The statistical analysis of compositional data: The Aitchison … · 2011-08-09 · V. Pawlowsky-Glahn and J. J. Egozcue CoDa historical remarks sample space Aitchison geometry ﬁnal

V. Pawlowsky-Glahn and

J. J. Egozcue

CoDa historical remarks sample space Aitchison geometry final comments

The statistical analysis ofcompositional data:

The Aitchison geometry

Prof. Dr. Vera Pawlowsky-GlahnProf. Dr. Juan Jose EgozcueAss. Prof. Dr. Rene Meziat

Instituto Colombiano del PetroleoPiedecuesta, Santander, Colombia

March 20–23, 2007


J. J. Egozcue


logo

IAMG Distinguished Lecturer – 2007

Prof. Dr. Vera Pawlowsky-Glahn

Department of Computer Science and Applied MathematicsUniversity of Girona, Spain


J. J. Egozcue


recall

compositional data are parts of some wholewhich only carry relative information

usual units of measurement: parts per unit,percentages, ppm, ppb, concentrations, ...

historically: data subject to a constant sumconstraint

examples: geochemical analysis; (sand, silt, clay)composition; proportions of minerals in a rock; ...


J. J. Egozcue


historical remarks: end of the XIXth century

Karl Pearson, 1897: “On a form of spurious correlationwhich may arise when indices are used in themeasurement of organs”

he was the first to point out dangers that may befall theanalyst who attempts to interpret correlations betweenratios whose numerators and denominators containcommon parts

the closure problem was stated within the framework ofclassical statistics, and thus within the framework ofEuclidean geometry in real space


J. J. Egozcue


the problem: negative bias & spurious correlation

example: scientists A and B record the composition of aliquots of soilsamples; A records (animal, vegetable, mineral, water) compositions,B records (animal, vegetable, mineral) after drying the sample; both areabsolutely accurate (adapted from Aitchison, 2005)

sample A x1 x2 x3 x4

1 0.1 0.2 0.1 0.62 0.2 0.1 0.2 0.53 0.3 0.3 0.1 0.3

sample B x ′1 x ′

2 x ′3

1 0.25 0.50 0.252 0.40 0.20 0.403 0.43 0.43 0.14

corr A x1 x2 x3 x4

x1 1.00 0.50 0.00 -0.98x2 1.00 -0.87 -0.65x3 1.00 0.19x4 1.00

corr B x ′1 x ′

2 x ′3

x ′1 1.00 -0.57 -0.05

x ′2 1.00 -0.79

x ′3 1.00


J. J. Egozcue


historical remarks: from 1897 to 1980 (and beyond)

the fact that correlations between closed data are inducedby numerical constraints caused Felix Chayes to attemptto separate the spurious part from the real correlation

(“On correlation between variables of constant sum”, 1960)

many studied the effects of closure on methods related tocorrelation and covariance analysis (principal componentanalysis, partial and canonical correlation analysis) ordistances (cluster analysis)

an exhaustive search was initiated within the frameworkof classical (applied) statistics


J. J. Egozcue


historical remarks: end of the XXth century

John Aitchison, 1982, 1986: “The statistical analysis ofcompositional data”

key idea: compositional data represent parts of somewhole; they only carry relative information

by analogy with the log-normal approach, Aitchisonprojected the sample space of compositional data,the D-part simplex SD, to real space RD−1 or RD,using log-ratio transformations

the log-ratio approach was born ...


J. J. Egozcue


compositional data: definition

definition: parts of some whole which carry only relativeinformation ⇐⇒ compositional data are equivalence classes

X2

1

1 X1

compositional data in R2 compositional data in R3

usual representation: subject to a constant sum constraint


J. J. Egozcue


compositional data: usual representation

definition: x = [x1, x2, . . . , xD] is a D-part composition

⇐⇒

xi > 0, for all i = 1, ..., DD∑

i=1xi = κ (constant)

κ = 1 ⇐⇒ measurements in parts per unitκ = 100 ⇐⇒ measurements in percent

other frequent units: ppm, ppb, ...

a subcomposition xs with s parts is obtained as the closure ofa subvector

[xi1 , xi2 , . . . , xis

]of x


J. J. Egozcue


the simplex as sample space

SD = {x = [x1, x2, . . . , xD]|xi > 0;D∑

i=1

xi = κ}

standard representation for D = 3:the ternary diagram

X1

X2

X3

x2

x1

x3


J. J. Egozcue


example 1: genetic hypothesis

MN

MM NN

data: genotyps in the MN system of blood groups; code: Ab = Aborigines;Ch = Chinese; In= Indian; AmIn = American Indian; Es = Eskimo;question: despite the high variability which can be observed, is there anyinherent stability in the data? do they follow any genetic law?


