Abstract:
An installation error estimating device has a predicted pattern acquirer which obtains a predicted positioning distribution pattern, which is obtained by computing a characteristic pattern of a predicted positioning distribution obtained by predicting a logical positioning distribution, at each observation point where a wireless tag is installed for positioning; and an observation data inputter to which positioning results obtained from the wireless tags by a tag reader are input as observation data. A dispersion pattern analyzer computes a characteristic pattern of a measured positioning distribution, which is obtained by statistical analysis of the applicable positioning result, as a measured positioning distribution pattern at each observation point based on the observation. An installation error estimator computes the installation error for the tag reader using the predicted positioning distribution patterns obtained and the measured positioning distribution patterns computed.

Description:
TECHNICAL FIELD 
     The present invention relates to an installation error estimating apparatus and installation error estimating method for performing installation support for a tag reader that performs radio positioning. 
     BACKGROUND ART 
     Although a potential demand for high-precision position management by means of wireless tags has hitherto become evident from the standpoint of distribution and security management and the like, a necessary level of measurement precision has not been attained, and therefore such position management has not become widely established. 
     However, with the development of UWB (Ultra Wide Band) technology in recent years, the trend of opening-up of frequencies has increased the possibility of making possible radio positioning on the order of several cm to several tens of cm. 
     For example, in Patent Literature 1, a position measurement method (positioning method) is disclosed whereby a position of a node is measured using a radio communication system having a node (corresponding to a wireless tag) provided with a function for transmitting a positioning signal, and a plurality of base stations (corresponding to tag readers). With this positioning method, at least one of a plurality of base stations transmits a reference signal after receiving a positioning signal. Also, at least two of the plurality of base stations measure a time at which a positioning signal is received, and a time at which a reference signal is received. With this positioning system, the position of a node is calculated using reception times of a positioning signal and reference signal measured by a base station having received the reference signal, and position information of a base station having received the positioning signal. 
     Also, in Non-Patent Literature 1, an implementation example is disclosed in which UWB is applied to the positioning method disclosed in Patent Literature 1. 
     CITATION LIST 
     Patent Literature 
     PTL 1 
     
         
         Japanese Patent Application Laid-Open No. 2005-140617 
       
    
     Non-Patent Literature 
     NPL 1 
     
         
         MIZUGAKI Ken&#39;ichi et al, “22-cm Accurate Location System with 1-cc Small Size Sensor Node: Practical Experiment of UWB Location System,” 2006 IEICE-ESS Fundamentals Review, pp. S-55 
       
    
     SUMMARY OF INVENTION 
     Technical Problem 
     However, in the above conventional positioning method, it is assumed that each base station is correctly installed at a predetermined position, and nothing is disclosed regarding a method of supporting correct installation of a base station at a predetermined position. More particularly, in a high-precision radio positioning system (generally called a “UWB positioning system”), extremely high positioning precision is expected at the same time as extremely high precision is required for base station (tag reader) installation. 
     In this regard, in wireless tag position detection, when a coordinate system defined by a tag reader and a coordinate system defined by a tag reader installation position and wireless tag movement range are associated, installing a tag reader at a predetermined position with a high degree of precision is time-consuming and expensive. For example, for tag reader installation, there is a method whereby deviation is found by comparing tag reader position and orientation measurement results with a design drawing, plan, or the like, by adding a measuring jig or measuring mark to a tag reader, and repeating deviation correction and measurement. With this kind of installation method, it is possible to achieve installation error convergence by repeating deviation correction and measurement. However, problems with this method are that it is extremely time-consuming and entails high implementation costs. 
     Furthermore, this method requires a high degree of expertise (special experience, know-how, and so forth) in jig handling and the like for measurement and adjustment that imposes constraints in terms of skill in executing installation and time required for installation. 
     Thus, a conventional installation method requires special installation structure, parts, and the like for a tag reader, and also requires a high degree of expertise in installation and measurement on the part of installation engineers. Consequently, there is a demand for a method to make possible simple installation that does not entail such requirements, while enabling high-precision positioning to be achieved. 
     It is therefore an object of the present invention to provide an installation error estimating apparatus and installation error estimating method capable of enabling error-free, high-precision positioning to be achieved with simple installation. 
     Solution to Problem 
     An installation error estimating apparatus of the present invention estimates installation error of a tag reader that positions a wireless tag, and has: a predicted pattern acquisition section that acquires a predicted positioning distribution pattern, which is obtained by calculating a characteristic pattern of a predicted positioning distribution obtained by predicting a positioning distribution, for each observation point where the wireless tag is installed and positioned; an observation data input section to which a positioning result of the tag reader with respect to the wireless tag is input as observation data; a variance pattern analysis section that calculates a characteristic pattern of a measured positioning distribution, which is obtained by means of statistical analysis of a positioning result, for each observation point, based on observation data input by means of the observation data input section; and an installation error estimating section that calculates installation error of the tag reader using a predicted positioning distribution pattern acquired by the predicted pattern acquisition section and a measured positioning distribution pattern calculated by the variance pattern analysis section. 
     An installation error estimating method of the present invention estimates installation error of a tag reader that positions a wireless tag, and has: a predicted pattern acquisition step of acquiring a predicted positioning distribution pattern, which is obtained by calculating a characteristic pattern of a predicted positioning distribution obtained by predicting a positioning distribution, for each observation point where the wireless tag is installed and positioned; an observation data input step of inputting a positioning result of the tag reader with respect to the wireless tag as observation data; a variance pattern analysis step of calculating a characteristic pattern of a measured positioning distribution, which is obtained by means of statistical analysis of a positioning result, for each observation point, based on observation data input by the observation data input step; and an installation error estimating step of calculating installation error of the tag reader using a predicted positioning distribution pattern acquired by the predicted pattern acquisition step and a measured positioning distribution pattern calculated by the variance pattern analysis step. 
     Advantageous Effects of Invention 
     The present invention is capable of enabling error-free, high-precision positioning to be achieved with simple installation. That is to say, when a tag reader is installed at a predetermined position and performs high-precision positioning, special installation structure, parts, and the like for the tag reader can be made unnecessary, and simple installation can be made possible that does not require a high degree of expertise in installation and measurement on the part of installation engineers, without sacrificing high-precision of positioning. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         FIG. 1  is a configuration diagram of a UWB positioning system that includes an installation error estimating apparatus according to Embodiment 1 of the present invention; 
         FIG. 2  is a schematic diagram showing an example of a positioning target area in this embodiment; 
         FIG. 3  is a plan of the positioning target area shown in  FIG. 2 ; 
         FIG. 4  is a block diagram showing the configuration of an installation error estimating apparatus according to this embodiment; 
         FIG. 5  is a drawing showing an example of observation point information in this embodiment; 
         FIG. 6  is a drawing showing an example of an operation screen display for setting observation point information in this embodiment; 
         FIG. 7  is a drawing showing an example of observation data in this embodiment; 
         FIG. 8  is a drawing showing an example of an operation screen display for starting and terminating positioning in this embodiment; 
         FIG. 9  is a drawing showing examples of log formats in this embodiment; 
         FIG. 10  is a drawing showing an example of coordinate management tables in this embodiment; 
         FIG. 11  is a drawing showing an example of a measured positioning distribution pattern list in this embodiment; 
         FIG. 12  is a drawing showing an example of an operation screen display for displaying estimated installation error in this embodiment; 
         FIG. 13  provides drawings for explaining the concept of a predicted positioning distribution pattern in this embodiment; 
         FIG. 14  provides drawings for explaining the concept of a predicted positioning distribution pattern when orientation deviates at the time of installation in this embodiment; 
         FIG. 15  provides drawings for explaining the concept of a predicted positioning distribution pattern when position deviates at the time of installation in this embodiment; 
         FIG. 16  provides drawings for explaining the concept of a predicted positioning distribution pattern when orientation and position deviate at the time of installation in this embodiment; 
         FIG. 17  is a schematic diagram showing an example of a predicted positioning distribution pattern at a plurality of observation points in this embodiment; 
         FIG. 18  is a drawing for explaining the concept of a case in which positioning results are dispersed in this embodiment; 
         FIG. 19  is a flowchart showing the operation of an installation error estimating apparatus according to this embodiment; 
         FIG. 20  is a drawing for explaining a case in which three-dimensional data is processed in this embodiment; 
         FIG. 21  is a block diagram showing the configuration of an installation error estimating apparatus according to Embodiment 2 of the present invention; 
         FIG. 22  is a drawing showing an example of convergence determination information in this embodiment; 
         FIG. 23  is a drawing showing an example of an estimation results list in this embodiment; 
         FIG. 24  is a flowchart showing the convergence determination processing procedure according to this embodiment; 
         FIG. 25  is a block diagram showing the configuration of an installation error estimating apparatus according to Embodiment 3 of the present invention; 
         FIG. 26  is a drawing showing an example of an operation screen display in which installation coordinates of a recommended additional observation point are input in this embodiment; 
         FIG. 27  is a flowchart showing the additional observation point decision processing procedure in this embodiment; and 
         FIG. 28  provides drawings for explaining the concept of an additional observation point in this embodiment. 
     
    
    
     DESCRIPTION OF EMBODIMENTS 
     Now, embodiments of the present invention will be described in detail with reference to the accompanying drawings. 
     The present invention relates to installation error estimation for estimating tag reader installation error when a user installs a tag reader that performs radio positioning. More particularly, in Embodiment 1, when a user installs a tag reader, a wireless tag is placed peripheral to the tag reader (at an observation point), the position of the wireless tag is measured (observed), and tag reader installation error is estimated based on obtained observation data; in Embodiment 2, when a user installs a tag reader, a wireless tag is placed peripheral to the tag reader (at an observation point), the position of the wireless tag is measured (observed), tag reader installation error is estimated based on obtained observation data, and the estimated installation error is evaluated; and in Embodiment 3, as an addition to Embodiment 2, when tag reader installation error is estimated, a position for newly placing a wireless tag as an observation object is further recommended if the estimated installation error does not satisfy a convergence condition. 
     The present invention can be applied to installation of various tag readers with different handled wireless tag frequency bands and types (active type or passive type), position detection methods, and so forth. In Embodiments 1 through 3, a method known as single-point positioning that uses a wireless tag utilizing UWB as a communication method (a UWB tag) is described by way of example. 
     A brief description of single-point positioning is given below. 
     Single-point positioning is a method whereby, after transmitting a UWB signal, a tag reader measures a delay time of a signal sent back from a wireless tag, and converts the measured delay time to a distance. Furthermore, after transmitting a UWB signal, a tag reader receives a reflected wave reflected by a wireless tag via a plurality of receiving antennas, and decides the direction of a wireless tag from a phase difference of received waves. Then the tag reader estimates (measures) the position of a wireless tag from the distance and wireless tag direction obtained in this way. 
     Embodiments 1 through 3 will now be described in order. 
     Embodiment 1 
       FIG. 1  is a configuration diagram of a UWB positioning system that includes an installation error estimating apparatus according to Embodiment 1 of the present invention. 
     The system shown in  FIG. 1  has installation error estimating apparatus  100 , tag reader  200 , and wireless tags  300 . In  FIG. 1 , three wireless tags  300   a ,  300   b , and  300   c  are shown as wireless tags  300 . Installation error estimating apparatus  100  estimates the installation position of tag reader  200 . The configuration and operation of installation error estimating apparatus  100  will be described in detail later herein. Tag reader  200  performs position detection for wireless tags  300  by means of single-point positioning. Wireless tags  300  are UWB tags, for example. 
     For example, a UWB positioning system sequentially detects wireless tag  300  positions by having wireless tags  300  attached to a person or object and performing radio communication with tag reader  200 . Consequently, a UWB positioning system can be applied to management of the entry and exit of people in an office, management of the location of medical supplies in a hospital, work efficiency improvement by ascertaining flow lines in a factory, and so forth. 
     Tag reader  200  and wireless tags  300  perform communication by means of radio (UWB). Installation error estimating apparatus  100  and tag reader  200  perform data exchange by means of communication via an IP (Internet Protocol) network, for example. 
       FIG. 2  is a schematic diagram showing an example of a positioning target area in this embodiment, and  FIG. 3  is a plan of the positioning target area shown in  FIG. 2 . Here, a positioning target area is a spatial area in which tag reader  200  is installed and performs wireless tag  300  position detection. 
     As shown in  FIG. 2 , positioning target area  400  in this embodiment is a rectangular parallelepiped in shape, being a simple room comprising floor  401 , ceiling  402 , four walls  403  through  406 , and door  407 . 
       FIG. 3  is a bird&#39;s-eye view of the room, looking from ceiling to floor. As shown in  FIG. 3 , the bottom-left corner of the room is designated origin (0, 0, 0)  410 . Positioning target area  400  has length m assigned to it in the x-axis direction, length m in the y-axis direction, and height h (not shown) in the z-axis direction (in the direction toward the viewer from the surface of the drawing). This coordinate system (xyz)  420  is a global coordinate system described later herein. 
     Also, as shown in  FIG. 3 , tag reader  200  is installed at installation position  411  in the center (m/2, m/2, Rz) of ceiling  402 . Internally held coordinate system  421  (this coordinate system (XYZ) being a local coordinate system described later herein) used in positioning is defined in tag reader  200 . Therefore, tag reader  200  is expected to be installed with an orientation such that the X-axis of local coordinate system  421  and the x-axis of global coordinate system  420  coincide, and the Y-axis of local coordinate system  421  and the y-axis of global coordinate system  420  coincide. 
     Consequently, in position detection of wireless tag  300 , it is necessary to make a coordinate system defined by tag reader  200  and a coordinate system whereby a tag reader  200  installation position and a wireless tag  300  movement range are defined associated. For example, a coordinate system whereby a tag reader  200  installation position and a wireless tag  300  movement range are defined is a coordinate system having a certain point in an office, factory, or the like as a reference point (hereinafter referred to as “global coordinate system”), as in a design drawing, plan, or the like. On the other hand, a coordinate system defined by tag reader  200  is a coordinate system having a certain point of tag reader  200  as a reference point (hereinafter referred to as “local coordinate system”). In this embodiment, as indicated above, a wireless tag  300  position is represented by (x, y, z) in global coordinate system  420 , and is represented by (X, Y, Z) in local coordinate system  421 . Hereinafter, coordinates (x, y, z) in a global coordinate system are called “global coordinates,” and coordinates (X, Y, Z) in a local coordinate system are called “local coordinates.” 
     If tag reader  200  is installed without error in the proper position and orientation in a design drawing, global coordinate system  420  and local coordinate system  421  can be made to coincide perfectly as shown in equation 1 below. In this case, a wireless tag  300  position measured in local coordinate system  421  by tag reader  200  can be represented in global coordinate system  420 . 
     
