# R/utilities.R In tk382/covarianceBP: Analysis of Continuously Varying Correlation

#### Documented in get_cor_rget_degreeget_est_Hget_Hget_p_from_degreeget_scorepost_score

```get_grad = function(x, y1, y2){
a = mean(y1^2)
b = mean(y2^2)
d = mean(y1*y2)
xmat = cbind(rep(1, length(y1)), x)
first = (a*b*d - d^3) * colSums(xmat)
second = b * d * colSums(xmat * (y1^2))
third = a * d * colSums(xmat * (y2^2))
fourth = (a*b+d^2)*colSums(xmat * y1 * y2)
out = (first - second - third + fourth)
return(out)
}

get_fisher = function(x, y1, y2){
a = mean(y1^2)
b = mean(y2^2)
d = mean(y1*y2)
xmat = cbind(rep(1, length(y1)), x)
first = (a*b+d^2)*(a*b-d^2)*(t(xmat) %*% xmat)
second = 4*a*b*d*d*
(colSums(xmat) %*% t(colSums(xmat)))/nrow(x)
return(solve(first-second))
}

get_score = function(x, y1, y2){
a = mean(y1^2)
b = mean(y2^2)
d = mean(y1*y2)
const = 1/(a*b-d^2)
return(const*
get_fisher(x,y1,y2) %*%
}

get_degree = function(x, y, Y){
d = 0
for (k in 1:ncol(Y)){
d = d + get_score(x, y, Y[,k])
}
return(d)
}

get_eta = function(rho12, rho23, rho13, rho11, rho22, rho33){
# num = (rho11*rho23 + rho12*rho13)
# denom = sqrt(rho11*rho22-rho12^2)*sqrt(rho11*rho33-rho13^2)
# return(num/denom)
a = rho11
b = rho22
c = rho33
d = rho12
e = rho13
f = rho23
num = 2*a^2*b*c*d*e + 3*a*c*d^3*e + 3*a*b*d*e^3-2*a^2*b^2*c*e
num = num - a^2*b*e^2*f - a*b^2*e^3 + a^3*b*c*f
num = num - 2*a*b*c*d^2*e - b*d^2*e^3 - a^2*c*d^2*f+2*d^3*e^3
num = num - a^2*b*c*d^2 - a*b*d^2*e^2 + 2*a^2*d*e*f^2
denom = sqrt((a*b+d^2)*(a*b-d^2)^2*(a*b+e^2)*(a*b-e^2)^2)
return(num/denom)
}

#' Estimate H
#'
#' @param Sigma True or estimated variance of all the variables of interest.
#' The first row / column must be the main variable of interest
#' @return Esitmated covariance matrix of the part of score statistics
#' @export
get_H = function(Sigma){
K = nrow(Sigma)
est_H = matrix(NA, K-1, K-1)
for (i in 2:(K-1)){
for (j in (i+1):K){
est_H[i-1, j-1] = get_eta(Sigma[1,i], Sigma[i,j], Sigma[j,1],
Sigma[1,1], Sigma[i,i], Sigma[j,j])
est_H[j-1, i-1] = est_H[i-1, j-1]
}
}
diag(est_H) = 1
return(est_H)
}

#' Estimate p-value from the degree statistic
#'
#' @param y vector: main variable of interest
#' @param Y Matrix: other variables of interest to measure the correlation with y
#' @param d double: computed degree statistic
#' @param numsim integer: number of simulations to draw the null distribution
#' @return p value of the degree statistic
#' @export
#' @examples
#' n = 30
#' k = 3
#' nullX = as.matrix(rnorm(30), ncol=1)
#' nullY = MASS::mvrnorm(n, rep(0,3), diag(3))
#' d = get_degree(as.matrix(nullX, ncol=1), nullY[,1], nullY[,2:3])
#' p = get_p_from_degree(nullY[,1], nullY[,2:3], d)
#' print(paste("The degree statistic is", d, "and the p-value is", p))
get_p_from_degree = function(y, Y, d, numsim = 5000){
bigy = cbind(y, Y)
K = ncol(bigy)
est_Sigma = stats::cor(bigy)
est_H = get_H(est_Sigma)
lambda = eigen(est_H)\$values
U      = eigen(est_H)\$vectors
null_d = matrix(NA, numsim, K-1)
for (k in 1:(K-1)){
null_d[,k] = rgamma(numsim, 1/2, 1/(2*lambda[k]))
}
p = sum(rowSums(null_d) > as.numeric(d))/numsim
return(p)
}

#' Computes the small sample correction
#'
#' @param score score statistic computed from get_score
#' @param coef coefficients for the cubic function
#' @return small-sample adjusted score statistic
#' @export
post_score = function(score, coef) {
roots = polyroot(c(-score, coef))
return(Re(roots)[abs(Im(roots)) < 1e-06][1])
}

#' Helper function for get_est_H
#' Returns each element of matrix H
#'
#' @param rho12 correlation between variable 1 and variable 2
#' @param rho23 correlation between variable 2 and variable 3
#' @param rho13 correlation between variable 1 and variable 3
#' @return element (j,k) of H from elements (i,j), (j,k), (i,k) of Sigma
#' @export
get_cor_r = function(rho12, rho23, rho13) {
num = (rho23 + 2 * rho12 * rho23) *
(rho12^2 + 1) * (rho13^2 + 1)
num = num + rho12 * rho13 * (6 + 2 *rho12 + 2 * rho13 + 2 * rho23)
num = num - rho12 * (rho13^2 + 1) * (3 * rho13 + rho13 + 2 * rho12 *rho23)
num = num - rho13 * (rho12^2 + 1) * (3 * rho12 + rho12 + 2 * rho13 * rho23)
denom = (1 - rho12^2) * (1 - rho13^2) * sqrt(1 + rho12^2) * sqrt(1 + rho13^2)
return(num/denom)
}

#' Computes the correlation matrix of the combined test statistics r
#' Takes input of the estimated or true covariance matrix
#'
#' @param Sigma estimate or true covariance matrix of \$K\$ variables
#' @return estimated H from the covariance matrix Sigma
#' @export
get_est_H = function(Sigma) {
K = nrow(Sigma)
est_H = matrix(0, K - 1, K - 1)
diag(est_H) = 1
for (i in 1:(K - 1)) {
for (j in (i + 1):K) {
eta = get_cor_r(Sigma[1, i],
Sigma[i, j], Sigma[j, 1])
est_H[i - 1, j - 1] = est_H[j -
1, i - 1] = eta
}
}
return(est_H)
}
```
tk382/covarianceBP documentation built on July 14, 2020, 4:01 p.m.