MICH Algorithm — mich

Implementation of Algorithms 1 & 2 in Berlind, Cappello, and Madrid Padilla (2025). This algorithm takes a sequence of \(T\) observations \(\mathbf{y}_{1:T}\), and iteratively fits \(L\) mean-scp models, \(K\) var-scp models, and \(J\) meanvar-scp models resulting in a variational approximation to the posterior distribution of the \(L\) mean, \(K\) variance, and \(J\) mean and variance change-points. The algorithm terminates once the percentage increase in the ELBO falls below tol.

Usage

mich_cpp(
  y,
  J,
  L,
  K,
  mu_0,
  lambda_0,
  fit_intercept,
  fit_scale,
  refit,
  max_iter,
  verbose,
  tol,
  omega_j,
  u_j,
  v_j,
  log_pi_j,
  pi_bar_j,
  log_pi_bar_j,
  b_bar_j,
  omega_bar_j,
  u_bar_j,
  v_bar_j,
  lgamma_u_bar_j,
  digamma_u_bar_j,
  omega_l,
  log_pi_l,
  pi_bar_l,
  log_pi_bar_l,
  b_bar_l,
  omega_bar_l,
  u_k,
  v_k,
  log_pi_k,
  pi_bar_k,
  log_pi_bar_k,
  u_bar_k,
  v_bar_k,
  lgamma_u_bar_k,
  digamma_u_bar_k
)

Arguments

y: A numeric vector. Length \(T\) vector of observations.
L, K, J: Integers. The number of mean, variance, and mean-variance change-points to include in the model.
mu_0: A scalar. Intercept parameter initialization.
lambda_0: A scalar. Baseline scale parameter initialization.
fit_intercept: A logical. If fit_intercept == TRUE, then an intercept is estimated and mu_0 gets updated.
fit_scale: A logical. If fit_scale == TRUE, then an initial scale is estimated and lambda_0 gets updated.
refit: A logical. If refit == TRUE, then the MICH algorithm is initialized using the provided posterior parameters, otherwise the null model \(\mu_t = 0\), \(\lambda_t = 1\) and \(\bar{\pi}_{\ell t} = \bar{\pi}_{k t} = \bar{\pi}_{j t} = 1/T\) is used as the initialization.
max_iter: An integer. Maximum number of iterations. If ELBO does not converge before max_iter is reached, then converged == FALSE in the returned fit object.
verbose: A logical. If verbose == TRUE, then value of the ELBO is printed every 5000th iteration.
tol: A scalar. Convergence tolerance for relative increase in ELBO.
omega_j, u_j, v_j: Scalars. Prior precision, shape, and rate parameters for meanvar-scp components of model.
log_pi_j: A numeric matrix. A \(T \times J\) matrix of prior log change-point location probabilities for each of the \(J\) mean-variance change-points.
pi_bar_j, log_pi_bar_j: Numeric matrices. \(T \times J\) matrices of initialized posterior change-point location probabilities and their log evaluations for the \(J\) mean-variance change-points.
b_bar_j: A numeric matrix. A \(T \times J\) matrix of initialized posterior mean parameters of the \(J\) mean-variance change-points.
omega_bar_j: A numeric matrix. A \(T\times J\) matrix of initialized posterior precision parameters of the \(J\) mean-variance change-points.
u_bar_j, lgamma_u_bar_j, digamma_u_bar_j: Numeric vectors. Length \(T\) vectors of posterior shape parameters for the meanvar-scp model components and their log-gamma and digamma evaluations.
v_bar_j: A numeric matrix. A \(T \times J\) matrix of initialized posterior rate parameters for the \(J\) mean-variance change-points.
omega_l: A scalar. Prior precision parameter for mean-scp components of model.
log_pi_l: A numeric matrix. A \(T \times L\) matrix of prior log change-point location probabilities for each of the \(L\) mean change-points.
pi_bar_l, log_pi_bar_l: Numeric matrices. \(T \times L\) matrices of initialized posterior change-point location probabilities and their log evaluations for the \(L\) mean change-points.
b_bar_l: A numeric matrix. A \(T \times L\) matrix of initialized posterior mean parameters of the \(L\) mean change-points.
omega_bar_l: A numeric matrix. A \(T\times L\) matrix of initialized posterior precision parameters of the \(L\) mean change-points.
u_k, v_k: Scalars. Prior shape and rate parameters for var-scp components of model.
log_pi_k: A numeric matrix. A \(T \times K\) matrix of prior log change-point location probabilities for each of the \(K\) variance change-points.
pi_bar_k, log_pi_bar_k: Numeric matrices. \(T \times K\) matrices of initialized posterior change-point location probabilities and their log evaluations for the \(K\) variance change-points.
u_bar_k, lgamma_u_bar_k, digamma_u_bar_k: Numeric vectors. Length \(T\) vectors of posterior shape parameters for the var-scp model components and their log-gamma and digamma evaluations.
v_bar_k: A numeric matrix. A \(T \times K\) matrix of initialized posterior rate parameters for the \(K\) variance change-points.

Value

A List. Parameters of the variational approximation the MICH posterior distribution, including:

y: A numeric vector. Original data set.
residual: A numeric vector. Residual \(\mathbf{r}_{1:T}\) after subtracting out each \(E[\mu_{\ell t}]\) and \(E[\lambda_{j t} \mu_{j t}]/E[\lambda_{j t}]\) from \(\mathbf{y}_{1:T}\).
mu: A numeric vector. Posterior estimate of \(\Sigma_{\ell=1}^L E[\mu_{\ell,1:T}|\mathbf{y}_{1:T}] + \Sigma_{j=1}^J E[\mu_{j,1:T}|\mathbf{y}_{1:T}]\).
lambda: A numeric vector. Posterior estimate of \(\Pi_{\\k=1}^K E[\lambda_{k,1:T}|\mathbf{y}_{1:T}] \times \Pi_{j=1}^J E[\lambda_{j,1:T}|\mathbf{y}_{1:T}]\).
delta: A numeric vector. Posterior variance correction term (see (B.6) or Berlind, Cappello, and Madrid Padilla (2025)).
converged: A logical. Indicates whether relative increase in the ELBO is less than tol.
elbo: A numeric vector. Value of the ELBO after each iteration.
mu_0: A scalar. Estimate of the intercept.
lambda_0: A scalar. Estimate of the initial precision.
J, L, K: Integers. number of mean, variance, and mean-variance components.
J_model: A list. A list of the posterior parameters for each of the \(J\) meanvar-scp components (only included in J > 0).
L_model: A list. A list of the posterior parameters for each of the \(L\) mean-scp components (only included in L > 0).
K_model: A list. A list of the posterior parameters for each of the \(K\) var-scp components (only included in K > 0).