ISO 11843-7:2018

Capability of detection — Part 7: Methodology based on stochastic properties of instrumental noise

Background noise exists ubiquitously in analytical instruments, whether or not a sample is applied to the instrument. This document is concerned with mathematical methodologies for estimating the minimum detectable value in case that the most predominant source of measurement uncertainty is background noise. The minimum detectable value can directly and mathematically be derived from the stochastic characteristics of the background noise. This document specifies basic methods to — extract the stochastic properties of the background noise, — use the stochastic properties to estimate the standard deviation (SD) or coefficient of variation (CV) of the response variable, and — calculate the minimum detectable value based on the SD or CV obtained above. The methods described in this document are useful for checking the detection of a certain substance by various types of measurement equipment in which the background noise of the instrumental output predominates over the other sources of measurement uncertainty. Feasible choices are visible and ultraviolet absorption spectrometry, atomic absorption spectrometry, atomic fluorescence spectrometry, luminescence spectrometry, liquid chromatography and gas chromatography.

Capacité de détection — Partie 7: Méthodologie basée sur les propriétés stochastiques du bruit instrumental

General Information

• Status: Published
• Publication Date: 04-Sep-2018
• ICS: 03.120.30 - Application of statistical methods
• ICS: 17.020 - Metrology and measurement in general
• Technical Committee: ISO/TC 69/SC 6 - Measurement methods and results
• Drafting Committee: ISO/TC 69/SC 6/WG 5 - Capability of detection
• Current Stage: 9092 - International Standard to be revised
• Completion Date: 16-Jun-2022

Relations

• Revises: ISO 11843-7:2012 - Capability of detection — Part 7: Methodology based on stochastic properties of instrumental noise
• Effective Date: 26-Jul-2014

Standards Content (Sample)

INTERNATIONAL ISO
STANDARD 11843-7
Second edition
2018-09
Capability of detection —
Part 7:
Methodology based on stochastic
properties of instrumental noise
Capacité de détection —
Partie 7: Méthodologie basée sur les propriétés stochastiques du bruit
instrumental
Reference number
ISO 11843-7:2018(E)
©
ISO 2018

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ISO 11843-7:2018(E)

COPYRIGHT PROTECTED DOCUMENT
© ISO 2018
All rights reserved. Unless otherwise specified, or required in the context of its implementation, no part of this publication may
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Published in Switzerland
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ISO 11843-7:2018(E)

Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 2
4 Quantitative analysis and background noise . 3
4.1 Error sources of analysis . 3
4.2 Random processes in background . 3
5 Theories for precision . 4
5.1 Theory based on auto-covariance function . 4
5.2 Theory based on power spectrum . 6
6 Practical use of FUMI theory . 9
6.1 Estimation of noise parameters by Fourier transform . 9
6.2 Estimation of noise parameters by autocovariance function .11
6.3 Procedures for estimation of SD .11
Annex A (informative) Symbols and abbreviated terms used in this document.14
Annex B (informative) Derivation of Formula (7) .15
Annex C (informative) Derivation of Formulae (14) to (16) .16
Bibliography .18
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ISO 11843-7:2018(E)

Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards
bodies (ISO member bodies). The work of preparing International Standards is normally carried out
through ISO technical committees. Each member body interested in a subject for which a technical
committee has been established has the right to be represented on that committee. International
organizations, governmental and non-governmental, in liaison with ISO, also take part in the work.
ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of
electrotechnical standardization.
The procedures used to develop this document and those intended for its further maintenance are
described in the ISO/IEC Directives, Part 1. In particular, the different approval criteria needed for the
different types of ISO documents should be noted. This document was drafted in accordance with the
editorial rules of the ISO/IEC Directives, Part 2 (see www .iso .org/directives).
Attention is drawn to the possibility that some of the elements of this document may be the subject of
patent rights. ISO shall not be held responsible for identifying any or all such patent rights. Details of
any patent rights identified during the development of the document will be in the Introduction and/or
on the ISO list of patent declarations received (see www .iso .org/patents).
Any trade name used in this document is information given for the convenience of users and does not
constitute an endorsement.
For an explanation of the voluntary nature of standards, the meaning of ISO specific terms and
expressions related to conformity assessment, as well as information about ISO's adherence to the
World Trade Organization (WTO) principles in the Technical Barriers to Trade (TBT) see www .iso
.org/iso/foreword .html.
This document was prepared by Technical Committee ISO/TC 69, Applications of statistical methods,
Subcommittee SC 6, Measurement methods and results.
This second edition cancels and replaces the first edition (ISO 11843-7:2012), which has been technically
revised.
The main changes compared to the previous edition are as follows:
— created a new 6.2;
— 6.2 of the first edition is renumbered 6.3.
A list of all parts in the ISO 11843 series can be found on the ISO website.
Any feedback or questions on this document should be directed to the user’s national standards body. A
complete listing of these bodies can be found at www .iso .org/members .html.
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ISO 11843-7:2018(E)

