JP2005341516A - Hierarchization maximum-likelihood estimation calculating method - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a method which hierarchizes the segmentation of a mesh, calculates a sequential operation, and obtains the position corresponding to a maximum likelihood with ultimate precision. <P>SOLUTION: The method hierarchizes the segmentation of a mesh, calculates the sequential operation, and estimates the position corresponding to a maximum likelihood in a designated space finally, when the position of a maximum value of the likelihood is asked in an object area. First, the likelihood of each mesh is calculated for a coarse mesh and the maximum likelihood is searched. The peripheral size of the next maximum likelihood is designated, and an interval of the meshes is broken into a small size in the designated area. The likelihood is asked regarding a fine mesh in the designated area, the position for obtaining the maximum likelihood is asked likewise sequentially, and the position corresponding to the maximum likelihood is decided by repeating this treatment, until the finally-designated mesh size is attained. <P>COPYRIGHT: (C)2006,JPO&NCIPI

Description

Detailed Description of the Invention

    The present invention relates to a calculation algorithm for obtaining a position where a likelihood is a function of a position and a likelihood is maximized in a specified region.

    As a method for obtaining the optimum value of the position divided into fine meshes, when applying a conventional method, the following two methods are known. The first method is a method of obtaining the likelihood of the entire fine mesh, searching for the maximum value, and setting the position corresponding to the maximum value as the optimal value.

    The second method is a method in which the position of the initial mesh is designated and the hill-climbing method is applied from that position.

Problems to be solved by the invention

    In the first method, when the mesh interval is reduced, the calculation time increases in a square order in the case of a plane. For example, in an area of 200 m square, 2500 meshes are required for a 4 m mesh, and 10,000 meshes for a 2 m mesh. If the accuracy is increased in this way, the number of calculations greatly increases.

    In the second method, the calculation time depends on the initial position. In addition, it takes time to search the periphery. Furthermore, if a local minimum occurs, it is also a problem that a final solution cannot be obtained.

Means to solve the problem

  In order to solve the conventional problem, the present invention proposes a method of obtaining a position corresponding to the maximum likelihood with final accuracy by hierarchizing mesh division methods and sequentially calculating. First, the likelihood of each mesh is calculated for a coarse mesh, and the maximum likelihood is searched. The size around the next maximum likelihood is designated, and the mesh interval is divided into small parts in the designated area. In the designated area, the likelihood is obtained for a fine mesh, and the position of the maximum value is set as the optimum position. In this case, the mesh separation method is hierarchized, and the optimum position is sequentially pursued. In this case, there is a mesh size that minimizes the number of calculations in each layer, and the number of calculations is minimized by sequentially performing calculations according to the size.

This will be described below with reference to the drawings and formulas. FIG. 1 shows a case of a square area. Even if it is not square, it can be obtained similarly. Let L be the vertical and horizontal lengths. Let l ₁ be the initial mesh spacing. With this size, the area is allocated for the time being, and the likelihood of all mesh intersections is obtained. An area surrounded by 2l ₁ is designated around the maximum value. Then it is assumed that delimit the area size of l _2. For the area surrounded by 21 ₂ , the likelihood is obtained, the position of the maximum value is obtained, and so on.

となる。Likelihood calculation time is
T = c · Mc: one element calculation time (constant), M: number of calculations.
The total number of calculations in the nth hierarchy is _Mn, and a general expression for the number of calculations is obtained.
Let the target area be L ₁ × L ₂ .
First, let M ₁ be the number of calculations when one layer is used. One layer corresponds to no layering. If the minimum mesh interval is Δl ₁ × Δl ₂ ,

It becomes.

である。計算最小回数の条件を求める。Then, when the two layers, when the first mesh size to l _1, the sum of the number of calculations and calculation times of the second layer of the first layer.

It is. Find the condition for the minimum number of calculations.

最小となる条件は、

これより、

であり、

以下同様に、ｎ階層まで計算すると、計算の最小数は、

で与えられる。

The minimum condition is

Than this,

And

Similarly, when calculating up to n levels, the minimum number of calculations is

Given in.

以降の階層化段数ｉにおいては、以下の漸化式で求められる。

ここで、分割数が整数倍となるための条件を求めると、Ｌ_１に対する最初の分割数をｋ、Ｌ_２に対する最初の分割数をｍとすると、

を満たすことが必要となる。その場合の最終段階の分割メッシュサイズは

最終分割メッシュサイズを指定した場合は、最初の分割数は必ずも整数とならないが、端の処理を適宜行うことにより実際は問題とならない。The initial dividing line segment is given by

The subsequent hierarchized stage number i is obtained by the following recurrence formula.

Here, when a condition for the number of divisions to be an integral multiple is obtained, if the first division number for L ₁ is k and the first division number for L ₂ is m,

It is necessary to satisfy. In that case, the final divided mesh size is

When the final division mesh size is designated, the initial division number is not necessarily an integer, but it does not actually cause a problem by appropriately performing the edge processing.

