被測物理量的測量不確定度

國

立

交

通

大

學

統計研究所

碩 士 論 文

被測物理量的測量不確定度

Measurement Uncertainty of Measurand

研 究 生：吳佩蓁

指導教授：陳鄰安 教授

被測物理量的測量不確定度

Measurement Uncertainty of Measurand

研 究 生：吳佩蓁 Student：Pei-Chen Wu

指導教授：陳鄰安 教授 Advisor：Dr. Lin-An Chen

國 立 交 通 大 學 理 學 院

統 計 研 究 所

碩 士 論 文

A Thesis

Submitted to Institute of Statistics

College of Science

National Chiao Tung University

in partial Fulfillment of the Requirements

for the Degree of Master in

Statistics

June 2009

被測物理量的測量不確定度

研究生: 吳佩蓁

指導教授: 陳鄰安教授

國立交通大學統計研究所

摘要

被測量物的不確定分析在量測科學上是一個重要的課題。然而，被測 量物的意義常被誤解，導致所求得的不確定性區間的意義造成混淆。被 測量物的真值應屬於參數，而測量結果應屬於隨機變數。通常，在測量 科學上估計參數會比預測變數的未來值有意義的多。古典不確定性分析 的理論都是將被測量物定義為隨機變數而發展的，但這會喪失不確定性 區間對於真值的準確性。我們在這篇文章中會討論被測量物的統計分 析。

Measurement Uncertainty of Measurand

Student: Pei-Chen Wu

Advisor: Dr. Lin-An Chen

Institute of Statistics

National Chiao Tung University

Hsinchu, Taiwan

Abstract

Uncertainty analysis of measurement of measurand is an important topic in metrology. However, vague statistical concept of measurand results in inefficient inference uncertainty for the true measurand. Measurand and the variable representing its measurement are completely different in probability concept; one is an unknown distributional parameter and the other is a random variable. Generally, a parameter may be estimated more efficiently than the prediction of the future observation of a random variable. The classical uncertainty analysis in literature is developed based on the structure that a measurand is a random variable. This misspecification of statistical model costs serious price of sacrificing efficiency in constructing uncertainty interval for gaining the knowledge of the true measurand. We formally formulate a statistical analysis for measurement of measurand.

致 謝

非常感謝陳鄰安教授不厭其煩及細心的指導，在此獻

上我最真心的感恩與祝福。老師對於教學的熱忱及真心對

待學生的態度深深影響我，我也期許自己能像老師一樣，

成為一個認真教學及教人的好老師。感謝所上的教授，老

師們專業的素養與親切的態度，讓我獲益匪淺。在此也感

謝洪慧念教授，蔡明田教授及吳柏林教授，在口試時提供

寶貴的見解。

謝謝我職場上工作夥伴們，由於他們在職務上的鼎力

相助，我才能在工作之餘完成碩士論文。

最後感謝我的家人，他們總是給我勇氣和動力，支持

我不斷往前進，感謝大家!

吳佩蓁

2009.06

Contents

Abstract (in Chinese) i

Abstract (in English) ii

Acknowledgements iii

Contents iv

1. Introduction 1

2. Uncertainty intervals for Random Measurand and Constant Measurand 4 3. Statistical Methods for Random Functional Measurand 8 4. Statistical Inferences for Statistical Model for Constant Measurand 11

5. Example 18 6. Conclusion 28 References 29

1. Introduction

An experiment for measuring the measurand, the quantity to be measured, is a method through a process that tries to gain or discover knowledge of the measurand. Measurements always have errors and therefore uncertainties. The practice of measurement science has made us realize that the comparisons of measured values require, in addition to the proper value, a statement of the reliability and quality of that value. General rules for evaluating and reporting uncertainty in measurement has been published by the most important and internationally widespread metrological publication-ISO (the International Standards Organization) Guide to the Expression of

Uncertainty in Measurement (GUM, 1993). According to the GUM, the measurement

result should be reported with a specified confidence as an uncertainty interval defining the range of values that could reasonably be attributed to the measurand. Unfortunately, uncertainty analysis of measurement is, in our opinion, totally inappropriate for missing the aim of gaining knowledge of real measurand due to the conceptual understanding of measurand.

Those conflicting views on statistical concepts of interpreting a measurand result in the inappropriateness. That is, a parameter and a random variable are misleadingly interchangeable used to represent the measurand. The efficiencies of predicting an unknown parameter and a random variable are remarkably different. In fact, it is more capable, from point of probability, in prediction of an unknown parameter than it of a future random variable. In GUM B.2.9, the measurand is defined as a particular quantity subject to measurement. On the other hand, GUM 3.3.1 also admits that measurand could have a true value. Baratto (2008) proposed a new but precise and comprehensive definition of measurand guiding that it is a specific quantity that one

intends to measure. The terms particularly and specially are to specify that a quantity is existed with restrictive conditions or assumptions. Hence, it is generally accepted that a measurand is an unknown constant to be predicted.

The uncertainty analysis becomes confusing for the fact that although a measurand is known as an unknown parameter, it is measurable. However, in classical statistical models, a parameter in the model is not measurable and is involved in the distribution of a measurable random variable. Hence, it is also generally accepted that the variable in measuring the unknown measurand be termed a measurand. Hence measurand or measurement quantity is used simultaneously to represent a parameter to be predicted and the variable for that its observations are used to predict the unknown measurand.

