Projected Pólya Treeallman.rhon.itam.mx/~lnieto/index_archivos/BNP2019.pdf · Introduction Projected tree Posterior inference Real data Contents 1 Introduction 2 Projected tree 3

Introduction Projected tree Posterior inference Real data

Projected Pólya Tree

Luis E. Nieto Barajas

(joint with G. Núñez-Antonio)

Department of StatisticsITAM

BNP Conference – June 27, 2019

Luis E. Nieto Barajas Projected Pólya Tree BNP Conference – June 27, 2019 1 / 19


Contents

1 Introduction

2 Projected tree

3 Posterior inference

4 Real data



Setting

Directional data : (Mardia, 1972) Unit vectors in k -dimensional space, i.e. k − 1

angles.

Examples : wind directions, orientation data, directions of bird migration,

mammalian activity patterns in ecological reserves, etc.

The most common case is k = 2 producing circular data.

One of the simplest ways to generate distributions on Sk is to radially project

distributions originally defined on Rk .

e.g. projected normal distribution (Núñez-Antonio & Gutiérrez-Peña, 2005).

Aim : Project a bivariate Pólya tree to the unit circle and study its properties.



Notation

Bivariate PT : F on (R2,B2) has a bivariate PT prior with pars (Π,A), where

Π = {Bm,j,k}, Bm,j,k = Bm,j × Bm,k and A = {αm,j,k}, j, k = 1, . . . , 2m, m = 1, 2, . . .,

if there exists r.v. Ym,j,k = (Ym+1,2j−1,2k−1,Ym+1,2j−1,2k ,Ym+1,2j,2k−1,Ym+1,2j,2k ) s.t.

1 Ym,j,k are independent

2 Ym,j,k ∼ Dir(αm,j,k ),

3 For every m = 1, 2, . . . and every j, k = 1, . . . , 2m,

F (Bm,j,k ) =m∏

l=1

Ym−l+1,jm,j,km−l+1,k

m,j,km−l+1

,

where j(m,j,k)l−1 =

⌈j(m,j,k)l

2

⌉and k (m,j,k)

l−1 =

⌈k(m,j,k)

l2

⌉are recursive decreasing

formulae, whose initial values are j(m,j,k)m = j and k (m,j,k)

m = k .



Notation

Centring : E{F (Bm,j,k )} = F0(Bm,j,k ) = 1/4m ifSince Bm,j,k = Bm,j × Bm,k , we take F0(x1, x2) = F10 (x1)F20 (x2).Match the partition with the dyadic quantiles of the marginals

Bm,j =

(F−1

10

(j − 12m

), F−1

10

(j

2m

)]and Bm,k =

(F−1

20

(k − 1

2m

), F−1

20

(k

2m

)]Define αm,j,k = (αρ(m + 1), . . . , αρ(m + 1)), α > 0 is the prec. par., ρ(m) = mδ withδ > 1 to define an abs. cont. tree.

Finite tree : stop partitioning the space at a finite level MAt the lowest level M, we spread probability according to f0Bivariate density at x = (x1, x2) ∈ R2

f (x) =

{M∏

m=1

Ym,j

(x1)m ,k

(x2)m

}4M f0(x),

Denote a finite bivariate PT as PTM (α, ρ, F0).



Notation

Centring : E{F (Bm,j,k )} = F0(Bm,j,k ) = 1/4m ifSince Bm,j,k = Bm,j × Bm,k , we take F0(x1, x2) = F10 (x1)F20 (x2).Match the partition with the dyadic quantiles of the marginals

Bm,j =

(F−1

10

(j − 12m

), F−1

10

(j

2m

)]and Bm,k =

(F−1

20

(k − 1

2m

), F−1

20

(k

2m

)]Define αm,j,k = (αρ(m + 1), . . . , αρ(m + 1)), α > 0 is the prec. par., ρ(m) = mδ withδ > 1 to define an abs. cont. tree.

Finite tree : stop partitioning the space at a finite level MAt the lowest level M, we spread probability according to f0Bivariate density at x = (x1, x2) ∈ R2

f (x) =

{M∏

m=1

Ym,j

(x1)m ,k

(x2)m

}4M f0(x),

Denote a finite bivariate PT as PTM (α, ρ, F0).



