixed model learning means according to the present invention. As shown in FIG. 3, the mixed model parameter learning unit according to the present invention includes an initial mixed model learning unit 301, a geospatial data dividing unit 302, a divided geospatial data storing unit 303, and a model learning unit 304. Yes.

  The initial mixed model learning means 301 assumes that there is no spatial dependence in each case when the value of the point state (FIG. 6B) is not fixed for each point, and the mixed model based on each case. The parameter is calculated, and the calculated mixed model parameter is stored in the mixed model parameter storage unit 203. At this time, the number of probability models and the type of probability model are specified in advance by the user. When ignoring the spatial dependence, it is possible to obtain model parameters of the mixed model by a general method such as an EM algorithm (Non-Patent Document 1). Note that the model parameters of the mixed model may be specified by the user.

  The geospatial data dividing unit 302 exclusively divides each case of the geospatial data (FIG. 4A) by the point state value when the value of the point state for each point is determined. That is, the geospatial data D is divided into {D1,..., Dk} according to the value ({1, .. k}) of the point state S. {D1,..., Dk} each correspond to divided geospatial data (group). The geospatial data dividing unit 302 stores each divided geospatial data in the divided geospatial data storage unit 303.

  The model learning means 304 determines model parameters (θi) by performing model learning using each (Di) of the divided geospatial data, and sets a set of model parameters obtained from each divided geospatial data as mixed model parameters. Is stored in the mixed model parameter storage means 203. That is, the model learning unit 304 corresponds to each divided geospatial data so as to maximize a standard (a model standard given in advance) according to the model learning algorithm for each divided geospatial data (group). Learn (optimize) model parameters of a probabilistic model. In this model learning, it is not necessary to consider spatial dependence.

  In this embodiment, since a normal distribution model is used as the probability model, for example, maximum likelihood estimation or Bayesian estimation can be used as the model learning algorithm. In the case of maximum likelihood estimation, maximizing the model criterion is to maximize the value (likelihood) of the likelihood function based on the normal distribution model (normal distribution function) for the training data (divided geospatial data). This is equivalent to

  As the probability model, in addition to the normal distribution model, a probability model using linear regression analysis, a decision tree, and a Bayesian network is also possible. In linear regression analysis, maximizing the model criterion corresponds to minimizing the square error between the learning data (divided geospatial data) and the output of the linear regression model. In a decision tree, maximizing a model criterion is equivalent to maximizing a value such as an information amount or a Gini value for learning data. In the Bayesian network, maximizing the model criterion corresponds to maximizing the posterior distribution (maximizing likelihood) for the learning data.

となるS*を求める。式(3)において地点状態Sは地理空間データDにおける各事例の地点状態の集合であり、地点状態の全ての組み合わせ中から、Prが最大になるSを求める。ただし、Sの取りうる値は非常に多いため局所最適解を求めることしかできないことが多い。式(3)の局所最適解の探索方法としては、MCMC法、ICM法など様々な手法が提案されている（非特許文献２）。地点状態最尤推定手段２０４は、各事例について推定した地点状態の集合を地点状態Sとして地点状態記憶手段２０５に格納する。 The point state maximum likelihood estimation means 204 uses the mixed model parameter θ, the point state MRF parameter λ, and the geospatial data D, and the likelihood of the geospatial data when the mixed model parameter and the point state MRF parameter are given. Estimate the point state S * such that is as high as possible. That is,

Find S * that becomes In Equation (3), the point state S is a set of point states of each case in the geospatial data D, and S that maximizes Pr is obtained from all combinations of the point states. However, since there are many possible values of S, it is often only possible to obtain a local optimal solution. Various methods such as the MCMC method and the ICM method have been proposed as a search method for the local optimum solution of Equation (3) (Non-Patent Document 2). The point state maximum likelihood estimating unit 204 stores the set of point states estimated for each case in the point state storage unit 205 as the point state S.

  The point state MRF parameter estimation unit 206 estimates the point state MRF parameter λ using the point state S in the point state storage unit 205, and stores the estimated point state MRF parameter λ in the point state MRF parameter storage unit 207.

  FIG. 7 is a block diagram showing an embodiment of the point state MRF parameter estimation means according to the present invention. As shown in FIG. 7, the spot state MRF parameter estimation unit according to the present invention includes a primary frequency calculation unit 701, a secondary frequency calculation unit 702, and a space-dependent parameter calculation unit 703.

