Variogram-derived measures for QC purposes

VARIOGRAM-DERIVED MEASURES FOR QC PURPOSES

Markku OhenojaControl Engineering group

University of Oulu

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05/02/2023Faculty of Technology / Control Engineering / Markku Ohenoja

02.05.2023

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Faculty of Technology / Control Engineering / Markku [email protected]

Petersen, L., Minkkinen, P. & Esbensen, K.H. 2005, "Representative sampling for reliable data analysis: Theory of Sampling", Chemometrics and Intelligent Laboratory Systems, vol. 77, no. 1–2, pp. 261-277.

Time

Mea

s.

https://s-media-cache-ak0.pinimg.com/236x/64/46/7f/64467fa3382ac08d567d36b6aef0513b.jpg

BACKGROUND

• All measurements retain some amount of uncertainty, but also sampling errors may affect on the result

• Utilization of different measurements collected with very different sampling rates requires evaluation of their information content

• Environmental measurements are often periodic, sparsely collected and from various sources

• Variographical analysis used for evaluating sampling errors and information content of the measurement

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OUTLINE

• What is Variogram and how it is calculated?

• Variogram-derived measures

• Examples within MMEA

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VARIOGRAM

• Tool for empirical estimation of sampling errors incl. analytical error

• Enables optimizing the sampling strategy with respect to variance of the sampling error and number of samples takes

• Provides an estimate of the standard error of the lot mean and the minimum possible error (MPE) of sampling

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Semi-variogramChrono-variogram

Variographical analysis

GeostatisticsKriging Variography Chronostatistics

VARIOGRAM

• Collection of the data• At least 30 samples with systematic sampling• 1/5 smaller sampling interval than routine samples• Flowrate/sample weight should be included

• 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) (=MPE)• Graphically, separate experiment…

• Auxiliary functions for comparing sampling strategies• Point-to-point calculation, algebraic modeling…

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ℎ𝑛 = 𝑎𝑛 −𝑎𝐿𝑎𝐿 ∙𝑀𝑛𝑀𝑛തതതത

𝑣ሺ𝑗ሻ= 12(𝑁− 𝑗) ൫ℎ𝑛+𝑗 −ℎ𝑗൯2𝑁/2𝑛=1 ≈ 12(𝑁− 𝑗)𝑎𝐿2 ൫𝑎𝑛+𝑗 −𝑎𝑗൯2

𝑁/2𝑛=1

VARIOGRAM

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0 10 20 300

5

10

15

20

25

30Variogram of 24h averaged online data

Sampling interval (days)

Rel

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rd d

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of th

e sa

mpl

ing

erro

r (%

)

0 10 20 300

5

10

15

20

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30Variogram of daily sample


Rel

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of th

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mpl

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)

VariogramSystematic samplingRandom sampling


σ2, σ

, 2σ,

...

VARIOGRAM

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0 10 20 300

5

10

15

20

25

30Variogram of 24h averaged online data


Rel

ativ

e st

anda

rd d

evia

tion

of th

e sa

mpl

ing

erro

r (%

)

0 10 20 300

5

10

15

20

25

30Variogram of daily sample


Rel

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rd d

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of th

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mpl

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r (%

)



3x

INDICES

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• Variogram-based indices applied for QC and PAT purposes

• Standard error of the mean• MPE/σProcess• v(1)/σProcess

INDICES

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• Variogram-based indices applied for QC and PAT purposes

• Standard error of the mean• MPE/σProcess• v(1)/σProcess

Process stability measure

Bisgaard & Kulahci, Quality Engineering, 17(2), 2005

Drift estimationPaakkunainen et al.,

Chemometrics and Intelligent Laboratory Systems, 88(1),

2007Fault diagnosisKouadri et al., ISA

Transactions, 51(3), 2012Temporal

uncertainty propagation

Jalbert et al., Journal of Hydrology, 397(1-2), 2011

DQOs for control charts

Minnit & Pitard, Journal of SAIMM, 108(2), 2008

STANDARD ERROR OF THE MEAN

• Variance estimate of the sampling attained from variogram

• Standard error of the mean calculated based on variance estimate and number of samples collected during a selected time frame

• Recursive calculation possible for online measurements moving average and its confidence intervals from the selected time frame

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Month 2M 3M 4M 5M HalfYear Year All0

0.5

1

1.5

2

2.5

3

2M

, %

Time frame for the lot mean

Online 17h averageOnline 12h averageOnline 8h averageOnline 6h averageOnline 4h averageOnline data

Month 2M 3M 4M 5M HalfYear Year All0

5

10

15

20

25

30

35

40

2M

, %

Time frame for the lot mean

LaboratoryCalibrated onlineRaw online x 10



02.05.2023TIEDEKUNTA TIEDEKUNTA / osasto osasto osaston osasto / Etuniminen Sukuniminen-Sukuniminen

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23-Nov-2009 12-Jan-2010 03-Mar-2010 22-Apr-2010 11-Jun-2010 31-Jul-2010 19-Sep-2010 08-Nov-2010 28-Dec-2010 16-Feb-20110

20

40

6031-Dec-2010

7.341 7.3415 7.342 7.3425 7.343 7.3435 7.344 7.3445 7.345 7.3455x 105

10

15

20

25Lot mean and 2

M (%) for Three day average

7.341 7.3415 7.342 7.3425 7.343 7.3435 7.344 7.3445 7.345 7.3455x 105

0

10

20

30

23-Nov-2009 12-Jan-2010 03-Mar-2010 22-Apr-2010 11-Jun-2010 31-Jul-2010 19-Sep-2010 08-Nov-2010 28-Dec-2010 16-Feb-20115

10

15

20

25

30Lot mean and confidence intervals for Three day average

DATA COMPARISON

• Multiple measurement sources with different sampling rates

• Data harmonization and comparison• Based on MPE• Comparable averaging of the dense data around sparse

samples,• Variographical analysis for whole averaged dense data

mimicking more densely collected laboratory measurements

• Information content evaluation based on v(1)/σProcess

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WHAT SPARSE CANNOT SEE?

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0 5 10 15 20 250.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16Variograms for collective samples

Sampling interval

Varia

nce

Variogram, Sparse meas.Variogram, Av. dense meas.

0 5 10 15 20 250

0.1

0.2Variogram of sparse measurement

Varia

nce

Sampling interval

0 200 400 600 800 10000

0.1

0.2Variogram of averaged dense measurement

Sampling interval

Varia

nce


WHEN DENSE IS NOT REPRESENTATIVE?

02.05.2023TIEDEKUNTA TIEDEKUNTA / osasto osasto osaston osasto / Etuniminen Sukuniminen-Sukuniminen

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26-May-2013 05-Jul-2013 14-Aug-2013 23-Sep-2013 02-Nov-2013 12-Dec-20130

10

20

30

40

Mea

s.

Time series

Dense meas.Sparse meas.

26-May-2013 05-Jul-2013 14-Aug-2013 23-Sep-2013 02-Nov-2013 12-Dec-20130

0.5

1

1.5es/P

Inde

x

Dense meas.Sparse meas.

26-May-2013 05-Jul-2013 14-Aug-2013 23-Sep-2013 02-Nov-2013 12-Dec-2013-1

-0.5

0

0.5

1Substracted index

Inde

x

SUMMARY

02.05.2023Faculty of Technology / Control Engineering / Markku [email protected]

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• Variogram can be utilized for1. Sampling error estimation2. Sampling optimization3. Moving average and confidence interval calculation4. Information content evaluation

• Recursive calculation enables e.g. monitoring, filtering, decision making

• Information content evaluation allows comparison of measurement sources