THEORY OF SAMPLING MMEA Certainty Seminar Markku Ohenoja 1 Markku Ohenoja / Control Engineering Laboratory

BACKGROUND Literature review on measurement uncertainty (MU) Including sampling process, composite samples, digital signal processing Importance of MU estimation well recognized ISO standards/guides present the well-established methods defined in metrology The methods basically assume a correct sampling process and unbiased measurements  Estimation of measurement uncertainty in the presence of sampling errors and systematic errors Markku Ohenoja / Control Engineering Laboratory 2

OUTLINE General approach to MU estimation Modeling approach Empirical approach Theory of sampling Motivation Sampling strategies Sampling errors Variographic analysis Other methods for estimating sampling uncertainty Uncertainty arising from systematic errors Markku Ohenoja / Control Engineering Laboratory 3

GENERAL APPROACH TO THE ESTIMATION OF MEASUREMENT UNCERTAINTY 4 Markku Ohenoja / Control Engineering Laboratory

MODELING APPROACH MATHEMATICAL/THEORETICAL/PREDICTIVE/BOTTOM-UP APPROACH Modeling the measurement Defining the measurement model (inputs of the model) Defining the ”spread” of input values Correlations and sensitivity Calculating combined (and expanded) uncertainty Well established for analytical measurements ISO guide/JCGM100/GUM Markku Ohenoja / Control Engineering Laboratory 5

MODELING APPROACH MATHEMATICAL/THEORETICAL/PREDICTIVE/BOTTOM-UP APPROACH Modeling the measurement Defining the measurement model (inputs of the model) Defining the ”spread” of input values Correlations and sensitivity Calculating combined (and expanded) uncertainty Well established for analytical measurements ISO guide/JCGM100/GUM, EURACHEM guide… Some drawbacks in the original guidance found and corrected Interpretation with complete Bayesian theory Non-Gaussian probability distributions: JCGM101 Other drawbacks found Cannot treat systematic effects Usually do not recognize sampling as a source of uncertainty Markku Ohenoja / Control Engineering Laboratory 6

EMPIRICAL APPROACH EXPERIMENTAL/RETROSPECTIVE/TOP-DOWN APPROACH Inter-organizational trials, internal methods validations, quality control Duplicate method + statistical analysis Well established for analytical measurements EURACHEM guide Cannot identify the sources of error components Sampling+analytical precision and analytical bias can be estimated Applicable for heterogeneous targets, field sampling… Estimating sampling bias is more challenging Other methods Sampling protocols Collaborative trial in sampling Sampling proficiency test Variographic analysis Markku Ohenoja / Control Engineering Laboratory 7

GY’S THEORY OF SAMPLING (TOS) 8 Markku Ohenoja / Control Engineering Laboratory

MOTIVATION OF TOS An analyst usually just analyses an object delivered to him  Defining the measurement target (that object or the lot from where it was taken) Heterogeneity of the sampling target ignored  Grab sampling  non-representative sample How to remove a sample from the lot? How much material should be selected? How often samples should be taken? Also applies for secondary sampling etc Markku Ohenoja / Control Engineering Laboratory 9

EXAMPLES OF POOR SAMPLING Markku Ohenoja / Control Engineering Laboratory 10 Grab sample Composite sample

SAMPLING ERRORS ACCORDING TO TOS Markku Ohenoja / Control Engineering Laboratory 11 Fundamental sampling errorGrouping and segregation error Incorrect sampling errors: Incremental delimitation error Increment extraction error Increment preparation error Point selection errors: Long-term PSE / Time fluctuation error Periodic PSE / Cyclic fluctuation error Sample weighting error

ESTIMATION OF SAMPLING ERRORS Modeling approach FSE estimation Mainly for particulate systems Expectance value for critical particles in the sample Gy’s formula Empirical approach  Variographical analysis / Variography Central 1-D TOS tool for practical sampling purposes FSE+GSE+TAE+IPE = minimum possible error Identifying PSEs Sampling intervals and sampling strategies Compositing of samples in spreadsheets Process Analysis Tool Markku Ohenoja / Control Engineering Laboratory 12

