Machine Learning Surrogate for Seismic Response of a Wooden House: A Comparison of SHAP, Sobol, and Morris Sensitivity Analyses

Abstract

Understanding the influence of structural parameters on the seismic response of wooden houses is essential for improving structural performance and model reliability. However, conducting extensive parametric studies using nonlinear time-history analysis is computationally expensive. To address this issue, this study proposes a machine learning (ML) surrogate framework for efficiently evaluating the seismic response of a wooden house and interpreting the importance of structural parameters. A dataset consisting of 289 nonlinear structural simulations was used to train the surrogate model, enabling efficient evaluation of parameter importance through multiple sensitivity analysis methods. A Gradient Boosting regression model was developed to approximate the results of nonlinear structural analyses. The surrogate model predicted the maximum inter-story drift with high accuracy, achieving a coefficient of determination of R² = 0.90. Using the trained surrogate model, six sensitivity analysis methods were applied: SHAP, Structural Perturbation, Drop-column Importance, Permutation Importance, Sobol sensitivity analysis, and the Morris method. The results showed that most sensitivity analysis methods consistently identified wall-related parameters, particularly W1, W3, and W4, as the dominant factors influencing structural response. This tendency was observed in both elastic and nonlinear response ranges, although the influence of these parameters became more pronounced under nonlinear conditions. While the Morris method produced slightly different sensitivity magnitudes due to its screening-based formulation, it still identified the same dominant parameters as the other approaches. The results demonstrate that the proposed ML surrogate framework, combined with explainable AI techniques, can effectively identify key structural parameters governing the seismic response of wooden structures. This approach provides a computationally efficient tool for structural sensitivity analysis and may support improved structural modeling and seismic performance evaluation.

1. Introduction

In recent years, achieving both sustainability and disaster resilience has become a critical challenge in the fields of architecture and structural engineering. In Japan, approximately 60% of newly constructed residential buildings are wooden houses [1], making the assurance of sufficient safety against external forces, such as earthquakes and climate-induced hazards, a pressing concern. At the same time, wood is a renewable material that has attracted significant attention for its potential to reduce environmental impacts throughout the building life cycle. Previous studies have quantitatively demonstrated the environmental advantages of wooden buildings, particularly in terms of life-cycle environmental loads [2].

The seismic performance of wooden structures is governed by numerous parameters, including material properties, geometric characteristics, boundary conditions, and seismic input. Many experimental and numerical studies have been conducted to investigate the seismic behavior of wooden houses. Accurate and quantitative evaluation of the influence of these parameters is therefore essential for understanding structural behavior and improving seismic design. Sensitivity analysis provides a systematic framework for quantifying the relative importance of input variables on structural response. Various sensitivity analysis methods have been proposed [3,4,5], including local perturbation approaches, variance-based methods, screening methods, and machine-learning-based importance measures. Because each method evaluates parameter influence from a different mathematical perspective, the resulting importance rankings may differ depending on the chosen approach.

Despite the availability of numerous sensitivity analysis techniques, systematic comparisons of these methods in structural engineering applications remain limited. In particular, the applicability and consistency of different sensitivity analysis approaches have not been sufficiently investigated for complex structural systems such as wooden houses. Because structural responses often involve nonlinear behavior and interactions among parameters, the reliability of parameter importance rankings may depend strongly on the selected analysis method.

In recent years, machine learning (ML) techniques have been increasingly applied in structural engineering. ML-based models have been widely used for predicting structural responses, identifying damage, and supporting structural health monitoring [6,7]. In particular, data-driven approaches have demonstrated strong potential for approximating complex structural behavior and improving the efficiency of structural analysis and monitoring systems. Among these approaches, machine-learning-based surrogate models have been increasingly introduced to efficiently approximate computationally expensive structural simulations. By learning the relationship between structural parameters and response quantities, surrogate models enable rapid prediction of structural behavior under various input conditions. As a result, they allow large-scale parametric studies and sensitivity analyses to be conducted without repeatedly performing nonlinear numerical simulations.

At the same time, explainable artificial intelligence (XAI) techniques have been proposed to improve the interpretability of machine learning models. Methods such as SHAP (SHapley Additive exPlanations) provide a theoretically grounded framework for quantifying the contribution of individual input variables to model predictions, thereby enhancing the transparency of data-driven structural analysis frameworks [8,9,10,11,12]. Several studies have applied SHAP to identify important structural parameters and to interpret machine-learning-based structural response prediction models.

