# Diversity Measures for Niching Algorithms

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## Abstract

**:**

## 1. Introduction

**Swarm diversity**which refers to the diversity with respect to the decision space, i.e., particle positions of the current swarm.**Niche diversity**which refers to the diversity with respect to the solution space, i.e., actual found solutions. Niching algorithms obtain multiple solutions by forming clusters of a few particles around the positions of optima. The cluster of particles is referred to as a niche while the fittest particle in the niche, the so called neighbourhood best (nbest), represents an optimum. Niche diversity refers to the spread of these neighbourhood bests in the solution space.**Phenotypical diversity**which refers to the diversity with respect to the objective space, i.e., the diversity with respect to the objective function values of the current particles or the nbests (depending on the research interest in question).

## 2. Diversity Measures Utilised in Population Based Algorithms

- Swarm, $S=\{{\mathbf{x}}_{1},\dots ,{\mathbf{x}}_{i},\dots {\mathbf{x}}_{n}\}(2<n<\infty )$, which refers to all particles in a PSO algorithm. In other complimentary techniques such as a GA, S refers to the population.
- Candidate solution, ${\mathbf{x}}_{i}$, which refers to a particle in a swarm, S. A candidate solution is incrementally adapted to find an optimum.
- Candidate niche, ${N}_{k}=\{{\mathbf{x}}_{i},\dots ,{\mathbf{x}}_{m}\}(m\le n)$, which is formed by a cluster of candidate solutions. A candidate niche can still be refined to find an actual solution, or optimum.
- Neighbourhood best (nbest), ${\widehat{\mathbf{x}}}_{i}\in {N}_{k}$, which is the particle with the best fitness in a candidate niche. This then represents an optimum.
- Solution(s): The final neighbourhood best or optimum (i.e., particle with the best fitness) in a particular identified niche.

#### 2.1. Swarm Diversity

#### 2.1.1. Sum of Distances

#### 2.1.2. Sum of Distances to Nearest Neighbour

#### 2.1.3. Average Distance around the Swarm Centre

#### 2.1.4. Average of the Mean Distance around All Candidate Solutions

#### 2.1.5. Entropy

#### 2.1.6. Solow–Polasky Diversity

#### 2.1.7. Swarm Diameter and Swarm Radius

#### 2.1.8. Other Measures of Diversity

#### 2.2. Niche Diversity

## 3. Materials and Methods

#### Determining Unique Solutions

## 4. Results and Discussion

#### 4.1. Sum of Distances

#### 4.2. Sum of Distances to the Nearest Neighbour

#### 4.3. Average Distance around the Swarm Centre

#### 4.4. Average of the Mean Distance around All Solutions

#### 4.5. Entropy

#### 4.6. Solow–Polasky Diversity

#### 4.7. Swarm Radius

#### 4.8. Swarm Diameter

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

PSO | Particle Swarm Optimisation |

DE | Differential Evolution |

GA | Genetic Algorithms |

ESPO | Enhanced Species-Based Particle Swarm Optimisation |

SD | Sum of Distances |

SDNN | Sum of Distances to Nearest Neighbour |

ADSC | Average Distance Around the Swarm Centre |

ADAA | Average of the Mean Distance around all Candidate Solutions |

SPD | Solow–Polasky Diversity |

SDM | Swarm Diameter |

SR | Swarm Radius |

## Appendix A. Branin

**Figure A1.**Branin Rcos at different iterations. (

**a**) ${f}_{1}$: iteration 50; (

**b**) ${f}_{1}$: iteration 100; (

**c**) ${f}_{1}$: iteration 250; (

**d**) ${f}_{1}$: iteration 500; (

**e**) ${f}_{1}$: iteration 750; (

**f**) ${f}_{1}$: iteration 1000.

## Appendix B. EggHolder

**Figure A2.**Inverted Egg Holder at different iterations. (

**a**) ${f}_{2}$: iteration 50; (

**b**) ${f}_{2}$: iteration 100; (

**c**) ${f}_{2}$: iteration 250; (

**d**) ${f}_{2}$: iteration 500; (

**e**); ${f}_{2}$: iteration 750.

## Appendix C. Inverted Schwefel Problem 2_26

**Figure A3.**Inverted Schwefel Problem 2_26 at different iterations. (

**a**) ${f}_{8}$: iteration 50; (

**b**) ${f}_{8}$: iteration 100; (

**c**) ${f}_{8}$: iteration 250; (

**d**) ${f}_{8}$: iteration 500; (

**e**) ${f}_{8}$: iteration 750.

## Appendix D. Test Functions in 2D

**Figure A4.**Test Functions shown in 2D. (

**a**) ${f}_{1}$; (

**b**) ${f}_{2}$; (

**c**) ${f}_{3}$; (

**d**) ${f}_{4}$; (

**e**) ${f}_{5}$.

## Appendix E. Test Functions in 2D

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**Figure 1.**Illustration of swarm diversity for niching algorithms. The crosses show the positions of optima. (

