Mathematical relation identification apparatus and method

• US 6,519,545 B1
• Filed: 01/31/2001
• Issued: 02/11/2003
• Est. Priority Date: 12/22/1997
• Status: Expired due to Fees

• Current Assignee: Amano KOKI Kabushiki Kaisha
• Original Assignee: Amano KOKI Kabushiki Kaisha
• Inventors: Amano, Kageaki
• Primary Examiner(s): Hilten, John S.
• Assistant Examiner(s): LE, JOHN H
• Application Number:
  US09/622,435
• Time in Patent Office: 741 Days
• Field of Search: 702/5, 702/36, 702/42, 702/168, 702/127, 708/146, 708/400, 708/401, 708/402, 708/404, 324/76.21, 324/300, 324/306, 324/307, 382/248, 382/251, 704/210, 704/215, 704/221, 704/229, 704/230, 704/240, 707/4
• US Class Current: 702/127
• CPC Class Codes: G06F 17/175 of multidimensional data
• CPC Class Codes: G06F 18/213 Feature extraction, e.g. by transforming the feature space; Summarisation; Mappings, e.g. subspace methods
• CPC Class Codes: Y10S 707/99937 Sorting

Abstract

A mathematical relation between base variables (x1, x2, . . . , xp), when a set of input data d is composed of p base variables (x1, x2, . . . , xp), and a plurality of data sets d (i) of such data set d is inputted, the data sets d(i) are distinguished by an input data distinguishing parameter i. When victors in a q dimensional space, mapped from the input data through q base functions (f1, . . . , fq), form a plane, a mathematical relation can be a linear combination of the base functions. A set of base functions (f1, . . . , fq) are prepared. Sets of the values F(i)=(F1i, . . . , Fqi) of the base functions corresponding to the input data d (i) are acquired. The sets F(i) are considered points in a q dimensional space. Direction cosine of a mapping plane is acquired through cofactors of a determinant of these points, or by solving an eigenvalue problem for determining a plane, the square sum of the perpendicular lines from these points to the plane is minimum. When the direction cosine of the plane is (L1, . . . , Lq), the following mathematical relation is outputted:

qΣ(Lk×fk)=0k=1.

Claims

1. An apparatus for identifying a mathematical relation between base variables (x1, x2, . . . ,xp), when a set of input data d is composed of p base variables (x1, x2, . . . ,xp), and a plurality of data set d(i) of such a data set d is inputted, the data sets d(i) are distinguished by an input data distinguishing parameter i, comprising:

a data acquisition means for acquiring values xji of base variables xj(j=1, . . . ,p) in sets of input data d(i) (i=1, 2, . . . ,n), which are distinguished by the input data distinguishing parameter i;

a base function defining means for selecting a mathematical function program (m=k), which is one of the mathematical function programs gm distinguished by a mathematical function specifying parameter m, the mathematical function programs output a function value corresponding to their input references, the base function defining means defines a base functions fk, by specifying a base variable to each of one or more than one input references of the selected mathematical function program fk;

a candidate mathematical relation specifying means, which specifies a candidate mathematical relation, by specifying a plurality of base functions fk (k=1, 2, . . . , q) from the base functions defined by the base function defining means;

a vector component acquisition means for sending the values of the base variables acquired by said data acquisition means, to each of the input references of base functions fk (k=1, . . . q) included in the candidate mathematical relation specifying means, which is defined by the candidate mathematical relation, for each data set d(i), and for acquiring and storing the output value Fki of the base functions fk into an array VC_ARRAY, in which the values of the functions Fki are stored in a form of a vector F(i) in a q dimensional space, for each input data specifying parameter i;

a direction cosine acquisition means for acquiring the direction cosine (L1, . . . , Lq) of a mapping plane spanned by the points P(i) in a vector space which correspond to the vectors stored in the vector component array VC_ARRAY, wherein the direction cosine acquisition means comprises;

a correlation sum calculating means for calculating the correlation sum <

a|b>

of the vector component array VE_ARRAY;

an eignevalue calculating means for calculating the minimum eigenvalue among the eigenvalues λ

(1), . . . , λ

(q) of a correlation sum matrix, the elements of which are correlation sums <

a|b>

;

and an eigenvector calculating means, which calculates the eigenvector corresponding to the minimum eignevalue, and outputs the eigenvector as the direction cosine (L1, . . . ,Lq);

and a mathematical relation outputting means for outputting a mathematical relation corresponding to the following expression or an mathematical expression or values, which can be deduced from the following expression, when the direction cosine of the plane corresponding to each base function is Lk(k=1, 2, . . . , q);

$\underset{k=1}{\overset{q}{\sum <br><br>}}<br><br>\text{\hspace{1em}<br><br>}<br><br>\left(\mathrm{Lk}\times <br><br>\mathrm{fk}\right)=0.$

2. An apparatus for identifying a mathematical relation according to claim 1, wherein the eigenvalue calculating means multiplies a weighting Wi² to the product of elements Fai and Fbi in the array VC_ARRAY, then obtains the sum of them in respect with the parameter i to obtain the correlation sum <

a|b>

3. An apparatus for identifying a mathematical relation according to claim 1, further comprising:

a validity estimating means for judging whether the obtained eigenvalue is larger than a predetermined value or not;

and a control program stored in a file which controls so that when the validity estimating means does not estimate that the eigenvalue is smaller than the predetermined value, the candidate mathematical relation specifying means selects another candidate mathematical relation, and when the validity estimating means estimates that the eigenvalue is smaller than the predetermined value, and the direction cosine corresponding to each of the base functions is Lk, the mathematical relation outputting means outputs a mathematical relation corresponding to the following mathematical relation of deduced from the mathematical relation $\underset{k=1}{\overset{q}{\sum <br><br>}}<br><br>\text{\hspace{1em}<br><br>}<br><br>\left(\mathrm{Lk}\times <br><br>\mathrm{fk}\right)=0.$

4. An apparatus for identifying a mathematical relation according to claim 1, wherein the apparatus is incorporated in a measuring apparatus, and used as an apparatus for interpolating or extrapolating data.

