SCIMs: Structural Causal Influence Models

• SCIMs are a class of models that combine structural equation processes, vector autoregressive methods, and graphical criteria to analyze causal effects in multivariate time series.
• They leverage process graphs and frequency-domain transfer functions to provide rigorous analysis of direct and indirect causal influences in dynamic systems.
• SCI metrics within SCIMs quantify sub-community causal impacts, offering actionable insights for intervention planning in applied fields like epidemiology.

Structural Causal Influence Models (SCIMs) are a class of models that formalize causal inference in multivariate time series by synthesizing structural equation models (SEMs), vector autoregressive (SVAR) processes, and modern graphical criteria. They provide process-level representations of dynamic systems, allowing rigorous analysis of direct and indirect causal effects both in the time and frequency domains. SCIMs are also foundational to contemporary approaches to structural causal influence (SCI) metrics, directly connecting graphical modeling with applied metrics for quantifying sub-community-level effects in epidemiological and other systems.

1. Structural Equation Process (SEP) Representation

At the core of SCIMs is the Structural Equation Process (SEP) representation, formulated for linear, stationary, Gaussian SVAR processes possibly containing latent components. Let $X(t)=(X_1(t), \ldots, X_n(t))^\top$ be a multivariate, zero-mean, real-valued process of order $p$, driven by independent white noise $\eta(t)$ with diagonal covariance $W$. Allowing $d$ latent processes $L(t)$, the full process vector is $X_{\text{total}} = (X_{\text{obs}}, L)$. For observed node $i \in O$,

$$X_i(t) = \sum_{j=1}^n \sum_{\tau=1}^p \phi_{ji}(\tau) X_j(t-\tau) + \sum_{h=1}^d \sum_{\tau=1}^p \phi_{h,i}(\tau) L_h(t-\tau) + \eta_i(t)$$

Equivalent matrix-filter formulations introduce SVAR coefficients $\Phi(k) \in \mathbb{R}^{(n+d)\times(n+d)}$ at each lag $k$ and partition convolutional filters $\Lambda(s)$ for observed variables, $\Gamma(s)$ for latent influences, and internal dynamics $X_O^I$, yielding

$$X_O = \Lambda * X_O + \Gamma * X_L + X_O^I = (\sum_{k=0}^\infty \Lambda^{*k}) * X_O^{LI}$$

where $*$ is discrete convolution and $X_O^{LI} \coloneqq X_O^I + \Gamma * X_L$. Stability ($\|\Lambda\|_1 < 1$) guarantees absolute summability of all filters (Reiter et al., 2023).

2. Process Graph Structure and Adjacency

The process graph, a finite abstraction, encodes causal relations at the process level. The full time-series graph $\mathcal{G}$ is an infinite DAG with nodes $V_i(t)$ and edges $V_j(t-k) \rightarrow V_i(t)$ whenever $\phi_{ji}(k) \neq 0$. The process graph $G$ compresses this by "forgetting" lags, yielding vertices $O \cup L$ with directed edges $U \rightarrow V$ if some $\phi_{U,V}(k) \neq 0$, and bidirected $U \leftrightarrow V$ if $U$ and $V$ are confounded via a latent trek through $L$.

Important adjacency measures:

• Lagged-adjacency filters $\Lambda(s)$
• Instantaneous adjacency $A = \Lambda(0)$
• Temporal-summed adjacency $A^+ = \sum_{k=1}^p \Phi(k)$

These structures collectively define the mixed graph $G' = (O, D, B)$ capturing both direct effects and latent confounding (Reiter et al., 2023).

3. Generalized Trek and Path Rules

SCIMs generalize classical trek and path rules to dynamic, process-level settings. For the auto-covariance sequence $C_{X_O}(h) = \text{Cov}[X_O(t), X_O(t+h)]$, all treks $\pi$ from $i$ to $j$ in $G'$ (including shared-source and bidirected root types) contribute via associated "trek-monomial filters" $C^{(\pi)}(h)$. The total covariance is then: $$C_{i,j}(h) = \sum_{\pi \in \mathcal{T}(i, j)} C^{(\pi)}(h)$$ For total causal effects, directed path filters $\Lambda^{(\pi)}$ are summed to yield the unconditioned causal-effect filter: $$H_{ij}(s) = \sum_{\pi \in P(i, j)} \Lambda^{(\pi)}(s)$$ In the Z-domain, this generalizes to $(\Lambda^\infty)_{ij} = (I - \Lambda)^{-1}_{ij}$ by path-expansion. Under interventions, edges into $X \cup Z$ are deleted, and only $Z$-avoiding paths define the controlled effect $H_{i \to j | do(Z) = 0}(s)$ (Reiter et al., 2023).