With the confusion, classically, the study of the uncertainty interval for measurement is based on a statistical model of random measurand. Let Y denotes the random measurand and predicted value yˆ computed from observations of its input quantities. ISO GUM proposes pooling estimated variance components for all sources of error with its square root, saying u_c(y) termed the combined uncertainty. It then

reports uncertainty interval in the form of expanded uncertainty, U, as

U y

Y = ˆ± (1.1) where U is termed as

U= k_pu_c(y) (1.2) and where kp is a coverage factor so that this uncertainty interval may cover the

distribution of the random measurand Y with a fixed confidence, saying 0.95. Is this type of uncertainty interval appropriate in gaining proper knowledge of uncertainty in prediction of the constant measurand? Or more specific, does an interval covering the

random measurand with probability 0.95 also cover the unknown constant measurand with the same probability?

For the common constant measurand, metrologists use different methods of measurement and analysis to define different random measurands. There are reported values and uncertainty intervals constructed from these different random measurands being communicated to other places and other times. These communications are not via the shared understanding or knowledge investigation of the common constant measurand, since each reported value and uncertainty interval is predictions of its corresponding random measurand determined by one method of measurement. The constant measurand should be the truth assumed to be unperturbed by variations in methods and instruments. Hence, in a course of discussion the perception of the method and analysis could differ from metrologist to metrologist, but they must talk about the same thing, the prediction of the measurand. Much information and knowledge must be lost if we use one reported value and uncertainty interval to explain the unknown constant measurand. Why shouldn’t the metrologists develop uncertainty interval to interpret the uncertainty of predicted value of the unknown measurand?

Here, in this paper, we want to express the other treatment of the measurand. Since the true value of it is a constant, we are supposed to look on it as a parameter. Then the uncertainty should be analyzed in terms of parameter. Of course, the outcomes are shorter and more meaningful than that of the random variable.

In section 2, we define the statistical model for random and constant measurand, and bring up an example to explain the difference of the uncertainty intervals between these two measurands. In section 3 and 4, we take input qualities into consideration. We define the statistical models for random and constant functional measurand,

respectively. Analyzing the uncertainty in different operations, such as addition, subtraction, multiplication and division, we find the variances of the random measurand and MSE’s of the constant measurand. In section 5, we carry out four examples. We rewrite the measurement function with constant measurand and find the uncertainty intervals. Besides, we also compare them with the intervals of random measurand method.

2. Uncertainty intervals for Random Measurand and Constant

Measurand

For the measurand, a particular quantity to be measured, its true value and the measurement of this true value are conceptually different in statistics. One is an unknown parameter and one is a random variable with a probability distribution and then their statistical inferences are with remarkably different efficiencies. We call the true value as the constant measurand and the measurement as the random measurand. It is supposed that we want to measure the amount of gas in a container. There is an unknown and fixed amount of gas contained in this specific container and it is the constant measurand. When we have made measurements several times with different results, the variable representing the measurement is the random measurand that also represents the amount of gas in this container. However, it is not a fixed number. We define statistical models for these two types of measurand.

In our opinion, the uncertainty analysis of the measurement of the constant measurand is more important for metrologists to analyze, not that of the random measurand. We study the uncertainty intervals for these two targets separately. The

simplest experiment of measurement is that we have a random measurand Y and we want to predict it with a random sampleY1,...,Yn.

Definition 2.1 The statistical model for random measurand includes:

(a) Random measurand: Y with distributionF_y,

(b) Probability model: Y1,...,Yn are random sample drawn from distributionFy.

For a random measurand Y with probability density function fy(y) , the aim in

developing uncertainty interval is to search an interval

( )

u,v , nonrandom or random, that satisfies

f_y(y)dy= 0.95 u

v

∫

. (2.1)

Unfortunately, the pdf fy is generally not (completely) known so that a

nonrandom uncertainty interval is not available. With the statistical model from random measurand, the observations y1,..., yn are used for computing uncertainty

interval for random measurand Y. This idea works for computing a random type or even approximate random type uncertainty interval.

The random measurand represents the measurement variable to measure the constant measurand. Next, we consider a model that deals with the true measurand value that has a sample of random measurand for prediction.

Definition 2.2. The statistical model for constant measurand includes:

(a) Constant measurand model: θy is an unknown parameter that is measurable,

(b) Probability model: Random variable Y measuring θ_y has distributionF_y, (c) Sampling model: There are random sample Y₁,...,Yn drawn from distributionFy.

The interest of uncertainty interval for constant measurand is to develop random interval

(

T₁,T₂

)

= t

(

₁(Y₁,...,Y_n),t₂(Y₁,...,Y_n)

)

such that

0.95= P_θ

y

{

T1≤θy ≤ T2

}

. (2.2)

This may be done by the classical statistical inferences of confidence interval.

Example 1. Suppose that there is a pencil on a table and we would like to measure its length. This pencil is the quantity to be measured. Along our definitions, the measured length is a random variable called the random measurand and the true length of this pencil is the constant measurand. It is supposed that we have random variables

Y₁,...,Y_n representing n measurements of the constant measurand (true length of the

pencil). We also assume that the instrument for measurement reveals that these random variables are independent and identically distributed with normal

distributionN

(

µ_y,σ2

)

. The best estimate of the random measurand Y is Y = 1

n _i=1Yi

n

∑

, and a 95% confidence interval for Y is Y

(

− t_0.025(n−1) ⋅ s,Y + t_0.025(n−1)⋅ s

)

where S is the sample standard deviation with S2= 1

n−1

(

Yi−Y

)

2

i=1 n

∑

and t_0.025(n−1) is the 0.975 quantile of the t-distribution T(n−1) where n−1 is degree of freedom. Suppose that we have a sequence of 5 measurements (mm) as

41.12, 41.08, 41.10, 41.14, 41.06.