5. An apparatus for identifying a mathematical relation according to claim 1, wherein the apparatus is incorporated in an apparatus for identifying trajectories of a moving object, and acquires input data of position as a function of a time parameter i, then outputs one or a plurality of mathematical relation(s), for identifying the trajectory of the moving object.

6. An apparatus for identifying a mathematical relation according to claim 1, wherein the apparatus is incorporated in an apparatus for identifying a form of a curve or a surface, and acquires input data of coordinates of infinite points on the curve or the surface, then outputs one or a plurality of mathematical relation(s), for identifying the form of the curve or the surface.

7. An apparatus for identifying a mathematical relation according to claim 6, wherein the form of curve or surface is a form of an object, and coordinates of points on the object are given as input data d(i).

8. An apparatus for identifying a mathematical relation according to claim 6, wherein the form of curve or surface is a form of curve or surface in an abstract space, which represents a mathematical relation of variables, and values of the variables are given as input data d(i).

9. An apparatus for identifying a mathematical relation according to claim 1, further comprising a memory media, which can be read out by a computer.

10. An apparatus for identifying a mathematical relation according to claim 9, wherein said memory media includes:

a data acquisition program stored in a file for acquiring sets of input data d(i);

a plurality of mathematical function program stored in a file which can output a function value corresponding to a set of its input variables;

a base function defining program stored in a file for defining base functions and/or defined base functions;

a candidate mathematical relation specifying program stored in a file for specifying a candidate mathematical relation and/or a set of base functions as a specified candidate mathematical relation;

a vector component acquisition program stored in a file, which sends the values of base variables to the input references of the base functions fk (k=1, . . . ,q) included in the candidate mathematical relation specified by the candidate mathematical relation specifying program, for each input data specifying parameter i, and acquires the value Fki of the base function to store a vector F(i), the component of which is the function values Fki (k=1, . . . ,q), into a vector component array VC_ARRAY, for each input data specifying parameter i;

a direction cosine acquiring program stored in a file, which acquires the direction cosine of a mapping plane corresponding to the vectors F(i) in the vector component array VC_ARRAY; and

a mathematical relation outputting program stored in file for outputting a mathematical relation, which corresponds to the following expression, when the direction cosine corresponding to each of the base functions fk is Lk (k=1, . . . , q);

$\underset{k=1}{\overset{q}{\sum <br><br>}}<br><br>\text{\hspace{1em}<br><br>}<br><br>\left(\mathrm{Lk}\times <br><br>\mathrm{fk}\right)=0.$

11. A method for identifying a mathematical relation between base variables (x1, x2, . . . ,xp), when a set of input data d is composed of p base variables (x1, x2, . . . ,xp), and a plurality of data set d(i) of such a data set d is inputted, the data sets d(i) are distinguished by an input data distinguishing parameter i, which comprises the steps of:

acquiring the values xji of base variables xj (j=1, . . . ,p) in sets of input data d(i) (i=1, 2, . . . ,n), which are distinguished by the input data distinguishing parameter i;

selecting a mathematical function program (m=k), which is one of the mathematical function programs gm distinguished by a mathematical function specifying parameter m, the mathematical function programs output a function value corresponding to their input references;

defining a base functions fk by specifying a base variable to each of one or more than one input references of the selected mathematical function program fk;

after defining a base functions fk, specifying a candidate mathematical relation, by specifying a plurality of base functions fk (k=1, 2, . . . ,q) from the base functions;

after acquiring the values of xji of base variables, sending the values of the base variables, to each of the input references of base functions fk (k=1, . . . q) included in the candidate mathematical relation, which is defined by the sending step, for each data set d(i), and for acquiring and storing the output value Fki of the base functions fk into an array VC_ARRAY, in which the values of the functions Fki are stored in a form of a vector F(i) in a q dimensional space, for each input data specifying parameter i;

obtaining the direction cosine (L1, . . . , Lq) of a mapping plane spanned by the points P(i) in a vector space which correspond to the vectors stored in the vector component array VC_ARRAY, wherein the direction cosine acquisition procedure calculates the correlation sum <

a|b>

of the vector component array VE_ARRAY, then calculates the minimum eigenvalue among the eigenvalues 1 (1), . . . , 1 (q) of a correlation sum matrix, the elements of which are correlation sums <

a|b>

, and finally calculates the eigenvector corresponding to the minimum eignevalue, and outputs the eigenvector as the direction cosine (L1, . . . ,Lq); and

outputting a mathematical relation corresponding to the following expression or an mathematical expression or values, which can be deduced from the following expression, when the direction cosine of the plane corresponding to each base function is Lk (k=1, 2, . . . ,q);

$\underset{k=1}{\overset{q}{\sum <br><br>}}<br><br>\text{\hspace{1em}<br><br>}<br><br>\left(\mathrm{Lk}\times <br><br>\mathrm{fk}\right)=0.$