  The primary frequency calculation means 701 calculates the frequency of each discrete value (a, b, c) from the point state, and the secondary frequency calculation means 702 calculates the frequency of the same or different set of discrete values. Then, the space-dependent parameter calculation means 703 calculates a spot state MRF parameter (dependency information) from the calculated primary frequency and secondary frequency, and stores it in the spot state MRF parameter storage means 207. Detailed operation description of the point state MRF parameter estimation means will be described later.

  FIG. 8 is a flowchart showing an execution procedure of processing performed by the geospatial mixed model construction device of FIG. As shown in FIG. 8, the execution procedure of this geospatial mixed model construction apparatus includes an initial mixed model learning step 801, a point state maximum likelihood estimation step 802, a mixed model learning step 803, a point state MRF parameter estimation step 804, and an end. A determination step 805 is provided. Hereinafter, the execution process of the flowchart of FIG. 8 will be described in detail using the geospatial data of FIG.

  In step 801, the mixed model learning unit 301 in the mixed model learning unit 202 calculates mixed model parameters on the assumption that there is no spatial dependency in each case. FIG. 5 shows an example of the mixed model parameter calculated in step 801. Model parameters {θa, θb, θc} of three normal distribution models (represented as {Ma, Mb, Mc}) labeled {a, b, c} are shown.

  In step 802, the point state maximum likelihood estimating means 204 calculates the maximum likelihood estimated value (practically approximate optimum value) of the point state S represented by the above-described equation (3).

式(4)を用いると、例えば、事例(ID=)1とモデルMaとの乖離値は、

と算出できる。 More specifically, first, the divergence value between the attribute variable value x and {Ma, Mb, Mc} is calculated for all points. For example, as the deviation value in the normal distribution model, equation (4) obtained by multiplying the log of equation (1) by -1 can be used. When the attribute variable value x is an average value, the divergence value is minimum.

Using equation (4), for example, the deviation value between case (ID =) 1 and model Ma is

And can be calculated.

  The result of calculating the divergence value between {Ma, Mb, Mc} and X (the set of x in each case) is shown in {Ma, Mb, Mc} in the table of FIG. Further, the identification value of the model having the smallest deviation value is shown as “Best” in the table of FIG.

  Next, an optimum point state estimated value is determined using the obtained divergence value and the point state MRF parameter λ. At this time, it is the first time in step 802, and λ is not determined, so that λ = 0 is assumed. In that case, the value of Best for each point is estimated as the optimum point state value. FIG. 6B shows the point state obtained in this way. FIG. 6C shows a spatial plot of the point state values of FIG. 6B, and FIG. 6D shows the point state values of FIG. The plot is shown with different fill patterns for b and c.

  In step 803, the model parameter θ is learned using the obtained point state S and geospatial data D. Since the calculation of the point state S in the first step 802 is considered as λ = 0, the same model parameter θ as that in the initial mixed model obtained in step 801 is obtained in the first step 803 (thus, in step 803). The first time may be skipped (assuming that it was skipped here).

  In step 804, the space state parameter λ (dependency information) regarding the point state is estimated from the point state S obtained in step 802 by the point state MRF parameter estimating means 206. For example, when the point states in FIGS. 6B to 6D are obtained, first, each label (a to c) in each lattice is obtained by the primary frequency calculating unit 701 in the point state MRF parameter estimating unit 206. The frequency π1 (primary frequency) is calculated. Next, the secondary frequency calculation means 702 calculates the frequency π2 (secondary frequency) of the label pair while avoiding duplication with respect to the pair of each lattice and the adjacent lattice. FIGS. 9A and 9B show an example of the primary frequency π1 and an example of the secondary frequency π2, respectively. In this example, it can be seen from FIG. 9B that there are 40 label pairs.

  By using the obtained primary frequency π 1 and secondary frequency π 2, the spatial dependence parameter calculation means 703 in the spot state MRF parameter estimation means 206 estimates the spatial dependence parameter λ related to the spot state, for example, according to the following procedure. To do.

  First, the difference between the label pair frequency π 2 and the expected value of the label pair calculated from the label frequency π 1 is calculated as λ ′.

