Practical applications of sampling theory
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Practical applications of sampling theory

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had to be treated as separate strata. Design requirements.
\ufffd Material balance for sulfur required annually
\ufffd A two-man team with portable instruments available for
atmospheric emission control measurements
\ufffd To avoid too much time spent in transporting and setting
up the instruments, a one-week measurement period is
carried out at each location, where the team is working;
that is, 52 periods/year are available and should be
allocated optimally
\ufffd For optimization, the emission sources were grouped into
six groups, which could be measured during 1 week from
one station in the field Acquisition of basic data. Existing records sup-
plemented with new experiments were used to estimate:
Fig. 7. To estimate the uncertainty of the annual sulfur balance in a pulp
factory, the standard deviations of the means of several material streams
have to be estimated. The streams have different sizes, contain sulfur in
many different compounds, and in many streams, high variability is
characteristic to the sulfur containing compounds. This is especially the
case for atmospheric emissions.).
\ufffd Relative contribution (size) of all major emission sources
Fig. 8. Heterogeneity (relative variation about the mean) of SO2 in
emissions from a soda recovery boiler during a measurement period.
Table 1
Relative sizes, relatives standard deviation of one sample/measurement
period, allocation of 32 samples (measurement periods) that can be made
during one week, and relative standard deviations of the mean of 1 week in
emission group #6
Stratum No Wi sri (%) ni sr(x¯) (%)
1 0.143 30 5 13.4
2 0.285 25 9 8.3
3 0.002 40 1 40
4 0.285 25 9 8.3
5 0.285 25 9 8.3
Total 1.000 32 srðx¯¯Þ ¼ 4:5%
P. Minkkinen / Chemometrics and Intelligent Laboratory Systems 74 (2004) 85\u201394 93
to the total emission of sulfur compounds
\ufffd Variance estimates for the different emission sources (e.g.,
from variogram or from Analysis of Variance). As an
example, Fig. 8 shows the SO2 emissions from the soda
recovery boiler as the process heterogeneity (relative
variation about the process mean). As can be seen, a high
variability is characteristic to emissions from this source.
In the peaks, the SO2 concentration is three times higher
than the process average. From the heterogeneity values,
variogram (Fig. 9) can be calculated for the process. Fig. 9
also shows the relative standard deviation estimates for
the sampling error in this process stream as function of the
sampling interval both for the systematic and the stratified
Fig. 9. Variogram calculated from the heterogeneities of Fig. 8 (upper part)
and relative standard deviation estimates for systematic and stratified
sample selection as function of the sampling interval calculated from the
variogram (lower part).
sample selection. The variogram and standard deviation
estimates were calculated by using Gy\u2019s method. With the
standard deviation estimates, the uncertainty of the
process mean during the measurement period can be
estimated. This measurement plan can be treated as a two-
level hierarchical sampling plan, where one sample is
analyzed from each of the N1 strata. By substituting
N1 = n1, n2 = 1 (and N2Hn2) and N3 = n3 = 0 in Eq. (8a),
the standard deviation of the mean of the measurement
period is obtained as
r2x¯ ¼
N1 \ufffd n1
N1 \ufffd 1
þ N2 \ufffd n2
N2 \ufffd 1
where s2
2 is the standard deviation estimate from Fig. 9 for
the used sampling frequency, and n1 is the number of
samples analyzed. Optimization procedure. Optimization was car-
ried out at two levels:
(1) For each group having more than one source, the
analytical resources were optimally allocated within the
one-week (5 days) measurement period between the
different sources (an example is given in Table 1).
(2) Of the 52 measurement weeks, two were allocated for
the insignificant sources, where only occasional
measurements need to be carried out, and 50 weeks
were allocated to the six most important groups. The
results of the final optimization are given in Table 2.
Table 2
Relative sizes, total relatives standard deviation between the mean of 1-
week measurement period, and allocation of 50 measurement weeks
between the different emission source groups
Emission group no. Wi (%) sri (%) ni
1 0.14 14.6 10
2 0.18 20.8 17
3 0.04 14.8 3
4 0.02 7.9 1
5 0.02 10.9 1
6 0.60 6.4 18
Standard deviation of the annual mean of sulfur emission, srðx¯¯Þ ¼ 1:5%
Table 2 shows that regardless of the high heterogeneity
of sulfur in gaseous emissions, the mean can be estimated
fairly accurate at an annual level. When the emission
measurements were carried out approximately according
to the optimized plan, the unaccounted sulfur in the annual
material balance was only about 1%, which is in a good
agreement with the theoretical calculation considering the
complexity of the process.
5. Conclusions
Sampling theory can, and should, be applied in all steps
directly to a paper factory through a pipeline. The pulp
factory had installed a process analyzer to the pipeline and
the amount of pulp pumped to the paper factory\u2014and the
bill for sold pulp\u2014was based on this measurement. When
the discrepancy in the mass-balances of paper produced
and pulp received was noticed, the sampling and analytical
chain was carefully checked. It turned out that partly due
to the sampling problems and partly due to the calibration
problems of the analyzer, the estimated amount of pulp
could be 10% too high. As the paper factory consumed
about 80000 tonne pulp per year, the analytical error was
really expensive. By improving the sampling and calibra-
tion procedure, the systematic errors could be removed and
[1] P.M. Gy, Sampling of Particulate Materials, Theory and Practice,
New York, 1986.
[7] W.G. Cochran, Sampling Techniques, Third edition, Wiley, New York,
P. Minkkinen / Chemometrics and Intelligent Laboratory Systems 74 (2004) 85\u20139494
of analytical procedures, from the planning to the analytical
measurements. At the moment, it is largely neglected. A
large amount of analytical resources is devoted for quality
control and environmental emission estimation in industries.
At national levels, large programs are devoted, e.g., to
monitor the state of environment and to guarantee the
quality and safety of food. Only seldom that the data quality
objectives are adequately defined before the sampling
campaigns, and consequently, the sampling theory is not
utilized when these campaigns are designed. Often, the
sampling plans have just evolved and their performance
has never been critically audited. This leaves a lot of room
for optimization in this field.
Before designing a sampling and analytical plan, one
should carefully consider what is the uncertainty level
tolerated by the user of the results. If the ambition level
is set too high, the investigation will be unnecessarily
expensive. In general, in the cost\u2013benefit relationship
considerations, the following rule applies: if the total
standard deviation is cut to half, the cost will increase
four times, and if only one quarter of original standard
deviation can be tolerated, the cost will be 16 times higher,
etc. But, what is important, this relationship only holds if
the analytical procedure has been optimized. If the resour-
ces are not optimally allocated, the required uncertainty
level may not be achieved at all. On the other hand,
optimization of existing procedures may become cheaper
and still give more reliable result than the original proce-
dure. An example: a pulp factory pumped the pulp slurry
[8] P. Minkkinen, SAMPEX\u2014a computer program for solving sampling
problems, Chemolab 7 (1989) 189\u2013194.
Elsevier, Amsterdam, 1982.
[2] P.M. Gy, Sampling of Heterogeneous and Dynamic Material Systems,
Elsevier, Amsterdam, 1992.
[3] P.M. Gy, Sampling for Analytical Purposes, Wiley, Chichester, 1998.
[4] F.F. Pitard, Pierre Gy\u2019s Sampling Theory and Sampling Practice,