Data processing: measuring – calibrating – or testing – Measurement system
Reexamination Certificate
2001-01-31
2003-02-11
Hilten, John S. (Department: 2863)
Data processing: measuring, calibrating, or testing
Measurement system
C382S141000, C356S394000, C600S300000, C600S558000, C702S032000, C707S793000
Reexamination Certificate
active
06519545
ABSTRACT:
TECHNICAL FIELD
The present invention relates to an apparatus for identifying a mathematical relation between variables of measured data. An apparatus for identifying a mathematical relation identifies a mathematical relation between numerical data of, for example, a form, trajectory, etc. The extrapolation, interpolation, storing of such data can be facilitated
For example, the apparatus can be used as follows:
1) Expression of a shape of an object
In a designing of a shape of an automobile, data of a shape of an object can be stored as a set of numerical data. However, when a shape can be expressed as a mathematical expression including a plurality of parameters, data can be easily extrapolated and interpolated. And new shape can be easily obtained, by varying the value of some parameters. And a computer analysis of the new shape, for example, aerodynamical resistance, etc, becomes easy. Further, a preparation of a mold for fabrication of an object having the corresponding shape, using a numerically controlled working machine, becomes easy.
In a shoe shop, even when a client found a pair of shoes, the size of which is identical to the size of his feet, and the design of which is acceptable, if the pair of shoes do not fit actually to his feet, he will not buy the pair of shoes. There are many variety in the shape of foot of human being. Therefore, sizes including small number of measured values of a foot are not sufficient to express a shape of a foot of human being. No apparatus or method is known, which can determine values of parameters of a mathematical expression, representing a complex shape of an object, for example, a shape of shoe or a shape of foot of human being.
2) Identification of a mathematical relation between variables of experimental data, when the mathematical relation shall be expressed by a non-linear function:
Even when a mathematical relation between obtained data is non-linear, data are often extrapolated and interpolated, using a linear approximation. However, when a mathematical relation between the variables of data can be identified, data can be exactly extrapolated and interpolated, without using linear approximation.
Assuming that a food is produced from stuffs A, B. And a characteristic value of the food X, for example, concentration Z of an amino acid, is a function of the concentration X, Y of the stuffs A, B, the pressure P and temperature T of a treatment. When the mathematical relation between these variables can be identified, it becomes easy to obtain the most preferable values of the X, Y. P, T, for getting the most appropriate concentration Z, using the mathematical relation. Thus the development of a new product can be facilitated.
For example, assuming that the relation between the variables X, Y, P, T, Z is as follow, no apparatus and method is known, which can determine the coefficients K
0
, K
1
, K
2
, K
3
, by small times of experiments.
Z/
(
X·Y
)=
K
0
+
K
1
·
P
1
·
K
2
T+K
3
·
X
3) Representation of a trajectory of a moving object: For example, it is said that the trajectory of golf club head of a professional golfer is approximately a plane, on the other hand, the trajectory of golf club head of an ordinal amateur is not a plane. The trajectory of golf club head is not a line, thus the mathematical expression of the trajectory of golf club head is not simple. An apparatus for identifying the mathematical relation identifies the mathematical relation of a trajectory of such a moving object so as to analyze the trajectory.
Trajectories of an airplane flying around over an airport, a ship sailing on the ocean, or a car running on the road, are not linear, but the changing rate of the direction of the movement is slow. When the mathematical relation of the motion of such moving objects can be identified, it is possible to estimate the possibility of a collision of such moving objects.
BACKGROUND
Prior Art
Some methods for identifying the mathematical relation between measured values are proposed. For example, a least square root method for determining the factors a and b in a linear relation y=ax+b is widely employed, under the assumption that a linear relation y=ax+b stands between the input data. Also a least square root method after a logarithmic transformation is widely employed.
Methods for identifying the mathematical relation between the measured data are explained compactly in “Statistics for analytical chemistry” by J. C. Miller/J. N. Miller, which is translated into Japanese by Munemori Makoto and published in Japan from Kyouritu Shuppan in 1991.
Japanese patent application JP-5-334431-A discloses an apparatus for giving a mathematical function, which approximates data of points on a line This apparatus is not applicable, when the data is not data of points on a line, or when the data is data of over three dimension.
Japanese patent application JP-5-266063-A discloses an apparatus for interpolating data in n dimension space, using a super surface in a (n+1) dimension space. This apparatus can interpolate data, but does not identify the mathematical relation of the data.
Object of the Invention
An object of the present invention is to propose an apparatus and method for outputting a mathematical relation between base variables (x
1
, x
2
, . . . , xp), when a set of input data d is comprised of p base variables (x
1
, x
2
, . . . , xp), and a plurality of data sets d(i) of such a data set d are inputted, where the symbols “i” is a parameter for distinguishing the sets of data set.
Disclosure of Invention
Glossary
The meaning of symbols and terms used in this Specification and the Claims are explained, before explaining the present invention. Symbol “{circumflex over ( )}” is a power (far example, (−2){circumflex over ( )}3=−8). Power is expressed also using a suffix (for example, (−2)
3
=−8).
Symbol “.” is a multiplication (for example 2·3=6). However, this symbol is abbreviated, when the meaning is obvious (for example, 2x+3y=0 means 2·x+3·y=0).
Symbol “<a·b>” is an inner product (scalar product) of vectors.
“Base variables x
1
, . . . , xp” are names of variables of data inputted into the apparatus according to the present invention (for example coordinates x, y, z pressure p, or time t).
“Input data d” is a set of base variables (x
1
, x
2
, . . . xp).
“d=(x
1
, . . . , xp)” means that the number of the base variables of the input data d is p, and the final base variable is xp.
“Input data d(i)” is the i-th set of the input data d. Symbol “i” is called “data specifying parameter”.
“xji” is the value of the j-th variable of the i-th set of data d(i).
“Mathematical function program” is a program for outputting a value corresponding to input reference values, after executing a mathematical calculation or using a table.
“An input reference and a function specifying reference”: For example, a function exp(k·x) can be considered as a two variable function, in such a case, the function exp(k·x) has two input variables k and x. However, this function can be considered also as a one variable function exp(k·x). In this case, the function has an input reference x and a reference k for specifying the function. References for specifying the function are called “function specifying reference”, and references as input values are called “input reference”.
“Function specifying parameter m” is a parameter for specifying a mathematical function program gm.
“Base function” is a function gk in a set of mathematical function programs gm stored in a mathematical function program storing memory, and specified by the function specifying parameter (m=k). The input variables to be inputted to the input reference of the mathematical function gk are specified. The function specifying references are specified when it is necessary. When the function is a constant function, which outputs a constant value irrespectively to the input, it is not necessary to specify the input references.
“Candidate mathematical relatio
Amano Koki Kabushiki Kaisha
Hilten John S.
Le John
McGlew and Tuttle , P.C.
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