What Are Dynamic Effects?
The Static Baseline
In the standard AME model for longitudinal data, the tie between actors \(i\) and \(j\) at time \(t\) is:
\[y_{ij,t} = \beta'x_{ij,t} + a_i + b_j + u_i'v_j + \epsilon_{ij,t}\]
The covariates (\(x_{ij,t}\)) can vary over time, but the sender effect \(a_i\), the receiver effect \(b_j\), and the latent positions \(u_i\) and \(v_j\) are constant: the model assumes a country’s position in the latent “sanctioning space” is the same in 1993 as in 2000.
Dynamic Latent Positions (dynamic_uv = TRUE)
When you set dynamic_uv = TRUE, each actor’s latent
position evolves over time according to an AR(1) process:
\[U_{i,k,t} = \rho_{uv} \, U_{i,k,t-1} + \epsilon_{i,k,t}, \quad \epsilon_{i,k,t} \sim N(0, \sigma_{uv}^2)\] \[V_{j,k,t} = \rho_{uv} \, V_{j,k,t-1} + \eta_{j,k,t}, \quad \eta_{j,k,t} \sim N(0, \sigma_{uv}^2)\]
Note the innovation parameterisation: \(\sigma_{uv}^2\) is the variance of the
period-to-period innovation, not the marginal/stationary variance of
\(U_{i,k,t}\) (which is \(\sigma_{uv}^2 / (1 - \rho_{uv}^2)\)), and
fit$sigma_uv holds posterior draws of the innovation
standard deviation. The autoregressive parameter \(\rho_{uv}\), estimated from the data,
controls persistence: near 1 means positions change slowly, near 0 means
they are essentially re-drawn each period. Use this when the underlying
community structure is evolving – alliances shift, trading blocs
realign.
Dynamic Additive Effects (dynamic_ab = TRUE)
When you set dynamic_ab = TRUE, the sender and receiver
effects evolve the same way:
\[a_{i,t} = \rho_{ab} \, a_{i,t-1} + \epsilon_{i,t}, \quad \epsilon_{i,t} \sim N(0, \sigma_{ab}^2)\] \[b_{j,t} = \rho_{ab} \, b_{j,t-1} + \eta_{j,t}, \quad \eta_{j,t} \sim N(0, \sigma_{ab}^2)\]
The same innovation parameterisation applies:
fit$sigma_ab holds draws of the innovation standard
deviation, and the implied stationary variance \(\sigma_{ab}^2 / (1 - \rho_{ab}^2)\) doubles
as the prior on the initial state \(a_{i,1}\). Substantively,
dynamic_ab captures changes in how much an actor
participates, not in who they connect with (that is
dynamic_uv’s territory). The two can be used alone or
together; dynamic_ab is often the place to start, since
changes in overall activity are common and relatively easy to
estimate.
Dynamic Regression Coefficients (dynamic_beta)
The third option lets the regression coefficients themselves evolve.
With dynamic_beta = TRUE (or a subset like
dynamic_beta = "dyad"), each selected coefficient follows
an AR(1):
\[\beta_{k,t} = \rho_{\beta,b(k)} \, \beta_{k,t-1} + \epsilon_{k,t}, \quad \epsilon_{k,t} \sim N(0, \sigma_{\beta,b(k)}^2 \Lambda_{kk})\]
where \(b(k)\) is the block of
coefficient \(k\) (intercept / dyad /
row / col), so coefficients in a block share \(\rho\) and \(\sigma\), while the design-based scale
\(\Lambda_{kk}\) (a \(g\)-prior analogue) keeps each innovation
variance commensurate with the data. The whole time path of \(\beta\) is drawn jointly via forward-filter
/ backward-sample (FFBS; Carter & Kohn 1994). One storage difference
from dynamic_uv / dynamic_ab:
fit$sigma_beta and fit$rho_beta hold
last-iteration scalars, while the full posterior draws live in
fit$SIGMA_BETA and fit$RHO_BETA – summarise
those (e.g. colMeans(fit$SIGMA_BETA)).
dynamic_beta accepts FALSE (default),
TRUE (all coefficients), block shortcuts
("intercept", "dyad", "row",
"col") or coefficient names, integer column indices, or a
length-p logical mask; see ?lame. When the
intercept (or a nodal coefficient) is dynamic, an automatic sum-to-zero
constraint on a (and b) keeps everything
jointly identified. dynamic_beta works for every family,
unipartite and bipartite, directed and symmetric, and combines with
dynamic_ab / dynamic_uv.