**a**) particles during search process; (

**b**) particles at convergence.

**Figure 6.**Quantification of diversity using the sum of distances to the nearest neighbour ($\mathit{Cont}.$).

**Figure 8.**Quantification of diversity using the average distance around the swarm centre measure ($\mathit{Cont}.$).

**Figure 10.**Quantification of diversity using the average of the mean distance around all solutions ($\mathit{Cont}.$).

Function & Name | Characteristics |
---|---|

Inverted Branin’s RCOS | 3 global peaks |

f_{1}$=-({({x}_{2}-\frac{5.1{x}_{1}^{2}}{4{\pi}^{2}}+\frac{5{x}_{1}}{\pi}-6)}^{2}+10(1-\frac{1}{8\pi})cos{x}_{1}+10)$ | ${x}_{1}\in [-5,10]$ |

${x}_{2}\in [0,15]$ | |

Inverted Egg Holder | 1 global peak |

f_{2}$=-{\sum}_{i=1}^{n-1}\left(-({x}_{i+1}+47)\alpha +\beta (-{x}_{i})\right)$ | Many local peaks |

$\alpha =sin\left(\sqrt{|{x}_{i+1}+{x}_{i}/2+47|}\right)$ | ${\mathbf{x}}_{\mathit{i}}\in {[-512,512]}^{2}$ |

$\beta =sin\left(\sqrt{|{x}_{i}-({x}_{i+1}+47|}\right))$ | |

Equal Maxima | ${5}^{n}$ global peaks |

f_{3}$={\sum}_{i=1}^{n}{sin}^{6}\left(5\pi {x}_{i}\right)$ | ${\mathbf{x}}_{\mathit{i}}\in {[0,1]}^{2}$ |

Modified Himmelblau | 4 global peaks |

f_{4}$=200-{\left({x}_{1}^{2}+{x}_{2}-11\right)}^{2}+{\left({x}_{1}+{x}_{2}^{2}-7\right)}^{2}$ | ${\mathbf{x}}_{\mathit{i}}\in {[-6,6]}^{2}$ |

Inverted Michalewicz | 1 global peak |

f_{5}$={\sum}_{i=1}^{n}[sin\left({x}_{i}\right).{sin}^{20}\left(\frac{i*{x}_{i}^{2}}{\pi}\right)]$ | $n!-1$ local peaks |

${\mathbf{x}}_{\mathit{i}}\in {[0,\pi ]}^{2}$ | |

Inverted Rastrigin | 1 global peak |

f_{6}$=-{\sum}_{i=1}^{n}[{x}_{i}^{2}-10cos\left(2\pi {x}_{i}\right)+10]$ | Many local peaks |

${\mathbf{x}}_{\mathit{i}}\in {[0,\pi ]}^{2}$ | |

Inverted Rosenbrock | 1 global peak |

f_{7}$\left(\overrightarrow{x}\right)=-{\sum}_{i=1}^{n-1}\left\{\right(100{({x}_{i+1}-{x}_{i}^{2})}^{2}$ | Many local peaks |

$+{({x}_{i}-1)}^{2}\left)\right\}$ | ${\mathbf{x}}_{\mathit{i}}\in {[-5,5]}^{4}$ |

Inverted Schwefel Problem 2_26 | 1 global peaks |

f_{8}$=-{\sum}_{i=1}^{n}{x}_{i}sin\left(\sqrt{|{x}_{i}|}\right)$ | ${8}^{n}-1$ local peaks |

${\mathbf{x}}_{\mathit{i}}\in {[-500,500]}^{2}$ | |

Inverted Six-Hump Camel Back | 2 global peaks |

f_{9}$={\sum}_{i=1}^{n-1}\{(4-2.1{x}_{i}^{2}+\frac{{x}_{i}^{4}}{3}){x}_{i}^{2}+{x}_{i}{x}_{i+1}+(-4+4{x}_{i+1}^{2}){x}_{i+1}^{2}\}$ | 4 local peaks |

