Estimation of Sampling Uncertainty in Process Analysis beyond the EURACHEM Guide

Transcription

1 S I R P E K A ANALYSDAGARNA, Uppsala, June, 01 Estimation of Sampling Uncertainty in Process Analysis beyond the EURACHEM Guide Pentti Minkkinen Lappeenranta University of Technology and SIRPEKA Oy (consulting) s: pentti.minkkinen@sirpeka.fi Pentti.Minkkinen@lut.fi Copyright: Pentti Minkkinen

2 Outline Introduction to TOS Sources of sampling error Definitions/characterising heterogeneity Correct and incorrect sampling Estimation of sampling uncertainty Optimization of sampling procedures Practical examples

3 LOT m Primary Secondary Analysis Result sample sample s 1 s s 3 s x (s) Propagation of errors: x s x s i Example: s x (5%) (%) (1%) 30(%) 5.5% GOAL: x = m Analytical process usually contains several sampling and sample preparation steps

4 SAMPLING Art of cutting a small portion of material from a large lot and transferring it to the analyzer Theory of sampling (theoretical) distributions with known properties SLOGANS The result is not better than the sample that it is based on Sample must be representative Theory that combines both technical and statistical parts of sampling has been developed by Pierre Gy: Sampling for Analytical Purposes, Wiley, 1998 Chemometrics and Intelligent Laboratory Systems (Special Issue: 50 Years or Pierre Gy s Sampling Theory) 74 (Issue1) 004

5 When sample is representative? Definition: sample is representative, if r ( SE) m ( SE) s ( SE) rref Bias Precision Required level Bias is minimized by using correct sampling procedures. Key is probabilistic sampling, i.e., all constituents of target material must have an equal chance to end up into the sample. s(se), standard deviation of sampling errors caused by random variation is estimated by using methods developed in sampling theory.

6 Heterogeneity Heterogeneity generates all sampling errors, except the preparation errors, PE Two types of heterogeneities defined by Gy: Constitution heterogeneity, CH SD of between smallest fragments of target material Distribution heterogeneity, DH Non-random distribution of the constituents in target (segregation, clustering, drift)

7 y y x x Two lots having the same mean a L, CH L, but different DH L

8 y y x Samples: x 8, s x Samples: x 5, s 1.

9 Experimental estimation of heterogeneity, h, at sample scale N individual fragments (if large enough), or groups of fragments, are collected and analyzed. Heterogeneity contribution of each sample is defined as where a i M s i s h a a M i L si i, al M s i 1,,, = analytical result (mass fraction) of sample i = the size of sample i M = mean sample size a L M si M a si i = estimate of the lot mean N

10 Standard deviation of the heterogeneity, h, is the estimate of the relative standard deviation of the constitution heterogeneity Characterization of the distribution heterogeneity of flowing streams is shown later

11 Global (Total) Estimation Error GEE Total Sampling Error TSE Total Analytical Error TAE Point Materialization Error PME Weighting Error SWE Increment Delimitation Error IDE Increment Extraction Error IEE Increment and Sample Preparation Error IPE Point Selection Error PSE Long Range Point Selection Error PSE 1 Periodic Point Selection Error PSE Fundamental Sampling Error FSE Grouping and Segregation Error GSE GEE=TSE +TAE TSE= (PSE+FSE+GSE)+(IDE+IEE+IPE)+SWE Error components of analytical determination according to P. Gy

12 DIMENSIONALITY Sample Ideal mixing 0-D Sampling target 0-dimensional, if 1) The whole target can be taken as the sample ) If the lot to be sampled can be mixed before sampling it can be treated as a 0-dimensional lot. Fundamental Sampling Error determines the correct sampling error of 0-D targets and it can be estimated by using binomial or Poisson distributions as models, or by using Gy s fundamental sampling error equations.

