Temporal Dependency to Causality (TD2C)

• TD2C is a method that transforms temporal dependencies in multivariate time series into causal claims using supervised learning and information-theoretic descriptors.
• It employs a wide range of features—including mutual information, transfer entropy, and residual-based measures—to capture lagged causal effects under explicit structural assumptions.
• TD2C and its related frameworks, such as Transformer-based and diffusion models, offer enhanced accuracy and zero-shot transfer in identifying causal relationships.

Temporal Dependency to Causality (TD2C) denotes both a specific supervised framework for causal discovery in multivariate time series and a broader methodological idea in which temporal dependencies are converted into lagged causal structure under explicit structural assumptions. In its explicit 2025 formulation, TD2C is a supervised, feature-based framework that reframes causal discovery from conditional-independence testing to pattern recognition over information-theoretic and statistical descriptors (Paldino et al., 3 Aug 2025). In parallel, several 2025 works use essentially the same transformation—temporal prediction or temporal dependency estimation followed by causal extraction—even when the acronym is not the primary method name, including Transformer-based forecasters, temporal structural causal simulation, and diffusion-based ordering methods (Huang et al., 21 Aug 2025, Gkorgkolis et al., 2 Jun 2025, Sanchez et al., 28 Oct 2025).

1. Definition and conceptual scope

The explicit TD2C framework is introduced as a method for discovering causal relationships in complex multivariate time series by learning discriminative signatures of temporal information flow rather than relying on a single conditional-independence test (Paldino et al., 3 Aug 2025). Its central hypothesis is that a causal link $X \to Y$ induces a persistent, learnable asymmetry in the flow of information through the temporal graph, even when clean conditional independencies are obscured. In that formulation, TD2C computes a large set of descriptors around each candidate lagged edge, summarizes them into a fixed-length feature vector, and trains a supervised classifier to distinguish true causal edges from non-causal pairs.

A broader reading of TD2C is also explicitly supported in later work. The Transformer-based framework in "Transforming Causality" is described as a direct instantiation of the TD2C idea: it first learns temporal predictive dependencies with a multi-layer Transformer forecaster and then converts those dependencies into a lagged causal graph by gradient-based sensitivity analysis (Huang et al., 21 Aug 2025). Temporal Causal-based Simulation (TCS) similarly operationalizes temporal dependency to causality by estimating a lagged graph, fitting structural mechanisms only on discovered parents, and then sampling from the resulting temporal SCM (Gkorgkolis et al., 2 Jun 2025). DOTS treats TD2C as the problem of turning temporal dependence into causality by imposing structural assumptions and recovering a temporal DAG from multiple diffusion-derived causal orderings (Sanchez et al., 28 Oct 2025).

This broader usage does not erase an older distinction: temporal order and temporal dependence are necessary ingredients for many causal interpretations, but they are not sufficient by themselves for causal dependence (Karimi, 2010). TD2C therefore belongs to a class of methods that add explicit assumptions, model structure, or learned asymmetry criteria to move from temporal dependence to causal claims.

2. Formal problem setting and assumptions

In the explicit TD2C framework, the observed data are a multivariate time series

$$X_t = (X_t^{(1)}, X_t^{(2)}, \dots, X_t^{(d)}) \in \mathbb{R}^d,\quad t = 1,\dots,T,$$

with candidate lagged edges

$$Z_i^{(t-\tau)} \to Z_j^{(t)}, \quad \tau \in \{1,\dots,L\},$$

where $L$ is the maximum lag (Paldino et al., 3 Aug 2025). The method assumes a temporal causal graph on time-indexed variables, with edges directed from past to future. Its temporal Markov Blanket is fixed as

$$\mathbf{MB}_{Z_i^{(t)}} = \{ Z_i^{(t-1)}, Z_i^{(t+1)}\},$$

using the assumption of first-order self-causality $Z_i^{(t-1)} \to Z_i^{(t)}$ (Paldino et al., 3 Aug 2025).

The assumptions supporting causal interpretation are distributed across the TD2C literature. The explicit feature-based TD2C formulation uses causal sufficiency, the Markov condition, faithfulness, causal stationarity, first-order self-causality, and the absence of contemporaneous edges in the temporal DAG used for training and evaluation (Paldino et al., 3 Aug 2025). The Transformer-based realization makes closely related assumptions explicit in a different vocabulary: temporal causal ordering or Markov property, approximate causal sufficiency, stationarity or stable mechanism during training, and a well-specified model class with sufficient capacity and data (Huang et al., 21 Aug 2025). DOTS places the problem under TiMINo-type SCM assumptions, including stationarity, causal sufficiency, additive noise with regularity, and temporal priority, optionally allowing contemporaneous edges while preserving temporal acyclicity (Sanchez et al., 28 Oct 2025).

