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


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apparently similar to
random fluctuations. In Fig. 5, we show both the curve of
hq and that of h2q.
We have here presented a simple means of analyzing the
descriptive function h(t), represented by the series of values
of hq and of breaking it down into two major components.
This result will be interpreted and completed in Section 4.
We must now see how the time-heterogeneity of L can be
characterized and quantified and how the sampling errors
3 The range of the moving average is preferably an odd number such as
11. It is respectively 9, 7, 5, 3 and 1 for the value of q smaller than 5 and
larger than 55.
4 We will see later on that the continuous trend can itself be further
P. Gy / Chemometrics and Intelligent Laboratory Systems 74 (2004) 39\u201347 41
apparently random fluctuations of the curve around this
continuous trend.
We therefore interpret this so that hq is the sum of two
components: h1q and h2q
hq ¼ h1q þ h2q ð3Þ
We have analyzed this type of functions h(q) or a(t) [16]
and proposed the following definitions.
n Continuous, quasi-functional relationship: For each
value of q there is a well-defined value of h(q). The
major property of this relationship is the fact that it is
continuous in all its constituent curve-points.
n Discontinuous quasi-random relationship: For each
value of q, the value of h can be regarded as extracted
at random from a population with mean m and variance
s2. The major property of this relationship is the fact
that it is discontinuous in all points.
n Stochastic relationship: There is a certain ambiguity
about the meaning of the adjective bstochasticQ. For
certain authors, it is merely a redundant synonymous of
brandomQ. In his bIntroduction to Stochastic ProcessesQ
(1960), M.S. Bartlett defines a stochastic process as ba
physical process in the real world that has some random
element involved in its structureQ. We have proposed
since 1979 [16] to re-formulate BartlettTs definition in
the following way: bA relationship is said to be
bstochasticQ when it has both a non-random (continu-
ous) element and a random (discontinuous) element
involved in its structureQ (the continuous element is
implicit in Bartlett\u2019s definition of ba physical process in
Fig. 3. Cement plant sample series of Q=60 increments, Iq, taken at a
constant time interval To=2 min. Smoothed results, h2q, (or hmq) of the
moving average operator.
the real worldQ). Thus the contribution hq of Iq is a
stochastic function of q.
In order to separate these two components of a stochastic
function, we have implemented the technique of the shifting
mean, described in Section 5.6.3 of Ref. [16], nowadays
better known as the moving average operator. As is well
known, when increasing the number of degrees of freedom,
we reduce the importance of the random component. This
moving mean hmq (m for mean) is defined as follows (with
a range covering 11 values,3 centred on hq):
hmq ¼
\ufffd
hq\ufffd5 þ hq\ufffd4 þ hq\ufffd3 þ hq\ufffd2 þ hq\ufffd1 þ hq þ hqþ1
þ hqþ2 þ hqþ3 þ hqþ4 þ hqþ5
\ufffd
=11 ¼ ~h2q ð4Þ
The moving average, or the shifting mean hmq, is a good
estimator of the continuous component h2q . Fig. 3
represents this moving mean, as applied to the series of
values of hq. It shows very clearly the continuous trend of
the series.4
The difference between the value hq and that of the
moving average, hmq, is an estimator of the discontinuous
component h1q. We shall write:
h1q ¼ hq \ufffd hmq or hq ¼ h2q þ hmq ¼ ~h1q þ h2q ð5Þ
Fig. 4 represents this difference alone. It shows well the
discontinuity part of the series.
Fig. 4. Cement plant sample series of Q=60 increments, Iq, taken at a
constant time interval To=2 min. Results of subtracting the smoothed trend
from the raw data, revealing the discontinuous component, h1q.
broken up into a sum of two components: a non-periodic component and a
periodic component.
We will thus use the notation a(tq), remembering that this is
an estimate only of the true, always unknown value(s).
nt La
The experimental (observed) variogram is designed to
characterize, to quantify the autocorrelation of the function
a(t), and to serve as a bridge between the set of values a(tq)
can be related to this quantified time-heterogeneity. This
will be the object of Section 3. The derivations below are to
a large degree analogous to those in Part II.
3. The one-dimensional model\u2014characterizing the
heterogeneity of a flowing lot L
3.1. The variogram
Autocorrelated series such as a(t) have been extensively
studied by Georges Matheron, the founder of the science
known as geostatistics, i.e. the science of spatial evaluation
of so-called bregionalized variablesQ, for example, mineral
deposits extending in three dimensions. Matheron devel-
oped a number of essential mathematical tools, including the
variogram, which is exactly needed to solve the present
one-dimensional problem also.
3.1.1. Definition of the variogram
We never know the function a(t): all we can do is to
obtain a time series of point values a(tq) with q=1,2,. . .Q.
Fig. 5. Cement plant sample series of Q=60 increments, Iq, taken at a
constant time interval To=2 min. Compound illustration of both the
smoothed trend component with the discontinuous component superposed.
P. Gy / Chemometrics and Intellige42
and the sampling variance involved when substituting the
series of point values for the complete curve a(t).
Suppose that Q=60 increments Iq have been taken at a
regular interval To during the passage of the lot between
time t=0 and t=TL. Each increment is (if needed, dried,
weighed and finally) assayed and its contribution hq=h(tq)
is calculated. We will define an auxilliary lag-parameter:
j : a number j ¼ 1; 2; N ; Jð Þ such that: jb ¼ Q\ufffd q
or qþ jb ¼ Q:
Dh(q,q\ufffdj)=hq\ufffdhq\ufffdj: Dh is thus the algebraic incre-
ment of h(t) between time tq and time tq\ufffdj. These two
points are separated on the time axis by the interval jTo,
which is called the time lag. The individual values, or the
sign of Dh(q,q\ufffdj) are unimportant\u2014we are only interested
progressive shifting of the second column, one line down
at a time, all values of the variogram for all lags
j=2,3,4. . .Q could also be calculated. In 1962, the
computation of a variogram involving 60\u2013100 data or
more could take from several hours to a couple of days.
Today, with modern computers and a convenient spread-
sheet program, it takes only seconds. Progress!
The variance s2(hq) of the population of values of hq is
called the sill of the variogram (which is a term stemming
from geology, meaning a bflat-lying slabQ).
We will now assume that the series of raw data has been
used to compute an experimental variogram. Nothing keeps
us from computing all values of the variogram down to
v(Q\ufffd1) but for various reasons (see Refs. [17\u201320]), it is
recommended to retain only the values of (Q\ufffdj) larger or
equal to Q/2 or to 30, whichever is greater.
Initially, the variogram was used only to estimate the
variance of the Point Selection Error (PSE). But the
in their statistical properties, quantified by their quadratic
mean for each time lag. This is why the function is called
the variogram v(jTo). v(j) is defined as the semi-mean
square of the algebraic increment (increase or decrease) of
h(t):
v jð Þ ¼ 1
2 Q\ufffd jð Þ
X
q
Dh q; q\ufffd jð Þ½ \ufffd2 variogram of h tð Þ
ð6Þ
Why use the semi-mean square and not just the mean
square? Suppose that the two values of hq and hq\ufffdj are
taken at random from a population with a zero mean and
variance s2. If the value of the mean square were used
instead of the semi-mean square, the variogram would be an
estimator of 2s2. By using the half-mean square, it is an
estimator of s2, which is consistent.
3.1.2. How to calculate the experimental variogram:
illustration for j=2
This illustrates the way the author used to calculate
variograms in the \u201960s, at a time computers were not yet
available\u2014unimaginable for today\u2019s scientists, but true!
The series of values of hq are written in two identical
columns, the second