${\mathbf{x}}_{\mathit{i}}\in {[0,\pi ]}^{2}$ |

Function | Mean | Stdev | Mean | Stdev | Mean | Stdev |
---|---|---|---|---|---|---|

(SD) | (SD) | (mSD) | (mSD) | (nSD) | (nSD) | |

${f}_{1}$ | $3.02\times {10}^{2}$ | $9.12\times {10}^{0}$ | $2.76\times {10}^{1}$ | $1.01\times {10}^{1}$ | $1.41\times {10}^{1}$ | $4.40\times {10}^{0}$ |

${f}_{2}$ | $3.62\times {10}^{3}$ | $4.87\times {10}^{2}$ | $7.93\times {10}^{0}$ | $8.11\times {10}^{0}$ | $1.31\times {10}^{3}$ | $4.47\times {10}^{2}$ |

${f}_{3}$ | $7.39\times {10}^{1}$ | $2.30\times {10}^{0}$ | $1.13\times {10}^{0}$ | $2.67\times {10}^{-1}$ | $2.15\times {10}^{1}$ | $2.57\times {10}^{0}$ |

${f}_{4}$ | $2.26\times {10}^{2}$ | $5.43\times {10}^{0}$ | $1.38\times {10}^{1}$ | $2.74\times {10}^{0}$ | $1.31\times {10}^{1}$ | $2.17\times {10}^{0}$ |

${f}_{5}$ | $3.32\times {10}^{1}$ | $2.50\times {10}^{1}$ | $0.00\times {10}^{0}$ | $0.00\times {10}^{0}$ | $3.32\times {10}^{1}$ | $2.50\times {10}^{1}$ |

${f}_{6}$ | $9.22\times {10}^{1}$ | $1.82\times {10}^{1}$ | $2.14\times {10}^{-1}$ | $3.37\times {10}^{-1}$ | $1.94\times {10}^{1}$ | $1.14\times {10}^{1}$ |

${f}_{7}$ | $4.30\times {10}^{2}$ | $5.75\times {10}^{1}$ | $4.28\times {10}^{1}$ | $2.19\times {10}^{1}$ | $1.70\times {10}^{1}$ | $8.44\times {10}^{0}$ |

${f}_{8}$ | $3.35\times {10}^{3}$ | $4.11\times {10}^{2}$ | $2.03\times {10}^{-1}$ | $2.03\times {10}^{-1}$ | $3.32\times {10}^{3}$ | $4.15\times {10}^{2}$ |

${f}_{9}$ | $1.41\times {10}^{2}$ | $1.15\times {10}^{1}$ | $8.32\times {10}^{0}$ | $3.47\times {10}^{0}$ | $1.09\times {10}^{1}$ | $3.45\times {10}^{0}$ |

Function | Mean | Stdev | Mean | Stdev | Mean | Stdev |
---|---|---|---|---|---|---|

(SDNN) | (SDNN) | (mSDNN) | (mSDNN) | (nSDNN) | (nSDNN) | |

${f}_{1}$ | $4.28\times {10}^{1}$ | $1.29\times {10}^{1}$ | $9.77\times {10}^{0}$ | $4.09\times {10}^{0}$ | $4.41\times {10}^{1}$ | $1.41\times {10}^{1}$ |

${f}_{2}$ | $7.29\times {10}^{3}$ | $3.03\times {10}^{3}$ | $7.52\times {10}^{1}$ | $1.13\times {10}^{2}$ | $6.54\times {10}^{3}$ | $2.37\times {10}^{3}$ |

${f}_{3}$ | $4.26\times {10}^{0}$ | $5.30\times {10}^{-1}$ | $2.35\times {10}^{-1}$ | $5.02\times {10}^{-2}$ | $5.58\times {10}^{0}$ | $7.37\times {10}^{-1}$ |

${f}_{4}$ | $1.89\times {10}^{1}$ | $3.73\times {10}^{0}$ | $4.22\times {10}^{0}$ | $1.39\times {10}^{0}$ | $2.97\times {10}^{1}$ | $5.04\times {10}^{0}$ |

${f}_{5}$ | $6.18\times {10}^{0}$ | $6.61\times {10}^{0}$ | $0.00\times {10}^{0}$ | $0.00\times {10}^{0}$ | $6.18\times {10}^{0}$ | $6.61\times {10}^{0}$ |

