Position detection method and position detector, exposure method and exposure apparatus, and device and device manufacturing method

Abstract

A parameter calculation unit statistically calculates the estimations and their certainty of a predetermined number of parameters, which uniquely specify any position on an object, for each of a plurality of measured sample sets on the basis of position information of marks composing the sample set. A valid value calculation unit calculates statistically valid values of the predetermined number of parameters on the basis of groups of the estimations and their certainty of the predetermined number of parameters for the respective sample sets. Furthermore, an evaluation unit statistically evaluates if the number of marks composing a sample set can be reduced. If it is determined that the number of marks can be reduced, the parameter calculation unit calculates values of the predetermined number of parameters by using a new sample set having reduced number of marks. Using the values of the predetermined number of parameters calculated in this manner, the position of any area on the object can be accurately detected.

What is claimed is:  
     
         1 . A position detection method for detecting position information of any area on an object provided with a plurality of position-measurement-points, the position detection method comprising: 
 selecting more position-measurement-points than a minimum number of measurements required to calculate values of a predetermined number of parameters, which uniquely specify position information of any area on the object, from the plurality of position-measurement-points and measuring pieces of position information of the respective selected position-measurement-points;    calculating respective positions of the selected position-measurement-points, based on the measurement results of the pieces of position information, and estimating probability density functions which each represent occurrence probability of the calculated position for respective one of the selected position-measurement-points;    calculating probability density of the calculated position of each of the position-measurement-points, based on respective one of the probability density functions; and    evaluating an error of each calculated position relative to respective reference position while using the respective calculated probability density's value as a piece of weight information and calculating values of the predetermined number of parameters, based on the evaluated errors.    
     
     
         2 . The position detection method according to  claim 1 , wherein the reference positions are determined in advance based on design information.  
     
     
         3 . The position detection method according to  claim 1 , wherein the error evaluation is performed by multiplying the error of each calculated position relative to the respective reference position by the respective probability density.  
     
     
         4 . The position detection method according to  claim 1 , wherein the probability density functions are probability density functions of normal distribution.  
     
     
         5 . The position detection method according to  claim 1 , wherein position measurement marks are formed at the position-measurement-points.  
     
     
         6 . The position detection method according to  claim 5 , wherein a plurality of position detection marks formed at the plurality of position-measurement-points include a first number of first marks of which surface states change in a first direction, and 
 wherein the piece of position information of each first mark measured in the step of selecting and measuring comprises pieces of position information at a plurality of feature portions of each first mark in the first direction.    
     
     
         7 . The position detection method according to  claim 6 , wherein for each of the selected first marks, the step of selecting and measuring measures the piece of position information at a plurality of positions lined in a direction crossing the first direction.  
     
     
         8 . The position detection method according to  claim 6 , wherein surface state of each of the first marks changes also in a second direction different from the first direction, and 
 wherein the piece of position information of each first mark measured in the step of selecting and measuring consists of position information, in the first direction, of a plurality of feature portions lined in the first direction and position information, in the second direction, of a plurality of feature portions lined in the second direction.    
     
     
         9 . The position detection method according to  claim 8 , wherein for each selected first mark, the step of selecting and measuring measures at least one of the position information, in the first direction, of a plurality of positions lined in a direction crossing the first direction and the position information, in the second direction, of a plurality of positions lined in a direction crossing the second direction.  
     
     
         10 . The position detection method according to  claim 6 , wherein the plurality of marks further include a second number of second marks, of which surface states change in a second direction different from the first direction, and 
 wherein for each second mark, the position information measured in the step of selecting and measuring is position information, in the second direction, of a plurality of feature portions.    
     
     
         11 . The position detection method according to  claim 10 , wherein for each selected second mark, the step of selecting and measuring measures the position information at a plurality of positions in a direction crossing the second direction.  
     
     
         12 . The position detection method according to  claim 5 , wherein a plurality of divided areas are arranged on the object, each of the divided areas being provided with the position measurement marks.  
     
     
         13 . The position detection method according to  claim 12 , wherein parameters associated with respective, representative points of the plurality of divided areas are included in the predetermined number of parameters.  
     