These observations are the sample realization of the random measurand. The average of these measurements is y = 41.10mm and sample standard deviation is s =

0.032mm. The uncertainty interval for the random measurand is

mm mm mm mm 2.776 0.032 41.10 0.089 10 . 41 ± × = ± .

This uncertainty interval indicates that the “next” realization of random measurand Y will between 41.011mm and 41.189mm with probability 0.95. This is not a direct

connection to the true length of the pencil. Isn’t it weird?

The constant measurand θ_y represents the true length of the pencil. With normal assumption, 100 1−

₍

α

₎

% confidence interval for θ_y is

n s t

y± _α2 . Hence, a 95% uncertainty interval for the constant measurand θ_y is

n s t y 0.025 ˆ _± θ . In this case, it is mm mm mm mm 41.10 0.040 5 032 . 0 776 . 2 10 . 41 ± × = ± .

The uncertainty intervals of random measurand and constant measurand are with the same center point yˆ =θˆy =41.10mm. However, the expanded uncertainty for the

constant measurand is 0.040mm, which is significant smaller than 0.089mm, the

expanded uncertainty for the random measurand. This uncertainty interval indicates that we have 95% confidence with true length of the pencil to be between 41.06mm

and 41. 14mm.

In this example, it is obviously that the primary interest is the true length of the pencil on desk, not the next measurement of the length. If we study the random measurand, we would stray from the main purpose. Therefore, we should be clear about what we are concerned.

3. Statistical Methods for Random Functional Measurand

The GUM was also developed under the assumption that the random measurand Y can not be measured directly, but is determined from several input (influence) qualities (also random variables) X1,..., Xk through a known functional relation as

Y = h X

(

₁,..., X_k

)

(3.1) where variables X_j’s are measurements of some other qualities. Any measurement for quantity Xj is subject to errors such as offset of a measuring instrument, drift in

its characteristics, and personal bias in reading. This random effect shows the variation in repeated measurements. Hence, this measurement function represents a relationship for measturement variable not only a physical law but also a measurement process.

It is assume that there are resultsXji,i=1,...,nj, a random sample drawn from the

distribution of variable X_j, that may be observed during the jth experiment. What have been done in literature in dealing with random measurand?

Definition 3.1. The statistical model for random functional measurand includes: (a) Measurement variables for measurand: Y = h X

(

1,..., Xk

)

,

(b) Probability model: X1,..., Xk are input quantities (variables) with joint distribution

function F_1,...,k

(

x₁,..., x_k

)

,

(c) Sampling model: For each j, j=1,...,k, X_j1,..., X_jn

jis a random sample

corresponding with random variable X_j.

the prediction estimate, classically it is the sample mean x _j, of variable X_j from the observations x_ji,i=1,...,n_j. The prediction of future random measurand is

yˆ =h

(

xˆ1,...,xˆk

)

. (3.2)

This provides a predictor of future observation of the random measurand Y, not an estimate of the unknown true value of the constant measurand. It is not complete to provide a predictor of Y without an indication of precision. This classical way in developing the uncertainty interval of predictor yˆ is stated below. Let’s denote Var(X_j)=σ2_j

, j=1,...,k . In the construction of uncertainty interval for the random measurand Y, it is generally assumed that xˆ is the expected value of the _j

distribution of input variable X_j, so that σ2_j =E[(X_j−xˆ_j)2] and this hold for all j’s. With predicted valueyˆ =h

(

xˆ₁,...,xˆ_k

)

, the first-order Taylor series approximation to the measurement variable Y about the estimates

(

xˆ1,...,xˆk

)

gives

∑

(

)

= − + ≈ k j j j j X x b y Y 1 ˆ ˆ (3.3) where _{x x h k} j k j _{h h} X x x h b ˆ _{, 1,...,} 1,..., ) ( = = = ∂ ∂

, called the uncertainty coefficient with respect to

influence quantity Xj. The combined standard uncertainty of the random measurand is

defined as the square root of its variance, which is approximated as

jl l k j j l j j j y b σ bbσ σ

∑

∑

= ≠ + ≈ 1 2 2 2 (3.4) where σjl = Cov(Xj, Xl) .

The uncertainty interval is defined as

) ( ˆ k u y y

with u_c(y)=σ_y and k_p is the coverage factor so that this uncertainty interval may cover the possible values of random measurand Y with a fixed probability, saying 1−α . Interpreted by Willink (2006), in a potential series of equally reliable independently-determined intervals, this uncertainty interval encloses the value of the random measurand Y, on an average, in 100(1−α) out of every 100 measurements.

Uncertainty in Sum and Differences

Suppose that we have the measurement variable as

k X X

Y = ₁+...+ .

The uncertainty of the random measurand is the square root of its variance. The variance is

∑

∑

= ≠ + = + + = k j j l l j j k

y Var X X Var X Cov X X

1 1 2 ) , ( ) ( ) ... ( σ

where Cov(X_j,X_l)=E[(X_j −xˆ_j)(X_l−xˆ_l)],j≠l,j,l=1,...,k , representing the covariance of variables.

Now, suppose that we have the measurement variable as

Y = X1+ ...+ Xk− (Z1+ ...+ Zm).

The variance of the random measurand is

∑

∑

∑

∑

∑

= = ≠ = ≠ = = ≠ = − + + + = m l k j l j l j m l j l j l j k j j l jl k l j m j j j y Z X Cov Z Z Cov X X Cov Z Var X Var ,..., 1 , ,..., 1 , ,..., 1 , , 1 1 , , 1,..., 2 ) , ( ) , ( ) , ( ) ( ) ( σ .

