Compositional analyses and other things

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Compositional analyses and other things

Rémi Losno

IPGP – Université de Paris

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Compositional Analysis

Statistics applied to geochemistry

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Composition of matter

● A given material is composed of parts. The sum of the weigh of each part if equal to the weigh of the material.

● Composition in a given part is the weigh of the part divided by the weigh of the material.

– w_total = Σ w_i

– C_i = w_i / w_total

● w is an extensive parameter, C an intensive one

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Closure condition

● If the given material contains n components, each w_i can vary independently and all

possible values make a D dimension space.

● The space determined by the C_i values has a dimension equal to D-1 because of the

closure condition constraint: Σ C_i = 100%

● All compositional values are therefore linked together and cannot vary independently. For example, if C_j increases, all the C_i decrease.

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Consequences

● So called "spurious correlations"

– Variations of C_j induce all C_i varying together

● Difficulties to decipher what is really varying, C_j ou C_k?

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"Spurious correlation"

A B C Sum

6.4 6.8 97 110

8.4 7.4 62 78

7.0 6.5 105 119

5.7 6.6 137 149

6.0 7.8 145 159

7.0 6.3 86 99

6.2 5.4 107 118

7.2 7.6 93 108

6.8 5.5 98 111

6.6 7.7 42 56

6.3 6.3 93 105

A B C

6% 6% 88%

11% 10% 80%

6% 6% 89%

4% 4% 92%

4% 5% 91%

7% 6% 87%

5% 5% 90%

7% 7% 86%

6% 5% 89%

12% 14% 74%

6% 6% 88%

3 components A, B and C, dilution by C

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5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 5.0

5.5 6.0 6.5 7.0 7.5 8.0

Quantities Data

A

B

5%

10%

15%

f(x) = 1.018 x − 0.001 R² = 0.868

Compositional data

B

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5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 5.0

5.5 6.0 6.5 7.0 7.5 8.0

Quantities Data

A

B

3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13%

0%

5%

10%

15%

f(x) = 1.018 x − 0.001 R² = 0.868

Compositional data

A

B

To get ride of such

behaviour issue, no way with a patch on

regression equation. The solution is coming with a modification of the

variable space

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New tools for compositional statistics

● Composition property

– do not depend on the size of the sample

– a minor component has no influence if a simple sum is used

● Performing elemental ratios

– log ratio rather than linear ratio

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Example: aerosols and soils

● Chemical composition

– Soils

– Sieved soils

– Generated aerosols

● 2 aerosol generation method comparison

– wind tunnel

– Sygavib

● 4 soils

Trabelsi et al. submitted to JGR

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Sygavib system

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CaO SiO2 Al2O3 Fe2O3 MgO K2O Na2O TiO2 SrO MnO

0.1 1.0 10.0

Syg Hsar WT Hsar FS Hsar

Classical approach

Ratio on bulk soil

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CaO SiO2 Al2O3 Fe2O3 MgO K2O Na2O TiO2 SrO MnO 0.0

1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

Classical approach: linear

scale

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14 / 33 CaO Al2O3 Fe2O3 MgO K2O Na2O TiO2 SrO MnO

0.100 1.000 10.000

What else? SiO2 removed

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CaO Al2O3 Fe2O3 MgO K2O Na2O TiO2 SrO MnO 0.000

1.000 2.000 3.000 4.000 5.000 6.000 7.000

What else? (linear = bad)

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16 / 33 CaO

Al2O3

Fe2O3

MgO K2O

Na2O

TiO2

SrO

MnO 0.100

1.000 10.000

Syg Attaya WT Attaya FS Attaya

CaO Al2O3

Fe2O3

MgO K2O

Na2O TiO2 SrO

MnO 0.100

1.000 10.000

Syg Cherarda WT Cherarda

CaO

Fe2O3

K2O

TiO2

MnO

0.100 1.000 10.000

CaO

Al2O3

Fe2O3

MgO K2O

Na2O

TiO2

SrO

MnO

0.100 1.000 10.000

Syg Ghraiba WT Ghraiba FS Ghraiba

What else?

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Compositional tools

● R, "compositions" package

● Variance and covariance analyses of log-ratio with

– PCA (Principal Component Analyse)

– Plot samples on a "compositional distance"

graph.

● Very sensitive to analytical uncertainties, fail if one zero is encountered.

– remove dubious variables

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Biplot including bulk and fine soils, and generated aerosol. Comp 1 and Comp 2 account together for ca. 86% of the total variance (56% and 30%, respectively). a:

Attaya, b: Cherrarda, c: Ghraiba, d: Hsar

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Biplot when Si and Na are removed.

Differences remain between bulk soil and fine fractions

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How to figure what is plotted

From John Aitchison, A concise Guide to Compositional Data Analysis, 2nd Compositional Data Analysis Workshop, CoDaWork’05, Girona Universitat de Girona

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What happens element by element?

● Comparison between two sampling heads

– One commercial

– One home made

● Field campaign in Tunisia

● Y. Xu et al., in preparation (hope to be submitted soon).

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Where we were

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Biplot on composition

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Biplot on composition

Projection of the perturbation vector

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Perturbation vector

● Compositional distance

● To see element by element

● probability of composition change

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REEs: subset of

compositional data

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REEs perturbation vector

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A new tool for REEs profile presentation

Crozet Kerguelen

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Instead of ...

La Ce Pr Nd Sm Eu Gd Tb Dy Ho Er Tm Yb Lu 0

0.5 1 1.5 2 2.5 3

3.5 Normalized REE profiles

Ker Cro

Who can see here that Ce will not discriminate Cro from Ker?

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Conclusions

● Compositional tools improve

– discussions on compositional data

– mathematics related to chemical compositions

– robustness of conclusion

● Compositional tools dramatically decrease the amount of word and sentences to

describe compositional evolution