SAEM.sclm: ML estimation of spatial censored linear models via the SAEM...

In RcppCensSpatial: Spatial Estimation and Prediction for Censored/Missing Responses

ML estimation of spatial censored linear models via the SAEM algorithm

Description

It fits the left, right, or interval spatial censored linear model using the Stochastic Approximation EM (SAEM) algorithm. It provides estimates and standard errors of the parameters and supports missing values on the dependent variable.

Usage

SAEM.sclm(y, x, ci, lcl = NULL, ucl = NULL, coords, phi0, nugget0,
  type = "exponential", kappa = NULL, lower = c(0.01, 0.01),
  upper = c(30, 30), MaxIter = 300, M = 20, pc = 0.2, error = 1e-04,
  show_se = TRUE)

Arguments

y	vector of responses of length n.
x	design matrix of dimensions n\times q, where q is the number of fixed effects, including the intercept.
ci	vector of censoring indicators of length n. For each observation: 1 if censored/missing, 0 otherwise.
lcl, ucl	vectors of length n representing the lower and upper bounds of the interval, which contains the true value of the censored observation. Default =NULL, indicating no-censored data. For each observation: lcl=-Inf and ucl=c (left censoring); lcl=c and ucl=Inf (right censoring); and lcl and ucl must be finite for interval censoring. Moreover, missing data could be defined by setting lcl=-Inf and ucl=Inf.
coords	2D spatial coordinates of dimensions n\times 2.
phi0	initial value for the spatial scaling parameter.
nugget0	initial value for the nugget effect parameter.
type	type of spatial correlation function: 'exponential', 'gaussian', 'matern', and 'pow.exp' for exponential, gaussian, matérn, and power exponential, respectively.
kappa	parameter for some spatial correlation functions. See CovMat.
lower, upper	vectors of lower and upper bounds for the optimization method. If unspecified, the default is c(0.01,0.01) for lower and c(30,30) for upper.
MaxIter	maximum number of iterations of the SAEM algorithm. By default =300.
M	number of Monte Carlo samples for stochastic approximation. By default =20.
pc	percentage of initial iterations of the SAEM algorithm with no memory. It is recommended that 50<MaxIter*pc<100. By default =0.20.
error	maximum convergence error. By default =1e-4.
show_se	logical. It indicates if the standard errors should be estimated by default =TRUE.

Details

The spatial Gaussian model is given by

Y = Xβ + ξ,

where Y is the n\times 1 response vector, X is the n\times q design matrix, β is the q\times 1 vector of regression coefficients to be estimated, and ξ is the error term which is normally distributed with zero-mean and covariance matrix Σ=σ^2 R(φ) + τ^2 I_n. We assume that Σ is non-singular and X has full rank \insertCitediggle2007springerRcppCensSpatial.

The estimation process is performed via the SAEM algorithm, initially proposed by \insertCitedelyon1999convergence;textualRcppCensSpatial. The spatial censored (SAEM) algorithm was previously proposed by \insertCitelachos2017influence;textualRcppCensSpatial and \insertCiteordonez2018geostatistical;textualRcppCensSpatial and is available in the package CensSpatial. These packages differ in the random number generation and optimization procedure.

This model is also a particular case of the spatio-temporal model defined by \insertCitevaleriano2021likelihood;textualRcppCensSpatial when the number of temporal observations is equal to one. The computing codes of the spatio-temporal SAEM algorithm are available in the package StempCens.

Value

An object of class "sclm". Generic functions print and summary have methods to show the results of the fit. The function plot can extract convergence graphs for the parameter estimates.

Specifically, the following components are returned:

Theta	estimated parameters in all iterations, θ = (β, σ^2, φ, τ^2).
theta	final estimation of θ = (β, σ^2, φ, τ^2).
beta	estimated β.
sigma2	estimated σ^2.
phi	estimated φ.
tau2	estimated τ^2.
EY	stochastic approximation of the first conditional moment.
EYY	stochastic approximation of the second conditional moment.
SE	vector of standard errors of θ = (β, σ^2, φ, τ^2).
InfMat	observed information matrix.
loglik	log-likelihood for the SAEM method.
AIC	Akaike information criterion.
BIC	Bayesian information criterion.
Iter	number of iterations needed to converge.
time	processing time.
call	RcppCensSpatial call that produced the object.
tab	table of estimates.
critFin	selection criteria.
range	effective range.
ncens	number of censored/missing observations.
MaxIter	maximum number of iterations for the SAEM algorithm.

Note

The SAEM final estimates correspond to the estimates obtained at the last iteration of the algorithm.

To fit a regression model for non-censored data, just set ci as a vector of zeros.

Author(s)

Katherine L. Valeriano, Alejandro Ordoñez, Christian E. Galarza, and Larissa A. Matos.

References

\insertAllCited

See Also

EM.sclm, MCEM.sclm, predict.sclm

Examples

# Example 1: 8% of right-censored observations
set.seed(1000)
n = 50   # Test with another values for n
coords = round(matrix(runif(2*n,0,15),n,2), 5)
x = cbind(rnorm(n), rnorm(n))
data = rCensSp(c(4,-2), 1, 3, 0.50, x, coords, "right", 0.08)

fit = SAEM.sclm(y=data$y, x=x, ci=data$ci, lcl=data$lcl, ucl=data$ucl,
                coords, phi0=2, nugget0=1, type="exponential", M=10,
                pc=0.18)
fit

# Example 2: censored and missing observations
set.seed(123)
n = 200
coords = round(matrix(runif(2*n,0,20),n,2), 5)
x = cbind(runif(n), rnorm(n), rexp(n))
data = rCensSp(c(1,4,-1), 2, 3, 0.50, x, coords, "left", 0.05, 0,
               "matern", 3)
data$y[c(10,120)] = NA
data$ci[c(10,120)] = 1
data$ucl[c(10,120)] = Inf

fit2 = SAEM.sclm(y=data$y, x=x, ci=data$ci, lcl=data$lcl, ucl=data$ucl,
                 coords, phi0=2, nugget0=1, type="matern", kappa=3,
                 M=10, pc=0.18)
fit2$tab
plot(fit2)

RcppCensSpatial documentation built on June 28, 2022, 1:07 a.m.