Recent advances in data-driven approaches have been increasingly applied to structural monitoring and early warning systems. For instance, machine-learning-based methods have been developed to monitor bridge cables using temperature and displacement data, as well as to assess bridge towers under strong wind conditions through multi-rate data fusion techniques [13]. These studies highlight the effectiveness of data-driven models in detecting abnormal structural behavior and enhancing the reliability of infrastructure monitoring systems. Furthermore, similar intelligent prediction methods have been proposed in related engineering fields, such as the use of deep-learning-based semantic segmentation techniques to predict rock core integrity [14]. These references have been incorporated to position our study within recent developments in data-driven modeling and to highlight the relevance of such approaches to structural monitoring and predictive analysis in engineering systems.

Previous research by Namba [15] demonstrated the applicability of SHAP-based importance analysis for evaluating the influence of seismic elements in a wooden house model. However, that study primarily focused on the application of SHAP for interpreting machine-learning predictions. The consistency of machine-learning-based importance measures with other sensitivity analysis methods has not yet been fully clarified.

Although the dataset and structural modeling framework used in the present study are based on the previous work by Namba [15], the objective of this research is fundamentally different. While the previous study focused mainly on the applicability of SHAP for influence analysis of structural responses, the present study systematically compares multiple sensitivity analysis techniques—including SHAP, structural perturbation analysis, drop-column importance, permutation importance, Sobol indices, and the Morris method—within a unified surrogate modeling framework.

Through this comparison, the study aims to clarify the consistency, advantages, and limitations of different sensitivity analysis methods when applied to nonlinear seismic response analysis of wooden structures and to provide insights into the evaluation of parameter importance in data-driven structural engineering applications.

2. Method

2.1. Building Model and Datasets

The target structure in this study is a single two-story wooden house. Figure 1 illustrates the overall flowchart of the research framework. The dataset used in this study was derived from the work of Namba [15].

First, the target building was defined, and the areas corresponding to structural elements were specified. For the seismic elements within these areas, backbone curves were parameterized, and their stiffness and strength were assumed to vary within a range of ±20%. This variation was introduced from a sensitivity analysis perspective to enable clear identification of the influence of each parameter on the seismic response.

Next, combinations of these parameters were systematically generated using an orthogonal array, allowing efficient and comprehensive sampling of the parameter space. By employing this design-of-experiments approach, the number of required structural analyses can be significantly reduced while maintaining sufficient coverage of interactions among parameters.

Structural analyses were then performed using the program wallstat, version 4.3.11 [16] for each parameter set, and the resulting response data, such as story drift, were obtained. These data were subsequently used to train a machine-learning-based surrogate model that approximates the structural response of the building. Finally, multiple sensitivity analysis methods were applied to the surrogate model, and the resulting importance rankings were compared.

In this study, the response variable was limited to the maximum inter-story drift of the first story in the X direction. This quantity was selected because inter-story drift is widely used as a key indicator of seismic performance and damage in wooden structures. While additional response quantities such as second-story drift, base shear, or acceleration responses could also be considered, the present study focuses on a single representative indicator in order to clarify the methodological comparison of sensitivity-analysis techniques.

To analyze the seismic responses of the wooden structure, the structural analysis program wallstat [16] was used. In wallstat, the analytical model consists of two primary components: beam–column elements and nonlinear lateral-load-resisting shear wall elements. Figure 2 shows the outline of the analytical model.

The beams and columns were modeled as beam–column elements with elastoplastic rotational springs at their ends. Each joint was modeled using an elastoplastic axial spring capable of representing both tensile and compressive behavior. The hysteretic characteristics of the tensile and compressive springs were defined as one-sided elastic and one-sided slip types, respectively.

The shear-resisting behavior of walls and braces was modeled using nonlinear spring elements, as illustrated in Figure 2b,c. For wall elements, bracing members were replaced by equivalent truss elements, whereas braces were represented by two truss elements with independent load–deformation relationships assigned to the tensile and compressive responses. The backbone curves of these elements were defined in the same manner as described in Ref. [15].

Figure 3 shows the backbone relationships used for the wall and brace elements and those assigned to the tensile and rotational springs representing the HD25 kN connection. The wall and brace components were modeled using a bilinear backbone combined with a slip component, which is frequently adopted as a simplified representation of the hysteretic response of wooden shear walls. Meanwhile, the tensile and rotational springs were described using slip-type hysteresis models in order to capture their nonlinear deformation characteristics.

