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# Sampling of discrete materials III quantitative approach sampling of one dimensional objects

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```n Characteristic parameters: TSY,TST and Q are the
characteristic parameters of these three selection modes.
n Comparison of selection variances: In examples
presented in our books, the Point Selection Variance
s2(PSE) is computed for all three selection modes presented
above. These examples show that:
B In the absence of cyclic fluctuations, the variance of the
mode SY is always slightly smaller than that of mode
ST, and much smaller than that of mode RA.
B In the presence of cyclic fluctuations, the variance of
the mode ST is Q times smaller than the maximum
of the variance of the mode SY, and smaller than that
of RA.
5.4. Error-generating functions W(j)
In our variogram approach, the error-generating func-
tions are meant to play the same role, in the one-dimensional
P. Gy / Chemometrics and Intellige46
model, as the variance in the zero-dimensional model
[18,20]. When one knows these functions, the sampling
variance\u2014here the point selection variance\u2014can be
obtained simply by dividing the error-generating function
by the number Q of increments. There are of course three
error-generating functions, one for each alternative selection
mode: WSY(j), WST(j), WRA(j). They are derived from the
auxiliary functions below.5 The actual derivations are not
given here as they are of a more technical nature, which is
not needed in order to fully appreciate their function: they
allow us to estimate the corresponding sampling errors
associated with the three alternative sampling strategies.
Systematic selection: WSY jð Þ ¼ 2w j=2ð Þ \ufffd wV jð Þ
ð16Þ
Stratified random selection: WST jð Þ ¼ wV jð Þ ð17Þ
Direct random selection: WRA jð Þ ¼ \ufffd2ðhqÞ ¼ DHL
¼ constant ð18Þ
5.5. Variance of the Point Selection Error (PSE)
Here again, we will distinguish the following:
Systematic selection: \ufffd2 PSEð ÞSY
¼ WSY jð ÞTSY=TL ð19Þ
Stratified random selection: \ufffd2 PSEð ÞST
¼ WST jð ÞTST=TL ð20Þ
Direct random selection: \ufffd2 PSEð ÞRA
¼ WRA jð Þ=Q ¼ DHL=Q ð21Þ
6. Recapitulation\u2014one-dimensional model
We have been able to develop a complete estimation
approach for the variance of the Point Selection Error s2
(PSE). This approach consists in:
n Taking from the stream a certain number Q (as large as
economically possible, but preferably always larger
than 60) of increments Iq, covering the entire domain
{0,TL}, with a constant time interval To
n (Drying, weighing\u2014if appropriate, and) assaying each
of these Q increments Iq.
n Achieving the following computations by means of a
simple computer program:
n Computing the value of hq for each increment Iq
5 This central result was obtained by Matheron.
WSY( j), WST( j), WRA( j)
n Computing the value of the sampling variances
nt Laboratory Systems 74 (2004) 39\u201347 47
7. Model and experimental variograms: sampling
variances computed at the end of a
variographic experiment
We must carefully distinguish between the model
variogramvmod( j), which assumes that the true values of
hq are known and implemented in our computations, and
the experimental variogramvexp( j). With the model vario-
gram vmod( j), Eqs. (19)\u2013(21) express the variances of the
Point Selection Error (PSE) proper. In practice, however, we
are never dealing with the true, unknown values of hq but
with the experimental estimates, always altered by sam-
pling, sample processing, weighing and analysis or assaying
errors. In this case, we must compute the experimental
variogram vexp( j); the corresponding auxiliary functions-
wexp( j) and wVexp( j) and the error-generating functions-
Wexp( j) related hereto. Expressions (19\u201321) then result in
the value of the variance of the Total Sampling Error (TSE)
which includes not only the Point Selection Error (PSE) but
also includes all the above experimental errors, recognised
as the bzero-dimensional errorsQ.
The most striking difference between model and exper-
imental variograms regards the value of v(0).