${f}_{6}$ | $1.88\times {10}^{1}$ | $1.78\times {10}^{1}$ | $1.52\times {10}^{-1}$ | $4.77\times {10}^{-1}$ | $9.62\times {10}^{0}$ | $3.96\times {10}^{0}$ |

${f}_{7}$ | $2.90\times {10}^{2}$ | $4.15\times {10}^{1}$ | $3.51\times {10}^{1}$ | $1.91\times {10}^{1}$ | $2.23\times {10}^{1}$ | $9.92\times {10}^{0}$ |

${f}_{8}$ | $5.47\times {10}^{3}$ | $1.67\times {10}^{3}$ | $6.40\times {10}^{0}$ | $7.23\times {10}^{0}$ | $5.59\times {10}^{3}$ | $1.63\times {10}^{3}$ |

${f}_{9}$ | $1.97\times {10}^{1}$ | $7.05\times {10}^{0}$ | $2.97\times {10}^{0}$ | $1.70\times {10}^{0}$ | $1.37\times {10}^{1}$ | $4.03\times {10}^{0}$ |

**Table 4.**Mean ($\mu $) and standard deviation ($\sigma $) for ADSC measure and its variants at the last iteration.

Function | Mean | Stdev | Mean | Stdev | Mean | Stdev |
---|---|---|---|---|---|---|

(ADSC) | (ADSC) | (mADSC) | (mADSC) | (nADSC) | (nADSC) | |

${f}_{1}$ | $4.55\times {10}^{-1}$ | $1.62\times {10}^{-1}$ | $3.42\times {10}^{-1}$ | $1.54\times {10}^{-1}$ | $0.00\times {10}^{0}$ | $0.00\times {10}^{0}$ |

${f}_{2}$ | $1.23\times {10}^{1}$ | $9.74\times {10}^{0}$ | $4.06\times {10}^{0}$ | $3.54\times {10}^{0}$ | $1.65\times {10}^{0}$ | $2.59\times {10}^{0}$ |

${f}_{3}$ | $6.54\times {10}^{-2}$ | $1.23\times {10}^{-2}$ | $3.52\times {10}^{-2}$ | $7.87\times {10}^{-3}$ | $5.95\times {10}^{-3}$ | $6.99\times {10}^{-3}$ |

${f}_{4}$ | $2.69\times {10}^{-1}$ | $6.93\times {10}^{-2}$ | $2.44\times {10}^{-1}$ | $1.32\times {10}^{-1}$ | $0.00\times {10}^{0}$ | $0.00\times {10}^{0}$ |

${f}_{5}$ | $8.45\times {10}^{-10}$ | $3.74\times {10}^{-9}$ | $8.45\times {10}^{-10}$ | $3.74\times {10}^{-9}$ | $8.45\times {10}^{-10}$ | $3.74\times {10}^{-9}$ |

${f}_{6}$ | $5.27\times {10}^{-3}$ | $9.36\times {10}^{-3}$ | $7.67\times {10}^{-3}$ | $1.21\times {10}^{-2}$ | $1.42\times {10}^{-3}$ | $5.35\times {10}^{-3}$ |

${f}_{7}$ | $6.81\times {10}^{-3}$ | $1.10\times {10}^{-2}$ | $1.03\times {10}^{-3}$ | $1.83\times {10}^{-3}$ | $1.41\times {10}^{-6}$ | $7.58\times {10}^{-6}$ |

${f}_{8}$ | $4.76\times {10}^{0}$ | $3.44\times {10}^{0}$ | $4.15\times {10}^{0}$ | $2.71\times {10}^{0}$ | $3.98\times {10}^{0}$ | $2.71\times {10}^{0}$ |

${f}_{9}$ | $1.13\times {10}^{-1}$ | $4.00\times {10}^{-2}$ | $6.54\times {10}^{-2}$ | $4.15\times {10}^{-2}$ | $6.34\times {10}^{-3}$ | $3.42\times {10}^{-2}$ |

**Table 5.**Mean ($\mu $) and standard deviation ($\sigma $) for ADAA measure and its variants at the last iteration.

Function | Mean | Stdev | Mean | Stdev | Mean | Stdev |
---|---|---|---|---|---|---|