Nonlinear time-history analysis was performed using the average-acceleration integration method with a time step of $1.0 \times { 10}^{-5}$ s. Viscous damping was incorporated by assuming a damping ratio of 2%, defined in proportion to the instantaneous stiffness of the system. To maintain numerical stability during the analysis, the viscous damping component was set to zero whenever the instantaneous stiffness became negative.

Figure 4 illustrates the characteristics of the input ground motion used in the analysis. The seismic input corresponds to the Japan Meteorological Agency (JMA) Kobe record obtained during the 1995 Osaka–Kobe earthquake, which is widely employed as a representative example of a severe seismic event. The peak ground acceleration of this record reaches approximately 0.87 G. Because the waveform contains strong shaking capable of inducing significant nonlinear structural responses, it is suitable for evaluating the performance and sensitivity. To investigate both elastic and nonlinear structural behavior, two different excitation levels were considered: 20% and 100% of the original amplitude of the ground motion. The main characteristics of the input motion, including peak ground acceleration, predominant period, duration, and spectral features, are summarized in

Table 1. Main characteristics of the JMA Kobe.

Property	Value
Maximum acceleration (PGA)	0.87 g
Excitation levels	20%, 100%
Duration	30 s

.

Figure 5 illustrates the relationship between shear force and inter-story drift obtained from the structural analysis. As shown in Figure 5a–d, the structural response remains within the elastic range under relatively small excitation levels. In contrast, Figure 5e exhibits distinct hysteretic loops, indicating the occurrence of plastic deformation under stronger ground motion.

These analytical responses correspond to the reference condition adopted in this study and are based on the modeling framework reported in Ref. [15]. The same modeling assumptions and parameter settings are employed here to ensure consistency with the previous study. As shown in Figure 1, the target parameters are varied within a range of ±20% around this reference condition. This variation range was selected based on common practices in sensitivity analysis and in accordance with Namba’s study [15]. A moderate variation, such as ±20%, allows the influence of structural parameters on the response to be evaluated while maintaining physically realistic parameter values. Moreover, this range is sufficiently large to capture nonlinear changes in structural behavior without introducing unrealistic structural configurations. Therefore, the structural responses obtained in the parametric analyses are expected to fall within the envelope defined by the elastic and inelastic behaviors illustrated in Figure 5.

The dataset used for training the surrogate model was generated based on an L289 orthogonal array design, resulting in a total of 289 structural simulation cases. Each case corresponds to a different combination of structural parameters determined by the experimental design. Nonlinear structural simulations were performed for all 289 cases to obtain the corresponding structural response data as described later.

2.2. Machine Learning Surrogate

In Ref. [15], several machine learning algorithms were evaluated for predicting the seismic response of the target wooden structure. The comparison showed that the optimal model differed depending on the excitation level. For the 20% excitation case, which corresponds to the elastic response range, linear regression achieved the smallest prediction error. In contrast, for the 100% excitation case, where nonlinear structural behavior becomes dominant, the Gradient Boosting Regressor provided the highest predictive accuracy. These were implemented using the Python (version 3.9.13) library scikit-learn (version 1.2.2) [17]. Based on these results, the models that exhibited the best performance for each excitation level were adopted in the present study. Specifically, a linear regression model was used for the 20% excitation condition, while a Gradient Boosting Regressor [18] was employed for the 100% excitation case. This selection allows the surrogate models to appropriately capture both linear and nonlinear characteristics of the structural response.

In this study, the Gradient Boosting Regressor was used with the following hyperparameters: n_estimators = 100, learning_rate = 0.1, max_depth = 3, min_samples_split = 2, min_samples_leaf = 1, subsample = 1.0, and random_state = 42. These default settings provided stable predictions for the current dataset while balancing computational cost and model complexity. However, performance is sensitive to hyperparameter selection, and computational cost may increase for large datasets.

The surrogate models were trained using the response dataset obtained from the structural analyses described before. The target areas are shown in Figure 6. The input variables consist of 18 structural parameters related to stiffness and strength variations in seismic elements, as shown in

Table 2. Input variables [11].

