Sampling of discrete materials quantitative approach sampling of zero dimensional objects
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Sampling of discrete materials quantitative approach sampling of zero dimensional objects

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de No
ised f
to be
also showing how this relates directly to sampling practise (materials, equipment and procedures). A highlight of the latter issue concerns
well as a theoretical standpoint, however, it may be
dimensional model. From a physical and math-
ematical standpoint, every element of the object
can be well represented by a one-dimensional
tolerance of, say, 20%): Here, conventional
statistics can be applied.
n No hypothesis of uniformity of the unit mass is
cannot be applied.
atory S
is represented by its projection on a plane (often
horizontal). We often have occasion to work on
useful to represent a physical object by a model of a
smaller number of dimensions.
o A three-dimensional model alone can represent bulky
lots L, e.g., an ore body and similar.
o Flat objects, such as a sheet of paper, a steel sheet, the
thickness of which is:
n small in comparison with the two dimensions of
its surface,
n practically uniform (with a tolerance of, say,
20%) can often be well represented by a two-
model. From a physical and mathematical stand-
point, the lot is here represented by its projection
on the axis of elongation.
o Discrete objects such as lots made up of a large
number of unspecified units, assumed to be inde-
pendent from one another; i.e., populations of non-
ordered units can, by extension and by convention, be
defined as zero-dimensional objects. There are two
n Unit masses are more or less uniform (with a
experimental estimation of the Fundamental Sampling Error (FSE). Part II is also fundamental for further developments in Part III, as it
presents a complete overview discussion of the basic sampling operation of the one-dimensional object as well.
D 2004 Elsevier B.V. All rights reserved.
Keywords: Discrete; Quantitative approach; Zero-dimensional objects; Sampling of Particulate matter; Theory of Sampling
1. Joint introduction of parts II and III: three-, two-,
one-, zero-dimensional models
o Strictly speaking, all material objects, lots L, occupy a
three-dimensional Cartesian space. From a practical as
o Elongated objects such as a rail, a cable or a flux of
matter whose length is:
n very large in comparison with the two dimensions
of its cross-section,
n practically uniform (with a tolerance of, say, 20%)
Parts II and III of this series are initiated by a joint discussion of
the Theory of Sampling for zero-dimensional objects. It is necessary
essential model rigour has been maintained. An attempt has been m
Sampling of d
II. Quantitative approach\u2014sam
Res. de Luynes, 14 Avenue Jean
Received 28 August 2003; received in rev
Chemometrics and Intelligent Labor
lots L, which can be considered as practically
0169-7439/$ - see front matter D 2004 Elsevier B.V. All rights reserved.
rete materials
ng of zero-dimensional objects
ailles, F-06400 Cannes, France
orm 6 April 2004; accepted 28 May 2004
res related to the lot. Part II then delineates the central elements of
brief within the limited format of the present tutorial series, but all
to focus on the central mathematical theoretical core of TOS while
ystems 74 (2004) 25\u201338
made. Conventional statistics
We shall here deal exclusively with this most
realistic case.
errors and quantified heterogeneities.
nt La
respected, and taught everywhere. In this series, we will
specifically study only the sampling of zero- and one-
dimensional objects.
Whereas the qualitative approach presented in Part I was
common to all multi-elemental objects susceptible of being
sampled (which excludes only compact solids), the present
quantitative approach involves two different mathematical
models corresponding to sampling of populations (zero-
dimensional objects) or to time series (one-dimensional
objects) respectively. It will be necessary to remind the
reader of a few salient definitions (extracted from Section 4
of Part I).
1.1. Definitions and notations
o LOT L: From a theoretical standpoint, a lot L of discrete
material can be regarded as a set of units. We must take
into consideration two kinds of sets.
o SET: The set can be:
! Either a population of non-ordered units (e.g.,
batch of stationary material),
! Or a series of ordered units (e.g., a sequence of
elementary cross-sections of a flowing stream.
The order is then chronological; ordered spatial
series can very often be treated with exactly the
same mathematical apparatus).
Different mathematical laws apply to these two kinds of
sets. It is a grave mistake, which is often very costly, to use
inappropriate laws or formulas!
A population of non-ordered units is, by convention,
described as a zero-dimensional object. The absence of
dimension expresses the absence of order (chronological,
spatial or otherwise). There may exist a concealed order, a
o Bulky, flat or elongated objects have a common
property: Their constituent elements are physically tied
or correlated to one another. They are therefore liable,
often likely, to show an essential spatial correlation
within their specific dimensional framework.
The prototype of a bulky three-dimensional compact
object is a mineral deposit. Following the works of
Krige, Sichel, de Wijs and more generally the South
African school of mathematicians who were particularly
interested in the first half of the 20th century in the gold
deposits found in their country, the French mining
engineer Georges Matheron (see Ref. [12]) founded a
new science known as bgeostatisticsQ: the science of
evaluation of mineral deposits. Michel David, another
French mining engineer living in Canada (see Ref. [13]),
helped significantly to diffuse the knowledge of geo-
statistics in English-speaking countries. In the present
work, we shall, however, not deal much with this science,
which today is widely known, acknowledged and
P. Gy / Chemometrics and Intellige26
correlation between units, but the model does not take this
into account and is not meant to detect it either. The zero-
2. Zero-dimensional model\u2014contribution made to the
heterogeneity of lot L by an unspecified unit Um
In the zero-dimensional model, the lot L can be regarded
in general terms as a population of unspecified units Um
which will be defined as the case requires. A is the
component of interest. The grade, or concentration, of
component A in a given object (lot L, unit U, sample S, etc.)
refers to its actual but unknown grade, not to some estimate
of it. The following definitions are made:
NU number of units Um in lot L,
Mm mass of unit Um (with wet materials, Mm is generally
the mass of dry solids)
Am mass of component A in unit Um
am mass proportion of component A in unit Um (i.e.,
grade of Um), defined by the identity
Am ¼ amMm ð1Þ
ML mass of lot L. ML is the sum Am of the masses Mm of
the NU units Um. Am is a sum extended from m=1 to
ML ¼
Mm ð2Þ
AL mass of component A in lot L
dimensional model presented here in Part II deals with this
A series of ordered units can be described as a one-
dimensional object: With a flowing stream, the dimension is
time: the order between consecutive units is chronological.
There usually exists a correlation between the rank of a unit
in the series and its composition. If such a correlation does
not exist, the mathematical model detects it. The one-
dimensional model presented in Part III of this article deals
with this problem.
Hence, the subdivision of the quantitative approach:
Part II. Sampling of zero-dimensional objects
Part III. Sampling of one-dimensional objects
In Section 5 of Part I, we defined the concepts of
homogeneity and heterogeneity. According to these defi-
nitions, sampling would be an exact process if the material
being sampled indeed was homogeneous. Unfortunately
(but fortunately for the theoretician), homogeneity is a
concept that can be defined mathematically