## 4 Materials and Methods
### Weighted Bayesian Kernel Machine Regression  
We consider a semiparametric Bayesian model based on the Bayesian kernel machine regression (BKMR), which was proposed in [PERSON] et al. (2015) [5], with the Gaussian process (GP) prior (see, e.g., [PERSON] and [PERSON], 2006 [7]). as a novel probabilistic approach for predicting the growth of indoor-cultured abalone. To describe the model in detail, let \(\hat{y}_{t}=n_{t}^{-1}\sum_{j=1}^{n_{t}}\log y_{ij},t=1,\ldots,T\) represent the average log-transformed growth measurements of indoor-cultured abalones at time \(t\). Let \(\mathbf{z}_{t}=(z_{t1},\ldots,z_{tp})^{\top}\) be a covariate vector, including dummy variables representing species, and let \(\mathbf{x}_{t}=(x_{t1},\ldots,x_{tM})^{\top}\) be a \(M\)-dimensional vector of features influencing prediction of the growth of the indoor-cultured abalone at time \(t\), such as water temperature, dissolved oxygen, power of hydrogen, and water salinity. We introduce a weighted BKMR model, which includes both linear and nonlinear effects of the predictors, as  
\[\hat{y}_{t}=\mathbf{z}_{t}^{\top}\mathbf{\beta}+f(\mathbf{x}_{t})+\epsilon_{t},\ \ \epsilon_{t}\ \overset{indep.}{\sim}\ \mathcal{N}(0,\sigma^{2}/n_{t}), \tag{2}\]  
where \(\mathbf{\beta}=(\beta_{1},\ldots,\beta_{p})^{\top}\) is a \(p\)-dimensional coefficient vector and \(f(\cdot):\mathbb{R}^{M}\to\mathbb{R}\) is an unknown and non-linear function modeled through a GP prior as  
\[\mathbf{f}=(f_{1},\ldots,f_{T})^{\top}\sim\mathcal{GP}(0,\sigma^{2}K),\ f_{t}=f( \mathbf{x}_{t}). \tag{3}\]  
Here, \(K=(K_{il})_{l,j=1}^{T}\) is an \(T\times T\) positive semi-definite covariance matrix, and \(K_{il}:\mathbb{R}^{M}\times\mathbb{R}^{M}\to\mathbb{R}\) is a kernel function that governs the smoothness of the realizations derived from the GP and determines the extent of shrinkage towards the mean ([PERSON] et al., 2014 [12]). In the paper, we use the squared exponential kernel for the kernel function \(K_{il}\), which is defined as  
\[K_{il}\equiv K(\mathbf{x}_{i},\mathbf{x}_{t})=\lambda_{f}\exp\left[-\sum_{m=1}^{M} \gamma_{m}(x_{im}-x_{lm})^{2}\right],\]  
where \(\gamma_{m}\geq 0,\ m=1,\ldots,M\), are inverse length-scale parameters, and \(\lambda_{f}\) is a positive real-valued scaling parameter. The parameter \(\lambda_{f}\) determines the overall magnitude of the covariance matrix. It effectively controls the amplitude of the realizations from the GP, allowing the model to adapt to the scale of the response variable. The inverse length-scale  
Figure 3: Correlation plot of weight and growth-related environmental factors. parameters, on the other hand, determine the degree of influence of the \(m\)-th environmental factor. A larger \(\gamma_{m}\) value indicates a stronger sensitivity to changes in \(x_{m}\), which results in a smoother function along that dimension. Conversely, smaller values of \(\gamma_{m}\) allow for greater flexibility and variability in the GP realizations with respect to the corresponding predictor. For the linear fixed effects, we assign the widely used noninformative flat prior  
\[\pi(\mathbf{\beta})\propto 1,\]  
We assume gamma priors for the variance parameters \(\lambda_{f}\) and \(\sigma^{2}\) as  
\[\pi(\lambda_{f})=\mathcal{G}amma(a_{\lambda},b_{\lambda})\ \ \ \text{and}\ \ \ \pi(\sigma^{-2})=\mathcal{G}amma(a_{\sigma},b_{\sigma}),\]  
where \(a_{\lambda}\) and \(a_{\sigma}\) denote the shape parameters, and \(b_{\lambda}\) and \(b_{\sigma}\) are the rate parameters. We set \(a_{\lambda}=100\), \(b_{\lambda}=1\), \(a_{\sigma}=0.001\), and \(b_{\sigma}=0.001\). In the context of abalone growth prediction, identifying the importance of growth-related farming environment factors is of paramount importance. To achieve this, we utilize the following spike-and-slab prior for the factor weights \(\gamma_{m}\), which enables variable selection by distinguishing between relevant and irrelevant factors.