Package 'condTruncMVN'

Conditional Truncated Multivariate Normal Parameters

Description

Suppose that Z = (X,Y) is from a fully-joint multivariate normal distribution of dimension n with mean and covariance matrix sigma truncated between lower and upper. This function provides the parameters for the conditional mean and covariance matrix of Y given X. See the vignette for more information.

Usage

condtMVN(
  mean,
  sigma,
  lower,
  upper,
  dependent.ind,
  given.ind,
  X.given,
  init = rep(0, length(mean))
)

Arguments

mean	the mean vector for Z of length of n
sigma	the symmetric and positive-definite covariance matrix of dimension n x n of Z.
lower	a vector of lower bounds of length n that truncate Z
upper	a vector of upper bounds of length n that truncate Z
dependent.ind	a vector of integers denoting the indices of dependent variable Y.
given.ind	a vector of integers denoting the indices of conditioning variable X. If specified as integer vector of length zero or left unspecified, the unconditional density is returned.
X.given	a vector of reals denoting the conditioning value of X. This should be of the same length as given.ind
init	initial value used for random generation of truncated multivariate normal in a Gibbs sampler. Default: A vector of zeros, equal to the number of components. For details, see tmvmixnorm::rtmvn().

Details

The first four arguments are the parameters of multivariate normal and the truncation space. dependent.ind, given.ind, X.given, init are all arguments that determines the conditional truncated MVN.

Using the full data Z, the conditional mean and conditional variance of Y|X are determined (Wang, 2006). Additionally, to reflect the reduced dimension of Y|X, the truncation limits are also adjusted.

See the vignette for more information.

Value

Returns a list of:

• condMean - conditional mean of Y|X

• condVar - conditional variance of Y|X

• condLower - the lower bound of Y|X

• condUpper - the upper bound of Y|X

• condInit - the initial values adjusted to match the dimension of Y|X. These are used to randomly generate the truncated multivariate normal rcmvtruncnorm.

Note

This function is based on condMVN from the condMVNorm package.

References

Wang, R. 2006. Appendix A: Marginal and conditional distributions of multivariate normal distribution. http://fourier.eng.hmc.edu/e161/lectures/gaussianprocess/node7.html.

See Also

cmvnorm, pmvnorm, Mvnorm

Examples

# Suppose X2,X3,X5|X2,X4 ~ N_3(1, Sigma) and truncated between -10 and 10.
d <- 5
rho <- 0.9
Sigma <- matrix(0, nrow = d, ncol = d)
Sigma <- rho^abs(row(Sigma) - col(Sigma))

# Conditional Truncated Normal Parameters
condtMVN(mean = rep(1, d),
  sigma = Sigma,
  lower = rep(-10, d),
  upper = rep(10, d),
  dependent.ind = c(2, 3, 5),
  given.ind = c(1, 4), X.given = c(1, -1)
)

---

Density of the Conditional Truncated Multivariate Normal

Description

Calculates the density of truncated conditional multivariate normal Y|X: $f(Y = y | X = X.given)$. See the vignette for more information.

Usage

dcmvtruncnorm(
  y,
  mean,
  sigma,
  lower,
  upper,
  dependent.ind,
  given.ind,
  X.given,
  log = FALSE
)

Arguments

y	vector or matrix of quantiles of Y. If a matrix, each row is taken to be a quantile. This is the quantity that the density is calculated from.
mean	the mean vector for Z of length of n
sigma	the symmetric and positive-definite covariance matrix of dimension n x n of Z.
lower	a vector of lower bounds of length n that truncate Z
upper	a vector of upper bounds of length n that truncate Z
dependent.ind	a vector of integers denoting the indices of dependent variable Y.
given.ind	a vector of integers denoting the indices of conditioning variable X. If specified as integer vector of length zero or left unspecified, the unconditional density is returned.
X.given	a vector of reals denoting the conditioning value of X. This should be of the same length as given.ind
log	logical; if TRUE, densities d are given as log(d).

References

Horrace, W.C. 2005. Some results on the multivariate truncated normal distribution. Journal of Multivariate Analysis, 94, 209–221. https://surface.syr.edu/cgi/viewcontent.cgi?article=1149&context=ecn

Examples

# Example 1: X2,X3,X5|X2,X4 ~ N_3(1, Sigma)
# truncated between -10 and 10.
d <- 5
rho <- 0.9
Sigma <- matrix(0, nrow = d, ncol = d)
Sigma <- rho^abs(row(Sigma) - col(Sigma))

# Log-density of 0
dcmvtruncnorm(
  rep(0, 3),
  mean = rep(1, 5),
  sigma = Sigma,
  lower = rep(-10, 5),
  upper = rep(10, d),
  dependent.ind = c(2, 3, 5),
  given.ind = c(1, 4), X.given = c(1, -1),
  log = TRUE
)

---

CDF for the Conditional Truncated Multivariate Normal

Description

Computes the distribution function for a conditional truncated multivariate normal random variate Y|X.

