Modeling compositional data

Some collaborators

Deformations: Paul SampsonWendy Meiring, Doris DamianSpace-time: Tilmann GneitingFrancesca BrunoDeterministic models: Montserrat Fuentes, Peter ChallenorMarkov random fields: Finn LindströmWavelets: Don PercivalBrandon Whitcher, Peter Craigmile, Debashis Mondal

Background

NAPAP, 1980’s

Workshop on biological monitoring, 1986

Dirichlet process: Gary Grunwald, 1987

Current framework: Dean Billheimer, 1995

Other co-workers: Adrian Raftery, Mariabeth Silkey, Eun-Sug Park

Compositional data

Vector of proportions

Proportion of taxes in different categories

Composition of rock samples

Composition of biological populations

Composition of air pollution

z =(z1,...,zk)T zi >0 zi =11

k

∑ z∈∇k−1

The triangle plot

Proportion 1

0

1

1

0

1

0

Proportion 2

Proportion 3

(0.55,0.15,0.30)

The spider plot

(0.40,0.20,0.10,0.05,0.25)

0.2

0.4

0.6

0.8

1.0

An algebra for compositions

Perturbation: For define

The composition acts as a zero, so .

Set so .

Finally define .

ξ,α ∈∇k−1

€

ξ ⊕ α =ξ1α1

ξ iα i1

k

∑,...,

ξkαk

ξ iα i1

k

∑

⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟∈∇k−1

ι =1k,...,

1k

⎛ ⎝ ⎜

⎞ ⎠ ⎟

ξ⊕ι =ξ

ξ−1 =1ξ1

,...,1ξk

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ ξ⊕ξ−1 =ι

ξ−η=ξ⊕η−1

The logistic normal

If

we say that z is logistic normal, in short Z ~ LN(,).

Other distributions on the simplex:

Dirichlet — ratios of independent gammas

“Danish” — ratios of independent inverse Gaussian

Both have very limited correlation structure.

alr(z)= logz1

zk,...,log

zk−1

zk

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

T

~MVN(μ,Σ)

Scalar multiplicationLet a be a scalar. Define

is a complete inner product space, with inner product given, e.g., by

N is the multinomial covariance N=I+jjT

j is a vector of k-1 ones.

is a norm on the simplex.

The inner product and norm are invariant to permutations of the components of the composition.

ξ⊗a=ξ1

a

ξia∑,...,

ξka

ξia∑

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

∇k−1,⊕,⊗( )

ξ,η =alr(ξ)TN−1alr(η)

€

ξ = ξ,ξ

Some models

Measurement error:

where j ~ LN(0,) .Regression:

Correspondence in Euclidean space:

ξj ξ uj

zj =ξ⊕εj

ξj =ξ⊕γ⊗uj

compositions

centeredcovariate

μj = β0 + β1 (xj −x )

alr−1(μj)=alr−1(β0)⊕alr−1(β1)⊗(xj −x )

Some regression lines

Time series (AR 1)

zk+1 =φ⊗ zk ⊕ k

A source receptor model

Observe relative concentration Yi of k species at a location over time.

Consider p sources with chemical profiles j. Let αi be the vector of mixing proportions of the different sources at the receptor on day i.

~ LN, αi ~ indep LN, i ~ zero mean LN

EYi = αiji=1

p

∑ θj =Θαi

Y =Θαi ⊕εi

Juneau air quality

50 observations of relative mass of 5 chemical species. Goal: determine the contribution of wood smoke to local pollution load.

Prior specification:

Inference by MCMC.

f(,α i, i,α ,Γ, ) =

f(α i α ,Γ) f( i )f(α )f(Γ)f( )

Wood smoke contribution

95% CL

50% CL

Source profiles

(fluoranthene)

(pyrene)

(benzo(a))

(chrysene)

(benzo(b))

State-space model

Space-time model of proportions

State-space model:

zj unobservable composition ~ LN(j,j)

yj k-vector of counts ~ Mult(

Inference using MCMC again

€

yj[ ]ii=1

k

∑ ,zj )

Stability of arthropod food webs

Omnivory thought to destabilize ecological communities

Stability: Capacity to recover from shock (relative abundance in trophic classes)

Mount St. Helens experiment: 6 treat-ments in 2-way factorial design; 5 reps.Predator manipulation (3 levels)Vegetation disturbance (2 levels)

Count anthropods, 6 wks after treatment. Divide into specialized herbivores, general herbivores, predators.

Specification of structure

is generated from independent observations at each treatment

mean depends only on treatment

Benthic invertebrates in estuary

EMAP estuaries monitoring program: Delaware Bay 1990. 25 locations, 3 grab samples of bottom sediment during summer

Invertebrates in samples classified into–pollution tolerant–pollution intolerant–suspension feeders (control group; mainly palp worms)

Site j, subsample t

j ~ CAR process z jt : LN( j +βxj,Ψ)

E( j −j ) = +λnj

(kk∈N( j)∑ −)

Var( j −j ) =Γnj

Effect of salinity