Package 'mvnfast' reference manual

Title:	Fast Multivariate Normal and Student's t Methods
Description:	Provides computationally efficient tools related to the multivariate normal and Student's t distributions. The main functionalities are: simulating multivariate random vectors, evaluating multivariate normal or Student's t densities and Mahalanobis distances. These tools are very efficient thanks to the use of C++ code and of the OpenMP API.
Authors:	Matteo Fasiolo [aut, cre], Thijs van den Berg [ctb]
Maintainer:	Matteo Fasiolo <[email protected]>
License:	GPL (>= 2.0)
Version:	0.2.8
Built:	2024-10-26 06:40:17 UTC
Source:	CRAN

Fast density computation for mixture of multivariate normal distributions.

Description

Fast density computation for mixture of multivariate normal distributions.

Usage

dmixn(X, mu, sigma, w, log = FALSE, ncores = 1, isChol = FALSE, A = NULL)
dmixn(X, mu, sigma, w, log = FALSE, ncores = 1, isChol = FALSE, A = NULL)

Arguments

`X`	matrix n by d where each row is a d dimensional random vector. Alternatively `X` can be a d-dimensional vector.
`mu`	an (m x d) matrix, where m is the number of mixture components.
`sigma`	as list of m covariance matrices (d x d) on for each mixture component. Alternatively it can be a list of m cholesky decomposition of the covariance. In that case `isChol` should be set to `TRUE`.
`w`	vector of length m, containing the weights of the mixture components.
`log`	boolean set to true the logarithm of the pdf is required.
`ncores`	Number of cores used. The parallelization will take place only if OpenMP is supported.
`isChol`	boolean set to true is `sigma` is the cholesky decomposition of the covariance matrix.
`A`	an (optional) numeric matrix of dimension (m x d), which will be used to store the evaluations of each mixture density over each mixture component. It is useful when m and n are large and one wants to call `dmixt()` several times, without reallocating memory for the whole matrix each time. NB1: `A` will be modified, not copied! NB2: the element of `A` must be of class "numeric".

Details

NB: at the moment the parallelization does not work properly on Solaris OS when ncores>1. Hence, dmixt() checks if the OS is Solaris and, if this the case, it imposes ncores==1.

Value

A vector of length n where the i-the entry contains the pdf of the i-th random vector (i.e. the i-th row of X).

Author(s)

Matteo Fasiolo <[email protected]>.

Examples

#### 1) Example use
# Set up mixture density
mu <- matrix(c(1, 2, 10, 20), 2, 2, byrow = TRUE)
sigma <- list(diag(c(1, 10)), matrix(c(1, -0.9, -0.9, 1), 2, 2))
w <- c(0.1, 0.9)

# Simulate
X <- rmixn(1e4, mu, sigma, w)

# Evaluate density
ds <- dmixn(X, mu, sigma, w = w)
head(ds)

##### 2) More complicated example
# Define mixture
set.seed(5135)
N <- 10000
d <- 2
w <- rep(1, 2) / 2
mu <- matrix(c(0, 0, 2, 3), 2, 2, byrow = TRUE) 
sigma <- list(matrix(c(1, 0, 0, 2), 2, 2), matrix(c(1, -0.9, -0.9, 1), 2, 2)) 

# Simulate random variables
X <- rmixn(N, mu, sigma, w = w, retInd = TRUE)

# Plot mixture density
np <- 100
xvals <- seq(min(X[ , 1]), max(X[ , 1]), length.out = np)
yvals <- seq(min(X[ , 2]), max(X[ , 2]), length.out = np)
theGrid <- expand.grid(xvals, yvals) 
theGrid <- as.matrix(theGrid)
dens <- dmixn(theGrid, mu, sigma, w = w)
plot(X, pch = '.', col = attr(X, "index")+1)
contour(x = xvals, y = yvals, z = matrix(dens, np, np),
        levels = c(0.002, 0.01, 0.02, 0.04, 0.08, 0.15 ), add = TRUE, lwd = 2)

#### 1) Example use
# Set up mixture density
mu <- matrix(c(1, 2, 10, 20), 2, 2, byrow = TRUE)
sigma <- list(diag(c(1, 10)), matrix(c(1, -0.9, -0.9, 1), 2, 2))
w <- c(0.1, 0.9)

# Simulate
X <- rmixn(1e4, mu, sigma, w)

# Evaluate density
ds <- dmixn(X, mu, sigma, w = w)
head(ds)

##### 2) More complicated example
# Define mixture
set.seed(5135)
N <- 10000
d <- 2
w <- rep(1, 2) / 2
mu <- matrix(c(0, 0, 2, 3), 2, 2, byrow = TRUE) 
sigma <- list(matrix(c(1, 0, 0, 2), 2, 2), matrix(c(1, -0.9, -0.9, 1), 2, 2)) 

# Simulate random variables
X <- rmixn(N, mu, sigma, w = w, retInd = TRUE)

# Plot mixture density
np <- 100
xvals <- seq(min(X[ , 1]), max(X[ , 1]), length.out = np)
yvals <- seq(min(X[ , 2]), max(X[ , 2]), length.out = np)
theGrid <- expand.grid(xvals, yvals) 
theGrid <- as.matrix(theGrid)
dens <- dmixn(theGrid, mu, sigma, w = w)
plot(X, pch = '.', col = attr(X, "index")+1)
contour(x = xvals, y = yvals, z = matrix(dens, np, np),
        levels = c(0.002, 0.01, 0.02, 0.04, 0.08, 0.15 ), add = TRUE, lwd = 2)

Fast density computation for mixture of multivariate Student's t distributions.

