# R/rftPval.R In Tokazama/iClass: More intuitive syntax for manipulating imaging data

#### Documented in eulerrftPval

#' RFT p-values
#'
#' Calculates p-values of a statistical field using random field theory
#'
#' @param D Image dimensions.
#' @param c Number of clusters.
#' @param k Spatial extent in resels
#' @param u Statistical threshold.
#' @param n Number of statistical field in conjunction.
#' @param resels Resel measurements of the search region.
#' @param df Degrees of freedom expressed as c(degrees of interest, degrees of error).
#' @param fieldType:
#' \itemize{
#' \item{"T"}{T-field}
#' \item{"F"}{F-field}
#' \item{"X"}{Chi-square field"}
#' \item{"Z"}{Gaussian field}
#' }
#'
#' @return The probability of obtaining the specified cluster
#' \itemize{
#' {"Pcor"}{"corrected p-value"}
#' {"Pu"}{"uncorrected p-value"}
#' {"Ec"}{"expected number of clusters"}
#' {"ek"}{"expected number of resels per cluster"}
#' }
#'
#' @details
#'
#' This function calculates p-values of a thresholded statistical field at various levels:
#'
#' set-level
#' rft.pval(D, c, k, u, n, resels, df, fieldType)
#'
#' cluster-level
#' rft.pval(D, 1, k, u, n, resels, df, fieldType)
#'
#' peak-level
#' rft.pval(D, 1, 0, u, n, resels, df, fieldType)
#'
#' Where set-level refers to obtaining the set of clusters, cluster-level refers to a specific
#' cluster, and peak-level refers to the maximum (or peak) of a single cluster.
#'
#' @references
#' Friston K.J., (1994) Assessing the Significance of Focal Activations Using Their Spatial Extent.
#' Friston K.J., (1996) Detecting Activations in PET and fMRI: Levels of Inference and Power.
#' Worlsey K.J., (1996) A Unified Statistical Approach for Determining Significant Signals in Images of Cerebral Activation.
#'
#' @author Zachary P. Christensen
#'
#' @seealso rftResults, resels
#'
#' @examples
#'
#' # generate some data as if we just fitted a linear regression
#' outimg1 <- makeImage(c(10, 10, 10), rt(1000))
#'
#' # create clusters using arbitrary threshold
#' clusters <- image2ClusterImages(outimg1, minClusterSize=1, minThresh = 2, maxThresh = Inf)
#' resels <- resels(mask, fwhm$fwhm) #' peak <- max(clusters[[1]]) #' peakP <- rftPval(3, 1, 0, 2, 1, resels, c(1, 1), fieldType="T") #' #' #' @export rftPval rftPval <- function(D, c, k, u, n, resels, df = c(idf, rdf), fieldType) { if (missing(fieldType)) { stop("Must specify fieldType") } else if (missing(df)) { stop("Must specify df") } else if (missing(resels)) { stop("Must specify resels") } else if (missing(u) && missing(k) && missing(c)) { stop("Must atleast specify one of u, k, or c") } G <- sqrt(pi) / gamma((1:(D + 1) / 2)) ec <- euler(u, df, fieldType) ec <- pmax(ec[1:(D + 1)], .Machine$double.eps)
P <- toeplitz(as.numeric(ec * G))
P[lower.tri(P)] <- 0
if (n != round(n)) {
n <- round(n)
warning("rounding exponent n' to", n)
}
phi <- diag(nrow = nrow(P))
pot <- P
while (n > 0) {
if (n %% 2)
phi <- phi %*% pot
n <- n %/% 2
pot <- pot %*% pot
}
P <- phi
P <- P[1,]
EM <- (resels[1:(D + 1)] / G) * P # maxima in all dimensions
Ec <- sum(EM) # number of overall expected maxima/clusters
EN <- P[1] * resels[D + 1] # number of resels in entire image
ek <- EN / EM[D + 1] # expected number of resels per cluster

rfB <- (gamma(D / 2 + 1) / ek) ^ (2 / D)
Punc <- exp( - rfB * (k ^ (2 / D))) # cumulative cluster-size distribution from which uncorrected P values are calculated