13 1-D Samples -D If the lot cannot be mixed before sampling the dimensionality of the lot depends on how the samples are delimited and cut from the lot. Autocorrelation has to be taken into account in sampling error estimation

14 Samples 3-D Lot is 3-dimensional, if none of the dimensions is completely included in the sample

15 Weighting error Weighting error is made in the estimation of simple arithmetic mean, when the sampling target consists of several sub-strata of different sizes (or densities) in process analysis, when the flow-rate varies

16 Lot Consisting of Strata of Different Sizes and Heterogeneities LOT M M Lk L1 M L N 1 n 1 s 1 N n s N k n k s k x 11 x 1 x 1 x k1 x k x k3 Lot consisting of k strata of different sizes and the quantities needed to optimize the sampling plan

17 W i M L i = Stratum weight = Relative size of the stratum i (1) M Li n i = No. of samples taken from stratum i n t = Total number of samples analyzed = ni x i n i j x i ij = Mean of stratum i () n x k i 1 W i xi = Grand mean of the lot (3) s s i x Wi i ni = Variance of the lot mean (4)

18 Weighting error in process analysis In process analysis the fluctuation of the flow-rate should be taken into account in estimating the mean over a time interval Sample weight can be used as stratum size in eq. 1 above to estimate the mean if the sample cut is proportional to the process flow-rate. Alternatively flow-rate at sampling time could be used, if reliable measurement is available (especially gaseous and liquid streams)

19 Proportional sampling Correctly executed proportional sampling eliminates the weighting error, if each sample is weighed and the mean is calculated as the weighted mean by using the sample masses as weights. Proportional sampling should be used, if the sub-samples are combined into a composite sample.

20 Effect of density If the density of the material varies within the lot and equal volumes are sampled the simple mean is erroneous

21 Example on weighting error: Drill core of stratified rock type Total mass of the drill core: M tot =15.8 kg Mass of the valuable mineral = 47.5 kg Density of the valuable mineral = 5 kg/dm 3 Density of the gangue =.6 kg/dm 3 Average density = kg/dm 3 True mass fraction of the mineral = 47.5kg/(15.8 kg) = = % SAMPLING PLAN: The drill core is divided into 100 slices of equal sizes, volume = 0,5 dm 3 and average mass, M s =1.58 kg

22 Example on weighting error(cont.) The drill core is divided into 100 slices of equal sizes, volume = 0,5 dm 3 and average mass, M s =1.58 kg Each sample is analyzed separately. The mean concentration as mass fraction, c m = Based on this result the average mass of the valuable mineral in the core is = c m M s 100=9.03 kg If every sample is weighed (mass M i ) and the weighted mean of the mass fraction is estimated the correct mean concentration is obtained: ci M i cw 0,3109 M i and the total mass of the valuable mineral M c M 100 min w 47,5 kg Relative weighting error is thus: (0.19-0,3109)/ = = -39,9 %

23 Process sampling: Simulation study on WE Three processes with 1000 data points were generated with low medium and high correlation between concentration and flow-rate

24 ai, Vi ai 8 r = Flow-rate Sample No. Weighted mean Simple mean Relat. Error (%) All sampled Every tenth sampled Every tenth sampled

25 ai, Vi ai 0 r = Flow-rate Sample No. Weighted mean Simple mean Relat. Error (%) All sampled Every tenth sampled Every tenth sampled

26 ai, Vi ai 80 r = Flow-rate Sample No. Weighted mean Simple mean Relat. Error (%) All sampled Every tenth sampled Every tenth sampled

27 RELATIVE ERROR (%) 10 Simulation results CORRELATION COEFFICIENT Weighting error is systematic with structurally (red) and circumstantially (blue) correlated data, but is random for uncorrelated (black) data. 10 % of data sampled.