A worked dynamic_beta example
Simulate a small longitudinal network where a single dyadic
covariate’s effect grows linearly over time, and fit with
dynamic_beta = "dyad".
library(lame)
set.seed(2026)
n_db <- 15
T_db <- 5
beta_t_true <- seq(-0.5, 1.0, length.out = T_db) # truth: rising effect
X_db <- replicate(T_db, matrix(rnorm(n_db * n_db), n_db, n_db), simplify = FALSE)
Y_db <- vector("list", T_db)
for (t in seq_len(T_db)) {
Yt <- beta_t_true[t] * X_db[[t]] + matrix(rnorm(n_db * n_db, 0, 0.4), n_db, n_db)
diag(Yt) <- NA
rownames(Yt) <- colnames(Yt) <- paste0("a", seq_len(n_db))
Y_db[[t]] <- Yt
}
names(Y_db) <- paste0("t", seq_len(T_db))
X_db_arr <- lapply(X_db, function(x) array(x, c(n_db, n_db, 1)))
fit_db <- lame(Y_db, Xdyad = X_db_arr,
family = "normal", R = 0,
nscan = 200, burn = 50, odens = 5,
dynamic_beta = "dyad", verbose = FALSE)
# fit$BETA is a 3-D array [n_stored x p x T]
dim(fit_db$BETA)
#> [1] 40 2 5
# coef() returns a p x T matrix of posterior means
coef(fit_db)
#> t1 t2 t3 t4 t5
#> intercept -0.00810071 -0.00810071 -0.00810071 -0.00810071 -0.00810071
#> X1_dyad -0.48797080 -0.12866840 0.23813546 0.63432234 0.99248965
# confint() returns one row per coef[t] with 95% credible interval
head(confint(fit_db), 8)
#> 2.5% 97.5%
#> intercept[t1] -0.03237708 0.01673393
#> X1_dyad[t1] -0.52464259 -0.44517349
#> intercept[t2] -0.03237708 0.01673393
#> X1_dyad[t2] -0.16936531 -0.08189014
#> intercept[t3] -0.03237708 0.01673393
#> X1_dyad[t3] 0.20885246 0.27956162
#> intercept[t4] -0.03237708 0.01673393
#> X1_dyad[t4] 0.58070472 0.68104416Migration warning for amen /
static-lame users. With dynamic_beta
active, fit$BETA becomes a 3-D
[n_stored, p, T] array rather than the 2-D
[n_stored, p] matrix. The amen-era idiom
apply(fit$BETA, 2, mean) still returns a
length-p vector – but it now silently averages across both
iterations and time, hiding the time variation you fit
the model to recover. Use coef(fit) instead: it returns a
[p, T] matrix for dynamic fits and a length-p
vector for static ones.
The per-period posterior mean for X1_dyad should track
the linear ramp
beta_t_true = c(-0.5, -0.125, 0.25, 0.625, 1.0), with the
AR(1) prior providing mild shrinkage relative to fitting a separate
ame() per period.
Choosing a prior family: dynamic_beta_kind
lame() exposes dynamic_beta_kind for
selecting the prior family on the time-varying coefficients:
Mean-reverting coefficient? Use the default
dynamic_beta_kind = "ar1". Permanent / unit-root drift (no mean reversion)? Use"rw1". Smooth with curvature (second-order smoothness)? Use"rw2". Smooth with a known length-scale of variation? Use"matern32"(set the scale viaprior = list(matern32_length_scale = ...); defaults tomax(2, T/4)). If the stationarity warning fires after fitting AR(1) – the posterior on \(\rho_\beta\) concentrated at or above 0.97 – refit with"rw1"(or"rw2"if acceleration matters).
"random_walk" is an alias for "rw1"; the
"rw2" and "matern32" samplers use a slower
R-level joint Gaussian update rather than the C++ FFBS. Refitting
fit_db under each kind and comparing the recovered
X1_dyad path against the truth:
kinds <- c("rw1", "rw2", "matern32")
paths <- sapply(kinds, function(k) {
fit_k <- lame(Y_db, Xdyad = X_db_arr, family = "normal", R = 0,
nscan = 200, burn = 50, odens = 5,
dynamic_beta = "dyad", dynamic_beta_kind = k,
prior = if (k == "matern32")
list(matern32_length_scale = 2) else NULL,
verbose = FALSE)
coef(fit_k)["X1_dyad", ]
})
round(rbind(ar1 = coef(fit_db)["X1_dyad", ], t(paths),
truth = beta_t_true), 3)
#> t1 t2 t3 t4 t5
#> ar1 -0.488 -0.129 0.238 0.634 0.992
#> rw1 -0.488 -0.126 0.230 0.638 0.988
#> rw2 -0.482 -0.123 0.224 0.624 0.999
#> matern32 -0.476 -0.124 0.223 0.627 0.989
#> truth -0.500 -0.125 0.250 0.625 1.000All four kinds recover essentially the same ramp – the point to take away, not a tie-breaker: on a smooth, gap-free trend the priors are nearly indistinguishable in-sample, and they diverge only when the data match their assumption. Choose with the decision tree above and the post-fit stationarity warning, not by eyeballing this table.