Uncertainty in Multiplication and Division

Y = X₁...X_k. The variance of the measurement variable is

) , ( ) ( 2 1 2 l j l j j l j k j j y Cov X X X X Y X Var X Y

∑

∑

≠ =         +         = σ .

Suppose that we have the measurement variable as

Y = 1

X₁...X_k.

Then, we have variance as

∑

∑

≠ =         +         − = l j l j l j k j j j y Cov X X X X Y X Var X Y ) ( ) ( 1 2 2 σ .

Suppose that we have the true measurand as

Y = X1...Xk

X_k+1...X_w.

Then, we have variance as

∑

∑

∑

∑

+ = = ≠ + = ≠ = ≠ =         − +               + +         = w k l k j l j l j l j l j l j w k l j l j k l j l j w j j j y X X Cov X X Y X X Cov X X Y X Var X Y ,..., 1 , ,..., 1 , ,..., 1 , , ,..., 1 , , 1 2 2 ) , ( ) , ( ) ( σ .

4. Statistical Inferences for Statistical Model for Constant

Measurand

For measurement of the unknown measurand, there are input variables X1,..., Xk

unknown parameters θ1,...,θk in their corresponding parameter spacesΘj’s, such that

the unknown measurand θ_y may be formulated as

θ_y = h

(

θ₁,...,θ_k

)

,θ_j ∈ Θ_j, j= 1,...,k (4.1) where h is a known specified function. Unknown parameter θ_y is the true value of the measurand, which is the target to be estimated. Input variables are measurements for Θ_j’s. They are random variables since the measurements are subject to measurement errors. This formulation serves direct way in studying estimation and uncertainty analysis for estimation of the constant measurand.

Definition 4.1 The statistical model for constant functional measurand includes: (a) Constant measurand model: θ_y = h

(

θ₁,...,θ_k

)

,θ_j ∈ Θ_j , where θ₁,...,θ_k are

measurable,

(b) Probability model: X1,..., Xk are input quantities representing the measurement

variables, respectively, for parameters θ₁,...,θ_k with joint distribution

F_1,...,k

(

x₁,..., x_k

)

,

(c) Sampling model: For each j , j=1,...,k, X_j1,..., X_jn

j is a random sample drawn

from distribution F_1,...,k

(

x₁,...,x_k

)

.

This statistical model needs to be further clarified as follows:

(1) In the measurand model, there exist true parameter values θ10,...,θk 0 such that the

true measurand θ_y isθ_{y 0}= h

(

θ₁₀,...,θ_{k 0}

)

. So, in the statistical inferences, we aim to predict trueθ1,...,θk. If there are efficient inferences for these unknown parameters,

classical statistical model. In the classical statistical models, unknown parameter is not measurable; but in this statistical model for the measurand, the parameters

θ₁,...,θ_k and even θ_y are measurable.

(3) The relationship between θ1,...,θk and joint distribution function F1,...,k

(

x1,...,xk

)

has to be practically investigated and specified.

The statistical model specifies the information for establishing theory of statistical inference procedures for measurandθ_y. Practically there are sample realizations

xj1,..., xjn_j

{

}

, j=1,...,k for drawing inference conclusions. What has been done in literature to deal withθ_y? And what has been done to deal with measurand θ_y by the use of these samples Xji,i=1,...,nj; j=1,...,k ?

The inference results depend crucially on the correct relation between parameters

θ1,...,θk and joint distribution functionF1,...,k

(

x1,..., xk

)

. Hence, without involving the

relation in the construction of inference techniques, any technique in influences of measurandθ_y, such as those proposed in literature, based on realization of random samples may provides very biased conclusions.

Let

(

)

j jn j j j ˆ x ,...,x ˆ 1 θ

θ = be appropriate estimate of θ_j based on observations

{

xji,i= 1,...,nj

}

, for j= 1,...,k . The point estimate of the unknown

measurand θy may then be constructed by estimate ofθj’s.

Definition 4.2 By letting

(

X X

)

j k j jn j j j ˆ ,..., , 1,..., ˆ 1 = =θ

θ , the point estimator of θy is

θˆy =h

(

θˆ1,...,θˆk

)

. (4.1) If we replace

(

)

j jn j j j ˆ X ,...,X ˆ 1 θ

θ = by its sample realization

(

)

j jn j j j ˆ x ,...,x ˆ 1 θ θ = ,

then this θˆ is called the point estimate of measurandy θy.

Two comments are needed to clarify the measurand estimator:

(a) The estimator θˆ is for estimation of parameter_y θ_y. On the other hand, the classical estimator h

(

Xˆ₁,...,Xˆ_k

)

represents a prediction of future variable Y. Suppose that we let Xˆ_j =θˆ_j for allj ’s. Then estimate θˆ and predictor yˆ are _y

identical. However, their roles are different, one is predicting a future observation and one is estimated value of a parameter and then their uncertainty intervals are also different.

