grain size distributions application nal comments · grain size distributions application nal...

grain size distributions appli ation �nal ommentsDoing statisti s of (grain size) distributions(or why doing it easy if it an be in�nitely ompli ated?)Raimon Tolosana-DelgadoSedi-Seminar6 November 2007

Raimon Tolosana-Delgado Doing statisti s of (grain size) distributions

grain size distributions appli ation �nal ommentsoutline1 grain-size distribution hara terization2 geometry and statisti s of distributions hara teristi sCoDa analysisHilbert spa e3 appli ationgeneralitiesa gla ial data set from the Aar-Gotthard massif (Alps)the Darss sill data set4 �nal omments Raimon Tolosana-Delgado Doing statisti s of (grain size) distributions

grain size distributions appli ation �nal ommentsa grain-size distribution example

sample RT1− 19

Den

sity

0 5 10 15

0.00

0.05

0.10

0.15

0.20



sample RT1− 19

Den

sity

0 5 10 15

0.00

0.05

0.10

0.15

0.20


two measurement typesquasi ontinuous LPS (x 5)binned sieve data


sample RT1− 19

Den

sity

0 5 10 15

0.00

0.05

0.10

0.15

0.20


distribution!in reasingbounded in [0,1℄

grain size distributions appli ation �nal ommentstraditional pro edure (Folk, 1979; Petrology of Sedimentary Ro ks)1 ompute the ummulative urve (distribution)2 read some quantiles form the line, and derive distribution measures: entral tenden ymode: most frequent φmedian: φ50

1

2(φ16 + φ84) ombined 1

3(φ16 + φ84 + φ50)sortinginterquartile range: 1

2(φ75 − φ25) (bad)

1

2(φ84 − φ16) ombined 1

4(φ84 − φ16) + 1

6.6(φ95 − φ5)Argument: statisti s require many al ulations ! omputers are here to do some dirty workRaimon Tolosana-Delgado Doing statisti s of (grain size) distributions

skewnesskurtosis

grain size distributions appli ation �nal omments hara teristi s CoDa analysis Hilbert spa ewhat must have a urve to be a density?it must be positiveits integral (area under the urve) must be one


grain size distributions appli ation �nal omments hara teristi s CoDa analysis Hilbert spa ewhat must have a urve to be a density?it must be positiveits integral (area under the urve) must be onewhat an have/be a distribution?several modes vs. one single maximum ontinuous vs. dis ontinuoussmooth urve vs. fra tal hara ter (extreme irregularity)symmetry vs. skewnessvirtually any shape (not just a normal one)Raimon Tolosana-Delgado Doing statisti s of (grain size) distributions

grain size distributions appli ation �nal omments hara teristi s CoDa analysis Hilbert spa e ompositional data: a brief summary (I) omposition: de�nitionsa omposition is . . . a ve tor x = [x1, . . . xD] withD positive omponents (xi ≥ 0)of onstant sum (P xi = 100)


grain size distributions appli ation �nal omments hara teristi s CoDa analysis Hilbert spa e ompositional data: a brief summary (I) omposition: de�nitionsa omposition is . . . a ve tor x = [x1, . . . xD] withD positive omponents (xi ≥ 0)of onstant sum (P xi = 100)its sample spa e is:the set of possible measurementsthe simplex (SD), e.g. if D = 3 the ternary diagram


grain size distributions appli ation �nal omments hara teristi s CoDa analysis Hilbert spa e ompositional data: a brief summary (I) omposition: de�nitionsa omposition is . . . a ve tor x = [x1, . . . xD] withD positive omponents (xi ≥ 0)of onstant sum (P xi = 100)its sample spa e is:the set of possible measurementsthe simplex (SD), e.g. if D = 3 the ternary diagramits s ale is:a subje tive assessment of the di�eren e between two valuesa relative one, an in rement from 100 ppm to 200 ppm is moresigni� ant that from 1000 ppm to 1100 ppm


grain size distributions appli ation �nal omments hara teristi s CoDa analysis Hilbert spa e ompositional data: a brief summary (II)the Simplex: a meaningful Eu lidean spa e stru turethe losure: C [·] : RD+ −→ S

D : x′ = xP

xi



D : x′ = xP

xi hange operation: perturbation x ⊕ y = C [x1 · y1, . . . , xD · yD]



