Order-q dependent stochastic processes in Bayesian applications …allman.rhon.itam.mx/~lnieto/index_archivos/ColoquioUNM2015.pdf · Confidence bound (95%) Kaplan-Meier KM Confidence

Contents Order-1 process Order-q process Extensions References

Order-q dependent stochastic processes inBayesian applications

Luis E. Nieto-Barajas

Department of Statistics, ITAM, Mexico

Statistics Colloquium, UNM, USA

October 2, 2015

Luis E. Nieto-Barajas Order-q stochastic processes


Contents

Order-1 process

Application in survival analysis

Order-q process

Application in time series modelingApplication in disease mapping

Extensions



Order−1 process

η1 η2 η3 η4 η5

θ1 θ2 θ3 θ4 θ5

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Dependence among {θk} is induced through latents {ηk}Close form expressions when use conjugate distributions

Want to ensure a given marginal distribution



Order−1 process

Nieto-Barajas & Walker (2001):

Beta process: {θk} ∼ BeP1(a, b, c)

θ1 ∼ Be(a, b), ηk | θk ∼ Bin(ck , θk),

θk+1 | ηk ∼ Be(a + ηk , b + ck − ηk)

⇒ θk ∼ Be(a, b) marginally

Gamma process: {θk} ∼ GaP1(a, b, c)

θ1 ∼ Ga(a, b), ηk | θk ∼ Po(ckθk),

θk+1 | ηk ∼ Ga(a + ck , b + ηk)

⇒ θk ∼ Ga(a, b) marginally



Order−1 process




θk+1 | ηk ∼ Be(a + ηk , b + ck − ηk)




θk+1 | ηk ∼ Ga(a + ck , b + ηk)




Order−1 process




θk+1 | ηk ∼ Be(a + ηk , b + ck − ηk)




θk+1 | ηk ∼ Ga(a + ck , b + ηk)




Survival Analysis

Hazard rate modelling

If T is a discrete r.v. with support on τk then

h(t) = θk I (t = τk)

with {θk} ∼ BeP1(a, b, c)

If T is a continuous r.v. and {τk} are a partition of IR+ then

h(t) = θk I (τk−1 < t ≤ τk)

with {θk} ∼ GaP1(a, b, c)

This is old stuff!, but what it is new is that there is anR-package called BGPhazard that implements these models



Survival Analysis

Hazard rate modelling

If T is a discrete r.v. with support on τk then

h(t) = θk I (t = τk)

with {θk} ∼ BeP1(a, b, c)

If T is a continuous r.v. and {τk} are a partition of IR+ then

h(t) = θk I (τk−1 < t ≤ τk)

with {θk} ∼ GaP1(a, b, c)

This is old stuff!, but what it is new is that there is anR-package called BGPhazard that implements these models



Survival Analysis

Example: Discrete survival model

We analyse the 6-MP clinical trial data which consists ofremission duration times (in months) for children with acuteleukemia.

The study consisted in comparing drug 6-MP versus placebo.We concentrate on the 21 patients who received placebo.

Observed time values range from 1 to 23 and there are nocensored observations.

To define the prior we took a = b = 0.0001 and ct = 50 forall t. We use command BeMRes to fit the model and thecommand BePloth to produce graphs.



Survival Analysis: Order-1 Beta process

0 5 10 15 20

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Estimate of hazard rates

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Hazard functionConfidence band (95%)Nelson−Aalen based estimate



Survival Analysis: Order-1 Beta process

0 5 10 15 20

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Estimate of Survival Function

times

Model estimateConfidence bound (95%)Kaplan−MeierKM Confidence bound (95%)



Survival Analysis

Example: Continuous survival model

We define a piecewise hazard function

The data are survival times of 33 leukemia patients. Timesare measured in weeks from diagnosis. Three of theobservations were censored.

The prior was defined by taking a = b = 0.0001 and

ck |ξiid∼ Ga(1, ξ) for k = 1, . . . ,K and ξ ∼ Ga(0.01, 0.01). We

took K = 10 intervals and chose the partition τk such thateach interval contains approximately the same number ofexact (not censored) observations.

We used the command GaMRes to fit the model andcommand GaPloth to produce graphs.