(b) In the attempt of gaining knowledge of the true measurandθ_y, an estimate θˆ _y is incomplete without an indicator of its precision since there is no confidence that we can say for θˆ to be equal to_y θ_y. Uncertainty interval for estimate of θ_y is an appropriate choice to explain the confidence of possible values ofθ_y. The mean square error (MSE) of estimator θˆ is y

MSEθ_y =E_

(

θˆy−θy

)

2_. (4.2)

In this constant measurand statistical model, the uncertainty in the result of constant measurand estimator θˆ is an estimate of the MSE of_y θˆ . Whenever _y θˆ is unbiased _y for true value θ_y of the measurand, the MSE of θˆ is equal to the variance of _y θˆ as _y

2

y y

MSE_θ =σ_θ . Under statistical model for measurand, the first order Taylor’s expansion for measurand function h on estimators

(

θˆ,...,θˆ

)

yields

(

)

∑

= + − + = k j j j j y y y R c 1 ˆ ˆ ˆ θ θ θ θ θ (4.3) where we let

(

)

_{i k} j k j i i h c ˆ _{, 1,...,} 1 ˆ ˆ ,..., ˆ = = = _{θ θ} θ ∂ θ θ ∂

, the sensitivity coefficient with respect to

influence parameter θ_j, and

y

R_θˆ is the remainder expressed by

(

)

₍

₎

(

)

₍

₎₍

₎

    − − +     − = − + = ≠ − + = =

∑

∑

l l j j l j j l k j j k j j k i i i i i i i i y h h R θ θ θ θ θ ∂ θ ∂ θ θ ∂ θ θ θ ∂ θ θ ∂ θ θ δ θ θ θ θ δ θ θ θ ˆ ˆ ˆ ˆ ˆ ,..., ˆ ˆ ˆ ˆ ,..., ˆ 2 1 ) ˆ ( ˆ 1 2 2 ) ˆ ( ˆ 1 2 1 2 ˆ

with0<δ< 1. When θˆ_i →θ_i, for all i, the remainder term approaches zero more

quickly than the first terms in (4.3) and all the higher terms are generally neglected, provided that the uncertainties in θˆ_i’s are small and θˆ_i's are, respectively, close to θi’s.

The MSE can be substituted into equation (4.3) to yield

(

)

(

)

(

)(

)

jl j y MSE c c MSE c c c c E c E MSE l j l j k j j l l j j l j l j j j k j j j j k j j θ θ θ θ θ θ θ θ θ θ θ

∑

∑

∑

∑

∑

≠ = ≠ = = + =       − − + − =               − ≈ 1 2 2 1 2 2 1 ˆ ˆ ˆ ˆ (4.4)

where MSEθ_j = E_

(

θˆj −θj

)

2_ and, MSEθ_jl =E[(θˆj −θj)(θˆl −θl)]called the co-mean

square error between estimates θ_j and θ_l which is not necessary to be nonnegative. This formulation introduces the uncertainty of estimateθˆ , as a linear combination of _y

MSE’s and co-MSE’s associated with parameter estimatesθˆ ’s. The expected values j

are considered to be the best estimate for parameters.

There are several comments on this decomposition of MSE of estimator of constant measurand:

(a) MSE_θ

j is the MSE of the constant measurand estimate contributed by the estimate

of parameter θj.

(b) MSE_θ

jl is the co-mean square error associated with estimates of parameters θj

and θl , and that contributes the MSE of the constant measurand estimate.

(c) From (4.4), the standard uncertainty of the estimate, θˆ , of the measurand that is _y attributed to the input quantity parameter estimates is a function of the estimated MSE_θ

j’s and their estimates ofMSEθjl’s.

Uncertainty in Sum and Differences

Suppose that we have the true meaurand as θy=θ1+ ...+θk.

We also have their estimator asθˆ_j,j=1,...,k. We then have

MSE_θ y = MSEθj j=1 k

∑

+ B_θ jBθl j≠l

∑

whereB E _{j j} j k j ], 1,..., ˆ [ − = = θ θ

θ , representing the bias of estimators θˆ ’s. j

Now, if we have true measurand as

θy=θ1+ ...+θk− (δ1+ ...+δm)

MSE_θ y = MSEθj j=1 k

∑

+ MSE_δ j j=1 m

∑

+ B_θ jBθl j≠l

∑

+ B_δ jBδl j≠l

∑

− B_θ jBδl j≠l

∑

.

Uncertainty in Multiplication and Division

Suppose that we have the true measurand as θ_y =θ₁...θ_k. Then, we have MSE_θ y = θ_y θ_j         2 MSE_θ j j=1 k

∑

+ θy θ_jθ_l BθjBθl j≠l

∑

.

Suppose that we have the true measurand as θy = 1 θ1...θk . Then, we have MSE_θ y = θ_y θ_j         2 MSE_θ j j=1 k

∑

+ θy θ_jθ_l BθjBθl j≠l

∑

.

Suppose that we have the true measurand as θy= θ1...θk θk+1...θw . Then, we have MSE_θ y = θ_y θ_j         2 MSE_θ j j=1 w

∑

+ j≠l, j,l=1,...,k

∑

+ j≠l, j,l= k+1,...,w

∑

        θ_y θ_jθ_l BθjBθl + −θy θjθl B_θ jBθl j≠l, j=1,...,k,l= k +1,...,w

∑

.

Here we list a table to compare the constant and random functional measurands.

5. Example

The measurement of Y is generally assumed to follow normal or t distribution in GUM. However, Sim and Lim (2008) claim that Y actually follows asymmetric distribution, not just normal or t distribution. The random measurand can be stated

(

)

_∑

(

)

∑

− = + −

+

≈ k k

MSE of the constant measurand Variance of the random variable

θ_y=θ₁+ ...+θ_k MSE_θ y = MSEθj j=1 k

∑

+ B_θ jBθl j≠l

∑

∑

∑

= ≠ + = k j j l l j j y Var X Cov X X 1 2 ) , ( ) ( σ Uncertainty in Sum and Differences ) ... ( ... 1 1 m k y δ δ θ θ θ + + − + + = _MSE θy= MSEθj j=1 k