     
         14 . The position detection method according to  claim 13 , wherein parameters associated with other points than the respective, representative points of the plurality of divided areas are further included in the predetermined number of parameters.  
     
     
         15 . A position detection method for detecting position information of any area on an object provided with a first number of position-measurement-points, the position detection method comprising: 
 selecting a plurality of measurement point subsets which each consist of a third number of position-measurement-points and are different from one another, the third number being larger than a second number and smaller than the first number, the second number being a minimum number of measurement points required to calculate a predetermined number of parameters that uniquely specify position information of any area on the object; and    statistically calculating, for each of the plurality of measurement point subsets, estimations of the predetermined number of parameters and certainty of the estimations, based on measurement results of the third number of position-measurement-points.    
     
     
         16 . The position detection method according to  claim 15 , further comprising: 
 obtaining statistically valid values of the predetermined number of parameters, based on the respective groups of estimations and certainty for the plurality of measurement point subsets calculated in the step of calculating.    
     
     
         17 . The position detection method according to  claim 16 , wherein the statistically valid value of each of the predetermined number of parameters is obtained by calculating average of the corresponding estimations weighted with the respective certainties, each of the certainties representing a piece of weight information for the respective estimation.  
     
     
         18 . The position detection method according to  claim 15 , wherein in the step of calculating, certainties of position measurement results of the position-measurement-points are taken in account to calculate the estimations and certainty thereof.  
     
     
         19 . The position detection method according to  claim 18 , wherein the step of calculating comprises: 
 calculating, for each of the plurality of measurement point subsets, respective positions of the third number of position-measurement-points based on measurement results of the third number of position-measurement-points and estimating probability density functions that each represent occurrence probability of the calculated position of the respective, selected position-measurement-point;    calculating respective probability densities of the calculated positions of the position-measurement-points, based on the probability density functions; and    evaluating an error of each of the calculated positions relative to respective reference position using the respective calculated probability density's value as a piece of weight information and calculating estimations of the predetermined number of parameters, based on the evaluated errors.    
     
     
         20 . The position detection method according to  claim 15 , wherein position measurement marks are formed at the position-measurement-points.  
     
     
         21 . The position detection method according to  claim 20 , wherein a plurality of divided areas, each of which is provided with the position measurement marks, are arranged on the object.  
     
     
         22 . A position detection method for detecting position information of any area on an object provided with a first number of position-measurement-points, the position detection method comprising: 
 selecting a first measurement point subset which consists of a third number of position-measurement-points, the third number being larger than a second number and smaller than the first number, the second number being a minimum number of measurement points required to calculate a predetermined number of parameters that uniquely specify position information of any area on the object;    selecting a plurality of second measurement point subsets which each consist of a fourth number of position-measurement-points and are different from one another, the fourth number being larger than the second number and smaller than the third number; and    statistically evaluating possibility of replacing the first measurement point subset by one of the plurality of second measurement point subsets, based on measurement results of the third number of position-measurement-points composing the first measurement point subset and measurement results of sets of the fourth number of position-measurement-points each composing one of the second measurement point subsets, the first measurement point subset being used to calculate the predetermined number of parameters.    
     
     
         23 . The position detection method according to  claim 22 , wherein the step of evaluating comprises: 
 statistically calculating estimations of the predetermined number of parameters and certainty of the estimations, based on measurement results of the third number of position-measurement-points composing the first measurement point subset;    statistically calculating estimations of the predetermined number of parameters and certainty of the estimations for each of the plurality of second measurement point subsets, based on measurement results of the fourth number of position-measurement-points; and    comparing the estimations and certainty of the first measurement point subset with the estimations and certainty for each of the plurality of second measurement point subsets and evaluating possibility of replacing the first measurement point subset by one of the plurality of second measurement point subsets, the first measurement point subset being used to calculate the predetermined number of parameters.    
     
     
         24 . The position detection method according to  claim 23 , wherein in the step of calculating estimations for the first measurement point subset, certainties of position measurement results of the position-measurement-points are taken into account upon calculating the estimations and certainty thereof.  
     