D : x′ = xP

xi hange operation: perturbation x ⊕ y = C [x1 · y1, . . . , xD · yD]�zero�, neutral element n = [1/D, . . . 1/D] omplement operation: power transformation λ⊙x = Cˆ

xλ1 , . . . , xλ

D

˜



D : x′ = xP


xλ1 , . . . , xλ

D

˜

Raimon Tolosana-Delgado Doing statisti s of (grain size) distributionsx2

x1

x3

0

-1

-2

1

2

-3

3


D : x′ = xP


xλ1 , . . . , xλ

D

˜Ait hison distan e: d2A(x,y) =

∑

i>j

(

ln xi

yi− ln

xj

yj

)2



D : x′ = xP


xλ1 , . . . , xλ

D


∑

i>j

(

ln xi

yi− ln

xj

yj

)2Ait hison s alar produ t 〈x,y〉A =P

i>j ln xi

yi· ln xj

yjAit hison norm ||x||A = d(n,x)



D : x′ = xP


xλ1 , . . . , xλ

D


∑

i>j

(

ln xi

yi− ln

xj

yj


i>j ln xi

yi· ln xj



parallel and orthogonal linesx2

x1

x3

n


D : x′ = xP


xλ1 , . . . , xλ

D


∑

i>j

(

ln xi

yi− ln

xj

yj


i>j ln xi

yi· ln xj



ellipses and ir les


D : x′ = xP


xλ1 , . . . , xλ

D


∑

i>j

(

ln xi

yi− ln

xj

yj


i>j ln xi

yi· ln xj

yjAit hison norm ||x||A = d(n,x)basis: a set of D− 1 ompositions (orthonormal) e1, . . . , eD−1 ∈ SD



D : x′ = xP


xλ1 , . . . , xλ

D


∑

i>j

(

ln xi

yi− ln

xj

yj


i>j ln xi

yi· ln xj

yjAit hison norm ||x||A = d(n,x)basis: a set of D− 1 ompositions (orthonormal) e1, . . . , eD−1 ∈ SD


2 systems of axisx y

z

x y

z


D : x′ = xP


xλ1 , . . . , xλ

D


∑

i>j

(

ln xi

yi− ln

xj

yj


i>j ln xi

yi· ln xj

yjAit hison norm ||x||A = d(n,x)basis: a set of D− 1 ompositions (orthonormal) e1, . . . , eD−1 ∈ SD oordinates: log-ratios (relative s ale) ξ1, . . . ξD−1 ∈ R su h that

x = ξ1 ⊙ e1 ⊕ ξ2 ⊙ e2 ⊕ · · · ⊕ ξD−1 ⊙ eD−1



D : x′ = xP


xλ1 , . . . , xλ

D


∑

i>j

(

ln xi

yi− ln

xj

yj

)2Ait hison s alar produ t 〈x,y〉A = 〈clr(x), clr(y)〉eucAit hison norm ||x||A = d(n,x)basis: a set of D− 1 ompositions (orthonormal) e1, . . . , eD−1 ∈ SD oordinates: log-ratios (relative s ale) ξ1, . . . ξD−1 ∈ R su h that

x = ξ1 ⊙ e1 ⊕ ξ2 ⊙ e2 ⊕ · · · ⊕ ξD−1 ⊙ eD−1 entered log-ratio transformation:clr(x) = ln x

D√

Q

i xi

= ln(x) − 1

D

P

i lnxiRaimon Tolosana-Delgado Doing statisti s of (grain size) distributions

grain size distributions appli ation �nal omments hara teristi s CoDa analysis Hilbert spa efrom ompositions to distributionsdistributions: ompositions of in�nite partsa distribution is . . . a urve f(x) withpositive value f(x) > 0 for all x


grain size distributions appli ation �nal omments hara teristi s CoDa analysis Hilbert spa efrom ompositions to distributionsdistributions: ompositions of in�nite partsa distribution is . . . a urve f(x) withpositive value f(x) > 0 for all xof sum equals 100% =⇒ integral equal to one R

f(x)dx = 1


grain size distributions appli ation �nal omments hara teristi s CoDa analysis Hilbert spa efrom ompositions to distributionsdistributions: ompositions of in�nite partsa distribution is . . . a urve f(x) withpositive value f(x) > 0 for all xof integral equal to one R

f(x)dx = 1its sample spa e is an in�nite-dimensional �simplex�, one of theAit hison spa es of measures (Egoz ue and Diaz-Barrero, 2003; vanden Boogaart, 2005)Raimon Tolosana-Delgado Doing statisti s of (grain size) distributions

grain size distributions appli ation �nal omments hara teristi s CoDa analysis Hilbert spa efrom ompositions to distributionsdistributions: ompositions of in�nite partsa distribution is . . . a urve f(x) withpositive value f(x) > 0 for all xof integral equal to one R

f(x)dx = 1its sample spa e is an in�nite-dimensional �simplex�, one of theAit hison spa es of measures (Egoz ue and Diaz-Barrero, 2003; vanden Boogaart, 2005)its s ale is?Raimon Tolosana-Delgado Doing statisti s of (grain size) distributions

grain size distributions appli ation �nal omments hara teristi s CoDa analysis Hilbert spa efrom ompositions to distributionshow do we ompare grain-size urves?sample RT1− 10