Hereinafter, specific examples will be described. The target area is a square of L = 200 [m]. Here, the final stage mesh size is designated and Δl = 3, 2.5, 2, 1.5, 1, 0.5 m. By substituting L and Δl into equation (8), n (number of hierarchies) having the minimum value of M _n is obtained and shown in Table 1. Table 1 shows that there is an optimum number of hierarchies that minimizes the number of calculations T depending on the value of Δl. The optimum number of layers differs depending on the minimum division size because the number of meshes is a discrete number.

  An embodiment of the present invention will be described with respect to a problem of estimating a position of a terminal moving within R, where R is a 200 m square area. FIG. 1 shows a target area of the embodiment. As shown in FIG. 1, it is assumed that a plurality of access points AP are arranged in advance in R. Further, it is assumed that the mobile device MS moves within the area. Radio waves are transmitted from the mobile station MS, and radio waves from the terminal are received by a plurality of access points AP arranged in R. The coordinates of each access point AP are known in advance. It is assumed that the position of the mobile terminal is calculated from reception level information of radio waves from the mobile device at the access point AP.

Since the reception level attenuates with distance, the position of the mobile terminal can be estimated from the magnitude of the reception level. If there is no obstacle in propagation between the mobile station MS and the access point AP and direct propagation, the position of the mobile station is uniquely determined by the reception levels of the three access points AP. In this case, there is no need to search. However, in general, there may be an obstacle in the propagation path. The reason why an electric wave arrives due to the presence of an obstacle is due to ambient reflection or diffraction. This is generally called multipath. In the case of a multipath propagation path, the reception level is accompanied by variations due to multipath in addition to distance attenuation. This variation varies depending on the location, and therefore varies depending on the position of the mobile station MS.

ここで、Ａは定数、αは距離減衰係数、Δは確率変数である。Δの確率分布が次式のようにデシベル正規分布に従うとする。

但し、σは標準偏差である。今、３つのアクセスポイントＡｐｉが受信する受信レベルをΓｉとする。受信するアクセスポイントは、４以上とすれば精度は向上する。逆に２以下の場合は、推定位置は点と成らず面となる。When the position of the mobile station MS is not known in advance, it is necessary to handle it probabilistically. If the distance between the mobile terminal and the access point is r, the reception level can be expressed as follows.

Here, A is a constant, α is a distance attenuation coefficient, and Δ is a random variable. Assume that the probability distribution of Δ follows a decibel normal distribution as shown in the following equation.

Where σ is the standard deviation. Now, let the reception level received by the three access points Api be Γi. If the number of access points to be received is 4 or more, the accuracy is improved. On the other hand, in the case of 2 or less, the estimated position is not a point but a surface.

となる。但し、位置および、受信レベルの確率は一様とすると、Ｃは常数となる。すなわち、位置を推定することは、位置を変えて確率の変化を求めることに対応する。Since Δ is a random variable, Γ is also a random variable. If the reception level at the access point Api is Γ _i and the position of the mobile station is x _s , the probability of receiving the reception level Γ _i is p (Γ _i | x _s ). Given the reception level, the probability of position is

It becomes. However, if the position and the probability of the reception level are uniform, C is a constant. That is, estimating the position corresponds to obtaining a change in probability by changing the position.

となる。λに対して対数をとったものを対数尤度という。これをΛと表すと

従って、式（５）により、位置の推定が可能となる。移動機ＭＳの位置の最尤推定は確率Λが最大となる位置である。 尤度を計算した例を図２に示す。縦軸が尤度の値で、尤も高い値が移動機の推定位置である。The position can be estimated by setting the estimated position as the estimated position. The probability of the estimated position when using a plurality of access points is the product of them, which is called likelihood. If received by n access points,

It becomes. The logarithm of λ is called log likelihood. If this is expressed as Λ

Therefore, the position can be estimated by the equation (5). The maximum likelihood estimation of the position of the mobile station MS is the position where the probability Λ is maximized. An example of calculating the likelihood is shown in FIG. The vertical axis is the likelihood value, and the highest likelihood value is the estimated position of the mobile device.

となる。このメッシュ間隔で区切る。端は、エリア外となるので、１６のメッシュの交点について尤度を計算する。尤度の最大のポイントをＰ１とする。Ｐ１を中心として２ｌ_１の区域を設定する。その区域を長さｌ_２でメッシュ分割を行う。ｌ_２は、次式で与えられる。

Next, a calculation method using a hierarchization method will be described. In order to estimate by changing the position of the mobile device, the region R is divided into meshes. Assuming that L = 200 m square, the final mesh interval is Δl ₁₁ = Δl ₁₂ = Δl = 0.5 m. The number of hierarchies is 8 which is the optimum value. The mesh spacing in the first layer is

It becomes. Divide by this mesh interval. Since the end is outside the area, the likelihood is calculated for the intersection of 16 meshes. Let P1 be the maximum likelihood point. An area of 2l ₁ is set around P1. The area is divided into meshes with length l ₂ . l ₂ is given by:

である。以下同様に計算を行う。尤度の計算は、メッシュ分割された交点に対して、式（１６）より順次求めることが可能である。Similarly, the likelihood is calculated and the maximum position is P2. Set the area around the P2, it is divided by _{l 3.} l ₃

It is. The same calculation is performed thereafter. The likelihood calculation can be sequentially obtained from the equation (16) for the mesh-divided intersections.