Introduction
The series of ISO 11843 is based on the probability distributions of the net state variable (measurand)
for both the linear and nonlinear calibration situations. The focus is implicitly, though sometimes
explicitly, on the uncertainty associated with an estimate of the measured response predominantly
coming from the baseline noise in instrumental analysis. In many, if not most, analytical instruments,
the baseline noise is considered the prime cause of uncertainty when the sample amount is as low as the
minimum detectable value. Within its domain of applicability, the method given in this document can
dispense with the repetition of real samples, thus helping to improve global environments by saving
time and energy that would be required by repetition.
The basic concept of ISO 11843-7 is the mathematical description of the probability distribution of
the response variable in terms of mathematically well-defined random processes. This description
straightforwardly leads to the minimum detectable value. As for the relation of the response and
measurand, linear and nonlinear calibration functions can be applied. In this manner, compatibility
with ISO 11843-2 and ISO 11843-5 is ensured.
The definition and applicability of the minimum detectable value are described in ISO 11843-1 and
ISO 11843-2; the definition and applicability of the precision profile are described in ISO 11843-5. The
precision profile expresses how the precision changes depending on the net state variable. ISO 11843-7
specifies the practical use of the fundamental concepts in ISO 11843 in case of the background noise
predominance in instrumental analysis.
The minimum detectable value, x , is generally expressed in the unit of the net state variable. If the
d
calibration function is linear, the SD or CV of the response variable estimated in this document can
linearly be transformed to the SD or CV of the net state variable, which in turn can be used for the
estimation of the minimum detectable value, x .
d
If the calibration function is nonlinear, the precision profile of the response variable in this document
needs to be transformed to the precision profile of the net state variable as shown in ISO 11843-5. In
this situation, the contents of ISO 11843-5 can be used for this purpose without modification.
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INTERNATIONAL STANDARD ISO 11843-7:2018(E)
Capability of detection —
Part 7:
Methodology based on stochastic properties of
instrumental noise
1 Scope
Background noise exists ubiquitously in analytical instruments, whether or not a sample is applied
to the instrument. This document is concerned with mathematical methodologies for estimating the
minimum detectable value in case that the most predominant source of measurement uncertainty is
background noise. The minimum detectable value can directly and mathematically be derived from the
stochastic characteristics of the background noise.
This document specifies basic methods to
— extract the stochastic properties of the background noise,
— use the stochastic properties to estimate the standard deviation (SD) or coefficient of variation (CV)
of the response variable, and
— calculate the minimum detectable value based on the SD or CV obtained above.
The methods described in this document are useful for checking the detection of a certain substance
by various types of measurement equipment in which the background noise of the instrumental
output predominates over the other sources of measurement uncertainty. Feasible choices are visible
and ultraviolet absorption spectrometry, atomic absorption spectrometry, atomic fluorescence
spectrometry, luminescence spectrometry, liquid chromatography and gas chromatography.
2 Normative references
The following documents are referred to in the text in such a way that some or all of their content
constitutes requirements of this document. For dated references, only the edition cited applies. For
undated references, the latest edition of the referenced document (including any amendments) applies.
ISO 3534-1, Statistics — Vocabulary and symbols — Part 1: General statistical terms and terms used in
probability
ISO 3534-2, Statistics — Vocabulary and symbols — Part 2: Applied statistics
ISO 3534-3, Statistics — Vocabulary and symbols — Part 3: Design of experiments
ISO 5725-1, Accuracy (trueness and precision) of measurement methods and results — Part 1: General
principles and definitions
ISO 11843-1, Capability of detection — Part 1: Terms and definitions
ISO 11843-2, Capability of detection — Part 2: Methodology in the linear calibration case
ISO 11843-5, Capability of detection — Part 5: Methodology in the linear and non-linear calibration cases
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ISO 11843-7:2018(E)

