dmnorm {mnorm}

Density of (conditional) multivariate normal distribution

Description

This function calculates and differentiates density of (conditional) multivariate normal distribution.

Usage

dmnorm(
  x,
  mean,
  sigma,
  given_ind = numeric(),
  log = FALSE,
  grad_x = FALSE,
  grad_sigma = FALSE,
  is_validation = TRUE,
  control = NULL,
  n_cores = 1L
)

Arguments

x

numeric vector representing the point at which density should be calculated. If x is a matrix then each row determines a new point.

mean

numeric vector representing expectation of multivariate normal vector (distribution).

sigma

positively defined numeric matrix representing covariance matrix of multivariate normal vector (distribution).

given_ind

numeric vector representing indexes of multivariate normal vector which are conditioned at values of x with corresponding indexes i.e. x[given_x] or x[, given_x] if x is a matrix.

log

logical; if TRUE then probabilities (or densities) p are given as log(p) and derivatives will be given respect to log(p).

grad_x

logical; if TRUE then the vector of partial derivatives of the density function will be calculated respect to each element of x. If x is a matrix then gradients will be estimated for each row of x.

grad_sigma

logical; if TRUE then the vector of partial derivatives (gradient) of the density function will be calculated respect to each element of sigma. If x is a matrix then gradients will be estimated for each row of x.

is_validation

logical value indicating whether input arguments should be validated. Set it to FALSE to get performance boost (default value is TRUE).

control

a list of control parameters. See Details.

n_cores

positive integer representing the number of CPU cores used for parallel computing. Currently it is not recommended to set n_cores > 1 if vectorized arguments include less then 100000 elements.

Details

Consider notations from the Details section of cmnorm. The function calculates density f(x^{(d)}|x^{(g)}) of conditioned multivariate normal vector X_{I_{d}} | X_{I_{g}} = x^{(g)}. Where x^{(d)} is a subvector of x associated with X_{I_{d}} i.e. unconditioned components. Therefore x[given_ind] represents x^{(g)} while x[-given_ind] is x^{(d)}.

If grad_x is TRUE then function additionally estimates the gradient respect to both unconditioned and conditioned components:

\nabla f(x^{(d)}|x^{(g)})= \left(\frac{\partial f(x^{(d)}|x^{(g)})}{\partial x_{1}} ,..., \frac{\partial f(x^{(d)}|x^{(g)})}{\partial x_{m}}\right),

where each x_{i} belongs either to x^{(d)} or x^{(g)} depending on whether i\in I_{d} or i\in I_{g} correspondingly. In particular subgradients of density function respect to x^{(d)} and x^{(g)} are of the form:

\nabla_{x^{(d)}}\ln f(x^{(d)}|x^{(g)}) = -\left(x^{(d)}-\mu_{c}\right)\Sigma_{c}^{-1}

\nabla_{x^{(g)}}\ln f(x^{(d)}|x^{(g)}) = -\nabla_{x^{(d)}}f(x^{(d)}|x^{(g)})\Sigma_{d,g}\Sigma_{g,g}^{-1}

If grad_sigma is TRUE then function additionally estimates the gradient respect to the elements of covariance matrix \Sigma. For i\in I_{d} and j\in I_{d} the function calculates:

\frac{\partial \ln f(x^{(d)}|x^{(g)})}{\partial \Sigma_{i, j}} = \left(\frac{\partial \ln f(x^{(d)}|x^{(g)})}{\partial x_{i}} \times \frac{\partial \ln f(x^{(d)}|x^{(g)})}{\partial x_{j}} - \Sigma_{c,(i, j)}^{-1}\right) / \left(1 + I(i=j)\right),

where I(i=j) is an indicator function which equals 1 when the condition i=j is satisfied and 0 otherwise.

For i\in I_{d} and j\in I_{g} the following formula is used:

\frac{\partial \ln f(x^{(d)}|x^{(g)})}{\partial \Sigma_{i, j}} = -\frac{\partial \ln f(x^{(d)}|x^{(g)})}{\partial x_{i}} \times \left(\left(x^{(g)}-\mu_{g}\right)\Sigma_{g,g}^{-1}\right)_{q_{g}(j)}-