	Name	Outline
1	J1	Joints (HD25kN) in area 1
2	J2	Joints (HD25kN) in area 2
3	J3	Joints (HD25kN) in area 3
4	J4	Joints (HD25kN) in area 4
5	J5	Joints (HD25kN) in area 5
6	J6	Joints (HD25kN) in area 6
7	J7	Joints (HD25kN) in area 7
8	J8	Joints (HD25kN) in area 8
9	W1	Walls (plywood wall) in area 1
10	W2	Walls (plywood wall) in area 2
11	W3	Walls (plywood wall) in area 3
12	W4	Walls (plywood wall) in area 4
13	W5	Walls (plywood wall) in area 5
14	W6	Walls (plywood wall) in area 6
15	W7	Walls (plywood wall) in area 7
16	W8	Walls (plywood wall) in area 8
17	F1	Floors in second story floor
18	F2	Floors in roof story

, while the target output is the maximum story drift in the first story of the building. The trained models were then used as surrogate models to efficiently evaluate structural responses required for the subsequent sensitivity analyses.

To generate the dataset efficiently, an orthogonal array table was adopted. Because multiple structural parameters with several levels were considered in this study, performing a full factorial design would require a prohibitively large number of simulations. Therefore, an orthogonal array was used to systematically sample representative combinations of parameters while maintaining balanced coverage of the parameter space. Among available orthogonal arrays, the L289 orthogonal array was selected because it is suitable for the number of parameters and levels considered in this study. This design enables efficient exploration of parameter interactions while limiting the total number of simulations to 289 cases.

To confirm the stability of the training process, the loss curves of the machine learning models were examined. 80% of the total dataset was used for model training, while the remaining 20% was reserved for validation. As shown in Figure 7, the training and validation losses converge smoothly without significant divergence, suggesting that overfitting did not occur during the training process. These results indicate that the training procedure was stable. A detailed evaluation of the predictive accuracy of the surrogate model is presented in the following section.

2.3. Sensitivity Analysis Methods

To quantify the influence of each input variable on the predicted structural response, six sensitivity analysis methods were applied. These methods evaluate parameter importance from different perspectives, including machine-learning-based explainability techniques and variance-based global sensitivity analyses.

2.3.1. SHAP

Computes Shapley values to determine local and global contributions of each feature to the model predictions. SHAP [19] is often used as XAI, which is proposed to solve the long-criticized black-box issue of ML models. SHAP is a collection of explainers based on a game theory approach that estimates Shapley values from an absolute average of the feature contributions over several simulations. Since the original ML model is complex, this approach uses additive feature importance measures based on a linear explanation model that is a linear combination of binary variables expressed by Equation (1).

$$f\left(x\right)\approx g\left(z^{\prime }\right)={\phi }_0+\sum _{i=1}^M{\phi }_i{z}^{\prime }$$

(1)

where $f\left(x\right)$ is the original model and $x$ is the original input feature. The explanation model uses $x^{\prime }$ as a simplified input feature and links it with a mapping function $x=h_x\left(x^{\prime }\right)$, while local methods attempt to guarantee that $g\left(z^{\prime }\right)\approx f\left(h_x\left(z^{\prime }\right)\right)$ whenever $z^{\prime }\approx x^{\prime }$. The value ${\phi }_i$ is the Shapley value, which is expressed by Equation (2).

$${\phi }_i\left(f,x\right)=\sum _{z^{\prime }\subseteq x^{\prime }}{\displaystyle \frac{\left|z^{\prime }\right|!\left(M-\left|z^{\prime }\right|-1\right)!}{M!}}(f_x\left(z^{\prime }\right)-f_x\left(z^{\prime }\setminus i\right))$$

(2)

When three desirable properties (local accuracy, missingness, and consistency) are satisfied, |z′| is the number of non-zero entries in z′, and all z′ vectors are a subset of x′. In this study, python library “SHAP” published in GitHub (version 0.47.2) was used [19].

2.3.2. Structural Perturbation

Structural perturbation analysis [20] is a sensitivity-analysis method that evaluates the influence of individual input variables on the output by introducing controlled variations in structural parameters. Unlike purely statistical approaches, structural perturbation directly reflects the physical relationship between model parameters and structural responses, making it particularly suitable for engineering applications.

In this study, the importance of each input parameter was evaluated by perturbing the parameter around a baseline structural configuration while keeping all other parameters fixed. The baseline configuration corresponds to the reference parameter set used in the surrogate model.