∑

+ MSEδj j=1 m

∑

+ BθjBθl j≠l

∑

+ BδjBδl j≠l

∑

− BθjBδl j≠l

∑

∑

∑

∑

∑

∑

= = ≠ = ≠ = ≠ = = − + + + = m l k j l j l j m l j l j l j k l j l j l j k j m j j j y Z X Cov Z Z Cov X X Cov Z Var X Var ,..., 1 , ,..., 1 , ,..., 1 , , ,..., 1 , , 1 1 2 ) , ( ) , ( ) , ( ) ( ) ( σ θ_y =θ₁...θ_k MSE_θ y = θy θj         2 MSE_θ j j=1 k

∑

+ θy θjθl B_θ jBθl j≠l

∑

( ) ( , ) 2 1 2 l j l j j l j k j j y Cov X X X X Y X Var X Y ∑ ∑ ≠ =         +         = σ θ_y = 1 θ1...θk MSE_θ y = θy θj         2 MSE_θ j j=1 k

∑

+ θy θjθl B_θ jBθl j≠l

∑

∑ ∑ ≠ =         +         − = l j l j l j k j j j y Cov X X X X Y X Var X Y ) ( ) ( 1 2 2 σ Uncertainty in Multiplicati on and Division θ_y = θ1...θk θ_k+1...θ_w

∑

∑

∑

∑

+ = = ≠ + = ≠ = ≠ = − +       + +         = w k l k j l j j l y l j y w k l j l j k l j l j w j j y l j l j j y B B B B MSE MSE ,..., 1 , ,..., 1 , ,..., 1 , , ,..., 1 , , 1 2 θ θ θ θ θ θ θ θ θ θ θ θ θ θ

∑

∑

∑

∑

+ = = ≠ + = ≠ = ≠ =         − +               + +         = w k l k j l j l j l j l j l j w k l j l j k l j l j w j j j y X X Cov X X Y X X Cov X X Y X Var X Y ,..., 1 , ,..., 1 , ,..., 1 , , ,..., 1 , , 1 2 2 ) , ( ) , ( ) ( σ

value of Xj. The distribution of

∑

(

)

= − = − k j j j j X x b y Y 1

ˆ is not always normal, if the

distribution of Xj may not be normal. Hence, Sim and Lim consider other

distributions and the coverage factors corresponding to different distribution. The uncertainty interval is Y = y ± kpuc(Y ) . The coverage factor, kp, can be obtained by

the coefficient of skewness and kurtosis in these distributions. Meanwhile, the central moments of random variable are involved to identify those two coefficients. So, Sim and Lim focus on coverage factors, corresponding to four asymmetric distributions: the Pearson family of distribution, the Tukey’s lambda- distribution, the Tukey’s gh-distribution, and the GS-distribution.

Here, we focus on the difference between parameter and variable, which lead to different standard uncertainties and uncertainty intervals. In the following examples, we’ll express the measurands as the constant ones, if they are actually constants, and express the constant measurand function by first-order Taylor expansion. We assume that the estimates are unbiased for true value of measurand. Therefore, with MSE of the estimate of the measurands, we can obtain a 95% uncertainty interval for the constant measurand.

Example 1.

The first example is drawn from Sim and Lim (2008), and Willink (2006). The measurand Y is the velocity of a type of wave in some medium and measurement function is

Y = X1

where X₁ is the distance from a transmitter to a receiver and X₂ is the time of flight. However, this measurand, velocity of a wave, is a particular quantity to be measured. The distance between a transmitter and a receiver should be fixed, and it is a constant to be measured. Besides, the time of the flight of the wave is a constant as well. Therefore, both can be expressed as parameter. We use θ and ₁ θ to replace ₂ X₁ and

X2 respectively, and θyreplace Y. So, the measurement function is

θy =

θ1

θ2

.

The first-order Taylor expansion of measurand function is

(

)

2

(

2 2

)

2 1 1 1 2 2 1 2 1 1 ˆ ˆ ˆ ˆ ˆ _{θ θ} θ θ θ θ θ θ θ θ θ θy = = + − − − .

It is supposed that θˆ1− is distributed as−2 ×10θ1 −4_θ

1V1, where V1 follows a chi-square

distribution; θˆ2− is distributed asθ2 U2V2, where U2follows a uniform distribution

U[4.5×10−6,5.5×10−6] and V₂ follows an exponential distribution with mean as 1. θ₁

is taken as 4.931 mm, and θ2 as 10.9×10 −3

s. The expected value is

(

)

2

(

2 2

)

2 1 1 1 2 2 1 1 ˆ ˆ ) ˆ ( θ θ θ θ θ θ θ θ θ θ = + E − − E − E _y while

(

θˆ₁−θ₁

)

=−2×10−4θ₁⋅E(V₁)=−2×10−4θ₁⋅1 E

(

)

1 2 10 5 . 5 10 5 . 4 ) ( ) ( ˆ 6 6 2 2 2 2 ⋅      × + × = ⋅ = − E U EV − − Eθ θ .

Hence, 087E(θˆ_y)=452. which is close to θ1

MSE_θ y = 1 θ2 2 MSEθ1 + θ₁ θ2 2       2 MSE_θ 2.

The MSE’s are:

2 1 4 2 ˆ ) 10 2 ( 2 1 1 θ σ_θ θ = = ⋅ − × − n n MSE .       + + = = 12 2 5 5 1 2 2 2 ˆ 2 2 αβ β α σ_θ θ n n MSE , where α = 4.5 ×10−6,β= 5.5 ×10−6.

Hence, a 95% uncertainty interval for the constant measurand θ_y is

      + − y y z MSE MSE z _{θ θ} θ θ θ θ 025 . 0 2 1 025 . 0 2 1 ˆ ˆ , ˆ ˆ .