Description

Fast density computation for mixture of multivariate Student's t distributions.

Usage

dmixt(X, mu, sigma, df, w, log = FALSE, ncores = 1, isChol = FALSE, A = NULL)
dmixt(X, mu, sigma, df, w, log = FALSE, ncores = 1, isChol = FALSE, A = NULL)

Arguments

`X`	matrix n by d where each row is a d dimensional random vector. Alternatively `X` can be a d-dimensional vector.
`mu`	an (m x d) matrix, where m is the number of mixture components.
`sigma`	as list of m covariance matrices (d x d) on for each mixture component. Alternatively it can be a list of m cholesky decomposition of the covariance. In that case `isChol` should be set to `TRUE`.
`df`	a positive scalar representing the degrees of freedom. All the densities in the mixture have the same `df`.
`w`	vector of length m, containing the weights of the mixture components.
`log`	boolean set to true the logarithm of the pdf is required.
`ncores`	Number of cores used. The parallelization will take place only if OpenMP is supported.
`isChol`	boolean set to true is `sigma` is the cholesky decomposition of the covariance matrix.
`A`	an (optional) numeric matrix of dimension (m x d), which will be used to store the evaluations of each mixture density over each mixture component. It is useful when m and n are large and one wants to call `dmixt()` several times, without reallocating memory for the whole matrix each time. NB1: `A` will be modified, not copied! NB2: the element of `A` must be of class "numeric".

Details

There are many candidates for the multivariate generalization of Student's t-distribution, here we use the parametrization described here https://en.wikipedia.org/wiki/Multivariate_t-distribution. NB: at the moment the parallelization does not work properly on Solaris OS when ncores>1. Hence, dmixt() checks if the OS is Solaris and, if this the case, it imposes ncores==1.

Value

A vector of length n where the i-the entry contains the pdf of the i-th random vector (i.e. the i-th row of X).

Author(s)

Matteo Fasiolo <[email protected]>.

Examples

#### 1) Example use
# Set up mixture density
df <- 6
mu <- matrix(c(1, 2, 10, 20), 2, 2, byrow = TRUE)
sigma <- list(diag(c(1, 10)), matrix(c(1, -0.9, -0.9, 1), 2, 2))
w <- c(0.1, 0.9)

# Simulate
X <- rmixt(1e4, mu, sigma, df, w)

# Evaluate density
ds <- dmixt(X, mu, sigma, w = w, df = df)
head(ds)

##### 2) More complicated example
# Define mixture
set.seed(5135)
N <- 10000
d <- 2
df = 10
w <- rep(1, 2) / 2
mu <- matrix(c(0, 0, 2, 3), 2, 2, byrow = TRUE) 
sigma <- list(matrix(c(1, 0, 0, 2), 2, 2), matrix(c(1, -0.9, -0.9, 1), 2, 2)) 

# Simulate random variables
X <- rmixt(N, mu, sigma, w = w, df = df, retInd = TRUE)

# Plot mixture density
np <- 100
xvals <- seq(min(X[ , 1]), max(X[ , 1]), length.out = np)
yvals <- seq(min(X[ , 2]), max(X[ , 2]), length.out = np)
theGrid <- expand.grid(xvals, yvals) 
theGrid <- as.matrix(theGrid)
dens <- dmixt(theGrid, mu, sigma, w = w, df = df)
plot(X, pch = '.', col = attr(X, "index")+1)
contour(x = xvals, y = yvals, z = matrix(dens, np, np),
        levels = c(0.002, 0.01, 0.02, 0.04, 0.08, 0.15 ), add = TRUE, lwd = 2)

#### 1) Example use
# Set up mixture density
df <- 6
mu <- matrix(c(1, 2, 10, 20), 2, 2, byrow = TRUE)
sigma <- list(diag(c(1, 10)), matrix(c(1, -0.9, -0.9, 1), 2, 2))
w <- c(0.1, 0.9)

# Simulate
X <- rmixt(1e4, mu, sigma, df, w)

# Evaluate density
ds <- dmixt(X, mu, sigma, w = w, df = df)
head(ds)

##### 2) More complicated example
# Define mixture
set.seed(5135)
N <- 10000
d <- 2
df = 10
w <- rep(1, 2) / 2
mu <- matrix(c(0, 0, 2, 3), 2, 2, byrow = TRUE) 
sigma <- list(matrix(c(1, 0, 0, 2), 2, 2), matrix(c(1, -0.9, -0.9, 1), 2, 2)) 

# Simulate random variables
X <- rmixt(N, mu, sigma, w = w, df = df, retInd = TRUE)

# Plot mixture density
np <- 100
xvals <- seq(min(X[ , 1]), max(X[ , 1]), length.out = np)
yvals <- seq(min(X[ , 2]), max(X[ , 2]), length.out = np)
theGrid <- expand.grid(xvals, yvals) 
theGrid <- as.matrix(theGrid)
dens <- dmixt(theGrid, mu, sigma, w = w, df = df)
plot(X, pch = '.', col = attr(X, "index")+1)
contour(x = xvals, y = yvals, z = matrix(dens, np, np),
        levels = c(0.002, 0.01, 0.02, 0.04, 0.08, 0.15 ), add = TRUE, lwd = 2)

Fast computation of the multivariate normal density.