Specifically, each input variable $X_i$ (e.g., joist stiffness, wall strength, or floor load) was independently varied by ±20% relative to the baseline value. The resulting change in the predicted response, defined as the maximum inter-story drift $Y$, was then evaluated using the trained surrogate model.

The sensitivity index for each parameter was calculated as:

$$S_i={\displaystyle \frac{\mid Y(X_i+\mathsf{\Delta }{X}_i)-Y\left(X_i\right)\mid }{\mathsf{\Delta }{X}_i/X_i}}$$

where $S_i$ represents the normalized sensitivity, $Y(X_i+\mathsf{\Delta }{X}_i)$ is the predicted response after perturbation, and $\mathsf{\Delta }{X}_i$ is the magnitude of the applied variation.

For each parameter, the surrogate model was used to compute the corresponding change in the predicted inter-story drift without rerunning the nonlinear structural simulations. This approach enables efficient evaluation of parameter influence while maintaining consistency with the surrogate-based sensitivity analysis framework.

2.3.3. Drop-Column Importance

Drop-column Importance [21] is a sensitivity analysis method that quantifies the contribution of each input variable to the predictive performance of a machine learning model. The method operates by systematically removing (or “dropping”) one feature at a time from the dataset, retraining the model without the excluded feature, and measuring the decrease in predictive accuracy.

Formally, let $R_{\mathrm{full}}^2$ denote the coefficient of determination for the model trained with all input features, and $R_{-i}^2$ denote the R² after dropping the $i$-th feature $X_i$. The drop-column importance for feature $X_i$ is defined as:

$$I_i=R_{\mathrm{full}}^2-R_{-i}^2$$

where a higher $I_i$ indicates greater importance of the variable in maintaining the model’s predictive performance.

In the context of structural analysis, drop-column importance provides a physically interpretable measure of the contribution of each structural variable (e.g., joist stiffness, wall strength, floor loads) to the prediction of maximum drift. Variables that, when removed, cause a large drop in model performance are considered critical for accurately modeling the structural response.

2.3.4. Permutation Importance

Permutation Importance [22] is a model-agnostic sensitivity analysis method that evaluates the contribution of each input feature to the predictive performance by randomly permuting its values across the dataset. Unlike drop-column importance, the model is not retrained; instead, the feature’s association with the output is disrupted, and the resulting decrease in prediction accuracy indicates its importance.

Formally, let $R_{\mathrm{orig}}^2$ be the model’s coefficient of determination on the original test data, and $R_{X_i\mathrm{-perm}}^2$ be the R² after permuting the $i$-th feature $X_i$. The permutation importance of $X_i$ is defined as:

$$P{I}_i=R_{\mathrm{orig}}^2-R_{X_i\mathrm{-perm}}^2$$

A higher $P{I}_i$ indicates that the variable has a stronger influence on the model’s predictions.

In the context of structural response analysis, permutation importance allows evaluation of the impact of structural variables such as joist properties, wall stiffness, and floor loads on maximum drift predictions. Since the surrogate model remains unchanged, this method is computationally efficient and can capture nonlinear interactions among variables.

By comparing permutation importance results with other methods such as SHAP values, structural perturbation, or drop-column importance, researchers can assess whether the surrogate model’s learned relationships align with physical expectations. Consistency across these methods provides confidence in the reliability and interpretability of the machine learning surrogate.

2.3.5. Sobol Sensitivity

Sobol Sensitivity Analysis [23] is a global, variance-based method for quantifying the contribution of each input variable and their interactions to the output variance. Unlike local methods, Sobol analysis evaluates sensitivity over the entire input space, making it suitable for nonlinear and non-additive models.

Let the model output be $Y=f(X_1,X_2,...,X_n)$, where $X_i$ are the input variables. Sobol indices decompose the total output variance $V\left(Y\right)$ into contributions from individual inputs and their interactions:

$$V(Y)=\sum _{i=1}^n{V}_i+\sum _{i<j}{V}_{ij}+\sum _{i<j<k}{V}_{ijk}+\cdots +V_{\mathrm{1,2},...,n}$$

• The first-order index $S_i=V_i/V\left(Y\right)$ measures the main effect of variable $X_i$ alone.
• Higher-order indices (e.g., $S_{ij}=V_{ij}/V\left(Y\right)$) capture interactions between variables.
• The total-order index $S_{T_i}$ includes both the main effect and all interaction effects involving $X_i$.

