Optimization of Fenoprofen solubility within green solvent through developing a novel and accurate GSO-GPR predictive model

3.1 Models

The Gaussian process (GP) is a collection of random parameters, a subset of which is associated with Gaussian distributions (Grbić et al., 2013). Covariance and mean functions characterize the GP. Data from the past must be linked in order for Gaussian process regression (GPR) models to work. In contrast to the GD, the GP is over functions. The predictive distribution of the test input is therefore understood by Gaussian process regression models (Rasmussen, 2003).

For GPR, it is not necessary to define the exact fitting function. The data collected in the field may be compared to a multi-dimensional Gaussian distribution sampled at random places (Quinonero-Candela and Rasmussen, 2005; Jiang et al., 2021).

y is exhibited as $f\left(\mathit{x}\right)$ , $D=\left\{\left({\mathit{x}}_i,y_i\right)|i=1,2,\cdots ,n\right\}$ , ${\mathit{x}}_i\in R^d$ as input matrix and $y_i\in R$ as output. $$y=f\left(\mathit{x}\right)$$ $$f\left(\mathit{x}\right)\sim GP\left(m\right(\mathit{x}),\mathbf{K})$$

K illustrates any covariance, that is explained through kernels and their corresponding, m(x) is the mean operator (Wu et al., 2020).

Many regression models employ the K-Nearest Neighbor (KNN) technique, which combines numerous supervised learning models. A basic regression and classification algorithm, this is the model under consideration (Aha et al., 1991; Ribeiro and dos Santos Coelho, 2020). Since it does not generalize from the training examples, it is called a “lazy algorithm.” Indeed, it keeps all of the data collected over the course of the testing process (Song et al., 2017). Using KNN regression is very simple. In order to perform that, it has to be calculated that the amount of neighbors with similar numerical target (Deng, 2020). Another approach is to weight the nearest neighbors to the center. To utilize regression, you use the same distance functions as KNN classification to analysis the samples. This is how you express the Euclidian (Euc_Distance) and the Manhattan distance (Man_Distance). Those equation which are shown below, demonstrate the determination of distance among × and y. $$Euc\_Distance=\sqrt{{\sum _{i=1}^k(x_i-y_i)}^2}$$ $$Man\_Distance=\sum _{i=1}^k|x_i-y_i|$$

KNN is trained through make a comparison among test data (X,y) and a training data $\text{D}=\left\{\left({\text{X}}_{\text{i}},{\text{y}}_{\text{i}}\right)\right\}$ . In regression, the last prediction of y is the mean over its k nearest neighbors' results, as shown in the equation below (Cheng and Ma, 2015; Devroye et al., 1994). $$\widehat{y}=\frac{1}{k}\sum _{i=1}^k{y}_i\left(X\right)$$

The other used model in this study is modular ANN (MANN). In order to develop MANN, the data points must be separated into multiple clusters, and then each learner is utilized to each cluster separately. In the current work, the fuzzy c-means clustering method is used (Bezdek, 2013; Wang et al., 2006). Soft or crisp clusters may be made. ANN (or equivalent methods) cannot extrapolate beyond the range of data provided for training. Under other conditions, when new data point exceeds the interval of those involved for model training, inaccurate predictions might be predicted. Fig. 1 shows the MANN diagrammatic model in which the training set is divided into four clusters. After input–output pairings are produced, the fuzzy c-means (FCM) technique divides them into four subgroups, and any subset is estimated by an ANN. The modular model's output comes from one of three local models.

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Fig. 1

Flowchart of MANN (https://theses.lib.polyu.edu.hk/bitstream/200/5912/1/b23930640.pdf, 2010).

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3.2 Model optimization

In this research, we employed glowworm swarm optimization (GSO) approach in order to tune hyper-parameters of models. Krishnanand and Ghose (Krishnanand and Ghose, 2005) created the GSO algorithm, which is an improvement of the ACO. It was driven by the glowworm metaphor and adapted to collaborative robots. Each artificial glowworm or agent in GSO has a local-decision range that lets it light up a two-dimensional work environment. The luciferin level is related to the agent's position's objective value. The brighter agent will fly to a better position. The number of neighbors influences the local decision range. When the density of neighbors is low, the range is widened to discover additional individuals; otherwise, the range is narrowed. The agent's movement direction is always determined by which adjacent individual is chosen. The higher the luciferin level of the neighbor, the greater the magnetism. Finally, most agents will congregate at various sites (Wu et al., 2012).