4. Frequency-Domain Analysis and Transfer Functions

Fourier-transforming the filters enables compact spectral representations and makes causal effects directly computable in the frequency domain: $$H(\omega) = \mathcal{F}\{\Lambda\}(\omega), \quad J(\omega) = \mathcal{F}\{\Gamma\}(\omega)$$ The observed spectral density satisfies

$$S_O(\omega) = (I - H(\omega))^{-T} S_O^{LI}(\omega) (I - H(\omega))^{-*}$$

For an unconfounded causal link $j \to i$, the cross-spectrum admits the direct factorization $C_{i,j}(\omega) = H_{ij}(\omega) S_{jj}(\omega)$. Causal-effect transfer functions in the presence of conditioning sets $Z$ are given as sums over path-products: $$H_{i \to j \mid do(Z)=0}(\omega) = \sum_{\pi \in P_{Z}(i, j)} \prod_{(u \to v) \in \pi} H_{uv}(\omega)$$ Algebraic analogues of the front-door and back-door criteria provide frequency-by-frequency recovery of causal transfer functions from spectral densities, e.g.,

$$H_{X \to W}(\omega) = S_{W,X}(\omega)/S_{X,X}(\omega)$$

(Reiter et al., 2023).

5. Graphical Identification, Separation Criteria, and Causal Discovery

SCIMs leverage and generalize d-separation and trek-separation to process graphs. D-separation at the process level yields rank or determinantal constraints on blocks of $S_O(\omega)$; trek-separation extends this to mixed graphs: the absence of a trek between $i$ and $j$ (avoiding a conditioning set $K$) implies vanishing minors in $S_O(\omega)$ for all frequencies. Algebraic front-door and back-door rules apply as in traditional SEMs, but operate globally in the frequency domain. Identifiability thus reduces to testing for rank-deficiency and spectral zeros, enabling direct graphical discovery of the mixed process graph $G'$ from observed time series.

Identifiability theory from SEMs (e.g., front-door, half-trek, instrumental variable criteria) transfers robustly to SEPs, facilitating extension to latent projection models and mixed observational/latent process graphs (Reiter et al., 2023).

6. Structural Causal Influence (SCI) Metrics in Applied Systems

The structural causal influence (SCI) metric operationalizes process-level causal inference in multi-unit dynamical systems such as epidemiological models. Consider communities $i=1, \ldots, m$ modeled by state variables $V(t)$ and deterministic structural equations, with epidemic transmission encoded in matrices $F$ (infection terms) and $V$ (transition terms). The basic reproduction number under status quo and under perfect intervention $\mathrm{do}(\Gamma_i)$ are, respectively,

$$\mathcal{R}_0^{\text{SD}} = \rho(F V^{-1}), \quad \mathcal{R}_0^{\mathrm{do}(\Gamma_i)} = \rho(F^{(-i)} V^{(-i)-1})$$

Defining the SCI metric: $$\mathcal{C}_i = \frac{\mathcal{R}_0^{\text{SD}} - \mathcal{R}_0^{\mathrm{do}(\Gamma_i)}}{\mathcal{R}_0^{\text{SD}}}$$ This quantifies the relative reduction in system-level $\mathcal{R}_0$ due to shielding community $i$—directly measuring the community's disproportionate causal influence. Key properties include scale-invariance, range $(-1, 1]$, additivity ($m = 2 \implies \mathcal{C}_1 + \mathcal{C}_2 = 1$), and sensitivity to transmission-affecting parameters (Surasinghe et al., 2024).

The metric applies both to ODE compartmental models (via next-generation matrix methods) and agent-based models (via simulation and exponential-growth fits to infer $\mathcal{R}_0$ and thus $\mathcal{C}_i$). Explicit pseudocode and estimation procedures are provided in (Surasinghe et al., 2024).

7. Applications, Assumptions, and Limitations

SCIMs are particularly advantageous where the researcher seeks a reduced representation of causal structure (fewer nodes, “no need to know exact lags”), graphical constraints on observed spectra, and direct interpretability of frequency-domain causal effects. Domains of application include—but are not limited to—climate teleconnections, neuroscience oscillations, econometrics, and computational epidemiology. In epidemiological modeling, SCI metrics allow quantification of how disparities among sub-groups (e.g., access to therapy, exposure differences) induce vulnerability at the population level, capturing the impact of social inequalities (Surasinghe et al., 2024).

Core assumptions underpinning SCIMs include:

• Linearity and Gaussianity
• Stationarity and stability (spectral radius outside unit circle)
• Process-level mixed-graph faithfulness

A plausible implication is that in multi-community models with cross-unit transmission, minimal between-group mixing can transfer vulnerability across communities even when some subpopulations would have subcritical epidemic dynamics in isolation. This highlights the system-level consequences of heterogeneity and the necessity of causal process models for robust intervention planning.

Table: Summary of Key Components in SCIMs

Component	Description	Reference
SEP Representation	Vector autoregressive with process-level SCM	(Reiter et al., 2023)
Process Graph $G'$	Mixed graph encoding direct and latent effects	(Reiter et al., 2023)
Frequency-Domain Transfer	Fourier domain causal effect filters	(Reiter et al., 2023)
Generalized Trek/Path Rule	Covariance and total effect decomposition	(Reiter et al., 2023)
SCI Metric	Proportional system-level causal influence	(Surasinghe et al., 2024)

SCIMs unify process-level graphical modeling, spectral analysis, and intervention-based causal quantification into a coherent framework, enabling causal discovery, identifiability, and influence assessment in dynamic, multivariate systems.