In this study, Sobol sensitivity analysis was applied to the Gradient Boosting surrogate model predicting maximum story drift in a single wooden house. The first-order and total-order indices were computed to identify the most influential structural variables (e.g., joist stiffness J3, wall W5, floor load F1) and assess the presence of variable interactions.

2.3.6. Morris Method

The Morris Method [24], also known as the Elementary Effects method, is a global, computationally efficient sensitivity analysis technique designed to identify the most influential input variables in nonlinear and non-additive models. Unlike variance-based methods such as Sobol, the Morris method provides a screening-level sensitivity measure with reduced computational cost.

The method works as follows:

• The input space is discretized into a grid of levels.
• Random trajectories are generated, where each input variable is incrementally perturbed one at a time along the trajectory.
• For each input variable $X_i$, the elementary effect (EE) is computed as:

$$E{E}_i={\displaystyle \frac{Y(X_1,...,X_i+\mathsf{\Delta },...,X_n)-Y(X_1,...,X_i,...,X_n)}{\mathsf{\Delta }}}$$

where Δ is the step size in the input space and Y is the model output.

4.

Across multiple trajectories, the mean absolute value of the elementary effects (${\mu }^{*}$) is calculated for each variable:

$${\mu }_i^{*}={\displaystyle \frac{1}{r}}\sum _{k=1}^r\mid E E_i^{\left(k\right)}\mid$$

where $r$ is the number of trajectories. A higher ${\mu }_i^{*}$ indicates a stronger influence of $X_i$ on the output. The standard deviation of the elementary effects can also be analyzed to identify variables that exhibit nonlinear or interaction effects.

3. Results and Discussion

3.1. Machine Learning Surrogate Performance

The Gradient Boosting surrogate model demonstrated high predictive accuracy for maximum story drift in the single wooden house. Figure 8 presents the predicted versus true drift values, with each point labeled by the predicted value. The strong agreement confirms that the surrogate model reliably captures the structural response across the range of input variables.

The trained model achieved a coefficient of determination of approximately R² ≈ 0.90 on the test dataset in JMA KOBE 100%, indicating that the surrogate model explains about 90% of the variance in the structural response obtained from the nonlinear time-history analyses. This result demonstrates that the model captures the dominant relationship between the structural parameters and the resulting seismic response. In addition, the model achieved a mean absolute error (MAE) of 2.517, indicating that the predicted maximum inter-story drift deviates from the reference nonlinear analysis results by approximately 2.5 mm on average. Considering that the drift values range roughly between 60 mm and 100 mm, this error magnitude is relatively small, suggesting that the surrogate model provides an accurate approximation of the structural response.

To further examine the predictive behavior of the surrogate model, residual plots were analyzed for both excitation levels, as shown in Figure 9a,b. In the 20% excitation case (Figure 9a), which corresponds to the elastic response range of the structure, the residuals are relatively small and distributed closely around zero. The magnitude of the residuals remains within approximately ±0.6 mm, indicating that the surrogate model provides highly accurate predictions in the elastic regime. The residuals appear randomly scattered without a systematic trend with respect to the predicted drift values, suggesting that the model does not exhibit significant bias in this range.

In contrast, the 100% excitation case (Figure 9b), representing the nonlinear structural response, shows a wider spread of residual values. This increase in residual variance is expected because the nonlinear behavior of structural elements, such as shear walls and braces, introduces stronger parameter interactions and more complex response characteristics. Despite this increased variability, the residuals remain approximately centered around zero and do not show a clear systematic pattern.

These results indicate that the surrogate model successfully captures the general relationship between the structural parameters and the seismic response across both elastic and nonlinear regimes. The slightly larger residual dispersion observed in the nonlinear case reflects the increased complexity of structural behavior rather than model bias. Combined with the obtained performance metrics (R² ≈ 0.90 and MAE = 2.517), the residual analysis confirms that the developed surrogate model provides a stable and sufficiently accurate approximation of the nonlinear structural analysis results.

3.2. Sensitivity Analysis Results

The sensitivity analysis results obtained from the six methods are summarized in Figure 10 and Figure 11, which represent the elastic and nonlinear response ranges, respectively. Although the absolute values differ among the methods, the overall ranking of influential parameters shows a consistent trend.

In the elastic range (Figure 10), most sensitivity analysis methods consistently identify the wall-related parameters, particularly W1, W3, and W4, as the dominant contributors to the structural response. This tendency is clearly observed in the results obtained using SHAP, Structural Perturbation, Drop-column Importance, Permutation Importance, and Sobol sensitivity indices.