All individuals (glowworms) have the identical value of luciferin, however it changes according on their response of function. The luciferin value at that site is well proportioned to the observed value of the sensor profile. Every glowworm increases its previous level of luciferin (Manimaran and Selladurai, 2014). Simultaneously, the glowworm's luciferin level is deducted from the prior luminescence measurement to imitate luminosity degradation. The rule of luciferin updating is: $$l_i(t+1)=(1-\rho )l_i\left(t\right)+\gamma J_i(t+1)$$

In the above equation, l_i(t) is the level of luciferin for the individual I based on the time t, ρ stands for the luciferin decay constant 0 < ρ < 1, γ reflects the luciferin enhancement constant, moreover, J_i denotes the objective function at individual i's position based on the time time, t.

The chance of traveling toward an adjacent j for each glowworm I is determined by (Muller, 2007): $$p_{\mathit{ij}}=\frac{l_j\left(t\right)-l_i\left(t\right)}{{\sum }_{k\in N_i\left(t\right)}{l}_k\left(t\right)-l_i\left(t\right)}$$

Where $j\in N_i\left(t\right)\text{,}{N}_i\left(t\right)=\left\{j:d_{i\text{,}j}\left(t\right)<r_d^i\left(t\right)\text{;}{l}_i\left(t\right)<l_j\left(t\right)\right\}$ stands for the neighborhood for the glowworm i in iteration t. d_i,j(t) shows the distance of individuals i and j at iteration or time t, moreover, rⁱ_d(t) stands for the variable neighborhood range amalgamated via individuals i at time t. Assume that individual i choose an individual j ∈ N_i(t) with p_ij(t) determined by the above equation. Then, motions of glowworms might be expressed as (He, 2022): $$x_i(t+1)=x_i\left(t\right)+s\left(\frac{x_j\left(t\right)-x_i\left(t\right)}{\Vert x_j\left(t\right)-x_i\left(t\right)\Vert }\right)$$

In this equation, s shows the step size described above and ∥∥ reflects the Euclidean norm operator.

The rule for neighborhood range updating: We connect with each individual i a neighborhood which radial range rⁱ_d is naturally dynamic 0 < rⁱ_d < rs.rs represents the radial range of the luciferin sensor (Krishnanand, 2009).

The neighborhood range is difficult to establish at a value that works well for diverse function landscapes since we suppose that a priori knowledge about the fitness function is unknown. A specified neighborhood range r_d, for example, would perform better on fitness functions where the optimum inter-peak distance is greater than r_d than on those where it is smaller than r_d. This optimization algorithm employs an adaptive neighborhood range to locate numerous peaks in a multimodal function landscape. The following rule may be observed to significantly boost performance (Abdullah, 2021; Zhou et al., 2013; Krishnanand, 2009): $$r_d^i(t+1)=\mathrm{m}\mathrm{i}\mathrm{n}\left\{rs\text{,}\mathrm{m}\mathrm{a}\mathrm{x}\left\{0\text{,}{r}_d^i\left(t\right)+\beta (nt-|Ni\left(t\right)\left|\right)\right\}\right\}$$

Where β stands for a constant value and nt is utilized to control the quantity of neighbors. The workflow of GSO is displayed in Fig. 2.

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Fig. 2

The Flowchart of the GSO algorithm.

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3.3 Performance evaluation

In our study, the accuracy of the utilized regression models are assessed using the criteria listed below. The following are the definitions of metrics: $$\mathit{MSE}=\frac{1}{N}\sum _{i=1}^N{(y_i-{\widehat{y}}_i)}^2$$ $$\mathit{RMSE}=\sqrt{\frac{1}{N}{\sum }_{i=1}^N{(y_i-{\widehat{y}}_i)}^2}$$ $$\mathit{MAPE}=\frac{1}{N}\sum _{i=1}^N\left|\frac{y_i-{\widehat{y}}_i}{y_i}\right|$$

Here, $y_i$ is the expected (observed) solubility, ${\widehat{y}}_i$ is the forecasted solubility, and N is the quntity of data points.