     
         25 . The position detection method according to  claim 24 , wherein the step of calculating estimations for the first measurement point subset comprises: 
 calculating respective positions of the third number of position-measurement-points, which compose the first measurement point subset, based on measurement results of the third number of position-measurement-points and estimating probability density functions that each represent occurrence probability of the calculated position of a respective point of the third number of position-measurement-points;    calculating respective probability densities of the calculated positions of the position-measurement-points, based on the probability density functions; and    evaluating an error of each of the calculated positions relative to respective reference position using the respective calculated probability density's value as a piece of weight information and calculating estimations of the predetermined number of parameters, based on the evaluated errors.    
     
     
         26 . The position detection method according to  claim 23 , wherein in the step of calculating estimations for each second measurement point subset, certainties of position measurement results of the position-measurement-points are taken into account upon calculating the estimations and certainty thereof.  
     
     
         27 . The position detection method according to  claim 26 , wherein the step of calculating estimations for each second measurement point subset comprises: 
 calculating respective positions of the fourth number of position-measurement-points, for each of the plurality of second measurement point subsets, based on measurement results of the fourth number of position-measurement-points and estimating probability density functions that each represent occurrence probability of the calculated position of a respective point of the fourth number of position-measurement-points;    calculating respective probability densities of the calculated positions of the position-measurement-points, based on the probability density functions; and    evaluating an error of each of the calculated positions relative to respective reference position using the respective calculated probability density's value as a piece of weight information and calculating estimations of the predetermined number of parameters, based on the evaluated errors.    
     
     
         28 . The position detection method according to  claim 22 , wherein the step of evaluating comprises: 
 statistically calculating, for each of the second measurement point subsets, estimations of the predetermined number of parameters and certainty of the estimations, based on measurement results of the fourth number of position-measurement-points;    statistically calculating position errors of the position-measurement-points of the first measurement point subset through use of the estimations of the predetermined number of parameters calculated in the step of calculating estimations and evaluating possibility of replacing the first measurement point subset by one of the plurality of second measurement point subsets.    
     
     
         29 . The position detection method according to  claim 28 , wherein in the step of calculating estimations, certainties of position measurement results of the position-measurement-points are taken into account upon calculating the estimations and certainty thereof.  
     
     
         30 . The position detection method according to  claim 29 , wherein the step of calculating estimations comprises: 
 calculating respective positions of the fourth number of position-measurement-points, for each of the plurality of second measurement point subsets, based on measurement results of the fourth number of position-measurement-points and estimating probability density functions that each represent occurrence probability of the calculated position of a respective point of the fourth number of position-measurement-points;    calculating respective probability densities of the calculated positions of the position-measurement-points, based on the probability density functions; and    evaluating an error of each of the calculated positions relative to respective reference position using the respective calculated probability density's value as a piece of weight information and calculating estimations of the predetermined number of parameters, based on the evaluated errors.    
     
     
         31 . The position detection method according to  claim 22 , further comprising a step which, when the step of evaluating finds second measurement point subsets that can replace the first measurement point subset, selects a most valid subset from the second measurement point subsets for replacement and adopts estimations of the predetermined number of parameters, calculated based on measurement results of the fourth number of position-measurement-points of the selected second measurement point subset, as values thereof, and 
 which, when the step of evaluating finds no second measurement point subsets that can replace the first measurement point subset, adopts estimations of the predetermined number of parameters, calculated based on measurement results of the second number of position-measurement-points of the first measurement point subset, as values thereof.    
     
     
         32 . The position detection method according to  claim 22 , wherein position measurement marks are formed at the position-measurement-points.  
     
     
         33 . The position detection method according to  claim 32 , wherein a plurality of divided areas, each of which is provided with the position measurement marks, are arranged on the object.  
     
     
         34 . A position detection method for detecting position information of any area on an object provided with a first number of position-measurement-points, the position detection method comprising: 
 selecting a plurality of first measurement point subsets which each consist of a third number of position-measurement-points and are different from one another, the third number being larger than a second number and smaller than the first number, the second number being a minimum number of measurement points required to calculate a predetermined number of parameters that uniquely specify position information of any area on the object;    selecting a plurality of second measurement point subsets which each consist of a fourth number of position-measurement-points and are different from one another, the fourth number being larger than the second number and smaller than the third number; and    statistically evaluating possibility of replacing the plurality of first measurement point subsets by one of the plurality of second measurement point subsets, as a measurement point set used to calculate the predetermined number of parameters.    
     
