Estimation of a bivariate hierarchical Ornstein-Uhlenbeck ... · Estimation of a bivariate...

IntroductionThe hierarchical latent OU model

Bayesian framework for statistical inferenceApplication of the OU model

Estimation of a bivariate hierarchicalOrnstein-Uhlenbeck model for longitudinal data

Zita Oravecz & Francis Tuerlinckx(E-mail to: [email protected])

IAP workshop, Ghent19th September, 2008

[email protected] Estimation of a hierarchical OU model



1 Introduction

2 The hierarchical latent OU modelThe latent OU modelThe hierarchical latent OU model

3 Bayesian framework for statistical inferenceSimulation methodsComputational detailsDerivation of some full conditional distributions

4 Application of the OU modelSimulation study 1Simulation study 2Application exampleConclusions




INTRODUCTION




Summary of the proposed model

Modeled data type: intensive longitudinal data (possiblyunstructured)

Observed data is a function of a latent continuous process

Bivariate SDE model for the changes in the latent true scorewithin the subject: an Ornstein-Uhlenbeck (OU) process

Hierarchically extended (multiple persons)

Time-varying and time-invariant covariate information can beincluded

Statistical inference: Bayesian framework




Typical area of applicability in psychology

Core affect measures

consciously accessible subjective aspect of feeling or emotion

compound of arousal (activation-deactivation) and hedonic(pleasantness-unpleasantness) values

when asked, people can report their CA

assumed to change continuously




The latent OU modelThe hierarchical latent OU model

THE HIERARCHICAL LATENT OUMODEL





Two-dimensional latent OU model

{dΘ(t) = B(µ−Θ(t))dt +σdW(t) (1)Y(t) = Θ(t)+ε(t) (2)

Eq. 1: transition equation: describes the change in the true score(dynamical aspect)

Eq. 2: observation equation: mapping of the true position on theobserved variable

dΘ(t): change in the variableB(µ−Θ(t))dt: deterministic partσdW(t): stochastic partinstantaneous covariance matrix: Σ = σσT

measurement error: ε(t)∼ N2(0,Σε)[email protected] Estimation of a hierarchical OU model




By integrating over the transition equation, we arrive at the followingposition equation:

Θ(t +d) = e−BdΘ(t)+µ(1− e−Bd)+σe−BdZ t+d

teBudW(u).

Solving the remaining stochastic integral based on Ito-calculus:

Θ(t +d) |Θ(t)∼ N2(µ+ e−Bd(Θ(t)−µ), Γ− e−BdΓe−B′d),

where Σ is re-parametrized as

Σ = BΓ+ΓB′.

The matrix Γ is called the stationary covariance matrix, based on thestationary distribution:

Θ(t)∼ N2(µ,Γ)





Hierarchical modeling

Individuals are random samples from a population

Each individual has person-specific OU parameters

AimsExplore general aspects of changes over timeExplain interindividual variability: regress random-effects ontocovariates

Notationp (p = 1, . . . ,P) is measured np times at: tp1, tp2, . . . , tps, . . . , tp,np .

Measured sequence of positions: Yp1, . . . ,Yps, . . . ,Yp,np





The hierarchical latent OU model

Yps = Θps +εps

εps ∼ N2(0,Σε)

Θps ∼ N2(Mps,Vps)

with

Mps ={

µps if s = 1µps + e−Bp(tps−tp,s−1)(Θp,s−1−µps) if s > 1

(1)

and covariance matrix

Vps ={

Γp if s = 1Γp− e−Bp(tps−tp,s−1)Γpe−B′p(tps−tp,s−1) if s > 1

(2)





Time-varying covariate information can be added through thehomebase parameter

µps = µp +(I2⊗ zps)δ

Time-varying effect on the homebase, e.g., the actual measurementtime





Decomposition of Bp and Γp

The positive definiteness has to be preserved:

Γp =

[γ1p

√γ1pγ2pργp√

γ1pγ2pργp γ2p

]

Bp =

[β1p

√β1pβ2pρβp√

β1pβ2pρβp β2p

]





Time-invariant covariate information

xjp : k covariates j= 1, ..., k, for person p

covariate scores in a vector with 1 for the intercept:

x′p = (1,x1p,x2p...,xkp)

vectors with regression coefficients

αµ1 ,αµ2 ,αγ1 ,αγ2 ,αβ1 ,αβ2 ,αργ,αρβ





Population distributions

Homebase (target level) intercept (µp)