Den

sity

0 5 10 15

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0.05

0.10

0.15

0.20

0.25

0.30

sample RT1− 16

Den

sity

0 5 10 15

0.0

0.1

0.2

0.3

0.4

sample RT1− 17

Den

sity

0 5 10 15

0.00

0.05

0.10

0.15

0.20


grain size distributions appli ation �nal omments hara teristi s CoDa analysis Hilbert spa efrom ompositions to distributionsthe Hilbert spa e of distributions: operationsHilbert spa e: �Eu lidean spa e of in�nite dimension� (but . . .)not unique!


grain size distributions appli ation �nal omments hara teristi s CoDa analysis Hilbert spa efrom ompositions to distributionsthe Hilbert spa e of distributions: operationsHilbert spa e: �Eu lidean spa e of in�nite dimension� (but . . .)not unique!perturbation (translation, sum), powering (multipli ation)the neutral element must be hosen: n(x) (normal, exponential,uniform)Raimon Tolosana-Delgado Doing statisti s of (grain size) distributions

grain size distributions appli ation �nal omments hara teristi s CoDa analysis Hilbert spa efrom ompositions to distributionsthe Hilbert spa e of distributions: operationsHilbert spa e: �Eu lidean spa e of in�nite dimension� (but . . .)not unique!perturbation (translation, sum), powering (multipli ation)the neutral element must be hosen: n(x) (normal, exponential,uniform)s alar produ t ⇒ angles, distan es, lengthsRaimon Tolosana-Delgado Doing statisti s of (grain size) distributions

grain size distributions appli ation �nal omments hara teristi s CoDa analysis Hilbert spa efrom ompositions to distributionsthe Hilbert spa e of distributions: operationsHilbert spa e: �Eu lidean spa e of in�nite dimension� (but . . .)not unique!perturbation (translation, sum), powering (multipli ation)the neutral element must be hosen: n(x) (normal, exponential,uniform)s alar produ t ⇒ angles, distan es, lengths〈f(x), g(x)〉 =

∫

dom(n)

clr (f(x)) · clr (g(x))n(x)dx


grain size distributions appli ation �nal omments hara teristi s CoDa analysis Hilbert spa efrom ompositions to distributionsthe Hilbert spa e of distributions: operationsHilbert spa e: �Eu lidean spa e of in�nite dimension� (but . . .)not unique!perturbation (translation, sum), powering (multipli ation)the neutral element must be hosen: n(x) (normal, exponential,uniform)s alar produ t ⇒ angles, distan es, lengths〈f(x), g(x)〉 =

∫

dom(n)

clr (f(x)) · clr (g(x))n(x)dx lr transformationclr (f(x)) = ln

f(x)

n(x)−

∫

dom(n)

lnf(x)

n(x)n(x)dxRaimon Tolosana-Delgado Doing statisti s of (grain size) distributions

grain size distributions appli ation �nal omments hara teristi s CoDa analysis Hilbert spa efrom ompositions to distributionsthe Hilbert spa e of distributions: basis and oordinatesin�nite dimension ⇒ in�nite oordinates, in�nite elements in a basisorthonormal:∫

d(n)

f∗(x) · g∗(x)n(x)dx = 0 and

∫

d(n)

(f∗(x))2· n(x)dx = 1



d(n)


∫

d(n)

(f∗(x))2· n(x)dx = 1series of polynomials (standard physi al-mathemati al tool)normal density: Hermite polynomialsexponential density: Laguerre polynomialsuniform density: Legendre polynomials (beta density: Ja obipolynomials)



d(n)


∫

d(n)

(f∗(x))2· n(x)dx = 1series of polynomials (standard physi al-mathemati al tool)normal density: Hermite polynomialsexponential density: Laguerre polynomialsuniform density: Legendre polynomials (beta density: Ja obipolynomials) oordinatesHermite polynomials: µ