These assumptions are precisely what prevent TD2C from collapsing into a purely predictive exercise. Temporal dependency alone can be produced by hidden common causes, undersampling, or regime changes. A common misconception is therefore that any lagged predictability is already causal. The general methodological literature reviewed in "A Brief Introduction to Temporality and Causality" separates temporal order, temporal dependence, and causal dependence, and treats causal dependence as stronger than temporal dependence (Karimi, 2010).

3. The supervised TD2C pipeline based on mutual-information featurization

The 2025 TD2C framework turns temporal causal discovery into supervised edge classification (Paldino et al., 3 Aug 2025). The pipeline begins by constructing a lagged representation of the multivariate time series and enumerating candidate edges $Z_i^{(t-\tau)} \to Z_j^{(t)}$. For each candidate edge, TD2C computes a feature vector from information-theoretic and statistical descriptors, pairs it with the ground-truth label, and trains a Balanced Random Forest to output

$$p_{ij\tau} = P(\text{edge } Z_i^{(t-\tau)} \to Z_j^{(t)} \mid \mathbf{x}_{ij\tau}).$$

The descriptor design is the core of the method. Mutual information and conditional mutual information are defined as

$$I(\mathbf{z}_i; \mathbf{z}_j) = \mathbb{E}_{\mathbf{z}_i,\mathbf{z}_j}\left[ \log \frac{p(z_i,z_j)}{p(z_i)p(z_j)}\right],$$

and

$$I(\mathbf{z}_i; \mathbf{z}_j \mid \mathbf{z}_k) = \mathbb{E}\left[ \log \frac{p(z_i,z_j \mid z_k)}{p(z_i \mid z_k)p(z_j \mid z_k)}\right].$$

TD2C estimates MI and CMI with the nonparametric KSG estimator, using scikit-learn’s KSG with default $$X_t = (X_t^{(1)}, X_t^{(2)}, \dots, X_t^{(d)}) \in \mathbb{R}^d,\quad t = 1,\dots,T,$$0 (Paldino et al., 3 Aug 2025).

A distinctive family of descriptors is based on generalized transfer-entropy asymmetry. For a candidate direction $$X_t = (X_t^{(1)}, X_t^{(2)}, \dots, X_t^{(d)}) \in \mathbb{R}^d,\quad t = 1,\dots,T,$$1,

$$X_t = (X_t^{(1)}, X_t^{(2)}, \dots, X_t^{(d)}) \in \mathbb{R}^d,\quad t = 1,\dots,T,$$2

$$X_t = (X_t^{(1)}, X_t^{(2)}, \dots, X_t^{(d)}) \in \mathbb{R}^d,\quad t = 1,\dots,T,$$3

and

$$X_t = (X_t^{(1)}, X_t^{(2)}, \dots, X_t^{(d)}) \in \mathbb{R}^d,\quad t = 1,\dots,T,$$4

In the reported experiments, the lag range for TE asymmetry is $$X_t = (X_t^{(1)}, X_t^{(2)}, \dots, X_t^{(d)}) \in \mathbb{R}^d,\quad t = 1,\dots,T,$$5 (Paldino et al., 3 Aug 2025).

The full descriptor set combines several families: mutual information and conditional mutual information between cause, effect, and Markov Blanket nodes; transfer-entropy-like quantities across multiple lags; residual-based correlations; linear regression coefficients; higher-order cross-moments; skewness; and kurtosis. The final set contains 63 features, of which 16 are new to TD2C (Paldino et al., 3 Aug 2025). Two residual-based descriptors are singled out as especially important: partial correlation of residuals and residual–input correlation asymmetry.