-\sum\limits_{k=1}^{n_{d}}(1+I(q_{d}(i)=k))\times (\Sigma_{d,g}\Sigma_{g,g}^{-1})_{k,q_{g}(j)}\times \frac{\partial \ln f(x^{(d)}|x^{(g)})}{\partial \Sigma_{i, q^{-1}_{d}(k)}},

where q_{g}(j)=\sum\limits_{k=1}^{m} I\left(I_{g,k} \leq j\right) and q_{d}(i)=\sum\limits_{k=1}^{m} I\left(I_{d,k} \leq i\right) represent the order of the i-th and j-th elements in I_{g} and I_{d} correspondingly i.e. x_{i}=x^{(d)}_{q_{d}(i)}=x_{I_{d, q_{d}(i)}} and x_{j}=x^{(g)}_{q_{g}(j)}=x_{I_{g, q_{g}(j)}}. Note that q_{g}(j)^{-1} and q_{d}(i)^{-1} are inverse functions. Number of conditioned and unconditioned components are denoted by n_{g}=\sum\limits_{k=1}^{m}I(k\in I_{g}) and n_{d}=\sum\limits_{k=1}^{m}I(k\in I_{d}) respectively. For the case i\in I_{g} and j\in I_{d} the formula is similar.

For i\in I_{g} and j\in I_{g} the following formula is used:

\frac{\partial \ln f(x^{(d)}|x^{(g)})}{\partial \Sigma_{i, j}} = -\nabla_{x^{(d)}}\ln f(x^{(d)}|x^{(g)})\times \left(x^{(g)}\times(\Sigma_{d,g} \times \Sigma_{g,g}^{-1} \times I_{g}^{*} \times \Sigma_{g,g}^{-1})^{T}\right)^T -

-\sum\limits_{k_{1}=1}^{n_{d}}\sum\limits_{k_{2}=k_{1}}^{n_{d}} \frac{\partial \ln f(x^{(d)}|x^{(g)})} {\partial \Sigma_{q_{d}(k_{1})^{-1}, q_{d}(k_{2})^{-1}}} \left(\Sigma_{d,g} \times \Sigma_{g,g}^{-1} \times I_{g}^{*} \times \Sigma_{g,g}^{-1}\times\Sigma_{d,g}^T\right)_{q_{d}(k_{1})^{-1}, q_{d}(k_{2})^{-1}},

where I_{g}^{*} is a square n_{g}-dimensional matrix of zeros except I_{g,(i,j)}^{*}=I_{g,(j,i)}^{*}=1.

Argument given_ind represents I_{g} and it should not contain any duplicates. The order of given_ind elements does not matter so it has no impact on the output.

More details on abovementioned differentiation formulas could be found in the appendix of E. Kossova and B. Potanin (2018).

Currently control has no input arguments intended for the users. This argument is used for some internal purposes of the package.

Value

This function returns an object of class "mnorm_dmnorm".

An object of class "mnorm_dmnorm" is a list containing the following components:

• den - density function value at x.

• grad_x - gradient of density respect to x if grad_x or grad_sigma input argument is set to TRUE.

• grad_sigma - gradient respect to the elements of sigma if grad_sigma input argument is set to TRUE.

If log is TRUE then den is a log-density so output grad_x and grad_sigma are calculated respect to the log-density.

Output grad_x is a Jacobian matrix which rows are gradients of the density function calculated for each row of x. Therefore grad_x[i, j] is a derivative of the density function respect to the j-th argument at point x[i, ].

Output grad_sigma is a 3D array such that grad_sigma[i, j, k] is a partial derivative of the density function respect to the sigma[i, j] estimated for the observation x[k, ].

References

E. Kossova., B. Potanin (2018). Heckman method and switching regression model multivariate generalization. Applied Econometrics, vol. 50, pages 114-143.

Examples

# Consider multivariate normal vector:
# X = (X1, X2, X3, X4, X5) ~ N(mean, sigma)

# Prepare multivariate normal vector parameters
  # expected value
mean <- c(-2, -1, 0, 1, 2)
n_dim <- length(mean)
  # correlation matrix
cor <- c(   1,  0.1,  0.2,   0.3,  0.4,
          0.1,    1, -0.1,  -0.2, -0.3,
          0.2, -0.1,    1,   0.3,  0.2,
          0.3, -0.2,  0.3,     1, -0.05,
          0.4, -0.3,  0.2, -0.05,     1)
cor <- matrix(cor, ncol = n_dim, nrow = n_dim, byrow = TRUE)
  # covariance matrix
sd_mat <- diag(c(1, 1.5, 2, 2.5, 3))
sigma <- sd_mat %*% cor %*% t(sd_mat)

# Estimate the density of X at point (-1, 0, 1, 2, 3)
x <- c(-1, 0, 1, 2, 3)
d.list <- dmnorm(x = x, mean = mean, sigma = sigma)
d <- d.list$den
print(d)

