JPS63142215A - Multivariable polynomial type measured value calibrator - Google Patents
Multivariable polynomial type measured value calibratorInfo
- Publication number
- JPS63142215A JPS63142215A JP29093186A JP29093186A JPS63142215A JP S63142215 A JPS63142215 A JP S63142215A JP 29093186 A JP29093186 A JP 29093186A JP 29093186 A JP29093186 A JP 29093186A JP S63142215 A JPS63142215 A JP S63142215A
- Authority
- JP
- Japan
- Prior art keywords
- polynomial
- measured value
- data
- calibration
- coefficients
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Granted
Links
- 239000011159 matrix material Substances 0.000 claims abstract description 13
- 238000000034 method Methods 0.000 claims abstract description 9
- 238000005259 measurement Methods 0.000 claims description 5
- 210000001508 eye Anatomy 0.000 description 5
- 238000004364 calculation method Methods 0.000 description 3
- 210000005252 bulbus oculi Anatomy 0.000 description 2
- 238000010586 diagram Methods 0.000 description 2
- 238000007476 Maximum Likelihood Methods 0.000 description 1
- 238000007405 data analysis Methods 0.000 description 1
- 230000007423 decrease Effects 0.000 description 1
- 238000005516 engineering process Methods 0.000 description 1
- 230000005484 gravity Effects 0.000 description 1
- 210000002784 stomach Anatomy 0.000 description 1
Landscapes
- Complex Calculations (AREA)
- Indication And Recording Devices For Special Purposes And Tariff Metering Devices (AREA)
Abstract
Description
【発明の詳細な説明】
[産業上の利用分野]
本発明は、多変数多項式型測定値を較正するための測定
値較正器に関するものである。DETAILED DESCRIPTION OF THE INVENTION [Industrial Field] The present invention relates to a measurement value calibrator for calibrating multivariate polynomial type measurements.
[従来の技術]
各種測定器は、本来測定しようとする量に対し、測定器
の出力が一対一対応し、しかも直線関係で結ばれている
のが望ましいが、そのような関係が実現されない場合の
方が多い。[Prior Art] It is desirable that the output of various measuring instruments has a one-to-one correspondence with the quantity to be measured, and that they are connected in a linear relationship. However, when such a relationship is not realized, There are more.
その場合、測定値を較正して初期の目的に合わせるので
あるが、この目的には、
y=ax+b ・・(0)は、
−σXシー
b=y−2x
(n:データ数、 X + V : xi’、 yl
の平均値)のようになる (石田正次著、データ解
析の基礎。In that case, the measured values are calibrated to match the initial purpose. For this purpose, y=ax+b...(0) is -σXc b=y-2x (n: number of data, X + V : xi', yl
(average value of ) (Masatsugu Ishida, Fundamentals of Data Analysis).
森北出版P35) 。Morikita Publishing P35).
同様に、
y)= 00+C,x、+C2x’、+・−+C,x7
* * (1)(iツ1,2.・・n)
のようにm次の多項式で表現される場合には、なる式を
解けば良いことが知られている (同書P92)。Similarly, y)=00+C,x,+C2x',+・-+C,x7
* * It is known that when expressed as an m-th degree polynomial such as (1) (i tsu 1, 2...n), it is sufficient to solve the following equation (P92 of the same book).
しかしながら、被測定変数が多数の測定値に影響される
場合の較正法は一般に知られておらず。However, a calibration method when a measured variable is influenced by a large number of measured values is generally unknown.
また(1)式の一般化は表現が複雑で計算も容易ではな
いと推測される。Furthermore, it is assumed that the generalization of equation (1) is complex in expression and calculation is not easy.
他方、行列論では、(1)式のように表現された関係に
対し、主としてn<mの場合の解法が一般逆行列として
知られており、パラメータの数がデータの数より多い場
合、最尤推定値になることがわかっている (土産 茂
ニ一般逆行列、数理科学5−5.68/72.1973
)。On the other hand, in matrix theory, for the relationship expressed as equation (1), the solution method mainly when n < m is known as the general inverse matrix, and when the number of parameters is greater than the number of data, the maximum (Souvenir Shigeru general inverse matrix, Mathematical Science 5-5.68/72.1973
).
