new calibration procedure in analytical chemistry in agreement to vim 3 miloslav suchanek ict prague...
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New calibration procedure in analytical chemistry in agreement
to VIM 3
Miloslav Suchanek
ICT Prague and EURACHEM
Czech Republic
Overview
- New definition of calibration
- Theoretical backround of various calibration methods
- Practial calculation with MS Excel
- Do we need measurement uncertainty?
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Terminology
x, independent variable c, concentration, content
y, dependent variable y, Y, indication, signal
Measurement in chemistry:
calibration of a measurement procedure
not calibration of an instrument
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Result : quantity value ± expanded measurement uncertainty
ISO/IEC Guide 99:2008 International vocabulary of metrology (VIM 3)
2.39 calibration
operation that, under specified conditions, in a first step,
1) established a relation between the quantity values with measurement uncertainties provided by measurement standards and corresponding indications with associated measurement uncertainties and, in a second step,
2) uses this information to establish a relation for obtaining a measurement result from an indication
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x u(x) y u(y)
Ordinary linear regression
Bivariate regression
Monte Carlo simulation
Bracketing
Calibration models
x – concentration, content; y – indication, signal
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Ordinary regression cannot be used!
underestimation of measurement uncertainty
Ordinary regression cannot be used!
underestimation of measurement uncertainty
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Solution:
1. Least square analysis with uncertainties in both variables - bivariate (bilinear) regression
2. Monte Carlo simulation (regression) (MCS)
3. Bracketing calibration
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Bivariate (bilinear) regression – theory (J.M. Lisy et.all: Computers Chem. 14, 189, 1990)
Task:
Estimate the parameters of linear equation y = b1 + b2.x
providing that experimental data have a structure:
xi u(xi) and yi u(yi)
(u(xi) and u(yi) are standard uncertainties)10T&M Conference 2010, SA
Solution:
N
jiiRi bxfywU 2)),((
j = 1,2; N is the number of experimental points
( , )i i i jR y f x b N
iRi RwU 2.
21/Ri Riw u 2 2 2 22 .Ri yi xiu u b u
Parameters of linear model are estimated iteratively
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See EXCEL calculations
1. Each calibration point is characterised by {xi u(xi), yi u(yi) } assumed to be normally distributed {N(xi, u2(xi)), N(yi, u2(yi)}
2. Replace each calibration point by a randomly selected point (j) {xi(j), yi(j)}
3. Perform a (simple) Linear Regression using the « new » calibration dataset (j)
4. Derive the slope and intercept of calibration (j): b2(j), b1(j)5. Repeat the sequence (e.g. 1000 times)6. Compute the average and standard deviation of all b2(j), b1(j) to obtain
the slope b2 and intercept b1, respectively.
The Monte Carlo steps
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The Monte Carlo calculation
provides reliable results
compliant with GUM (ISO/IEC Guide 98-3:2008)
easy to implement in a spreadsheet
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See EXCEL calculations
Bracketing calibration
Model equation
2 1 1 2
2 1
.( ) .( )
( )x x
xc Y Y c Y Y
cY Y
concentration of analyte in sample cx
concentration of analyte in standards c1, c2
(one below and one above concn. in sample)signals corresponding to the analyte concns. Y1, Y2, Yx
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See EXCEL calculations
c[mg/L] u( c) A u(A) Rsc RsA10 0,3 0,117 0,005 3,0% 4,3%20 0,6 0,208 0,005 3,0% 2,4%30 0,9 0,304 0,007 3,0% 2,3%40 1,2 0,403 0,005 3,0% 1,2%50 1,5 0,506 0,006 3,0% 1,2%
sample 0,252 0,007
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10 20 30 40 50
0,1
0,2
0,3
0,4
0,5
Ab
so
rba
nce
(a
.u.)
concentration (mg/L)
5 points calibration
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BIVARIATE REGRESSIONGOTO EXCEL
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X(sample) u (k=1) Rsu 24,25 0,75 3,10%
RESULT
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Monte Carlo simulationGOTO EXCEL
X(sample) u (k=1) Rsu 24,28 0,83 3,40%
RESULT
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0 10 20 30 40 50 600,0
0,1
0,2
0,3
0,4
0,5
0,6
Ab
sorb
an
ce (
a.u
.)
concentration (mg/L)
The simulated dataset
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GOTO EXCEL
Bracketing
X(sample) u (k=1) Rsu 24,58 1,00 4,05%
RESULT
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Conclusions
Sample value, c u(c) RsuOrdinary linear regression 24,29 0,48 2,0%Bivariate linear regression 24,25 0,75 3,1%Monte Carlo simulation 24,28 0,84 3,4%Bracketing 24,58 1,00 4,1%
Measurement uncertainty is the most important in decision making process!
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L
u u
L-1.64*u L+1.64*u
u is the procedure characterization!
acceptance area rejection area
Measurement result with 95% probability
below limit
Measurement result with 95% probability
over limit
¿ grey zone ?
5 %
5 %
results
3.28 * u