       
         
           
             
               
                 
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     Here, (a 1 , a 2 , a 3 ) is known information in a design drawing as a difference between origin  410  in global coordinate system  420  and position  411  at which tag reader  200  should be installed, and comprises fixed values. 
       FIG. 4  is a block diagram showing the configuration of installation error estimating apparatus  100  according to this embodiment. 
     Installation error estimating apparatus  100  shown in  FIG. 4  has observation point setting section  110 , predicted pattern calculation section  120 , observation data input section  130 , variance pattern analysis section  140 , installation error estimating section  150 , and installation error output section  160 . 
     Installation error estimating apparatus  100  comprises, for example, a computer system (personal computer, workstation, or the like) having a communication function. Although not shown in the drawing, this computer system broadly comprises an input apparatus, computer main unit, output apparatus, and communication apparatus. The input apparatus is a keyboard, mouse, or the like, for example. The output apparatus is a display, printer, or the like, for example. The communication apparatus is a communication interface capable of connecting with an IP network or the like, for example. The computer main unit mainly comprises a CPU (Central Processing Unit) and storage apparatus, for example. The CPU has a control function and computation function. The storage apparatus has, for example, ROM that stores a program and data, and RAM that stores data temporarily. The ROM may be flash memory whose contents can be electrically rewritten. 
     The configuration (including input and output) of sections  110  through  160  making up installation error estimating apparatus  100  will now be described, followed by a description of the operation (internal processing) of installation error estimating apparatus  100 . 
     Observation point setting section  110  sets information relating to an observation point at which wireless tag  300  is installed and positioned as observation point information—that is, stores a user observation point specification result. 
     Specifically, observation point setting section  110  stores observation point information input by a user using an input apparatus (not shown) in a predetermined format. 
     Observation point information includes a wireless tag  300  identifier (hereinafter referred to as “tag ID”), coordinates (x, y, z) of an observation point at which wireless tag  300  is placed (hereinafter referred to as “observation point coordinates”), and a name of an observation point at which wireless tag  300  is placed (hereinafter referred to as “observation point name”). 
       FIG. 5  is a drawing showing an example of observation point information. 
     As shown in  FIG. 5 , tag ID  501 , observation point coordinates  502 , and observation point name  503  are stored in observation point information  500  in a mutually associated fashion. 
     When installation error estimating apparatus  100  is started by a user, observation point setting section  110  displays an operation screen for setting observation point information. 
       FIG. 6  is a drawing showing an example of an operation screen display for setting observation point information. 
     Operation screen  510  (hereinafter also referred to as “screen  1 ”) shown in  FIG. 6  is provided with tag ID input column  511 , observation point coordinate input column  512 , and observation point name input column  513 , according to observation point information  500  items. Operation screen  510  is displayed until setting completed button  514  is pressed. 
     Predicted pattern calculation section  120  calculates a characteristic pattern of a predicted positioning distribution obtained by predicting a theoretical positioning distribution for each observation point as a predicted positioning distribution pattern, based on observation point information set by observation point setting section  110 . Specifically, predicted pattern calculation section  120  acquires observation point information  500  (tag ID  501 , observation point coordinates  502 , and observation point name  503 ) from observation point setting section  110 . Then predicted pattern calculation section  120  predicts a positioning distribution (theoretical values) for observation point coordinates  502 , and creates predicted positioning distribution pattern L described later herein. For example, assume that for tag reader  200  installation error, orientation (rotation direction) deviation is designated θ, x-axis direction position deviation is designated a, and y-axis direction position deviation is designated b. At this time, predicted positioning distribution pattern L for each observation point can be represented as function f in equation 2 below, using an affine transformation of a wireless tag  300  theoretical positioning distribution characteristic. The function f derivation method—that is, the predicted positioning distribution pattern creation method used by predicted pattern calculation section  120 —will be described later herein.
 
[2]
 
 L =ƒ(θ, a,b )  (Equation 2)
 
     In this embodiment, it has been assumed that observation point setting section  110  and predicted pattern calculation section  120  are provided, and a predicted positioning distribution pattern is calculated each time a user inputs observation point information, but the present invention is not limited to this. For example, it is possible for a predicted positioning distribution pattern to be calculated beforehand for each observation point, and to be held inside or outside installation error estimating apparatus  100 , associated with observation point information. The predicted positioning distribution pattern holding means may be a storage apparatus incorporated in installation error estimating apparatus  100 , an external storage apparatus connected to installation error estimating apparatus  100 , or any of various kinds of apparatus in a network capable of communication with installation error estimating apparatus  100 . 
     Observation data input section  130  has tag reader  200  observation results for wireless tags  300  as input as observation data—that is, stores tag reader  200 &#39;s wireless tag  300  observation results. Specifically, observation data input section  130  receives from  200 , and stores, observation data obtained by tag reader  200  by performing positioning a plurality of times (for example, 100 times) for each wireless tag  300 . The observation data may be received from tag reader  200  in real time, or may be received together after observation ends (after observation has been performed a plurality of times for each wireless tag  300 ). In this embodiment, a case in which observation data is received in real time is described as an example. 
       FIG. 7  is a drawing showing an example of observation data. 
     Observation data  520  shown in  FIG. 7  includes, as data types (items), sequence number  521 , tag ID  522 , positioning position coordinates (positioning result)  523  of positioned wireless tag  300 , observation point name  524 , and positioning reliability  525 . Positioning position coordinates  523  of wireless tag  300  are identified by at least one of tag ID  522  and observation point name  524 . Positioning reliability  525  is assigned a high rank, for example, if communication conditions are determined to be good. In this embodiment, it is assumed that all wireless tags  300  are installed in places in clear view, and therefore communication conditions are good and the highest rank of “A” is assigned to each measurement. 
     Observation data input section  130  displays an operation screen for starting and terminating positioning. 
       FIG. 8  is a drawing showing an example of an operation screen display for starting and terminating positioning. 
     Operation screen  530  (hereinafter also referred to as “screen  2 ”) shown in  FIG. 8  is provided with log file input field  531 , observation data display field  532 , positioning start button  533 , and Positioning Termination button  534 . 
     In an observation data  520  positioning start, when a user inputs a filename in log file input field  531  and presses positioning start button  533 , a “start positioning” command is passed to tag reader  200 , and tag reader  200  starts wireless tag  300  positioning. On receiving observation data  520  from tag reader  200 , observation data input section  130  sequentially displays data included in received observation data  520  in observation data display field  532 , and also writes a positioning result to the file specified by the user. 
       FIG. 9  is a drawing showing examples of log formats. More particularly,  FIG. 9A  shows an example of a log file displayed in observation data display field  532 , and  FIG. 9B  shows an example of a file output log. 
     As shown in  FIG. 9A , in log  540  displayed in observation data display field  532 , a sequence number, tag ID, x-coordinate, y-coordinate, and z-coordinate of the tag identified by the tag ID, observation point name, and estimated reliability, are displayed sequentially. Display of log  540  is omitted in order from a past log (data with a small sequence number) due to screen size limitations. 
     As shown in  FIG. 9B , in file output log  541 , the same kind of data as in log  540  displayed on the screen is added sequentially to a file from left to right, delimited by commas, without an item name. 
     Wireless tag  300  positioning is terminated automatically when a predetermined number of measurements (for example, 100 measurements per wireless tag) has been reached, or is forcibly terminated by depression of Positioning Termination button  534  by the user. In this embodiment, the number of measurements is set as 100 per wireless tag, for example. Therefore, for example, when three wireless tags  300  are positioned, positioning is terminated automatically when the sequence number reaches 300. Although not shown in the drawings, a button or input field relating to setting of the number of measurements may be added to operation screen  530 . 
     Variance pattern analysis section  140  calculates a characteristic pattern of a measured positioning distribution, which is obtained by means of statistical analysis of positioning results, for each observation point, based on observation data input by means of observation data input section  130 . Specifically, variance pattern analysis section  140  receives observation data  520  from observation data input section  130 , and performs statistical analysis of a positioning distribution (measured values) for each wireless tag  300  identified by tag ID  522 . Then variance pattern analysis section  140  passes an obtained measured positioning distribution characteristic pattern (measured positioning distribution pattern) to installation error estimating section  150  as “variance pattern S.” 
     More specifically, on receiving observation data  520  (log file  541 ) from observation data input section  130 , variance pattern analysis section  140  creates coordinate management tables  550  shown in  FIG. 10 , for example. Coordinate management tables  550  comprise a tag ID, an observation point name, and coordinate information. Coordinate information comprises a serial number, x-coordinate value, y-coordinate value, and reliability. In  FIG. 10 , coordinate management table  551  is a coordinate management table in which the tag ID is managed as “1” and the observation point name as “1.” Similarly, coordinate management table  552  is a coordinate management table in which the tag ID is managed as “2” and the observation point name as “2,” and coordinate management table  553  is a coordinate management table in which the tag ID is managed as “3” and the observation point name as “3.” In this embodiment, for convenience of explanation, z-axis direction coordinates are the same for tag reader  200  and wireless tags  300  and are not required in the processing described later herein, and are therefore omitted. 
     Variance pattern analysis section  140  creates measured positioning distribution pattern list  560  shown in  FIG. 11 , for example. Measured positioning distribution pattern list  560  records results of measured positioning distribution pattern analysis. As shown in  FIG. 11 , measured positioning distribution pattern list  560  comprises, for example, a tag ID, observation result x-coordinate or y-coordinate average value μ and variance σ, and pattern length P based on a computational formula described later herein. The actual analytical method used by variance pattern analysis section  140  will be described in detail later herein. 
     Installation error estimating section  150  calculates installation error of tag reader  200  using a predicted positioning distribution pattern calculated by predicted pattern calculation section  120  and variance pattern S calculated by variance pattern analysis section  140  based on observation data  520 . Specifically, installation error estimating section  150  receives predicted positioning distribution pattern L represented by equation 2 from predicted pattern calculation section  120 , and variance pattern S from variance pattern analysis section  140 , for each observation point at which wireless tag  300  is placed. Then, using transmitted predicted positioning distribution pattern L and variance pattern S, installation error estimating section  150  solves equation 3 below so that predicted positioning distribution pattern L becomes a good approximation of variance pattern S at each observation point. By this means, installation error estimating section  150  estimates θ, a, and b comprising tag reader  200  installation error. Installation error estimating section  150  passes estimated installation error (θ, a, b) to installation error output section  160 . The actual processing performed by installation error estimating section  150  will be described in detail later herein.
 
