Multi-verse optimizer (MVO)MVO, short for Multi-Verse Optimizer, draws inspiration from three cosmological phenomena: white holes, black holes, and wormholes. This algorithm applies the concepts of black holes and white holes for search space exploration, with wormholes utilized to exploit these spaces. Initially, a collection of random universes is created21,22. During each iteration, entities from high-inflation universes have a tendency to shift to low-inflation universes through white and black holes. Simultaneously, random teleportations occur towards the optimal universe through wormholes. The algorithm computes two parameters to control the extent and frequency of solution alterations22:$$\text{Probability of Wormhole Existence (PWE)}=a+t\left(\frac{b-a}{T}\right)$$
(3)
$$\text{Rate of Travelling Distance (RTD)}=1-{t}^{1/p}\left(\frac{1}{{T}^{1/p}}\right)$$
(4)
where a refers to the minimum value, b denotes the maximum value, t indicates the current iteration, T stands for the total number of iterations, and p defines the accuracy of exploitation. The updated positions of solutions are derived by subtracting the computed values of PWE and RTD into the following equation22:$${x}_{ji}= \left\{\begin{array}{c}{x}_{j}+RTD+\left(\left({u}_{j}-{l}_{j}\right)\cdot {r}_{4}+{l}_{j}\right) {r}_{3}<0.5\\ {x}_{j}-RTD+\left(\left({u}_{j}-{l}_{j}\right)\cdot {r}_{4}+{l}_{j}\right) {r}_{3}\ge 0.5\end{array}\right.$$$${x}_{ji}={x}_{j}^{{\text{Ro}}\text{ulette Wheel}}\hspace{1em}\text{if }{r}_{2}\ge PWE$$where \({x}_{j}\) represents the j-th element of the best solution, \({l}_{j}\) and \({u}_{j}\) are the lower and upper bounds of the j-th element, respectively. Moreover, \({r}_{2}\), \({r}_{3}\), \({r}_{4}\) denote random numbers sampled from the range [0, 1], \({x}_{ji}\) denotes the j-th parameter in the i-th solution, and \({x}_{j}^{Roulette wheel}\) is the j-th element of a solution selected using the roulette wheel selection method. The algorithm facilitates exploration and exploitation by varying \({r}_{2}\), \({r}_{3}\), and \({r}_{4}\). Initially, a set of random solutions is generated, and their respective objectives are computed. The positions of the solutions are then updated using the above equations. This process is repeated until a termination criterion is met.Bagging ensemble modelBootstrap Aggregation, commonly referred to as bagging, stands as a widely utilized ensemble technique. Bagging Regression culminates in the final prediction by aggregating the predictions of individual models derived from a randomized selection of the original data through averaging (for regression) or voting (for classification). This meta-estimator is proficient in substantially diminishing the variance it produces by incorporating a randomization mechanism in its prediction formulation. Furthermore, this approach proves beneficial in mitigating overfitting concerns associated with complex algorithms. Boosting regressions, on the other hand, yield more precise results through the utilization of weak (base) models23,24. Figure 4 illustrates the comprehensive architecture of the bagging process.Fig. 4Fig. 5Predicted and reference concentration values using BAG-KNN model.Fig. 6Predicted and reference concentration values using BAG-PR model.Fig. 7Predicted and reference concentration values using BAG-TWR model.Polynomial regression (PR)Polynomial regression (PR) is beneficial when there is evidence to suggest that the relationship between two variables is not linear, but instead follows a curved pattern. This approach represents the correlation between output parameter and multiple input parameters by employing a polynomial model25,26. Considering the dependent parameter as y and an independent parameter x, the equation for polynomial regression is given by27:$$y={a}_{0}+{a}_{1}x+{a}_{2}{x}^{2}+\dots +{a}_{n}{x}^{n}$$
(5)
Here, x denotes the independent variable, y stands for the dependent variable, \({a}_{0},{a}_{1},{a}_{2},\dots ,{a}_{n}\) are the coefficients, and n stands for the polynomial’s degree. The degree of the polynomial controls the form of the curve most suitable for the data.K-nearest neighbors (KNN)KNN regression works by determining the k closest neighbors to a new unseen data point within the feature space. Following this step, the model predicts the output by calculating the average or weighted average of the output values derived from those k neighbors. The algorithm can be summarized in the following steps28:
1.
Using a distance metric, estimate how far the new data is from the rest of the training set:$$d\left({x}_{i},{x}_{j}\right)=\sqrt{{\sum }_{k=1}^{n}{\left({x}_{ik}-{x}_{jk}\right)}^{2}}$$
(6)
where \({x}_{i}\) and \({x}_{j}\) are two data points in the feature space, n denoted the count of features, and \({x}_{ik}\) and \({x}_{jk}\) are the values of k-th feature for i–th and j-th data points.
2.
Choose the k-nearest neighbors by considering the calculated distances.
3.
Take the mean or weighted average of the outputs of the KNN to arrive at the output value for the new data point:$$\widehat{y}=\frac{{\sum }_{i=1}^{k}{w}_{i}{y}_{i}}{{\sum }_{i=1}^{k}{w}_{i}}$$
(7)
Here, \(\widehat{y}\) stands for the predicted output value, \({y}_{i}\) is the output value of the i-th nearest neighbor, and \({w}_{i}\) stands for the weight of the i-th closest neighbor. The weights can be assigned according to the inverse of the distance to the new data point.
KNN regression has several advantages over other regression algorithms. It is effective for handling linear and non-linear connections between input features and the target variable, and it can be implemented quickly29,30.Tweedie regression (TWR)The Tweedie regression model is a specific form of Generalized Linear Model (GLM) that is particularly suitable for analyzing data that is non-negative, significantly right-skewed, and exhibits both symmetric and heavy-tailed characteristics. It is particularly useful for continuous data that may have a probability mass at zero31. A random variable Y is considered to follow a Tweedie distribution if its density function is a member of the class of exponential dispersion models (EDM). The density function can be represented by the following expression32:$${f}_{Y}\left(y;\upmu ,\upphi ,p\right)=a\left(y,\upphi ,p\right)\text{exp}\left\{\frac{y\uppsi -k\left(\uppsi \right)}{\upphi }\right\}$$
(8)
where \(\upmu =E\left(Y\right)={k}{\prime}\left(\uppsi \right)\) denotes the mean, \(\upphi > 0\) signifies the dispersion parameter, \(\uppsi\) represents the canonical parameter, and \(k\left(\uppsi \right)\) corresponds to the cumulant function. Here is the formula for the variance of Y32:$${\text{Var}}\left(Y\right)=\upphi V\left(\upmu \right)=\upphi {\upmu }^{p}$$
(9)
Here, p is the power parameter, which determines the form of the variance function and thus the specific distribution within the Tweedie family. Tweedie regression models are extensively used in various fields, including insurance (for modeling claims data), biology, ecology, and econometrics. The flexibility of the Tweedie distribution makes it an excellent choice for modeling data that exhibit both continuous and discrete characteristics, such as insurance claims that have a large number of zeros and positive continuous values. Estimation of the parameters in Tweedie regression models can be challenging due to the complex form of the density function. The paper discusses two alternative estimation methods: quasi-likelihood and pseudo-likelihood. These methods are computationally simpler and faster than the traditional maximum likelihood estimation, especially when the power parameter p falls within complex ranges.