(ADAA) | (ADAA) | (mADAA) | (mADAA) | (nADAA) | (nADAA) | |

${f}_{1}$ | $8.49\times {10}^{0}$ | $3.56\times {10}^{-1}$ | $1.21\times {10}^{0}$ | $4.91\times {10}^{-1}$ | $7.02\times {10}^{0}$ | $7.98\times {10}^{-1}$ |

${f}_{2}$ | $6.92\times {10}^{2}$ | $2.47\times {10}^{1}$ | $9.98\times {10}^{0}$ | $1.16\times {10}^{1}$ | $6.15\times {10}^{2}$ | $5.42\times {10}^{1}$ |

${f}_{3}$ | $5.06\times {10}^{-1}$ | $2.43\times {10}^{-2}$ | $6.83\times {10}^{-2}$ | $1.38\times {10}^{-2}$ | $5.17\times {10}^{-1}$ | $2.58\times {10}^{-2}$ |

${f}_{4}$ | $4.83\times {10}^{0}$ | $1.94\times {10}^{-1}$ | $5.59\times {10}^{-1}$ | $1.96\times {10}^{-1}$ | $4.89\times {10}^{0}$ | $1.67\times {10}^{-1}$ |

${f}_{5}$ | $1.36\times {10}^{-1}$ | $1.15\times {10}^{-1}$ | $0.00\times {10}^{0}$ | $0.00\times {10}^{0}$ | $1.36\times {10}^{-1}$ | $1.15\times {10}^{-1}$ |

${f}_{6}$ | $7.53\times {10}^{-1}$ | $2.75\times {10}^{-1}$ | $6.40\times {10}^{-3}$ | $1.38\times {10}^{-2}$ | $9.98\times {10}^{-1}$ | $2.63\times {10}^{-1}$ |

${f}_{7}$ | $5.16\times {10}^{0}$ | $3.39\times {10}^{-1}$ | $8.99\times {10}^{-1}$ | $4.54\times {10}^{-1}$ | $2.40\times {10}^{0}$ | $6.37\times {10}^{-1}$ |

${f}_{8}$ | $5.96\times {10}^{2}$ | $1.96\times {10}^{1}$ | $1.64\times {10}^{0}$ | $1.82\times {10}^{0}$ | $5.96\times {10}^{2}$ | $2.00\times {10}^{1}$ |

${f}_{9}$ | $1.60\times {10}^{0}$ | $2.02\times {10}^{-1}$ | $2.54\times {10}^{-1}$ | $1.31\times {10}^{-1}$ | $1.45\times {10}^{0}$ | $1.41\times {10}^{-1}$ |

**Table 6.**Mean ($\mu $) and standard deviation ($\sigma $) for the Entropy measure and its variants at the last iteration.

Function | Mean | Stdev | Mean | Stdev | Mean | Stdev |
---|---|---|---|---|---|---|

(${\widehat{\mathbf{E}}}_{20}$) | (${\widehat{\mathbf{E}}}_{20}$) | (m${\widehat{\mathbf{E}}}_{20}$) | (m${\widehat{\mathbf{E}}}_{20}$) | (n${\widehat{\mathbf{E}}}_{20}$) | (n${\widehat{\mathbf{E}}}_{20}$) | |

${f}_{1}$ | $6.29\times {10}^{-1}$ | $3.71\times {10}^{-2}$ | $2.14\times {10}^{-1}$ | $7.44\times {10}^{-2}$ | $2.82\times {10}^{-1}$ | $3.24\times {10}^{-2}$ |

${f}_{2}$ | $6.58\times {10}^{-1}$ | $4.72\times {10}^{-2}$ | $7.55\times {10}^{-3}$ | $7.01\times {10}^{-3}$ | $5.76\times {10}^{-1}$ | $6.93\times {10}^{-2}$ |

${f}_{3}$ | $9.12\times {10}^{-1}$ | $2.35\times {10}^{-2}$ | $1.38\times {10}^{-1}$ | $1.98\times {10}^{-2}$ | $7.31\times {10}^{-1}$ | $3.20\times {10}^{-2}$ |

${f}_{4}$ | $5.97\times {10}^{-1}$ | $3.31\times {10}^{-2}$ | $1.63\times {10}^{-1}$ | $4.16\times {10}^{-2}$ | $4.29\times {10}^{-1}$ | $2.91\times {10}^{-2}$ |

${f}_{5}$ | $1.29\times {10}^{-1}$ | $2.92\times {10}^{-2}$ | $0.00\times {10}^{0}$ | $0.00\times {10}^{0}$ | $1.29\times {10}^{-1}$ | $2.92\times {10}^{-2}$ |

