Type:	Package
Title:	Spatial Estimation and Prediction for Censored/Missing Responses
Version:	1.0.0
Description:	It provides functions for estimating parameters in linear spatial models with censored or missing responses using the Expectation-Maximization (EM), Stochastic Approximation EM (SAEM), and Monte Carlo EM (MCEM) algorithms. These methods are widely used to obtain maximum likelihood (ML) estimates in the presence of incomplete data. The EM algorithm computes ML estimates when a closed-form expression for the conditional expectation of the complete-data log-likelihood is available. The MCEM algorithm replaces this expectation with a Monte Carlo approximation based on independent simulations of the missing data. In contrast, the SAEM algorithm decomposes the E-step into simulation and stochastic approximation steps, improving computational efficiency in complex settings. In addition, the package provides standard error estimation based on the Louis method. It also includes functionality for spatial prediction at new locations. References used for this package: Galarza, C. E., Matos, L. A., Castro, L. M., & Lachos, V. H. (2022). Moments of the doubly truncated selection elliptical distributions with emphasis on the unified multivariate skew-t distribution. Journal of Multivariate Analysis, 189, 104944 <doi:10.1016/j.jmva.2021.104944>; Valeriano, K. A., Galarza, C. E., & Matos, L. A. (2023). Moments and random number generation for the truncated elliptical family of distributions. Statistics and Computing, 33(1), 32 <doi:10.1007/s11222-022-10200-4>.
License:	GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]
Encoding:	UTF-8
RoxygenNote:	7.3.3
Imports:	ggplot2, gridExtra, MomTrunc, mvtnorm, Rcpp, Rdpack, relliptical, stats, StempCens
RdMacros:	Rdpack
LinkingTo:	RcppArmadillo, Rcpp, RcppProgress, roptim
Depends:	R (≥ 2.10)
LazyData:	true
NeedsCompilation:	yes
Packaged:	2026-03-31 02:37:22 UTC; 55199
Author:	Katherine A. L. Valeriano [aut, cre], Christian Galarza Morales [ctb], Larissa Avila Matos [ctb]
Maintainer:	Katherine A. L. Valeriano <katandreina@gmail.com>
Repository:	CRAN
Date/Publication:	2026-03-31 22:40:42 UTC

Covariance matrix for spatial models

Description

It computes the spatial variance-covariance matrix using exponential, gaussian, matérn, or power exponential correlation function.

Usage

CovMat(phi, tau2, sig2, coords, type = "exponential", kappa = NULL)

Arguments

Details

The spatial covariance matrix is given by

\Sigma = [Cov(s_i, s_j )] = \sigma^2 R(\phi) + \tau^2 I_n,

where \sigma^2 > 0 is the partial sill, \phi > 0 is the spatial scaling parameter, \tau^2 > 0 is known as the nugget effect in the geostatistical framework, R(\phi) is the n\times n correlation matrix computed from a correlation function, and I_n is the n\times n identity matrix.

The spatial correlation functions available are:

Exponential:

Corr(d) = exp(-d/\phi),

Gaussian:

Corr(d) = exp(-(d/\phi)^2),

Matérn:

Corr(d) = \frac{1}{2^{(\kappa-1)}\Gamma(\kappa)}\left(\frac{d}{\phi}\right)^\kappa K_\kappa \left( \frac{d}{\phi} \right),

Power exponential:

Corr(d) = exp(-(d/\phi)^\kappa),

where d \geq 0 is the Euclidean distance between two observations, \Gamma(.) is the gamma function, \kappa is the smoothness parameter, and K_\kappa(.) is the modified Bessel function of the second kind of order \kappa.

Value

An n\times n spatial covariance matrix.

Author(s)

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

See Also

dist2Dmatrix, EM.sclm, MCEM.sclm, SAEM.sclm

Examples

set.seed(1000)
n = 20
coords = round(matrix(runif(2*n, 0, 10), n, 2), 5)
Cov = CovMat(phi=5, tau2=0.8, sig2=2, coords=coords, type="exponential")

ML estimation of spatial censored linear models via the EM algorithm

Description

It fits a spatial linear model with left-, right-, or interval-censored responses using the Expectation-Maximization (EM) algorithm. The function provides parameter estimates and their standard errors, and supports missing values in the response variable.

