Transcript Weighting error

WSC 5
Weighting Error – the Often
Neglected Component of the Total
Sampling Error
Pentti Minkkinen
Lappeenranta University of Technology
e-mail: [email protected]
Global Estimation Error
GEE
Total Sampling Error
TSE
Total Analytical Error
TAE
Point Materialization Error
Weighting Error
PME
SWE
Increment Delimitation Error
Increment Extraction
Error
Increment and Sample
Preparation Error
IDE
IXE
IPE
Point Selection Error
PSE
Long Range
Point Selection Error
Periodic
Point Selection Error
PSE1
PSE2
Fundamental
Sampling Error
Grouping and
Segregation Error
FSE
GSE
GEE=TSE +TAE
TSE= (PSE+FSE+GSE)+(IDE+IXE+IPE)+SWE
Error components of analytical determination according to P.Gy
Weighting error
Weighting error is made in the estimation of simple
arithmetic mean, when
• the sampling target consists of several sub-strata of
different sizes
• in process analysis, when the flow-rate varies
Lot Consisting of Strata of Different Sizes and
Heterogeneities
LOT
ML1
MLk
ML2
N1
n1
1
x11
N2
n2
2
x12
x21
Nk
nk
k
xk1
xk2
xk3
Lot consisting of k strata of different sizes and the quantities
needed to optimize the sampling plan
M Li
Wi 
 M Li
= Relative size of the stratum i
(1)
MLi = sizes of strata (e.g. as mass or volume), (i = 1,2, …, k)
Ni = Relative size of stratum i expressed as the number of potential
samples that could be taken from a strata = N i  M Li , MSi is the size
M Si
of samples taken from stratum i.
i = Standard deviation of one sample taken from stratum i
ci = Cost of one sample analyzed from stratum i
ct = Total cost of the estimation of the grand mean of the lot
ni = No. of samples taken from stratum i
nt = Total number of samples analyzed =  ni
ni
xi 
x
j
ni
ij
= Mean of stratum i
(2)
k
x  Wi x i = Grand mean of the lot
(2)
2
N

n

2
2
i
i
i


W
Variance of the lot mean =
 i N 1 n
x
i
i
(3a)
i 1
If the samples taken are small in comparison to the stratum size (as is
usually the case) this equation simplifies to
2

 2  Wi 2 i , if in all strata ni << Ni and Ni>> 1
x
ni
(3b)
k
Total cost of the investigation in general case is
c t   ni c i
(4a)
i 1
Usually the costs are independent of strata, and
ct  nt c * ,
if
c1  c2 ,..., ck  c *
(4b)
a) Optimal allocation of samples, if only
the relative sizes Wi of the strata are
known
If only the sizes of strata are known and the total cost ct of the
investigation is fixed, then the best strategy is to allocate the samples
proportionally to the sizes of strata:
ni  Wi nt
, where
ct
nt 
c*
(5)
Both nt and ni have to be rounded to integers so that the total cost
will not be exceeded.
b) Optimal allocation of samples, when the target
value, cT, is given to the total cost and the
variance of the lot mean has to be minimized
Wi i cT
ni  k
c*
Wi i
i 1
Here, too, ni’s have to rounded into integers so that the
target cost is not exceeded.
(6)
c)
Optimal allocation of samples, when the
target value, T, is given to the standard
deviation of the lot mean and total cost has to be
minimized
Wi i
ni  2
T
k
W 
i 1
i
i
(7)
Again, ni’s have to be rounded to integers so that the required
standard deviation of the lot mean will not be exceeded.
Sampling error in process analysis
• In process analysis the fluctuation of the
flow-rate should be taken into account in
estimating the mean over a time
Incorrect sample delimitation
Cutter movement
Incorrect sample profile
Correct sample delimitation
Cutter movement
Correct sample profiles
v = constant  0.6 m/s
if d > 3 mm, b  3d = b0
if d < 3 mm, b  10 mm = b0
a b
c
v
d = diameter of largest particles
b0 = minimum opening of the
sample cutter
Correct design for proportional
sampler:
correct increment extraction
Proportional sampling
• Correctly executed proportional sampling
eliminates the weighting error, if each
sample is weighed and the mean is
calculated as weighted mean by using the
sample masses as weights.
• Subsamples have to be sampled
proportionally, if they are combined into a
composite sample
Effect of density
• If the density of the material varies within
the lot and equal volumes are sampled the
simple mean is erroneous
Example on weighting error: Drill core of
stratified rock type
Total mass of the drill core: Mtot=152.8 kg
Mass of the valuable mineral = 47.5 kg
Density of the valuable mineral = 5 kg/dm3
Density of the gangue = 2.6 kg/dm3
Average density = 3.056 kg/dm3
True mass fraction of the mineral = 47.5kg/(152.8 kg) = 0.3109
= 31.09 %
SAMPLING PLAN:
The drill core is divided into 100 slices of equal sizes,
volume = 0,5 dm3 and average mass, Ms =1.528 kg
Example on weighting error(cont.)
The drill core is divided into 100 slices of equal sizes,
volume = 0,5 dm3 and average mass, Ms =1.528 kg
 Each sample is analyzed separately. The mean concentration as
mass fraction, cm = 0.190
 Based on this result the average mass of the valuable mineral in the
core is = cm · Ms ·100=29.03 kg
 If every sample is weighed (mass Mi) and the weighted mean of the
mass fraction is estimated the correct mean concentration is
ci  M i