3 Terms and definitions
For the purposes of this document, the terms and definitions given in ISO 3534-1, ISO 3534-2,
ISO 3534-3, ISO 5725-1, ISO 11843-1, ISO 11843-2, ISO 11843-5 and the following apply. A list of symbols
and abbreviated terms used in this document is provided in Annex A.
ISO and IEC maintain terminological databases for use in standardization at the following addresses:
— ISO Online browsing platform: available at https: //www .iso .org/obp
— IEC Electropedia: available at http: //www .electropedia .org/
3.1
precision profile
mathematical description of the standard deviation (SD) of the response variable
[σ (X)] or net state variable [σ (X)] as a function of the net state variable
Y X
Note 1 to entry: The coefficient of variation (CV) of the response variable or net state variable as a function of the
net state variable is also referred to as a precision profile.
Note 2 to entry: Precision means the SD or CV of the observed response variable or SD or CV of the net state
variable when estimated by the calibration function (see ISO 11843-5).
[SOURCE: ISO 11843-5:2008, 3.4, modified — “coefficient of variation” has been removed and Note 1 to
entry has been added instead. Note 2 to entry has also been added.]
3.2
minimum detectable value of the net state variable
x
d
value of the net state variable in the actual state that will lead, with probability 1 – β, to the conclusion
that the system is not in the basic state
Note 1 to entry: Under the assumption that the SD, σ (X), of the net state variable is constant [(σ (X) = σ ], the
X X X
minimum detectable value, x , is defined as
d
xk=+ k σ (1)
()
dc d X
where
k denotes a coefficient to specify the probability of an error of the first kind;
c
k is a coefficient to specify the probability of an error of the second kind.
d
If the SD, σ , of the response variable is assumed to be constant [σ (X) = σ ], then the minimum detectable value
Y Y Y
can be calculated by the following Formula (2):
xk=+ k σ //ddYX (2)
()()
dc d Y
where |dY/dX| denotes the absolute value of the slope of the linear calibration function and is constant.
Note 2 to entry: If the net state variable is normally distributed, the coefficients k = k = 1,65 specify the
c d
probabilities of an error of the first and second kinds (= 5 %) and Formula (1) can simply be written as x = 3,30σ .
d X
Note 3 to entry: If k = k = 1,65, Formula (1) takes the form that σ / x = 1/3,30 = 30 %. Therefore, x can be
c d X d d
found in the precision profile (3.1). x is located at X, the CV of which is 30 %.
d
Note 4 to entry: Different types of precision profiles (3.1) are defined, but they can be transformed to each other.
For example, the SD, σ (X), of the response variable can be transformed to the SD, σ (X), of the net state variable
Y X
by means of the absolute value of the derivative, |dY/dX|, of the calibration function [Y = f(X)]: σ (X) = ⌠ (X)/|dY/
X Y
dX| (see ISO 11843-5). This treatment is an approximation, the extent of which depends on local curvature,
2 2
involving d Y/dX .
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ISO 11843-7:2018(E)