Usage

EM.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, error = 1e-04, show_se = TRUE)

Arguments

Details

The spatial Gaussian model is given by

Y = X\beta + \xi,

where Y is the n\times 1 response vector, X is the n\times q design matrix, \beta is the q\times 1 vector of regression coefficients to be estimated, and \xi is the error term, assumed to follow a normal distribution with zero-mean and covariance matrix \Sigma=\sigma^2 R(\phi) + \tau^2 I_n. We assume that \Sigma is non-singular and that X has a full rank (Diggle and Ribeiro 2007).

The estimation is carried out using the EM algorithm, originally proposed by Dempster et al. (1977). The conditional expectations required in the E-step are computed using the function meanvarTMD from the package MomTrunc.

Value

An object of class "sclm". Generic functions print and summary are available to display the fitted results. The plot method can be used to visualize convergence diagnostics of the parameter estimates.

Specifically, the following components are returned:

Note

The final EM estimates correspond to the parameter values obtained at the last iteration of the EM algorithm.

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

Author(s)

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

References

Dempster AP, Laird NM, Rubin DB (1977). “Maximum likelihood from incomplete data via the EM algorithm.” Journal of the Royal Statistical Society: Series B (Methodological), 39(1), 1–38.

Diggle P, Ribeiro P (2007). Model-based Geostatistics. Springer.

See Also

MCEM.sclm, SAEM.sclm, predict.sclm

Examples

# Simulated example: 10% of left-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), runif(n))
data = rCensSp(c(-1,3), 2, 4, 0.5, x, coords, "left", 0.10, 0, "gaussian")

fit = EM.sclm(y=data$y, x=x, ci=data$ci, lcl=data$lcl, ucl=data$ucl,
              coords=coords, phi0=3, nugget0=1, type="gaussian")
fit

ML estimation of spatial censored linear models via the MCEM algorithm

Description

It fits a spatial linear model with left-, right-, or interval-censored responses using the Monte Carlo EM (MCEM) algorithm. The function provides parameter estimates and their standard errors, and supports missing values in the response variable.

Usage

MCEM.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 = 500, nMin = 20, nMax = 5000,
  error = 1e-04, show_se = TRUE)

Arguments

Details

The spatial Gaussian model is given by

Y = X\beta + \xi,

where Y is the n\times 1 response vector, X is the n\times q design matrix, \beta is the q\times 1 vector of regression coefficients to be estimated, and \xi is the error term, assumed to follow a normal distribution with zero-mean and covariance matrix \Sigma=\sigma^2 R(\phi) + \tau^2 I_n. We assume that \Sigma is non-singular and that X has a full rank (Diggle and Ribeiro 2007).

Parameter estimation is carried out using the MCEM algorithm, originally proposed by Wei and Tanner (1990). The Monte Carlo (MC) approximation starts with a sample of size nMin. At each iteration, the sample size increases by (nMax - nMin) / MaxIter, reaching nMax at the final iteration. The random samples are generated using the slice sampling algorithm implemented in the relliptical package.

Value

An object of class "sclm". Generic functions print and summary are available to display the fitted results. The plot method can be used to visualize convergence diagnostics of the parameter estimates.

Specifically, the following components are returned:

Note

The MCEM final estimates correspond to the mean of the estimates obtained at each iteration after deleting the half and applying a thinning of 3.

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

Author(s)

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

References

Diggle P, Ribeiro P (2007). Model-based Geostatistics. Springer.

Wei G, Tanner M (1990). “A Monte Carlo implementation of the EM algorithm and the poor man's data augmentation algorithms.” Journal of the American Statistical Association, 85(411), 699–704. doi:10.1080/01621459.1990.10474930.

