Jellyfish Optimizer (Joe)
This study employs several numerical optimization methods described in the next section. Jo is a metaheuristic algorithm motivated by the collective behavior of jellyfish. This algorithm is inspired by the natural behavior of jellyfish herds, particularly foraging strategies and movement dynamics, to address complex optimization tasks. Designed to effectively navigate the solution space and converge to optimal results16.
The algorithm repeatedly improves solutions through a combination of exploitation and exploitation strategies. The individual's position update for each jellyfish is mathematically represented by the following equation16:
$$\:{x}_{ij}\left(t+1\right)={x}_{ij}\left(t\right)+{\delta\:}{x}_{ij}\left(t\right)$$
here, \(\:{x}_{ij}\left(t+1\right)\) Represents the new position of the jellyfish \(\:I\) In dimensions \(\:j \) On time \(\:t+1 \), \(\:{x}_{ij}\left(t\right)\) It's the current location \(\:{\delta \:}{x}_{ij}\left(t\right)\) Represents the displacement vector17.
Automatic Relationship Determining Regression (ARD)
ARD regression is an advanced Bayesian modeling approach that performs simultaneous function selection and coefficient estimation by inferring the importance of predictors from the data. This model is based on Bayesian principles and provides a principled approach to feature selection and normalization.18.
In ARD regression, the goal is to predict the target variable \(\:y \) Based on a specific input vector x. The model is written as follows:11:
$$ y = w^{t} x + \varepsilon $$
ARD inference and learning is carried out within the Bayesian framework. Variable Bayesian techniques have been employed to estimate the backward distribution of both model weights and related hyperparameters10. These techniques aim to reduce the discrepancy between the exact posterior distribution and its approximate counterpart. The normalization parameter (α) is usually learned by maximizing the evidence or posterior probability of the model and uses methods such as type II maximum likelihood methods and empirical Bayesian estimation.
ARD regression offers the advantages of automated feature selection by assigning individual association weights to input features, effectively reducing model complexity and improving interpretability. However, high-dimensional data may be computationally intensive and may require careful adjustment of hyperparameters to avoid overfitting or underfitting.
Linear regression (LR)
LR assumes a linear relationship between input functions and target variables and aims to find the optimal linear equation that explains this relationship19,20.
Consider a dataset containing \(\:n \) Data points, each sample is shown as a vector \(\:xi = \left({x}_{i1}, {x}_{i2}, \dots\:, {x}_{im}\right)\) of \(\:m \) Input variables and corresponding target variables are \(\:{y} _{i} \). The purpose of LR is to find a set of coefficients \(\:{\upbeta\:}=\left({{\upbeta\:}}_{0}), {{\upbeta\:}}_{1}, \dots\:,{{{\upbeta\:}}}\ Like a linear equation12:
$$y_{i} = \beta_{0} + \beta_{1} x _{{i1}} + \beta_{2} x _{{i2}} + \ldots + \beta_{m} x _{im}} + \_^^^^{im}}
It provides the best approximation of the relationship between the input function and the target output. here, \(\:{{\upbeta \:}} _ {0} \) It represents an intercept term, \(\:{\upbeta \:}} _ {j} \) (for \(\:j \:= \:1, \:2, \:\dots \:, \:m \)) shows the coefficients related to \(\:j \)-th input variable, and \(\varepsilon _{i}\) Means an error term that captures unexplained variability of a target variabletwenty one.
The usual least squares (OLS) method is frequently used in LR to estimate coefficients. The purpose of OLS is to minimize the sum of squared mismatches between the expected target variable and the model predicted value derived from the linear equation. Mathematically, this can be formulated as12:
$$\:\anderset {{\upbeta \:}}{\text {min}}{\sum \:}_{i=1}^{n}{\left({y}_{i} – \left({{\upbeta \:}}}_{0}+{{\upbeta \:}}_{1}{x}_{i1}+{{\upbeta \:}}_{2}{x}_{i2}+\dots \:+{{\upbeta \:}}_{m}{x}_{im}\right)\right)}^{2}$$
Once the coefficients are estimated, the linear regression model provides a mechanism for estimating the target variable \(\:\widehat {y} \) For new observations based on predictor values \(\:x \). The predicted value is calculated as12,22:
$$\:\widehat {y}={{\upbeta \:}}_{0}+{{\upbeta \:}}_{1}{x}_{1}+{{\upbeta \:}}_{2}{x}_{2}+\dots \:+{{\upbeta \:}}_{m}{x}_{m}$$
where \(\:\varvec{x}=\left({x}_{1}, {x}_{2}, \dots\:, {x}_{m}\right)\) Represents the input value for the new observation.
LR is simple, fast, easy to interpret, and a powerful baseline for many prediction tasks. Its main limitation is the assumption of linear relationships between variables, which can reduce the performance of complex and nonlinear data. Additionally, it can be made sensitive to multicollinearity and outliers.
