Uncertainty Estimation Using Monte-Carlo Method Constrained by Correlations of the Data

Simple and easy uncertainty estimation method is proposed. Provided that specification or simple experimental result is available, possible variance and covariance in error are estimated and Monte-Carlo simulation reflecting constraint caused by the covariance can be performed. Comparison between uncertainties obtained by the proposed method and that by actual measurements on real CMM shows good agreement within 1 m μ over-estimation. Introduction In recent years, the concept of traceability of measurement is spreading and the importance of uncertainty comes to be recognized[1]. Reality, however, is that uncertainty can be analytically calculated only in simple model case. For example, thinking about measurement with coordinate measurement machine (CMM), it has not only the complex structure but also the user may arbitrarily configure measurement task and the procedure for evaluating measured result. It requires enormous effort for users to evaluate uncertainty of the measurement, and the difficulty prevents the application on complicate measurement. In order to solve this problem, uncertainty estimation by Monte-Carlo simulation is becoming mainstream. This method is firstly realized as Virtual CMM (VCMM) by PTB[2]. However, VCMM also requires times and effort for the simulation, because it needs a lot of experimental data. In this report, very easy and simple uncertainty estimation method “Constrained Monte-Carlo Simulation (CMS) method” is proposed. This method is intended to estimate measurement uncertainty quickly with reasonable reliability and reduce user’s burden for it. Constrained Monte-Carlo Simulation In this method, Monte-Carlo simulation reflecting constraint caused by possible correlations in error is performed. The method consists of three steps. At first a covariance matrix of measurement errors is estimated. Secondly the matrix is decomposed into eigen vectors and eigen values. In the last step the eigen vectors are linearly coupled with random coupling coefficients, where variance of coefficients correspond to eigen values respectively, and we can obtain a trial measurement’s error. Derivation of the covariance matrix, the first step, is a key point of this method[3]. We may derive it from Machine’s specifications. For example, a specification of CMM is represented as a maximum permissible error in form of MPE [m] E a b l = + ⋅ , (1) where a and b are constant terms and l is measuring length. We consider the MPEE has a information of variance and covariance of the measurement. This means the machine potentially Key Engineering Materials Vols. 381-382 (2008) pp 587-590 online at http://www.scientific.net © (2008) Trans Tech Publications, Switzerland All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of the publisher: Trans Tech Publications Ltd, Switzerland, www.ttp.net. (ID: 125.54.119.145-29/04/08,01:58:54) shows error up to max a b l + ⋅ , where max l is maximum measurement range, but shorter the length than max l the error is reduced due to covariance effect. For ease of explanation, we consider one dimensional length measurement. Two coordinate values 1 x and 2 x result the size measurement 2 1 l x x = − . ( ) Var l , the variance of the size measurement, may be written as Eq. (2). Here, let the variance of 1 x and 2 x be independent of allocation of the coordinate values and the covariance be a function of l . Furthermore, assuming that when the length between them is longer than max l the covariance becomes 0, we have Eq. (3). 2 1 1 2 ( ) ( ) ( ) 2 ( , ) Var l Var x Var x Cov x x = + − 2 2 ( ) , x Var Cov l = − (2) max 1 ( ) , 2 x Var Var l = (3) Using Eqs. (2) and (3), ( ) Cov l is represented as Eq. (4). Here, let us simplify discussion by interpreting MPEE be comparable to 95% probability limit of normal distribution. Then, ( ) Var l may also be written as Eq. (5) from Eq. (1). { } max 1 ( ) ( ) ( ) , 2 Cov l Var l Var l = − (4) ( ) { }2 ( ) / 2 , Var l a b l = + ⋅ (5) From Eq. (4) covariance can be calculated as a function of length l . Suppose our measurement strategy have a series of n point coordinates, covariance matrix C with n n × dimension is filled. Having the variance and covariance information, a unique Monte-Carlo simulation fully reflecting the given statistical characteristics is performed. In the second step, the covariance matrix C is decomposed into the eigen vectors V and the corresponding eigen values D . In the last step, trial value $ x is obtained by recomposing with random number i ε as Eq. (7), where i ε is a random number satisfying ( ) 0 i E ε = and ( ) i i Var ε λ = . It utilizes technique of principal component analysis. This step is repeated up to desired number of times and same number of trial values are produced. , T = C VDV (6) where [ ] [ ] [ ] [ ] [ ] 1 1 , diag , 2 n n λ λ = = V v v v D L L 1 1 2 2 ˆ , n n ε ε ε = + + + x v v v L (7) Integrating Effect Caused by Probe In the previous chapter, we propose a procedure for deriving covariance matrix from MPEE . However, constant term of the equation includes probing directional error which would be represented as a function of azimuth angle φ and elevation angle θ of probing direction. We have to reflect the effect separately. 588 Measurement Technology and Intelligent Instruments VIII