View source: R/bivariate-marginal-density.R
dtmvnorm.marginal2 | R Documentation |
This function computes the bivariate marginal density function f(x_q, x_r) from a k-dimensional Truncated Multivariate Normal density function (k>=2). The bivariate marginal density is obtained by integrating out (k-2) dimensions as proposed by Tallis (1961). This function is basically an extraction of the Leppard and Tallis (1989) Fortran code for moments calculation, but extended to the double truncated case.
dtmvnorm.marginal2(xq, xr, q, r, mean = rep(0, nrow(sigma)), sigma = diag(length(mean)), lower = rep(-Inf, length = length(mean)), upper = rep(Inf, length = length(mean)), log = FALSE, pmvnorm.algorithm=GenzBretz())
xq |
Value x_q |
xr |
Value x_r |
q |
Index position for x_q within mean vector to calculate the bivariate marginal density for. |
r |
Index position for x_r within mean vector to calculate the bivariate marginal density for. |
mean |
Mean vector, default is |
sigma |
Covariance matrix, default is |
lower |
Vector of lower truncation points,
default is |
upper |
Vector of upper truncation points,
default is |
log |
Logical; if |
pmvnorm.algorithm |
Algorithm used for |
The bivariate marginal density function f(x_q, x_r) for x \sim TN(μ, Σ, a, b) and q \ne r is defined as
F_{q,r}(x_q=c_q, x_r=c_r) = \int^{b_1}_{a_1}...\int^{b_{q-1}}_{a_{q-1}}\int^{b_{q+1}}_{a_{q+1}}...\int^{b_{r-1}}_{a_{r-1}}\int^{b_{r+1}}_{a_{r+1}}...\int^{b_{k}}_{a_{k}} \varphi{_{α}}_{Σ}(x_s, c_q, c_r) dx_s
Stefan Wilhelm <Stefan.Wilhelm@financial.com>, Manjunath B G <bgmanjunath@gmail.com>
Tallis, G. M. (1961). The moment generating function of the truncated multinormal distribution. Journal of the Royal Statistical Society, Series B, 23, 223–229
Leppard, P. and Tallis, G. M. (1989). Evaluation of the Mean and Covariance of the Truncated Multinormal Applied Statistics, 38, 543–553
Manjunath B G and Wilhelm, S. (2009). Moments Calculation For the Double Truncated Multivariate Normal Density. Working Paper. Available at SSRN: https://www.ssrn.com/abstract=1472153
lower = c(-0.5, -1, -1) upper = c( 2.2, 2, 2) mean = c(0,0,0) sigma = matrix(c(2.0, -0.6, 0.7, -0.6, 1.0, -0.2, 0.7, -0.2, 1.0), 3, 3) # generate random samples from untruncated and truncated distribution Y = rmvnorm(10000, mean=mean, sigma=sigma) X = rtmvnorm(500, mean=mean, sigma=sigma, lower=lower, upper=upper, algorithm="gibbs") # compute bivariate marginal density of x1 and x2 xq <- seq(lower[1], upper[1], by=0.1) xr <- seq(lower[2], upper[2], by=0.1) grid <- matrix(NA, length(xq), length(xr)) for (i in 1:length(xq)) { for (j in 1:length(xr)) { grid[i,j] = dtmvnorm.marginal2(xq=xq[i], xr=xr[j], q=1, r=2, sigma=sigma, lower=lower, upper=upper) } } plot(Y[,1], Y[,2], xlim=c(-4, 4), ylim=c(-4, 4), main=expression("bivariate marginal density ("*x[1]*","*x[2]*")"), xlab=expression(x[1]), ylab=expression(x[2]), col="gray80") points(X[,1], X[,2], col="black") lines(x=c(lower[1], upper[1], upper[1], lower[1], lower[1]), y=c(lower[2],lower[2],upper[2],upper[2],lower[2]), lty=2, col="red") contour(xq, xr, grid, add=TRUE, nlevels = 8, col="red", lwd=2) # scatterplot matrices for untruncated and truncated points require(lattice) splom(Y) splom(X)
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