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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