For example, from FIG. 9A, the probability of occurrence of label a is 12/25, and therefore the probability of occurrence of label pair aa is (12/25) ² . On the other hand, the actual occurrence probability of the label pair aa is 10/40 from FIG. 9B. Therefore, taking the log of the ratio of these probabilities, λ ′ (a, a) = log ((10/40) / (12/25) ² ) ≈0.035. The fact that λ ′ takes a positive value means that the actual secondary frequency π2 is larger than the expected value of the secondary frequency calculated from the primary frequency π1, so that the next to a is likely to be a. It can be estimated that positive autocorrelation is working.

  Further, since the probability that the label pair a-c occurs is 2 * (12/25) * (4/25) from FIG. 9A, λ ′ (a, c) ≈−0.311. The fact that λ ′ takes a negative value means that the actual secondary frequency π2 is smaller than the expected value of the secondary frequency calculated from the primary frequency π1, so that the next to a is less likely to be c. It can be estimated that negative cross-correlation is working.

  Λ ′ is calculated in the same manner for other label pairs, and all the calculated λ ′ are summarized in FIG.

  Here, regarding the autocorrelation, λ ′> 0 is +1, otherwise is 0, and λ ′ <0 is −1, and otherwise is 0. In other words, autocorrelation does not consider negative correlations, only considers whether positive correlations work, and does not consider positive correlations for cross-correlation, only considers whether negative correlations work . When the value of λ ′ is changed in this way, a space-dependent parameter λ is obtained for each label pair as shown in FIG.

  Various variations of the calculation method of λ ′ and λ can be considered. For example, λ '= λ, λ can take a value other than {0, +1, -1}, or the value of {0, + α, -α} Or take it.

  In step 805, it is determined whether or not the end condition is satisfied. If the end condition is satisfied, the process ends. If not, the process returns to step 802. The termination condition may be that the number of loops in the flow in FIG. 8 has reached a predetermined number or that the change in θ or λ has been eliminated. In this case (in the case of the first loop), it is assumed that the process ends if λ = 0 for all label pairs, but continues even if one label pair has λ ≠ 0 (the end condition is not satisfied). Therefore, as shown in FIG. 10B, since there exists a pair of λ ≠ 0, the process returns to step 802.

  In the second time of step 802, first, a deviation value between each model and data X is calculated using the latest model parameter θ. Since the first time in step 803 is skipped, the calculated divergence value is the same as the divergence value calculated in the first time in step 802 (shown in FIG. 6A). However, since λ is obtained in the first step 804 this time, the optimum point state S must be searched in consideration of spatial dependence. Here, an example using ICM (Non-Patent Document 2) known as an approximate search method is shown. ICM repeats the process of calculating a penalty when a certain state value (label) is taken at a randomly selected point and replacing it with the label with the lowest penalty. As the penalty, a deviation value between the model and the data, or a value obtained by multiplying λ by a negative sign can be considered.

  When the divergence value in FIG. 6A and λ in FIG. 10B are used, for example, consider that the label value of the point 20 (ID = 20) is a. At this time, the deviation value increases by (2.9-0.4 =) 2.5. Also, since the number of a-b pairs is reduced by three and the number of a-a pairs is increased by three, the penalty related to spatial dependence (change in dependence in the whole space) is reduced by -3. Therefore, the penalty is reduced by -0.5. Therefore, the label of the point 20 is changed from b to a. That is, the label is changed because the total value (calculated value) is smaller than the threshold value (here, zero). In the case where the label value at the point 20 (ID = 20) is changed to the label c, the penalty reduction may be calculated in the same manner, and the label may be changed to the label with the larger penalty reduction.

  The above processing is performed by selecting some other points (examples). That is, some cases are performed instead of all cases from the viewpoint of processing efficiency. The results of searching for the spot state S in this way are shown as an example in FIGS. 11 (A) to 11 (C).

  In step 803, the model parameter θ is learned using the obtained point state S and geospatial data D. First, the geospatial data dividing unit 302 in the mixed model learning unit 202 divides the geographic learning data D according to the discrete values (label values) of the point states. FIG. 12 shows learning data Da for the model Ma when the point state of FIG. 11A is used. The model parameter θa is learned by the model learning means 304 in the mixed model learning means 202 using this learning data. Specifically, the average and standard deviation of X are calculated from the learning data Da in FIG. This calculation is equivalent to obtaining the maximum likelihood estimation value of the parameter in the likelihood function (the value of the parameter that maximizes the likelihood function) in the maximum likelihood estimation method. For the learning data Db and Dc for the models Ma and Mb, the average and standard deviation of X are calculated in the same manner. The model parameter θ thus obtained is shown in FIG.