28 EXAMPLES ON TRUE PROCESS DATA Atmospheric emissions Wastewater Solid wastes

29 FLOWRATE (m3/h) NOx (mg/m3) Weighting error: calculation of mean when flow-rate varies: true Process data x Time (h) NO x concentrations and total gas flow-rate measured as onehour averages from a power plant during one month

30 CALCULATION OF TOTAL MONTHLY NO x EMISSION Mean of NO x concentrations: 9.5 mg/m 3 Mean of gas flow-rate : m 3 /h Total gas flow: m 3 Total NO x emitted (unweighted): kg Weighted mean of NO x concentration: c i V i = 37.5 mg/m V 3 i Total NO x emitted (weighted): kg Weighting error of simple mean: in mean concentration = mg/m 3 in total monthly emission = kg

31 COD in industrial wastewater 10 COD in WASTEWATER; r= a i, FLOW-RATE SAMPLING TIME In this process COD concentration in the discharge from a treatment plant and the flow-rate are circumstantially correlated. Data scaled for display.

32 COD Results: Weighting error Mean of COD = 159. mg O /dm 3 Mean of flow-rate = m 3 /d Total volume of water = m 3 Annual COD discharge (not weighted) = kg O Weighted mean of COD: = kg O /dm 3 Total COD discharged (weighted) = kg O Weighting error of simple mean: as mean COD = 4.06 mg O /dm 3 as total annual discharge = kg O In this case the simple mean is.6 % higher than the weighted mean.

33 Minimization of weighting error in process analysis, when proportional cross-steam sampling cannot be used Flow-rate is measured simultaneously with sampling and used as weight in calculating the mean. Sampling system is coupled to a flow meter so that a fixed volume is taken when the required total volume has passed the sampling point. In this case the simple average can be used as the mean concentration.

34 CONCLUSIONS on weighting errors Weighting error is often a significant component of sampling errors and has to taken into account when the average value, mean concentration or total mass of analyte in the sampling target is estimated. Increasing the No. of samples does not necessarily reduce the weighting error, if the flow-rate and concentration are correlated.

35 SAMPLE MATERIALIZATION: Incorrect sample delimitation Cutter movement Incorrect sample profile

36 Correct sample delimitation Cutter movement Correct sample profiles

37 v = constant 0.6 m/s if d > 3 mm, b 3d = b 0 if d < 3 mm, b 10 mm = b 0 c a b v d = diameter of largest particles b 0 = minimum opening of the sample cutter Correct design for proportional sampler: correct increment extraction

38 Correct design (Vezin cutter) movement

39 Delimitation of a sample from a process stream Correctly delimited sample is a complete slice of equal thickness from the whole process stream Examples from incorrectly delimited samples

40 Determination of the fat content of chocolate The sample was taken from melted chocolate mass with a sampling cup depicted above. After the sample solidified the analytical sample was taken from it by drilling through the solidified mass. Why are the results biased?

41 The analytical sample is incorrectly delimited, because before the melt solidifies the solid particles have time to sediment Correct delimitation with the cup used Correctly designed cup for the drill used for taking the analytical sample

42 Process analyzers often have sample delimitation problems

43 Incorrect Increment and Sample Preparation Errors Contamination (extraneous material in sample) Losses (adsorption, condensation, precipitation, etc.) Alteration of chemical composition (preservation) Alteration of physical composition (agglomeration, breaking of particles, moisture, etc.) Involuntary mistakes (mixed sample numbers, lack of knowledge, negligence) Deliberate faults (salting of gold ores, deliberate errors in increment delimitation, forgery, etc.)