σ2 , σ2−1

√

2σ2, et . (deviation from normality)Laguerre polynomials: λ − 1Raimon Tolosana-Delgado Doing statisti s of (grain size) distributions

grain size distributions appli ation �nal omments hara teristi s CoDa analysis Hilbert spa eexample of Hermite polynomialspolynomials: H0 = 1, H1 = x, H2 = 1√2(1 − x2)de omposition: φ(x; µ, σ) = φ(x; 0, 1) ⊕ µσ2 ⊙ eH1 ⊕ σ2−1√

2σ2⊙ eH2

−15 −10 −5 0 5

0.00

0.04

0.08

0.12

mean: −5

−15 −10 −5 0 5

050

100

var: 10

−8 −6 −4 −2

0.0

0.1

0.2

0.3

0.4

mean: −5

−8 −6 −4 −2

010

2030

var: 1

−4 −2 0 2 4 6

0.00

0.10

0.20

0.30

mean: 0

−4 −2 0 2 4 6

02

46

810

var: 2

−1.0 −0.5 0.0 0.5 1.0

0.0

0.4

0.8

1.2

mean: 0

−1.0 −0.5 0.0 0.5 1.0

−4

−3

−2

−1

01

var: 0.1

2 3 4 5 6 7

0.0

0.2

0.4

0.6

mean: 5

2 3 4 5 6 7

05

1020

var: 0.5

0 5 10

0.00

0.10

0.20 mean: 5

0 5 10

020

4060

80

var: 5


grain size distributions appli ation �nal omments hara teristi s CoDa analysis Hilbert spa eexample of Hermite polynomials

−10 −5 0 5 10

0.00

0.02

0.04

0.06

0.08

X

Y

0, −0.25, 0, 0, −0.050, −0.25, 0.25, 0, −0.050, −0.25, 0.5, 0, −0.050, −0.25, 0.75, 0, −0.050, −0.25, 1, 0, −0.05


grain size distributions appli ation �nal omments generalities Aar-Gotthard Darssalgorithm1 look at the data2 hoose a reasonable �zero� distribution, referen e distribution


grain size distributions appli ation �nal omments generalities Aar-Gotthard Darssalgorithm1 look at the data2 hoose a reasonable �zero� distribution, referen e distributiondi�usion ⇒ normal distribution omminution ⇒ fra tal, exponential distributionbounded phenomena ⇒ uniform distribution


grain size distributions appli ation �nal omments generalities Aar-Gotthard Darssalgorithm1 look at the data2 hoose a reasonable �zero� distribution, referen e distributiondi�usion ⇒ normal distribution omminution ⇒ fra tal, exponential distributionbounded phenomena ⇒ uniform distribution3 ompute the �standardized� density:ln( your histogram/zero density )=Y4 �x (whi h, how many) polynomials to use, orthonormal with respe tto the zero density (X)Raimon Tolosana-Delgado Doing statisti s of (grain size) distributions

grain size distributions appli ation �nal omments generalities Aar-Gotthard Darssalgorithm1 look at the data2 hoose a reasonable �zero� distribution, referen e distributiondi�usion ⇒ normal distribution omminution ⇒ fra tal, exponential distributionbounded phenomena ⇒ uniform distribution3 ompute the �standardized� density:ln( your histogram/zero density )=Y4 �x (whi h, how many) polynomials to use, orthonormal with respe tto the zero density (X)5 Y = X · β =⇒ regression =⇒ β̂ new �data� (meaningless β0!)Raimon Tolosana-Delgado Doing statisti s of (grain size) distributions

grain size distributions appli ation �nal omments generalities Aar-Gotthard Darssalgorithm1 look at the data2 hoose a reasonable �zero� distribution, referen e distributiondi�usion ⇒ normal distribution omminution ⇒ fra tal, exponential distributionbounded phenomena ⇒ uniform distribution3 ompute the �standardized� density:ln( your histogram/zero density )=Y4 �x (whi h, how many) polynomials to use, orthonormal with respe tto the zero density (X)5 Y = X · β =⇒ regression =⇒ β̂ new �data� (meaningless β0!)6 apply the method you want to β̂prin ipal omponent analysis, non-supervised lassi� ationmapping Raimon Tolosana-Delgado Doing statisti s of (grain size) distributions

grain size distributions appli ation �nal omments generalities Aar-Gotthard Darssa gla ial data set from the Aar-Gotthard massif (Alps)goal and lo ation hara terize the e�e t of (weathering and) omminution ongrain-size/geo hemistry/mineralogy ompositionsfo us: omminution ∼ gla ial omminution (�uvial sorting)Aar-Gotthard graniti massif, entral Alps (Switzerland); RhoneGlets her, Damma Glets her, Tiefer Glets her�uvial sediments, re ent entral morraines, older lateral morrainesgrain size analyses:laser parti le sizer (LPS), 116 lasses of φ ∈ [−0.865, 14.61]sieves, 11 lasses from φ < −1 to φ > 8analyses of major oxides and tra e element geo hemistry (not now)Raimon Tolosana-Delgado Doing statisti s of (grain size) distributions

grain size distributions appli ation �nal omments generalities Aar-Gotthard Darssmeasured densitiessample RT1− 1b