     
         35 . The position detection method according to  claim 34 , wherein the step of evaluating comprises: 
 statistically calculating, for each of the plurality of first measurement point subsets, estimations of the predetermined number of parameters and certainty of the estimations, based on measurement results of the third number of position-measurement-points;    calculating statistically valid estimations of the predetermined number of parameters and certainty of the estimations, based on groups of the estimations and certainty thereof for the plurality of first measurement point subsets, calculated in the step of calculating estimations for each first measurement point subset;    statistically calculating estimations of the predetermined number of parameters and certainty of the estimations for each of the plurality of second measurement point subsets, based on measurement results of the fourth number of position-measurement-points; and    comparing the statistically valid estimations and certainty with the estimations and certainty for each of the plurality of second measurement point subsets and evaluating possibility of adopting one of the plurality of second measurement point subsets as a measurement point set used to calculate the predetermined number of parameters.    
     
     
         36 . The position detection method according to  claim 35 , wherein in the step of calculating estimations for each first measurement point subset, certainties of position measurement results of the position-measurement-points are taken into account upon calculating the estimations and certainty thereof.  
     
     
         37 . The position detection method according to  claim 36 , wherein the step of calculating estimations for each first measurement point subset comprises: 
 calculating respective positions of the third number of position-measurement-points, for each of the plurality of first measurement point subsets, based on measurement results of the third number of position-measurement-points and estimating probability density functions that each represent occurrence probability of the calculated position of a respective point of the third number of position-measurement-points;    calculating respective probability densities of the calculated positions of the position-measurement-points, based on the probability density functions; and    evaluating an error of each of the calculated positions relative to respective reference position using the respective calculated probability density's value as a piece of weight information and calculating estimations of the predetermined number of parameters, based on the evaluated errors.    
     
     
         38 . The position detection method according to  claim 35 , wherein in the step of calculating estimations for each second measurement point subset, certainties of position measurement results of the position-measurement-points are taken into account upon calculating the estimations and certainty thereof.  
     
     
         39 . The position detection method according to  claim 38 , wherein the step of calculating estimations for each second measurement point subset comprises: 
 calculating respective positions of the fourth number of position-measurement-points, for each of the plurality of second measurement point subsets, based on measurement results of the fourth number of position-measurement-points and estimating probability density functions that each represent occurrence probability of the calculated position of a respective point of the fourth number of position-measurement-points;    calculating respective probability densities of the calculated positions of the position-measurement-points, based on the probability density functions; and    evaluating an error of each of the calculated positions relative to respective reference position using the respective calculated probability density's value as a piece of weight information and calculating estimations of the predetermined number of parameters, based on the evaluated errors.    
     
     
         40 . The position detection method according to  claim 35 , further comprising a step which, when the step of evaluating finds second measurement point subsets that can replace the first measurement point subset, selects a most valid subset from the second measurement point subsets for replacement and adopts estimations of the predetermined number of parameters, calculated based on measurement results of the fourth number of position-measurement-points of the selected second measurement point subset, as values thereof, and 
 which, when the step of evaluating finds no second measurement point subsets that can replace the first measurement point subset, adopts the statistically valid estimations as values of the predetermined number of parameters.    
     
     
         41 . The position detection method according to  claim 34 , wherein the step of evaluating comprises: 
 statistically calculating, for each of the second measurement point subsets, estimations of the predetermined number of parameters and certainty of the estimations, based on measurement results of the fourth number of position-measurement-points;    calculating position errors of all the position-measurement-points of the plurality of first measurement point subsets through use of the estimations of the predetermined number of parameters calculated for each of the second measurement point subsets and evaluating possibility of replacing the plurality of first measurement point subsets by one of the plurality of second measurement point subsets.    
     
