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)

Arguments

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 <matteo.fasiolo@gmail.com>.

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)

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)

Arguments

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 <matteo.fasiolo@gmail.com>.

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)

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)

Arguments

Value

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

Author(s)

Matteo Fasiolo <matteo.fasiolo@gmail.com>

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)

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)

Arguments

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 <matteo.fasiolo@gmail.com>

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)

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)

Arguments

Value

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

Author(s)

Matteo Fasiolo <matteo.fasiolo@gmail.com>

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)

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)

Arguments

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 <matteo.fasiolo@gmail.com>.

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 
}

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
)

Arguments

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 <matteo.fasiolo@gmail.com>, 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

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
)

Arguments

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

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 <matteo.fasiolo@gmail.com>, 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

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)

Arguments

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

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.

Author(s)

Matteo Fasiolo <matteo.fasiolo@gmail.com>, 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

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)

Arguments

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

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.

Author(s)

Matteo Fasiolo <matteo.fasiolo@gmail.com>, 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