As a second embodiment, a processing method when the target area is not rectangular will be described. An example is shown in FIG. Set the minimum rectangular area that covers the target area. The rectangular region may be divided into meshes and the hierarchization method may be applied in the same manner. At this time, the likelihood of the mesh intersection deviating from the target area may be zero. In the case of log likelihood, a negative enough large value may be set.

Regarding the setting of the area in the hierarchization, in the above description, the method of setting the double square of the new mesh size centering on the maximum likelihood has been described. However, the setting of this area is more than twice or 2 Even within double is possible. If the value is increased, the final error is reduced, but it takes time. If the value is reduced, the time is shortened, but the error is increased, resulting in a tradeoff between the two.

The invention's effect

表より、１階層に比較し、最適な８階層で行うと１４３倍計算時間が短縮される。更に、移動機の台数を１０台として同様なシミュレーションを行った結果を表３に示す。

表３からわかるように、階層化を行なった場合移動機数を１０台に増やしてもシミュレーション時間が０．１２４［秒］に抑えられることが可能である。Table 2 shows the calculation time of the computer when the simulation is performed five times and the hierarchy is not performed and when the hierarchy is eight. However, the parameters are A = 130 and α = 3.4σ = 2.

From the table, the calculation time is reduced by a factor of 143 when the optimal eight layers are used compared to the first layer. Furthermore, Table 3 shows the results of a similar simulation with 10 mobile units.

As can be seen from Table 3, when the hierarchization is performed, the simulation time can be suppressed to 0.124 [seconds] even if the number of mobile devices is increased to 10.

Table 4 shows a comparison between the non-hierarchization and the optimum hierarchization with the final mesh interval as a parameter for the number of calculations.

From the table, when the mesh interval is 3 m, the number of calculations is reduced by 846 times when it is 36 times and 0.5 m. Furthermore, in the case of hierarchization, when the mesh interval is halved, the number of calculations is about four times that in non-hierarchy, but in the case of optimum hierarchization, it is only an increase of about 1.2 times. . The smaller the final mesh interval, the greater the effect.

  Fig. 7 compares the accuracy of position estimation with and without 8-level hierarchization. Not hierarchizing corresponds to a case where calculation is performed for all of the minimum mesh sizes. From the figure, there is no difference in accuracy, indicating that the estimation obtained by hierarchization is correct.

Location of target area R and access point AP Likelihood calculation example, x-axis and y-axis represent areas, and the vertical axis represents likelihood values. How to separate meshes in hierarchical algorithms Flow chart of calculation of layering method State of change of layered area In the first layer, l ₁₁ and l ₁₂ indicate mesh intervals in the first layer. P1 indicates the position of the maximum likelihood obtained in the calculation of the first hierarchy. In the second layer, l ₂₁ and l ₂₂ indicate mesh intervals in the second layer. P2 indicates the position of the maximum likelihood obtained in the second layer calculation. The number of layers and the number of calculations when the mesh size is used as a parameter Δl indicates the mesh size at the final stage. Number of layers and number of calculations when mesh size is used as a parameter. The number of calculations is linearly expanded and displayed. Comparison of accuracy when the maximum likelihood position is estimated with the minimum mesh when the layering method is applied and not applied. For the estimation accuracy, the distance between the true position and the estimated position is displayed in m. An example of region setting and mesh division when the target region is a non-rectangular region.

Claims (4)

For the problem of finding the position where the likelihood is a function of the position in the designated area and the likelihood becomes the maximum, the designated area is divided by the first mesh calculated in advance, and the intersection of the mesh A likelihood is calculated, a position where the likelihood becomes maximum is obtained, a new area is specified around the obtained position, the area is divided at a second mesh interval, and the likelihood for each mesh intersection is calculated. In order to calculate the likelihood until the final accurate mesh interval is obtained, the optimal number of hierarchies is calculated so that the number of calculations until the final mesh size is calculated is minimized. A mesh size in each layer corresponding to the number of hierarchies is obtained, likelihoods are sequentially calculated according to the above hierarchization calculation, calculation is performed up to the final mesh interval, and the maximum likelihood and position are obtained. Estimation calculation method. The hierarchical maximum likelihood estimation calculation method according to claim 1, wherein the region is a rectangular region. The hierarchized maximum likelihood estimation calculation method according to claim 1, wherein a region having an arbitrary shape is covered with a minimum rectangular region including the region, and the maximum likelihood and position of the rectangular region are calculated by hierarchization. A hierarchy in which the number of hierarchized stages is designated in advance with respect to claim 1-3, the optimum mesh size of each hierarchy in the designated stage number is calculated in advance, and the maximum likelihood and position are hierarchically determined according to the mesh size Maximum likelihood estimation calculation method.

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