     
         42 . The position detection method according to  claim 41 , wherein in the step of calculating estimations for each second measurement point subset, certainties of position measurement results of the position-measurement-points are taken into account upon calculating the estimations and certainty thereof.  
     
     
         43 . The position detection method according to  claim 42 , wherein the step of calculating estimations for each second measurement point subset comprises: 
 calculating respective positions of the fourth number of position-measurement-points, for each of the plurality of second measurement point subsets, based on measurement results of the fourth number of position-measurement-points and estimating probability density functions that each represent occurrence probability of the calculated position of a respective point of the fourth number of position-measurement-points;    calculating respective probability densities of the calculated positions of the position-measurement-points, based on the probability density functions; and    evaluating an error of each of the calculated positions relative to respective reference position using the respective calculated probability density's value as a piece of weight information and calculating estimations of the predetermined number of parameters, based on the evaluated errors.    
     
     
         44 . The position detection method according to  claim 34 , further comprising a step which, when the step of evaluating finds second measurement point subsets that can replace the first measurement point subset, selects a most valid one, for replacement, of the second measurement point subsets and adopts estimations of the predetermined number of parameters, calculated based on measurement results of the fourth number of position-measurement-points of the selected second measurement point subset, as values thereof, and 
 which, when the step of evaluating finds no second measurement point subsets that can replace the first measurement point subset, statistically calculates estimations of the predetermined number of parameters and certainty of the estimations, based on measurement results of the third number of position-measurement-points for each of the plurality of first measurement point subsets, and adopts as values of the predetermined number of parameters statistically valid estimations thereof calculated based on groups of the estimations and certainty thereof for the plurality of first measurement point subsets.    
     
     
         45 . The position detection method according to  claim 44 , wherein the statistically valid value of each of the predetermined number of parameters is obtained by calculating average of the corresponding estimations weighted with the respective certainties, each of the certainties representing a piece of weight information for the respective estimation.  
     
     
         46 . The position detection method according to  claim 34 , wherein position measurement marks are formed at the position-measurement-points.  
     
     
         47 . The position detection method according to  claim 46 , wherein a plurality of divided areas each of which is provided with the position measurement marks are arranged on the object.  
     
     
         48 . A position detector that detects position information of any area on an object provided with a plurality of position-measurement-points, the position detector comprising: 
 a measurement unit that measures pieces of position information of more position-measurement-points than a minimum number of measurements required to calculate values of a predetermined number of parameters, which uniquely specify position information of any area on the object, the position-measurement-points being selected from the plurality of position-measurement-points;    an estimation unit that is electrically connected to the measurement unit and that detects respective positions of the selected position-measurement-points, based on the measurement results of the pieces of position information, estimates probability density functions which each represent occurrence probability of the detected position for respective one of the selected position-measurement-points, and calculates probability density of the detected position of each of the position-measurement-points; and    a parameter calculation unit that is electrically connected to the measurement unit and the estimation unit and that evaluates detection error of each of the detected positions while using the respective calculated probability density's value as a piece of weight information and calculates such values of the predetermined number of parameters that the detection errors become statistically minimum as a whole, based on the detection errors.    
     
     
         49 . The position detector according to  claim 48 , wherein said measurement unit comprises an image pickup unit that picks up images of marks formed on the object.  
     
     
         50 . A position detector that detects position information of any area on an object provided with a first number of position-measurement-points, the position detector comprising: 
 a measurement unit that measures positions of the position-measurement-points;    a set-selection unit that selects a plurality of measurement point subsets which each consist of a third number of position-measurement-points and are different from one another, the third number being larger than a second number and smaller than the first number, the second number being a minimum number of measurement points required to calculate a predetermined number of parameters that uniquely specify position information of any area on the object; and    an estimation computing unit that is electrically connected to the measurement unit and the set-selection unit and that statistically calculates, for each of the plurality of measurement point subsets, estimations of the predetermined number of parameters and certainty of the estimations, based on measurement results of the third number of position-measurement-points.    
     
