This study employs a “phase-action” coding method to convert continuous surgical videos into structured temporal sequences: the phase dimension is defined by the continuous execution of standard operational states together with sudden abnormal events; while the action dimension represents the executable operations of the robotic system at the current moment. This integrated approach aligns with surgical cognitive patterns, clearly distinguishing routine operations from abnormal events.

Let there be N standard operational states S i , M random events E j , W executable operations Z k , which are defined as follows:

At any time t:

S i ( t ) ( i = 1 , 2 , ⋯ , N ) ∈ { 0 , 1 } denotes whether the i-th standard operational state is active. If S 1 =‘Needle Holding’, S 1 ( t ) = 1 indicates that the step ‘Needle Holding’ is being performed, and S 1 ( t ) = 0 indicates it is not.

E j ( t ) ( j = 1 , 2 , ⋯ , M ) ∈ { 0 , 1 } denotes whether the j-th random event occurs. If E 1 = ‘Needle Dropping’, E 1 ( t ) = 1 indicates that the abnormal event ‘Needle Dropping’ has occurred, and E 1 ( t ) = 0 indicates it has not.

Z k ( t ) ( k = 1 , 2 , ⋯ , W ) ∈ { 0 , 1 } denotes whether the k-th executable action is triggered or recommended by the system. If Z 1 = ‘Needle Retrieval’, Z 1 ( t ) = 1 indicates that the action is triggered, and Z 1 ( t ) = 0 indicates it is not.

Accordingly, the surgical phase and action at time t can be described as: V ( t ) = [ S 1 ( t ) , S 2 ( t ) , ⋯ , S N ( t ) ; E 1 ( t ) , E 2 ( t ) , ⋯ , E M ( t ) ] T Z ( t ) = [ Z 1 ( t ) , Z 2 ( t ) , ⋯ , Z W ( t ) ] T

Where, V ( t ) represents the stage variable at time t , integrating both standard operational states and random events. Z ( t ) denotes the action variable at time t , comprising W executable operational states Z 1 ( t ) , Z 2 ( t ) , ⋯ , Z W ( t ) .

The core objective of this study is to learn a dynamic decision function from surgical video time-series data that can accurately describe the mapping from historical phase information to the current execution action. This function represents the conditional probability distribution of the system taking a specific action given the historical surgical context, which can be generally expressed as: P ( Z ( t ) ∣ V ( t − 1 ) , V ( t − 2 ) , ⋯ , V ( t − p ) )

Where p denotes the length of the historical time window influencing the current decision. This conditional probability distribution essentially defines a state transition process, mapping the sequence of past p phase states { V ( t − 1 ) , V ( t − 2 ) , ⋯ , V ( t − p ) } to the action variable Z ( t ) at time t.

To characterize dynamic interactions within the ‘phase’ variable V ( t ) and its potential influence on the ‘action’ variable ( t ) , this study employs a Vector Autoregressive (VAR) model as the core temporal modeling tool. The VAR model enables simultaneous estimation of the interdependencies among multiple endogenous variables and captures the persistent effects of historical states on the current system through the introduction of lag terms. This provides an ideal structural foundation for subsequently identifying the Granger causal relationships between the “phase” and “action” variables.

The VAR-based phase-action model can be expressed as follows: Z ( t ) = f ( V ( t − 1 ) , V ( t − 2 ) , ⋯ , V ( t − p ) ; Θ ) + ε ( t )

Where, f ( ⋅ ) is a linear function, Θ is the parameter matrix to be estimated, and ε ( t ) is a random disturbance term. This model represents action decisions as a linear combination of historical phase variables, providing interpretable parameter estimates for subsequent analysis.