${f}_{6}$ | $3.24\times {10}^{-1}$ | $6.80\times {10}^{-2}$ | $1.39\times {10}^{-2}$ | $1.38\times {10}^{-2}$ | $3.47\times {10}^{-1}$ | $5.91\times {10}^{-2}$ |

${f}_{7}$ | $8.12\times {10}^{-1}$ | $3.21\times {10}^{-2}$ | $1.45\times {10}^{-1}$ | $7.05\times {10}^{-2}$ | $3.76\times {10}^{-1}$ | $5.05\times {10}^{-2}$ |

${f}_{8}$ | $5.34\times {10}^{-1}$ | $4.06\times {10}^{-2}$ | $1.29\times {10}^{-3}$ | $1.20\times {10}^{-3}$ | $5.27\times {10}^{-1}$ | $3.66\times {10}^{-2}$ |

${f}_{9}$ | $5.14\times {10}^{-1}$ | $5.78\times {10}^{-2}$ | $9.38\times {10}^{-2}$ | $3.82\times {10}^{-2}$ | $3.18\times {10}^{-1}$ | $3.11\times {10}^{-2}$ |

**Table 7.**Mean ($\mu $) and standard deviation ($\sigma $) for the SPD measure and its variants at the last iteration.

Function | Mean | Stdev | Mean | Stdev | Mean | Stdev |
---|---|---|---|---|---|---|

(SPD) | (SPD) | (mSPD) | (mSPD) | (nSPD) | (nSPD) | |

${f}_{1}$ | $1.11\times {10}^{0}$ | $6.63\times {10}^{-3}$ | $1.03\times {10}^{0}$ | $1.05\times {10}^{-2}$ | $1.08\times {10}^{0}$ | $1.30\times {10}^{-4}$ |

${f}_{2}$ | $1.03\times {10}^{1}$ | $1.35\times {10}^{0}$ | $1.04\times {10}^{0}$ | $3.99\times {10}^{-2}$ | $9.52\times {10}^{0}$ | $1.27\times {10}^{0}$ |

${f}_{3}$ | $1.01\times {10}^{0}$ | $3.34\times {10}^{-4}$ | $1.00\times {10}^{0}$ | $1.88\times {10}^{-4}$ | $1.01\times {10}^{0}$ | $1.91\times {10}^{-4}$ |

${f}_{4}$ | $1.06\times {10}^{0}$ | $4.19\times {10}^{-3}$ | $1.01\times {10}^{0}$ | $3.49\times {10}^{-3}$ | $1.05\times {10}^{0}$ | $5.76\times {10}^{-5}$ |

${f}_{5}$ | $1.00\times {10}^{0}$ | $3.49\times {10}^{-3}$ | $1.00\times {10}^{0}$ | $0.00\times {10}^{0}$ | $1.00\times {10}^{0}$ | $3.49\times {10}^{-3}$ |

${f}_{6}$ | $1.01\times {10}^{0}$ | $1.87\times {10}^{-3}$ | $1.00\times {10}^{0}$ | $2.41\times {10}^{-4}$ | $1.01\times {10}^{0}$ | $1.47\times {10}^{-3}$ |

${f}_{7}$ | $1.09\times {10}^{0}$ | $6.42\times {10}^{-3}$ | $1.01\times {10}^{0}$ | $6.84\times {10}^{-3}$ | $1.03\times {10}^{0}$ | $6.30\times {10}^{-3}$ |

${f}_{8}$ | $1.21\times {10}^{1}$ | $1.31\times {10}^{0}$ | $1.01\times {10}^{0}$ | $5.51\times {10}^{-3}$ | $1.19\times {10}^{1}$ | $1.28\times {10}^{0}$ |

${f}_{9}$ | $1.03\times {10}^{0}$ | $4.96\times {10}^{-3}$ | $1.00\times {10}^{0}$ | $1.96\times {10}^{-3}$ | $1.02\times {10}^{0}$ | $6.19\times {10}^{-4}$ |

**Table 8.**Mean ($\mu $) and standard deviation ($\sigma $) for SR measure and its variants at the last iteration.

Function | Mean | Stdev | Mean | Stdev | Mean | Stdev |
---|---|---|---|---|---|---|