[3]
 
ƒ(θ ,a,b )= S   (Equation 3)
 
     Estimating the three variables θ, a, and b using equation 3 requires at least variance patterns S 1 , S 2 , and S 3  relating to three observation points. Thus, in this embodiment, three observation points  431 ,  432 , and  433  (observation point  1 , observation point  2 , and observation point  3 , respectively) are provided, as shown in  FIG. 2 . 
     Installation error output section  160  reports installation error (θ, a, b) estimated by installation error estimating section  150  to the user via a predetermined operation screen. 
       FIG. 12  is a drawing showing an example of an operation screen display for displaying estimated installation error (θ, a, b). 
     Operation screen  570  (hereinafter also referred to as “screen  3 ”) shown in  FIG. 12  is provided with rotation direction error display field  571 , X-axis direction error display field  572 , Y-axis direction error display field  573 , redo button  574 , and terminate button  575 . Angle error θ is displayed in rotation direction error display field  571 , X-axis direction distance error a in X-axis direction error display field  572 , and Y-axis direction distance error b in Y-axis direction error display field  573 . Redo button  574  is a button to allow the user to start over from the beginning. When redo button  574  is pressed, a “screen transition” command is sent to observation point setting section  110 , and screen  1  (operation screen  510  shown in  FIG. 6 ) is displayed. When Terminate button  575  is pressed, installation error output section  160  terminates display of screen  3 , and also terminates operation of installation error estimating apparatus  100 . 
     Estimated installation error (θ, a, b) is used to find a transformation matrix that performs coordinate conversion between a local coordinate system and global coordinate system. For example, in this embodiment, a local coordinates to global coordinates transformation matrix is created using estimated installation error (θ, a, b), and tag reader  200  local coordinate system positioning data is converted to global coordinates. By this means, results can be obtained that are equivalent to those when tag reader  200  is installed in a correct position and orientation by repeating precise measurement and correction. This shows that, even if the installation precision of tag reader  200  is inadequate, high-precision positioning is possible by estimating installation error (θ, a, b) accurately. Therefore, high-precision positioning can be implemented even if tag reader  200  installation is simple. 
     The description here refers to the predicted positioning distribution pattern creation method used by predicted pattern calculation section  120 , which will be described in detail using  FIG. 13  through  FIG. 17 . 
     First, a predicted positioning distribution pattern will be described. 
     In this embodiment, positioning results are acquired by placing wireless tag  300  in a place in clear view and without intervening obstructions vis-a-vis tag reader  200  (observation point  420 ), and performing measurement a plurality of times (for example, 100 times). A measurement result distribution extends in an arc of angle error a on spherical surface  600  with line segments linking tag reader  200  to two installation positions  411  and  430  as radius r (see  FIG. 13A ). Below, this distribution is called “arc-shaped positioning distribution”  601 . Here, radius r is calculated using tag reader  200  installation position coordinates and observation point information  500  observation point coordinates  502 . Also, angle error a is calculated based on a theoretical positioning distribution characteristic for each observation point coordinate  502 , where a theoretical positioning distribution characteristic is derived from a result of positioning a wireless tag in an experimental environment beforehand. 
       FIG. 13B  is a drawing in which the vicinity of observation point  430  shown in  FIG. 13A  is enlarged. In this embodiment, arc-shaped positioning distribution  601  is projected in a tangential direction at observation point  430 . The reason for performing projection is that a positioning result is a multidimensional vector, and therefore dimensions are reduced by projection in a direction in which a characteristic emerges. Below, a projected distribution is called simply “positioning distribution”  602 . 
     For example, if angle error a is small and radius r is sufficiently large, arc-shaped positioning distribution  601  can approximate positioning distribution  602  (the same applies when one or other of the conditions is satisfied). This means, for example, that if arc-shaped positioning distribution  601  represents a normal distribution, positioning distribution  602  on a straight line (tangent) can also be treated as a normal distribution. 
     Also, in order to differentiate between a predicted positioning distribution and a positioning distribution obtained from actual measurement results here, the former is called a “predicted positioning distribution” and the latter a “measured positioning distribution” below. 
     If positioning distribution width L is considered as a characteristic pattern of a predicted positioning distribution, width L of positioning distribution  602  (hereinafter referred to as “predicted positioning distribution pattern L”) can be represented by equation 4 below, where “r” and “α” are the radius and angle error respectively in arc-shaped positioning distribution  601 , as stated above.
 
[4]
 
 L= 2 ·r ·tan α  (Equation 4)
 
     In this embodiment, positioning distribution width (error distribution width) L is used as a positioning distribution pattern for both theoretical values and observed values, but the present invention is not limited to this. For example, for a positioning distribution pattern, a peak value of a positioning distribution or the like may be used as a positioning distribution pattern instead of positioning distribution width (error distribution width) L. Furthermore, in this embodiment it is also possible to use results of a statistical method such as principal component analysis as a positioning distribution pattern. 
     Next, a predicted positioning distribution pattern creation method will be described for a case in which tag reader  200  installation deviates when tag reader  200  is installed at (m/2, m/2, Rz) on the plan (see  FIG. 3 ), for example. For convenience of explanation, separate descriptions will be given here for three possible cases. 
     First, a predicted positioning distribution pattern creation method will be described for a case in which tag reader  200  has been installed at the correct position, and in which the orientation of tag reader  200  nevertheless deviates. 
     Secondly, a predicted positioning distribution pattern creation method will be described for a case in which tag reader  200  is installed with the expected orientation of, and in which the position of tag reader  200  nevertheless deviates. 
     Thirdly, a predicted positioning distribution pattern creation method will be described for a case in which tag reader  200  is installed with deviation of both orientation and position. 
     (1) When Orientation Deviates 
     The first example of predicted positioning distribution pattern creation is for a case in which installation has been achieved at the correct position, but with deviation of orientation, when attempting to install tag reader  200  at (m/2, m/2, Rz) on the plan, for example. The predicted positioning distribution pattern creation method in this case will be described using  FIG. 14 . 
     Solid-line coordinate system  610  shown in  FIG. 14A  results from parallel translation of the coordinate system assigned to the plan (global coordinate system  420 ) to position  411  at which installation of tag reader  200  is expected (hereinafter referred to as a “virtual global coordinate system”). Also, the dotted-line coordinate system shown in  FIG. 14A  is the coordinate system used in positioning (local coordinate system  421 ), held internally by tag reader  200 . In this case, local coordinate system  421  of tag reader  200  is assumed to be rotated through angle θ relative to coordinate system  610 . For the sake of simplicity, Z-axis (z-axis) direction coordinates are here assumed to be the same for tag reader  200  and wireless tag  300 , and a description is given in terms of two-dimensional coordinates. Also, coordinates in a virtual global coordinate system are called “virtual global coordinates” below. 
     When, for example, both coordinate systems completely coincide, and wireless tag  300  is positioned at observation point  1  (m/2, m) and is measured a plurality of times, a positioning result in local coordinate system  421  output by tag reader  200  is the pattern represented by equation 4 in the x-axis direction. 
     However, in this case, the coordinates of observation point  1  are (X 1 ,Y 1 ), as represented by equation 5 below, as local coordinates (X, Y) deviating by angle θ from virtual global coordinates (x, y). In this case, r=m/2. This radius r frequently occurs in the following equations, but is the same as in equation 5, and a description is omitted where it occurs. 
     
       
         
           
             
               
                 
                   [ 
                   5 
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     ( 
                     
                       
                         
                           
                             X 
                             1 
                           
                         
                       
                       
                         
                           
                             Y 
                             1 
                           
                         
                       
                     
                     ) 
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               θ 
                             
                           
                           
                             
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               θ 
                             
                           
                         
                         
                           
                             
                               
                                 - 
                                 sin 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               θ 
                             
                           
                           
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               θ 
                             
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             0 
                           
                         
                         
                           
                             r 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     5 
                   
                   ) 
                 
               
             
           
         
       
     
     Therefore, when a positioning result output by tag reader  200  is mapped directly to global coordinate system  420 , observation point  1  coordinates become coordinates (x 2 , y 2 ) deviating as represented by equation 7 from original coordinates (x 1 , y 1 ) represented by equation 6 below. 
     
       
         
           
             
               
                 
                   [ 
                   6 
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     ( 
                     
                       
                         
                           
                             x 
                             1 
                           
                         
                       
                       
                         
                           
                             y 
                             1 
                           
                         
                       
                     
                     ) 
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             
                               m 
                               / 
                               2 
                             
                           
                         
                         
                           
                             m 
                           
                         
                       
                       ) 
                     
                     = 
                     
                       
                         ( 
                         
                           
                             
                               0 
                             
                           
                           
                             
                               r 
                             
                           
                         
                         ) 
                       
                       + 
                       
                         ( 
                         
                           
                             
                               
                                 m 
                                 / 
                                 2 
                               
                             
                           
                           
                             
                               
                                 m 
                                 - 
                                 r 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     6 
                   
                   ) 
                 
               
             
             
               
                 
                   [ 
                   7 
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     ( 
                     
                       
                         
                           
                             x 
                             2 
                           
                         
                       
                       
                         
                           
                             y 
                             2 
                           
                         
                       
                     
                     ) 
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             
                               X 
                               1 
                             
                           
                         
                         
                           
                             
                               Y 
                               1 
                             
                           
                         
                       
                       ) 
                     
                     + 
                     
                       ( 
                       
                         
                           
                             
                               m 
                               / 
                               2 
                             
                           
                         
                         
                           
                             
                               m 
                               - 
                               r 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     7 
                   
                   ) 
                 
               
             
           
         
       
     
       FIG. 14D  is a drawing showing the positional relationship of these two pairs of coordinates (x 1 , y 1 ) and (x 2 , y 2 ). Terms (m/2, m−r) T  (where T represents transposition) common to equation 6 and equation 7 represent the positional relationship between origin  410  in global coordinate system  420  and expected tag reader  200  installation position  411 , and are fixed values. 
     At this time, predicted positioning distribution pattern L 0  deviates from predicted positioning distribution pattern L ideal  represented by equation 4 when tag reader  200  is assumed to have been installed as expected (see  FIG. 14B ). 
       FIG. 14C  is a principal-part enlarged view of  FIG. 14B , showing the nature of predicted positioning distribution pattern deviation. 
     In this case, even if it is assumed that the coordinate system has been rotated, there is no change in the distance between tag reader  200  and wireless tag  300  (that is, the radius), and therefore there is no change in size between predicted positioning distribution pattern L ideal  and predicted positioning distribution pattern L 0 . Deviating predicted positioning distribution pattern L 0  has a shape resulting from rotating predicted positioning distribution pattern L ideal  through an angle of −θ. 
     Next, deviating predicted positioning distribution pattern L 0  is projected in a linear direction parallel to predicted positioning distribution pattern L ideal , creating predicted positioning distribution pattern L 1 . Predicted positioning distribution pattern L 1  can be represented by equation 8 below. The reason for performing projection will be explained when the analytical method used by variance pattern analysis section  140  is described later herein.
 
[8]
 
 L   1 =2 ·r ·tan α·cos θ  (Equation 8)
 
     (2) When Position Deviates 
     The second example of predicted positioning distribution pattern creation is for a case in which installation has been achieved with the expected orientation, but with deviation of position, when attempting to install tag reader  200  at (m/2, m/2, Rz) on the plan, for example. The predicted positioning distribution pattern creation method in this case will be described using  FIG. 15 . 
     As in  FIG. 14 , the solid-line coordinate system shown in  FIG. 15A  is virtual global coordinate system  610 , and the dotted-line coordinate system is local coordinate system  421 . It is assumed that local coordinate system  421  of tag reader  200  deviates by a in the x-axis direction and by b in the y-axis direction with respect to virtual global coordinate system  610 . For the sake of simplicity, here also, Z-axis (z-axis) direction coordinates are assumed to be the same for tag reader  200  and wireless tag  300 , and a description is given in terms of two-dimensional coordinates. 
     When, for example, both coordinate systems fully coincide, and wireless tag  300  is positioned at observation point  1  (m/2, m) and is measured a plurality of times, a positioning result in local coordinate system  421  output by tag reader  200  is the pattern represented by equation 4 in the x-axis direction. 
     However, in this case, the distance between tag reader  200  and wireless tag  300  (that is, the radius) changes in local coordinate system  421  through the parallel translation and deviation of local coordinate system  421  with respect to virtual global coordinate system  610 . Consequently, when the position deviates, a major change between predicted positioning distribution pattern L ideal  and predicted positioning distribution pattern L 0  is caused by the change in radius r. 
     This is due to the fact that, as a characteristic of single-point positioning, angle error α shown in  FIG. 13A  is always the same for observation points aligned on the same radius. Therefore, as is clear from equation 4, if radius r increases, predicted positioning distribution pattern L also increases. 
     Here, if observation point  1  is viewed in local coordinate system  421 , radius r 1  can be represented by equation 9 below.
 
[9]
 
 r   1 =√{square root over ( a   2 +( b−r ) 2 )}  (Equation 9)
 
     Also, a predicted positioning distribution pattern exists in a direction perpendicular to this radius r 1 . 
     Furthermore, in local coordinate system  421  of tag reader  200  deviating by a in the x-axis direction and by b in the y-axis direction with respect to virtual global coordinate system  610 , observation point  1  becomes (X 1 , Y 1 ) as represented by equation 10 below. 
     
       
         
           
             
               
                 
                   [ 
                   10 
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     ( 
                     
                       
                         
                           
                             X 
                             1 
                           
                         
                       
                       
                         
                           
                             Y 
                             1 
                           
                         
                       
                     
                     ) 
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
                             - 
                             a 
                           
                         
                       
                       
                         
                           
                             r 
                             - 
                             b 
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     10 
                   
                   ) 
                 
               
             
           
         
       
     
     Therefore, when a positioning result output by tag reader  200  is mapped directly to a virtual global coordinate system, predicted positioning distribution pattern L 0  deviates from predicted positioning distribution pattern L ideal  represented by equation 4 (see  FIG. 15B ). 
     Thus, deviating predicted positioning distribution pattern L 0  is projected in a linear direction parallel to predicted positioning distribution pattern L ideal , and predicted positioning distribution pattern L 1  is created. Predicted positioning distribution pattern L 1  can be represented by equation 11 below. 
     
