Stem.Bootstrap: Parametric bootstrap

Description Usage Arguments Details Value Author(s) References See Also Examples

Description

This functions performs the spatio-temporal parametric bootstrap for computing the parameter standard errors.

Usage

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Stem.Bootstrap(StemModel, B)

Arguments

StemModel

an object of class “Stem.Model” given as output by the Stem.Estimation function.

B

number of bootstrap iterations.

Details

The spatio-temporal bootstrap is used for parameter uncertainty assessment. The resampling scheme is based on the estimated model: each bootstrap sample is drawn directly from the Gaussian distributions which define the model with the parameter vector replaced by the corresponding ML estimates. For each of the B bootstrap samples, the ML estimates are computed (using the procedure of Stem.Estimation function). Then the B bootstrap replications are returned in a list.

Value

The function returns a list of elements called “boot.output”. Each element of the list is an object of class “Stem.Model” and so it is composed by the following elements:

skeleton

a list with components phi, p, K where the phi vector is given by
the StemModel$estimates$phi.hat vector.

data

a list with components z, coordinates, covariates, r, n and d where the z matrix is given by the simulated data matrix.

estimates

A list of four objects: phi.hat, y.smoothed, loglik, convergence.par as the output of the Stem.Estimation function.

Author(s)

Michela Cameletti [email protected]

References

Amisigo, B.A., Van De Giesen, N.C. (2005) Using a spatio-temporal dynamic state-space model with the EM algorithm to patch gaps in daily riverflow series. Hydrology and Earth System Sciences 9, 209–224.

Fasso', A., Cameletti, M., Nicolis, O. (2007) Air quality monitoring using heterogeneous networks. Environmetrics 18, 245–264.

Fasso', A., Cameletti, M. (2007) A general spatio-temporal model for environmental data. Tech.rep. n.27 Graspa - The Italian Group of Environmental Statistics - http://www.graspa.org.

Fasso', A., Cameletti, M. (2009) A unified statistical approach for simulation, modelling, analysis and mapping of environmental data. Accepted for publication by Simulation: transaction of the Society for Modeling and Simulation International.

Xu, K., Wikle, C.K. (2007) Estimation of parameterized spatio-temporal dynamic models. Journal of Statistical Inference and Planning 137, 567–588.

Mc Lachlan, G.J., Krishnan, T. (1997) The EM Algorithm and Extensions. Wiley, New York.

Shumway, R.H., Stoffer, D.S. (2006) Time Series Analysis and Its Applications: with R Examples. Springer, New York.

See Also

See Also pm10, Stem.Model and Stem.Estimation

Examples

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#load the data
data(pm10)

#extract the data
coordinates <- pm10$coords
covariates <- pm10$covariates
z <- pm10$z

#build the parameter list 
#(the phi list is used for the algorithm starting values)
phi <- list(beta=matrix(c(3.65,0.046,-0.904),3,1),
			sigma2eps=0.1,
			sigma2omega=0.2,
			theta=0.01,
			G=matrix(0.77,1,1),
			Sigmaeta=matrix(0.3,1,1),
			m0=as.matrix(0),
			C0=as.matrix(1))

K <- matrix(1,ncol(z),1)

mod1 <- Stem.Model(z=z,covariates=covariates,
	coordinates=coordinates,phi=phi,K=K)
class(mod1)
is.Stem.Model(mod1)

#mod1 is given as output by the Stem.Model function
mod1.est <- Stem.Estimation(mod1)

#it is computer intensive
mod1.boot <- Stem.Bootstrap(StemModel=mod1.est, B=3)
names(mod1.boot)
#the first element of the output list
names(mod1.boot$boot.output[[1]])


#check if there is no convergence for some bootstrap iteration
B <- length(mod1.boot$boot.output)
n.null <- sum (unlist (lapply(mod1.boot$boot.output, 
function(x) length(x$StemModel)==1)) )
pos.null <- which((unlist (lapply(mod1.boot$boot.output, 
function(x) length(x$StemModel)==1)) ))
cat("------------B non null =", B - n.null,"\n")
if(length(pos.null)>0) boot1.mod = mod1.boot[-pos.null]

#put the bootstrap output in a matrix
npar <- length(unlist((mod1.boot$boot.output[[1]]$estimates$phi.hat)))-1
boot.estimates <- matrix(NA, nrow = (B - n.null), ncol = npar)

for(b in 1:(B - n.null)) {
phi.estimated <- mod1.boot$boot.output[[b]]$estimates$phi.hat
boot.estimates[b,] <-  c(phi.estimated$beta,
			phi.estimated$sigma2eps,
      		phi.estimated$sigma2omega,
			phi.estimated$theta,
      		phi.estimated$G,
			phi.estimated$Sigmaeta,
			phi.estimated$m0)
}

#compute the parameter standard errors
se <- sqrt(diag(var(na.omit(boot.estimates))))

