pivmet (version 0.5.0)

piv_sim: Generate Data from a Gaussian Nested Mixture

Description

Simulate \(N\) observations from a nested Gaussian mixture model with \(k\) pre-specified components under uniform group probabilities \(1/k\), where each group is in turn drawn from a further level consisting of two subgroups.

Usage

piv_sim(
  N,
  k,
  Mu,
  stdev,
  Sigma.p1 = diag(2),
  Sigma.p2 = 100 * diag(2),
  W = c(0.5, 0.5)
)

Value

y

The \(N\) simulated observations.

true.group

A vector of integers from \(1:k\) indicating the values of the latent variables \(Z_i\).

subgroups

A \(k \times N\) matrix where each row contains the index subgroup for the observations in the \(k\)-th group.

Arguments

N

The desired sample size.

k

The desired number of mixture components.

Mu

The input mean vector of length \(k\) for univariate Gaussian mixtures; the input \(k \times D\) matrix with the means' coordinates for multivariate Gaussian mixtures.

stdev

For univariate mixtures, the \(k \times 2\) matrix of input standard deviations, where the first column contains the parameters for subgroup 1, and the second column contains the parameters for subgroup 2.

Sigma.p1

The \(D \times D\) covariance matrix for the first subgroup. For multivariate mixtures only.

Sigma.p2

The \(D \times D\) covariance matrix for the second subgroup. For multivariate mixtures only.

W

The vector for the mixture weights of the two subgroups.

Details

The functions allows to simulate values from a double (nested) univariate Gaussian mixture:

$$ (Y_i|Z_i=j) \sim \sum_{s=1}^{2} p_{js}\, \mathcal{N}(\mu_{j}, \sigma^{2}_{js}), $$

or from a multivariate nested Gaussian mixture:

$$ (Y_i|Z_i=j) \sim \sum_{s=1}^{2} p_{js}\, \mathcal{N}_{D}(\bm{\mu}_{j}, \Sigma_{s}), $$

where \(\sigma^{2}_{js}\) is the variance for the group \(j\) and the subgroup \(s\) (stdev is the argument for specifying the k x 2 standard deviations for univariate mixtures); \(\Sigma_s\) is the covariance matrix for the subgroup \(s, s=1,2\), where the two matrices are specified by Sigma.p1 and Sigma.p2 respectively; \(\mu_j\) and \(\bm{\mu}_j, \ j=1,\ldots,k\) are the mean input vector and matrix respectively, specified by the argument Mu; W is a vector of dimension 2 for the subgroups weights.

Examples

# Bivariate mixture simulation with three components

N  <- 2000
k  <- 3
D <- 2
M1 <- c(-45,8)
M2 <- c(45,.1)
M3 <- c(100,8)
Mu <- rbind(M1,M2,M3)
Sigma.p1 <- diag(D)
Sigma.p2 <- 20*diag(D)
W   <- c(0.2,0.8)
sim <- piv_sim(N = N, k = k, Mu = Mu, Sigma.p1 = Sigma.p1,
Sigma.p2 = Sigma.p2, W = W)
graphics::plot(sim$y, xlab="y[,1]", ylab="y[,2]")