  In the second time of step 804, the point state MRF parameter λ is estimated using the point state of FIG. 11B as in the first time. FIG. 14 shows the calculation result of the point state MRF parameter λ in the second loop.

  In the second time of step 805, it is assumed that the end condition is not satisfied and the process returns to step 802.

  In the third time of step 802, the optimal spot state S is calculated using the model parameters of FIG. 13 and the spot state MRF parameters of FIG. First, the deviation value between the model and the data X is calculated for each point using the latest model parameter θ. Of the divergence values calculated for each point, the divergence values relating only to point 3 and point 22 are shown in FIG. Next, the optimum point state S is searched. For example, consider that the label value of point 3 is b. At this time, the deviation value increases by 2.4. Also, since the number of a-b pairs is reduced by 3 and the number of b-b pairs is increased by 3, the penalty for spatial dependence is reduced by -3. Therefore, the penalty is reduced by -0.6. Therefore, the label of point 3 is changed from a to b. Similarly, since the spatial dependency of the point 22 is reduced by setting the label value to a, the label is changed from b to a. FIG. 16 shows the point state S obtained in the third loop.

  FIG. 17A shows the third calculation result in step 803, and FIG. 17B shows the third calculation result in step 804. Assuming that an end condition for ending the loop at the third loop is used, FIGS. 17A and 17B correspond to the parameters θ and λ of the finally obtained geospatial mixed model.

  Finally, in this example, as can be seen from FIG. 16, the space is divided into one a area, two b areas, and two c areas, and the probability distribution of the failure index X is the same model for each area. Represented by a mixed model that takes parameters. Therefore, it can be expected that by investigating carefully the same area, it will be possible to discover hidden factors affecting the failure index.

  In this embodiment, a geospatial mixed model using MRF related to the discrete point state S instead of MRF related to the model parameter θ is adopted, and the calculation cost of the quasi-Newton method or the like is obtained by following the above procedure. It is possible to construct a mixed geospatial model without using the necessary methods.

  In this embodiment, since X is one-dimensional, the normal distribution parameter is n = 2 (two of μ and σ). However, when X is a four-dimensional continuous value vector, for example, the multi-dimensional normal distribution parameter is Maximum n = 4 + 4 * 4 = 20 is required. At this time, in the present embodiment, the number of parameters λ increases in a square while the number of model parameters n increases only in a linear manner, while the number of parameters λ increases in a square. It is efficient when used in many cases.

The block diagram showing the hardware constitutions of this invention. The block diagram of the geospatial mixed model construction apparatus in connection with one Embodiment of this invention. The block diagram of the mixed model learning means in connection with one Embodiment of this invention. The figure which shows the example of geospatial data. The figure which shows the example (the 1) of a mixed model parameter. The figure which shows the example (the 1) of the estimation result of a point state. The block diagram of the point state MRF parameter estimation means in connection with one Embodiment of this invention. The figure which shows the flowchart of the geospatial mixed model construction apparatus in connection with one Embodiment of this invention. The figure which shows the example of a primary frequency and a secondary frequency. The figure which shows the example (the 1) of the calculation result of the space dependence parameter calculation means 703. The figure which shows the example (the 2) of the estimation result of a point state. The figure which shows the example of division | segmentation geospatial data. The figure which shows the example (the 2) of a mixed model parameter. The figure which shows the example (the 2) of the calculation result of the space dependence parameter calculation means 703. The figure which shows the example of the calculation result of the deviation value of a model and data. The figure which shows the example (the 3) of the estimation result of a point state. The figure which shows the example of model parameter (theta) and geographic state MRF parameter (lambda).

Claims (7)