Survival Analysis: Order-1 Gamma process

0 20 40 60 80 100 120 140

0.0

0.2

0.4

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0.8

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Estimate of Survival Function

times

Model estimateConfidence bound (95%)Kaplan−MeierKM Confidence bound (95%)



Order−2 process

η1 η2 η3 η4 η5

θ1 θ2 θ3 θ4 θ5

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Throw more arrows to induce higher order dependence

There is no way to obtain a given marginal distribution:say beta or gamma

Unless we include an extra latent (layer)



Order−2 process

η1 η2 η3 η4 η5

θ1 θ2 θ3 θ4 θ5

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Order−2 process

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Order−2 process

ω

η1 η2 η3 η4 η5

θ1 θ2 θ3 θ4 θ5

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With this common ancestor ω we can through more arrowsand still ensure a given marginal



Space and time process

This idea can be use to induce time and/or spatial dependence

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t = 1 t = 2 t = 3

θ1,1(η1,1)

θ2,1(η2,1)

θ3,1(η3,1)

θ1,2(η1,2)

θ2,2(η2,2)

θ3,2(η3,2)

θ1,3(η1,3)

θ2,3(η2,3)

θ3,3(η3,3)

θ1,4(η1,4)

θ2,4(η2,4)

θ3,4(η3,4)

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ω



Order-q beta process

Jara & al. (2013):

Order−q (AR) beta process: {θt} ∼ BePq(a, b, c)

ω ∼ Be(a, b) ηt | ωind∼ Bin(ct , ω)

θt | η ∼ Be

a +

q∑j=0

ηt−j , b +

q∑j=0

(ct−j − ηt−j)

θt ∼ Be(a, b) marginally




Properties:

Corr(θt , θt+s) =(a + b)

(∑q−sj=0 ct−j

)+(∑q

j=0 ct−j

)(∑qj=0 ct+s−j

)(a + b +

∑qj=0 ct−j

)(a + b +

∑qj=0 ct+s−j

) ,

for s ≥ 1.

If ct = c for all t then {θt} becomes strictly stationary with

Corr(θt , θt+s) =(a + b) max{q − s + 1, 0}c + (q + 1)2c2

{a + b + (q + 1)c}2.



Autocorrelation in {θt}

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lag

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0lag




Example: Unemployment rate in Chile

Bimonthly data from 1980 to 2010

Use our BePq as likelihood for the data {Yt}Took priors for (a, b, c): a ∼ Un(0, 1000), b ∼ Un(0, 1000)

and ct | λiid∼ Po(λ) and λ ∼ Un(0, 1000)



Time series: Yt = Unemployement in Chile

1980 1985 1990 1995 2000 2005 2010

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ploym

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te in C

hile

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q=3q=4q=5q=6q=7q=8



Time series: Yt = Unemployement in Chile

1980 1990 2000 2010 2020

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Year

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BePBDM



Spatial process



Spatial process

Nieto-Barajas & Bandyopadhyay (2013):

Spatial gamma process: {θt} ∼ SGaP(a, b, c)

ω ∼ Ga(a, b) ηij | ωind∼ Ga(cij , ω)

θi | η ∼ Ga

a +∑j∈∂i

cij , b +∑j∈∂i

ηij

∂i is the set of neighbours of region i

θt ∼ Ga(a, b) marginally



Disease mapping

Study: Mortality in pregnant women due to hypertensive disorderin Mexico in 2009. Areas are the States

Yi = Number of deaths in region iEi = At risk: Number of births (in thousands)λi = Maternity mortality rate

Zero-inflated model

f (yi ) = πi I (yi = 0) + (1− πi )Po(yi | λiEi )

λi = θi exp(β′xi ) πi =ξie

δ′zi

1 + ξieδ′zi

β is a vector of reg. coeff. s.t. βk ∼ N(0, σ20)

θi ∼ SGaP(a, a, c)ξi ∼ Ga(b, b)



Disease mapping

Six explanatory variables:

X1 number of medical units (hospitals + clinics)

X2 proportion of pregnant women with soc. sec.

X3 prop. of pregnant women who were seen by a physician inthe first trimester of pregnancy

X4 public expenditure in health per capita in thousands of MX

Z1 poverty index

Z2 proportion of births in clinics and hospitals



Estimated mortality rate λi

[3.05,6.33)[6.33,6.67)[6.67,7.38)[7.38,8.73)[8.73,21.07]

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Estimated zero inflated prob. πi

[0,0.01)[0.01,0.04)[0.04,0.06)[0.06,0.5)[0.5,0.6]

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Extensions

Construct dependent Bayesian nonparametric priors (DDP,dPT)

Use same ideas with stochastic processes instead of randomvariables

Dependent Dirichlet processes using multinomial processes aslatents

Dependent gamma processes using Poisson processes aslatents

These constructions are currently under study



References

Jara, A., Nieto-Barajas, L. E. & Quintana, F. (2013). A time series model forresponses on the unit interval. Bayesian Analysis 8, 723–740.

Nieto-Barajas, L. E. & Bandyopadhyay, D. (2013). A zero-inflated spatialgamma process model with applications to disease mapping. Journal ofAgricultural, Biological and Environmental Statistics 18, 137–158.

Nieto-Barajas, L. E. & Walker, S. G. (2002). Markov beta and gammaprocesses for modelling hazard rates. Scandinavian Journal of Statistics 29,413–424.


Documents

Order-q dependent stochastic processes in Bayesian applications …allman.rhon.itam.mx/~lnieto/index_archivos/ColoquioUNM2015.pdf · Confidence bound (95%) Kaplan-Meier KM Confidence