# Initialisation Approaches for Population-Based Metaheuristic Algorithms: A Comprehensive Review
## 3 Major Initialisation Methods
### Probability Distributions  
The probability distribution describes the possible values and likelihood that a random number is effective within a defined interval. Different probability distributions and their rigorous statistical properties can be used to initialise the population of metaheuristic algorithms. [PERSON], [PERSON], and [PERSON] [13] used variants of Beta distribution, uniform distribution, normal distribution, logarithmic normal distribution, exponential distribution, Rayleigh distribution, Weibull distribution, and Latin hypercube sampling [31] to form 22 different initialisation schemes in order to evaluate PSO, CS, DE, ABC, and GA. The variants of the probability distributions are as follows:  
* Beta distribution  
The Beta distribution is a continuous probability distribution over the interval (0,1). It can be written as \(X\sim Be(a,b)\). Varying the values of \(a\) and \(b\) resulted in a variant of \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Reference** & **Year** & **Initialisation Scheme** & **Optimisation Problem** \\ \hline
[21] & 2007 & [PERSON], [PERSON], and Faure & Benchmark test functions \\ \hline
[8] & 2008 & Van [PERSON] and Sobol sequences & Benchmark test functions \\ \hline
[25] & 2018 & A quasirandom sequence Torus & Benchmark test function \\ \hline
[22] & 2020 & [PERSON] Faure and Sobol sequences & Benchmark test function \\ \hline
[26] & 2005 & Halton low-discrepancy sequence & Benchmark function \\ \hline
[27] & 2005 & Sobol and Halton sequences & Benchmark function \\ \hline
[23] & 2002 & Faure sequences & Benchmark functions \\ \hline
[28] & 2009 & Sobol sequence & Benchmark functions \\ \hline
[29] & 2021 & Halton, Sobol, and Torus & Benchmark functions \\ \hline
[30] & 2012 & Sobol sequences & Image Segmentation \\ \hline
[31] & 2017 & Latin hypercube sampling (LHS) & Optimisation of structural components under Fatigue \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of quasirandom methods. Figure 9: Halton sequence after 1000 th iteration. the Beta distribution, generating sequences with different behaviours in the search space. Three variants of the Beta distribution were used,  
* Uniform distribution A uniform distribution is defined over the interval [a, b], and it is usually written as \(X\sim U(a,b)\). One variant of the normal distribution was used. * Normal distribution The Gaussian Normal distribution is usually written as \(X\sim N\big{(}\mu,\sigma^{2}\big{)}\). In addition, varying the values of \(\mu\) and \(\sigma^{2}\) resulted in three (3) variants of the normal distribution, which generates sequences with different behaviours in the search space. * Logarithmic normal distribution The logarithmic normal distribution is often written as \(lnX\sim N\big{(}\mu,\sigma^{2}\big{)}\). Four (4) variants of the logarithmic normal distribution were created by varying the values of \(\mu\) and \(\sigma^{2}\). * Exponential distribution An exponential distribution is asymmetric with a long tail and can be written as \(X\sim\exp(\lambda)\). Varying \(\lambda\), resulted in three variants of the distribution which were used to initialise the population of the five algorithms. * Rayleigh distribution The Rayleigh distribution can be written as \(X\sim Rayleigh(\sigma)\). Three (3) variants of the distribution were created by varying the value of \(\sigma\). * Weibull distribution This distribution can be considered as a generalisation of a few other distributions. It can be written as \(X\sim Weibull(\lambda,k)\). For example, \(k=1\) corresponds to an exponential distribution, while \(k=2\) leads to the Rayleigh distribution. In the same vein, three variants of the distribution were created. The convergence and accuracy of five metaheuristic optimizers were evaluated on the benchmark test functions and the CEC2020 test functions. These optimisers are then initialised using 22 different initialization schemes [13].