# R/huge.inference.R In huge: High-Dimensional Undirected Graph Estimation

#### Documented in huge.inference

#-----------------------------------------------------------------------#
# Package: High-dimensional Undirected Graph Estimation                 #
# huge.inference(): graph inference                                     #
#-----------------------------------------------------------------------#

#' Graph inference
#'
#' Implements the inference for high dimensional graphical models, including Gaussian and Nonparanormal graphical models
#' We consider the problems of testing the presence of a single edge and the hypothesis is that the edge is absent.
#'
#' For Nonparanormal graphical model we provide Score test method and Wald Test. However it is really slow for inferencing on Nonparanormal model, especially for large data.
#'
#'
#' @param data The input \code{n} by \code{d} data matrix(\code{n} is the sample size and \code{d} is the dimension).
#' @param T The estimated inverse of correlation matrix of the data.
#' @param alpha The significance level of hypothesis.The default value is \code{0.05}.
#' @param type The type of input data. There are 2 options: \code{"Gaussian"} and \code{"Nonparanormal"}. The default value is \code{"Gaussian"}.
#' @param method When using nonparanormal graphical model. Test method with 2 options: \code{"score"} and \code{"wald"}. The default value is \code{"score"}.
#' @return
#' An object is returned:
#' \item{data}{
#'   The \code{n} by \code{d} data matrix from the input.
#' }
#' \item{p}{
#'   The \code{d} by \code{d} p-value matrix of hypothesis.
#' }
#' \item{error}{
#'   The type I error of hypothesis at alpha significance level.
#' }
#' @examples
#' #generate data
#' L = huge.generator(n = 50, d = 12, graph = "hub", g = 4)
#'
#' #graph path estimation using glasso
#' est = huge(L$data, method = "glasso") #' #' #inference of Gaussian graphical model at 0.05 significance level #' T = tail(est$icov, 1)[[1]]
#' out1 = huge.inference(L$data, T, L$theta)
#'
#' #inference of Nonparanormal graphical model using score test at 0.05 significance level
#' T = tail(est$icov, 1)[[1]] #' out2 = huge.inference(L$data, T, L$theta, type = "Nonparanormal") #' #' #inference of Nonparanormal graphical model using wald test at 0.05 significance level #' T = tail(est$icov, 1)[[1]]
#' out3 = huge.inference(L$data, T, L$theta, type = "Nonparanormal", method = "wald")
#'
#' #inference of Nonparanormal graphical model using wald test at 0.1 significance level