[SOURCE: ISO 11843-5:2008, 3.2, modified — Notes to entry 1 to 4 have been added.]
4 Quantitative analysis and background noise
4.1 Error sources of analysis
The quantitative analysis to obtain a measurand from a sample is generally considered to consist
of preparation, instrumental analysis, data handling and calibration. These steps of analysis are
mechanically independent of each other and so are probabilistically independent as well.
This document applies only to instrumental analysis. However, the errors from the other steps affect
the error of the final value of the measurand, as well. That is, the combined uncertainty associated with
an estimate of the measurand depends on the propagation of all uncertainties relating to the relevant
steps. The following conditions are necessary for the use of ISO 11843-7.
At concentrations near the minimum detectable value in chromatography, the error from the sample
injection into a chromatograph is even less important (e.g. CV = 0,3 % in a recent apparatus) than the
background noise (CV = 30 % by definition). If the importance of a factor other than noise is comparable
to that of the noise, the methodologies of this document are not applicable.
Data handling is usually a process to extract a signal component from noisy instrumental output such
as peak height or area, which is a relative height of a summit of a peak-shaped signal or integration of
intensities over a signal region, respectively. The statistical influences of this process are the major
concern of this document. The use of a digital or analogue filter can also be taken into account, if the
noise after the filtration is analysed for this purpose.
4.2 Random processes in background
Typical examples of the response variable are area and height measured in chromatography. In this
document, intensity difference [Formula (6)] and area [Formulae (10) and (11)] are taken as the
difference and summation of intensities Y of instrumental output. The response variables are usually
i
independent of each other even if they are obtained from consecutive measurement by the same
instrument. On the other hand, the consecutive intensities Y are formulated as a time-dependent
i
[1]
random process, and in many cases, can be considered 1/f noise .
The power spectrum, P( f ), of 1/f noise has a slope inversely proportional to frequency, f:
1
Pf ∝ (3)
()
f
when f is near zero.
The simplest model of random processes is the white noise. Let w denote the random variable of the
i

white noise at point i. By definition, the mean of the white noise is zero and the SD, w , of the white noise
is constant at every point i. A prominent feature of the white noise is that the noise intensities, w and
i
w , are independent of each other, if i ≠ j.
j
The autoregressive process [AR(1)-process] of first order is a mathematical model in which the
intensities, M and M , are not independent of each other (i ≠ j). The AR(1)-process is treated as a major
i j
component of time-dependent changes of instrumental output [see Formula (9)]. The AR(1)-process at
point i is defined to take the form
MM=+ρ m (4)
ii−1 i
where
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ISO 11843-7:2018(E)

m denotes the random variable of the white noise at point i;
i
ρ is a constant parameter (−1 < ρ < 1).
5 Theories for precision
5.1 Theory based on auto-covariance function
[2][3][4]
A theory has been proposed based on an auto-covariance function:
 
ψ τ ≡ EY Y (5)
()
st +τ t
 
 00s 
where E[·] denotes the mean of a random variable inside the square brackets over t .
0
Figure 1 — Signal (upper line) and noise (lower line) with intensity difference
The upper part of Figure 1 depicts the signal as (an approximation to) a rectangular pulse. Noise
(constituting background) on the signal is depicted as the oscillatory curve in the lower part of the
figure. t denotes a time value on the background portion of the signal and t + τ denotes a time value
0 0 s
on the signal itself. The measurement (signal reading) is the difference in intensities at times t and
0
t + τ . The value of the signal would be zero at t in the absence of background noise. The signal has a
0 s 0
finite value at t + τ when a sample is measured. In the ISO 11843-7 measurement model, the signal and
0 s
noise are superimposed, and this mixed random process takes the value Y at time t . The intensities at
i i
times t and t + τ are described as Y and Y , respectively, and the intensity difference is given
0 0 s
t t +τ
0 0 s
by Formula (6).
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ISO 11843-7:2018(E)

Figure 2 — Auto-covariance function of noise
The difference between the values of the auto-covariance function, ψ(τ), at 0 and τ gives the right side
s
of Formula (7).
Near the minimum detectable value, which is dictated by the background fluctuation, intensity
difference often applies in analytical optical spectrometry. The definition of intensity difference, e.g.
[2][3][4]
signal reading corrected for background, is :
ΔYY=−Y (6)
tt+τ
00s
Here, ΔY corresponds to the response variable Y. The variance of the intensity difference is written as
[2][3][4]
shown in Formula (7) . [For the derivation of Formula (7), see Annex B.]
2
 