     
         51 . The position detector according to  claim 50 , wherein the estimation computing unit comprising: 
 an estimation unit that, for each of the plurality of measurement point subsets, detects respective positions of the third number of position-measurement-points, based on the measurement results of the pieces of position information of the third number of position-measurement-points, estimates probability density functions which each represent occurrence probability of the detected position for the respective point of the third number of position-measurement-points, and calculates probability density of the detected position of each of the position-measurement-points; and    a parameter calculation unit that is electrically connected to the estimation unit and that evaluates detection error of each of the detected positions while using the respective calculated probability density's value as a piece of weight information and calculates such values of the predetermined number of parameters that the detection errors become statistically minimum as a whole, based on the detection errors.    
     
     
         52 . The position detector according to  claim 50 , further comprising: 
 a parameter value determining unit that is electrically connected to the estimation computing unit and that calculates statistically valid estimations of the predetermined number of parameters based on groups of the estimations and certainty thereof, calculated by the estimation computing unit, for the plurality of measurement point subsets.    
     
     
         53 . A position detector that detects position information of any area on an object provided with a first number of position-measurement-points, the position detector comprising: 
 a measurement unit that measures positions of the position-measurement-points;    a set-selection unit that selects a first measurement point subsets, which each consist of a third number of position-measurement-points, and a plurality of second measurement point subsets which each consist of a fourth number of position-measurement-points and are different from one another, the third number being larger than a second number and smaller than the first number, the second number being a minimum number of measurement points required to calculate a predetermined number of parameters that uniquely specify position information of any area on the object, the fourth number being larger than the second number and smaller than the third number; and    an evaluation computing unit that is electrically connected to the set-selection unit and that evaluates possibility of replacing the first measurement point subset by one of the plurality of second measurement point subsets, the first measurement point subset being used to calculate the predetermined number of parameters.    
     
     
         54 . The position detector according to  claim 53 , wherein the evaluation computing unit comprises: 
 an estimation calculation unit that is electrically connected to the measurement unit and that statistically calculates, for the specific measurement point subset, estimations of the predetermined number of parameters and certainty of the estimations, based on measurement results of position information of position-measurement-points composing the specific measurement point subset which is selected from the first measurement point subset and the plurality of second measurement point subsets; and    an evaluation unit that is electrically connected to the estimation calculation unit and that compares the estimations and certainty of the first measurement point subset with the estimations and certainty for each of the plurality of second measurement point subsets and evaluates possibility of replacing the first measurement point subset by one of the plurality of second measurement point subsets, the first measurement point subset being used to calculate the predetermined number of parameters.    
     
     
         55 . The position detector according to  claim 54 , wherein the estimation calculation unit comprises: 
 an estimation unit that detects respective positions of position-measurement-points composing the specific measurement point subset, based on the measurement results of position information of position-measurement-points composing the specific measurement point subset, estimates probability density functions which each represent occurrence probability of the detected position for respective one of the position-measurement-points of the specific measurement point subset, and calculates probability density of the detected position of each of the position-measurement-points; and    a parameter calculation unit that is electrically connected to the estimation unit and that evaluates detection error of each of the detected positions while using the respective calculated probability density's value as a piece of weight information and calculates such values of the predetermined number of parameters that the detection errors become statistically minimum as a whole, based on the detection errors.    
     
     
         56 . The position detector according to  claim 53 , wherein the evaluation computing unit comprises: 
 an estimation calculation unit that is electrically connected to the measurement unit and that statistically calculates, for the specific measurement point subset, estimations of the predetermined number of parameters and certainty of the estimations, based on measurement results of position information of position-measurement-points composing the specific measurement point subset, which is selected from the plurality of second measurement point subsets;    an evaluation unit that is electrically connected to the estimation calculation unit and that calculates position errors of the position-measurement-points, composing the first measurement point subset, through use of estimations of the predetermined number of parameters for each of the polarity of second measurement point subsets and evaluates possibility of replacing the first measurement point subset by one of the plurality of second measurement point subsets.    
     