The dominance of these parameters can be explained by the structural mechanism of the model. In the elastic range, the global stiffness of the structure is primarily governed by the lateral stiffness of the shear walls. Therefore, parameters associated with wall stiffness and load-transfer mechanisms naturally exhibit higher sensitivity compared with joint parameters (J1–J8) or floor parameters (F1–F2). The Sobol analysis also confirms this trend, indicating that the variance of the output response is largely attributed to the variation in these wall-related parameters. These results indicate that wall groups W1, W3, and W4 have the largest influence on the predicted inter-story drift. This tendency can be explained by the structural load-transfer mechanism of wooden shear-wall systems. In wooden houses, the lateral deformation during seismic excitation is typically dominated by the shear deformation of the walls aligned with the loading direction. Therefore, the overall lateral stiffness of the building is strongly related to the effective wall length in the excitation direction. Because the wall groups W1, W3, and W4 represent major wall segments in this direction, their structural properties significantly influence the global seismic response. Consequently, the high importance values identified by the sensitivity analysis are considered structurally reasonable.

In contrast, in the nonlinear range (Figure 11), the sensitivity values of the dominant parameters increase significantly. In particular, W1 remains the most influential parameter across nearly all methods, while W3 and W4 also maintain relatively high importance.

This result reflects the physical behavior of the structure under strong excitation. When the structural response enters the nonlinear regime, plastic deformation occurs mainly in the shear-resisting components, such as braces and shear walls. Consequently, parameters related to these elements exert a stronger influence on the inter-story drift response.

At the same time, the influence of other parameters becomes relatively smaller compared with the dominant wall parameters. This indicates that once nonlinear behavior develops, the structural response becomes increasingly controlled by the key lateral-resisting elements.

Overall, five methods—SHAP, Structural Perturbation, Drop-column Importance, Permutation Importance, and Sobol analysis—show broadly consistent sensitivity rankings. These methods consistently highlight the same dominant parameters, suggesting that the surrogate model successfully captures the underlying structural behavior.

However, the Morris method shows slightly different trends compared with the other approaches. In particular, the Morris sensitivity indices tend to produce relatively larger values for several parameters that appear less influential in the other methods.

This difference can be attributed to the methodological characteristics of the Morris screening approach. Unlike variance-based methods such as Sobol, the Morris method evaluates sensitivity based on elementary effects along discrete sampling trajectories in the parameter space. As a result, the Morris indices are more sensitive to local nonlinearities and interaction effects between parameters. In structural systems where nonlinear behavior emerges, these local effects may lead to higher sensitivity estimates for certain parameters.

Despite these differences in magnitude, the Morris method still identifies the same dominant wall-related parameters, indicating that the overall interpretation of influential variables remains consistent across the different sensitivity analysis techniques.

The agreement among multiple sensitivity analysis methods provides confidence in the reliability of the ML surrogate model. The results demonstrate that the proposed explainable AI framework can successfully identify structurally meaningful parameters that govern the seismic response of the investigated wooden house model. Furthermore, the analysis highlights that the importance of wall-related parameters becomes even more pronounced in the nonlinear response range, which is consistent with the expected structural mechanics.

It should be noted that some structural parameters, particularly wall and joint properties, may exhibit structural dependence. In such cases, importance measures based on feature permutation or feature removal may be affected by multicollinearity, potentially leading to biased estimates of individual contributions.

To quantitatively evaluate the agreement among different sensitivity analysis methods, Spearman rank correlation coefficients were calculated based on the importance rankings obtained from each method. The results are shown in Figure 12.

In the 20% excitation case (Figure 12a), the Spearman correlation coefficients between most method pairs are extremely high, ranging from 0.94 to 1.00. This indicates that the importance rankings produced by different sensitivity analysis methods are almost identical in the elastic response regime. In particular, SHAP, structural perturbation, Sobol indices, and the Morris method show nearly perfect agreement, suggesting that the dominant structural parameters are consistently identified regardless of the analysis technique used.

In the 100% excitation case (Figure 12b), which corresponds to the nonlinear response range, the correlation values are slightly lower but still remain relatively high. The Spearman coefficients generally range between 0.56 and 0.97, indicating moderate to strong agreement among the methods. The reduction in correlation is considered reasonable because nonlinear structural behavior introduces stronger parameter interactions and more complex response characteristics, which can lead to differences in how sensitivity analysis methods evaluate parameter importance.