Description

Fast computation of the multivariate normal density.

Usage

dmvn(X, mu, sigma, log = FALSE, ncores = 1, isChol = FALSE)
dmvn(X, mu, sigma, log = FALSE, ncores = 1, isChol = FALSE)

Arguments

`X`	matrix n by d where each row is a d dimensional random vector. Alternatively `X` can be a d-dimensional vector.
`mu`	vector of length d, representing the mean of the distribution.
`sigma`	covariance matrix (d x d). Alternatively it can be the cholesky decomposition of the covariance. In that case isChol should be set to TRUE.
`log`	boolean set to true the logarithm of the pdf is required.
`ncores`	Number of cores used. The parallelization will take place only if OpenMP is supported.
`isChol`	boolean set to true is `sigma` is the cholesky decomposition of the covariance matrix.

Value

A vector of length n where the i-the entry contains the pdf of the i-th random vector.

Author(s)

Matteo Fasiolo <[email protected]>

Examples

N <- 100
d <- 5
mu <- 1:d
X <- t(t(matrix(rnorm(N*d), N, d)) + mu)
tmp <- matrix(rnorm(d^2), d, d)
mcov <- tcrossprod(tmp, tmp)  + diag(0.5, d)
myChol <- chol(mcov)

head(dmvn(X, mu, mcov), 10)
head(dmvn(X, mu, myChol, isChol = TRUE), 10)

## Not run: 
# Performance comparison: microbenchmark does not work on all
# platforms, hence we need to check whether it is installed
if( "microbenchmark" %in% rownames(installed.packages()) ){
library(mvtnorm)
library(microbenchmark)

a <- cbind(
      dmvn(X, mu, mcov),
      dmvn(X, mu, myChol, isChol = TRUE),
      dmvnorm(X, mu, mcov))
      
# Check if we get the same output as dmvnorm()
a[ , 1] / a[, 3]
a[ , 2] / a[, 3]

microbenchmark(dmvn(X, mu, myChol, isChol = TRUE), 
               dmvn(X, mu, mcov), 
               dmvnorm(X, mu, mcov))
               
detach("package:mvtnorm", unload=TRUE)
}

## End(Not run)

N <- 100
d <- 5
mu <- 1:d
X <- t(t(matrix(rnorm(N*d), N, d)) + mu)
tmp <- matrix(rnorm(d^2), d, d)
mcov <- tcrossprod(tmp, tmp)  + diag(0.5, d)
myChol <- chol(mcov)

head(dmvn(X, mu, mcov), 10)
head(dmvn(X, mu, myChol, isChol = TRUE), 10)

## Not run: 
# Performance comparison: microbenchmark does not work on all
# platforms, hence we need to check whether it is installed
if( "microbenchmark" %in% rownames(installed.packages()) ){
library(mvtnorm)
library(microbenchmark)

a <- cbind(
      dmvn(X, mu, mcov),
      dmvn(X, mu, myChol, isChol = TRUE),
      dmvnorm(X, mu, mcov))
      
# Check if we get the same output as dmvnorm()
a[ , 1] / a[, 3]
a[ , 2] / a[, 3]

microbenchmark(dmvn(X, mu, myChol, isChol = TRUE), 
               dmvn(X, mu, mcov), 
               dmvnorm(X, mu, mcov))
               
detach("package:mvtnorm", unload=TRUE)
}

## End(Not run)

Fast computation of the multivariate Student's t density.

Description

Fast computation of the multivariate Student's t density.

Usage

dmvt(X, mu, sigma, df, log = FALSE, ncores = 1, isChol = FALSE)
dmvt(X, mu, sigma, df, log = FALSE, ncores = 1, isChol = FALSE)

Arguments

`X`	matrix n by d where each row is a d dimensional random vector. Alternatively `X` can be a d-dimensional vector.
`mu`	vector of length d, representing the mean of the distribution.
`sigma`	scale matrix (d x d). Alternatively it can be the cholesky decomposition of the scale matrix. In that case isChol should be set to TRUE. Notice that ff the degrees of freedom (the argument `df`) is larger than 2, the `Cov(X)=sigma*df/(df-2)`.
`df`	a positive scalar representing the degrees of freedom.
`log`	boolean set to true the logarithm of the pdf is required.
`ncores`	Number of cores used. The parallelization will take place only if OpenMP is supported.
`isChol`	boolean set to true is `sigma` is the cholesky decomposition of the covariance matrix.

Details

There are many candidates for the multivariate generalization of Student's t-distribution, here we use the parametrization described here https://en.wikipedia.org/wiki/Multivariate_t-distribution. NB: at the moment the parallelization does not work properly on Solaris OS when ncores>1. Hence, dmvt() checks if the OS is Solaris and, if this the case, it imposes ncores==1.

Value

A vector of length n where the i-the entry contains the pdf of the i-th random vector.

Author(s)

Matteo Fasiolo <[email protected]>

Examples

N <- 100
d <- 5
mu <- 1:d
df <- 4
X <- t(t(matrix(rnorm(N*d), N, d)) + mu)
tmp <- matrix(rnorm(d^2), d, d)
mcov <- tcrossprod(tmp, tmp)  + diag(0.5, d)
myChol <- chol(mcov)

head(dmvt(X, mu, mcov, df = df), 10)
head(dmvt(X, mu, myChol, df = df, isChol = TRUE), 10)

N <- 100
d <- 5
mu <- 1:d
df <- 4
X <- t(t(matrix(rnorm(N*d), N, d)) + mu)
tmp <- matrix(rnorm(d^2), d, d)
mcov <- tcrossprod(tmp, tmp)  + diag(0.5, d)
myChol <- chol(mcov)

head(dmvt(X, mu, mcov, df = df), 10)
head(dmvt(X, mu, myChol, df = df, isChol = TRUE), 10)

Fast computation of squared mahalanobis distance between all rows of `X` and the vector `mu` with respect to sigma.