µp ∼ N2

( (x′pαµ1

x′pαµ2

),Σµ

)Volatility (Γp)

γ12p =√

γ1p×√

γ2p×ρp

γ1p ∼ LN(x′pαγ1 ,σ2γ1) γ2p ∼ LN(x′pαγ2 ,σ

2γ2)

Fisher-transformed cross-correlation

F(ργp) =12

log(

1+ργp

1−ργp

)∼ N(x′ργ

αργ,σ2

Fργ)

Centralizing tendency (Bp) follows the same decomposition as thevolatility




Simulation methodsComputational detailsDerivation of some full conditional distributions

BAYESIAN FRAMEWORK FORSTATISTICAL INFERENCE





Bayesian statistical inference

Bayesian methods make explicit use of probability for quantifyinguncertainty

uncertainty rules of probability are used directly to makeinferences about the parameters

the inference is based on the actual occurring data (not allpossible data sets that might have occurred)

Bayesian parameter estimation: exploring the posterior density of theparameters

p(parameters|data) ∝ p(data|parameters)p(parameters)





Simulation methods

Goal: generating samples from the posterior probability distribution

Direct simulation: prohibitively expensive (high-dimensionality)

Markov chain Monte Carlo simulation:

iterative sampling: drawing values from approximatedistributions, then correcting them

improved in each step: convergence to the target distribution

a sufficiently large number of iterations results in a Markov chainwith the posterior distribution as its equilibrium distribution





Gibbs-sampler

Markov chain algorithm efficient for multidimensional problems

alternating conditional sampling: the parameter vector is dividedinto subparts

Full conditional distribution: the probability distribution of asubelement/subvector of the parameter vector, given the valuesof all other parameters as obtained in the previous iteration, aswell as the data

one iteration: the algorithm cycles through these subvectors,drawing each subset conditional on the other values





Metropolis-Hastings (M-H) step

No known distributional form: a Metropolis-Hastings (M-H) step

adaptation of a random walk that uses acceptance/rejection ruleto converge to the target distribution

candidate is sampled from a proposal distribution

the decision of whether the new candidate is accepted is based onthe comparison of the density ratios (candidate vs. previouslyaccepted)





Computational details

Sampling algorithm: custom-written MATLAB program

Subroutines of the program have been written in C (usingMATLAB’s MEX facility to interface)

Parallel computing can be applied (different sample chains areindependent and can be computed on separate processors)

Estimation of all model parameters by 15000 iterations in a 100subjects × 100 observations × 2 dimensions set-up on acomputing node with an AMD Opteron250 processor and 2Gb ofRAM takes approximately 5 hours





DERIVATION OF SOME FULLCONDITIONAL DISTRIBUTIONS





Derivation of the full conditional for Θps

Prior for Θps:

Θps ∼ N2(Θ0,ΦΘ).

Uninformative prior: Θ0 = 0 and ΦΘ = 1000× I2

Likelihood for Θps given other parameters are known:

L(µps,Bp,Γp,Σε,Yps,Θp,s−1,Θp,s+1 |Θps) =

= f (Θps | Yps,Σε) × f (Θps |Θp,s−1)× f (Θp,s+1 |Θps)





Derivation of the full conditional for Θps

The values of the latent state Θps are drawn sequentially.

Full conditional for the Θps (except for s = np and s = 1 )

(Θps | µps,Bp,Γp,Σε,Yps,Θp,s−1,Θp,s+1) ∝

e− 1

2

{(Θps−Θ0

)TΦ−1

Θ

(Θps−Θ0

)}e− 1

2

{(Yps−Θps

)TΣ−1

εp

(Yps−Θps

)}e− 1

2

{(Θps−µps−e−BpdpsΘp,s−1+e−Bpdpsµps

)TV−1

ps

(Θps−µps−e−BpdpsΘp,s−1+e−Bpdpsµps

)}

e− 1

2

{(Θp,s+1−µp,s+1−e−Bpdp,s+1Θps+e−Bpdp,s+1µp,s+1

)TV−1

p,s+1(Θp,s+1−µp,s+1−e−Bpdp,s+1Θps+e−Bpdp,s+1µp,s+1

)}

which results in a normal distribution . . .





