Bayesian nonparametric estimation based on normalized measures driven mixtures for locations.
MixNRMI1cens(
xleft,
xright,
probs = c(0.025, 0.5, 0.975),
Alpha = 1,
Kappa = 0,
Gama = 0.4,
distr.k = "normal",
distr.p0 = "normal",
asigma = 0.5,
bsigma = 0.5,
delta_S = 3,
delta_U = 2,
Meps = 0.01,
Nx = 150,
Nit = 1500,
Pbi = 0.1,
epsilon = NULL,
printtime = TRUE,
extras = TRUE,
adaptive = FALSE
)The function returns a list with the following components:
Numeric vector. Evaluation grid.
Numeric array. Matrix
of dimension \(\texttt{Nx} \times (\texttt{length(probs)} + 1)\) with the posterior mean and the desired quantiles input
in probs.
Numeric vector of length(x) with
conditional predictive ordinates.
Numeric vector of
length(Nit*(1-Pbi)) with the number of mixtures components
(clusters).
Numeric vector of length(Nit*(1-Pbi)) with the
values of common standard deviation sigma.
Numeric vector of
length(Nit*(1-Pbi)) with the values of the latent variable U.
List of length(Nit*(1-Pbi)) with the clustering
allocations.
List of length(Nit*(1-Pbi)) with the
cluster means (locations). Only if extras = TRUE.
List of
length(Nit*(1-Pbi)) with the mixture weights. Only if extras = TRUE.
List of length(Nit*(1-Pbi)) with the unnormalized weights
(jump sizes). Only if extras = TRUE.
Integer constant. Number of jumps of the continuous component of the unnormalized process.
Integer constant. Number of grid points for the evaluation of the density estimate.
Integer constant. Number of MCMC iterations.
Numeric constant. Burn-in period proportion of Nit.
Numeric vector with execution time provided by
proc.time function.
Integer corresponding to the kernel chosen for the mixture
Data used for the fit
A named list with the parameters of the NRMI process
Numeric vector. Lower limit of interval censoring. For exact data the same as xright
Numeric vector. Upper limit of interval censoring. For exact data the same as xleft.
Numeric vector. Desired quantiles of the density estimates.
Numeric constant. Total mass of the centering measure. See details.
Numeric positive constant. See details.
Numeric constant. \(0\leq \texttt{Gama} \leq 1\). See details.
The distribution name for the kernel. Allowed names are "normal", "gamma", "beta", "double exponential", "lognormal" or their common abbreviations "norm", "exp", or an integer number identifying the mixture kernel: 1 = Normal; 2 = Gamma; 3 = Beta; 4 = Double Exponential; 5 = Lognormal.
The distribution name for the centering measure. Allowed names are "normal", "gamma", "beta", or their common abbreviations "norm", "exp", or an integer number identifying the centering measure: 1 = Normal; 2 = Gamma; 3 = Beta.
Numeric positive constant. Shape parameter of the gamma prior
on the standard deviation of the mixture kernel distr.k.
Numeric positive constant. Rate parameter of the gamma prior
on the standard deviation of the mixture kernel distr.k.
Numeric positive constant. Metropolis-Hastings proposal variation coefficient for sampling sigma.
Numeric positive constant. Metropolis-Hastings proposal variation coefficient for sampling the latent U.
Numeric constant. Relative error of the jump sizes in the continuous component of the process. Smaller values imply larger number of jumps.
Integer constant. Number of grid points for the evaluation of the density estimate.
Integer constant. Number of MCMC iterations.
Numeric constant. Burn-in period proportion of Nit.
Numeric constant. Extension to the evaluation grid range. See details.
Logical. If TRUE, prints out the execution time.
Logical. If TRUE, gives additional objects: means, weights and Js.
Logical. If TRUE, uses an adaptive MCMC strategy to sample the latent U (adaptive delta_U).
The function is computing intensive. Be patient.
Barrios, E., Kon Kam King, G. and Nieto-Barajas, L.E.
This generic function fits a normalized random measure (NRMI) mixture model for density estimation (James et al. 2009) with censored data. Specifically, the model assumes a normalized generalized gamma (NGG) prior for the locations (means) of the mixture kernel and a parametric prior for the common smoothing parameter sigma, leading to a semiparametric mixture model.
This function coincides with MixNRMI1 when the lower (xleft)
and upper (xright) censoring limits correspond to the same exact value.
The details of the model are: $$X_i|Y_i,\sigma \sim k(\cdot
|Y_i,\sigma)$$ $$Y_i|P \sim P,\quad
i=1,\dots,n$$ $$P \sim \textrm{NGG(\texttt{Alpha,
Kappa, Gama; P\_0})}$$ $$\sigma \sim
\textrm{Gamma(asigma, bsigma)}$$ where
\(X_i\)'s are the observed data, \(Y_i\)'s are latent (location)
variables, sigma is the smoothing parameter, k is a parametric
kernel parameterized in terms of mean and standard deviation, (Alpha,
Kappa, Gama; P_0) are the parameters of the NGG prior with P_0 being
the centering measure whose parameters are assigned vague hyper prior
distributions, and (asigma,bsigma) are the hyper-parameters of the
gamma prior on the smoothing parameter sigma. In particular:
NGG(Alpha, 1, 0; P_0) defines a Dirichlet process; NGG(1,
Kappa, 1/2; P_0) defines a Normalized inverse Gaussian process; and
NGG(1, 0, Gama; P_0) defines a normalized stable process.
The evaluation grid ranges from min(x) - epsilon to max(x) +
epsilon. By default epsilon=sd(x)/4.
1.- Barrios, E., Lijoi, A., Nieto-Barajas, L. E. and Prünster, I. (2013). Modeling with Normalized Random Measure Mixture Models. Statistical Science. Vol. 28, No. 3, 313-334.
2.- James, L.F., Lijoi, A. and Prünster, I. (2009). Posterior analysis for normalized random measure with independent increments. Scand. J. Statist 36, 76-97.
3.- Kon Kam King, G., Arbel, J. and Prünster, I. (2016). Species Sensitivity Distribution revisited: a Bayesian nonparametric approach. In preparation.
MixNRMI2, MixNRMI1cens,
MixNRMI2cens, multMixNRMI1
### Example 1
if (FALSE) {
# Data
data(acidity)
x <- acidity
# Fitting the model under default specifications
out <- MixNRMI1cens(x, x)
# Plotting density estimate + 95% credible interval
plot(out)
}
if (FALSE) {
### Example 2
# Data
data(salinity)
# Fitting the model under default specifications
out <- MixNRMI1cens(xleft = salinity$left, xright = salinity$right, Nit = 5000)
# Plotting density estimate + 95% credible interval
attach(out)
plot(out)
# Plotting number of clusters
par(mfrow = c(2, 1))
plot(R, type = "l", main = "Trace of R")
hist(R, breaks = min(R - 0.5):max(R + 0.5), probability = TRUE)
detach()
}
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