一般逆行列の理論では、n > mの場合は十分に調べ
られておらず、同様に多変数で表現される場合も十分調
べられていない。In the theory of general inverse matrices, the case where n > m has not been sufficiently investigated, and similarly the case where it is expressed in multiple variables has not been sufficiently investigated.
[発明が解決しようとする問題点]
−゛1本発明の目的は、多数のパラメータに影響さ容易
に得られる係数を使って、上記多項式に基づく測定値の
較正を簡単に実施可能にすることにある。[Problems to be Solved by the Invention] -゛1 The purpose of the present invention is to make it possible to easily calibrate measured values based on the above polynomial using easily obtained coefficients that are influenced by a large number of parameters. It is in.
[問題点を解決するための手段]
上記目的を達成するための本発明の測定値較正器は、上
述した多変数多項式の演算を行う演算装置を備え、上記
演算装置におけるデータ較正のための係数として、一般
逆行列の計算法の一般化した方法によって計算される係
数を設定し、これによって測定値の較正を行うことを特
徴とするものである。[Means for Solving the Problems] A measured value calibrator of the present invention for achieving the above object includes an arithmetic device that performs the arithmetic operation of the multivariate polynomial described above, and a coefficient for data calibration in the arithmetic device. The method is characterized in that coefficients calculated by a generalized method of calculating a general inverse matrix are set, and measured values are calibrated using these coefficients.
[作用及び効果]
演算装置におけるデータ較正のための係数として、一般
逆行列の計算法の一般化した方法によって計算される係
数を用いているので、多変数の多項式の係数が容易に得
られ、その係数を使って非常に簡単な回路構成により、
測定値を簡単に較正”+−Cto+C++”t+C+z
ff++C+3”t”z”+Ft+C+sYt”CH6
X FCI 7!イ!i◆018”i’で参〇、、yj
+−−bi= C20”21”i+C2241+C23
”i”24”1ffi+C25!i+C26”i+02
7”1ffi+Cz8”iYi+C29F ”・・(2
)
となるが、これを行列表現すると以下のようになる。[Operation and Effect] Since coefficients calculated by a generalized method of calculating a general inverse matrix are used as coefficients for data calibration in the arithmetic device, coefficients of multivariate polynomials can be easily obtained. With a very simple circuit configuration using those coefficients,
Easily calibrate measured values “+-Cto+C++”t+C+z
ff++C+3"t"z"+Ft+C+sYt"CH6
X FCI 7! stomach! i◆018"i' 〇,,yj
+--bi=C20"21"i+C2241+C23
"i"24"1ffi+C25!i+C26"i+02
7”1ffi+Cz8”iYi+C29F”...(2
), but when expressed as a matrix, it becomes as follows.
C: l×2 。C: l×2.
!L:パラメータの項数)・・(4) と表せる。! L: Number of parameter terms)...(4) It can be expressed as
(4)式の両辺にX”(T:転置行列を示す)を掛けて
逆行列を取ると、
C= (XT X)−1XT A 、
−(5)になる。Multiplying both sides of equation (4) by X'' (T: indicates the transposed matrix) and taking the inverse matrix, C= (XT X)-1XT A,
−(5).
ここで、データ数nがパラメータの散文より大きい場合
は、一般にXtX(文×立行列)の逆tj列は存在する
。何らかの事情で十分データを取ねなかったときは、そ
の逆行列は存在しないが、その場合は一般逆行列(XT
T)4″(前述参考文献)を用いれば良い。Here, when the number of data n is larger than the prose of the parameter, there generally exists an inverse tj column of XtX (sentence×column). If sufficient data cannot be collected for some reason, the inverse matrix does not exist, but in that case, the general inverse matrix (XT
T) 4″ (the above-mentioned reference) may be used.
