rsdc_simulate: Simulate Multivariate Regime-Switching Data (TVTP)

Description

Simulates a multivariate time series from a regime-switching model with time-varying transition probabilities (TVTP) driven by covariates X. Transition probabilities are generated via a multinomial logistic (softmax) link; observations are drawn from regime-specific Gaussian distributions.

Usage

rsdc_simulate(n, X, beta, mu, sigma, N, seed = NULL)

Value

A list with:

states: Integer vector of length n; the simulated regime index at each time.
observations: Numeric matrix $n \times K$; the simulated observations.
transition_matrices: Array $N \times N \times n$; the transition matrix $P_t$ used at each time step (with $P_1$ undefined by construction; see Details).

Arguments

n: Integer. Number of time steps to simulate.
X: Numeric matrix $n \times p$ of covariates used to form the transition probabilities. Row X[t, ] corresponds to covariates available at time t. Only rows 1:(n-1) are used to transition from t-1 to t.
beta: Numeric array $N \times N \times p$. Softmax coefficients for the multinomial transition model; beta[i, j, ] parameterizes the transition from state $i$ to state $j$.
mu: Numeric matrix $N \times K$. Regime-specific mean vectors.
sigma: Numeric array $K \times K \times N$. Regime-specific covariance (here, correlation/variance) matrices; each $K \times K$ slice must be symmetric positive definite.
N: Integer. Number of regimes.
seed: Optional integer. If supplied, sets the RNG seed for reproducibility.

Details

Initial state and first draw: The initial regime $S_1$ is sampled uniformly; the first observation $y_1$ is drawn from $\mathcal{N}(\mu_{S_1}, \Sigma_{S_1})$.
TVTP via softmax: For $t \ge 2$, the row $i$ of $P_t$ is $$P_t(i, j) = \frac{\exp\!\big(X_{t-1}^\top \beta_{i,j}\big)} {\sum_{h=1}^N \exp\!\big(X_{t-1}^\top \beta_{i,h}\big)}\,,$$ computed with log-sum-exp stabilization.
Sampling: Given $S_{t-1}$, draw $S_t$ from the categorical distribution with probabilities $P_t(S_{t-1}, \cdot)$ and $y_t \sim \mathcal{N}(\mu_{S_t}, \Sigma_{S_t})$.

Examples

Run this code

set.seed(123)
n <- 200; K <- 3; N <- 2; p <- 2
X <- cbind(1, scale(seq_len(n)))

beta <- array(0, dim = c(N, N, p))
beta[1, 1, ] <- c(1.2,  0.0)
beta[2, 2, ] <- c(1.0, -0.1)

mu <- rbind(c(0, 0, 0),
            c(0, 0, 0))
rho <- rbind(c(0.10, 0.05, 0.00),
             c(0.60, 0.40, 0.30))
Sig <- array(0, dim = c(K, K, N))
for (m in 1:N) {
  R <- diag(K); R[lower.tri(R)] <- rho[m, ]; R[upper.tri(R)] <- t(R)[upper.tri(R)]
  Sig[, , m] <- R
}
sim <- rsdc_simulate(n = n, X = X, beta = beta, mu = mu, sigma = Sig, N = N, seed = 99)