When time_index is supplied with unequal gaps, the AR(1)
/ RW1 conditional variance is scaled by the gap (suppressing \(\Lambda_{kk}\) for readability): \[q_t = \sigma_\beta^2 \cdot \frac{1 -
\rho_\beta^{2\Delta_t}}{1 - \rho_\beta^2}\] so
quarterly-then-annual data gets the correct prior; "rw2"
and "matern32" support unequal gaps through their joint
precision-matrix construction.
# observations at t = 1, 2, 4, 8 (a doubling gap structure)
fit_gaps <- lame(Y_db, Xdyad = X_db_arr, family = "normal", R = 0,
nscan = 200, burn = 50, odens = 5,
dynamic_beta = "dyad",
time_index = c(1, 2, 4, 8, 16)[seq_len(T_db)],
verbose = FALSE)
fit_gaps$time_index
#> [1] 1 2 4 8 16
round(coef(fit_gaps), 3)
#> t1 t2 t3 t4 t5
#> intercept -0.011 -0.011 -0.011 -0.011 -0.011
#> X1_dyad -0.485 -0.124 0.225 0.625 0.997Hierarchical pooling: dynamic_beta_pool
When two or more coefficient blocks are dynamic, each block has its
own AR(1) hyperparameters by default, and short panels (\(T \le 5\)) can leave them badly
under-identified. dynamic_beta_pool shrinks block-level
hyperparameters toward a shared mean: "rho" for common
persistence, "sigma" for a common innovation scale,
"both" for full pooling –
e.g. lame(..., dynamic_beta = c("intercept", "dyad"), dynamic_beta_pool = "both").
With a single dynamic block the hierarchical layer is skipped entirely
and the fit is identical to the default "none". See
?lame for the diagnostic comparison of pooled vs
independent hyperparameter posteriors.
Per-actor time-varying slopes:
dynamic_beta_per_actor
For questions about whether one actor responds to a covariate
differently than another – and whether that gap evolves – set
dynamic_beta_per_actor = "row" (or "col") and
pick the covariate via per_actor_covariate_idx,
e.g. lame(..., dynamic_beta_per_actor = "row", per_actor_covariate_idx = 1L).
Deviations are sum-to-zero centred at every period, so the population
coefficient stays in coef(fit) while the per-actor
deviations land in fit$THETA_ACTOR (and their posterior
means in fit$theta_actor_mean, labelled by actor and
period). See ?per_actor_slopes.
Diagnosing your dynamic fit
summary(fit) prints the Dynamic coefficients per
period block and the per-block posterior-mean
rho_beta, and warns when a block’s posterior on \(\rho_\beta\) is concentrated at or above
0.97 – the signal to refit with
dynamic_beta_kind = "rw1".
detect_change_point(fit, threshold_bf = 5) is a
heuristic regime-switch diagnostic: it scales each coefficient’s largest
one-period jump by \(\sigma_\beta\) and
compares it to a prior-null reference, reporting a bounded tail-ratio
score in the bf column (not a formal Bayes factor). On the
smooth linear ramp above there is no break, so we expect a score of
0:
detect_change_point(fit_db, threshold_bf = 5)
#> coef bf m_post_mean m_prior_q95 t_hat warn
#> 1 X1_dyad 0 0.7297556 2.767291 4 FALSERead bf below 3 as weak, 3 to 10 as worth inspecting,
and above 10 as a jump the AR(1) prior struggles to generate;
t_hat is a prompt for visual inspection, not a formal
break-time estimate. The diagnostic is deliberately conservative (only
stark breaks trip it reliably at short \(T\)); see
?detect_change_point. For prior elicitation before
fitting, dynamic_beta_prior_summary(T = 10, kind = "ar1")
simulates prior paths so you can check the implied roughness and range
for your T.
Finally, the Gibbs analog of HMC divergences: every fit exposes
per-block Metropolis failure counts as fit$mh_counters, and
a warning fires automatically when any block exceeds a 10% failure rate.
Treat anything above 5% as suspicious.
Temporal-trend posterior-predictive check:
gof_temporal()
Where detect_change_point() targets
coefficients,
gof_temporal(fit, stat = "density", n_rep = 200) targets
the network statistic itself: it fits a linear time trend to a
chosen statistic (density / mean / reciprocity / transitivity), computes
the same slope on simulate(fit) replicates, and returns a
two-sided posterior-predictive p-value; small p_pp (below
0.05) means the observed trend is incompatible with the fitted model.
Reach for it when the substantive statistic drifts across the panel; the
informative worked example lives in the overview vignette; see
?gof_temporal.
