Time-indexed graphs

Discrete-time CDMs often share a time-invariant lag structure. Unroll that structure to a static DAG over occasions t = 1:T, then apply standard identification on the unrolled graph.

CausalDynamics.LaggedEdge — Type

LaggedEdge(parent, child, lag)

Directed edge from parent at occasion t - lag to child at occasion t.

lag == 0 is contemporaneous (same occasion). lag ≥ 1 is a lagged parent.

source

CausalDynamics.TemporalDAGSpec — Type

TemporalDAGSpec(variables, edges)

Time-invariant lag structure over named endogenous variables.

Arguments

variables: symbols appearing in the model (documentation / validation)
edges: vector of LaggedEdge or (parent, child, lag) tuples

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CausalDynamics.TemporalUnrolling — Type

TemporalUnrolling

Result of unrolling a TemporalDAGSpec for T occasions.

Fields

T: number of occasions
spec: source specification
graph: unrolled DiGraph
node_index: (variable, t) => node id
index_node: node id => (variable, t) (1-based indexing)

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CausalDynamics.unroll_temporal_dag — Function

unroll_temporal_dag(spec::TemporalDAGSpec, T::Int)

Unroll spec to a static DAG over occasions t = 1:T.

Edges (parent, child, lag) become parent[t-lag] → child[t] for each valid t.

source

CausalDynamics.temporal_node — Function

temporal_node(unrolling, variable, t)

Return the node index for variable at occasion t in unrolling.

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CausalDynamics.temporal_node_label — Function

temporal_node_label(unrolling, node)

Human-readable label "var[t]" for a node index in unrolling.

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CausalDynamics.d_separated_temporal — Function

d_separated_temporal(unrolling, treatment, t_treat, outcome, t_outcome, conditioned)

d_separated on the unrolled graph for (treatment, t_treat) and (outcome, t_outcome). conditioned is a vector of (variable, t) pairs.

source

CausalDynamics.temporal_backdoor_adjustment_set — Function

temporal_backdoor_adjustment_set(unrolling, treatment, t_treat, outcome, t_outcome)

Backdoor adjustment set for the effect of treatment at t_treat on outcome at t_outcome in the unrolled DAG. Returns a Set of node indices into unrolling.graph.

source

CausalDynamics.temporal_backdoor_adjustment_nodes — Function

temporal_backdoor_adjustment_nodes(unrolling, treatment, t_treat, outcome, t_outcome)

Like temporal_backdoor_adjustment_set but returns Set of (variable, t) pairs.

source

Plotting an unrolling

With DAGMakie.jl loaded, dagplot_temporal places occasions left→right and variables as rows (same grid as DAGMakie.dagplot_time_indexed):

using CausalDynamics, DAGMakie, CairoMakie

spec = TemporalDAGSpec(
    [:x, :y],
    [(:x, :x, 1), (:y, :y, 1), (:x, :y, 1)],
)
u = unroll_temporal_dag(spec, 3)

fig, ax, p = dagplot_temporal(u; figure_size = (520, 260))
fig

Confounded treatment (book Ch. 28)

using CausalDynamics

spec = TemporalDAGSpec(
    [:x, :y, :a, :c],
    [
        (:c, :c, 1), (:a, :c, 1), (:c, :a, 0),  # confounder dynamics + confounding
        (:x, :x, 1), (:a, :x, 1), (:c, :x, 1),  # state evolution
        (:x, :y, 0),                             # measurement
    ],
)
u = unroll_temporal_dag(spec, 10)

# Effect of A_{t-1} on X_t: adjust for C_{t-1}
adj = temporal_backdoor_adjustment_nodes(u, :a, 2, :x, 2)
# Set containing (:c, 1)

See also Utilities, Discrete-time CDMs, and the CDCS book Ch. 28.