Usage

pcmvtruncnorm(
  lowerY,
  upperY,
  mean,
  sigma,
  lower,
  upper,
  dependent.ind,
  given.ind,
  X.given,
  ...
)

Arguments

lowerY	the vector of lower limits for Y|X. Passed to tmvtnorm::ptmvnorm().
upperY	the vector of upper limits for Y|X. Must be greater than lowerY. Passed to tmvtnorm::ptmvnorm().
mean	the mean vector for Z of length of n
sigma	the symmetric and positive-definite covariance matrix of dimension n x n of Z.
lower	a vector of lower bounds of length n that truncate Z
upper	a vector of upper bounds of length n that truncate Z
dependent.ind	a vector of integers denoting the indices of dependent variable Y.
given.ind	a vector of integers denoting the indices of conditioning variable X. If specified as integer vector of length zero or left unspecified, the unconditional density is returned.
X.given	a vector of reals denoting the conditioning value of X. This should be of the same length as given.ind
...	Additional arguments passed to tmvtnorm::ptmvnorm(). The CDF is calculated using the Genz algorithm based on these arguments: maxpts, abseps, and releps.

Details

Calculates the probability that Y|X is between lowerY and upperY. Z = (X, Y) is the fully joint multivariate normal distribution with mean equal mean and covariance matrix sigma, truncated between lower and upper. See the vignette for more information.

Note

For one-dimension conditionals Y|X, this function uses the ptruncnorm() function in the truncnorm package. Otherwise, this function uses tmvtnorm::ptmvnorm().

Examples

# Example 1: Let X2,X3,X5|X2,X4 ~ N_3(1, Sigma)
# truncated between -10 and 10.
d <- 5
rho <- 0.9
Sigma <- matrix(0, nrow = d, ncol = d)
Sigma <- rho^abs(row(Sigma) - col(Sigma))

# Find P(-0.5 < X2,X3,X5 < 0 | X2,X4)
pcmvtruncnorm(rep(-0.5, 3), rep(0, 3),
  mean = rep(1, d),
  sigma = Sigma,
  lower = rep(-10, d),
  upper = rep(10, d),
  dependent.ind = c(2, 3, 5),
  given.ind = c(1, 4), X.given = c(1, -1)
)

# Example 2: Let X1| X2 = 1, X3 = -1, X4 = 1, X5 = -1 ~ N(1, Sigma) truncated
# between -10 and 10. Find P(-0.5 < X1 < 0 | X2 = 1, X3 = -1, X4 = 1, X5 = -1).
pcmvtruncnorm(-0.5, 0,
  mean = rep(1, d),
  sigma = Sigma,
  lower = rep(-10, d),
  upper = rep(10, d),
  dependent.ind = 1,
  given.ind = 2:5, X.given = c(1, -1, 1, -1)
)

---

Random Sample from Conditional Truncated Multivariate Normal

Description

Randomly samples from conditional truncated multivariate normal distribution variate, Y|X, where Z = (X, Y) is the fully joint multivariate normal distribution with mean, covariance matrix sigma, and truncated between lower and upper. See the vignette for more information.

Usage

rcmvtruncnorm(
  n,
  mean,
  sigma,
  lower,
  upper,
  dependent.ind,
  given.ind,
  X.given,
  init = rep(0, length(mean)),
  burn = 10L,
  thin = 1
)

Arguments

n	number of random samples desired (sample size).
mean	the mean vector for Z of length of n
sigma	the symmetric and positive-definite covariance matrix of dimension n x n of Z.
lower	a vector of lower bounds of length n that truncate Z
upper	a vector of upper bounds of length n that truncate Z
dependent.ind	a vector of integers denoting the indices of dependent variable Y.
given.ind	a vector of integers denoting the indices of conditioning variable X. If specified as integer vector of length zero or left unspecified, the unconditional density is returned.
X.given	a vector of reals denoting the conditioning value of X. This should be of the same length as given.ind
init	initial value used for random generation of truncated multivariate normal in a Gibbs sampler. Default: A vector of zeros, equal to the number of components. For details, see tmvmixnorm::rtmvn().
burn	the burn-in, which is the number of initial iterations to be discarded. Default: 10. Passed to rtmvn().
thin	thinning lag (default as 1).

Note

Uses rtmvn from the tmvmixnorm package to find the random variate.

Examples

# Generate 2 random numbers from X2,X3,X5|X2,X4 ~ N_3(1, Sigma)
# truncated between -10 and 10.
d <- 5
rho <- 0.9
Sigma <- matrix(0, nrow = d, ncol = d)
Sigma <- rho^abs(row(Sigma) - col(Sigma))

set.seed(2342)
rcmvtruncnorm(2,
  mean = rep(1, d),
  sigma = Sigma,
  lower = rep(-10, d),
  upper = rep(10, d),
  dependent.ind = c(2, 3, 5),
  given.ind = c(1, 4), X.given = c(1, -1)
)

# Example 2: Generate two random numbers from
# X1|X2, X3, X4, X5 ~ N(1, Sigma) truncated between -10 and 10.
set.seed(2342)
rcmvtruncnorm(2,
  mean = rep(1, d),
  sigma = Sigma,
  lower = rep(-10, d),
  upper = rep(10, d),
  dependent.ind = 1,
  given.ind = 2:5, X.given = c(1, -1, 1, -1)
)

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