GB/T 27419-2018Guide to the evaluation and expression of uncertainty in measurement -- Supplement 1: Propagation of distributions using a Monte Carlo method (English PDF)

Foreword

This standard was drafted in accordance with the rules given in GB/T 1.1-2009.

This standard uses the translation method equivalent to ISO /IEC Guide 98-3/Suppl.1.2008 "Measurement uncertainty - Part 3. Measurement

Guide to Uncertainty Representation (GUM..1995) Supplementary Document 1. Distribution Propagation Based on the Monte Carlo Method.

This standard has the following editorial changes compared to ISO /IEC Guide 98-3/Suppl.1.2008.

--- Modified the standard name.

This standard is proposed and managed by the National Certification and Accreditation Standardization Technical Committee (SAC/TC261).

This standard was drafted. China Metrology University, China Institute of Metrology, Zhejiang Institute of Metrology, Shanghai Metrology and Testing

Research Institute, Guangzhou Institute of Metrology and Measurement Technology, Hangzhou Quality and Technology Supervision and Inspection Institute, Binzhou College, Shaanxi Institute of Metrology, China

Measurement (Beijing) Institute of Inspection and Testing.

Introduction

1 Scope

This standard provides a general numerical method for the measurement uncertainty assessment, which is consistent with the GUM general principle.

[ISO /IEC Guide 98-3.2008, G.1.5]. Suitable for multiple inputs and single outputs that can be characterized by specific PDFs

Volume model [ISO /IEC Guide 98-3.2008, G.1.4, G.5.3].

As in GUM, this standard mainly deals with physically defined quantities—that is, measured values that can be characterized by unique values.

Representation of uncertainty [ISO /IEC Guide 98-3..2008, 1.2].

This standard is also unsatisfied or unable to determine whether the GUM uncertainty framework condition is met [ISO /IEC Guide 98-3.2008,

G.6.6] provides guidance on the uncertainty assessment. It can be used to make the GUM uncertainty framework difficult to apply, such as the complexity of the model.

Case. This standard gives a method guide for computer execution.

This standard obtains a PDF of the output and determines the following parameters.

a) an estimate of the output;

b) the standard uncertainty of the estimate;

c) The inclusion interval of the output corresponding to the given inclusion probability.

Known (i) the relationship model between input and output, (i) input quantity PDFs, then the output has a unique PDF. Usually output

The amount of PDF cannot be determined analytically. Therefore, the purpose of this method in this standard is to. without introducing a non-quantitative approximation,

The above a), b) and c) are determined within the specified numerical tolerances.

For a given inclusion probability, this standard can be used to determine the corresponding inclusion interval, including the interval of the probability symmetry and the shortest

Contains intervals.

This standard applies to two types of inputs, one of which is independent of each other and can be characterized by an appropriate PDF;

The other type is that the inputs are not independent of each other, ie some or all of these inputs can be characterized by a joint PDF.

This standard applies to the uncertainty of assessing the following typical conditions.

--- The magnitude of each uncertainty component is not similar [ISO /IEC Guide 98-3.2008, G.2.2];

--- When applying the uncertainty propagation law, it is difficult or inconvenient to calculate the partial derivative of the model [ISO /IEC Guide 98-3.2008, Chapter 5];

--- The output of the PDF is not a normal distribution or t distribution [ISO /IEC Guide98-3.2008, G.6.5];

--- The estimated value of the output is approximately the same as its standard uncertainty [ISO /IEC Guide 98-3.2008, G.2.1];

--- Model is complex [ISO /IEC Guide98-3.2008, G.1.5];

--- Asymmetry of PDFs for each input [ISO /IEC Guide 98-3.2008, G.5.3].

This standard provides a verification method to check whether the GUM uncertainty framework is applicable. Apparently suitable in the GUM uncertainty framework

In the case of use, it is still the main method of uncertainty assessment.

It is usually sufficient to keep one or two significant figures for the uncertainty report. This standard gives a calculation guide, reasonable assurance basis

The valid figures reported by the provided information are correct.

This standard provides a detailed case description.