Training is performed only on synthetic nonlinear autoregressive processes. The paper uses 19 processes minus one unstable, three noise distributions, $$X_t = (X_t^{(1)}, X_t^{(2)}, \dots, X_t^{(d)}) \in \mathbb{R}^d,\quad t = 1,\dots,T,$$6 variables, and time length $$X_t = (X_t^{(1)}, X_t^{(2)}, \dots, X_t^{(d)}) \in \mathbb{R}^d,\quad t = 1,\dots,T,$$7. Training uses 9 odd-index processes $$X_t = (X_t^{(1)}, X_t^{(2)}, \dots, X_t^{(d)}) \in \mathbb{R}^d,\quad t = 1,\dots,T,$$8 3 noises $$X_t = (X_t^{(1)}, X_t^{(2)}, \dots, X_t^{(d)}) \in \mathbb{R}^d,\quad t = 1,\dots,T,$$9 120 series each, yielding 3240 series and 243,000 candidate pairs; the unseen-dynamics test uses 9 even-index processes $$Z_i^{(t-\tau)} \to Z_j^{(t)}, \quad \tau \in \{1,\dots,L\},$$0 3 noises $$Z_i^{(t-\tau)} \to Z_j^{(t)}, \quad \tau \in \{1,\dots,L\},$$1 40 series each, yielding 1080 series and 81,000 pairs (Paldino et al., 3 Aug 2025). The decision threshold is selected by leave-one-process-out cross-validation, and the final threshold is $$Z_i^{(t-\tau)} \to Z_j^{(t)}, \quad \tau \in \{1,\dots,L\},$$2, chosen by minimizing ROC distance to $$Z_i^{(t-\tau)} \to Z_j^{(t)}, \quad \tau \in \{1,\dots,L\},$$3.

This formulation is explicitly architecture-agnostic at inference time. The learned classifier is not fitting one causal graph jointly; it classifies lagged edges individually from featurized local temporal structure. That design explains the reported zero-shot transfer to unseen dynamics and realistic benchmarks with different numbers of variables.

4. Realizations and neighboring formulations of the TD2C idea

The term TD2C also functions as an organizing principle across several adjacent lines of work. The common pattern is that temporal dependency is first learned or estimated, then reinterpreted as causality under additional structure.

Formulation	Core mechanism	Reported role or result
Supervised TD2C	MI/CMI, TE-like, residual, HOC, and linear descriptors with a Balanced Random Forest	State-of-the-art zero-shot causal discovery on unseen dynamics and realistic benchmarks (Paldino et al., 3 Aug 2025)
Transformer-based realization	Multi-layer Transformer forecaster, finite-difference gradients, attention masks for forbidden links	12.8% improvement in F1-score for causal discovery and 98.9% lag estimation accuracy (Huang et al., 21 Aug 2025)
TCS	Estimate lagged graph, fit structural functions $$Z_i^{(t-\tau)} \to Z_j^{(t)}, \quad \tau \in \{1,\dots,L\},$$4, learn noise, ancestral sampling	Model-agnostic pipeline for generating realistic temporal causal data (Gkorgkolis et al., 2 Jun 2025)
DOTS	Diffusion score matching, multiple causal orderings, transitive-closure aggregation, pruning	Mean window-graph $$Z_i^{(t-\tau)} \to Z_j^{(t)}, \quad \tau \in \{1,\dots,L\},$$5 from $$Z_i^{(t-\tau)} \to Z_j^{(t)}, \quad \tau \in \{1,\dots,L\},$$6 to $$Z_i^{(t-\tau)} \to Z_j^{(t)}, \quad \tau \in \{1,\dots,L\},$$7 on synthetic benchmarks (Sanchez et al., 28 Oct 2025)
TDRL	Recover latent temporal causal variables from nonlinear mixtures under fixed and changing dynamics	Time-delayed latent causal influences are reliably identified (Yao et al., 2022)

In the Transformer-based realization, temporal prediction is primary. A causality-aware forecaster learns

$$Z_i^{(t-\tau)} \to Z_j^{(t)}, \quad \tau \in \{1,\dots,L\},$$8

and the causal graph extractor then computes finite-difference gradients

$$Z_i^{(t-\tau)} \to Z_j^{(t)}, \quad \tau \in \{1,\dots,L\},$$9

with respect to each input variable and time index. Edges are declared by thresholding the maximal normalized gradient over time, and causal lags are read off by the argmax time index. Prior knowledge enters through attention masks that hard-enforce forbidden links across all Transformer layers, with explicit source–target separation used to block indirect propagation (Huang et al., 21 Aug 2025).

TCS places TD2C inside a temporal SCM pipeline. It estimates the lagged causal graph $L$0, fits structural mechanisms

$L$1

learns noise distributions, and then performs temporal ancestral sampling. Its implemented causal-discovery modules include PCMCI, DYNOTEARS, and Causal Pretraining, and its model-selection layer uses a min–max optimization over C2ST detectors and TCS configurations (Gkorgkolis et al., 2 Jun 2025).