44 Estimation of Fundamental Sampling Error by Using Poisson Distribution Poisson distribution describes the random distribution of rare events in a given interval. If m s is the number of critical particles in sample, the standard deviation expressed as the number of particles is s n m n (1) The relative standard deviation is s r 1 m n ()

45 If mn 0 confidence interval can be approximated by using normal distribution N m, m ) otherwise by using e.g. Poisson ( n n distribution tables or theory If the analytical sample is prepared by serial dilution (e.g., often, when bacterial count is made), the total relative standard deviation of the sampling and dilution chain is s T s r1 sr... srk... mn 1 mn m nk (3)

46 Example Plant Manager: I am producing fine-ground limestone that is used in paper mills for coating printing paper. According to their specification my product must not contain more than 5 particles/tonne particles larger than 5 mm. How should I sample my product? Sampling Expert: That is a bit too general a question. Let s first define our goal. Would 0 % relative standard deviation for the coarse particles be sufficient? Plant Manager: Yes. Sampling Expert: Well, let s consider the problem. We could use the Poisson distribution to estimate the required sample size. Let s see:

47 The maximum relative standard deviation s r = 0 % = 0.. From equation we can estimate how many coarse particles there should be in the sample to have this standard deviation n 1 1 s r 0. 5 If 1 tonne contains 5 coarse particles this result means that the primary sample should be 5 tonnes. This is a good example of an impossible sampling problem. Even though you could take a 5 tonne sample there is no feasible technology to separate and count the coarse particles from it. You shouldn t try the traditional analytical approach in controlling the quality of your product. Instead, if the specification is really sensible, you forget the particle size analyzers and maintain the quality of your product by process technological means, that is, you take care that all equipment are regularly serviced and their high performance maintained to guarantee the product quality. Plant Manager: Thank you

48 PROCESS ANALYSIS: Estimation of point selection error, PSE PSE is the error of the mean of a continuous lot estimated by using discrete samples. PSEdepends on sample selection mode, if consecutive values are autocorrelated. Selection options: random stratified random stratified systematic. Point selection error has two components: PSE = PSE 1 + PSE PSE 1... error component caused by random drift PSE... error component caused by cyclic drift Statistics of correlated series is needed to evaluate the sampling variance.

49 CONCENTRATION CONCENTRATION CONCENTRATION Random selection TIME TIME Stratified selection Systematic selection TIME

50 s x N i n i s 1 s Mean in stratified sampling N1 N 1 n1 s1 N n s 1 n N 1 n n 1 = size of stratum (= No. of potential samples) = NO. of samples taken = between strata variances = within-strata variances determination variances) 1 s x s N if, and N n N n N 1 1 n, 1 1 If every stratum is sampled the between-strata variance is eliminated from the lot mean:

51 When sampling autocorrelated series the same number of samples gives different uncertainties for the mean depending on selection strategy Random sampling: Stratified sampling: Systematic sampling: s s s x x x Normally s p > s str > s sys, s s s p n str n sys except in periodic processes, where s sys may be the largest n s p is the process standard deviation, s str and s sys standard deviation estimates where the autocorrelation has been taken into account.

52 Systematic sampling from periodic process a i TIME a L = 0 a sample = a i TIME a L = 0 a sample = If too low sampling frequency is used in sampling periodic processes there is always a danger that the mean is biased

53 Estimation of PSE by variography Variogaphic experiment: N samples collected at equal distances and analyzed, ai, M s, M are analytical results, i sample sizes and mean sample size, respectively. a a M i Heterogeneity of the i L h s i 1,,, N i, process: al M Mean of the process: a L Variogram of heterogeneity as function of sampling interval j : M si M a si i V j 1 ( N j) N j i 1 ( h h ) i j i, j 1,,, N To estimate variances the variogram has to be integrated (numerically in Gy s method)

54 V V V V A B C D SAMPLE INTERVAL Shapes of variograms: A. Random process; B. Process with non-periodic drift; C. Periodic process; D. Complex process

55 Interpretation of the variogram V s p Range Sill V(drift) V(cyclic) V 0 Random effects (sampling, preparation, analysis) V intercept 0 s p Process (nugget effect) variance SAMPLE LAG, j

56 Example 1: Estimation of sulfur in 1 wastewater effluent 0.5 h i DAYS Standard deviation of the heterogeneity of the process, s p = 0.8 =8. %