SAMPLING STRATEGIES Markku Ohenoja / Control Engineering Laboratory 13

VARIOGRAPHY - MATHEMATICS Collection of the data At least 30 samples with systematic sampling 1/5 smaller sampling interval than routine samples Calculation of “the heterogeneity” of the data Calculation of the experimental variogram v(j) Relationship between the samples and the lag distance j Estimation of the intercept v(0) (nugget) Graphically, separate experiment… Auxiliary functions for comparing sampling strategies Point-to-point calculation, algebraic modeling… Markku Ohenoja / Control Engineering Laboratory 14

VARIOGRAPHY - VARIOGRAM Markku Ohenoja / Control Engineering Laboratory 15 Sill Maximum variance of the data series investigated Time series is no longer correlated Range Lag distance where variogram levels off Lags below this are more and more autocorrelated and TSE is smaller Nugget, v(0) Variance in 0-D situation Minimum possible error

VARIOGRAPHY - EXAMPLES Markku Ohenoja / Control Engineering Laboratory 16 Analytical results Sample mass Variogram Simulated cross-stream sampling example

VARIOGRAPHY - EXAMPLES Markku Ohenoja / Control Engineering Laboratory 17 Random Stratified Systematic Sampling variance 3 * Standard deviation (%) Simulated cross-stream sampling example

VARIOGRAPHY - EXAMPLES Markku Ohenoja / Control Engineering Laboratory 18 Sulfur discharge in wastewater in one month period Confidence interval for average sulfur measurement over 1-year period: 1 sample/week  52 samples  U=±1.1% (Systematic sampling) 2 samples/week  104 samples  U=±0.57% (Systematic sampling) Assuming normally distributed measurements and U=±1.1% acceptable  657 samples needed (Random sampling)

VARIOGRAPHY - EXAMPLES Markku Ohenoja / Control Engineering Laboratory 19 Monitoring of the daily biogas yield (CH 4 ) in a three month period

OTHER APPROACHES FOR ESTIMATING THE UNCERTAINTY ARISING FROM SAMPLING 20 Markku Ohenoja / Control Engineering Laboratory

MODELING APPROACH Gy’s formula Fish-bone diagrams Markku Ohenoja / Control Engineering Laboratory 21 Heterogeneity invariant Constitutional parameter Liberation parameter Particle shape parameter Size range parameter Top particle size

EMPIRICAL APPROACH Duplicate method Sampling and analytical precision Analytical bias from certified reference materials Sampling bias if reference target available, or inter-organizational sampling trials Markku Ohenoja / Control Engineering Laboratory 22

UNCERTAINTY ARISING FROM SYSTEMATIC ERRORS 23 Markku Ohenoja / Control Engineering Laboratory

SOURCES OF SYSTEMATIC ERRORS Calibration Sampling Digital signal processing Bias from processing algorithms Rounding phenomena (finite word length) A/D conversions Gain and its temperature effect, long-term stability and temperature drift of onboard calibration reference, integral nonlinearity, cross-talk, time jitter Usually systematic errors are (assumed to be) corrected Some errors cannot be identified or quantified Markku Ohenoja / Control Engineering Laboratory 24

ESTIMATION OF SYSTEMATIC ERRORS Empirical approach of MU estimation Reference targets, inter-organizational sampling trials Increment of expanded uncertainty to include bias Possibility theory / Theory of evidence Random-fuzzy variables Markku Ohenoja / Control Engineering Laboratory 25 Systematic Random

SUMMARY 26 Markku Ohenoja / Control Engineering Laboratory

SUMMARY General approach to MU estimation Theory of sampling Other methods for estimating sampling uncertainty Uncertainty arising from systematic errors Markku Ohenoja / Control Engineering Laboratory 27

SUMMARY General approach to MU estimation  Sampling is seldom accounted in MU estimation Theory of sampling  TOS provides tools for estimating sampling errors  Variography for 1-D objects (process streams) Estimation of minimum possible error Comparison of sampling intervals and strategies Process analytical tool Other methods for estimating sampling uncertainty Uncertainty arising from systematic errors  Random-fuzzy approach for systematic errors Markku Ohenoja / Control Engineering Laboratory 28