       
         
           
             
               
                 
                   [ 
                   11 
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       L 
                       1 
                     
                     = 
                     
                       
                         2 
                         · 
                         
                           r 
                           1 
                         
                         · 
                         tan 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         α 
                         · 
                         cos 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         β 
                         1 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     where 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     cos 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     β 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     1 
                   
                   = 
                   
                     
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         β 
                         1 
                       
                     
                     = 
                     
                       
                         r 
                         - 
                         b 
                       
                       
                         
                           
                             a 
                             2 
                           
                           + 
                           
                             
                               ( 
                               
                                 b 
                                 - 
                                 r 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     11 
                   
                   ) 
                 
               
             
           
         
       
     
     (3) When Orientation and Position Deviate 
     The third example of predicted positioning distribution pattern creation is for a case in which both orientation and position deviate when attempting to install tag reader  200  at (m/2, m/2, Rz) on the plan, for example. Predicted positioning distribution pattern creation in this case will be described using  FIG. 16 . 
     As in  FIG. 14 , the solid-line coordinate system shown in  FIG. 16A  is virtual global coordinate system  610 , and the dotted-line coordinate system is local coordinate system  421 . It is assumed that local coordinate system  421  of tag reader  200  deviates by a in the x-axis direction, by b in the y-axis direction and by θ in the rotation direction, with respect to virtual global coordinate system  610 . 
     For the sake of simplicity, here also, Z-axis (z-axis) direction coordinates are assumed to be the same for tag reader  200  and wireless tag  300 , and a description is given in terms of two-dimensional coordinates. 
     A case in which both the orientation and position of tag reader  200  deviate can be represented by a linear sum of a first rotation and a second parallel translation. 
     Therefore, when both the orientation and position of tag reader  200  deviate, a rotation component should be added to equation 11. That is to say, deviating predicted positioning distribution pattern L 0  shown in  FIG. 15B  should be rotated through an angle of −θ as shown in  FIG. 14C  (see deviating predicted positioning distribution pattern L 0 ′ shown in  FIG. 16B ). 
     Therefore, x-axis-direction predicted positioning distribution pattern L 1  for observation point  1  can be represented by equation 12 below.
 
[12]
 
 L   1 =2 ·r   1 ·tan α·cos  T   1   (Equation 12)
 
where T 1 =T 1 =β 1 −θ
 
     Predicted positioning distribution patterns for the three observation points  1  through  3  will now be described, with schematic diagrams thereof shown in  FIG. 17 . 
     As shown in  FIG. 17 , for observation point  1 , projection is performed using projection angle T 1  for pre-projection deviating predicted positioning distribution pattern L 0 ′ (represented here by L 01 ′ to differentiate it from other observation points), and predicted positioning distribution pattern L 1  is acquired. 
     The above concept can be applied to other observation points—for example, observation point  2  (m, m/2) and observation point  3  (m/2, 0). In this case, y-axis-direction predicted positioning distribution pattern L 2  for observation point  2  is represented by equation 13, and x-axis-direction predicted positioning distribution pattern L 3  for observation point  3  is represented by equation 14. 
     
       
         
           
             
               
                 
                   [ 
                   13 
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       
                         L 
                         2 
                       
                       = 
                       
                         
                           2 
                           · 
                           
                             r 
                             2 
                           
                           · 
                           tan 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           α 
                           · 
                           cos 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           T 
                           2 
                         
                       
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                       where 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         β 
                         2 
                       
                     
                     = 
                     
                       
                         r 
                         - 
                         a 
                       
                       
                         
                           
                             
                               ( 
                               
                                 r 
                                 - 
                                 a 
                               
                               ) 
                             
                             2 
                           
                           + 
                           
                             b 
                             2 
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       r 
                       2 
                     
                     = 
                     
                       
                         
                           
                             ( 
                             
                               r 
                               - 
                               a 
                             
                             ) 
                           
                           2 
                         
                         + 
                         
                           b 
                           2 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       T 
                       2 
                     
                     = 
                     
                       
                         β 
                         2 
                       
                       + 
                       θ 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     13 
                   
                   ) 
                 
               
             
             
               
                 
                   [ 
                   14 
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       L 
                       3 
                     
                     = 
                     
                       
                         2 
                         · 
                         
                           r 
                           3 
                         
                         · 
                         tan 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         α 
                         · 
                         cos 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         T 
                         3 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       where 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         β 
                         3 
                       
                     
                     = 
                     
                       
                         r 
                         + 
                         b 
                       
                       
                         
                           
                             a 
                             2 
                           
                           + 
                           
                             
                               ( 
                               
                                 r 
                                 + 
                                 b 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       r 
                       3 
                     
                     = 
                     
                       
                         
                           a 
                           2 
                         
                         + 
                         
                           
                             ( 
                             
                               b 
                               + 
                               r 
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     T 
                     3 
                   
                   = 
                   
                     
                       β 
                       3 
                     
                     + 
                     θ 
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     14 
                   
                   ) 
                 
               
             
           
         
       
     
     As shown in  FIG. 17 , for observation point  2 , projection is performed using projection angle T 2  for pre-projection deviating predicted positioning distribution pattern L 02 ′, and predicted positioning distribution pattern L 2  is acquired. Also, for observation point  3 , projection is performed using projection angle T 3  for pre-projection deviating predicted positioning distribution pattern L 03 ′, and predicted positioning distribution pattern L 3  is acquired. 
     In this way, predicted pattern calculation section  120  creates predicted positioning distribution patterns L 1  through L 3  for observation points  1  through  3  specified by observation point information  500 . 
     A detailed description of the analytical method used by variance pattern analysis section  140  will now be described, with additional reference to  FIG. 18 . 
     On receiving log file  541  (observation data  520 ) from observation data input section  130 , variance pattern analysis section  140  creates coordinate management tables  550  shown in  FIG. 10 . The actual procedure is as follows. 
     First, variance pattern analysis section  140  reads log file  541  received from observation data input section  130 , one line at a time. In the case of the example shown in  FIG. 9B , variance pattern analysis section  140  reads the first line for which the sequence number starts at “1,” and on confirming that the tag ID is “1” and the observation point name is “observation point  1 ,” determines whether or not these are already known. At this time, tag ID “1” and observation point name “1” are not known, and therefore variance pattern analysis section  140  creates coordinate management table  551  in which the tag ID is managed as “1” and the observation point name as “1.” 
     At this time, variance pattern analysis section  140  stores “1” as the tag ID and “observation point  1 ” as the observation point name when coordinate management table  551  is created. Serial numbers start from 1, and the serial number is incremented by 1 each time new coordinate information is added. For the x-coordinate value, y-coordinate value, and reliability, log file  541  data is copied. 
     Variance pattern analysis section  140  processes the second line and third line of log file  541  in the same way as the first line, creating coordinate management tables  552  and  553  respectively, and copying the respective x-coordinate value, y-coordinate value, and reliability. 
     When the fourth line of log file  541  is read, since tag ID “1” and observation point name “1” are already known, variance pattern analysis section  140  updates previously created coordinate management table  551 . Specifically, variance pattern analysis section  140  adds serial number “2,” x-coordinate value “x 41 ,” y-coordinate value “x 41 ,” and reliability “A.” Thereafter, variance pattern analysis section  140  repeats the same kind of processing until the last line of the file. 
     Next, variance pattern analysis section  140  creates measured positioning distribution pattern list  560  shown in  FIG. 11 , using created coordinate management tables  550 . 
     Since actually measured results are distributed in two dimensions, different straight lines  620  and  621  may be candidates for checking a positioning distribution characteristic, as shown in  FIG. 18 . Therefore, it is difficult to uniquely identify measured positioning distribution pattern S. 
     Thus, in this embodiment, for example, a measurement result distribution is projected in a direction perpendicular to predicted positioning distribution pattern L, and measured positioning distribution pattern S is extracted. In the example shown in  FIG. 18 , measured positioning distribution pattern S exists on straight line  621  parallel to predicted positioning distribution pattern L. 
     When projection is performed, the advantage of comparison in the same dimension can be applied to a characteristic comparison between predicted positioning distribution pattern L and measured positioning distribution pattern S. Also, if the projection angle is small, it is possible to approximate a characteristic of pre-projection data directly. For example, data resulting from projecting normally distributed pre-projection data can also be regarded as normally distributed. 
     That is to say, variance pattern analysis section  140  extracts an x-axis direction pattern as measured positioning distribution pattern S 1  of wireless tag  300  installed at observation point  1  (m/2, m) on the y-axis of the global coordinate system; variance pattern analysis section  140  extracts a y-axis direction pattern as measured positioning distribution pattern S 2  of wireless tag  300  installed at observation point  2  (m, m/2) on the x-axis of the global coordinate system; and variance pattern analysis section  140  extracts an x-axis direction pattern as measured positioning distribution pattern S 3  of wireless tag  300  installed at observation point  3  (m/2, 0) on the y-axis of the global coordinate system. 
     Variance pattern analysis section  140  records the results of performing measured positioning distribution pattern analysis in this way in the form of measured positioning distribution pattern list  560  shown in  FIG. 11 , for example. As explained above, measured positioning distribution pattern list  560  comprises, for example, a tag ID, observation result x-coordinate or y-coordinate average value μ and variance σ, and pattern length P. 
     Here, pattern length P is extracted, for example, by means of a computational formula decided beforehand according to a normal distribution characteristic (see equation 15 below). Here, N is a standardized score in a standard normal probability table.
 
[15]
 
 P= 2* N*σ   (Equation 15)
 
     If this computational formula is applied to the case of tag ID “1” projected in the x-axis direction, when N=2, this means that approximately 95% of data is included in μ−2σ≦x≦μ+2σ. 
     Variance pattern analysis section  140  can calculate pattern length P by applying N that has been defined beforehand in this way. 
     The actual processing performed by installation error estimating section  150  will now be described in detail. 
     Installation error estimating section  150  acquires, in advance, predicted positioning distribution pattern L for each observation point at which wireless tag  300  is installed from predicted pattern calculation section  120 , and measured positioning distribution pattern S based on observation data  520  from variance pattern analysis section  140 . Then installation error estimating section  150  estimates installation error (θ, a, b) by solving an equation resulting from substituting these in equation 3. 
     Specifically, equation 16 below holds true for wireless tag  300   a  placed at global coordinate system observation point  1  (m/2, m). 
     
       
         
           
             
               
                 
                   [ 
                   16 
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       S 
                       1 
                     
                     = 
                     
                       
                         2 
                         · 
                         
                           r 
                           1 
                         
                         · 
                         tan 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         α 
                         · 
                         cos 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         T 
                         1 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       where 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         β 
                         1 
                       
                     
                     = 
                     
                       
                         r 
                         - 
                         b 
                       
                       
                         
                           
                             a 
                             2 
                           
                           + 
                           
                             
                               ( 
                               
                                 b 
                                 - 
                                 r 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       r 
                       1 
                     
                     = 
                     
                       
                         
                           a 
                           2 
                         
                         + 
                         
                           
                             ( 
                             
                               b 
                               - 
                               r 
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     T 
                     1 
                   
                   = 
                   
                     
                       β 
                       1 
                     
                     - 
                     θ 
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     16 
                   
                   ) 
                 
               
             
           
         
       
     
     Also, equation 17 below holds true for wireless tag  300   b  placed at global coordinate system observation point  2  (m, m/2). 
     
       
         
           
             
               
                 
                   [ 
                   17 
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       S 
                       2 
                     
                     = 
                     
                       
                         2 
                         · 
                         
                           r 
                           2 
                         
                         · 
                         tan 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         α 
                         · 
                         cos 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         T 
                         2 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       where 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         β 
                         2 
                       
                     
                     = 
                     
                       
                         r 
                         - 
                         a 
                       
                       
                         
                           
                             
                               ( 
                               
                                 r 
                                 - 
                                 a 
                               
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     Furthermore, equation 18 below holds true for wireless tag  300   c  placed at global coordinate system observation point  3  (m/2, 0). 
     