Inspecting the priors actually in effect:
prior_summary()
prior_summary(fit) prints the hyperparameters actually
used, defaults filled in – for a dynamic fit that includes the per-block
AR(1) priors alongside the variance components and the g-prior on
beta. Run it once per fit to catch typos in a
prior = list(...) override; see ?prior_summary
and the cross-sectional vignette for
the general tour.
Multi-chain \(\hat R\) via
lame_parallel() and rhat_dynamic_beta()
A single chain only ever buys you within-chain split-\(\hat R\). For a between-chain diagnostic on
the per-period coefficient path, fit several chains with
lame_parallel(..., n_chains = 4, combine_method = "list")
and call rhat_dynamic_beta() on the resulting list. Require
both rhat_mvt < 1.01 (the multivariate statistic catches
chains that agree per period but disagree on the trajectory) and
rhat_max_univariate < 1.01 before quoting a path; short
vignette-length chains typically fail this bar, so use production
lengths. combine_ame_chains() pools chains and routes
through posterior::summarise_draws() for per-(coefficient,
period) ESS. ame_parallel() is the cross-sectional twin;
see ?rhat_dynamic_beta, ?lame_parallel, and
the cross-sectional vignette.
Forecasting
When at least one dynamic component is active,
predict(fit, h = K) propagates the state-space model
forward \(K\) periods, sampling \((\rho_\beta, \sigma_\beta)\) from the
posterior so the forecast carries hyperparameter uncertainty. The forecasting vignette is the full worked
reference (forecast scales, per-draw arrays, credible intervals,
counterfactual covariates, exposure offsets, near-unit-root warning).
One kind-specific note lives here: under "rw2" /
"matern32" the forward recursion pins \(\rho_\beta\) at 1, so forecast variance
grows linearly in \(h\) as under RW1 –
the smoothing structure shapes the in-sample path only.
Visualising the coefficient path: autoplot.lame
For a fit with dynamic_beta active,
autoplot(fit) returns a ggplot ribbon plot per
coefficient (posterior median line, 95% interval, faceted by
coefficient). Under dynamic_beta = "dyad" the intercept
remains static, so pass coefs = to restrict the plot to the
genuinely time-varying coefficients:
library(ggplot2)
autoplot(fit_db, coefs = "X1_dyad", probs = c(0.025, 0.5, 0.975)) +
labs(title = "Posterior coefficient path",
subtitle = "Median line, 95% credible interval ribbon",
x = "Time period", y = "Coefficient value")
LOO / WAIC via save_log_lik = TRUE
Pass save_log_lik = TRUE and lame attaches
a [n_stored, n_obs] pointwise log-likelihood matrix to
fit$log_lik, which registered S3 methods hand directly to
loo::loo(fit) and loo::waic(fit):
fit_ll <- lame(Y_db, Xdyad = X_db_arr, family = "normal", R = 0,
nscan = 2000, burn = 500, odens = 25,
dynamic_beta = "dyad",
save_log_lik = TRUE,
verbose = FALSE)
dim(fit_ll$log_lik) # [n_stored, n_observations]
loo_db <- loo::loo(fit_ll)
print(loo_db)The code is left unevaluated because a useful PSIS-LOO calculation
needs more draws than a vignette should generate. Run it with a
converged fit before using elpd_loo. If the matrix would
not fit in RAM, save_log_lik = "chunked" streams it to disk
(see the forecasting vignette).
On the log-likelihood scale: for normal,
binary, cbin, poisson, and
ordinal the stored pointwise log-likelihood is the
exact family-specific Y density on the response scale,
so elpd_loo is directly comparable to a Stan / brms
loo() on the same family and data (modulo satisfactory
\(\hat R\) / ESS). For the rank
likelihood frn, the exact marginal needs GHK Monte Carlo
(log_lik_method = "observed_ghk", longitudinal path only);
the default falls back to the augmented-Z normal density with a one-time
warning. Inspect fit$log_lik_method for the branch
used.
Prior Specifications
The AR(1) coefficients and innovation variances have sensible default priors:
- \(\rho_{uv} \sim \text{TruncNormal}(0.9, 0.1, 0, 1)\), centered on high persistence
- \(\rho_{ab} \sim \text{TruncNormal}(0.8, 0.15, 0, 1)\), slightly less persistent
- \(\rho_{\beta} \sim \text{TruncNormal}(0.8, 0.15, 0, 0.999)\)
- \(\sigma_{uv}^2, \sigma_{ab}^2, \sigma_{\beta}^2 \sim \text{InverseGamma}(2, 1)\)
All can be customized via the prior argument
(e.g. prior = list(rho_beta_mean = ..., sigma_beta_scale = ...))
if you have strong beliefs about the rate of change.
