Covariance Matrices {MixfMRI}

Covariance Matrices

Description

These functions compute posterior probabilities, Fisher information with covariance matrix of parameters, covariance matrix of posterior probabilities, and covariance matrix of logit posterior probabilities.

Usage

  post.prob(x, fcobj)
  cov.param(x, fcobj, post.z, drop.ETA1 = FALSE)
  cov.post.z(x, fcobj, post.z, cov.param = NULL, all.x = FALSE,
                 drop.ETA1 = FALSE)
  cov.logit.z(x, fcobj, post.z, cov.param = NULL, cov.post.z = NULL,
                  all.x = FALSE, drop.ETA1 = FALSE)

Arguments

x

an input list of two elements X.gbd and PV.gbd.

fcobj

a fclust object.

post.z

a matrix of dim = N * K for posterior probabilities, which is also the return value of post.prob().

cov.param

a covariance matrix of dim = d * d for parameters, which is also a return of cov.param(). d is total number of parameters which is dependent on data and models.

cov.post.z

a covariance list of length equal to number of active voxels, which is also a return of cov.post.z().

all.x

all cov matrices for all observations are returned if TRUE, while for only active observations (those of class ids are greater than 1) if FALSE.

drop.ETA1

if drop the ETA[1] from the cov matrix due to the min.1st.prop constrain.

Details

These functions are required to compute covariance matrices of parameters and posterior probabilities.

Use post.prob() to get the posterior probabilities.

Input the returns of post.prob() to cov.param() to obtain the cov matrix for parameters (inversed Fisher information obtained from inner product of gradient of log observed data likelihood). A list is returned with I for Fisher information, and cov for the covariance matrix which is inverted by ginv().

Input the returns of post.prob() and cov.param() to cov.post.z() to obtain the cov matrix for posterior probabilities by the multivariate delta method on the cov matrix for parameters.

Input the returns of post.prob(), cov.param(), and cov.post.z() to cov.logit.z() to obtain cov matrix for logit posterior probabilities by the multivariate delta method on cov matrix of posterior probabilities.

Value

A matrix or a list is returned.

The cov.param() will return a list containing two elements I for the Fisher information, and cov for the covariance matrix by generalized inversed of the Fisher information. The dimension of both elements are d * d where d = K * 7 - 4 for 2D data and d = K * 9 - 4 for 3D data if drop.ETA1 = TRUE, otherwise they are d = K * 7 - 3 and d = K * 9 -4, respectively.

The cov.post.z() will return a list containing cov matrices of posterior probabilities for each valid/selected voxel.

The cov.logit.z() will return a list containing cov matrices of logit posterior probabilities for each valid/selected voxel.

Author(s)

Wei-Chen Chen and Ranjan Maitra.

References

Chen, W.-C. and Maitra, R. (2021) “A Practical Model-based Segmentation Approach for Accurate Activation Detection in Single-Subject functional Magnetic Resonance Imaging Studies”, arXiv:2102.03639.

See Also

EMCluster::lmt(), lmt.I().

Examples

library(MixfMRI, quietly = TRUE)
library(EMCluster, quietly = TRUE)
.FC.CT$model.X <- "I"
.FC.CT$CONTROL$debug <- 0
K <- 3
  

.rem <- function(){

  ### Fit toy1.
  set.seed(1234)
  X.gbd <- toy1$X.gbd
  X.range <- apply(X.gbd, 2, range)
  X.gbd <- t((t(X.gbd) - X.range[1,]) / (X.range[2,] - X.range[1,]))
  PV.gbd <- toy1$PV.gbd
  fcobj <- fclust(X.gbd, PV.gbd, K = K, min.1st.prop = 0.5)
  
  ### Test cov matrix of posterior z and logit posterior z.
  x <- list(X.gbd = X.gbd, PV.gbd = PV.gbd)
  post.z <- post.prob(x, fcobj)
  cov.param <- cov.param(x, fcobj, post.z = post.z)
  cov.post.z <- cov.post.z(x, fcobj, post.z = post.z,
                               cov.param = cov.param$cov)
  cov.logit.z <- cov.logit.z(x, fcobj, post.z = post.z,
                                 cov.param = cov.param$cov,
                                 cov.post.z = cov.post.z)
  
  ### Compute cov matrix of log odds ratio for all k > 1.
  A <- cbind(rep(-1, K - 1), diag(1, K - 1))
  logit.p <- log(post.z[fcobj$class != 1,] / (1 - post.z[fcobj$class != 1,]))
  log.or <- logit.p %*% t(A)
  cov.log.or <- lapply(cov.logit.z, function(x) A %*% x %*% t(A))
  
  ### Check if 0 vector covered by 95% confidence ellipsoid.
  id <- 1
  plot(log.or[id,],
       xlim = log.or[id, 1] + c(-5, 5),
       ylim = log.or[id, 2] + c(-5, 5),
       main = "1st observation", xlab = "x", ylab = "y")
  plotBN(log.or[id,], cov.log.or[[id]])
  points(0, 0, col = 2)

}