ISO 11843-7:2012
(Main)Capability of detection — Part 7: Methodology based on stochastic properties of instrumental noise
Capability of detection — Part 7: Methodology based on stochastic properties of instrumental noise
Background noise ubiquitously exists in analytical instruments whether or not a sample is applied to the instrument. ISO 11843-7:2012 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. It 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 ISO 11843-7:2012 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
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Standards Content (Sample)
INTERNATIONAL ISO
STANDARD 11843-7
First edition
2012-06-01
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 11843-7:2012(E)
©
ISO 2012
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ISO 11843-7:2012(E)
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ISO 11843-7:2012(E)
Contents Page
Foreword .iv
Introduction . v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Quantitative analysis and background noise . 2
4.1 Error sources of analysis . 2
4.2 Random processes in background . 3
5 Theories for precision . 3
5.1 Theory based on auto-covariance function . 3
5.2 Theory based on power spectrum . 5
6 Practical use of FUMI theory . 9
6.1 Estimation of noise parameters . 9
6.2 Procedures for estimation of SD .10
Annex A (informative) Symbols and abbreviated terms used in this part of ISO 11843 .13
Annex B (informative) Derivation of Formula (7) .14
Annex C (informative) Derivation of Formulae (14) to (16) .15
Bibliography .17
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ISO 11843-7:2012(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.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2.
The main task of technical committees is to prepare International Standards. Draft International Standards
adopted by the technical committees are circulated to the member bodies for voting. Publication as an
International Standard requires approval by at least 75 % of the member bodies casting a vote.
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.
ISO 11843-7 was prepared by Technical Committee ISO/TC 69, Application of statistical methods, Subcommittee
SC 6, Measurement methods and results.
ISO 11843 consists of the following parts, under the general title Capability of detection:
— Part 1: Terms and definitions
— Part 2: Methodology in the linear calibration case
— Part 3: Methodology for determination of the critical value for the response variable when no calibration
data are used
— Part 4: Methodology for comparing the minimum detectable value with a given value
— Part 5: Methodology in the linear and non-linear calibration cases
— Part 6: Methodology for the determination of the critical value and the minimum detectable value in Poisson
distributed measurements by normal approximations
— Part 7: Methodology based on stochastic properties of instrumental noise
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ISO 11843-7:2012(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 part of ISO 11843 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 calibration
d
function is linear, the SD or CV of the response variable estimated in this part of ISO 11843 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 part of ISO 11843
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 the slightest modification.
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INTERNATIONAL STANDARD ISO 11843-7:2012(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 part of ISO 11843 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.
It 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 part of ISO 11843 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 referenced documents are indispensable for the application 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 11843-1:1997, Capability of detection — Part 1: Terms and definitions
ISO 11843-2:2000, Capability of detection — Part 2: Methodology in the linear calibration case
ISO 11843-5:2008, Capability of detection — Part 5: Methodology in the linear and non-linear calibration cases
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
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.
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ISO 11843-7:2012(E)
3.1
precision profile
mathematical description of the standard deviation (SD) of the response variable [σ (X)]
Y
or net state variable [σ (X)] as a function of the net state variable.
X
[ISO 11843-5:2008, 3.4]
NOTE 1 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 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 (ISO 11843-5).
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 Under the assumption that the SD, σ (X), of the net state variable is constant [(σ (X) = σ ], the minimum
X X X
detectable value, x , is defined as
d
x = (k + k )σ (1)
d c 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 can be
Y Y Y
calculated by the following equation:
x = (k + k )(σ /|dY/dX|) (2)
d c d Y
where |dY/dX| denotes the absolute value of the slope of the linear calibration function and is constant.
NOTE 2 If the net state variable is normally distributed, the coefficients k = k = 1,65 specify the probabilities of an error
c d
of the first and second kinds (= 5 %) and Formula (1) can simply be written as x = 3,30σ .
d X
NOTE 3 If k = k = 1,65, Formula (1) takes the form that σ / x = 1/3,30 = 30 %. Therefore, x can be found in the
c d X d d
precision profile. x is located at X, the CV of which is 30 %.
d
NOTE 4 Different types of precision profiles 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) = σ (X)/|dY/dX|
X Y
2 2
(ISO 11843-5). This treatment is an approximation, the extent of which depends on local curvature, involving d Y/dX .
NOTE 5 Adapted from ISO 11843-5:2008.
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 part of ISO 11843 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
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ISO 11843-7:2012(E)
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 part of ISO 11843 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 part of
ISO 11843. 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 part of
ISO 11843, 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 independent of each
i
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 random process, and in many cases, can be
i
[1]
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.
In mathematical theory, the simplest model of random processes is the white noise. Let w denote the random
i
variable of the 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
i
and w , are independent of each other, if i ≠ j.
j
The Markov process is a mathematical model in which the intensities, M and M , are not independent of each
i j
other (i ≠ j). The Markov process is treated as a major component of time-dependent changes of instrumental
output [see Formula (9)]. The Markov process at point i is defined to take the form:
MM=+ρ m (4)
ii−1 i
where
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 which was proposed by Winefordner and his co-workers is 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
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ISO 11843-7:2012(E)
Y
t
t
s
t t + t
0 0 s
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
0
value on the background portion of the signal and t + τ denotes a time value on the signal itself. The
0 s
measurement (signal reading) is the difference in intensities at times t and t + τ . The value of the signal
0 0 s
would be zero at t in the absence of background noise. The signal has a finite value at t + τ when a sample
0 0 s
is measured. In the ISO 11843-7 measurement model, the signal and noise are superimposed, and this mixed
random process takes the value Y at time t . The intensities at times t and t + τ are described as Y and
i i 0 0 s
t
0
Y , respectively, and the intensity difference is given by Formula (6).
t +τ
0 s
Ψ
t
t
s
0 t
s
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 of Formula (7).
s
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ISO 11843-7:2012(E)
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. signal reading corrected
[2][3][4]
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 shown
[2][3][4]
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 from
s
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.
5.2 Theory based on power spectrum
[6]
A theory based on the power spectrum of the baseline, which is called FUMI (function of mutual information)
[7][8]
, 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.
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