Pcor <- 1 - ppois(c - 1, lambda = (Ec + .Machine$double.eps) * Punc) z <- list(Pcor = Pcor, Punc = Punc, Ec = Ec, ek = ek) z } # Euler # # Calculates the euler characteristic at a threshold level # # @param u Statistical value (typically the maxima of a cluster or statistical field). # @param df Degrees of freedom expressed as c(degrees of interest, degrees of error). # @param fieldType: # \itemize{ # \item{'T'}{T-field} # \item{'F'}{F-field} # \item{'X'}{Chi-square field'} # \item{'Z'}{Gaussian field} # } # @return A vector of estimated euler characteristics for dimensions 0:D. # # @references # Worlsey K.J., (1996) A Unified Statistical Approach for Determining Significant Signals in Images of Cerebral Activation. # @author Zachary P. Christensen # # @seealso \code{\link{rftPval}}, \code{\link{resels}} # @examples # # ## generate some data as if we just fitted a linear regression # outimg1 <- makeImage(c(10,10,10), rt(1000)) # maskimg <- getMask(outimg1) # fwhm <- estSmooth(outimg1, maskimg) # resels <- resels(maskimg, fwhm$fwhm)
# ec <- euler(max(outimg1), c(1,10), fieldType='T')
# pvox <- sum(ec*resels)
#
# @export euler
euler <- function(u, df, fieldType) {
if (missing(fieldType))
stop("Must specify fieldType")
else if (missing(df))
stop("Must specify df")
else if (missing(u))
stop("Must specify u")

ec <- c(0, 0, 0, 0)
if (fieldType == "T") {
ec[1] <- 1 - pt(u, df[2])
ec[2] <- (((4 * log(2))^(1/2))/(2 * pi)) * ((1 + ((u^2)/df[2]))^(-1/2 * (df[2] - 1)))
ec[3] <- (4 * log(2))/((2 * pi)^(3/2)) * ((1 + u^2/df[2])^((1 - df[2])/2)) * u/((df[2]/2)^(1/2)) *
exp(lgamma((df[2] + 1)/2) - lgamma(df[2]/2))
ec[4] <- (((4 * log(2))^(3/2))/((2 * pi)^2)) * ((1 + ((u^2)/df[2]))^(-1/2 * (df[2] - 1))) *
((((df[2] - 1)/df[2]) * (u^2)) - 1)
} else if (fieldType == "F") {
ec[1] <- 1 - pf(u, df[1], df[2])
ec[2] <- ((4 * log(2))/(2 * pi))^(1/2) * exp(lgamma((df[2] + df[1] - 1)/2) - (lgamma(df[2]/2) +
lgamma(df[1]/2))) * 2^(1/2) * (df[1] * u/df[2])^(1/2 * (df[1] - 1)) * (1 + df[1] *
u/df[2])^(-1/2 * (df[2] + df[1] - 2))
ec[3] <- ((4 * log(2))/(2 * pi)) * exp(lgamma((df[2] + df[1] - 2)/2) - (lgamma(df[2]/2) +
lgamma(df[1]/2))) * (df[1] * u/df[2])^(1/2 * (df[1] - 2)) * (1 + df[1] * u/df[2])^(-1/2 *
(df[2] + df[1] - 2)) * ((df[2] - 1) * df[1] * u/df[2] - (df[1] - 1))
ec[4] <- ((4 * log(2))/(2 * pi))^(3/2) * exp(lgamma((df[2] + df[1] - 3)/2) - (lgamma(df[2]/2) +
lgamma(df[1]/2))) * 2^(-1/2) * (df[1] * u/df[2])^(1/2 * (df[1] - 3)) * (1 + df[1] *
u/df[2])^(-1/2 * (df[2] + df[1] - 2)) * ((df[2] - 1) * (df[2] - 2) * (df[1] * u/df[2])^2 -
(2 * df[2] * df[1] - df[2] - df[1] - 1) * (df[1] * u/df[2]) + (df[1] - 1) * (df[1] -
2))
} else if (fieldType == "X") {
ec[1] <- 1 - pchisq(u, df[2])
ec[2] <- ((4 * log(2))/(2 * pi))^(1/2) * (u^(1/2 * (df[2] - 1)) * exp(-u/2 - lgamma(df[2]/2))/2^((df[2] -
2)/2))
ec[3] <- ((4 * log(2))/(2 * pi)) * (u^(1/2 * (df[2] - 1)) * exp(-u/2 - lgamma(df[2]/2))/2^((df[2] -
2)/2)) * (u - (df[2] - 1))
ec[4] <- ((4 * log(2))/(2 * pi))^(3/2) * (u^(1/2 * (df[2] - 1)) * exp(-u/2 - lgamma(df[2]/2))/2^((df[2] -
2)/2)) * (u^2 - (2 * df[2] - 1) * u + (df[2] - 1) * (df[2] - 2))
} else if (fieldType == "Z") {
ec[1] <- 1 - pnorm(u, df[2])
ec[2] <- (4 * log(2))^(1/2)/(2 * pi) * exp(-u^2/2)
ec[3] <- (4 * log(2))/((2 * pi)^(3/2)) * exp(-u^2/2) * u
ec[4] <- (4 * log(2))^(3/2)/((2 * pi)^2) * exp(-u^2/2) * (u^2 - 1)
}
ec
}
`
Tokazama/iClass documentation built on Aug. 18, 2017, 1:12 a.m.