     
         57 . The position detector according to  claim 56 , wherein the estimation calculation unit comprises: 
 an estimation unit that detects respective positions of position-measurement-points composing the specific measurement point subset, based on the measurement results of position information of position-measurement-points composing the specific measurement point subset, estimates probability density functions which each represent occurrence probability of the detected position for respective one of the position-measurement-points of the specific measurement point subset, and calculates probability density of the detected position of each of the position-measurement-points; and    a parameter calculation unit that is electrically connected to the estimation unit and that evaluates detection error of each of the detected positions while using the respective calculated probability density's value as a piece of weight information and calculates such values of the predetermined number of parameters that the detection errors become statistically minimum as a whole, based on the detection errors.    
     
     
         58 . The position detector according to  claim 53 , further comprising a parameter value determining unit that is electrically connected to the evaluation computing unit and that calculates values of the predetermined number of parameters, based on evaluation results of the evaluation computing unit.  
     
     
         59 . A position detector that detects position information of any area on an object provided with a first number of position-measurement-points, the position detector comprising: 
 a measurement unit that measures positions of the position-measurement-points;    a set-selection unit that selects a plurality of first measurement point subsets, which each consist of a third number of position-measurement-points and are different from one another, and a plurality of second measurement point subsets which each consist of a fourth number of position-measurement-points and are different from one another, the third number being larger than a second number and smaller than the first number, the second number being a minimum number of measurement points required to calculate a predetermined number of parameters that uniquely specify position information of any area on the object, the fourth number being larger than the second number and smaller than the third number; and    an evaluation computing unit that is electrically connected to the set-selection unit and that evaluates possibility of adopting one of the plurality of second measurement point subsets as a measurement point subset to calculate the predetermined number of parameters.    
     
     
         60 . The position detector according to  claim 59 , wherein the evaluation computing unit comprises: 
 an estimation calculation unit that is electrically connected to the measurement unit and that statistically calculates, for the specific measurement point subset, estimations of the predetermined number of parameters and certainty of the estimations, based on measurement results of position information of position-measurement-points composing the specific measurement point subset which is selected from the plurality of first measurement point subset and the plurality of second measurement point subsets, and calculates statistically valid estimations of the predetermined number of parameters and certainty of the estimations, based on groups of estimations of the predetermined number of parameters and certainty of the estimations for the plurality of first measurement point subsets;    an evaluation computing unit that is electrically connected to the estimation calculation unit and that compares the statistically valid estimations and certainty of the first measurement point subset with the estimations and certainty for each of the plurality of second measurement point subsets, and evaluates possibility of adopting one of the plurality of second measurement point subsets as a measurement point subset to calculate the predetermined number of parameters.    
     
     
         61 . The position detector according to  claim 60 , wherein the estimation calculation unit comprises: 
 an estimation unit that is electrically connected to the measurement unit and that detects respective positions of position-measurement-points composing the specific measurement point subset, based on the measurement results of position information of position-measurement-points composing the specific measurement point subset, estimates probability density functions which each represent occurrence probability of the detected position for respective one of the position-measurement-points of the specific measurement point subset, and calculates probability density of the detected position of each of the position-measurement-points; and    a parameter calculation unit that is electrically connected to the estimation unit and that evaluates detection error of each of the detected positions while using the respective calculated probability density's value as a piece of weight information and calculates such values of the predetermined number of parameters that the detection errors become statistically minimum as a whole, based on the detection errors.    
     
     
         62 . The position detector according to  claim 59 , wherein the evaluation computing unit comprises: 
 an estimation calculation unit that is electrically connected to the measurement unit and that statistically calculates, for the specific measurement point subset, estimations of the predetermined number of parameters and certainty of the estimations, based on measurement results of position information of position-measurement-points composing the specific measurement point subset which is selected from the plurality of second measurement point subsets; and    an evaluation unit that is electrically connected to the estimation calculation unit and that calculates errors of all the position-measurement-points of the plurality of first measurement point subsets through use of the estimations of the predetermined number of parameters calculated for each of the second measurement point subsets, and evaluates possibility of replacing the plurality of first measurement point subsets by one of the plurality of second measurement point subsets.    
     
