SmoothEst            package:AnalyzeFMRI            R Documentation

_E_s_t_i_m_a_t_e _t_h_e _v_a_r_i_a_n_c_e-_c_o_v_a_r_i_a_n_c_e _m_a_t_r_i_x _o_f _a _G_a_u_s_s_i_a_n _r_a_n_d_o_m _f_i_e_l_d

_D_e_s_c_r_i_p_t_i_o_n:

     Estimate the variance-covariance matrix of a Gaussian random field

_U_s_a_g_e:

     SmoothEst(mat, mask, voxdim, method = "Forman")

_A_r_g_u_m_e_n_t_s:

     mat: 3D array that is the Gaussian Random Field.

    mask: 3D mask array.

  voxdim: Vector of length 3 containing the voxel dimensions.

  method: The estimator to use. method = "Forman" (the default) uses
          the estimator proposed in [1]. method = "Friston" uses the
          estimator proposed in [2, 3], but tis can be biased when the
          amount of smoothing is small compared to the size of each
          voxel (see [1] for more details and example below)

_D_e_t_a_i_l_s:

     Calculates the varaince-covariance matrix using the variance
     covariance matrix of partial derivatives.

_V_a_l_u_e:

     A (3x3) diagonal matrix.

_A_u_t_h_o_r(_s):

     J. L. Marchini

_R_e_f_e_r_e_n_c_e_s:

     [1] Stephen D. Forman et al. (1995) Improved assessment of
     significant activation in functional magnetic resonance imaging
     (fMRI): Use of a cluster-size threshold. Magnetic Resonance in
     Medicine, 33:636-647.

     [2] Karl J. Friston et al. (1991) Comparing functional (PET)
     images: the assessment of significant change. J. Cereb. Blood Flow
     Metab. 11:690-699. 

     [3] Stefan J. Kiebel et al. (1999) Robust smoothness estimation in
     statistical parametric maps using standardized residuals from the
     general linear model. NeuroImage, 10:756-766.

_E_x_a_m_p_l_e_s:

     ###############
     ## EXAMPLE 1 ##
     ###############
     ## example that illustrates the bias of the Friston
     ## method when smoothing is small compared to voxel size
     ## NB. The presence of bias becomes clearer if the
     ##     simulations below are run about 100 times and
     ##     the results averaged

     ksize <- 13
     d <- c(64, 64, 64)
     voxdim <- c(1, 1, 1)
     FWHM <- 2 ## using a small value of FWHM (=2) compared to voxel size (=1)
     sigma <- diag(FWHM^2, 3) / (8 * log(2))
     mask <- array(1, dim = d)
     num.vox <- sum(mask)

     grf <- Sim.3D.GRF(d = d, voxdim = voxdim, sigma = sigma, ksize = ksize, mask = mask, type = "field")$mat
                    
     sigma
     SmoothEst(grf, mask, voxdim, method = "Friston")
     SmoothEst(grf, mask, voxdim, method = "Forman") ## compared to sigma
     ##the Forman estimator is better (on average) than the Friston estimator

     ###############
     ## EXAMPLE 2 ##
     ###############
     ## increasing the amount of smoothing decreases the bias of the Friston estimator

     ksize <- 13
     d <- c(64, 64, 64)
     voxdim <- c(1, 1, 1)
     FWHM <- 5 ## using a large value of FWHM (=5) compared to voxel size (=1)
     sigma <- diag(FWHM^2, 3) / (8 * log(2))
     mask <- array(1, dim = d)
     num.vox <- sum(mask)

     grf <- Sim.3D.GRF(d = d, voxdim = voxdim, sigma = sigma, ksize = ksize, mask = mask, type = "field")$mat

     SmoothEst(grf, mask, voxdim, method = "Friston")
     SmoothEst(grf, mask, voxdim, method = "Forman")                
     sigma