Definition

Let X = (X1,X2) s.t. X | f ∼ f and f ∼ PTM (α, ρ,F0).

We project X to the unit circle by using polar coordinates (X1,X2)→ (Θ,R),

where Θ is the angle and R = ||X|| is the resultant.

The inverse transf. becomes X1 = R cos Θ and X2 = R sin Θ. Thus, the Jacobian

is J = R.

The projected Pólya tree, denoted by PPTM (α, ρ, f0), has density

f (θ) =

∫ ∞0

{ M∏m=1

Ym,j(r cos θ)

m ,k(r sin θ)m

}4M f0(r cos θ, r sin θ) r dr .



Properties

Smooth : at the boundaries of the partitions. The marginalisation can also be seen

as a mixture

f (θ) =

∫f (θ | r)f (r)dr .

Easily centred : Say on the projected normal by taking f0(x) = N2(x | µ, I) a

bivariate normal density with µ = (µ1, µ2) and var-cov I. The projected Pólya tree

becomes

f (θ) =

∫ ∞0

{ M∏m=1

Ym,j(r cos θ)

m ,k(r sin θ)m

}4M (2π)−1e−

12 µ′µ r

× exp

[−

12

{r2 − 2r (µ1 cos θ + µ2 sin θ)

}]I(0,2π](θ)dr .



Properties

Numerically approximated : There is no analytical expression. Use MC or

quadrature. If 0 = r (0) < r (1) < · · · < r (L) <∞ is a partition, then

f (θ) ≈L∑

l=1

f (r (l) cos θ, r (l) sin θ) |J|(

r (l) − r (l−1))

Moments : Circular densities are periodic, f (θ + 2π) = f (θ), require trigonometric

moments ϕp = E(eipΘ) = ap + ibp , where ap = E(cos pΘ) and bp = E(sin pΘ),

νθ = arctan(b1/a1), %θ =√

a21 + b2

1 ,

where %θ ∈ [0, 1].



Properties

Posterior consistency : Let f ∼ PPT(α, ρ, f0) and f∗(θ) be an arbitrary density s.t.KL(f∗, f0) <∞. Then, if

∑∞m=1 ρ(m)−1/2 <∞, as n→∞ f achieves weak

posterior consistency.In particular, if ρ(m) = mδ , we need δ > 2.

Examples : M = 4, α = 1, ρ(m) = mδ with δ = 1.1, and different values of µ.

Used L = 100 points.



Examples

0 1 2 3 4 5 6

0.0

0.5

1.0

1.5

theta

f(th

eta)

0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

0.8

theta

f(th

eta)

0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

0.8

1.0

1.2

theta

f(th

eta)

0 1 2 3 4 5 6

0.0

0.5

1.0

1.5

2.0

theta

f(th

eta)

FIGURE – Ten simulated densities µ : (0, 1) (top left), (1, 0) (top right), (0,−1) (bottom left) and(−1, 0) bottom right.



Examples

0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

0.8

theta

f(th

eta)

0 1 2 3 4 5 6

0.0

0.5

1.0

1.5

theta

f(th

eta)

0 1 2 3 4 5 6

0.0

0.5

1.0

1.5

2.0

2.5

3.0

theta

f(th

eta)

0 1 2 3 4 5 6

01

23

45

theta

f(th

eta)

FIGURE – Ten simulated densities µ : (0, 0) (top left), (1, 1) (top right), (2, 2) (bottom left) and (5, 5)bottom right.



Examples

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●

●

(0,1) (1,0) (0,−1) (−1,0)

−2

02

4

●

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●

●

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●

●

●

●

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(0,1) (1,0) (0,−1) (−1,0)

0.2

0.4

0.6

0.8

1.0

●

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●

(0,0) (1,1) (2,2) (5,5)

−3

−2

−1

01

23

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(0,0) (1,1) (2,2) (5,5)

0.0

0.2

0.4

0.6

0.8

1.0

FIGURE – Prior distribution of moments. νθ (first column), and %θ (second column).



Posterior

Let θ1, θ2, . . . , θn a sample of size n s.t. θi | f ∼ f , independently, and

f ∼ PPTM (α, ρ, f0).