J. J. Egozcue


requirements for a proper analysis

scale invariance: the analysis should not depend on theclosure constant κ

permutation invariance: the order of the parts should beirrelevant

subcompositional coherence: studies performed onsubcompositions should not stand in contradiction withthose performed on the full composition


J. J. Egozcue


why a new geometry on the simplex?

in real space we add vectors, we multiply them by a constant, welook for orthogonality between vectors, we look for distancesbetween points, ...

possible because <D is a linear vector space

BUT Euclidean geometry is not a proper geometry for compositionaldata because

results might not be in the simplex when we addcompositional vectors, multiply them by a constant, or computeconfidence regions

Euclidean differences are not always reasonable: from0.05% to 0.10% the amount is doubled; from 50.05% to 50.10%the increase is negligible


J. J. Egozcue


basic operations

closure of z = [z1, z2, . . . , zD] ∈ <D+

C [z] =

[κ · z1∑D

i=1 zi,

κ · z2∑Di=1 zi

, · · · ,κ · zD∑D

i=1 zi

]

perturbation of x ∈ SD by y ∈ SD

x⊕ y = C [x1y1, x2y2, . . . , xDyD]

powering of x ∈ SD by α ∈ <

α� x = C [xα1 , xα

2 , . . . , xαD ]


J. J. Egozcue


interpretation of perturbation and powering

A

B C

A

B C

left: perturbation of initial compositions (◦) by p = [0.1, 0.1, 0.8]resulting in compositions (?)

right: powering of compositions (?) by α = 0.2 resulting incompositions (◦)


J. J. Egozcue


comments

closure = projection of a point in <D+ on SD

points on a ray are projected onto the same point

a ray in <D+ is an equivalence class

the point on SD is a representant of the class

a generalization to other representants is possible

for z ∈ <D+ and x ∈ SD, x⊕ (α� z) = x⊕ (α� C [z])


J. J. Egozcue


vector space structure of (SD,⊕,�)

commutative group structure of (SD,⊕)1 commutativity: x⊕ y = y⊕ x2 associativity: (x⊕ y)⊕ z = x⊕ (y⊕ z)3 neutral element: e = C [1, 1, . . . , 1] = barycentre of SD

4 inverse of x: x−1 = C[x−1

1 , x−12 , . . . , x−1

D

]⇒ x⊕ x−1 = e and x⊕ y−1 = x y

properties of powering1 associativity: α� (β � x) = (α · β)� x;2 distributivity 1: α� (x⊕ y) = (α� x)⊕ (α� y)3 distributivity 2: (α + β)� x = (α� x)⊕ (β � x)4 neutral element: 1� x = x


J. J. Egozcue


inner product space structure of (SD,⊕,�)

inner product : 〈x, y〉a =1

2D

D∑i=1

D∑j=1

lnxi

xjln

yi

yj, x, y ∈ SD

norm : |x|a =

√√√√ 12D

D∑i=1

D∑j=1

(ln

xi

xj

)2

, x ∈ SD

distance : da(x, y) =

√√√√ 12D

D∑i=1

D∑j=1

(ln

xi

xj− ln

yi

yj

)2

, x, y ∈ SD

Aitchison geometry on the simplex


J. J. Egozcue


properties of the Aitchison geometry

distance and perturbation: da(p⊕ x, p⊕ y) = da(x, y)

distance and powering: da(α� x, α� y) = |α|da(x, y)

compositional lines: y = x0 ⊕ (α� x)(x0 = starting point, x = leading vector)

orthogonal lines: y1 = x0 ⊕ (α1 � x1), y2 = x0 ⊕ (α2 � x2),

y1 ⊥y2 ⇐⇒ 〈x1, x2〉a = 0

(the inner product of the leading vectors is zero)parallel lines: y1 = x0 ⊕ (α� x) ‖ y2 = p⊕ x0 ⊕ (α� x)


J. J. Egozcue


orthogonal compositional lines

x y

z

x y

z

orthogonal grids in S3, equally spaced, 1 unit in Aitchisondistance; the right grid is rotated 45o with respect to the left grid


J. J. Egozcue


circles and other geometric figures

x2

x1

x3

n


J. J. Egozcue


advantages of Euclidean spaces

orthonormal basis can be constructed: {e1, . . . , eD−1}

coordinates obey the rules of real Euclidean space:

x ∈ SD ⇒ y = [y1, . . . , yD−1] ∈ RD−1, with yi = 〈x, ei〉astandard methods can be directly applied to coordinates

expressing results as compositions is easy:

if h : SD 7→ RD−1 assigns to each x ∈ SD its coordinates,i.e. h(x) = y, then

h−1(y) = x =D−1⊕i=1

yi � ei


J. J. Egozcue


conclusions

the Aitchison geometry of the simplex offers a new tool toanalyse CoDa

the geometry is apparently complex, but it is completelyequivalent to standard Euclidean geometry in real space

the key is to use a proper representation in coordinates

Documents

The statistical analysis of compositional data: The Aitchison … · 2011-08-09 · V. Pawlowsky-Glahn and J. J. Egozcue CoDa historical remarks sample space Aitchison geometry ﬁnal