(Θps | µps,Bp,Γp,Σε,Yps,Θp,s−1,Θp,s+1)∼ N2(MΘps ,VΘps)

where

VΘps = (Φ−1Θ

+Σ−1ε +V−1

ps +(e−Bpdp,s+1)TV−1p,s+1(e

−Bpdp,s+1))−1

and

MΘps = VΘps(Φ−1Θ

Θ0 +Σ−1εp

Yps +V−1ps µps +V−1

ps e−BpdpsΘp,s−1

−V−1ps e−Bpdpsµps +(e−Bpdp,s+1)TV−1

p,s+1Θp,s+1

−(e−Bpdp,s+1)TV−1p,s+1µp,s+1 +(e−Bpdp,s+1)TV−1

p,s+1e−Bpdp,s+1µp,s+1)





Full conditional for Θp1

(Θp1 | µp1,Γp,Σε,Yp1)∼ N2(MΘp1 ,VΘp1)

with covariance matrix

VΘp1 = (Φ−1Θ

+Σ−1ε +Γ−1

p )−1

and mean

MΘp1 = VΘp1(Φ−1Θ

Θ0 +Σ−1ε Yp1 +Γ−1

p µp1)





Full conditional for Θp,np

(Θp,np | µps,Bp,Γp,Σε,Yp,np)∼ N2(MΘp,np,VΘp,np

)

with covariance matrix

VΘp,np= (Φ−1

Θ+Σ−1

ε +V−1p,np

)−1

and mean

MΘp,np= VΘp,np

(Φ−1Θ

Θ0 +Σ−1εp

Yp,np +V−1p,np

µp,np+V−1

p,npe−Bpdp,np Θp,np−1

−V−1p,np

e−Bpdp,np µp,np).





Derivation of the full conditional for γ1p

Prior for γ1p:γ1p ∼ LN(x′pαγ1 ,σ

2γ1)

where the density function is:

f (γ1p) =1

γ1p

√2πσ2

γ1

e− 1

2(log(γ1p)−x′pαγ1 )2

σ2γ1

Full conditional for γ1p:

f (γ1p | {Θps}nps=1,µp,γ2p,ργp,Bp) ∝ f (γ1p)

np

∏s=1|Vps|−

12 e−

12

(MT

psV−1ps Mps

)




Simulation study 1Simulation study 2Application exampleConclusions

APPLICATION OF THE OUMODEL





Study 1

Ten data sets were simulated based on the results of a real lifeapplication

ten measurements per day, for ten days, for 100 subjects

semi-random measurement time-points

two longitudinal variables

measurement time-points were used as linear and quadratictime-effects

Estimationfor each dataset 3 chains were run15 000 iterations per chain5 000 discarded burn-incomputation time: 5 hours per chain





Model Simulated Mean posterior SD of the posteriorParameter value estimate estimates

αµ1 6.00 5.99 0.04αµ2 5.00 4.97 0.07σ2

µ10.40 0.42 0.07

σµ1µ2 0.05 0.04 0.04σ2

µ20.30 0.27 0.05

δLµ1 1.00 1.00 0.12δQµ1 0.00 -0.01 0.12δLµ2 4.00 3.96 0.12δQµ2 -4.00 -4.00 0.14αγ1 0.80 0.77 0.07σ2

γ10.40 0.38 0.04

αγ2 1.00 1.01 0.10σ2

γ20.20 0.20 0.03

αγρ 0.10 0.09 0.03σ2

γρ0.10 0.08 0.01

αβ1-4.20 -4.21 0.04

σ2β1

0.40 0.39 0.10

αβ2-4.00 -3.93 0.09

σ2β2

0.50 0.49 0.18

αβρ-0.10 -0.11 0.03

σ2βρ

0.10 0.08 0.02

σ21ε

0.20 0.21 0.04σ2

2ε1.00 0.88 0.13





Posterior density estimates





MCMC trace plots





Study 2

Ten data sets were simulated based on a more arbitrary design

the levels of measurement error were increased

in the first dimension: higher measurement error (σ21ε

= 4) thanstochastic variance (αγ1 = 1, expected value appr.: 3.5)

cross-effects were substantial (σ2γρ

= 0.5 and αβρ= 0.5)





Model Simulated Mean posterior SD of the posteriorParameter value estimate estimates

αµ1 0.00 -0.04 0.12αµ2 0.00 -0.01 0.14σ2

µ12.00 2.14 0.32

σµ1µ2 0.70 0.71 0.23σ2

µ21.00 0.93 0.17

δLµ1 2.00 1.86 0.22δQµ1 0.00 0.10 0.22δLµ2 0.00 -0.01 0.26δQµ2 4.00 3.99 0.26αγ1 1.00 1.32 0.11σ2

