## 5 Results  
In this section, we present the results of analyzing the growth data of indoor-cultured abalone, as introduced in Section 4.1. We begin by applying the proposed weighted Bayesian Kernel Machine Regression (WBKMR) to the data and discussing the findings. Then, we assess the effectiveness of the model by comparing its predictive performance with that of the well-known BKMR model. The BKMR model assumes equal variance of errors. In all experiments, we draw 10,000 samples from the Markov chain Monte Carlo (MCMC) algorithm (Appendix A), discarding the first 5000 as burn-in, and use the remaining 5000 as posterior samples for inference. The experiments are conducted on an iMac Pro (Apple Inc., Cupertino, CA, USA) with 128 GB of 2666 MHz DDR4 memory and a 3 GHz 10-Core Intel Xeon W CPU. Before presenting the results of the data analysis, we first examine the convergence of the MCMC algorithm using trace plots for selected parameters, as shown in Figure 4. The trace plots indicate that the MCMC chain has reached convergence, as evidenced by the stable fluctuations around a central value with no clear trends. This suggests good mixing and effective exploration of the parameter space. Below, we present the results of the data analysis. We start by examining the importance of key environmental factors in predicting abalone growth. Then, we provide the estimated effects of these factors on abalone growth. Figure 5 provides the Posterior Inclusion Probability (PIP) for each key environmental factor, including time, DO, salinity, NS, pH, and water temperature. The PIP is derived from the spike-and-slab prior, which enables the identification of influential variables by quantifying the probability of each factor being included in the model. As shown in the figure, variables such as time, DO, salt, and NS exhibit high PIP values, underscoring their significant roles in predicting abalone growth. Specifically, the high PIP for time reflects the natural progression of growth over the study period, while the elevated PIP for DO, salt, and NS suggests their influence on the environmental conditions that support abalone growth. In contrast, the low PIP value for water temperature suggests that its influence on growth is minimal within the observed environmental conditions. The intermediate PIP for pH suggests a moderate, yet less pronounced, role compared to the other variables. Figure 4: Trace plots of the linear effects \(\beta\) for checking the convergence of the MCMC algorithm. Figure 6 illustrates the estimated effects of key environmental factors selected based on PIP on abalone growth. The blue line in each plot represents the relationship between each factor and the growth function \(f(x)\), while the shaded regions indicate the 95% credible intervals, reflecting the uncertainty in the estimates. The effect of DO shows a negative trend, suggesting that higher DO levels may slightly inhibit growth, though the effect remains uncertain. The influence of pH appears minimal, as evidenced by a nearly flat relationship, indicating a neutral effect on growth. Salt concentration shows a positive effect on growth, whereas nutrient supply has a non-linear relationship, with growth increasing as supply rises but eventually leveling off. Time exhibits a strong positive influence on growth, reflecting the natural progression of abalone growth over the study period. Figure 5: Barplot of Posterior Inclusion Probability (PIP) representing the Importance of key environmental factors. Figure 6: _Cont._  
Next, we present the estimated joint effects of pairs of environmental factors (DO, supply, salt) on abalone growth in Figure 7. In the figure, the first plot on the left shows the combined effect of DO and supply, with red areas indicating higher estimated growth values and blue areas indicating lower growth. This suggests that higher supply levels paired with lower DO values are associated with increased growth.