Gaussian Process Regression (GPR)
GPR is a stochastic regression model that defines Gaussian processes using a collection of random variables. The GPR model represents the target variable as a function of the input variable and aims to estimate the underlying function based on observed data. The GPR model is defined as follows:16:
$$ \:y = f \left(x \right) + \varepsilon $$
where y Represents a target variable. x Indicates the input variable. f It is the underlying function \(\varepsilon \) Represents a residual error17,18.
k-nearest Neighbors (knn)
KNN is a robust non-parametric ML model that estimates target points by utilizing collective information from K-recent neighbors in the feature space. KNN uses similarity between adjacent data instances to examine local characteristics of data sets for predictive purposes. If a new input is displayed, the algorithm finds the most comparable entries for K in the training data and determines the predicted value by averaging the results related to these nearby instances19,20.
The KNN model determines the predicted value y For specific test inputs x As13:
$$ \:y = \frac {1}{k}\sum \limits_ {i = 1}^{k}{y} _{i} $$
where \(\:{y} _{i} \) represents the target value of k Neighbor recently.
Multilayer Perceptron (MLP)
MLPs represent a type of artificial neuron architecture that is distinguished by several hierarchical layers of mutual link processing units, often referred to as artificial neurons. The architecture of MLP utilizes a nonlinear activation mechanism at the input received by individual neurons21,22.
The output of the MLP model is calculated by forward propagation, which involves passing input to the network layer. Predicted value y For specified input x It can be expressed as23,24:
$$\:y=f\left({w}_{2}\cdot\:{\upsigma\:}\left({w}_{1}\cdot\:x+{b}_{1}\right)+{b}_{2}\right)$$
In this equation, \(\:{w} _{1} \) and \(\:{w} _{2} \) Represents a weight matrix, \(\:{b} _{1} \) and \(\:{b} _{2} \) Indicates the bias vector \(\:{\upsigma \:} \) Shows the activation functions applied for each element.
Data Preprocessing
The following data preprocessing steps were performed before applying the regression model:
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Outlier detection: Anomalous data (outlier) identification was made through the application of Z-score technology. This approach measures the extent to which individual observations diverge from the central tendency of the distribution, expressed in standard deviation units compared to the mean. By determining the Z-score value for each data point, outliers that deviate significantly from the average can be identified. Typically, a threshold is set to determine whether a data point is considered an outliertwenty five.
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Normalization: In this study, normalization of the data was performed using the MIN-MAX scaler. By using the MIN-MAX scaler, the values for each feature can be effectively converted to a given range, typically ranging from 0 to 1. During the scaling process, the minimum value of the feature is subtracted and the result is divided by the range between the maximum and minimum values.26.
Hyperparameter tuning
In this study, we employed a Political Optimizer (PO) algorithm to effectively adjust the hyperparameters. PO represents an optimization strategy that draws conceptual inspiration from political dynamics and social interactions. This algorithm simulates political deliberations in which agents exchange perspectives, engage in mutual influence, and collaboratively advance towards shared solutions.27. The PO algorithm examines the hyperparameter domain through successive changes in positions related to a phoric political party, in which each party embody the unique composition of the hyperparameter. Through processes similar to opinion formation, mutual influence, and inter-party dynamics, algorithms systematically guide searches towards optimal configurations that strengthen targeted performance criteria.28,29.