     
         63 . The position detector according to  claim 62 , wherein the estimation calculation unit comprises: 
 an estimation unit that detects respective positions of position-measurement-points composing the specific measurement point subset, based on the measurement results of position information of position-measurement-points composing the specific measurement point subset, estimates probability density functions which each represent occurrence probability of the detected position for respective one of the position-measurement-points of the specific measurement point subset, and calculates probability density of the detected position of each of the position-measurement-points; and    a parameter calculation unit that is electrically connected to the estimation unit and that evaluates detection error of each of the detected positions while using the respective calculated probability density's value as a piece of weight information and calculates such values of the predetermined number of parameters that the detection errors become statistically minimum as a whole, based on the detection errors.    
     
     
         64 . The position detector according to  claim 59 , further comprising a parameter value determining unit that is electrically connected to the parameter calculation unit and that calculates values of the predetermined number of parameters, based on evaluation results of the evaluation computing unit.  
     
     
         65 . An exposure method for transferring a predetermined pattern onto divided areas on a substrate, comprising: 
 calculating a predetermined number of parameters that pertain to positions of the divided areas by a position detection method according to  claim 1  and calculating arrangement information of the divided areas on the substrate; and    transferring the pattern onto the divided areas while aligning the substrate based on the arrangement information of the divided areas calculated in the step of calculating arrangement information.    
     
     
         66 . An exposure apparatus that transfers a predetermined pattern onto divided areas on a substrate, comprising: 
 a stage unit that moves the substrate along a movement plane; and    a position detector according to  claim 48  that calculates arrangement information of the divided areas on the substrate mounted on the stage unit.    
     
     
         67 . A device manufacturing method including a lithography process, 
 wherein the lithography process transfers a predetermined pattern onto divided areas on a substrate using an exposure method according to  claim 65 .    
     
     
         68 . An exposure method for transferring a predetermined pattern onto divided areas on a substrate, comprising: 
 calculating a predetermined number of parameters that pertain to positions of the divided areas by a position detection method according to  claim 15  and calculating arrangement information of the divided areas on the substrate; and    transferring the pattern onto the divided areas while aligning the substrate based on the arrangement information of the divided areas calculated in the step of calculating arrangement information.    
     
     
         69 . A device manufacturing method including a lithography process, 
 wherein the lithography process transfers a predetermined pattern onto divided areas on a substrate using an exposure method according to  claim 68 .    
     
     
         70 . An exposure method for transferring a predetermined pattern onto divided areas on a substrate, comprising: 
 calculating a predetermined number of parameters that pertain to positions of the divided areas by a position detection method according to  claim 22  and calculating arrangement information of the divided areas on the substrate; and    transferring the pattern onto the divided areas while aligning the substrate based on the arrangement information of the divided areas calculated in the step of calculating arrangement information.    
     
     
         71 . A device manufacturing method including a lithography process, wherein the lithography process transfers a predetermined pattern onto divided areas on a substrate using an exposure method according to claim  70 .  
     
     
         72 . An exposure method for transferring a predetermined pattern onto divided areas on a substrate, comprising: 
 calculating a predetermined number of parameters that pertain to positions of the divided areas by a position detection method according to  claim 34  and calculating arrangement information of the divided areas on the substrate; and    transferring the pattern onto the divided areas while aligning the substrate based on the arrangement information of the divided areas calculated in the step of calculating arrangement information.    
     
     
         73 . A device manufacturing method including a lithography process, 
 wherein the lithography process transfers a predetermined pattern onto divided areas on a substrate using an exposure method according to claim  72 .    
     
     
         74 . An exposure apparatus that transfers a predetermined pattern onto divided areas on a substrate, comprising: 
 a stage unit that moves the substrate along a movement plane; and    a position detector according to  claim 50  that calculates arrangement information of the divided areas on the substrate mounted on the stage unit.    
     
     
         75 . An exposure apparatus that transfers a predetermined pattern onto divided areas on a substrate, comprising: 
 a stage unit that moves the substrate along a movement plane; and    a position detector according to  claim 53  that calculates arrangement information of the divided areas on the substrate mounted on the stage unit.    
     
     
         76 . An exposure apparatus that transfers a predetermined pattern onto divided areas on a substrate, comprising: 
 a stage unit that moves the substrate along a movement plane; and    a position detector according to  claim 59  that calculates arrangement information of the divided areas on the substrate mounted on the stage unit.