σψ= 20 −ψτ (7)
() ()
ΔY s
 
Formula (7) is of practical use when the actual auto-covariance functions, ψ(0) and ψ(τ ), are known
s
from the observation of background noise as shown in Figure 2. The substitution of Formula (7) for
Formula (2) (σ = σ ) leads to the minimum detectable value.
Y ΔY
[5]
Use can be made of the Wiener-Khintchine theorem , which relates the auto-covariance function to
the power spectral density through the Fourier transform:
2
∞
ψ ττ= Sf Gf cos 2πffd (8)
() () () ()
sb s
∫
0
where
S ( f ) denotes the power spectrum of the observed background noise;
b
G( f ) is the frequency response of the (linear) read-out system.
Formula (8) indicates the estimation of the measurement SD, Formula (7), through the noise power
spectrum.
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ISO 11843-7:2018(E)

5.2 Theory based on power spectrum
A theory based on the power spectrum of the baseline, which is called FUMI (function of mutual
[6][7][8]
information) , provides the SD values of the measured area and height in instrumental analysis.
These measured values are the integration of the instrumental output over the integration region, as
illustrated in Figure 3. If the signal (shape and size) is invariant, the error of measured area or height,
as long as it comes from the noise alone, is equal to the area created by the noise over the integration
region. That is, the measurement error is the same as the noise-created area. The SD of the noise-
created areas coincides with the SD of measured heights or areas.
Key
Z zero window
S signal region
I integration region
NOTE Additional symbols are explained in Annex A.
Figure 3 — Signal and noise with zero window and integration area
The number of data points over the integration region is k − k .
f c
In the FUMI theory, the noise intensity, Y , at point i is described as the mixed random processes of the
i
white noise and AR(1)-process:
Yw=+M (9)
ii i
The purpose of the FUMI theory is to estimate the SD of the noise-created areas, A , over the integration
F
region (see Figure 3).
In practice, especially chromatography, different modes of integration are adopted as illustrated in
Figure 4. The measurement is of the integrated intensities above the baseline, which is horizontal or
oblique within the domain [k + 1, k ]. The horizontal baseline is horizontally drawn from the intensity
c f
(corrected) at the zero point, and the oblique line is drawn between the intensities at the edges of
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ISO 11843-7:2018(E)

the signal region [0, k ]. The latter is often useful for a slowly changing background, called “drift” in
e
chromatography.
Figure 4 — Integration modes over horizontal line (dashed line) and oblique line (solid line)
between t and t
k +1 k
c f
The noise-created area, which is the area between the random path and horizontal line over the
integration region without signal, can be written as
k
f
AY= Δt (10)
∑
Fi
ik=+1
c
where
Y the noise intensity is described by Formula (9);
i
Δt is the time interval between consecutive data.
Here, A means the response variable Y.
F
If the oblique line is used as the baseline, the noise-created area takes the form:
k
 
f
 
AY= ΔtA− (11)
∑
Fi T
 
 
ik=+1
c
 
where A denotes the area of the trapezoid created by the oblique line, horizontal line and vertical
T
lines at the edges of the integration region from k + 1 to k (see Figure 4). The area of the trapezoid in
c f
Figure 4 is taken with a negative sign. The area is taken with a positive sign if the oblique line lies above
the horizontal line in the integration region and with a negative sign otherwise.
The general expression of the SD, σ , of noise-created areas is:
F
12/
 2 
 
σ = EA (12)
 
FF
 
 
 
where E[·] denotes the ensemble mean of a random variable inside the square brackets. It should
be noted that E[A ] = 0, since, by definition, the ensemble mean of the noise-created area over the
F
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ISO 11843-7:2018(E)

horizontal line [the first term in the right side of Formula (11)] is zero and the ensemble mean of the
trapezoid, A , is zero (also see Annex C).
T
The above derivation is based on the assumption that there exists no uncertainty of the zero level,
i.e. Y = 0. In practice, however, this type of uncertainty exists and should be taken into account. The
0
measurement in chromatography is usually performed relative to the zero level, which is the average of
background intensities over a region referred to here as a zero window (see Figure 3).
2
The squared SD, σ , of measured areas (noise-created areas) within the zero window takes the
Y
[6][7][8]
form :
22 2
σσ=+σ (13)
YZ F
2 2
where σ denotes the variance originating from the zero window and σ is the variance from the
Z F
measured area [σ given by Formula (12)]. The summary of the derivation of Formulae (14) and (15) is
F
given in Annex C. The minimum detectable value can be obtained by the substitution of Formula (13)
for Formula (2).
2 [6][7][8]
The variance, σ , can be described as
Z
2 2
bb2
 