このように、(5)式より得られる係数Cを用いて、任
意の次数、任意の変数に対して、(2)式のることは既
に知られており (武田常広二三次元位置計測装置と原
反力計による重心測定、製品科学研究所研究報告No、
94.37/48)、一般逆行列が誤差の二乗和が最小
になる最尤推足値になることから、(5)式は一般に良
いデータ較正係数を表わすものと期待できる。このこと
を後述の実験例によって示す。In this way, using the coefficient C obtained from equation (5), it is already known that equation (2) can be used for any order and arbitrary variable (Tsunehiro Takeda, Three-dimensional position measurement Center of gravity measurement using device and original reaction force meter, Product Science Research Institute Research Report No.
94.37/48), and since the general inverse matrix becomes the maximum likelihood estimate that minimizes the sum of squared errors, equation (5) can be expected to generally represent a good data calibration coefficient. This will be illustrated by the experimental examples described below.
なお、変数の数および次数は他の先験的知識によって決
定される。Note that the number and order of variables are determined by other a priori knowledge.
このデータ較正器は第1図に示した回路構成によって実
現できる。同図は、2変数2次多項式による測定値較正
器の回路図で、前記(2)式の演算を行うように各種演
算要素を配列している0図中に示したXは、乗算を行う
演算要素を示し、同じく+は加算を行う演算要素を示し
ている。データ次に1本発明に関する実験例を示す。This data calibrator can be realized by the circuit configuration shown in FIG. The figure is a circuit diagram of a measured value calibrator using a two-variable quadratic polynomial, in which various calculation elements are arranged to perform the calculation of equation (2). It shows an arithmetic element, and + also shows an arithmetic element that performs addition. Data Next, an experimental example related to the present invention will be shown.
先に本発明者らが特願昭60−146227号として出
願した「眼球屈折力測定装置」において、眼球の回転角
は2つのミラーの回転角により測定されるのであるが、
回転ミラーを球面ミラーの中心に置くことができないた
めに、それぞれのミラーの回転角が、眼の左右、上下の
回転角に直接対応しない。In the "eyeball refractive power measuring device" previously filed by the present inventors as Japanese Patent Application No. 146227/1980, the rotation angle of the eyeball is measured by the rotation angle of two mirrors.
Since the rotating mirror cannot be placed at the center of the spherical mirror, the rotation angle of each mirror does not directly correspond to the horizontal and vertical rotation angles of the eye.
即ち、模型眼を左右(X軸)、上下(Y軸)に回転でき
るステージに乗せて、5度きざみに回転したときの2つ
のミラーの出力値は、第2図のようになり、滑らかな関
数ではあるが、とても−次式で表現できるような関係に
はなっていない、そこで、
aI :模型眼の水平方向の回転角、
bl :模型眼の上下方向の回転角、
xl :ミラーXの回転角、
yI :ミラーYの回転角、
と定義して、
と表現し、n=35(左右±15°、上下±10’、5
度きざみにデータをとる。)とし、1,2,3゜4.5
次の多項式で表現したときの二乗誤差の和は、第1表の
ように、次数を増加させるに従い単調減少し、次数を上
げる毎に良い近似を与えていることがわかる。−例とし
て、3次式で近似したときの第2図のデータの較正結果
を第3図に示す。In other words, when the model eye is placed on a stage that can be rotated left and right (X axis) and up and down (Y axis), and rotated in 5 degree increments, the output values of the two mirrors will be as shown in Figure 2, and will be smooth. Although it is a function, it does not have a relationship that can be expressed by the following equation. Therefore, aI: horizontal rotation angle of the model eye, bl: vertical rotation angle of the model eye, xl: rotation angle of the mirror Rotation angle, yI: Rotation angle of mirror Y, defined as, and expressed as, n = 35 (left and right ±15°, vertical ±10', 5
Take data in increments. ), 1,2,3°4.5
It can be seen that the sum of squared errors when expressed by the following polynomial equation monotonically decreases as the degree increases, as shown in Table 1, giving better approximation as the degree increases. - As an example, FIG. 3 shows the calibration result of the data in FIG. 2 when approximated by a cubic equation.