       
         
           
             
               
                 
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                       where 
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                     18 
                   
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     Therefore, installation error (θ, a, b) can be calculated by solving three-way simultaneous equations comprising equation 16, equation 17, and equation 18. 
     In the following description, the operation of installation error estimating apparatus  100  having the above configuration will be described using  FIG. 19 . 
     First, prior to error estimation processing by installation error estimating apparatus  100 , the user performs positioning in order to install tag reader  200 . In this embodiment, there is no particular restriction on the positioning method. For example, positioning can be performed by using a tape measure to mark positions m/2 in the x-axis direction and m/2 in the y-axis direction from origin  410  of floor  401 . At this time, there may be deviation from the expected-value position (m/2, m/2, Rz) and orientation. 
     Next, the user fixes tag reader  200  at the decided position. In this embodiment, there is no particular restriction on the fixing method. A known method whereby a radio LAN (Local Area Network) access point is fixed to ceiling  402 , for example, can be used. Here too, as in the case of positioning, there may be deviation from the expected-value position (m/2, m/2, Rz) and orientation. 
     The user selects a minimum of three places—for example, observation point  431  (observation point  1 ) having coordinates (m/2, m, Rz), observation point  432  (observation point  2 ) having coordinates (m, m/2, Rz), and observation point  433  (observation point  3 ) having coordinates (m/2, 0, Rz). Then the user installs wireless tags  300  with previously known tag IDs at selected observation points  1  through  3 . As actual examples of wireless tag  300  installation, for instance, wireless tag  300   a  with a tag ID of “1” is installed at observation point  1 , wireless tag  300   b  with a tag ID of “2” is installed at observation point  2 , and wireless tag  300   c  with a tag ID of “3” is installed at observation point  3 . 
     It is assumed that tag reader  200  has already been installed in the vicinity of expected-value position (m/2, m/2, Rz) in this way. Next, with the three wireless tags  300   a  through  300   c  installed in the vicinity of the three observation points  1  through  3 , the user starts installation error estimating apparatus  100 . That is to say, the processing in the flowchart shown in  FIG. 19  is started. 
     First, in step S 1000 , observation point setting section  110  performs display of screen  1 . Specifically, when the user starts installation error estimating apparatus  100 , observation point setting section  110  displays operation screen  510  (screen  1 ) shown in  FIG. 6 . By this means, the user can input data necessary for setting observation point information  500 . 
     Then, in step S 1010 , the user performs input of data necessary for setting observation point information. Specifically, the user inputs observation point information data items (tag ID, observation point coordinates, and observation point name) in tag ID input column  511 , observation point coordinate input column  512 , and observation point name input column  513  respectively in screen  1  displayed in step S 1000 . For example, observation point  1 , observation point  2 , and observation point  3  coordinates are input in observation point coordinate input column  512 . An observation point name may be input by the user, or a default value may be displayed. 
     Then, in step S 1020 , observation point setting section  110  determines whether or not setting completed button  514  has been pressed. If the result of this determination is that setting completed button  514  has been pressed (S 1020 : YES), observation point setting section  110  proceeds to step S 1030 . On the other hand, if setting completed button  514  has not been pressed (S 1020 : NO), observation point setting section  110  returns to step S  1010 . That is to say, observation point setting section  110  continues to display screen  1  until setting completed button  514  is pressed. 
     In step S 1030 , observation point setting section  110  performs observation point information setting. Specifically, on receiving a “set” command by means of setting completed button  514 , observation point setting section  110  mutually associates the data (tag ID, observation point coordinates, and observation point name) input in step S 1010 , and creates and stores observation point information  500  shown in  FIG. 5 . 
     Then, in step S  1040 , predicted pattern calculation section  120  performs predicted positioning distribution pattern creation for all the observation points. Specifically, predicted pattern calculation section  120  acquires observation point information  500  (tag ID  501 , observation point coordinates  502 , and observation point name  503 ) set in step S 1030  from observation point setting section  110 . Next, predicted pattern calculation section  120  predicts a positioning distribution (theoretical values) and creates predicted positioning distribution pattern L for all observation points specified by observation point information  500 . The predicted positioning distribution pattern L creation method is as described above. For example, predicted pattern calculation section  120  creates predicted positioning distribution patterns L 1  through L 3  for observation points  1  through  3  specified by observation point information  500  in accordance with the above-described creation method. On completing the creation of predicted positioning distribution patters L for all the observation points, predicted pattern calculation section  120  sends a “screen transition” command to observation data input section  130 . 
     Then, in step S 1050 , observation data input section  130  performs display of screen  2 . Specifically, on receiving a “screen transition” command from predicted pattern calculation section  120 , observation data input section  130  displays operation screen  530  (screen  2 ) shown in  FIG. 8 . 
     Then, in step S 1060 , observation data input section  130  determines whether or not positioning start button  533  has been pressed. If the result of this determination is that positioning start button  533  has been pressed (S 1060 : YES), observation data input section  130  proceeds to step S 1070 . On the other hand, if positioning start button  533  has not been pressed (S 1060 : NO), observation data input section  130  returns to step S 1050 , and continues to display screen  2 . For example, when the user inputs a filename in log file input field  531  and presses positioning start button  533  in order to perform file output of observation data  520 , observation data input section  130  passes a “start positioning” command to tag reader  200 . 
     In step S 1070 , observation data input section  130  executes wireless tag  300  positioning. Specifically, observation data input section  130  passes a “start positioning” command to tag reader  200 . On receiving a “start positioning” command from observation data input section  130 , tag reader  200  starts wireless tag  300  positioning, and passes observation data  520  to observation data input section  130 . 
     On receiving observation data  520  from tag reader  200 , observation data input section  130  sequentially displays data included in received observation data  520  in observation data display field  532  of screen  2  (see  FIG. 9A ). At the same time, observation data input section  130  writes a positioning result to a file specified by the user (see  FIG. 9B ). If, for example, 100 observations (positioning operations) have been recommended per wireless tag, observation data input section  130  checks the sequence number, and automatically terminates positioning when sequence number  300  is reached. In this embodiment, the number of observations is set at 100 per wireless tag, for example. 
     Then, in step S 1080 , observation data input section  130  determines whether or not the number of observations has reached a predetermined number. If the result of this determination is that the number of observations has reached the predetermined number (S 1080 : YES), observation data input section  130  proceeds to step S 1090 . On the other hand, if the number of observations has not reached the predetermined number (S 1080 : NO), observation data input section  130  returns to step S 1070 , and continues wireless tag  300  positioning. For example, in this embodiment three wireless tags  300   a ,  300   b , and  300   c  are installed, and therefore the total number of observations is 300. Therefore, observation data input section  130  checks the sequence number, and when the sequence number reaches 300, determines that the number of observations has reached the predetermined number (300). When wireless tag  300  positioning terminates, observation data input section  130  passes log file  541 , to which observation data  520  has been written, to variance pattern analysis section  140 . 
     In step S 1090 , variance pattern analysis section  140  performs measured positioning distribution pattern creation for all the observation points. Specifically, variance pattern analysis section  140  acquires observation data  520  (log file  541 ) from observation data input section  130 , and performs analysis of observation data  520 . Next, variance pattern analysis section  140  stores the results in measured positioning distribution pattern list  560 , and passes measured positioning distribution pattern list  560  to installation error estimating section  150 . For example, variance pattern analysis section  140  creates coordinate management tables  550  shown in  FIG. 10  by means of the above-described analytical method, and creates measured positioning distribution pattern list  560  shown in  FIG. 11  using created coordinate management tables  550 . 
     Then, in step S 1100 , installation error estimating section  150  performs installation error estimation. Specifically, installation error estimating section  150  acquires predicted positioning distribution pattern L of each wireless tag  300  observation point created by predicted pattern calculation section  120  in step S 1040 , and acquires measured positioning distribution pattern S based on observation data  520  created by variance pattern analysis section  140  in step S 1090 . Next, installation error estimating section  150  estimates installation error (θ, a, b) by solving an equation resulting from substituting the above in equation 3. For example, equation 16 holds true for wireless tag  300   a  placed at global coordinate system observation point  1  (m/2, m), equation 17 holds true for wireless tag  300   b  placed at global coordinate system observation point  2  (m, m/2), and equation 18 holds true for wireless tag  300   c  placed at global coordinate system observation point  3  (m/2, 0). Therefore, installation error (θ, a, b) can be calculated by solving three-way simultaneous equations comprising equation 16, equation 17, and equation 18. On completing installation error (θ, a, b) estimation, installation error estimating section  150  passes estimation results (θ, a, b) to installation error output section  160 . 
     Then, in step S 1110 , installation error output section  160  performs display of the installation error estimation results in screen  3 . Specifically, on receiving installation error (θ, a, b) estimated in step S 1100  from installation error estimating section  150 , installation error output section  160  displays operation screen  570  (screen  3 ) shown in  FIG. 12 . In  FIG. 12 , angle error θ is displayed in rotation direction error display field  571 , X-axis direction distance error a is displayed in X-axis direction error display field  572 , and Y-axis direction distance error b is displayed in Y-axis direction error display field  573 . 
     Then, in step S 1120 , installation error output section  160  determines whether or not redo button  574  has been pressed. If the result of this determination is that redo button  574  has been pressed (S 1120 : YES), installation error output section  160  returns to step S 1000 . On the other hand, if redo button  574  has not been pressed (S 1120 : NO), installation error output section  160  proceeds to step S 1130 . For example, when the user presses redo button  574 , installation error output section  160  sends a “screen transition” command to observation point setting section  110 . On receiving a “screen transition” command from installation error output section  160 , observation point setting section  110  displays screen  1 . Therefore, from this point onward, the user can redo the processing from the start. 
     In step S 1130 , installation error output section  160  determines whether or not Terminate button  575  has been pressed. If the result of this determination is that Terminate button  575  has been pressed (S 1130 : YES), installation error output section  160  terminates display of screen  3 , and also terminates operation of installation error estimating apparatus  100 . On the other hand, if Terminate button  575  has not been pressed (S 1130 : NO), installation error output section  160  returns to step S 1110 , and continues to display screen  3 . 
     In the above description, a case has been described by way of example in which wireless tags  300  and tag reader  200  are installed at the same height, but the present invention is not limited to this, and it is also possible for wireless tags  300  to be installed at a different height from tag reader  200 . 
     A case will now be described in which wireless tags  300  and tag reader  200  are installed at different heights, taking a case in which tag reader  200  has been installed at the correct position but with deviation of its orientation, as an example. The same kind of method can also be applied to a case in which tag reader  200  is installed with the expected orientation of but with deviation of its position, and a case in which tag reader  200  is installed with deviation of both orientation and position, and therefore descriptions of these cases will be omitted here. 
       FIG. 20  is a conceptual diagram for explaining a case in which three-dimensional data is processed.  FIG. 20  shows a case in which wireless tag  300  is installed and positioned at observation point  434  (observation point  1 ′) deviating in the z-axis direction from observation point  431  (observation point  1 ) having coordinates (m/2, m, Rz). It is assumed that the coordinates of observation point  1 ′ are (m/2, m, Rz′). 
       FIG. 20A  is an explanatory drawing when the line of sight is positioned in a positive-to-negative direction on the x-axis, and  FIG. 20B  is an explanatory drawing when the line of sight is positioned in a positive-to-negative direction on the z-axis from directly above tag reader  200 . Here, a description is given with A (x A , y A , z A ) and B (x B , y B , z B ) extracted from a plurality of wireless tag  300  positioning results. 
     First, for predicted positioning distribution pattern L, radius r may be replaced by radius r′ shown in  FIG. 20A . Radius r′ can be represented by equation 19 below.
 
[19]
 
 r ′=√{square root over (( m/ 2) 2 +( R   z   −R   z ′) 2 )}{square root over (( m/ 2) 2 +( R   z   −R   z ′) 2 )}  (Equation 19)
 