0 5 10 15

0.0

0.2

0.4

0.6

0.8

1.0

sample RT1− 2

0 5 10 15

0.0

0.2

0.4

0.6

0.8

1.0

sample RT1− 3

0 5 10 15

0.0

0.2

0.4

0.6

0.8

1.0

sample RT1− 7a

0 5 10 15

0.0

0.2

0.4

0.6

0.8

1.0

sample RT1− 7b

0 5 10 15

0.0

0.2

0.4

0.6

0.8

1.0

sample RT1− 7c

0 5 10 15

0.0

0.2

0.4

0.6

0.8

1.0

sample RT1− 8

0 5 10 15

0.0

0.2

0.4

0.6

0.8

1.0

sample RT1− 9

0 5 10 15

0.0

0.2

0.4

0.6

0.8

1.0

sample RT1− 10

0 5 10 15

0.0

0.2

0.4

0.6

0.8

1.0

sample RT1− 11

0 5 10 15

0.0

0.2

0.4

0.6

0.8

1.0

sample RT1− 16

0 5 10 15

0.0

0.2

0.4

0.6

0.8

1.0

sample RT1− 17

0 5 10 15

0.0

0.2

0.4

0.6

0.8

1.0

sample RT1− 19

0 5 10 15

0.0

0.2

0.4

0.6

0.8

1.0

sample RT1− 20

0 5 10 15

0.0

0.2

0.4

0.6

0.8

1.0


grain size distributions appli ation �nal omments generalities Aar-Gotthard Darssmeasured densitiessummary of hara teristi ssample gla ier pla ement oarsest modes �nest1b Rhone side yes 2 yes2 Rhone side yes 1 no3 Rhone side yes 1 no7a Damma side (older) yes 3 yes7b Damma side (older) yes 3 no7 Damma side (older) yes many yes8 Damma side (older) yes 2 yes9 Damma front no 1 yes10 Damma front yes 1 no11 Damma front yes 1 no16 Rhone washed no 1 yes17 Rhone washed yes 2 yes19 Tiefer front yes 2 yes20 Tiefer front yes 2 noRaimon Tolosana-Delgado Doing statisti s of (grain size) distributions

grain size distributions appli ation �nal omments generalities Aar-Gotthard Darss lr-transformed densities�tted model: 11 Laguerre polynomials (exponential referen e)0 5 10 15

−6

−4

−2

02

RT1−1b

phi

rel.d

ens.

0 5 10 15

−6

−4

−2

02

4

RT1−2

phi

rel.d

ens.

0 5 10 15

−8

−6

−4

−2

02

4

RT1−3

phi

rel.d

ens.

0 5 10 15

−6

−4

−2

02

RT1−7a

phi

rel.d

ens.

0 5 10 15

−6

−4

−2

02

RT1−7b

phi

rel.d

ens.

0 5 10 15

−6

−4

−2

02

RT1−7c

phi

rel.d

ens.

0 5 10 15

−6

−4

−2

02

RT1−8

phi

rel.d

ens.

0 5 10 15

−6

−4

−2

02

RT1−9

phi

rel.d

ens.

0 5 10 15

−6

−4

−2

02

RT1−10

phi

rel.d

ens.

0 5 10 15

−6

−4

−2

02

RT1−11

phi

rel.d

ens.

0 5 10 15

−8

−6

−4

−2

02

RT1−16

phi

rel.d

ens.

0 5 10 15

−6

−4

−2

02

RT1−17

phi

rel.d

ens.

0 5 10 15

−6

−4

−2

02

RT1−19

phi

rel.d

ens.

0 5 10 15

−6

−4

−2

02

4

RT1−20

phi

rel.d

ens.