# Estimate the density of X at points
# x=(-1, 0, 1, 2, 3) and y=(-1.2, -1.5, 0, 1.2, 1.5)
y <- c(-1.5, -1.2, 0, 1.2, 1.5)
xy <- rbind(x, y)
d.list.1 <- dmnorm(x = xy, mean = mean, sigma = sigma)
d.1 <- d.list.1$den
print(d.1)

# Estimate the density of Xc=(X2, X4, X5 | X1 = -1, X3 = 1) at 
# point xd=(0, 2, 3) given conditioning values xg=(-1, 1)
given_ind <- c(1, 3)
d.list.2 <- dmnorm(x = x, mean = mean, sigma = sigma, 
                   given_ind = given_ind)
d.2 <- d.list.2$den
print(d.2)

# Estimate the gradient of density respect to the argument and 
# covariance matrix at points 'x' and 'y'
d.list.3 <- dmnorm(x = xy, mean = mean, sigma = sigma,
                   grad_x = TRUE, grad_sigma = TRUE)
# Gradient respect to the argument
grad_x.3 <- d.list.3$grad_x
   # at point 'x'
print(grad_x.3[1, ])
   # at point 'y'
print(grad_x.3[2, ])
# Partial derivative at point 'y' respect 
# to the 3-rd argument
print(grad_x.3[2, 3])
# Gradient respect to the covariance matrix
grad_sigma.3 <- d.list.3$grad_sigma
# Partial derivative respect to sigma(3, 5) at 
# point 'y'
print(grad_sigma.3[3, 5, 2])

# Estimate the gradient of the log-density function of 
# Xc=(X2, X4, X5 | X1 = -1, X3 = 1) and Yc=(X2, X4, X5 | X1 = -1.5, X3 = 0)
# respect to the argument and covariance matrix at 
# points xd=(0, 2, 3) and yd=(-1.2, 0, 1.5) respectively given
# conditioning values xg=(-1, 1) and yg=(-1.5, 0) correspondingly
d.list.4 <- dmnorm(x = xy, mean = mean, sigma = sigma,
                   grad_x = TRUE, grad_sigma = TRUE,
                   given_ind = given_ind, log = TRUE)
# Gradient respect to the argument
grad_x.4 <- d.list.4$grad_x
   # at point 'xd'
print(grad_x.4[1, ])
   # at point 'yd'
print(grad_x.4[2, ])
# Partial derivative at point 'xd' respect to 'xg[2]'
print(grad_x.4[1, 3])
# Partial derivative at point 'yd' respect to 'yd[5]'
print(grad_x.4[2, 5])
# Gradient respect to the covariance matrix
grad_sigma.4 <- d.list.4$grad_sigma
# Partial derivative respect to sigma(3, 5) at 
# point 'yd'
print(grad_sigma.4[3, 5, 2])

# Compare analytical gradients from the previous example with
# their numeric (forward difference) analogues at point 'xd'
# given conditioning 'xg'
delta <- 1e-6
grad_x.num <- rep(NA, 5)
grad_sigma.num <- matrix(NA, nrow = 5, ncol = 5)
for (i in 1:5)
{
  x.delta <- x
  x.delta[i] <- x[i] + delta
  d.list.delta <- dmnorm(x = x.delta, mean = mean, sigma = sigma,
                         grad_x = TRUE, grad_sigma = TRUE,
                         given_ind = given_ind, log = TRUE)
  grad_x.num[i] <- (d.list.delta$den - d.list.4$den[1]) / delta
   for(j in 1:5)
   {
     sigma.delta <- sigma
     sigma.delta[i, j] <- sigma[i, j] + delta 
     sigma.delta[j, i] <- sigma[j, i] + delta 
     d.list.delta <- dmnorm(x = x, mean = mean, sigma = sigma.delta,
                            grad_x = TRUE, grad_sigma = TRUE,
                            given_ind = given_ind, log = TRUE)
     grad_sigma.num[i, j] <- (d.list.delta$den - d.list.4$den[1]) / delta
   }
}
# Comparison of gradients respect to the argument
h.x <- cbind(analytical = grad_x.4[1, ], numeric = grad_x.num)
rownames(h.x) <- c("xg[1]", "xd[1]", "xg[2]", "xd[3]", "xd[4]")
print(h.x)
# Comparison of gradients respect to the covariance matrix
h.sigma <- list(analytical = grad_sigma.4[, , 1], numeric = grad_sigma.num)
print(h.sigma)