Predictive simulations and intervals

Draws parameter samples using the sandwich variance V_theta, simulates predictive counts with zero-inflation and returns per-cell prediction intervals. Two types of intervals are available: conditional (latent variables drawn from their variational posterior, after one VE step per draw) and marginal (latent variables drawn from their prior).

Usage

Predictions(
  Y,
  X,
  fit,
  MC,
  type = c("conditional", "marginal"),
  probs = c(0, 0.9),
  output = "intervals",
  cl = NULL,
  verbose = TRUE
)

Arguments

Y

An n x p count matrix (NAs allowed).

X

An (n*p) x d design matrix (stacked by blocks of size n).

fit

Estimator output list (from Miss.ZIPLNPCA).

MC

Integer, number of Monte Carlo draws.

type

"conditional" (default) or "marginal".

conditional

for each parameter draw, a VE step gives \(M\), \(S\) and \(\xi\); then \(U \sim \mathcal{B}(\xi)\) and \(Z_i \sim \mathcal{N}(X\beta + M_i C^\top, C\,\mathrm{diag}(S_i)\,C^\top)\).

marginal

no VE step; \(U \sim \mathcal{B}(\mathrm{logit}^{-1}(X\gamma))\) and \(Z_i \sim \mathcal{N}(X\beta, C C^\top)\).

In both cases \(U = 1\) means presence and the prediction is \(U \times \mathcal{P}(\exp(Z))\).

probs

Numeric vector of length 2, the lower and upper quantile levels of the interval. Default c(0, 0.9).

output

Character vector, what to return besides the intervals. One or several of "intervals" (always returned), "draws", "latent", "params", or "all".

cl

Parallelisation of the VE steps (conditional type only), passed to pblapply. NULL (default): sequential. An integer: number of forked processes (not available on Windows). A cluster from makeCluster: works on all platforms. All random draws are made in the main process, so results for a given seed do not depend on cl.

verbose

Logical. If TRUE (default), prints the current step and shows progress bars for the VE steps and the simulation of the latent variables. Progress bars can also be controlled globally with pboptions.

Value

A list with elements:

• lower, upper: n x p matrices of predictive quantiles at levels probs.

• draws (if requested): list with count (Poisson draws) and presence (Bernoulli draws \(U\)), arrays n x p x MC. The prediction is count * presence.

• latent (if requested): list with Z, array n x p x MC of Gaussian latent draws, and for the conditional type M, S (arrays n x q x MC) and xi (array n x p x MC) from the VE steps.

• params (if requested): list with gamma, beta (matrices MC x d) and C (matrix MC x (p*q), column-major vectorisation of C).

• settings: list with type, probs and MC.

See also

V_theta, Miss.ZIPLNPCA_VE

Examples

set.seed(1)
n <- 30; p <- 5; d <- 2; q <- 2
X <- cbind(1, rnorm(n * p))                       # (n*p) x d design
theta <- list(B = matrix(c(1, 0.5)), D = matrix(c(1, -0.5)),
              C = matrix(rnorm(p * q, sd = 0.5), p, q))
Y <- Simul(X, theta, list(n = n, p = p, d = d, q = q))$Y
Y[sample(n * p, 20)] <- NA                        # missing entries
fit <- Miss.ZIPLNPCA(Y, X, q)

# \donttest{
# Conditional intervals (default), 90% upper bound
pi_cond <- Predictions(Y, X, fit, MC = 20)
#> Computing the sandwich variance...
#> Running 20 VE steps:
#> Simulating the latent variables:
#> Simulating the counts and computing the intervals...
pi_cond$upper[1:3, ]
#>       [,1] [,2] [,3] [,4] [,5]
#> [1,] 103.4  8.9 63.4  0.1  1.0
#> [2,]   6.7  7.1  7.3 34.6  4.2
#> [3,]   7.0  1.0  8.1  0.0  6.3

# Marginal intervals at other levels, with the draws
pi_marg <- Predictions(Y, X, fit, MC = 20, type = "marginal",
                       probs = c(0.05, 0.95), output = "draws")
#> Computing the sandwich variance...
#> Simulating the latent variables:
#> Simulating the counts and computing the intervals...
dim(pi_marg$draws$count)
#> [1] 30  5 20
# }