Description

Fast computation of squared mahalanobis distance between all rows of X and the vector mu with respect to sigma.

Usage

maha(X, mu, sigma, ncores = 1, isChol = FALSE)
maha(X, mu, sigma, ncores = 1, isChol = FALSE)

Arguments

`X`	matrix n by d where each row is a d dimensional random vector. Alternatively `X` can be a d-dimensional vector.
`mu`	vector of length d, representing the central position.
`sigma`	covariance matrix (d x d). Alternatively is can be the cholesky decomposition of the covariance. In that case `isChol` should be set to `TRUE`.
`ncores`	Number of cores used. The parallelization will take place only if OpenMP is supported.
`isChol`	boolean set to true is `sigma` is the cholesky decomposition of the covariance matrix.

Value

a vector of length n where the i-the entry contains the square mahalanobis distance i-th random vector.

Author(s)

Matteo Fasiolo <[email protected]>

Examples

N <- 100
d <- 5
mu <- 1:d
X <- t(t(matrix(rnorm(N*d), N, d)) + mu)
tmp <- matrix(rnorm(d^2), d, d)
mcov <- tcrossprod(tmp, tmp)
myChol <- chol(mcov)

rbind(head(maha(X, mu, mcov), 10),
      head(maha(X, mu, myChol, isChol = TRUE), 10),
      head(mahalanobis(X, mu, mcov), 10))

## Not run: 
# Performance comparison: microbenchmark does not work on all
# platforms, hence we need to check whether it is installed
if( "microbenchmark" %in% rownames(installed.packages()) ){
library(microbenchmark)

a <- cbind(
  maha(X, mu, mcov),
  maha(X, mu, myChol, isChol = TRUE),
  mahalanobis(X, mu, mcov))
  
# Same output as mahalanobis
a[ , 1] / a[, 3]
a[ , 2] / a[, 3]

microbenchmark(maha(X, mu, mcov),
               maha(X, mu, myChol, isChol = TRUE),
               mahalanobis(X, mu, mcov))
}

## End(Not run)
N <- 100
d <- 5
mu <- 1:d
X <- t(t(matrix(rnorm(N*d), N, d)) + mu)
tmp <- matrix(rnorm(d^2), d, d)
mcov <- tcrossprod(tmp, tmp)
myChol <- chol(mcov)

rbind(head(maha(X, mu, mcov), 10),
      head(maha(X, mu, myChol, isChol = TRUE), 10),
      head(mahalanobis(X, mu, mcov), 10))

## Not run: 
# Performance comparison: microbenchmark does not work on all
# platforms, hence we need to check whether it is installed
if( "microbenchmark" %in% rownames(installed.packages()) ){
library(microbenchmark)

a <- cbind(
  maha(X, mu, mcov),
  maha(X, mu, myChol, isChol = TRUE),
  mahalanobis(X, mu, mcov))
  
# Same output as mahalanobis
a[ , 1] / a[, 3]
a[ , 2] / a[, 3]

microbenchmark(maha(X, mu, mcov),
               maha(X, mu, myChol, isChol = TRUE),
               mahalanobis(X, mu, mcov))
}

## End(Not run)

Mean-shift mode seeking algorithm

Description

Given a sample from a d-dimensional distribution, an initialization point and a bandwidth the algorithm finds the nearest mode of the corresponding Gaussian kernel density.

Usage

ms(X, init, H, tol = 1e-06, ncores = 1, store = FALSE)
ms(X, init, H, tol = 1e-06, ncores = 1, store = FALSE)

Arguments

`X`	n by d matrix containing the data.
`init`	d-dimensional vector containing the initial point for the optimization.
`H`	Positive definite bandwidth matrix representing the covariance of each component of the Gaussian kernel density.
`tol`	Tolerance used to assess the convergence of the algorithm, which is stopped if the absolute values of increments along all the dimensions are smaller then tol at any iteration. Default value is 1e-6.
`ncores`	Number of cores used. The parallelization will take place only if OpenMP is supported.
`store`	If `FALSE` only the latest iteration is returned, if `TRUE` the function will return a matrix where the i-th row is the position of the algorithms at the i-th iteration.

Value

A list where estim is a d-dimensional vector containing the last position of the algorithm, while traj is a matrix with d-colums representing the trajectory of the algorithm along each dimension. If store == FALSE the whole trajectory is not stored and traj = NULL.

Author(s)

Matteo Fasiolo <[email protected]>.

Examples

set.seed(434)

# Simulating multivariate normal data
N <- 1000
mu <- c(1, 2)
sigma <- matrix(c(1, 0.5, 0.5, 1), 2, 2)
X <- rmvn(N, mu = mu, sigma = sigma)

# Plotting the true density function
steps <- 100
range1 <- seq(min(X[ , 1]), max(X[ , 1]), length.out = steps)
range2 <- seq(min(X[ , 2]), max(X[ , 2]), length.out = steps)
grid <- expand.grid(range1, range2)
vals <- dmvn(as.matrix(grid), mu, sigma)

contour(z = matrix(vals, steps, steps),  x = range1, y = range2, xlab = "X1", ylab = "X2")
points(X[ , 1], X[ , 2], pch = '.')
 