More specifically, each action variable (in Section 2.2) can be written as: Z k ( t ) = ∑ q = 1 p ∑ i = 1 N γ k , i q S i ( t − q ) + ∑ q = 1 p ∑ j = 1 M δ k , j q E j ( t − q ) + ε k ( t )

Where, Z k ( t ) ( k = 1 , 2 , ⋯ , W ) denotes the k-th action at time t , p is the lag order, representing the length of historical information influencing the current action, γ k , i q represents the influence of the i-th state at lag q on action Z k ( t ) , S i ( t − q ) ( i = 1 , 2 , ⋯ , N ) is the value of the i-th phase variable at past time t − q , δ k , j q represents the influence of the j-th random event at lag q on action Z k ( t ) , E j ( t − q ) ( j = 1 , 2 , ⋯ , M ) is the value of the j-th random event at past time t − q , ε k ( t ) is a random noise term at time t .

This formulation captures both the influence of historical phase states and abnormal events on current actions, providing a clear, interpretable framework for modeling surgical decision-making. These actions may be triggered by abnormal events (e.g., “Bleeding” triggering “Electrocoagulation”) or arise naturally from routine surgical logic (e.g., the “Suturing” state leading to “Knot Tying”). Granger causality, which will be introduced in Section 2.4, is later incorporated into the time-series modeling framework to identify and remove spurious correlations, enabling the construction of a decision model with explicit causal interpretability and strong generalization.

Building upon the VAR-based phase-action model introduced in Section 2.3, Granger causality tests are employed to identify temporal causal relationships between surgical phases and robotic actions. Herein, the “phase component” refers to V ij ( t ) = ( S i ( t ) , E j ( t ) ) , which denotes the standard operational state variable S i ( t ) or individual random event variable E j ( t ) extracted from the integrated stage variable V ( t ) (defined in Section 2.2). In a non-Markovian surgical environment, if incorporating historical information from past phase variables S i ( t − 1 ) , S i ( t − 2 ) , ⋯ , S i ( t − q ) and random events E j ( t − 1 ) , E j ( t − 2 ) , ⋯ , E j ( t − q ) significantly improves the prediction of the current action Z k ( t ) , these variables are considered Granger causes of Z k ( t ) . The testing process, detailed in Algorithm 1, is performed independently for each potential (phase component, action component) causal pair. Algorithm 1

Element-wise Granger causality testing in temporal sequences. Flowchart outlines a statistical testing procedure using surgical video data, detailing inputs, output, and stepwise methods for encoding observational data, building restricted and unrestricted models, calculating sums of squares, computing an F statistic, comparing to a critical value, and determining the presence or absence of causal links.

Here, α ¯ 0 denotes the constant term, p represents the lag order, β ¯ k q signifies the autoregressive coefficient of Z k ( t ) at lag q.

(2) Unrestricted model: Incorporates lagged terms of all historical phase variables to explicitly capture the respective causal influences of past states and events on the current action. Z k ( t ) = α 0 + ∑ q = 1 p β k q Z k ( t − q ) + ∑ q = 1 p ∑ i = 1 N γ k , i q S i ( t − q ) + ∑ q = 1 p ∑ j = 1 M δ k , j q E j ( t − q ) + ε k ( t )

The coefficient γ k , i q and δ k , j q represent the average effect of the state S i ( t − q ) and E j ( t − q ) has on the action Z k ( t ) at lag q, e.g., the driving effect of a G6 operation on the picking up needle action, or the triggering effect of dropping needle on the picking up needle action.

(3) Causal significance F

The likelihood ratio statistic is used to assess whether the predictive improvement of the unrestricted model relative to the restricted model is statistically significant, which can be expressed as follows: F = ( SS R r − SS R ur ) / Q SS R ur / ( n − p − 1 )

Here, SS R r and SS R ur represent the sums of squared residuals of the restricted and unrestricted models, respectively; SS R r reflects the prediction error without causal variables, while SS R ur reflects the prediction error after introducing causal variables. n denotes the sample size, Q represents the total number of lagged terms of the causal variables, p is the lag order (Q = 2p).