obtained:
c 
 0,3109
w
M
i
 and the total mass of the valuable mineral
M min  cw  M 100  47,5 kg
Relative weighting error is thus:
(0.19-0,3109)/0.3109 = -0.389 = -39,9 %
20 samples, 1 dm3 by volume analysed
Sample
#
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
MEAN
SUM
Mass
fraction
Sample mass
kg
Mineral content
kg
1
0
0
0
0,6579
0
0
0
0
0,6579
0
0
0
0
0
0
0,6579
0,6579
0
0
0,1816
5,0
2,6
2,6
2,6
3,8
2,6
2,6
2,6
2,6
3,8
2,6
2,6
2,6
2,6
2,6
2,6
3,8
3,8
2,6
2,6
2,96
59,2
5,0
0
0
0
2,5
0
0
0
0
2,5
0
0
0
0
0
0
2,5
2,5
0
0
15,0
Weighted mean= 15 kg/(59,2 kg) = 0,2534
Correct mean =
0.3109 = 31,09 %
Process sampling: Simulation study
Three processes with 1000 data points were
generated with
low
medium
and high
correlation between concentration and flow-rate
r = 0.011
8
ai
6
4
2
0
25
30
35
40
45
50
55
60
65
Flow-rate
4
ai, Vi
2
0
-2
-4
0
100
200
300
400
500
600
700
800
900
1000
Sample No.
Weighted mean Simple mean
All sampled
45.3669
45.3429
Every tenth sampled 45.5452
Every tenth sampled
45.4100
Relat. Error (%)
-0.053
0.3929
0.0948
r = -0.558
20
ai
15
10
5
0
25
30
35
40
45
50
55
60
65
Flow-rate
ai, Vi
2
0
-2
-4
0
100
200
300
400
500
600
700
800
900
1000
Sample No.
Weighted mean Simple mean
All sampled
43.3210
45.3429
Every tenth sampled 43.4187
Every tenth sampled
45.4100
Relat. Error (%)
4.67
0.225
4.82
r = 0.972
80
ai
60
40
20
25
30
35
40
45
50
55
60
65
Flow-rate
4
ai, Vi
2
0
-2
-4
0
100
200
300
400
500
600
700
800
900
1000
Sample No.
Weigted mean Simple mean Relat. Error (%)
All sampled
47.1692
45.3429
-3.87
Every tenth sampled 47.2751
0.225
Every tenth sampled
45.4100
-3.73
Second simulation
Correl.
coefficient
0.0248
Weighted
mean
36.1798
Simple
mean
36.1346
Rel. Error
%
-0.125
0.169
36.6219
36.1346
-1.33
0.971
37.5777
36.1346
-3.84
Weighting error: calculation of mean when flow-rate varies
NOx (mg/m3)
350
300
250
200
150
FLOWRATE (m3/h)
100
0
3.5
x 10
100
200
300
400
500
600
700
800
100
200
300
400
Time (h)
500
600
700
800
5
3
2.5
2
1.5
1
0
NOx concentrations and total gas flow-rate measured as onehour averages from a power plant during one month
CALCULATION OF TOTAL MONTHLY NOx EMISSION
Mean of NOx concentrations:
Mean of gas flow-rate :
229.5 mg/m3
2.327 ·105 m3/h
Total gas flow:
1.718·108 m3
Total NOx emitted (unweighted): 39400 kg
Weighted mean of NOx concentration:
ciVi
 Vi
= 237.5 mg/m3
Total NOx emitted (weighted):
40800 kg
Weighting error of simple mean:
in mean concentration =
in total monthly emission =
-7.97 mg/m3
-1400 kg
Minimization of weighting error in
Process analysis, when proportional
cross-steam sampling cannot be used
• Flow-rate is measured simultaneously with
sampling and used as weight in calculating the
mean.
• Sampling system is coupled to a flow meter so
that a fixed volume is taken when the required
total volume has passed the sampling point. In
this case the simple average can be used as the
mean concentration.
CONCLUSIONS
Weighting error is often a significant component of
sampling errors and has to taken into account when the
average value, mean concentration or total mass of analyte
in the sampling target is estimated.
Increasing the No. of samples does not necessarily
reduce the sampling error, if the flow-rate and
concentration are correlated.
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