DOTS approaches the same problem through score-based diffusion and multiple causal orderings. It learns the score $L$2 of lag-embedded time series, estimates Hessians via score Jacobians, uses Hessian-diagonal variance for leaf detection, aggregates multiple orderings to approximate the transitive closure, and then prunes indirect edges. Under its assumptions, temporal dependencies are translated into causal assertions through the score/Hessian structure of a TiMINo-type SCM (Sanchez et al., 28 Oct 2025).

A more formal, trace-property line of work arrives at a different but related notion of temporal causality. "Synthesis of Temporal Causality" proves that omega-regular causes for omega-regular effects are themselves omega-regular and can be computed as nondeterministic Büchi automata (Finkbeiner et al., 2024). "Closure and Complexity of Temporal Causality" shows that safety, reachability, and recurrence properties are closed under causal inference, while persistence and obligation properties are not (Carelli et al., 15 May 2025). These papers do not present TD2C as a time-series classifier, but they formalize the same move from temporal property to causal explanation at the level of trace languages.

5. Empirical profile and computational characteristics

The explicit TD2C framework reports strong zero-shot performance on held-out synthetic processes. Over 1080 unseen-dynamics runs, it achieves Accuracy $L$3, Balanced Accuracy $L$4, F1-Score $L$5, Precision $L$6, and Recall $L$7; Balanced Accuracy, F1-Score, and Recall are the best among the compared methods (Paldino et al., 3 Aug 2025).

On realistic benchmarks, the same paper reports that TD2C is either best or competitive on all tested datasets. On DREAM3-50, it reaches F1 $L$8, Precision $L$9, and Balanced Accuracy $$\mathbf{MB}_{Z_i^{(t)}} = \{ Z_i^{(t-1)}, Z_i^{(t+1)}\},$$0, with the best F1 and Precision among compared methods. On NetSim-5, it reaches F1 $$\mathbf{MB}_{Z_i^{(t)}} = \{ Z_i^{(t-1)}, Z_i^{(t+1)}\},$$1, and on NetSim-10, F1 $$\mathbf{MB}_{Z_i^{(t)}} = \{ Z_i^{(t-1)}, Z_i^{(t+1)}\},$$2, both the best reported F1 values in the comparison (Paldino et al., 3 Aug 2025). Friedman plus Wilcoxon-Holm tests show that the recall advantage is statistically significant; for F1 and Balanced Accuracy, TD2C has the best average rank but belongs to a top cluster with PCMCI, MVGC, and PCMCI-GPDC.

The internal importance ranking is also informative. The top 15 features include residual–input correlation, partial correlation of residuals, mean MI from effect’s MB, linear coefficients, mean MI from cause’s MB, MB–MB information interaction, higher-order cross moments, and TE asymmetry (Paldino et al., 3 Aug 2025). This indicates that the learned decision rule is not reducible to a single statistic.

The reported computational complexity of explicit TD2C is

$$\mathbf{MB}_{Z_i^{(t)}} = \{ Z_i^{(t-1)}, Z_i^{(t+1)}\},$$3

where $$\mathbf{MB}_{Z_i^{(t)}} = \{ Z_i^{(t-1)}, Z_i^{(t+1)}\},$$4 is the number of variables, $$\mathbf{MB}_{Z_i^{(t)}} = \{ Z_i^{(t-1)}, Z_i^{(t+1)}\},$$5 the maximum lag, and $$\mathbf{MB}_{Z_i^{(t)}} = \{ Z_i^{(t-1)}, Z_i^{(t+1)}\},$$6 the number of observations. Runtime per series is about $$\mathbf{MB}_{Z_i^{(t)}} = \{ Z_i^{(t-1)}, Z_i^{(t+1)}\},$$7s for $$\mathbf{MB}_{Z_i^{(t)}} = \{ Z_i^{(t-1)}, Z_i^{(t+1)}\},$$8 with one job and $$\mathbf{MB}_{Z_i^{(t)}} = \{ Z_i^{(t-1)}, Z_i^{(t+1)}\},$$9s with 50 jobs; for $Z_i^{(t-1)} \to Z_i^{(t)}$0, about $Z_i^{(t-1)} \to Z_i^{(t)}$1s with one job and $Z_i^{(t-1)} \to Z_i^{(t)}$2s with 50 jobs; for $Z_i^{(t-1)} \to Z_i^{(t)}$3, about $Z_i^{(t-1)} \to Z_i^{(t)}$4s with 50 jobs (Paldino et al., 3 Aug 2025).