57 V i Sample interval (d) Variogram of sulfur in wastewater stream

58 s r (%) 5 0 s str 15 s sys Sample interval (d) Relative standard deviation estimates, which take autocorrelation into account

59 Estimate the uncertainty of the annual mean, if one sample/ week is analyzed by using systematic sample selection Sampling interval = 7 d s sys = 7.8 % Number of samples/y = n =5 ssys 7.8% Standard deviation of the annual mean = sx 1.1% n 5 Expanded uncertainty = U s.% 0.95 x Process standard deviation was 8. %. If the number of samples is estimated by using normal approximation (or samples are selected completely randomly) the required number of samples is for the same uncertainty: s p (8.%) n 657 s (1.1%) x

60 Example : Confidence intervals for COD in wastewater effluent 35 COD 30 5 COD, (mg/l) DAYS Process data: a L = 1.9 mg/l, s(process)= 3.8 mg/l, s r = 9.5 %

61 VARIOGRAM OF h 0.1 COD, variogram 0.1 s p LAG (d) Variogram of the process data

62 sr(syst),sr(strat), (%) 30 s r (Process) Rel. standard deviations LAG (d) Relative standard deviation estimates as function sampling interval for systematic and stratified sampling modes

63 COD, (mg/l) mean COD, (mg/l) Confidence intervals assuming normal distribution 18 Confidence interval of point estimates 18 Confidence interval of developing average DAYS DAYS 95 % confidence intervals (blue) of the daily COD measurements (red) Development of the 95 % confidence intervals (blue) of the mean during 30 days (red)

64 MEASUREMENT OF AVERAGE GMO CONTENT OF SHIPLOADS OF SOY BEANS: Development of sampling protocol During unloading 100 samples (~0.5 kg) were collected from each of 15 shiploads entering different European ports by using systematic sampling mode (KeLDA Project). The shipments presented wide ranges of heterogeneities and GMO concentrations. Evaluation of the uncertainty of the analytical procedure: From each primary sample duplicate extraction of the GMO material was made; Analytical measurement (PCR method) was duplicated from 53 extracts. A shipload closest to nominal mean concentration of 1 % of GMO was used as the Model Lot to develop a fit-for-purpose sampling protocol for correct classification (Labeling) of the Lot.

65 s r (%) s r(tae) s r(extr) s r(pcr) GMO CONTENT (%) Dependence of the relative standard deviation of the Total Analytical Error TAE and its two components on GMO-content given as mass %.

66 s r (sy), s r (st) (%) VARIOGRAM GMO CONTENT (%) 0 MODEL LOT, a L = 4 % SAMPLE No SAMPLING INTERVAL The heterogeneity of the Model Lot was characterized taking 100 samples from a shipload of soy beans for GMO determination.

67 CUMULTIVE PROBABILITY GMO RESULT (%) GMO RESULT (%) Operation characteristics for different incremental sampling schemes if the true mean a L = 1 %: 100, 50, 0, 10, 8, and 5 increments respectively. Y-axis = probability that the result obtained for the mean of the lot is smaller or equal of the value given in X-axis

68 Lower conf. interval (%) SAMPLES/LOT Lower 95 % confidence interval of GMO measurement, if the mean GMO content a L = 1 % and the shipment has the heterogeneity characteristic of the Model Lot

69 CONCLUSIONS Sampling uncertainty can be, and should be estimated If the sampling uncertainty is not known it is questionable whether the sample should be analyzed at all Sampling nearly always takes a significant part of the total uncertainty budget Optimization of sampling and analytical procedures may result significant savings, or better results, including scales from laboratory procedures and process sampling to large national surveys

70 ACKNOWLEDGEMENTS: Special thanks go to my colleague Prof. Kim Esbensen for long-time cooperation and Dr. Claudia Paoletti for cooperation and providing the GMO data. TACK THANK YOU