By definition: vmod 0ð Þ ¼ 0 while
vexp 0ð Þ ¼ v0 > 0 never 0Þ ð22Þð
To all intents and purposes, v0 represents the variance of
all experimental errors; more specifically the variance of the
zero-dimensional sampling errors CSE+ISE as well as the
Total Analytical Errors (TAE) (which, however, are usually
negligible, Part I). The Point Selection Error (PSE), by
definition, is not included in v0 but is given by Eqs. (19)\u2013(21).
8. Errors committed during one-dimensional sampling\u2014
breaking up the Total Sampling Error, TSE
The practical sampling operation can be broken up into
two independent phases:
Phase 1: Selection of a certain number Q of immaterial
point-increments I. Generally speaking, this first phase
n Computing the consecutive values of the variogram
v( j)
n Estimating the value of v(0), thus completing the series
v(0), v(1), v(2), etc. (Fig. 9)
n Computing the value of all necessary auxiliary func-
tions S( j), w( j), SV( j) and wV( j)
n Computing the value of the error-generating functions
generates the Point Selection Error (PSE). This selection is
usually correct. The Point Selection Error (PSE) is
archetypical of the one-dimensional model. In the most
general case PSE can be broken up into a sum of two, and
only two, components:
o Time Fluctuation Error, TFE (non-cyclic)
o Cyclic Fluctuation Error, CFE
PSE ¼ TFEþ CFE ð23Þ
Phase 2: Extraction of the corresponding material incre-
ments about the Q selected points. This operation, which is
called bmaterialization of the point-incrementsQ, takes place
at the scale of groups of elements, where the zero-dimen-
sional model is applicable. The materialization of incre-
ments can be either probabilistic or non-probabilistic (Part I,
Section 7, Fig. 6).
When this sampling is non-probabilistic, there is no
means of forecasting the errors generated as there is no
possible theoretical approach to non-probabilistic sampling.
When it is probabilistic, the zero-dimensional modelde-
veloped in Part II is fully applicable.The same errors are
involved here, namely:
o Correct Sampling Errors (CSE): CSE is the sum of two,
and only two, components:
n Fundamental Sampling Error, FSE
n Grouping and Segregation Error, GSE
CSE ¼ FSEþ GSE ð24Þ
o Incorrect Sampling Errors (ISE): ISE is the sum of
three, and only three, components:
n Incorrect Delimitation Error, IDE
n Incorrect Extraction Error, IEE
n Incorrect Processing Errors, IPE
ISE ¼ IDEþ IEEþ IPE ð25Þ
Phases 1+2 for the one-dimensional model: The Total
Sampling Error (TSE) involved when the sampling is
probabilistic is the sum of seven, and only seven, components
divided into three groups according to Eqs. (23)\u2013(25):
TSE ¼ PSEþ CSEþ ISE
¼ TFEþ CFEð Þþ FSEþGSEð Þþ IDEþIEEþIPEð Þ
ð26Þ
This completes our tutorial introduction to one-dimen-
sional sampling. The format allotted does not allow further
elaborations here. However, the introduction in Part III ties
in reasonably with the augmented presentations in [20],
which contain much further detail.
P. Gy / Chemometrics and Intellige
Sampling of discrete materials
Interrelationship between zero- and one-dimensional objects
One-dimensional model-heterogeneity of a lot L flowing from time t=0 to time t=T
Description and quantification
Descriptive function a(t)
Qualitative properties of the function a(t)
Autocorrelation of the series-breaking up h
The one-dimensional model-characterizing the heterogeneity of a flowing lot L
The variogram
Definition of the variogram
How to calculate the experimental variogram: illustration for j=2
Examples of variograms
First example: variogram of the feed to a uranium ore processing plant (U content)
Second example: variogram of the feed to a cement kiln (CaO %).
Third example: variogram of the same feed to the cement kiln as in Fig. 7 (for the mass Mq of the increments I
One-dimensional model-breaking up hq and the variagram v(h
Breaking up h
Breaking up v(j)
One-dimensional model-from the variogram to the variance of the Point Selection Error,```