Since sample size is needed in the uncertainty interval, we assume that n = 1, 5, 10, 30 and 50. The uncertainty interval of the method for constant measurand is named as UM; that of the method for random measurand is named as USL . In the table below is

the comparison of USL from Sim and Lim (2008) and UM corresponding to different

sample size. Methods for uncertainty interval for random measurand USL, 95% uncertainty interval (n is not involved) Methods for uncertainty interval for constant measurand UM, 95% uncertainty interval (n is involved) GUF (GUM uncertainty framework) (451.91, 452.86) n=1 (451.9063, 452.8643) Pearson family of distribution (451.96, 452.93) n=5 (452.1711, 452.5995) Tukey’s lambda-distribution (451.95, 452.93) n=10 (452.2339, 452.5368)

Tukey’s gh-distribution (451.96, 452.93) n=30 (452.2979, 452.4728) GS-distribution (Generalized S-distribution) (451.95, 452.93) n=50 (452. 3176, 452.4531) MCM (Monte Carlo method) (452.06, 453.09)

In left columns shows few different methods of uncertainty intervals for random measurand. Besides GUM framework, Sim and Lim have tried five other methods. There are only slight differences between uncertainty intervals from five other methods. In the right columns shows the uncertainty interval for constant measurand. We have some finding as below. Firstly, when rounding to hundredth digit, UM, with n as 1, is the same as USL of GUF. This clearly indicates that, the

sample size is not taken into consideration in Sim and Lim’s method. Secondly, in our method, the 95% uncertainty interval for constant measurand θ_y is shorter than the results of Sim and Lim.

Example 2.

Another example is from Sim and Lim (2008) and Willink (2006). The measurand Y is an intensity measured by a circular sensor from a simple measurement function

Y =exp(X1)

X₂2

where X1 is the ideal reading of the sensor which follows a normal N (1.044, 0.0225)

distribution and X2 is the diameter of the sensor that follows a uniform U (0.57, 0.61)

one, and it should be a constant. The diameter of the sensor is fixed, and it’s a constant as well. Since the reading of the sensor and the diameter of the sensor are particular quantities that can be measured. So, we use θ₁ and θ₂to replace X1 and X2

respectively, and θyto replace Y. So, the measurement function is

θ_y=exp(θ1)

θ₂2 .

The first-order Taylor expansion of measurand function is

(

)

3

(

2 2

)

2 1 1 2 2 2 2 2 2 ˆ ˆ 2 ˆ ˆ 1 1 1 1 θ θ θ θ θ θ θ θ θ θ θ θ − − − + = e e e e .

The MSE of the estimate of measurand is

MSE_θ y = eθ1 θ₂2       2 MSE_θ 1 + 2 eθ1 θ₂3       2 MSE_θ 2 where MSE_θ 1 = σ_θ 1 2 n and MSEθ2 = σ_θ 2 2 n . We can obtain σθ1 2 =0.0225, and σ_θ 2 2 =

(

0.61− 0.57

)

2

12 from the given distribution. The question doesn’t indicate the value for θ1 and θ2. So, we assume they are unbiased, and estimate them with their

expected value. We take θˆ1 as E(θ1)=1.044 and θˆ2 as E(θ2)=

1.044+ 0.0225

2 = 0.59.

Hence, a 95% uncertainty interval for the constant measurand θ_y is

        + − y y z MSE e MSE z e θ θ θ θ θ θ 2 0.025 2 ˆ 025 . 0 2 2 ˆ ˆ , ˆ 1 1 .

The 95% uncertainty interval for the measurand, [6.001, 11.025], is obtained from Wiilink(2006). In the table below is the comparison of USL from Sim and Lim (2008)

and UM corresponding to different sample size n = 1, 5, 10, 30 and 50

Methods for uncertainty interval for random measurand USL, 95% uncertainty interval (n is not involved) Methods for uncertainty interval for constant measurand UM, 95% uncertainty interval (n is involved) GUF (GUM uncertainty framework) (5.7143, 10.7437) n=1 ( 5.680745, 10.63960) Pearson family of distribution (6.0020, 11.0242) n=5 (7.05134, 9.269008) Tukey’s lambda-distribution (5.9948, 11.0111) n=10 (7.37611, 8.944238) Tukey’s gh-distribution (6.0016, 11.0164) n=30 (7.707494, 8.612854) GS-distribution (Generalized S-distribution) (5.9835, 11.0143) n=50 ( 7.80953, 8.510818) MCM (Monte Carlo method) (5.9980, 11.0210)

In the left columns are uncertainty intervals of random variables. Among them, USL of Tukey’s lambda-distribution and USL of Pearson family are shorter than others.

In the right columns is the uncertainty interval of constant measurand method. Obviously, the 95% uncertainty interval for constant measurand θy is shorter than the

Example 3.

An example comes from Semyon Rabinovich (1995). The current I, generated by γ rays from standards of unit radium mass, is defined by the expression

τ

CU I = . C is

the capacitance of the capacitor which helps the ionization current compensate, U is the initial voltage on the capacitor, and τ is the compensation time (seconds). The measurement is performed by making 27 repeated observations in Semyon Rabinovich (1995), with certain fixed conditions: C =4006.3 pF and U=7V is established, and compensation time is measured. Under same circumstances, the compensation time should be fixed, and it should be a constant. The current, the measurand, generated by γ rays should be constant as well. So, we replace τ withθ_τ, and I with θI. The formula of the current is expressed asθI =

CU θ_τ . The first-order Taylor expansion of measurand function is

(

τ τ

)

τ τ τ θ θ θ θ θ θ _ −      − + = = ˆ ˆ ˆ 2 CU CU CU I .