     Also, in the above description, when the z-coordinate of the observation point at which wireless tag  300  is placed is Rz, measured positioning distribution pattern S can be interpreted as being the result of analyzing the result of projecting the positioning result onto an x-y plane represented by z=Rz. 
     Therefore, even when positioning results are represented by three-dimensional coordinates, A (x A , y A , z A ) is projected onto an x-y plane represented by z=Rz′, becoming a (x a , y a , z a ), and B (x B , y B , z B ) is projected onto the same x-y plane, becoming b (x b , y b , z b ). As explained above, three-dimensional data processing can perform analysis on a two-dimensional plane as shown in  FIG. 20B  in the same way as in  FIG. 18 . 
     By this means, even when positioning results are represented by three-dimensional coordinates, installation error (θ, a, b) can be estimated by calculating predicted positioning distribution pattern L and measured positioning distribution pattern S, as explained above. 
     In this embodiment, a case has been described in which, when a user installs tag reader  200 , wireless tags  300  are placed peripheral to tag reader  200 , and an installation error estimating apparatus estimates tag reader  200  installation error based on relevant observation data (positioning results). 
     Therefore, an installation error estimating apparatus of this embodiment does not require special installation structure, parts, and the like for tag reader  200 , and does not require a high degree of expertise in installation and measurement on the part of installation engineers. Consequently, time for measurement and adjustment using a jig or tool requiring expertise can be eliminated when installing tag reader  200 . Also, a person with no special experience, know-how, or the like can easily perform installation of tag reader  200 . Thus, the effort required for tag reader  200  installation can be greatly reduced in terms of equipment, personnel, and time, enabling implementation costs to be greatly reduced. 
     In short, according to this embodiment, error-free, high-precision positioning can be achieved with simple installation, without a high degree of expertise, special skills, or the like. 
     In this embodiment, the description has assumed initial installation work, but the present invention is not limited to this. For example, it is also possible to apply the present invention to periodic maintenance as well as initial installation work. 
     In this embodiment, the description has assumed a case in which installation error is in the rotation direction, the x-axis direction, and the y-axis direction, but the present invention is not limited to this. For example, it is also possible to apply the present invention to a case involving only the rotation direction, or a case involving only the x-axis direction and y-axis direction. 
     For example, in a case involving only the rotation direction, “0” should be substituted for a and b in equation 3. In this case, only a single variable, θ, is found, and therefore a minimum of one observation point is sufficient. 
     Also, in a case involving only the x-axis direction and y-axis direction, “0” should be substituted for θ in equation 3. In this case, only two variables, a and b, are found, and therefore a minimum of two observation points are sufficient. 
     In this embodiment, the description has assumed a scenario in which three wireless tags— 300   a ,  300   b , and  300   c —are installed peripheral to tag reader  200 , and are measured simultaneously, but the present invention is not limited to this. For example, if there are marked individual differences between wireless tags, measurement times for individual wireless tags may be staggered. 
     In this embodiment, a single-point positioning method has been described whereby tag reader  200  is installed at one place, a signal transmitted by tag reader  200  is reflected by wireless tag  300 , and a distance and direction of arrival are estimated based on the reflected signal, and are converted to a position. However, the present invention is not limited to this. For example, it is also possible to apply the present invention to a TDOA (Time Difference of Arrival) method whereby three or more tag readers are installed, and position calculation is performed using differences in times of reception from wireless tags. 
     In this embodiment, a description has been given of a predicted positioning distribution pattern that utilizes an arc-shaped positioning distribution characteristic whereby the distance from tag reader  200  to an observation point at which wireless tag  300  is installed is taken as the radius, with tag reader  200  as the center, but the present invention is not limited to this. For example, a predicted positioning distribution pattern may also utilize a radial-direction distribution characteristic. 
     Embodiment 2 
     Embodiment 2 is a case in which, when a tag reader is installed, a wireless tag is placed peripheral to the tag reader (at an observation point), the position of the wireless tag is measured, tag reader installation error is estimated based on obtained observation data, and the estimated installation error is evaluated. 
     This embodiment will now be described using  FIG. 21  through  FIG. 24 . 
       FIG. 21  is a block diagram showing the configuration of an installation error estimating apparatus according to Embodiment 2 of the present invention. This installation error estimating apparatus  700  has a similar basic configuration to installation error estimating apparatus  100  corresponding to Embodiment 1 shown in  FIG. 4 , and configuration elements in  FIG. 21  identical to those in  FIG. 4  are assigned the same reference codes as in  FIG. 4  and descriptions thereof are omitted here. 
     Installation error estimating apparatus  700  shown in  FIG. 21  has convergence determination section  720  in addition to observation point setting section  110 , predicted pattern calculation section  120 , observation data input section  130 , variance pattern analysis section  140 , installation error estimating section  710 , and installation error output section  160 . 
     The following description centers in detail upon convergence determination section  720 , which is the main area of difference between installation error estimating apparatus  100  according to Embodiment 1 shown in  FIG. 4  and installation error estimating apparatus  700  according to this embodiment shown in  FIG. 21 . 
     In Embodiment 1, wireless tags  300   a ,  300   b , and  300   c  were installed in places in clear view of tag reader  200 , and therefore communication conditions were good. This is shown by the fact that reliability continues to be evaluated at the highest rank of “A” in observation data  520  shown in  FIG. 7 . The quality (reliability) of communication conditions can be acquired by measuring signal level, for example. 
     However, if furniture or fixtures are present within positioning target area  400  in which wireless tag  300  position detection is performed, reflected waves in response to a transmission wave from tag reader  200  increase, and positioning precision of wireless tag  300  may decline. 
     In this case, it is not evident at the time of installation at which observation points positioning precision of wireless tag  300  is good or bad beforehand. Thus, in this embodiment, provision is made for three or more wireless tags  300  to be installed peripheral to tag reader  200 , and for an absolute number of observation points for which positioning precision is good to be increased. 
     A characteristic of this embodiment is that an observation point that should be involved in installation error estimation is selected based on reliability at the time of wireless tag  300  position detection. Also, a characteristic of this embodiment is that convergence determination section  720  estimates maximum-likelihood installation error allowing the error distribution of all selected observation points to be best approximated. 
     Installation error estimating section  710  has a similar basic configuration to installation error estimating section  150  in Embodiment 1, but differs in the following respect. Installation error estimating section  150  in Embodiment 1 estimates tag reader  200  installation error (θ, a, b) by simultaneous solution of equation 16, equation 17, and equation 18 on receiving variance pattern S from variance pattern analysis section  140 . In contrast, on receiving variance pattern S from variance pattern analysis section  140 , installation error estimating section  710  in Embodiment 2 first issues a request to convergence determination section  720  for narrowing down of calculation objects. Then installation error estimating section  710  estimates installation error (θ, a, b) based on the calculation objects narrowed down by convergence determination section  720 . 
     As explained above, convergence determination section  720  selects an observation point that should be involved in installation error estimation based on reliability at the time of wireless tag  300  position detection, and estimates maximum-likelihood installation error allowing the error distribution of all selected observation points to be best approximated. Consequently, when requested by installation error estimating section  710  to narrow down calculation objects, convergence determination section  720  creates convergence determination information  730  shown in  FIG. 22  and estimation results list  740  shown in  FIG. 23 , for example. 
     Convergence determination information  730  includes observation data information  731  including information used by convergence determination section  720  for convergence determination, and maximum-likelihood combination information  732  indicating a combination of observation points for which maximum-likelihood installation error allowing the error distribution of all selected observation points to be best approximated has been calculated. Observation data information  731  further includes tag ID  733 , observation point name  734 , observation point coordinates  735 , and average reliability  736 . 
     Estimation results list  740  comprises observation point combination  741  and installation error  742 . Observation point combination  741  is a combination of observation points for which the simultaneous equations described in Embodiment 1 hold true, selected by convergence determination section  720 . Installation error  742  is installation error resulting from solution of simultaneous equations by installation error estimating section  710  by means of an observation point combination. 
     In this embodiment, convergence determination section  720  estimates maximum-likelihood installation error using a least-squares method. Specifically, convergence determination section  720  calculates an error distance that is the square of the difference between a measured positioning distribution pattern of a determination observation point that is an observation point other than an observation point not involved in installation error estimation, and a value resulting from assigning estimated installation error to a predicted positioning distribution pattern. Next, convergence determination section  720  finds a total value resulting from adding together obtained error distances equal in number to the number of determination observation points, finds estimated installation error that minimizes this total value, and takes this to be maximum-likelihood installation error. 
     The actual processing performed by convergence determination section  720  will be described in detail later herein. 
     The operation of installation error estimating apparatus  700  having the above configuration will now be described using  FIG. 24 . 
     Here, a case will be described by way of example in which wireless tag  300  positioning is performed at five places, and installation error (θ, a, b) is estimated from the obtained observation data. The processing shown in  FIG. 24  replaces step S 1100  shown in  FIG. 19  in Embodiment 1, and should be inserted between step S 1090  and step S 1110  shown in  FIG. 19 . That is to say, of the steps in the flowchart shown in  FIG. 19 , steps other than step S 1100  are the same in this embodiment, and therefore a description of parts common to Embodiment 1 is omitted here. 
     At the stage of completion of step S 1090  shown in  FIG. 19 , installation error estimating section  710  holds function f representing predicted positioning distribution pattern L, and variance pattern S, for each of five observation points (not shown). 
     Then, in step S 2000  in  FIG. 24 , installation error estimating section  710  performs creation of convergence determination information  730  for selecting observation points that are to be calculation objects. Specifically, on receiving variance pattern S from variance pattern analysis section  140 , installation error estimating section  710  issues a request to convergence determination section  720  for narrowing down of calculation objects. When requested to narrow down calculation objects by installation error estimating section  710 , convergence determination section  720  creates convergence determination information  730  shown in  FIG. 22 . 
     The actual procedure is as follows, for example. 
     After first receiving a request to narrow down calculation objects from installation error estimating section  710 , convergence determination section  720  next receives coordinate management tables  550  (all coordinate management tables for observation point  1  through observation point  5 ) shown in  FIG. 5  from variance pattern analysis section  140 . Then convergence determination section  720  copies tag IDs and observation point names of received coordinate management tables  550  to tag ID  733  and observation point name  734  respectively of convergence determination information  730 . Also, convergence determination section  720  calculates the average of coordinate management tables  550  reliabilities, and records this in average reliability  736  of convergence determination information  730 . For example, 3 points are assigned to reliability “A” and 2 points to reliability “B,” averaging is performed, and if the average is 2.5 points or above, an average reliability of “A” is assigned. 
     Next, convergence determination section  720  receives observation point information  500  shown in  FIG. 5  from predicted pattern calculation section  120 . Then convergence determination section  720  extracts observation point coordinates from an observation point information  500  line for which observation point name  734  of convergence determination information  730  and observation point name  503  of observation point information  500  match, and records these in observation point coordinates  735  of convergence determination information  730 . 
     In Embodiment 2, as in Embodiment 1, for convenience of explanation, a z-coordinate is omitted and a description is given in terms of two-dimensional coordinates. 
     Then, in step S 2010 , convergence determination section  720  performs selection of observation points that are to be calculation objects. Specifically, convergence determination section  720  extracts an observation point that is to be a calculation object by holding a selection criterion internally, and comparing average reliability  736  of convergence determination information  730  with that selection criterion. For example, in the example shown in  FIG. 22 , if the selection criterion is “average reliability is to be A,” convergence determination section  720  compares average reliability  736  of convergence determination information  730  with that selection criterion. Through this comparison, convergence determination section  720  selects observation point  1  through observation point  4  as observation points that are to be calculation objects, and excludes observation point  5 . 
     Then, in step S 2020 , convergence determination section  720  performs creation of estimation results list  740  in which observation point combinations are recorded. Specifically, convergence determination section  720  finds an observation point combination for performing installation error estimation from observation points selected in step S 2010 , and records an obtained observation point combination in observation point combination  741  of estimation results list  740 . For example, if four observation points are selected in step S 2010 , the number of combinations is four ( 4 C 3 =4). Of the four combinations,  FIG. 23  illustrates only combination  1  comprising observation point  1 , observation point  2 , and observation point  3 , and combination  2  comprising observation point  1 , observation point  2 , and observation point  4 . Convergence determination section  720  requests installation error estimating section  710  to perform installation error estimation for each of the observation point combinations recorded in observation point combination  741 . 
     Then, in step S 2030 , installation error estimating section  710  receives a request from convergence determination section  720 , and performs installation error estimation for each observation point combination. Specifically, installation error estimating section  710  estimates installation error (θ, a, b) from a predicted positioning distribution pattern and measured positioning distribution pattern of a specified observation point for each observation point combination recorded in step S 2020 . The actual estimation method is similar to the method described in Embodiment 1, and therefore a description thereof will be omitted here. Installation error estimating section  710  passes estimated installation error to convergence determination section  720 . 
     Then, in step S 2040 , convergence determination section  720  performs recording of installation error in estimation results list  740 . Specifically, convergence determination section  720  receives installation error (θ, a, b) estimated in step S 2030  from installation error estimating section  710 , and records received installation error at the corresponding combination location in installation error column  742 . 
     Then, in step S 2050 , convergence determination section  720  determines whether or not installation error estimation has been completed for all the combinations—that is, whether or not recording in estimation results list  740  has been completed. If the result of this determination is that installation error estimation has been completed for all the combinations (that is, recording in estimation results list  740  has been completed) (S 2050 : YES), convergence determination section  720  proceeds to step S 2060 . On the other hand, if installation error estimation has not been completed for all the combinations (that is, recording in estimation results list  740  has not been completed) (S 2050 : NO), convergence determination section  720  returns to step S 2030 , and performs installation error estimation for an unprocessed combination. 
     For example, in the example shown in  FIG. 22 , when installation error estimation has been completed, and recording in estimation results list  740  has been completed, for all the combinations, four sets of estimated installation error are present. 
     In step S 2060 , convergence determination section  720  performs evaluation of estimated installation error. Specifically, convergence determination section  720  applies installation error estimated by installation error estimating section  710  in step S 2030  to an observation point (hereinafter referred to as “determination observation point”) other than an observation point relating to simultaneous equations, and performs evaluation of this case. Installation error evaluation is performed using a least-squares method on determination observation points, for example. 
     More specifically, in the example shown in  FIG. 22  and  FIG. 23 , a determination observation point of combination  1  (observation point  1 , observation point  2 , and observation point  3 ) is only observation point  4 , and a determination observation point of combination  2  (observation point  1 , observation point  2 , and observation point  4 ) is only observation point  3 . The same principle also applies to the other combinations. 
     At this time, convergence determination section  720  finds, for each combination, installation error that minimizes the sum of error distances D k   2  (hereinafter referred to as “total value”) defined by equation 20 below for determination observation point k (one or more) of that combination. Error distance D k   2  is defined as the square of the difference between measured positioning distribution pattern S k  and values substituted for installation error (θ, a, b) estimated by function f k  representing predicted positioning distribution pattern L (|S k −f k | 2 ).
 
[20]
 
Σ D   k   2   =Σ|S   k −ƒ k | 2   (Equation 20)
 
     For example, to describe a case in which a determination observation point is observation point  3 , the following applies to error distance D k   2 . When a determination observation point is observation point  3 , error distance D 3   2  can be represented by equation 21 below using values substituted for installation error (θ (2) , a (2) , b (2) ) estimated from positioning results of combination  2  (observation points  1 ,  2 , and  4 ) in equation 18. In the example shown in  FIG. 22  and  FIG. 23 , there is only one determination observation point for each combination. Consequently, in the final analysis, installation error (θ (2) , a (2) , b (2) ) estimated from positioning results of combination  2  (observation point  1 , observation point  2 , and observation point  4 ) is evaluated using this equation 21. 
     