This standard is a supplement to GUM and should be used in conjunction with GUM. Other methods that are basically consistent with GUM can also be used instead. this

The standard user is the same as the GUM user.

Note 1. This standard does not consider the case of a model that is not a single output (for example, a solution involving a quadratic equation, and no root is specified).

Note 2. This standard does not consider the situation in which the output a priori PDF can be obtained, but the treatment method of this standard is applicable to this case [16].

2 Normative references

The following documents are indispensable for the application of this document. For dated references, only dated versions apply to this document. For undated references, the latest edition (including all amendments) applies to this document.

ISO /IEC Guide 98-3.2008 Measurement uncertainty - Part 3. Guidance guide to measurement uncertainty (GUM..1995)

[Uncertaintyofmeasurement-Part 3. Guidetotheexpressionofuncertaintyin measurement

(GUM..1995)]

ISO /IEC Guide 99.2007 International Metrology Vocabulary Basic and General Concepts and Related Terminology (VIM) [Internationalvo-

cabularyofmetrology-Basicandgeneralconceptsandassociatedterms(VIM)]

3 Terms and definitions

Unless otherwise stated, this standard uses the terms and definitions in ISO /IEC Guide 98-3 and ISO /IEC Guide 99. among them

The definitions most relevant to this standard in the above documents are given below (see 4.2). More definitions are also given below, including from other collars.

An definition adopted by the domain and important to this standard.

A summary of the main symbols is given in Appendix G.

3.1

Probability distribution probabilitydistribution

Gives a (random variable) function that takes a random variable to take any given value or a probability that takes a given set.

Note. The probability that a random variable takes all the values in the set is equal to 1.

[quoted from ISO 3534-1..19931.3; ISO /IEC Guide 98-3..2008, C.2.3]

Note 1. When the probability distribution is related to a single (scalar) random variable, it is called a univariate probability distribution; if the probability distribution is related to multiple random variables, it is called

Multivariate probability distribution. Multivariate probability distributions can also be referred to as joint distributions.

Note 2. A probability distribution can be in the form of a distribution function or a probability density function.

3.2

Distribution function distributionfunction

For each value xi, a function is given for the probability that the random variable X is less than or equal to xi, which is.

GX(xi)=Pr(X<=xi)

[quoted from ISO 3534-1..19931.4; GUM..1995 C.2.4]

3.3

Probability density function probabilitydensityfunction

The derivative of the distribution function, if the derivative exists, the function is.

gX(xi)=dGX(xi)/dxi

Note. gX(xi)dxi is the "probability unit".

gX(xi)dxi=Pr(xi \u003cX\u003cxi dxi)

[quoted from ISO 3534-1..19931.5; ISO /IEC Guide 98-3..2008, C.2.5]

3.4

Normal distribution normaldistribution

The probability distribution of the continuous random variable X, whose probability density function is.

gX(xi)=

sigma 2pi

Exp -

(xi-µ

2e

Êê

Note. µ is the expectation of the random variable X, and sigma is the standard deviation of the random variable X.

[quoted from ISO 3534-1..19931.37; ISO /IEC Guide 98-3..2008, C.2.14]

Note. Normal distribution is also known as Gaussian distribution.

3.5

t distribution t-distribution

The probability distribution of the continuous random variable X, whose probability density function is.

gX(xi)=

Gamma((nu 1)/2)

pinuGamma(nu/2)

1 xi

-(nu 1)/2

Tz-1e-tdt,z >0

3.6

Expectation expectation

The nature of random variables. For a continuous random variable X characterized by PDF with gX(xi), the expectation is.

xigX(xi)dxi

Note 1. Not all random variables have expected values.

Note 2. For a given function F(X), the expectation of the random variable Z=F(X) is.

F(xi)gX(xi)dxi.

3.7

Variance variance

The nature of random variables. For a continuous random variable X characterized by PDF with gX(xi), the variance is.

[xi-E(X)] 2gX(xi)dxi

Note. Not all random variables have a variance.

3.8

Standard deviation standarddeviation

The square root of the variance [V(X)] 1/2.