第1表Table 1
第1図は本発明に係る多変数多項式型測定値較正器の回
路構成図、第2図は模型眼を5度きざみで、左右15度
上下10度だけ動かしたとき(格子の交点に相当する)
に測定された2つのミラーのふれ角(0印で示した)に
関する説明図、第3図は第2図のデータを(5)式を用
いて35個のデータから較正した結果(0印)を示す説
明図である。
指足代理人
工業技術院製品科学研究所長
高僑教司Fig. 1 is a circuit diagram of the multivariate polynomial type measured value calibrator according to the present invention, and Fig. 2 shows the case where the model eye is moved by 15 degrees left and right and 10 degrees up and down in 5 degree increments (corresponding to the intersection points of the grid). )
Figure 3 is the result of calibrating the data in Figure 2 from 35 pieces of data using equation (5) (marked 0). FIG. Professor Takashi, Director, Product Science Research Institute, Agency of Industrial Science and Technology
Claims (1)
ているが、滑らかな変化をしているデータを、多変数の
多項式で表現して、この多項式の演算を行う演算装置を
備え、上記演算装置におけるデータ較正のための係数と
して、一般逆行列の計算法の一般化した方法によって計
算される係数を設定し、これによって測定値の較正を行
うことを特徴とする多変数多項式型測定値較正器。1. Equipped with an arithmetic device that expresses data that has a nonlinear relationship affected by many parameters but changes smoothly using a multivariate polynomial, and performs operations on this polynomial, and performs the above operations. A multivariable polynomial type measurement value calibration characterized in that coefficients calculated by a generalized method of calculating a general inverse matrix are set as coefficients for data calibration in the device, and measurement values are calibrated using this coefficient. vessel.
Priority Applications (1)
Application Number | Priority Date | Filing Date | Title |
---|---|---|---|
JP61290931A JP2507875B2 (en) | 1986-12-05 | 1986-12-05 | Multivariate polynomial measurement calibrator |
Applications Claiming Priority (1)
Application Number | Priority Date | Filing Date | Title |
---|---|---|---|
JP61290931A JP2507875B2 (en) | 1986-12-05 | 1986-12-05 | Multivariate polynomial measurement calibrator |
Publications (2)
Publication Number | Publication Date |
---|---|
JPS63142215A true JPS63142215A (en) | 1988-06-14 |
JP2507875B2 JP2507875B2 (en) | 1996-06-19 |
Family
ID=17762360
Family Applications (1)
Application Number | Title | Priority Date | Filing Date |
---|---|---|---|
JP61290931A Expired - Lifetime JP2507875B2 (en) | 1986-12-05 | 1986-12-05 | Multivariate polynomial measurement calibrator |
Country Status (1)
Country | Link |
---|---|
JP (1) | JP2507875B2 (en) |
Cited By (1)
Publication number | Priority date | Publication date | Assignee | Title |
---|---|---|---|---|
JPH0317511U (en) * | 1989-07-01 | 1991-02-21 |
Citations (2)
Publication number | Priority date | Publication date | Assignee | Title |
---|---|---|---|---|
JPS54142470A (en) * | 1978-04-27 | 1979-11-06 | Tohoku Electric Power Co | Forecast monitor |
JPS5968620A (en) * | 1982-10-13 | 1984-04-18 | Toshiba Corp | Correcting device of measured value |
-
1986
- 1986-12-05 JP JP61290931A patent/JP2507875B2/en not_active Expired - Lifetime
Patent Citations (2)
Publication number | Priority date | Publication date | Assignee | Title |
---|---|---|---|---|
JPS54142470A (en) * | 1978-04-27 | 1979-11-06 | Tohoku Electric Power Co | Forecast monitor |
JPS5968620A (en) * | 1982-10-13 | 1984-04-18 | Toshiba Corp | Correcting device of measured value |
Cited By (1)
Publication number | Priority date | Publication date | Assignee | Title |
---|---|---|---|---|
JPH0317511U (en) * | 1989-07-01 | 1991-02-21 |
Also Published As
Publication number | Publication date |
---|---|
JP2507875B2 (en) | 1996-06-19 |
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