# Estimating the mode from "nrep" starting points
nrep <- 10
index <- sample(1:N, nrep)
for(ii in 1:nrep) {
  start <- X[index[ii], ]
  out <- ms(X, init = start, H = 0.1 * sigma, store = TRUE)
  lines(out$traj[ , 1], out$traj[ , 2], col = 2, lwd = 2) 
  points(out$final[1], out$final[2], col = 4, pch = 3, lwd = 3) # Estimated mode (blue)
  points(start[1], start[2], col = 2, pch = 3, lwd = 3)         # ii-th starting value 
}
set.seed(434)

# Simulating multivariate normal data
N <- 1000
mu <- c(1, 2)
sigma <- matrix(c(1, 0.5, 0.5, 1), 2, 2)
X <- rmvn(N, mu = mu, sigma = sigma)

# Plotting the true density function
steps <- 100
range1 <- seq(min(X[ , 1]), max(X[ , 1]), length.out = steps)
range2 <- seq(min(X[ , 2]), max(X[ , 2]), length.out = steps)
grid <- expand.grid(range1, range2)
vals <- dmvn(as.matrix(grid), mu, sigma)

contour(z = matrix(vals, steps, steps),  x = range1, y = range2, xlab = "X1", ylab = "X2")
points(X[ , 1], X[ , 2], pch = '.')
 
# Estimating the mode from "nrep" starting points
nrep <- 10
index <- sample(1:N, nrep)
for(ii in 1:nrep) {
  start <- X[index[ii], ]
  out <- ms(X, init = start, H = 0.1 * sigma, store = TRUE)
  lines(out$traj[ , 1], out$traj[ , 2], col = 2, lwd = 2) 
  points(out$final[1], out$final[2], col = 4, pch = 3, lwd = 3) # Estimated mode (blue)
  points(start[1], start[2], col = 2, pch = 3, lwd = 3)         # ii-th starting value 
}

Fast simulation of r.v.s from a mixture of multivariate normal densities

Description

Fast simulation of r.v.s from a mixture of multivariate normal densities

Usage

rmixn(
  n,
  mu,
  sigma,
  w,
  ncores = 1,
  isChol = FALSE,
  retInd = FALSE,
  A = NULL,
  kpnames = FALSE
)
rmixn(
  n,
  mu,
  sigma,
  w,
  ncores = 1,
  isChol = FALSE,
  retInd = FALSE,
  A = NULL,
  kpnames = FALSE
)

Arguments

`n`	number of random vectors to be simulated.
`mu`	an (m x d) matrix, where m is the number of mixture components.
`sigma`	as list of m covariance matrices (d x d) on for each mixture component. Alternatively it can be a list of m cholesky decomposition of the covariance. In that case `isChol` should be set to `TRUE`.
`w`	vector of length m, containing the weights of the mixture components.
`ncores`	Number of cores used. The parallelization will take place only if OpenMP is supported.
`isChol`	boolean set to true is `sigma` is the cholesky decomposition of the covariance matrix.
`retInd`	when set to `TRUE` an attribute called "index" will be added to the output matrix of random variables. This is a vector specifying to which mixture components each random vector belongs. `FALSE` by default.
`A`	an (optional) numeric matrix of dimension (n x d), which will be used to store the output random variables. It is useful when n and d are large and one wants to call `rmvn()` several times, without reallocating memory for the whole matrix each time. NB: the element of `A` must be of class "numeric".
`kpnames`	if `TRUE` the dimensions' names are preserved. That is, the i-th column of the output has the same name as the i-th entry of `mu` or the i-th column of `sigma`. `kpnames==FALSE` by default.

Details

Notice that this function does not use one of the Random Number Generators (RNGs) provided by R, but one of the parallel cryptographic RNGs described in (Salmon et al., 2011). It is important to point out that this RNG can safely be used in parallel, without risk of collisions between parallel sequence of random numbers. The initialization of the RNG depends on R's seed, hence the set.seed() function can be used to obtain reproducible results. Notice though that changing ncores causes most of the generated numbers to be different even if R's seed is the same (see example below). NB: at the moment the RNG does not work properly on Solaris OS when ncores>1. Hence, rmixn() checks if the OS is Solaris and, if this the case, it imposes ncores==1.

Value

If A==NULL (default) the output is an (n x d) matrix where the i-th row is the i-th simulated vector. If A!=NULL then the random vector are store in A, which is provided by the user, and the function returns NULL. Notice that if retInd==TRUE an attribute called "index" will be added to A. This is a vector specifying to which mixture components each random vector belongs.

Author(s)

Matteo Fasiolo <[email protected]>, C++ RNG engine by Thijs van den Berg <http://sitmo.com/>.

References

John K. Salmon, Mark A. Moraes, Ron O. Dror, and David E. Shaw (2011). Parallel Random Numbers: As Easy as 1, 2, 3. D. E. Shaw Research, New York, NY 10036, USA.