Geospatial data storage means for storing geospatial data having a plurality of cases including at least one variable representing the property of the evaluation object by numerical value and position data indicating a position in geospatial space;
Parameter storage means for storing parameter information representing parameters of a plurality of probability models obtained by modeling the probability distribution of the variables;
Application model information storage means for storing application model information representing the probability model to be applied for each position in the geographic space;
A probability model to be applied to each position in the geospace, and a probability model applied to one or more neighboring positions included in a predefined neighborhood range for each position in the geospace. Model dependence calculation means for calculating model dependence information in which a dependence between the two probability models is numerically expressed for each set of the same or different two probability models based on a relationship;
Model dependence information storage means for storing the model dependence information calculated by the model dependence calculation means;
The probability model to be applied for each position in the geospatial is selected from the plurality of probability models so that the likelihood of the geospatial data with respect to the set of the parameter information and the model dependency information is high. A probability model selection means for updating the applied model information to indicate the probability model selected for each position;
Based on the updated application model information, each of the plurality of groups is configured to divide the geospatial data into a plurality of groups to which the same probability model is applied, and to maximize a predetermined model criterion. Parameter learning means for learning parameters of the probability model corresponding to and updating the parameter information to indicate the learned parameters of each probability model;
Model building device with The model dependence calculation means further calculates the model dependence information based on the updated applied model information, and indicates the model dependence information in the model dependence information storage means to indicate the calculated model dependence information. Update dependency information,
The probability model selection unit is configured to apply the probability to be applied for each position in the geospatial so that the likelihood of the geospatial data with respect to a set of updated parameter information and updated model dependency information is high. Select a model,
The model construction apparatus according to claim 1. The at least one variable includes L (L is an integer of 2 or more) variables,
The probability model models a probability distribution of the remaining S variables when LS (S is an integer of 1 or more) number of the variables is given.
The model construction apparatus according to claim 1, wherein the model construction apparatus is a model construction apparatus. The model dependence calculation means includes
Calculating the frequency of each probability model from the applied model information as primary frequency information;
By obtaining a set of a probability model of each position in the geospace and the probability model of the vicinity position included in the vicinity range of each position, each set of the two probability models of the same or different Calculate the frequency as secondary frequency information,
The model construction apparatus according to any one of claims 1 to 3, wherein the model dependence information is calculated using the primary frequency information and the secondary frequency information. The model dependence calculation means includes
From the primary frequency information, calculate an expected value of the frequency of each set of the two probability models that are the same or different,
Calculating the model dependency information based on the difference between the expected value of the frequency of each set and the frequency of each set indicated in the secondary frequency information;
The model construction device according to claim 4 characterized by things. Geospatial data storage means for storing geospatial data having a plurality of cases including at least one variable representing the property of the evaluation object by numerical value and position data indicating a position in geospatial space;
Parameter storage means for storing parameter information representing parameters of a plurality of probability models obtained by modeling the probability distribution of the variables;
Application model information storage means for storing application model information representing the probability model to be applied for each position in the geographic space;
Preparation steps, and
A probability model to be applied to each position in the geospace, and a probability model applied to one or more neighboring positions included in a predefined neighborhood range for each position in the geospace. A model dependency information calculating step for calculating model dependency information that represents numerically the dependency between the two probability models for each set of the same or different two probability models based on the relationship;
Storing the model dependency information in a model dependency information storage means;
The probability model to be applied for each position in the geospatial is selected from the plurality of probability models so that the likelihood of the geospatial data with respect to the set of the parameter information and the model dependency information is high. A probability model selection step of updating the applied model information to indicate the probability model selected for each position;
Based on the updated application model information, each of the plurality of groups is configured to divide the geospatial data into a plurality of groups to which the same probability model is applied, and to maximize a predetermined model criterion. Learning a parameter of the probability model corresponding to and updating the parameter information to indicate the learned parameter of each probability model; and
Model building method with Accessing geospatial data storage means for storing geospatial data having a plurality of cases including at least one variable representing the property to be evaluated numerically and position data indicating a position in geospatial;
Accessing parameter storage means for storing parameter information representing a parameter of each of a plurality of probability models modeling the probability distribution of the variable;
Accessing application model information storage means for storing application model information representing the probability model to be applied for each position in the geographic space;
A probability model to be applied to each position in the geospace, and a probability model applied to one or more neighboring positions included in a predefined neighborhood range for each position in the geospace. A model dependency calculating step for calculating model dependency information that represents numerically the dependency between the two probability models for each set of the same or different two probability models based on the relationship;
Storing the model dependency information in a model dependency information storage means;
The probability model to be applied for each position in the geospatial is selected from the plurality of probability models so that the likelihood of the geospatial data with respect to the set of the parameter information and the model dependency information is high. A probability model selection step of updating the applied model information to indicate the probability model selected for each position;
Based on the updated application model information, each of the plurality of groups is configured to divide the geospatial data into a plurality of groups to which the same probability model is applied, and to maximize a predetermined model criterion. Learning a parameter of the probability model corresponding to and updating the parameter information to indicate the learned parameter of each probability model; and
Model building program with

Images