# Predictive Modeling and Experimental Analysis of Cyclic Shear Behavior in Sand-Fly Ash Mixtures
## 5 Data Analysis
The data used for this model is derived from the fifth cycle of loading, which represents the final loading cycle in a constant load shear test setup. With 222 data points in total, 85% were allocated for training and the remaining 15% for testing, ensuring the highest prediction performance. This Nonlinear Autoregressive Model with External Input (NARX) architecture leverages time series data and is specifically structured to handle scenarios where the output depends on both past values of itself and external inputs. Here, the external input represents the fly ash content in the mixture and horizontal strain, while the model predicts the shear stress based on both current and past values, incorporating a loading step delay. The graph depicts a hysteresis loop, which is a phenomenon commonly observed in materials that exhibit nonlinear elastic behavior in which the loop is related to shear stress and shear strain. The network's architecture is straightforward yet robust, featuring a hidden layer with 10 neurons to allow for sufficient complexity in learning nonlinear relationships between the input and output (Figure 10). Bayesian regularization was employed as the training algorithm, a choice that helps prevent over fitting by adding a penalty for overly complex models thus improving generalization performance even with a relatively small dataset. After 289 epochs, the training process reached convergence with an impressively low Mean  
Figure 8: Schematical demonstration of discretization of hysteresis loop. Squared Error (MSE) of 1.7321, along with a correlation coefficient (R) of 0.9997, indicating that the model accurately fits the training data and captures the underlying relationships between input parameters and shear stress outputs. For the testing phase, which included 33 data points, the model maintained high predictive accuracy, with an R value of 0.9696, though MSE rose to 150.2853. This increase in MSE for the test set, while still indicative of good performance, suggests a minor degree of error, which could result from limited data diversity or sample size in the testing dataset. Additionally, the model was evaluated on an additional test set comprising the full 222 data points, achieving an R value of 0.9957 and an MSE of 24.0151. This further  
Figure 10: The architecture of the NARX model. Figure 9: The flowchart of the analysis. validation underscores the model's capability to generalize well, though the slight rise in MSE across non-training data highlights that some error is introduced when predicting new or slightly different conditions, possibly indicating a degree of over fitting. The model, with its optimized structure and use of Bayesian regularization, demonstrates an effective solution for capturing the cyclic shear behavior of fly ash-sand mixtures. The high R values across training and testing phases reflect the model's strong predictive capabilities, and the architecture's reliance on past shear stress values with a time delay of two allows it to accurately model the cyclic nature of the material's response. Apparently, this setup positions the model as a powerful tool for predicting cyclic behavior in geotechnical applications, even though minor improvements in generalization might further enhance performance in future studies. The performance plots of the NARX model provide insights into its ability to capture the measured stress response of plain sand and sand-fly ash mixtures during cyclic loading, as evaluated through the cyclic direct shear test. In this model, the output values represent the measured shear stress during testing, while the target data represents the stress values recorded during the fifth (and final) cycle of loading, reflecting the model's goal of predicting behavior at this ultimate cycle. In Figure 11, the plot shows the Mean Squared Error (MSE) over 289 epochs, highlighting error convergence during training and testing.