Examples

# Create mixture of two components
mu <- matrix(c(1, 2, 10, 20), 2, 2, byrow = TRUE)
sigma <- list(diag(c(1, 10)), matrix(c(1, -0.9, -0.9, 1), 2, 2))
w <- c(0.1, 0.9)

# Simulate
X <- rmixn(1e4, mu, sigma, w, retInd = TRUE)
plot(X, pch = '.', col = attr(X, "index"))

# Simulate with fixed seed
set.seed(414)
rmixn(4, mu, sigma, w)

set.seed(414)
rmixn(4, mu, sigma, w)

set.seed(414)  
rmixn(4, mu, sigma, w, ncores = 2) # r.v. generated on the second core are different

###### Here we create the matrix that will hold the simulated random variables upfront.
A <- matrix(NA, 4, 2)
class(A) <- "numeric" # This is important. We need the elements of A to be of class "numeric". 

set.seed(414)
rmixn(4, mu, sigma, w, ncores = 2, A = A) # This returns NULL ...
A                                         # ... but the result is here

# Create mixture of two components
mu <- matrix(c(1, 2, 10, 20), 2, 2, byrow = TRUE)
sigma <- list(diag(c(1, 10)), matrix(c(1, -0.9, -0.9, 1), 2, 2))
w <- c(0.1, 0.9)

# Simulate
X <- rmixn(1e4, mu, sigma, w, retInd = TRUE)
plot(X, pch = '.', col = attr(X, "index"))

# Simulate with fixed seed
set.seed(414)
rmixn(4, mu, sigma, w)

set.seed(414)
rmixn(4, mu, sigma, w)

set.seed(414)  
rmixn(4, mu, sigma, w, ncores = 2) # r.v. generated on the second core are different

###### Here we create the matrix that will hold the simulated random variables upfront.
A <- matrix(NA, 4, 2)
class(A) <- "numeric" # This is important. We need the elements of A to be of class "numeric". 

set.seed(414)
rmixn(4, mu, sigma, w, ncores = 2, A = A) # This returns NULL ...
A                                         # ... but the result is here

Fast simulation of r.v.s from a mixture of multivariate Student's t densities

Description

Fast simulation of r.v.s from a mixture of multivariate Student's t densities

Usage

rmixt(
  n,
  mu,
  sigma,
  df,
  w,
  ncores = 1,
  isChol = FALSE,
  retInd = FALSE,
  A = NULL,
  kpnames = FALSE
)
rmixt(
  n,
  mu,
  sigma,
  df,
  w,
  ncores = 1,
  isChol = FALSE,
  retInd = FALSE,
  A = NULL,
  kpnames = FALSE
)

Arguments

`n`	number of random vectors to be simulated.
`mu`	an (m x d) matrix, where m is the number of mixture components.
`sigma`	as list of m covariance matrices (d x d) on for each mixture component. Alternatively it can be a list of m cholesky decomposition of the covariance. In that case `isChol` should be set to `TRUE`.
`df`	a positive scalar representing the degrees of freedom. All the densities in the mixture have the same `df`.
`w`	vector of length m, containing the weights of the mixture components.
`ncores`	Number of cores used. The parallelization will take place only if OpenMP is supported.
`isChol`	boolean set to true is `sigma` is the cholesky decomposition of the covariance matrix.
`retInd`	when set to `TRUE` an attribute called "index" will be added to the output matrix of random variables. This is a vector specifying to which mixture components each random vector belongs. `FALSE` by default.
`A`	an (optional) numeric matrix of dimension (n x d), which will be used to store the output random variables. It is useful when n and d are large and one wants to call `rmvn()` several times, without reallocating memory for the whole matrix each time. NB: the element of `A` must be of class "numeric".
`kpnames`	if `TRUE` the dimensions' names are preserved. That is, the i-th column of the output has the same name as the i-th entry of `mu` or the i-th column of `sigma`. `kpnames==FALSE` by default.

Details

There are many candidates for the multivariate generalization of Student's t-distribution, here we use the parametrization described here https://en.wikipedia.org/wiki/Multivariate_t-distribution.

Notice that this function does not use one of the Random Number Generators (RNGs) provided by R, but one of the parallel cryptographic RNGs described in (Salmon et al., 2011). It is important to point out that this RNG can safely be used in parallel, without risk of collisions between parallel sequence of random numbers. The initialization of the RNG depends on R's seed, hence the set.seed() function can be used to obtain reproducible results. Notice though that changing ncores causes most of the generated numbers to be different even if R's seed is the same (see example below). NB: at the moment the parallelization does not work properly on Solaris OS when ncores>1. Hence, rmixt() checks if the OS is Solaris and, if this the case, it imposes ncores==1

Value

Author(s)

Matteo Fasiolo <[email protected]>, C++ RNG engine by Thijs van den Berg <http://sitmo.com/>.

References

John K. Salmon, Mark A. Moraes, Ron O. Dror, and David E. Shaw (2011). Parallel Random Numbers: As Easy as 1, 2, 3. D. E. Shaw Research, New York, NY 10036, USA.