#create a summary table with Estimates, Standard Errors (SE) and T-statistics.
phi.hat <- mod1.est$estimates$phi.hat
MLE <- c(phi.hat$beta, phi.hat$sigma2eps, phi.hat$sigma2omega,
phi.hat$theta, phi.hat$G, phi.hat$Sigmaeta,phi.hat$m0)
output1 <- cbind(MLE, se, MLE/se)
colnames(output1)<- c("Estimate", "SE", "T-stat.")
output1

#compute the 95% confidence intervals
IC <- matrix(NA,nrow=npar,ncol=2)
for(i in 1 : npar) {
      IC[i,] <- c(quantile(boot.estimates[,i],0.025),
quantile(boot.estimates[,i],0.975))
}


#create a summary table with Estimates, Standard Errors (SE) 
#and T-statistics and confidence intervals.
output2 <- cbind(output1,IC)
colnames(output2) <- c("Estimate", "SE", "T-stat.", "IC_inf", "IC_sup")
output2

Example output

Loading required package: mvtnorm
Loading required package: MASS
[1] "Stem.Model"
[1] TRUE
****************EM Algorithm - iteration n. 1 
***NR Algorithm - iteration n. 1 
***NR Algorithm - iteration n. 2 
***NR Algorithm - iteration n. 3 
***NR Algorithm - iteration n. 4 
***NR Algorithm - iteration n. 5 
****************EM Algorithm - iteration n. 2 
***NR Algorithm - iteration n. 1 
***NR Algorithm - iteration n. 2 
***NR Algorithm - iteration n. 3 
***NR Algorithm - iteration n. 4 
****************EM Algorithm - iteration n. 3 
***NR Algorithm - iteration n. 1 
***NR Algorithm - iteration n. 2 
***NR Algorithm - iteration n. 3 
***NR Algorithm - iteration n. 4 
***NR Algorithm - iteration n. 5 
****************EM Algorithm - iteration n. 4 
***NR Algorithm - iteration n. 1 
***NR Algorithm - iteration n. 2 
****************EM Algorithm - iteration n. 5 
***NR Algorithm - iteration n. 1 
***NR Algorithm - iteration n. 2 
****************EM Algorithm - iteration n. 6 
***NR Algorithm - iteration n. 1 
***NR Algorithm - iteration n. 2 
****************EM Algorithm - iteration n. 7 
=========================== BOOTSTRAP ITERATION N. 1 ===========================  
****************EM Algorithm - iteration n. 1 
***NR Algorithm - iteration n. 1 
***NR Algorithm - iteration n. 2 
****************EM Algorithm - iteration n. 2 
***NR Algorithm - iteration n. 1 
****************EM Algorithm - iteration n. 3 
***NR Algorithm - iteration n. 1 
****************EM Algorithm - iteration n. 4 
=========================== BOOTSTRAP ITERATION N. 2 ===========================  
****************EM Algorithm - iteration n. 1 
***NR Algorithm - iteration n. 1 
****************EM Algorithm - iteration n. 2 
***NR Algorithm - iteration n. 1 
****************EM Algorithm - iteration n. 3 
=========================== BOOTSTRAP ITERATION N. 3 ===========================  
****************EM Algorithm - iteration n. 1 
***NR Algorithm - iteration n. 1 
****************EM Algorithm - iteration n. 2 
***NR Algorithm - iteration n. 1 
****************EM Algorithm - iteration n. 3 
[1] "boot.output"
[1] "skeleton"  "data"      "estimates"
------------B non null = 3 
          Estimate           SE    T-stat.
 [1,]  4.783145941 1.562749e-02 306.072608
 [2,]  0.002709126 8.723504e-05  31.055474
 [3,] -0.947914091 3.846030e-02 -24.646557
 [4,]  0.084248655 2.542421e-03  33.137183
 [5,]  0.173456019 1.713464e-03 101.231229
 [6,]  0.007540571 3.808402e-04  19.799832
 [7,]  0.975095649 7.088034e-03 137.569271
 [8,]  0.067555184 1.922474e-03  35.139718
 [9,] -1.255130285 4.492622e-01  -2.793759
          Estimate           SE    T-stat.       IC_inf       IC_sup
 [1,]  4.783145941 1.562749e-02 306.072608  4.777880916  4.805210495
 [2,]  0.002709126 8.723504e-05  31.055474  0.002715966  0.002870147
 [3,] -0.947914091 3.846030e-02 -24.646557 -1.003994089 -0.940250813
 [4,]  0.084248655 2.542421e-03  33.137183  0.081054309  0.085879295
 [5,]  0.173456019 1.713464e-03 101.231229  0.174186044  0.177415693
 [6,]  0.007540571 3.808402e-04  19.799832  0.007524053  0.008226455
 [7,]  0.975095649 7.088034e-03 137.569271  0.972485276  0.984340910
 [8,]  0.067555184 1.922474e-03  35.139718  0.068947486  0.072599600
 [9,] -1.255130285 4.492622e-01  -2.793759 -0.922237838 -0.072558155

Stem documentation built on May 29, 2017, 11:12 p.m.