       
         
           
             
               
                 
                   [ 
                   21 
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       D 
                       3 
                       2 
                     
                     = 
                     
                       
                          
                         
                           
                             S 
                             3 
                           
                           - 
                           
                             
                               2 
                               · 
                               
                                 r 
                                 3 
                               
                             
                             ⁢ 
                             tan 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               α 
                               · 
                               cos 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               T 
                               3 
                             
                           
                         
                          
                       
                       2 
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       where 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         β 
                         3 
                       
                     
                     = 
                     
                       
                         r 
                         + 
                         
                           b 
                           
                             ( 
                             2 
                             ) 
                           
                         
                       
                       
                         
                           
                             a 
                             
                               ( 
                               2 
                               ) 
                             
                             2 
                           
                           + 
                           
                             
                               ( 
                               
                                 r 
                                 + 
                                 
                                   b 
                                   
                                     ( 
                                     2 
                                     ) 
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       r 
                       3 
                     
                     = 
                     
                       
                         
                           a 
                           
                             ( 
                             2 
                             ) 
                           
                           2 
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 b 
                                 
                                   ( 
                                   2 
                                   ) 
                                 
                               
                               + 
                               r 
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     T 
                     3 
                   
                   = 
                   
                     
                       β 
                       3 
                     
                     + 
                     
                       θ 
                       
                         ( 
                         2 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     21 
                   
                   ) 
                 
               
             
           
         
       
     
     Then, in step S 2070 , convergence determination section  720  determines whether or not installation error has been evaluated for all the combinations. If the result of this determination is that installation error has been evaluated for all the combinations (S 2070 : YES), convergence determination section  720  proceeds to step S 2080 . On the other hand, if installation error has not been evaluated for all the combinations (S 2070 : NO), convergence determination section  720  returns to step S 2060 , and performs installation error evaluation for a remaining combination. 
     In step S 2080 , convergence determination section  720  performs a maximum-likelihood installation error decision. Specifically, convergence determination section  720  uses step S 2060  evaluation results (that is, a total value represented by equation 20) to find a combination that minimizes the total value, and finds installation error (θ, a, b) estimated in this combination. This installation error (θ, a, b) is installation error that minimizes the total value (the sum of error distances D k   2 ), and is “maximum-likelihood installation error.” The obtained maximum-likelihood installation error is passed to installation error output section  160 . 
     For example, in the example shown in  FIG. 22  and  FIG. 23 , it is assumed that evaluation has been performed on installation error estimated for all combinations, using an expression such as represented by equation 22, and the relationship shown in equation 22 below has been obtained.
 
[22]
 
D 3   2 ≦D 1   2 ≦D 2   2 ≦D 4   2   (Equation 22)
 
     At this time, the minimum total value (error distance D k   2 ) is error distance D 3   2  using installation error (θ (2) , a (2) , b (2) ) estimated from the positioning results of combination  2  (observation point  1 , observation point  2 , and observation point  4 ). Therefore, in this example, installation error (a (2) , b (2) , θ (2) ) in combination  2  corresponding to error distance D 3   2  is decided upon as maximum-likelihood installation error. 
     Installation error estimating apparatus  700  operation then proceeds to step S 1110  shown in  FIG. 19 . In this embodiment, convergence determination section  720  estimates maximum-likelihood installation error, and passes the estimated maximum-likelihood installation error to installation error output section  160 . Consequently, installation error output section  160  displays this maximum-likelihood installation error in operation screen  570  (screen  3 ) shown in  FIG. 12 . 
     Therefore, according to this embodiment, since convergence determination section  720  is provided and performs evaluation of estimated installation error, the same kind of effect can be obtained as in Embodiment 1 even in an environment in which communication conditions are not necessarily good. 
     Also, in the installation of tag reader  200  and wireless tags  300 , there is a possibility of error being included not only in the tag reader  200  installation position, but also in the wireless tag  300  installation positions. If an observation point for which wireless tag  300  installation position error is large is included in an observation point combination, error distance Dk 2  increases due to the application of installation error estimated from positioning results of that combination to a determination convergence point. It is thus possible to exclude the possibility of an observation point with large wireless tag  300  installation position error being included in installation error estimation. 
     In this embodiment, a least-squares method is applied to determination observation points when estimating maximum-likelihood installation error, but the present invention is not limited to this. For example, averaging of installation error of each combination may also be used in maximum-likelihood installation error estimation. 
     Embodiment 3 
     Embodiment 3 is a case in which, in addition to the provisions of Embodiment 1, a position for newly placing a wireless tag as an observation object is further recommended if estimated installation error does not satisfy a convergence condition. 
     This embodiment will now be described using  FIG. 25  through  FIG. 28 . 
       FIG. 25  is a block diagram showing the configuration of an installation error estimating apparatus according to Embodiment 3 of the present invention. This installation error estimating apparatus  800  has a similar basic configuration to installation error estimating apparatus  100  corresponding to Embodiment 1 shown in  FIG. 4  and installation error estimating apparatus  700  corresponding to Embodiment 2 shown in  FIG. 21 . Configuration elements in  FIG. 25  identical to those in  FIG. 4  and  FIG. 21  are assigned the same reference codes as in  FIG. 4  and  FIG. 21 , and descriptions thereof are omitted here. 
     Installation error estimating apparatus  800  shown in  FIG. 25  has observation point recommendation section  830  in addition to the configuration of Embodiment 2. 
     The following description centers in detail upon observation point recommendation section  830 , which is the main area of difference between installation error estimating apparatus  700  according to Embodiment 2 shown in  FIG. 21  and installation error estimating apparatus  800  according to this embodiment shown in  FIG. 25 . 
     In Embodiment 2, a mode was described in which installation error approximating with the highest degree of precision a predicted positioning distribution pattern and measured positioning distribution pattern of observation data from among an estimated plurality of installation errors is decided upon as maximum-likelihood installation error. 
     In contrast, in this embodiment, a mode is described in which a threshold value is provided for precision of approximation, and wireless tag  300  is newly positioned, and installation error estimation is repeated, until high precision greater than or equal to the threshold value is attained. 
     More particularly, a characteristic of installation error estimating apparatus  800  according to this embodiment is that observation point recommendation section  830  decides a position of an observation point to be added, and observation point setting section  810  reports to the user that wireless tag  300  re-installation and re-measurement should be carried out. 
     Convergence determination section  820  has the following functions in addition to the same kind of function as convergence determination section  720  in Embodiment 2. Namely, convergence determination section  820  has a function of holding threshold value T internally, and comparing a value obtained by dividing a total value for which distance error is minimal by the number of determination observation points with threshold value T in all current combinations, and also a function of determining whether or not distance error variance V satisfies a threshold value condition. 
     Convergence determination section  820  determines that the threshold value condition is satisfied if distance error variance V is less than or equal to threshold value T (V≦T), and determines that the threshold value condition is not satisfied if distance error variance V is greater than threshold value T (V&gt;T). In the event of determining that the threshold value condition is not satisfied, convergence determination section  820  passes convergence determination information  730  shown in  FIG. 22  and installation error corresponding to the minimum distance error total value (maximum-likelihood installation error in Embodiment 2) to observation point recommendation section  830 . 
     For example, in Embodiment 2, if variance V of distance error calculated from error distance D 3   2  that is minimal in equation 22 is greater than threshold value T (T&lt;D 3   2 ), convergence determination section  820  determines that distance error variance V does not satisfy the threshold value condition. In this case, distance error variance V matches error distance D 3   2  since the number of determination observation points is one. Then convergence determination section  820  passes current convergence determination information  730  and installation error (a (2) , b (2) , θ (2) ) corresponding to distance error variance V to observation point recommendation section  830  according to the determination. 
     On receiving convergence determination information  730  from convergence determination section  820 , observation point recommendation section  830  decides an observation point at which wireless tag  300  is to be newly placed and undergo positioning, and passes the coordinates of the decided observation point to observation point setting section  810 . The actual processing performed by observation point recommendation section  830  will be described in detail later herein. 
     Observation point setting section  810  has the following functions in addition to the same kind of function as observation point setting section  110  in Embodiment 1 and Embodiment 2. Namely, observation point setting section  810  differs from observation point setting section  110  in receiving new observation point coordinates from observation point recommendation section  830 , and in displaying a predetermined operation screen for reporting to the user those new observation point coordinates after they have been received. The operation screen for reporting to the user observation point recommendation section  830  output (added new observation point coordinates) has, for example, the same kind of screen configuration as screen  1  shown in  FIG. 6 , as shown in  FIG. 26 . The operation screen is a screen in which received observation point coordinates are input in observation point coordinate input column  512 . 
       FIG. 26  shows a screen when observation point recommendation section  830  has decided upon the addition of two observation points, and recommends installation and positioning of wireless tags  300  at (x 6 , y 6 , R z ) and (x 7 , y 7 , R z ). 
     Next, the method whereby observation point recommendation section  830  decides an additional observation point for newly placing and positioning wireless tag  300  will first be outlined. 
     Observation point recommendation section  830  decides a new observation point based on predetermined decision rules. The decision rules are held by observation point recommendation section  830 , and are defined as shown below, for example. 
     Rule 1: The reliability of a nearby observation point should be high. 
     Rule 2: The distance from another observation point should be a fixed distance or more. 
     Rule 3: An additional observation point should be on a circle concentric with, or on the same radius as, an observation point with high reliability in an estimated local coordinate system. 
     Here, the basis for rule 1 is that “the probability of good communication conditions is high in the vicinity of an observation point with high reliability”; rule 2 is based on “achieving approximation in the entire positioning area rather than local approximation”; and the basis for rule 3 is that, when calculating a measured positioning distribution pattern base on the following three bases, an improvement in calculation precision can be expected by comparing reliabilities mutually. A basis for rule 3 is, first, that observation points present on concentric circles in a local coordinate system have a high probability of having the same variance pattern. A basis for rule 3 is, secondly, that observation points on the same radius have a high probability of being in clear view of a tag reader, and being close in terms of communication environment, such as the effects of reflected waves. Thirdly, a basis for rule 3 is that, due to the characteristic whereby “a measurement result distribution extends in an arc of angle error a on a spherical surface with line segments linking a tag reader to wireless tags as radius r,” as illustrated by  FIG. 13 , the same angle error α can be expected. 
     Observation point recommendation section  830  decides upon a point that satisfies all of rules 1 through 3 as a new observation point. 
     However, the greater the number of new observation points, the higher is the cost of installation work for wireless tags  300 , and the higher is the calculation cost due to an increase in the number of combinations of observation points for which installation error estimation is performed, making it desirable for the number of new observation points to be kept within limits. 
     The processing procedure up to a decision on a new observation point by observation point recommendation section  830  will now be described using  FIG. 27  and  FIG. 28 .  FIG. 27  is a flowchart showing additional observation point decision processing, and  FIG. 28  shows conceptual diagrams explaining the additional observation point decision method. 
     Here, a presupposition for the processing is that it has been decided beforehand that two new observation points are to be selected. It is also assumed that predetermined value d has been decided upon beforehand as the distance from another observation point in rule 2. 
     First, in step S 3000 , observation point recommendation section  830 , on receiving convergence determination information  730  from convergence determination section  820 , creates a map such as shown in  FIG. 28A , for example, based on received convergence determination information  730 . Although it is not actually necessary to draw the coordinate axes and circle represented by dotted lines, they are drawn here for convenience of explanation. 
     In  FIG. 28 , observation points are displayed by means of symbols that differ according to average reliability  736  of observation data information  731 . For example, symbol “⊚” means that the reliability of most data is “A,” with average reliability of 2.75 or above; symbol “◯” means that reliability is registered as “A” on average although “B” is partially included, with average reliability of 2.5 or above and less than 2.75; and symbol “Δ” means that the reliability of most data is “B,” with average reliability of 2.5 or below. Positions and observation point names on the map can be mutually associated using observation point name  734  and observation point coordinates  735  of observation data information  731 . 
     Then, in step S 3010 , observation point recommendation section  830  performs observation point selection. Specifically, observation point recommendation section  830  selects one observation point from observation points for which average reliability is high. For example, based on rule 1, observation point recommendation section  830  selects one observation point from among observation point  1 , observation point  2 , and observation point  3 , for which average reliability is high, and narrows down candidates to the vicinity of the selected observation point. Here, it is assumed that single observation point  3  has been randomly selected from the three observation points  1  through  3  to which the same average reliability has been assigned. In this case, the selected observation point may, of course, also be observation point  1  or observation point  2 . 
     Then, in step S 3020 , observation point recommendation section  830  makes a selection decision for the rule 3 application criterion. Specifically, observation point recommendation section  830  decides whether to select a point present on a concentric circle or a point on a straight line linking a local coordinate origin to an existing observation point as a rule 3 application criterion. Here, a decision can be made on a case-by-case basis as to random or alternate selection of which point is to be prioritized and the like in accordance with rule 3. 
     Then, in step S 3030 , observation point recommendation section  830  determines whether or not selection of a point on a straight line has been decided in step S 3020 . If the result of this determination is that selection of a point on a straight line has been decided (S 3030 : YES), observation point recommendation section  830  proceeds to step S 3040 . On the other hand, if selection of a point on a straight line has not been decided—that is, if selection of a point on a concentric circle has been decided—(S 3030 : NO), observation point recommendation section  830  proceeds to step S 3090 . 
     Step S 3040  through step S 3080  show processing when (1) it has been decided to prioritize a point on a straight line linking a local coordinate origin to an already observed existing observation point as a result of the determination in step S 3030 . Step S 3090  through step S 3140  show processing when (2) it has been decided to prioritize a point on a concentric circle as a result of the determination in step S 3030 . Each of these cases is described below. 
     (1) When it has been decided to prioritize a point on a straight line linking a local coordinate origin to an already observed existing observation point 
     In step S 3040 , observation point recommendation section  830  calculates an intersection point of a straight line linking a currently selected observation point to a local coordinate origin, and a circle with a local coordinate system origin as its center and for which the distance from a currently selected observation point is predetermined value d. 
     A specific example will now be described, referring to  FIG. 22  and  FIG. 23 . From installation error (a (2) , b (2) , θ (2) ) corresponding to minimal error distance D 3   2 , observation point recommendation section  830  represents the relationship between local coordinates (X, Y) and virtual global coordinates (x, y) by means of equation 23 and equation 24 below, where equation 24 is derived from equation 23. 
     