The MSE of the estimated measurand is

MSE_θ I = − CU θ_τ2       2 MSE_θτ where 2 2 89 0.00039041 s n MSE = τ = τ θ θ σ , 2 τ θ σ =0.01054131s , and 2 τ = 74.41481 s as

best estimate for θ_τ. So, MSE 0.01001328 10-24A2

I = ×

θ .

(

)

A MSE z CU MSE z CU I I 12 12 025 . 0 025 . 0 , _ˆ 376.6657 10 ,377.0579 10 ˆ − − _× × =       + − _θ τ θ τ θ θ =

(

3.766657×10−10,3.770579×10−10

)

A.

The 95% uncertainty interval from Semyon Rabinovich (1995) is

(

)

A

I =(3.761±0.009)×10−10 = 3.752×10−10,3.77×10−10 .

Example 4.

An example from Semyon Rabinovich (1995). The density of a solid body is

measured by

V m

=

ρ , where m is the mass of the body and V is the volume of the body. Under the same circumstances, the mass and the volume of the solid body wouldn’t be changed no matter how many times we measured. They are both particular quantities to be measured, so they are constant measurand. We should replace m with θm, v with θv and ρ with θρ, in order to show them in terms of parameter. Then we

express the formula as

v m

θ θ

θ_ρ = . Please see the 11 repeated measurements in Semyon

Rabinovich (1995). We haveθm = 252.9120 ×10

−3

kg, andθv=195.3798 ×10

−6

m3.

The first-order Taylor expansion of the measurand function is

(

)

(

v v

)

v m m m v v m v m θ θ θ θ θ θ θ θ θ θ θ θ_{ρ } −      − + −       + = = 1 ˆ ˆ ˆ ˆ ˆ 2 .

MSE_θρ = ∂θρ ∂θm       2 MSE_θ m + ∂θ_ρ ∂θv       2 MSE_θ v = 1 θ_v       2 MSE_θ m + − θm θv 2       2 MSE_θ v

where θ_m and θ_vare regarded as the best estimate of θ_m and θ_v respectively.

MSE_θ m = σ_θ m 2 n , 2 -12 2 10 2.130545 kg m = × θ σ and MSE_θ v = σ_θ v 2 n , 6 -18 2 10 1.802727 m v = × θ σ , and 3 3 10 1.294463 ˆ ˆ ˆ m kg v m = × = θ θ θ_ρ .

Therefore, a 95% uncertainty interval for the constant measurand θ_ρis

3 3 3 025 . 0 025 . 0 m kg ) 10 1.294470 , 10 (1.294456 ˆ ˆ , ˆ ˆ × × =       + − ρ ρ θ θ _θ θ θ θ MSE z MSE z v m v m _.

In our method, expanded uncertainty isz_0.025 MSE_θ

ρ = 0.007 = 7 ×10 −3kg

m3.

Now, let’s compare the result with those in Semyon Rabinovich (1995). In this case, Semyon Rabinovich tried two different ways to find the expanded uncertainty. One is under student t distribution with degree of freedom as 10: the relative error is6.0×10-4%, which means the expanded uncertainty is

3 3 -3 4 -10 7.76678 10 1.294463 % 10 6.0 m kg × = × × × .

The other is under student t distribution with the effective estimate of the degree of freedom, 19v_eff = : the relative error is5.7×10-4%, which means the expanded uncertainty is 3 3 -3 4 -10 7.37844 10 1.294463 % 10 7 . 5 m kg × = × × × .

With the method of constant measurand, the expanded uncertainty 7×10−3 is smaller than the other two. So, the uncertainty interval of the measurand is shorter. With this method, we can provide shorter and more meaningful interval. Why don’t we use it?

6. Conclusion

When the measurand is an unknown, but measurable constant, it should be regarded as a parameter. We consider the true value of the measurand. With first-Taylor expansion and Central limit theorem, we have MSE’s of the input quantities, and obtain an uncertainty interval for the constant measurand, which covers the true value of the measurand.

In this paper, we clarify the misinterpretation of the constant and variable measurand. Before analyzing uncertainty, we should know what we are concerned? The prediction of the variable or the estimation of the parameter? If we are dealing with an unknown parameter, then we have a more efficient way to analyze the true value. In our method, the uncertainty interval of the true value is shorter than that of the random measurand method.

However, criticizing others is not what we want to do; we just have different idea about the treatment of the measurand. We just provide a meaningful and thoughtful concept about the measurand. Hope this will be helpful to get more precise knowledge about the quantity to be measured.

References

Baratto (2008). Measruand: a cornerstone concept in metrology. Metrologia, 45, 299- 307.

BIPM, IEC, IFCC, IUPAC, IUPAP and OIML (1993). Guide to the Expression of

Uncertainty in Measurement. Geneva, Switzerland: International Organization for

Standardization.

C.H. Sim and M.H. Lim (2008). Evaluating expanded uncertainty in measurement with a fitted distribution. Metrologia, 45, 178-184

Gaetano Inculano, Andrea Zanobini, Annaarita Lazzari, and Gabriella Pellegrini Gualtieri (2003). Measurement uncertainty in a multivariate model: a novel approach. IEEE transactions on instrumentation and measurement, 52, 1573-1580. John R.Taylor(1982). An instruction to error analysis. University science book.

Semyon Rabinovich (1995). Measurement errors: theory and practice. American institute of Physics.

Patrick F. Dunn (2005). Measurement and data analysis for engineering and science. McGraw-Hill.

Willink, R (2006). Uncertainty analysis by moments for asymmetric variables.