Other TD2C-style realizations report complementary empirical gains. The Transformer-based realization reports a 12.8% improvement in F1-score for causal discovery and 98.9% accuracy in estimating causal lags, with dataset-specific lag-estimation figures of 97.8%, 99%, and 100% (Huang et al., 21 Aug 2025). DOTS improves mean synthetic window-graph $Z_i^{(t-1)} \to Z_i^{(t)}$5 from about $Z_i^{(t-1)} \to Z_i^{(t)}$6 to about $Z_i^{(t-1)} \to Z_i^{(t)}$7, and on the CausalTime benchmark attains the highest average summary-graph $Z_i^{(t-1)} \to Z_i^{(t)}$8 while roughly halving runtime relative to graph-optimisation methods (Sanchez et al., 28 Oct 2025). TDRL reports Mean Correlation Coefficient values of $Z_i^{(t-1)} \to Z_i^{(t)}$9, $Z_i^{(t-\tau)} \to Z_j^{(t)}$0, and $Z_i^{(t-\tau)} \to Z_j^{(t)}$1 in three synthetic settings designed to evaluate recovery of latent temporal causal processes (Yao et al., 2022).

6. Assumptions, misconceptions, and open technical issues

TD2C methods derive their force from assumptions rather than from temporal ordering alone. The explicit feature-based TD2C paper states causal sufficiency, Markov condition, faithfulness, causal stationarity, first-order self-causality, and no contemporaneous edges in the training temporal DAG (Paldino et al., 3 Aug 2025). The Transformer-based realization requires temporal causal ordering, approximate causal sufficiency, stable mechanisms during training, and sufficient model fit for gradients to encode causal structure (Huang et al., 21 Aug 2025). DOTS relies on stationarity, additive noise, and no latent confounders (Sanchez et al., 28 Oct 2025). If these fail, temporal predictive patterns can remain highly informative yet lose causal validity.

Several limitations recur across the literature. Explicit TD2C does not provide theoretical identifiability guarantees; it is an empirical pattern learner whose reliability is supported by synthetic and benchmark performance rather than by formal proofs (Paldino et al., 3 Aug 2025). It also does not enforce acyclicity, so cycles can appear in predicted graphs. Scalability remains quadratic in the number of variables and linear in the maximum lag up to the $Z_i^{(t-\tau)} \to Z_j^{(t)}$2 factor from KSG estimation. The Transformer-based realization is sensitive to patching, perturbation size $Z_i^{(t-\tau)} \to Z_j^{(t)}$3, threshold $Z_i^{(t-\tau)} \to Z_j^{(t)}$4, and attention-mask design, and its hard prior mechanism mainly supports forbidden rather than required edges (Huang et al., 21 Aug 2025). TCS explicitly concludes that realistic causal generation of temporal data remains an unsolved problem, with real-data C2ST AUCs still around $Z_i^{(t-\tau)} \to Z_j^{(t)}$5–$Z_i^{(t-\tau)} \to Z_j^{(t)}$6 on several datasets (Gkorgkolis et al., 2 Jun 2025).

A common misconception is that TD2C simply rebrands Granger causality. The overlap is real—both rely on directed predictability from the past—but the explicit TD2C framework deliberately replaces single-test decisions with a learned classifier over heterogeneous descriptors, while other realizations replace linear VAR structure with Transformers, diffusion score models, or latent nonlinear state-space models (Paldino et al., 3 Aug 2025, Sanchez et al., 28 Oct 2025, Yao et al., 2022). Another misconception is that any temporally asymmetric dependency suffices. The broader literature on temporality and causality repeatedly warns that hidden common causes and alternative explanations can produce temporal dependencies without genuine causal influence (Karimi, 2010).

The current state of the field suggests a layered interpretation. TD2C is not one theorem or one architecture. It is a family of methods that treat temporal dependence as raw material for causal inference, then constrain, regularize, or reinterpret that dependence through information-theoretic asymmetry, structural priors, temporal SCMs, diffusion orderings, or latent conditional-independence structure. This suggests that TD2C is best understood as a research program for converting temporal structure into causal knowledge, with the decisive technical question always being which assumptions make that conversion trustworthy.