γ10.50 0.28 0.05

αγ2 2.00 1.95 0.10σ2

γ20.10 0.10 0.02

αγρ 0.50 0.41 0.05σ2

γρ0.10 0.05 0.01

αβ1-4.00 -3.75 0.24

σ2β1

0.50 0.40 0.26

αβ2-3.50 -3.44 0.20

σ2β2

0.10 0.18 0.13

αβρ0.50 0.38 0.07

σ2βρ

0.10 0.06 0.03

σ21ε

4.00 3.20 0.24σ2

2ε2.00 2.13 0.73





APPLICATION EXAMPLE





Data collection

Participants: 80 students from the University of Leuven

Method:Experience sampling

wristwatch

beeps 9 times perday for 7 days

semi-random beeps

Core Affect Grid −→

Descriptivesaverage 60 (SD=3.4) measurements (missingness: MAR)age 21 (SD=4.7)60% women





Summary - The modeled dataset

Pooled dataset of 80 individuals

From each individual:

1 Chain of measurement coordinates

2 Chain of measurement times

3 Big Five Personality questionnaire scores (5-point scale):neuroticismextraversionopenness to experienceagreeablenessconscientiousness





Details of the Bayesian inference

non-informative priors were assigned

posterior estimates are based on 10000 iterations with 6 chains

burn-in: 5000 iterations

computation time: fewer than 2 hours per chain

convergence checked (R statistics, visual assessment)





Model Posterior 95% posterior Posteriorparameter Description Mean credibility interval SD

Pleasantnessαµ1 Average homebase 6.00 5.85 6.15 0.07σ2

µ1Variance of the average homebase 0.39 0.26 0.57 0.07

δµ11 Linear time-effect 0.40 -0.04 0.85 0.22δµ12 Quadratic time-effect -0.15 -0.61 0.29 0.23αγ1 Average log-variability 0.84 0.70 0.98 0.07σ2

γ1Variance of the log-variability 0.35 0.23 0.50 0.06

αβ1Average log-centralizing tendency -4.22 -4.41 4.03 0.09

σ2β1

Variance of the log-centralizing tendency 0.48 0.28 0.76 0.12

Activationαµ2 Average homebase 5.30 5.16 5.45 0.07σ2

µ2Variance of the average homebase 0.33 0.22 0.49 0.07

δµ21 Linear time-effect 4.29 3.79 4.80 0.25δµ22 Quadratic time-effect -4.18 -4.71 -3.67 0.26αγ2 Average log-variability 1.05 0.93 1.17 0.06σ2

γ2Variance of the log-variability 0.35 0.23 0.50 0.06

αβ2Average log-centralizing tendency -4.12 -4.31 -3.92 0.09

σ2β2

Variance of the log-centralizing tendency 0.49 0.29 0.79 0.12

Cross-effectsσµ1µ2 Covariance between the homebases 0.05 -0.04 0.16 0.05αργ Average Fisher transformed cross-correlation 0.02 -0.04 0.09 0.03αρ

βAverage Fisher transformed off-diagonal of B -0.04 -0.13 0.03 0.04





Time-varying effect estimates





Person-specific autocorrelation estimates





Model Posterior 95% Posterior Posteriorparameter Description Covariate mean credibility interval SD

Pleasantnessαµ1N Homebase Neuroticism -0.32 -0.58 -0.07 0.13α

γ1N Variability Neuroticism 0.26 0.01 0.51 0.12Cross-effects

αργN Cross-correlation Neuroticism 0.13 0.01 0.25 0.06

αργC Cross-correlation Conscientiousness -0.18 -0.30 -0.05 0.06

αρ

βA Off-diagonal of B Agreeableness -0.25 -0.45 -0.07 0.09

Note. Model parameters refer to the regression weights. For example αµ1N is the regression weight for neuroticism relating tothe homebase in the pleasantness dimension (µ1).





CONCLUSIONS





Conclusions

Recovery was sufficient, better for the simulation with smallermeasurement error (less noisy measurements )

Bayesian methodology can be efficiently applied to estimateSDE models with known probability density function of thestates

Extension to more dimensions is possible

Model testing can be done in the Bayesian framework, e.g., DIC





THANK YOUFOR YOUR ATTENTION!


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