Overall, these results quantitatively support the statement that the sensitivity analysis methods show broadly consistent trends, particularly in identifying the most influential structural parameters. The strong correlations observed among multiple methods increase confidence in the robustness of the obtained parameter importance rankings.

The results of this study also have practical implications for structural engineering practice. The identification of dominant parameters influencing seismic response can support more efficient structural design and assessment. For example, the importance rankings consistently indicate that certain wall groups have a strong influence on inter-story drift. This suggests that the arrangement and stiffness of walls in the loading direction play a critical role in controlling seismic deformation of wooden houses.

From an engineering perspective, these findings can contribute to the prioritization of structural reinforcement strategies. By identifying the parameters that most strongly influence seismic response, engineers can focus on improving the stiffness and strength of critical structural components, thereby achieving more effective seismic performance improvements.

Furthermore, the use of a machine-learning-based surrogate model combined with multiple sensitivity analysis techniques provides a practical framework for evaluating parameter importance in complex structural systems. This approach can help engineers better understand the influence of design variables and support decision-making in performance-based seismic design.

4. Conclusions

This study presented a machine learning surrogate framework for evaluating the seismic response of a wooden house and investigated the effectiveness of multiple sensitivity analysis methods for identifying influential structural parameters. Linear regression and Gradient Boosting surrogate models were developed to approximate the nonlinear time-history analysis results of the structural model. The surrogate demonstrated high predictive accuracy, achieving a coefficient of determination of $R^2=0.90$ for the maximum inter-story drift.

Using the trained surrogate model, six sensitivity analysis approaches—SHAP, structural perturbation analysis, drop-column importance, permutation importance, Sobol sensitivity analysis, and the Morris method—were applied to evaluate the influence of structural parameters. The results showed that most methods consistently identified wall-related parameters, particularly W1, W3, and W4, as dominant contributors to the seismic response. This tendency was observed in both the elastic and nonlinear response ranges. From a structural perspective, this result is consistent with the mechanical behavior of wooden houses, where the seismic response is largely governed by the stiffness and deformation capacity of shear-resisting wall elements aligned with the loading direction.

In the nonlinear response range, the influence of these dominant parameters became even more pronounced. This behavior reflects the structural mechanism under strong ground motion, where plastic deformation concentrates in the shear-resisting components such as braces and shear walls. As a result, the global structural response becomes increasingly governed by a limited number of key parameters.

A comparison among the sensitivity analysis methods showed that SHAP, Structural Perturbation, Drop-column Importance, Permutation Importance, and Sobol analysis produced broadly consistent rankings of parameter importance. In contrast, the Morris method showed slightly different sensitivity magnitudes due to its screening-based formulation, which is more sensitive to local nonlinearities and interaction effects in the parameter space. Nevertheless, the Morris method still identified the same dominant wall-related parameters, indicating that the overall interpretation of influential variables remained consistent across the different approaches.

The agreement among multiple sensitivity analysis methods confirms the reliability of the ML surrogate model and demonstrates the potential of explainable AI techniques for structural response analysis. The proposed framework provides an efficient approach for interpreting complex structural simulations and identifying critical parameters governing seismic behavior. Such insights may contribute to improved structural modeling, parameter calibration, and future studies on the application of machine learning in earthquake engineering.

This study focuses on the maximum inter-story drift of the first story as a representative indicator of seismic performance. Although this metric is widely used in the seismic assessment of wooden houses, future studies should consider multiple response quantities, including upper-story drift, bidirectional responses, and additional structural performance indicators in order to further extend the applicability of the proposed framework.

In addition, the present analysis is based on a single two-story wooden house model under a specific set of simulation conditions. Therefore, the findings should be interpreted within the scope of the analyzed structural configuration and seismic input. Further studies involving different building layouts, structural systems, and earthquake records are necessary to verify the general applicability of the proposed framework.

It should also be noted that some structural parameters may exhibit correlations due to physical relationships within the structural system. Such feature dependence can influence importance measures based on variable permutation or removal, such as permutation importance and drop-column importance. In this study, the use of multiple sensitivity analysis methods with different theoretical foundations helps mitigate this issue by enabling cross-validation of parameter importance. Nevertheless, further investigation of feature dependence and its impact on interpretability methods remains an important topic for future research.