WassFIntegral {frechet}

Generalized Fréchet integrals of 1D distribution

Description

Calculating generalized Fréchet integrals of 1D distribution (equipped with Wasserstein distance)

Usage

WassFIntegral(phi, t_out, Q, Qout)

Arguments

phi

An eigenfunction along which we want to project the distribution

t_out

Support of phi

Q

A length(t_out) X length(Qout) matrix whose jth row corresponds to the quantile function on grid Qout for the jth time point.

Qout

Support of the quantile valued process

Value

A list of the following:

f

Quantile function corresponding to the frechet integral of Q along phi

References

Dubey, P., & Müller, H. G. (2020). Functional models for time‐varying random objects. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 82(2), 275-327.

Examples

#simulation as in the paper Dubey, P., & Müller, H. G. (2020).
# Functional models for time‐varying random objects. 
# JRSSB, 82(2), 275-327.

n <- 100
N <- 50
t_out <- seq(0,1,length.out = N)

phi1 <- function(t){
  (t^2-0.5)/0.3416
}
phi2 <- function(t){
  sqrt(3)*t
}
phi3 <- function(t){
  (t^3 - 0.3571*t^2 - 0.6*t + 0.1786)/0.0895
}

Z <- cbind(rnorm(n)*sqrt(12), rnorm(n), runif(n)*sqrt(72), runif(n)*sqrt(9))
mu_vec <- 1 + Z[,1] %*% t(phi1(t_out)) + Z[,2] %*% t(phi3(t_out))
sigma_vec <- 3 + Z[,3] %*% t(phi2(t_out)) + Z[,4] %*% t(phi3(t_out))

# grids of quantile function
Nq <- 40
eps <- 0.00001
Qout <- seq(0+eps,1-eps,length.out=Nq)

# I: four dimension array of n x n matrix of squared distances 
# between the time point u of the ith process and 
# process and the time point v of the jth object process, 
# e.g.: I[i,j,u,v] <- d_w^2(X_i(u) X_j(v)).
I <- array(0, dim = c(n,n,N,N))
for(i in 1:n){
  for(j in 1:n){
    for(u in 1:N){
      for(v in 1:N){
        #wasserstein distance between distribution X_i(u) and X_j(v) 
        I[i,j,u,v] <- (mu_vec[i,u] - mu_vec[j,v])^2 + (sigma_vec[i,u] - sigma_vec[j,v])^2
      }
    }
  }
}

# check ObjCov work 
Cov_result <- ObjCov(t_out, I, 3)
#Cov_result$lambda #12 6 1.75

# calculate Q 
i <- 6 # for the ith subject
Q <- t(sapply(1:N, function(t){
  qnorm(Qout, mean = mu_vec[i,t], sd = sigma_vec[i,t])
}))

score_result <- WassFIntegral(Cov_result$phi[,1], t_out, Q, Qout)
score_result$f