RT

1−17

RT

1−10

RT

1−7c

RT

1−8

RT

1−20

RT

1−1b

RT

1−19

RT

1−3

RT

1−7a

RT

1−7b

RT

1−2

RT

1−11

RT

1−9

RT

1−16

020

060

010

00

Cluster Dendrogram

hclust (*, "ward")dist(t(beta))

Hei

ght


grain size distributions appli ation �nal omments generalities Aar-Gotthard Darssoriginal densities�tted model: 11 Laguerre polynomials (exponential referen e)0 5 10 15

0.00

00.

010

0.02

0

RT1−1b

phi

dens

ity

0 5 10 15

0.00

0.01

0.02

0.03

0.04

0.05 RT1−2

phi

dens

ity

0 5 10 15

0.00

0.01

0.02

0.03

0.04

0.05

0.06

RT1−3

phi

dens

ity

0 5 10 15

0.00

0.01

0.02

0.03

0.04

RT1−7a

phi

dens

ity

0 5 10 15

0.00

00.

010

0.02

00.

030

RT1−7b

phi

dens

ity

0 5 10 15

0.00

00.

005

0.01

00.

015

0.02

0

RT1−7c

phi

dens

ity

0 5 10 15

0.00

00.

010

0.02

0

RT1−8

phi

dens

ity

0 5 10 15

0.00

0.01

0.02

0.03

0.04

RT1−9

phi

dens

ity

0 5 10 15

0.00

0.01

0.02

0.03

0.04

RT1−10

phi

dens

ity

0 5 10 15

0.00

0.01

0.02

0.03

0.04

0.05

RT1−11

phi

dens

ity

0 5 10 15

0.00

0.01

0.02

0.03

0.04

0.05

0.06

RT1−16

phi

dens

ity

0 5 10 15

0.00

00.

010

0.02

00.

030

RT1−17

phi

dens

ity

0 5 10 15

0.00

00.

005

0.01

00.

015

0.02

00.

025

RT1−19

phi

dens

ity

0 5 10 15

0.00

0.01

0.02

0.03

0.04

0.05

0.06

RT1−20

phi

dens

ity

−0.6 −0.4 −0.2 0.0 0.2 0.4 0.6

−0.

6−

0.4

−0.

20.

00.

20.

40.

6

explained var: 0.97

Comp.1

Com

p.2 RT1−1b

RT1−2

RT1−3

RT1−7aRT1−7b

RT1−7c

RT1−8

RT1−10

RT1−11

RT1−17

RT1−19

RT1−20

−3 −2 −1 0 1 2 3

−3

−2

−1

01

23

C2

C3 C4C5

C6

C7

C8

C9

C10

C11


grain size distributions appli ation �nal omments generalities Aar-Gotthard Darss omparing densities: luster analysis of β̂'snon-supervised lassi� ation: Ward luster analysisR

T1−

17

RT

1−10

RT

1−7c

RT

1−8

RT

1−20

RT

1−1b

RT

1−19

RT

1−3

RT

1−7a

RT

1−7b

RT

1−2

RT

1−11

RT

1−9

RT

1−16

040

080

012

00

cluster analysis, Ward method


Hei

ght


grain size distributions appli ation �nal omments generalities Aar-Gotthard Darss omparing densities: 9 and 16 against the other?sample RT1− 9

Den

sity

0 5 10 15

0.0

0.1

0.2

0.3

sample RT1− 16

Den

sity

0 5 10 15

0.0

0.1

0.2

0.3

0.4

sample RT1− 17

Den

sity

0 5 10 15

0.00

0.05

0.10

0.15

0.20

sample RT1− 20

Den

sity

0 5 10 15

0.0

0.1

0.2

0.3

0.4


grain size distributions appli ation �nal omments generalities Aar-Gotthard Darss omparing densities: a �map� of di�eren es

Raimon Tolosana-Delgado Doing statisti s of (grain size) distributions−0.6 −0.4 −0.2 0.0 0.2 0.4 0.6

−0.

6−

0.4

−0.

20.

00.

20.

40.

6

explained var: 0.97

Comp.1

Com

p.2

RT1−1b

RT1−2

RT1−3

RT1−7aRT1−7b

RT1−7c

RT1−8

RT1−10

RT1−11

RT1−17

RT1−19

RT1−20

−3 −2 −1 0 1 2 3

−3

−2

−1

01

23

C2

C3 C4

C5

C6

C7

C8

C9

C10

C11

hara teristi smorraines:old side: 7(a,b, ), 8re ent side: 1b, 2, 3front: (9) 10, 11, 19,20washed sediment:(16) 17unimodal:2, 3, 10, 11tail to �nest:heavy: 7 , (16), 17,19yes: 1b, 7a, 8, (9)no: 2, 3, 7b, 10, 11,20C2: pure exponential

grain size distributions appli ation �nal omments generalities Aar-Gotthard Darss omparing densities using the normal referen e


RT

1−7c

RT

1−8

RT

1−3

RT

1−10

RT

1−11

RT

1−2

RT

1−17

RT

1−7a

RT

1−7b

RT

1−20

RT

1−1b

RT

1−19

RT

1−9

RT

1−16

050

000

1000

0015

0000

Cluster Dendrogram


Hei

ght

−0.6 −0.4 −0.2 0.0 0.2 0.4 0.6

−0.