3.9

R moment momentmentoforderr

The expectation that the random variable is r-th power, ie.

xirgX(xi)dxi

Note 1. The r-th order center distance is the expectation of the random variable Z=[XE(X)]r.

Note 2. It is expected that E(X) is the first moment and the variance V(X) is the second order central moment.

3.10

Covariance covariance

The characteristics of two random variables. For joint (multivariate) PDF is two consecutive random variables x1 and X2 of gX(xi), where X=

(X1,X2)T, xi=(xi1,xi2)T, whose covariance is.

[xi1-E(X1)][xi2-E(X2)]gX(xi)dxi1dxi2

Note. Not all variables have covariance between them.

3.11

Uncertainty matrix uncertaintymatrix

N × N dimensional matrix, the elements on the diagonal are the square of the standard uncertainty of the estimated components of the N-dimensional vector, other non-diagonal

The elements on the line are the covariance between the two estimates.

Note 1. The N × N dimensional uncertainty matrix Ux of the N-dimensional vector X estimate x can be expressed as.

Ux=

u(x1,x1) u(x1,xN)

u(xN,x1)u(xN,xN)

Where u(xi,xi)=u2(xi) is the variance of xi (the square of the standard uncertainty), and u(xi,xj) is the covariance of xi and xj. If the component of X

Xi is not related to Xj, then u(xi, xj)=0.

Note 2. Covariance is also known as mutual uncertainty.

Note 3. The uncertainty matrix is also called the covariance matrix or the variance-covariance matrix.

3.12

Contains interval coverageinterval

A range of values for a given probability based on the information available.

Note 1. The inclusion interval is sometimes called the confidence interval or the Bayesian interval.

Note 2. In general, there are more than one inclusion interval at a given probability.

Note 3. The inclusion interval should not be called a 'confidence interval' to avoid confusion with statistical concepts [ISO /IEC Guide 98-3.2008, 6.2.2].

Note 4. This definition differs from the definition in ISO /IEC Guide 99.2007 because the concept of 'true value' is not used in this standard. The reason is given in GUM.

[ISO /IEC Guide 98-3.2008, E.5].

3.13

Contains probability coverageprobability

The probability of including a certain amount of value within a specified inclusion interval.

Note. The inclusion probability is sometimes referred to as "confidence" [ISO /IEC Guide 98-3.2008, 6.2.2].

3.14

Contains the length of the interval lengthofacoverageinterval

Within a containment interval, the maximum is subtracted from the minimum.

3.15

Probability symmetry contains interval probabilisticalysymmetriccoverageinterval

The inclusion interval of a quantity whose probability is less than the minimum value in the interval is equal to the probability that the value is greater than the maximum value of the interval.

3.16

The shortest contains the interval shortestcoverageinterval

Among all the inclusion intervals of one quantity having the same inclusion probability, there is an inclusion section having the shortest inclusion interval length.

3.17

Distributed propagation propagationofdistribution

A method of determining an output probability distribution from a probability distribution of each input quantity on which the output depends.

Note. This method can be analytical or numerical, accurate or approximate.

3.18

GUM uncertainty framework GUMuncertaintyframework

The uncertainty propagation law is applied and the output is characterized by a normal distribution or a t distribution to determine the inclusion interval.

3.19

Monte Carlo method MonteCarlomethod

A method of distributed propagation by random sampling from a probability distribution.

3.20

Numerical tolerance

Contains the half width of the shortest interval of all values, which can be expressed in terms of the specified number of significant digits.

Example. All numbers greater than 1.75 and less than 1.85 can be represented by two significant digits of 1.8 with a numerical tolerance of (1.85-1.75)/2=0.05.

Note. See 7.9.2 for the method of calculating the numerical tolerance.

4 conventions and symbols

The following conventions and symbols apply to this document.

4.1 The mathematical model of univariate (scalar) measurements [ISO /IEC Guide 98-3.2008, 4.1] can be represented by the functional relationship f.

Y=f(X) (1)

Where. Y is a single (scalar) output and X is N inputs (X1,, XN)T. Xi is a random variable, which may have a value of xii,

The expectation is xi. Y is a random variable, which may take the value eta and expect y.

Note 1. The same symbol can be used to re...