Based on the results of the Granger causality test, a dynamic correction mechanism for surgical procedures is established. When the system detects an abnormal event, it automatically triggers the corresponding corrective action. For example, “pick up needle + re-execute G6” can be determined by integrating its causal association “dropped needle” with the preceding state “G6 operation.” Unlike conventional predefined workflows, this mechanism generates adaptive response strategies through data-driven causal inference, enabling process reconstruction in non-sequential or unexpected surgical scenarios. The corresponding pseudocode is presented as follows:

Due to the scarcity of annotated data on abnormal events during actual surgical procedures, coupled with ethical and privacy constraints, this study employs synthetic data to validate a surgical process modeling approach based on Granger causality testing. The dataset design focuses on the causal relationship between “abnormal events” and “countermeasures,” comprising three core temporal variables:

S i (original gesture, e.g., G6 “left hand pulling thread”).

E j (abnormal event, e.g., “dropped needle”).

Z k (recovery action, e.g., “pick up needle + re-execute G6”).

A variable value of 1 indicates that an event has occurred, while 0 indicates that it has not.

This study employs both positive and negative examples as samples. Positive examples are constructed based on clinical logic, establishing explicit causal relationships such as “needle drop → needle retrieval” and “inaccurate positioning → repositioning.” This ensures that Z k is driven by historical data from S i and E j , thereby simulating the anomaly handling procedures encountered in actual surgical procedures. Negative examples are generated by randomly producing sequences of unrelated variables ( S i , E j , Z k )to eliminate temporal correlation interference, thereby validating the model’s discriminative capability in scenarios lacking genuine causal relationships.

To enhance the clinical plausibility of synthetic data, three senior consultants reviewed all predefined causal relationships, yielding a Kappa consistency score of 0.92. Furthermore, to better simulate real surgical environments, we introduced clinical noise in 20% of samples (specifically simulating gesture recognition bias caused by tissue occlusion) to bolster both data authenticity and model robustness. In this study, two datasets of different scales were constructed: one containing 5,000 samples (2,500 positive and 2,500 negative), and the other containing 10,000 samples (5,000 positive and 5,000 negative). The goal is to evaluate the model’s stability and generalization ability under varying data scales.

When constructing the VAR model, the selection of the lag order p is crucial to model performance. To effectively capture the influence of prior actions on the current recovery operation while avoiding excessive noise, p must be properly determined. Considering that intraoperative abnormal events typically span about two action steps from occurrence to response (for example, after a “needle drop,” the surgeon must first “pick up the needle” and then “rethread”), this study preliminarily sets the lag order to p = 2.

To further optimize the selection of the lag order and improve both reproducibility and transparency, the Akaike Information Criterion (AIC) was adopted for validation. As a widely recognized information-theoretic metric for model selection, the AIC effectively balances goodness of fit with model parsimony. Specifically, lower AIC values indicate a more optimal trade-off between capturing meaningful temporal dependencies and minimizing the risk of overfitting, thereby ensuring the model remains both robust and computationally efficient. By comparing AIC values under different p settings, the results show that when p = 2, the model achieves the smallest residual (AIC = −3.2), outperforming p = 1 (AIC = −2.8) and p = 3 (AIC = −2.9). Therefore, the optimal lag order was finally determined to be p = 2, ensuring an accurate representation of the surgical dynamics while avoiding overfitting.

The experimental parameters are set as follows: the sample size n = 5,000, the lag order p = 2, the generation probabilities for the variables S i (normal gesture) and E j (abnormal event) are 0.3 and 0.2, respectively, and the weighting coefficient for the lagged term S and that for the lagged term E are both 0.5. The regression coefficients (including α 0 , β i , γ j , and δ l ) are estimated using Ordinary Least Squares (OLS), with a total of Q = 4 lag terms.

Table 1 and Figure 3 demonstrate the performance of the proposed method across varying sample sizes. The model achieved an accuracy exceeding 95.6% on both 5,000 and 10,000 samples, with the Matthews correlation coefficient stabilizing at 0.912. This indicates excellent discriminative capability and generalization performance.

It is worth noting that the model’s recall consistently exceeds its precision, indicating a preference for minimizing the rate of missed true causative events—that is, reducing false negatives. In safety-critical surgical scenarios, this bias toward “better to err on the side of false positives than false negatives” aligns with the safety-first clinical principle. It helps ensure all critical anomalies are effectively identified and trigger appropriate response mechanisms.