6−

0.4

−0.

20.

00.

20.

40.

6

explained var: 1

Comp.1

Com

p.2

RT1−1b

RT1−2

RT1−3

RT1−7aRT1−7b

RT1−7cRT1−8

RT1−10RT1−11

RT1−17

RT1−19

RT1−20

−10000 −5000 0 5000 10000

−10

000

−50

000

5000

1000

0

C2C3

C4

C5

C6

C7C8

C9 C10C11

hara teristi sold side morraines: 7(a,b, ), 8; re ent side: 1b, 2, 3; front: (9) 10, 11, 19, 20;washed sediment: (16) 17heavy tail to �nest: 7 , (16), 17, 19; some �ne: 1b, 7a, 8, (9); no �ne: 2, 3,7b, 10, 11, 20unimodal: 2, 3, 10, 11; C2: mean; C3: inverse varian e

grain size distributions appli ation �nal omments generalities Aar-Gotthard Darssthe Darss sill data setDrogden Sill

Darss Sill

Darss

peninsula

Falster

Moen

PlantagenetGround

Arkona Basin

Meck

lenburg

Bay

Fehmarn Belt

Oere

so

un

d

Gre

at

Belt

Little

Belt

Kattegat

Kadet C

hannel

Lolland

Germany

Denmark Sweden

Investigation areaRaimon Tolosana-Delgado Doing statisti s of (grain size) distributions10000 20000 30000 40000 50000 60000 70000

4180

000

4200

000

4220

000

4240

000

Easting

Nor

thin

g

grain size distributions appli ation �nal omments generalities Aar-Gotthard Darssthe Darss sill data setthe Darss sill data setBalti Sea (412,560 km2): world large bra kish water bodieso eanographi onditions ontrolled by: (i) large river freshwaterinput, (ii) restri ted o ean onne tions (Danish Straits)∼ 73% of water ex hange is via the Darss Sill (minimum waterdepth: 18 m bsl)prevailing out�ow of bra kish Balti waters in the upper part of thewater olumn usually along the Danish oastin�ow of more saline water in the southern part of the area and inthe deeper part of the Kadet Channeltidal urrents are negligible; sediment-transporting bottom urrentsare intermittentout ropping till geneti ally related to an i e marginal zone, shortre-advan e of the retreating Late Wei hselian i e sheet (∼ 13,500years BP); thin over of lag sediments (lo ally, stones and blo ks >1 m) Raimon Tolosana-Delgado Doing statisti s of (grain size) distributions

grain size distributions appli ation �nal omments generalities Aar-Gotthard Darss onventional analysis: average (µ) and sorting (σ2)10000 20000 30000 40000 50000 60000 70000

4180

000

4200

000

4220

000

4240

000

Easting

Nor

thin

g

mean phi:

0.18−0.810.85−1.51.51−2.17

2.17−2.832.83−3.453.49−4.15

10000 20000 30000 40000 50000 60000 70000

4180

000

4200

000

4220

000

4240

000

Easting

Nor

thin

g

phi variance:

0.03−1.121.13−2.192.23−3.31

3.37−4.314.46−4.836.63−6.63Kadet Channel, Danish oast (↑ φ̄; ↑ σ2

φ;)Kadet Channel, German oast (∼ φ̄; ↓ σ2φ;)Ground, Plantagenet basin? (↑ φ̄; ↓ σ2

φ;)Bar, Darss Sill itself? (↓ φ̄; ↓ σ2φ;)Raimon Tolosana-Delgado Doing statisti s of (grain size) distributions

grain size distributions appli ation �nal omments generalities Aar-Gotthard Darssproposed analysis (I): average (β1 = µσ2 ) and sorting (β2 = σ2−1√

2σ2)

10000 20000 30000 40000 50000 60000 70000

4180

000

4200

000

4220

000

4240

000

Easting

Nor

thin

g

first coordinate:

0.04−18.8218.91−37.2238.04−54.22

57.1−75.1276.05−83.44113.55−113.55

10000 20000 30000 40000 50000 60000 70000

4180

000

4200

000

4220

000

4240

000

Easting

Nor

thin

g

second coordinate:

−26.81 :: −26.81−19.58 :: −17.83−17.63 :: −13.13

−12.58 :: −8.72−8.51 :: −3.96−3.94 :: 0.6two areas with extremely low varian e (↓↓ σ2

φ;)undis riminated ground (∼ σ2φ;)Raimon Tolosana-Delgado Doing statisti s of (grain size) distributions

grain size distributions appli ation �nal omments generalities Aar-Gotthard Darssproposed analysis (II): a non-redundant hara terization−0.2 −0.1 0.0 0.1 0.2

−0.

2−

0.1

0.0

0.1

0.2

Comp.1

Com

p.2

−400 −200 0 200 400

−40

0−

200

020

040

0

b1 b2

β1 and β2 almost proportional (inverse)Raimon Tolosana-Delgado Doing statisti s of (grain size) distributions

grain size distributions appli ation �nal omments generalities Aar-Gotthard Darssproposed analysis (II): a non-redundant hara terization−0.2 −0.1 0.0 0.1 0.2

−0.

2−

0.1

0.0

0.1

0.2

Comp.1

Com

p.2

−400 −200 0 200 400

−40

0−

200

020

040

0

b1 b2

10000 20000 30000 40000 50000 60000 70000

4180

000

4200

000

4220

000

4240

000

Easting

Nor

thin

g

second PC:

−0.86 :: −0.26−0.26 :: 0.350.35 :: 0.93

0.96 :: 1.53Inf :: −Inf2.33 :: 2.76

β1 and β2 almost proportional (inverse) omplementary dire tion to β1||β2 (non-normality?)Raimon Tolosana-Delgado Doing statisti s of (grain size) distributions

grain size distributions appli ation �nal omments generalities Aar-Gotthard Darssproposed analysis (II): a non-redundant hara terization10000 20000 30000 40000 50000 60000 70000

4180

000

4200

000

4220

000

4240

000

Easting

Nor

thin

g

first coordinate:

0.04−18.8218.91−37.2238.04−54.22

57.1−75.1276.05−83.44113.55−113.55

10000 20000 30000 40000 50000 60000 70000

4180

000

4200

000

4220

000

4240

000

EastingN

orth

ing

second PC:

−0.86 :: −0.26−0.26 :: 0.350.35 :: 0.93

0.96 :: 1.53Inf :: −Inf2.33 :: 2.76Kadet Channel, Danish oast (low departure)Kadet Channel, German oast (average departure)two singular areas (extreme sorting, average-high departure)undis riminated ground (low departure) ∼ Danish oastRaimon Tolosana-Delgado Doing statisti s of (grain size) distributions

grain size distributions appli ation �nal omments�nal omments1 grain size urves an be seen as (in�nite-dimensional) ompositionshigh-dimensionalfurther stru ture (ordered bins, smoothness)relative s ale


grain size distributions appli ation �nal omments�nal omments1 grain size urves an be seen as (in�nite-dimensional) ompositionshigh-dimensionalfurther stru ture (ordered bins, smoothness)relative s ale2 grain size information an be summarized in a few parametersdepend on a referen e urveorthogonal =⇒ independentin reasingly omplexRaimon Tolosana-Delgado Doing statisti s of (grain size) distributions

grain size distributions appli ation �nal omments�nal omments1 grain size urves an be seen as (in�nite-dimensional) ompositionshigh-dimensionalfurther stru ture (ordered bins, smoothness)relative s ale2 grain size information an be summarized in a few parametersdepend on a referen e urveorthogonal =⇒ independentin reasingly omplex3 these parameters may be statisti ally treatedto un over groups ( luster)to display patterns of variation (biplot)Raimon Tolosana-Delgado Doing statisti s of (grain size) distributions

grain size distributions appli ation �nal omments�nal omments1 grain size urves an be seen as (in�nite-dimensional) ompositionshigh-dimensionalfurther stru ture (ordered bins, smoothness)relative s ale2 grain size information an be summarized in a few parametersdepend on a referen e urveorthogonal =⇒ independentin reasingly omplex3 these parameters may be statisti ally treatedto un over groups ( luster)to display patterns of variation (biplot)to explain them (regression, anova)to make several measures ompatible (weighted average)Raimon Tolosana-Delgado Doing statisti s of (grain size) distributions

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