#' T = tail(est$icov, 1)[[1]] #' out4 = huge.inference(L$data, T, L$theta, 0.1, type = "Nonparanormal", method = "wald") #' @references #' 1.Q Gu, Y Cao, Y Ning, H Liu. Local and global inference for high dimensional nonparanormal graphical models.\cr #' 2.J Jankova, S Van De Geer. Confidence intervals for high-dimensional inverse covariance estimation. \emph{Electronic Journal of Statistics}, 2015.\cr #' @export huge.inference = function(data, T, adj, alpha = 0.05, type = "Gaussian", method = "score"){ d = ncol(data) n = nrow(data) if(type == "Gaussian") { x=scale(data) U=cor(x) W<-2*T - T%*%U%*%T W_n<-matrix(0,d,d) sigma<-matrix(0,d,d) for(j in 1:d) { for (k in 1:d) { sigma[j, k] = sqrt(T[j, j]*T[k, k]+T[j, k]^2) W_n[j, k] = sqrt(n)*W[j, k]/sigma[j, k] } } p<-matrix(0,d,d) p<-apply(W_n, 1, function(x) 2*(1 - pnorm(abs(x)))) rm(sigma,W_n) } if(type == "Nonparanormal") { x=data U<-matrix(0,d,d) G=list() Test<-matrix(0,d,d) for(i in 1:n) G[[i]]<-matrix(0,d,d) Temp_jk<-matrix(0,n,n) for(j in 1:d) { for(k in 1:d) { if(j==k) { U[j, k] = 1 next } for(i1 in 1:n) { for(i2 in 1:n) { Temp_jk[i1, i2] = sign((x[i1, j] - x[i2, j])*(x[i1, k] - x[i2, k])) G[[i1]][j, k] = G[[i1]][j, k] - pi/2*Temp_jk[i1, i2] } } U[j, k] = sin(pi/2*sum(Temp_jk)/((n-1)*n)) for(i in 1:n) G[[i]][j, k] = G[[i]][j, k]/(n-1) + asin(U[j, k]) } } #F F<-apply(U,1,function(x) sqrt(1-x^2)) #R R<-matrix(0,d^2,d^2) for (i in 1:n) R<-R + as.matrix(as.vector(F*G[[i]]))%*%as.vector(F*G[[i]]) R<-R/n #kronecker product of T T_k<-kronecker(T, T) if(method == "score") { S<-matrix(0,d,d) sigma<-matrix(0,d,d) #ST_n ST_n<-matrix(0,d,d) for(j in 1:d) { for(k in 1:d) { #S ej<-matrix(0,d,1) ek<-matrix(0,d,1) ej[j] = 1 ek[k] = 1 T_h<-T T_h[j, k]=0 S[j, k] = t(ej)%*%t(T_h)%*%U%*%T_h%*%ek/(T[j, j]*T[k, k]) #w temp1<-T_k[,j*k] temp1<-temp1[-j*k] w<-as.matrix((-temp1)/T_k[j*k, j*k]) #sigma temp2<-R[j*k,] temp2<-as.matrix(temp2[-j*k]) sigma[j, k] = sqrt(R[j*k, j*k] - 2*t(temp2)%*%w + t(w)%*%R[-j*k, -j*k]%*%w) ST_n[j, k] = S[j, k]*sqrt(n)/(2*sigma[j, k]) } } #p-value p<-matrix(0,d,d) p<-apply(ST_n, 1, function(x) 2*(1 - pnorm(abs(x)))) rm(temp1,temp2,ST_n,S) } if(method == "wald") { T_W<-matrix(0,d,d) sigma<-matrix(0,d,d) #W_n W_n<-matrix(0,d,d) temp1<-T%*%U temp2<-U%*%T for(j in 1:d) { for(k in 1:d) { #w temp3<-T_k[,j*k] temp3<-temp3[-j*k] w<-as.matrix((-temp3)/T_k[j*k,j*k]) #sigma temp4<-R[j*k,] temp4<-as.matrix(temp4[-j*k]) sigma[j, k] = sqrt(R[j*k, j*k] - 2*t(temp4)%*%w + t(w)%*%R[-j*k, -j*k]%*%w) #T_W ej<-matrix(0,d,1) ek<-matrix(0,d,1) ej[j] = 1 ek[k] = 1 T_W[j ,k] = (T[j,k]*t(ej)%*%temp1%*%ek + T[j, k]*t(ej)%*%temp2%*%ek - t(ej)%*%t(T)%*%temp2%*%ek)/(t(ej)%*%temp1%*%ek + t(ej)%*%temp2%*%ek - 1) W_n[j, k] = T_W[j, k]*sqrt(n)/(2*sigma[j, k]*T[j, j]*T[k, k]) } } #p-value p<-matrix(0,d,d) p<-apply(W_n, 1, function(x) 2*(1 - pnorm(abs(x)))) rm(temp1,temp2,temp3,temp4,T_W,W_n) } rm(R,F,G,w,T_k) } error=0 for(j in 1:d) { for(k in 1:d) { if(j==k) next if(p[j, k]<alpha && adj[j, k]==0){ error=error+1 } } } error=error/d^2 inf = list() inf$data = data
inf$p = p inf$error = error

rm(U,p)
return(inf)
}


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huge documentation built on July 1, 2021, 1:06 a.m.