       
         
           
             
               
                 
                   [ 
                   23 
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     ( 
                     
                       
                         
                           X 
                         
                       
                       
                         
                           Y 
                         
                       
                     
                     ) 
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               θ 
                             
                           
                           
                             
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               θ 
                             
                           
                         
                         
                           
                             
                               
                                 - 
                                 sin 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               θ 
                             
                           
                           
                             
                               cos 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               θ 
                             
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               x 
                               - 
                               a 
                             
                           
                         
                         
                           
                             
                               y 
                               - 
                               b 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     23 
                   
                   ) 
                 
               
             
             
               
                 
                   [ 
                   24 
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     ( 
                     
                       
                         
                           x 
                         
                       
                       
                         
                           y 
                         
                       
                     
                     ) 
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 θ 
                               
                             
                             
                               
                                 
                                   - 
                                   sin 
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 θ 
                               
                             
                           
                           
                             
                               
                                 sin 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 θ 
                               
                             
                             
                               
                                 cos 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 θ 
                               
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               X 
                             
                           
                           
                             
                               Y 
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       ( 
                       
                         
                           
                             a 
                           
                         
                         
                           
                             b 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     24 
                   
                   ) 
                 
               
             
           
         
       
     
     Next, observation point recommendation section  830  converts the respective observation point coordinates to global coordinates using an estimated transformation matrix (see equation 24). 
     Since the coordinates of observation point  3  in the virtual global coordinate system are known to be (x 3 , y 3 ) from observation point coordinates  735  of convergence determination information  730 , coordinates (X 3 , Y 3 ) of observation point  3  in the local coordinate system can be obtained by substitution in equation 23. 
     To be precise, (x 3 , y 3 ) are coordinates in a global coordinate system, and the parallel translation relationship represented by equation 6 and equation 7 applies between a global coordinate system and a virtual global coordinate system. However, in this embodiment, the two are treated as the same for convenience of explanation. 
     Therefore, additional observation point  6  (X 6 , Y 6 ) is a point of intersection of straight line  841  and circle  842  (see  FIG. 28B ). Straight line  841  is a straight line represented by equation 25 below, linking local coordinate origin  840  and observation point  3 . Circle  842  is a circle represented by equation 26 below, with local coordinate system origin  840  as its center and for which the distance from observation point  3  is predetermined value d. 
     
       
         
           
             
               
                 
                   [ 
                   25 
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   Y 
                   = 
                   
                     
                       
                         Y 
                         3 
                       
                       
                         X 
                         3 
                       
                     
                     ⁢ 
                     X 
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     25 
                   
                   ) 
                 
               
             
             
               
                 
                   [ 
                   26 
                   ] 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       X 
                       2 
                     
                     + 
                     
                       Y 
                       2 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           
                             
                               X 
                               3 
                               2 
                             
                             + 
                             
                               Y 
                               3 
                               2 
                             
                           
                         
                         - 
                         d 
                       
                       ) 
                     
                     2 
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     26 
                   
                   ) 
                 
               
             
           
         
       
     
     Therefore, virtual global coordinates (x 6 , y 6 ) of additional observation point  6  are obtained by substitution in equation 24, which is an inverse transform of equation 23. 
     Then, in step S 3050 , observation point recommendation section  830  determines whether or not the point calculated in step S 3040  (additional observation point  6 ) satisfies rule 1. If the result of this determination is that the calculated point (additional observation point  6 ) satisfies rule 1 (S 3050 : YES), observation point recommendation section  830  proceeds to step S 3060 . On the other hand, if the calculated point (additional observation point  6 ) does not satisfy rule 1 (S 3050 : NO), observation point recommendation section  830  returns to step S 3040 . 
     Also, in step S 3060 , observation point recommendation section  830  determines whether or not the point calculated in step S 3040  (additional observation point  6 ) satisfies rule 2. If the result of this determination is that the calculated point (additional observation point  6 ) satisfies rule 2 (S 3060 : YES), observation point recommendation section  830  proceeds to step S 3070 . On the other hand, if the calculated point (additional observation point  6 ) does not satisfy rule 2 (S 3060 : NO), observation point recommendation section  830  returns to step S 3040 . 
     In step S 3070 , observation point recommendation section  830  decides upon the point calculated in step S 3040  (additional observation point  6 ) as a new observation point (additional observation point). Also, observation point recommendation section  830  updates the currently selected observation point to the additional observation point. For example, since additional observation point  6  satisfies rule 1 and rule 2, additional observation point  6  is decided upon as a new observation point (additional observation point). 
     Then, in step S 3080 , observation point recommendation section  830  determines whether or not the number of additional observation points has reached a predetermined value (here, two). If the result of this determination is that the number of additional observation points has reached the predetermined value (S 3080 : YES), observation point recommendation section  830  terminates the series of processing steps. On the other hand, if the number of additional observation points has not reached the predetermined value (S 3080 : NO), observation point recommendation section  830  returns to step S 3040 . Then observation point recommendation section  830  finds additional observation point  7  (x 7 , y 7 ) by means of the same kind of procedure, for example. 
     Step S 3040  through step S 3080  are repeated in this way until the number of new observation points reaches the predetermined value. 
     (2) When it has been decided to prioritize a point on a concentric circle 
     On the other hand, in step S 3090 , observation point recommendation section  830  calculates a circle with a local coordinate system origin as its center and for which the distance from a currently selected observation point is predetermined value d. For example, observation point recommendation section  830  calculates circle  842  with local coordinate system origin  840  as its center and for which the distance from observation point  3  is predetermined value d (see  FIG. 28C ). This circle  842  is represented by equation 26. 
     Then, in step S 3100 , observation point recommendation section  830  randomly selects a point on the circle calculated in step S 3090 . For example, observation point recommendation section  830  selects additional observation point  6  (X 6 , Y 6 ) that satisfies equation 26. 
     Specifically, for example, as described above, provision can be made for a point of intersection of straight line  841  represented by equation 25 and circle  842  represented by equation 26 to be calculated, and for this point of intersection to be selected as first additional observation point candidate (X 6 , Y 6 ). As explained above, straight line  841  represented by equation 25 is a straight line linking the currently selected observation point (observation point  3 ) to local coordinate origin  840 , and circle  842  represented by equation 26 is a circle with local coordinate system origin  840  as its center and for which the distance from the currently selected observation point (observation point  3 ) is predetermined value d. 
     Then, in step S 3110 , observation point recommendation section  830  determines whether or not the point calculated in step S 3100  (additional observation point  6 ) satisfies rule 1. Specifically, for example, observation point recommendation section  830  determines whether virtual global coordinates (x 6 , y 6 ) of additional observation point  6  obtained by adding (X 6 , Y 6 ) to equation 24 satisfy rule 1. If the result of this determination is that the calculated point (additional observation point  6 ) satisfies rule 1 (S 3110 : YES), observation point recommendation section  830  proceeds to step S 3120 . On the other hand, if the calculated point (additional observation point  6 ) does not satisfy rule 1 (S 3110 : NO), observation point recommendation section  830  returns to step S 3100 . 
     Also, in step S 3120 , observation point recommendation section  830  determines whether or not the point calculated in step S 3100  (additional observation point  6 ) satisfies rule 2. Specifically, for example, observation point recommendation section  830  determines whether virtual global coordinates (x 6 , y 6 ) of additional observation point  6  obtained by adding (X 6 , Y 6 ) to equation 24 satisfy rule 2. If the result of this determination is that the calculated point (additional observation point  6 ) satisfies rule 2 (S 3120 : YES), observation point recommendation section  830  proceeds to step S 3130 . 
     On the other hand, if the calculated point (additional observation point  6 ) does not satisfy rule 2 (S 3120 : NO), observation point recommendation section  830  returns to step S 3100 . 
     In step S 3130 , observation point recommendation section  830  decides upon the point calculated in step S 3100  (additional observation point  6 ) as a new observation point (additional observation point). Also, observation point recommendation section  830  updates the currently selected observation point to the additional observation point. For example, since additional observation point  6  satisfies rule 1 and rule 2, additional observation point  6  is decided upon as a new observation point (additional observation point). 
     Then, in step S 3140 , observation point recommendation section  830  determines whether or not the number of additional observation points has reached a predetermined value (here, two). If the result of this determination is that the number of additional observation points has reached the predetermined value (S 3140 : YES), observation point recommendation section  830  terminates the series of processing steps. On the other hand, if the number of additional observation points has not reached the predetermined value (S 3140 : NO), observation point recommendation section  830  returns to step S 3100 . At this time, local coordinates of additional observation point  7  can be decided, for example, as point (X 7 , Y 7 ) obtained by rotating additional observation point  6  through previously decided angle β in the local coordinate system. 
     Step S 3100  through step S 3140  are repeated in this way until the number of new observation points reaches the predetermined value. 
     The two additional observation points (X 6 , Y 6 , R z ) and (X 7 , Y 7 , R z ) decided by observation point recommendation section  830  in this way are passed to observation point setting section  810 . On receiving the new observation point coordinates from observation point recommendation section  830 , observation point setting section  810  displays operation screen  840  (screen  4 ) in which the received observation point coordinates are input in observation point coordinate input column  512  (see  FIG. 26 ). As described above,  FIG. 26  recommends installation and positioning of wireless tags  300  at two additional observation points (x 6 , y 6 , R z ) and (x 7 , y 7 , R z ). 
     The user then inputs the tag IDs of wireless tags  300  to be installed at the additional observation points in tag ID input column  511 . After the user has installed wireless tags  300  at the additional observation points, positioning is started for the two wireless tags  300  installed at the additional observation points by depression of setting completed button  514  by the user. 
     Thus, according to this embodiment, in addition to provision of the effects of Embodiment 1 and Embodiment 2, a threshold value is provided for precision of approximation, and wireless tag  300  is newly positioned, and installation error estimation is repeated, until high precision greater than or equal to the threshold value is attained. Consequently, this embodiment enables the number of times re-measurement and re-installation are performed until a threshold value is met to be reduced. 
     More particularly, according to this embodiment, when observation point recommendation section  830  decides a position of an observation point to be added, observation point recommendation section  830  considers variation in observation points, observation point communication conditions, selection of observation points for which there is a high probability of obtaining the same kind of positioning distribution, and so forth. Consequently, this embodiment makes it all the more possible to achieve high-precision positioning with simple installation, irrespective of observation point installation conditions or communication conditions. 
     The disclosure of Japanese Patent Application No. 2008-302492, filed on Nov. 27, 2008, including the specification, drawings and abstract, is incorporated herein by reference in its entirety. 
     INDUSTRIAL APPLICABILITY 
     An installation error estimating apparatus and installation error estimating method of the present invention are suitable for use as an installation error estimating apparatus and installation error estimating method capable of enabling error-free, high-precision positioning to be achieved with simple installation, and can be effectively applied to installation support for an apparatus having an antenna, for example. 
     REFERENCE SIGNS LIST 
     
         
           100 ,  700 ,  800  Installation error estimating apparatus 
           110 ,  810  Observation point setting section 
           120  Predicted pattern calculation section 
           130  Observation data input section 
           140  Variance pattern analysis section 
           150 ,  710  Installation error estimating section 
           160  Installation error output section 
           200  Tag reader 
           300 ,  300   a ,  300   b ,  300   c  Wireless tag 
           500  Observation point information 
           510  Operation screen (screen  1 ) 
           520  Observation data 
           530  Operation screen (screen  2 ) 
           540 ,  541  Log 
           550 ,  551 ,  552 ,  553  Coordinate management table 
           560  Measured positioning distribution pattern list 
           570  Operation screen (screen  3 ) 
           720 ,  820  Convergence determination section 
           730  Convergence determination information 
           740  Estimation results list 
           830  Observation point recommendation section 
           840  Operation screen (screen  4 )