Examples

# Create mixture of two components
df <- 6
mu <- matrix(c(1, 2, 10, 20), 2, 2, byrow = TRUE)
sigma <- list(diag(c(1, 10)), matrix(c(1, -0.9, -0.9, 1), 2, 2))
w <- c(0.1, 0.9)

# Simulate
X <- rmixt(1e4, mu, sigma, df, w, retInd = TRUE)
plot(X, pch = '.', col = attr(X, "index"))

# Simulate with fixed seed
set.seed(414)
rmixt(4, mu, sigma, df, w)

set.seed(414)
rmixt(4, mu, sigma, df, w)

set.seed(414)  
rmixt(4, mu, sigma, df, w, ncores = 2) # r.v. generated on the second core are different

###### Here we create the matrix that will hold the simulated random variables upfront.
A <- matrix(NA, 4, 2)
class(A) <- "numeric" # This is important. We need the elements of A to be of class "numeric". 

set.seed(414)
rmixt(4, mu, sigma, df, w, ncores = 2, A = A) # This returns NULL ...
A                                             # ... but the result is here

# Create mixture of two components
df <- 6
mu <- matrix(c(1, 2, 10, 20), 2, 2, byrow = TRUE)
sigma <- list(diag(c(1, 10)), matrix(c(1, -0.9, -0.9, 1), 2, 2))
w <- c(0.1, 0.9)

# Simulate
X <- rmixt(1e4, mu, sigma, df, w, retInd = TRUE)
plot(X, pch = '.', col = attr(X, "index"))

# Simulate with fixed seed
set.seed(414)
rmixt(4, mu, sigma, df, w)

set.seed(414)
rmixt(4, mu, sigma, df, w)

set.seed(414)  
rmixt(4, mu, sigma, df, w, ncores = 2) # r.v. generated on the second core are different

###### Here we create the matrix that will hold the simulated random variables upfront.
A <- matrix(NA, 4, 2)
class(A) <- "numeric" # This is important. We need the elements of A to be of class "numeric". 

set.seed(414)
rmixt(4, mu, sigma, df, w, ncores = 2, A = A) # This returns NULL ...
A                                             # ... but the result is here

Fast simulation of multivariate normal random variables

Description

Fast simulation of multivariate normal random variables

Usage

rmvn(n, mu, sigma, ncores = 1, isChol = FALSE, A = NULL, kpnames = FALSE)
rmvn(n, mu, sigma, ncores = 1, isChol = FALSE, A = NULL, kpnames = FALSE)

Arguments

`n`	number of random vectors to be simulated.
`mu`	vector of length d, representing the mean.
`sigma`	covariance matrix (d x d). Alternatively is can be the cholesky decomposition of the covariance. In that case `isChol` should be set to `TRUE`.
`ncores`	Number of cores used. The parallelization will take place only if OpenMP is supported.
`isChol`	boolean set to true is `sigma` is the cholesky decomposition of the covariance matrix.
`A`	an (optional) numeric matrix of dimension (n x d), which will be used to store the output random variables. It is useful when n and d are large and one wants to call `rmvn()` several times, without reallocating memory for the whole matrix each time. NB: the element of `A` must be of class "numeric".
`kpnames`	if `TRUE` the dimensions' names are preserved. That is, the i-th column of the output has the same name as the i-th entry of `mu` or the i-th column of `sigma`. `kpnames==FALSE` by default.

Details

Notice that this function does not use one of the Random Number Generators (RNGs) provided by R, but one of the parallel cryptographic RNGs described in (Salmon et al., 2011). It is important to point out that this RNG can safely be used in parallel, without risk of collisions between parallel sequence of random numbers. The initialization of the RNG depends on R's seed, hence the set.seed() function can be used to obtain reproducible results. Notice though that changing ncores causes most of the generated numbers to be different even if R's seed is the same (see example below). NB: at the moment the RNG does not work properly on Solaris OS when ncores>1. Hence, rmvn() checks if the OS is Solaris and, if this the case, it imposes ncores==1.

Value

Author(s)

Matteo Fasiolo <[email protected]>, C++ RNG engine by Thijs van den Berg <http://sitmo.com/>.

References

John K. Salmon, Mark A. Moraes, Ron O. Dror, and David E. Shaw (2011). Parallel Random Numbers: As Easy as 1, 2, 3. D. E. Shaw Research, New York, NY 10036, USA.

Examples

d <- 5
mu <- 1:d

# Creating covariance matrix
tmp <- matrix(rnorm(d^2), d, d)
mcov <- tcrossprod(tmp, tmp)

set.seed(414)
rmvn(4, 1:d, mcov)

set.seed(414)
rmvn(4, 1:d, mcov)

set.seed(414)  
rmvn(4, 1:d, mcov, ncores = 2) # r.v. generated on the second core are different

###### Here we create the matrix that will hold the simulated random variables upfront.
A <- matrix(NA, 4, d)
class(A) <- "numeric" # This is important. We need the elements of A to be of class "numeric". 

set.seed(414)
rmvn(4, 1:d, mcov, ncores = 2, A = A) # This returns NULL ...
A                                     # ... but the result is here

d <- 5
mu <- 1:d

# Creating covariance matrix
tmp <- matrix(rnorm(d^2), d, d)
mcov <- tcrossprod(tmp, tmp)

set.seed(414)
rmvn(4, 1:d, mcov)

set.seed(414)
rmvn(4, 1:d, mcov)

set.seed(414)  
rmvn(4, 1:d, mcov, ncores = 2) # r.v. generated on the second core are different

###### Here we create the matrix that will hold the simulated random variables upfront.
A <- matrix(NA, 4, d)
class(A) <- "numeric" # This is important. We need the elements of A to be of class "numeric". 

set.seed(414)
rmvn(4, 1:d, mcov, ncores = 2, A = A) # This returns NULL ...
A                                     # ... but the result is here

Fast simulation of multivariate Student's t random variables

Description

Fast simulation of multivariate Student's t random variables

Usage

rmvt(n, mu, sigma, df, ncores = 1, isChol = FALSE, A = NULL, kpnames = FALSE)
rmvt(n, mu, sigma, df, ncores = 1, isChol = FALSE, A = NULL, kpnames = FALSE)