kk− kk−
() ()
1 − ρ 1 − ρ
fc fc
2 2 2 2
 
 
σ = w + b −2ρ + ρ m (14)
Z
2 2
 
b 1 − ρ
2
1 − ρ
b 1 − ρ  
()
[6][7][8]
and the influence of the signal integration over the signal region takes the form :
22

σ =−kk w (first term, 15)
()
F fc
 2 kk− 
kk− ()
fc
fc
1 1 − ρ 1 − ρ
2 2
 

+ kk−−2ρ + ρ m (second term, 15)
fc
2  2 
1 − ρ
()1 − ρ 1 − ρ
 
 
2
2kk −k
 
cf c
1 − ρ 1 − ρ
2 2
  
+ρ m (third term, 15)
2
 
1 − ρ
1 − ρ
 
22

+aw (fourth term, 15)
 kk−+1 −i 
 
kk−  
fc
2kk −k −2k
   
fc
kk−−i
ef c c
 
1− ρ  1− ρ kk+−1 1− ρ 1− ρ ec 
 
2 2
  ec   
+ α −2α ρ + ρ m
 
∑
  
2 −2
   
1− ρ 1− ρ
 1− ρ 1− ρ   
 i=1 
   
 
 
 
 (fifth term, 15)
where
kk− kk++1
()()
fc fc
α = (16)
2k
e
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ISO 11843-7:2018(E)

denotes the SD of the white noise;

w
is the SD of the white noise included in the autoregressive process of first order;

m
is the constant parameter of the autoregressive process of first order;

p
k , k and k are defined in Figures 3 and 4;
c f e
b denotes the number of consecutive data points in the zero window [−b + 1, 0].
The five terms in Formula (15) denote the following stochastic contributions to the measurement
uncertainty:
— first term: the error from the white noise in the integration domain (k − k data points);
f c
— second term: the error from the AR(1)-process in the integration domain (k − k data points);
f c
— third term: the influence of k data points between the zero point and the starting point of integration;
c
— fourth term: the effect of the white noise in the oblique baseline;
— fifth term: the effect of the AR(1)-process in the oblique baseline.
NOTE The applicability of the FUMI theory is rather wide, but there are two typical situations where it is not.
One is that the predominant error source is not background noise. In mass spectroscopy, if the ionization process
produces much more error than the noise, the FUMI theory underestimates the SD of measured areas.
The other situation is where actual instrumental noise includes the noise that cannot successfully be
approximated by the mixed processes of the white noise and AR(1)-process. An example is spike noise of high
intensity.
Baseline shifts of fixed patterns which are beyond the stationary assumption are often observable
in gradient chromatography, but the integration mode of the oblique baseline can assure, though
restricted, the robustness of the approach.
6 Practical use of FUMI theory
6.1 Estimation of noise parameters by Fourier transform
All the parameters necessary for applying the FUMI theory, i.e. Formulae (13) to (16), can be uniquely
determined from the experimental data. The signal parameters (b, k , k , k ) can be set according to the
c f e
 
shape of a target peak, as shown in Figure 3. On the other hand, the noise parameters ()wm,,ρ are
automatically determined from the power spectral density of the noise, as described below.
The noise power spectral density results from the Fourier transform of noise data, Y . The Fourier and
i
inverse Fourier transforms are:
N−1
ki
ˆ
YY= W (17)
∑
ki
i=0
N−1
1
−ki
ˆ
Y = YW (18)
∑
ik
N
k=0
where
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N denotes the number of data points involved in the region of the Fourier transform;
W exp[−j(2π/N)];
j is the imaginary unit.
The power spectral density, P(k), of the random process, Y , is defined as
i
ˆˆ
YY
kk
Pk = (19)
()
N
ˆ ˆ
where Y is the conjugate number of Y .
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