See Also

EM.sclm, SAEM.sclm, predict.sclm

Examples

# Example 1: left censoring data
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(2,-1), 2, 3, 0.70, x, coords, "left", 0.08, 0, "matern", 1)

fit = MCEM.sclm(y=data$y, x=x, ci=data$ci, lcl=data$lcl, ucl=data$ucl,
                coords, phi0=2.50, nugget0=0.75, type="matern",
                kappa=1, MaxIter=30, nMax=1000)
fit$tab

# Example 2: left censoring and missing data
yMiss = data$y
yMiss[20] = NA
ci = data$ci
ci[20] = 1
ucl = data$ucl
ucl[20] = Inf

fit1 = MCEM.sclm(y=yMiss, x=x, ci=ci, lcl=data$lcl, ucl=ucl, coords,
                 phi0=2.50, nugget0=0.75, type="matern", kappa=1,
                 MaxIter=300, nMax=1000)
summary(fit1)
plot(fit1)

TCDD concentration data

Description

The level of dioxin (2,3,7,8-tetrachlorodibenzo-p-dioxin or TCDD) data was collected in November 1983 by the U.S. Environmental Protection Agency (EPA) in several areas of a highway in Missouri, USA. The TCDD measurement was subject to a limit of detection (cens); thereby, the TCDD data is left-censored. Only the locations used in the geostatistical analysis by Zirschky and Harris (1986) are shown.

Usage

data("Missouri")

Format

A data frame with 127 observations and five variables:

xcoord

x coordinate of the start of each transect (ft).

ycoord

y coordinate of the start of each transect (ft).

TCDD

TCDD concentrations (mg/kg).

transect

transect length (ft).

cens

indicator of censoring (left-censored observations).

Source

Zirschky JH, Harris DJ (1986). “Geostatistical analysis of hazardous waste site data.” Journal of Environmental Engineering, 112(4), 770–784.

See Also

EM.sclm, MCEM.sclm, SAEM.sclm

Examples

data("Missouri")
y = log(Missouri$TCDD)
cc = Missouri$cens
coord = cbind(Missouri$xcoord/100, Missouri$ycoord)
x = matrix(1, length(y), 1)
lcl = rep(-Inf, length(y))
ucl = y

## SAEM fit
set.seed(83789)
fit1 = SAEM.sclm(y, x, cc, lcl, ucl, coord, 5, 1, lower=c(1e-5,1e-5),
                 upper=c(50,50))
fit1$tab

## MCEM fit
fit2 = MCEM.sclm(y, x, cc, lcl, ucl, coord, 5, 1, lower=c(1e-5,1e-5),
                 upper=c(50,50), MaxIter=300, nMax=1000)
fit2$tab

## Imputed values
cbind(fit1$EY, fit2$EY)[cc==1,]

ML estimation of spatial censored linear models via the SAEM algorithm

Description

It fits a spatial linear model with left-, right-, or interval-censored responses using the Stochastic Approximation EM (SAEM) algorithm. The function provides parameter estimates and their standard errors, and supports missing values in the response 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

Details

The spatial Gaussian model is given by

Y = X\beta + \xi,

where Y is the n\times 1 response vector, X is the n\times q design matrix, \beta is the q\times 1 vector of regression coefficients to be estimated, and \xi is the error term, assumed to follow a normal distribution with zero-mean and covariance matrix \Sigma=\sigma^2 R(\phi) + \tau^2 I_n. We assume that \Sigma is non-singular and that X has a full rank (Diggle and Ribeiro 2007).

Parameter estimation is carried out using the SAEM algorithm, originally proposed by Delyon et al. (1999). The spatial censored SAEM approach has been previously developed by Lachos et al. (2017) and Ordoñez et al. (2018), and is implemented in the CensSpatial package. Differences among implementations mainly arise from the random number generation schemes and optimization procedures.

This model can also be viewed as a particular case of the spatio-temporal model proposed by Valeriano et al. (2021), when the number of temporal observations is equal to one. The corresponding SAEM implementation for the spatio-temporal setting is available in the StempCens package.

Value

An object of class "sclm". Generic functions print and summary are available to display the fitted results. The plot method can be used to visualize convergence diagnostics of the parameter estimates.

Specifically, the following components are returned:

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, Christian E. Galarza, and Larissa A. Matos.

