Penalised ALS time-varying coefficient estimate

Computes a fast point estimate of a time-varying coefficient vector $\beta_t$ for a longitudinal network model by solving the first-difference-penalised least-squares problem. No MCMC, no multiplicative effects, no random effects – purely a regression point estimate with a smoothing penalty on the coefficient path. Useful for rapid exploration and for generating starting values for a full lame(dynamic_beta = ...) fit.

Usage

als_dynamic_beta(Y, Xdyad, lambda = 0, intercept = TRUE)

Arguments

Y: A list of $T$ response matrices (or a 3-D array with third dimension time).
Xdyad: A list of $T$ dyadic covariate arrays (each $n \times n \times p$, or a single $n \times n$ matrix if $p=1$).
lambda: Non-negative smoothing parameter. Default 0 (no penalty, per-period OLS).
intercept: Logical; include an intercept (default TRUE).

Value

A list with beta (a $p \times T$ matrix of estimates), lambda, residual_ss (sum of squared residuals across periods), intercept, and call. Class "als_dynamic_beta".

Details

$$\min_{\beta_{1:T}} \sum_t \|y_t - X_t \beta_t\|^2 + \lambda \sum_{t=2}^{T} \|\beta_t - \beta_{t-1}\|^2$$

At $\lambda = 0$ this is $T$ independent per-period OLS fits. As $\lambda \to \infty$ the solution converges to a single pooled $\beta$ (constant over time).

Relation to other entry points

als_dynamic_beta is a regression-only smoother; the returned object carries time-varying $\beta_t$ alone, no additive ($a, b$) or multiplicative ($U, V$) AME components. It is therefore not a special case of lame_als (which always estimates the full AME decomposition) and not a special case of lame(dynamic_beta = TRUE) (which is Bayesian with an AR(1) / RW1 / RW2 / Matern 3/2 state-space prior). Use it when you want a fast, point-only, deterministic smoother for the coefficient path; use lame(dynamic_beta = ...) when you need full posterior uncertainty and additive / multiplicative effects.

Examples

set.seed(1)
n <- 10; Tn <- 4
X <- replicate(Tn, array(rnorm(n*n*2), c(n, n, 2)), simplify = FALSE)
beta_true <- rbind(seq(-1, 1, length.out = Tn), seq(0.5, -0.5, length.out = Tn))
Y <- vector("list", Tn)
for (t in seq_len(Tn)) {
  Yt <- X[[t]][, , 1] * beta_true[1, t] + X[[t]][, , 2] * beta_true[2, t] +
        matrix(rnorm(n*n, 0, 0.2), n, n)
  diag(Yt) <- NA
  Y[[t]] <- Yt
}
fit_als <- als_dynamic_beta(Y, X, lambda = 0)     # per-period LS
fit_smooth <- als_dynamic_beta(Y, X, lambda = 10) # smoother path