SM.MAP.MixReparametrized {Ultimixt}

summary of the output produced by K.MixReparametrized

Description

Label switching in a simulated Markov chain produced by K.MixReparametrized is removed by the technique of Marin et al. (2004). Namely, component labels are reorded by the shortest Euclidian distance between a posterior sample and the maximum a posteriori (MAP) estimate. Let \theta_i be the i-th vector of computed component means, standard deviations and weights. The MAP estimate is derived from the MCMC sequence and denoted by \theta_{MAP}. For a permutation \tau \in \Im_k the labelling of \theta_i is reordered by

\tilde{\theta}_i=\tau_i(\theta_i)

where \tau_i=\arg \min_{\tau \in \Im_k} \mid \mid \tau(\theta_i)-\theta_{MAP}\mid \mid.

Angular parameters \xi_1^{(i)}, \ldots, \xi_{k-1}^{(i)} and \varpi_1^{(i)}, \ldots, \varpi_{k-2}^{(i)}s are derived from \tilde{\theta}_i. There exists an unique solution in \varpi_1^{(i)}, \ldots, \varpi_{k-2}^{(i)} while there are multiple solutions in \xi^{(i)} due to the symmetry of \mid\cos(\xi) \mid and \mid\sin(\xi) \mid. The output of \xi_1^{(i)}, \ldots, \xi_{k-1}^{(i)} only includes angles on [-\pi, \pi].

The label of components of \theta_i (before the above transform) is defined by

\tau_i^*=\arg \min_{\tau \in \Im_k}\mid \mid \theta_i-\tau(\theta_{MAP}) \mid \mid.

The number of label switching occurrences is defined by the number of changes in \tau^*.

Usage

SM.MAP.MixReparametrized(estimate, xobs, alpha0, alpha)

Arguments

estimate

Output of K.MixReparametrized

xobs

Data set

alpha0

Hyperparameter of Dirichlet prior distribution of the mixture model weights

alpha

Hyperparameter of beta prior distribution of the radial coordinate

Details

Details.

Value

MU

Matrix of MCMC samples of the component means of the mixture model

SIGMA

Matrix of MCMC samples of the component standard deviations of the mixture model

P

Matrix of MCMC samples of the component weights of the mixture model

Ang_SIGMA

Matrix of computed \xi's corresponding to SIGMA

Ang_MU

Matrix of computed \varpi's corresponding to MU. This output only appears when k > 2.

Global_Mean

Mean, median and 95\% credible interval for the global mean parameter

Global_Std

Mean, median and 95\% credible interval for the global standard deviation parameter

Phi

Mean, median and 95\% credible interval for the radius parameter

component_mu

Mean, median and 95\% credible interval of MU

component_sigma

Mean, median and 95\% credible interval of SIGMA

component_p

Mean, median and 95\% credible interval of P

l_stay

Number of MCMC iterations between changes in labelling

n_switch

Number of label switching occurrences

Note

Note.

Author(s)

Kate Lee

References

Marin, J.-M., Mengersen, K. and Robert, C. P. (2004) Bayesian Modelling and Inference on Mixtures of Distributions, Handbook of Statistics, Elsevier, Volume 25, Pages 459–507.

See Also

K.MixReparametrized

Examples

#data(faithful)
#xobs=faithful[,1]
#estimate=K.MixReparametrized(xobs,k=2,alpha0=0.5,alpha=0.5,Nsim=1e4)
#result=SM.MAP.MixReparametrized(estimate,xobs,alpha0=0.5,alpha=0.5)