mixedsde (version 1.0)

BayesianNormal: Bayesian Estimation In Mixed Stochastic Differential Equations

Description

Gibbs sampler for Bayesian estimation of the random effects $(\alpha_j, \beta_j)$ in the mixed SDE $dX_j(t)= (\alpha_j- \beta_j X_j(t))dt + \sigma a(X_j(t)) dW_j(t)$.

Usage

BayesianNormal(times, X, model = c("OU", "CIR"), prior, start, random, nMCMC = 1000, propSd = 0.2)

Arguments

times

vector of observation times

X

matrix of the M trajectories (each row is a trajectory with $N= T/\Delta$ column).

model

name of the SDE: 'OU' (Ornstein-Uhlenbeck) or 'CIR' (Cox-Ingersoll-Ross).

prior

list of prior parameters: mean and variance of the Gaussian prior on the mean mu, shape and scale of the inverse Gamma prior for the variances omega, shape and scale of the inverse Gamma prior for sigma

start

list of starting values: mu, sigma

random

random effects in the drift: 1 if one additive random effect, 2 if one multiplicative random effect or c(1,2) if 2 random effects.

nMCMC

number of iterations of the MCMC algorithm

propSd

proposal standard deviation of $\phi$ is $|\mu|*$propSd$/\log(N)$ at the beginning, is adjusted when acceptance rate is under 30% or over 60%

Value

alpha

posterior samples (Markov chain) of $\alpha$

beta

posterior samples (Markov chain) of $\beta$

mu

posterior samples (Markov chain) of $\mu$

omega

posterior samples (Markov chain) of $\Omega$

sigma2

posterior samples (Markov chain) of $\sigma^2$

References

Hermann, S., Ickstadt, K. and C. Mueller (2016). Bayesian Prediction of Crack Growth Based on a Hierarchical Diffusion Model. Appearing in: Applied Stochastic Models in Business and Industry.

Rosenthal, J. S. (2011). 'Optimal proposal distributions and adaptive MCMC.' Handbook of Markov Chain Monte Carlo (2011): 93-112.