Arguments

`n`	number of random vectors to be simulated.
`mu`	vector of length d, representing the mean of the distribution.
`sigma`	scale matrix (d x d). Alternatively it can be the cholesky decomposition of the scale matrix. In that case isChol should be set to TRUE. Notice that ff the degrees of freedom (the argument `df`) is larger than 2, the `Cov(X)=sigma*df/(df-2)`.
`df`	a positive scalar representing the degrees of freedom.
`ncores`	Number of cores used. The parallelization will take place only if OpenMP is supported.
`isChol`	boolean set to true is `sigma` is the cholesky decomposition of the covariance matrix.
`A`	an (optional) numeric matrix of dimension (n x d), which will be used to store the output random variables. It is useful when n and d are large and one wants to call `rmvn()` several times, without reallocating memory for the whole matrix each time. NB: the element of `A` must be of class "numeric".
`kpnames`	if `TRUE` the dimensions' names are preserved. That is, the i-th column of the output has the same name as the i-th entry of `mu` or the i-th column of `sigma`. `kpnames==FALSE` by default.

Details

There are in fact many candidates for the multivariate generalization of Student's t-distribution, here we use the parametrization described here https://en.wikipedia.org/wiki/Multivariate_t-distribution.

Notice that rmvt() does not use one of the Random Number Generators (RNGs) provided by R, but one of the parallel cryptographic RNGs described in (Salmon et al., 2011). It is important to point out that this RNG can safely be used in parallel, without risk of collisions between parallel sequence of random numbers. The initialization of the RNG depends on R's seed, hence the set.seed() function can be used to obtain reproducible results. Notice though that changing ncores causes most of the generated numbers to be different even if R's seed is the same (see example below). NB: at the moment the RNG does not work properly on Solaris OS when ncores>1. Hence, rmvt() checks if the OS is Solaris and, if this the case, it imposes ncores==1.

Value

Author(s)

Matteo Fasiolo <[email protected]>, C++ RNG engine by Thijs van den Berg <http://sitmo.com/>.

References

John K. Salmon, Mark A. Moraes, Ron O. Dror, and David E. Shaw (2011). Parallel Random Numbers: As Easy as 1, 2, 3. D. E. Shaw Research, New York, NY 10036, USA.

Examples

d <- 5
mu <- 1:d
df <- 4

# Creating covariance matrix
tmp <- matrix(rnorm(d^2), d, d)
mcov <- tcrossprod(tmp, tmp) + diag(0.5, d)

set.seed(414)
rmvt(4, 1:d, mcov, df = df)

set.seed(414)
rmvt(4, 1:d, mcov, df = df)

set.seed(414)  
rmvt(4, 1:d, mcov, df = df, ncores = 2) # These will not match the r.v. generated on a single core.

###### Here we create the matrix that will hold the simulated random variables upfront.
A <- matrix(NA, 4, d)
class(A) <- "numeric" # This is important. We need the elements of A to be of class "numeric". 

set.seed(414)
rmvt(4, 1:d, mcov, df = df, ncores = 2, A = A) # This returns NULL ...
A                                     # ... but the result is here

d <- 5
mu <- 1:d
df <- 4

# Creating covariance matrix
tmp <- matrix(rnorm(d^2), d, d)
mcov <- tcrossprod(tmp, tmp) + diag(0.5, d)

set.seed(414)
rmvt(4, 1:d, mcov, df = df)

set.seed(414)
rmvt(4, 1:d, mcov, df = df)

set.seed(414)  
rmvt(4, 1:d, mcov, df = df, ncores = 2) # These will not match the r.v. generated on a single core.

###### Here we create the matrix that will hold the simulated random variables upfront.
A <- matrix(NA, 4, d)
class(A) <- "numeric" # This is important. We need the elements of A to be of class "numeric". 

set.seed(414)
rmvt(4, 1:d, mcov, df = df, ncores = 2, A = A) # This returns NULL ...
A                                     # ... but the result is here

Package 'mvnfast'

Help Index

Fast density computation for mixture of multivariate normal distributions.

Description

Usage

Arguments

Details

Value

Author(s)

Examples

Fast density computation for mixture of multivariate Student's t distributions.

Description

Usage

Arguments

Details

Value

Author(s)

Examples

Fast computation of the multivariate normal density.

Description

Usage

Arguments

Value

Author(s)

Examples

Fast computation of the multivariate Student's t density.

Description

Usage

Arguments

Details

Value

Author(s)

Examples

Fast computation of squared mahalanobis distance between all rows of X and the vector mu with respect to sigma.

Description

Usage

Arguments

Value

Author(s)

Examples

Mean-shift mode seeking algorithm

Description

Usage

Arguments

Value

Author(s)

Examples

Fast simulation of r.v.s from a mixture of multivariate normal densities

Description

Usage

Arguments

Details

Value

Author(s)

References

Examples

Fast simulation of r.v.s from a mixture of multivariate Student's t densities

Description

Usage

Arguments

Details

Value

Author(s)

References

Examples

Fast simulation of multivariate normal random variables

Description

Usage

Arguments

Details

Value

Author(s)

References

Examples

Fast simulation of multivariate Student's t random variables

Description

Usage

Arguments

Details

Value

Fast computation of squared mahalanobis distance between all rows of `X` and the vector `mu` with respect to sigma.