Consider a data augmentation approach (Tanner, 1991) : define latent resultant

lengths R1,R2, . . . ,Rn s.t. (Θi ,Ri ) and (X1i ,X2i ) are 1 :1.

Posterior conditionals :Ym,j,k | data ∼ Dir(αm,j,k + Nm,j,k ), where

Nm,j,k = (Nm+1,2j−1,2k−1,Nm+1,2j−1,2k ,Nm+1,2j,2k−1,Nm+1,2j,2k ).

f (ri | Y, θi ) ∝{∏M

m=1 Ym,j

(ri cos θi )m ,k

(ri sin θi )m

}f0(ri cos θi , ri sin θi ) ri ,

Require MH step with random walk proposal Ga(κ, κ/r (t)) with variation

coefficient 1/√κ.



Simulated data

Sampled from a projected bivariate normal f (x) =∑4

j=1 πj N2(x | ηj , I), with

π = (0.1, 0.2, 0.4, 0.3) and η1 = (1.5, 1.5), η2 = (−1, 1), η3 = (−1,−2),

η4 = (1.5,−1.5).

n = 50 and n = 500

Fitted PPTM (α, ρ, f0), with f0 = N2(µ, I), µ ∈ {(0, 0), (1, 1), (2, 2)}, ρ(m) = mδ

with δ = 1.1, α ∈ {0.5, 1, 2} and M = 4.

MCMC : 10, 000 iterations, 1, 000 of burn-in, 5th thinning.



Simulated data

theta

f(th

eta)

0 1 2 3 4 5 6

0.0

0.1

0.2

0.3

0.4

0.5

theta

f(th

eta)

0 1 2 3 4 5 6

0.0

0.1

0.2

0.3

0.4

theta

f(th

eta)

0 1 2 3 4 5 6

0.0

0.1

0.2

0.3

0.4

0.5

0.6

theta

f(th

eta)

0 1 2 3 4 5 6

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

FIGURE – Posterior estimates with n = 500. Columns α = 0.5 and α = 2. Rows µ = (0, 0) andµ = (2, 2).



El Triunfo Reserve

Data : Temporal activity (time of the day) for three animal species : peccary, tapir

and deer.

Data sizes are : 16, 35 and 115

Fitted PPTM (α, ρ, f0), with f0 = N2(µ, I), µ = (0, 0), ρ(m) = m1.1 and M = 4.

Tried α ∈ {0.5, 1, 2} to compare. Placed hyper-prior α ∼ Ga(1, 2).

LPML gof statistics

α Peccary Tapir Deer0.5 −23.05 −61.02 −208.311 −23.22 −60.20 −206.922 −24.10 −59.57 −205.68

Ga(1, 2) −23.40 −60.15 −206.77Proj.Normal −26.52 −59.43 −207.54

DPM Proj.Normal −24.64 −59.56 −204.31



El Triunfo Reserve

theta

f(th

eta)

0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

0.8

1.0

theta

f(th

eta)

0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

theta

f(th

eta)

0 1 2 3 4 5 6

0.0

0.1

0.2

0.3

0.4

0.5

FIGURE – Posterior estimates. Peccary (top left), tapir (top right) and deer (bottom).Peccaries are seen from 6 :00 to 18 :00 hrs.



El Triunfo Reserve

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peccary tapir deer

−2

02

4

FIGURE – Posterior distribution of νθ . Preferred activity-time :peccaries (midday), tapirs (20 :30 hours) and deer (19 :00 hours).



References

Mardia, K.V. (1972). Statistics of Directional Data. London, Academic press.

Nuñez-Antonio, G. and Gutiérrez-Peña, E. (2005). A Bayesian analysis of

directional data using the projected normal distribution. Journal of Applied

Statistics 32, 995–1001.

Tanner, M.A. (1991). Tools for statistical inference : Observed data and data

augmentation methods. Springer, New York.


Documents

Projected Pólya Treeallman.rhon.itam.mx/~lnieto/index_archivos/BNP2019.pdf · Introduction Projected tree Posterior inference Real data Contents 1 Introduction 2 Projected tree 3