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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 accordingly (see also Section 7 below). 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,