References

Delyon B, Lavielle M, Moulines E (1999). “Convergence of a stochastic approximation version of the EM algorithm.” The Annals of Statistics, 27(1), 94–128.

Diggle P, Ribeiro P (2007). Model-based Geostatistics. Springer.

Lachos VH, Matos LA, Barbosa TS, Garay AM, Dey DK (2017). “Influence diagnostics in spatial models with censored response.” Environmetrics, 28(7).

Ordoñez JA, Bandyopadhyay D, Lachos VH, Cabral CRB (2018). “Geostatistical estimation and prediction for censored responses.” Spatial Statistics, 23, 109–123. doi:10.1016/j.spasta.2017.12.001.

Valeriano KL, Lachos VH, Prates MO, Matos LA (2021). “Likelihood-based inference for spatiotemporal data with censored and missing responses.” Environmetrics, 32(3).

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)

Distance matrix computation

Description

It computes the Euclidean distance matrix for a set of coordinates.

Usage

dist2Dmatrix(coords)

Arguments

Value

An n\times n distance matrix.

Author(s)

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

Examples

set.seed(1000)
n = 100
x = round(runif(n,0,10), 5)     # X coordinate
y = round(runif(n,0,10), 5)     # Y coordinate
Mdist = dist2Dmatrix(cbind(x, y))

Prediction in spatial models with censored/missing responses

Description

It performs spatial prediction at a set of new S spatial locations.

Usage

## S3 method for class 'sclm'
predict(object, locPre, xPre, ...)

Arguments

Details

This function performs prediction under the mean squared error (MSE) criterion, where the conditional expectation E(Y | X) is used as the optimal predictor.

Value

The function returns a list with:

Author(s)

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

See Also

EM.sclm, MCEM.sclm, SAEM.sclm

Examples

set.seed(1000)
n = 120
coords = round(matrix(runif(2*n,0,15),n,2), 5)
x = cbind(rbinom(n,1,0.50), rnorm(n), rnorm(n))
data = rCensSp(c(1,4,-1), 2, 3, 0.50, x, coords, "left", 0.10, 20)

## Estimation
data1 = data$Data

# Estimation: EM algorithm
fit1 = EM.sclm(y=data1$y, x=data1$x, ci=data1$ci, lcl=data1$lcl,
               ucl=data1$ucl, coords=data1$coords, phi0=2.50, nugget0=1)

# Estimation: SAEM algorithm
fit2 = SAEM.sclm(y=data1$y, x=data1$x, ci=data1$ci, lcl=data1$lcl,
                 ucl=data1$ucl, coords=data1$coords, phi0=2.50, nugget0=1)

# Estimation: MCEM algorithm
fit3 = MCEM.sclm(y=data1$y, x=data1$x, ci=data1$ci, lcl=data1$lcl,
                 ucl=data1$ucl, coords=data1$coords, phi0=2.50, nugget0=1,
                 MaxIter=300)
cbind(fit1$theta, fit2$theta, fit3$theta)

# Prediction
data2 = data$TestData
pred1 = predict(fit1, data2$coords, data2$x)
pred2 = predict(fit2, data2$coords, data2$x)
pred3 = predict(fit3, data2$coords, data2$x)

# Cross-validation
mean((data2$y - pred1$predValues)^2)
mean((data2$y - pred2$predValues)^2)
mean((data2$y - pred3$predValues)^2)

Censored spatial data simulation

Description

It simulates censored spatial data under a linear model for a specified censoring rate.

Usage

rCensSp(beta, sigma2, phi, nugget, x, coords, cens = "left", pcens = 0.1,
  npred = 0, cov.model = "exponential", kappa = NULL)

Arguments

Value

If npred > 0, it returns two lists: Data and TestData; otherwise, it returns a list with the simulated data.

Data

TestData

Author(s)

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

Examples

set.seed(1000)
n = 100
coords = round(matrix(runif(2*n,0,15),n,2), 5)
x = cbind(1, rnorm(n))
data = rCensSp(beta=c(5,2), sigma2=2, phi=4, nugget=0.70, x=x,
               coords=coords, cens="left", pcens=0.10, npred=10,
               cov.model="gaussian")
data$Data
data$TestData