(SR) | (SR) | (mSR) | (mSR) | (nSR) | (nSR) | |

${f}_{1}$ | $1.67\times {10}^{1}$ | $1.44\times {10}^{0}$ | $3.81\times {10}^{0}$ | $1.63\times {10}^{0}$ | $1.54\times {10}^{1}$ | $1.40\times {10}^{0}$ |

${f}_{2}$ | $1.25\times {10}^{3}$ | $4.04\times {10}^{1}$ | $2.62\times {10}^{1}$ | $2.99\times {10}^{1}$ | $1.24\times {10}^{3}$ | $3.19\times {10}^{1}$ |

${f}_{3}$ | $9.95\times {10}^{-1}$ | $1.41\times {10}^{-1}$ | $1.50\times {10}^{-1}$ | $3.32\times {10}^{-2}$ | $9.84\times {10}^{-1}$ | $1.31\times {10}^{-1}$ |

${f}_{4}$ | $8.96\times {10}^{0}$ | $7.36\times {10}^{-1}$ | $2.10\times {10}^{0}$ | $5.99\times {10}^{-1}$ | $8.46\times {10}^{0}$ | $2.15\times {10}^{-1}$ |

${f}_{5}$ | $8.13\times {10}^{-1}$ | $6.34\times {10}^{-1}$ | $0.00\times {10}^{0}$ | $0.00\times {10}^{0}$ | $8.13\times {10}^{-1}$ | $6.34\times {10}^{-1}$ |

${f}_{6}$ | $1.44\times {10}^{0}$ | $2.01\times {10}^{-1}$ | $2.27\times {10}^{-2}$ | $4.81\times {10}^{-2}$ | $1.39\times {10}^{0}$ | $1.08\times {10}^{-1}$ |

${f}_{7}$ | $7.62\times {10}^{0}$ | $6.12\times {10}^{-1}$ | $1.47\times {10}^{0}$ | $7.20\times {10}^{-1}$ | $4.40\times {10}^{0}$ | $1.07\times {10}^{0}$ |

${f}_{8}$ | $1.17\times {10}^{3}$ | $6.37\times {10}^{1}$ | $3.30\times {10}^{0}$ | $3.64\times {10}^{0}$ | $1.17\times {10}^{3}$ | $6.37\times {10}^{1}$ |

${f}_{9}$ | $3.00\times {10}^{0}$ | $5.31\times {10}^{-1}$ | $6.07\times {10}^{-1}$ | $3.04\times {10}^{-1}$ | $2.33\times {10}^{0}$ | $7.54\times {10}^{-2}$ |

**Table 9.**Mean ($\mu $) and standard deviation ($\sigma $) for SDM measure and its variants at the last iteration.

Function | Mean | Stdev | Mean | Stdev | Mean | Stdev |
---|---|---|---|---|---|---|

(SDM) | (SDM) | (mSDM) | (mSDM) | (nSDM) | (nSDM) | |

${f}_{1}$ | $1.86\times {10}^{1}$ | $5.09\times {10}^{-1}$ | $4.82\times {10}^{0}$ | $1.97\times {10}^{0}$ | $1.59\times {10}^{1}$ | $3.25\times {10}^{-2}$ |

${f}_{2}$ | $1.28\times {10}^{3}$ | $4.07\times {10}^{1}$ | $2.63\times {10}^{1}$ | $3.00\times {10}^{1}$ | $1.25\times {10}^{3}$ | $2.79\times {10}^{1}$ |

${f}_{3}$ | $1.17\times {10}^{0}$ | $6.38\times {10}^{-2}$ | $1.65\times {10}^{-1}$ | $3.62\times {10}^{-2}$ | $1.13\times {10}^{0}$ | $2.36\times {10}^{-2}$ |

${f}_{4}$ | $9.54\times {10}^{0}$ | $9.28\times {10}^{-1}$ | $2.38\times {10}^{0}$ | $6.42\times {10}^{-1}$ | $8.59\times {10}^{0}$ | $1.04\times {10}^{-2}$ |

${f}_{5}$ | $8.45\times {10}^{-1}$ | $6.72\times {10}^{-1}$ | $0.00\times {10}^{0}$ | $0.00\times {10}^{0}$ | $8.45\times {10}^{-1}$ | $6.72\times {10}^{-1}$ |

${f}_{6}$ | $2.43\times {10}^{0}$ | $3.61\times {10}^{-1}$ | $2.27\times {10}^{-2}$ | $4.81\times {10}^{-2}$ | $2.38\times {10}^{0}$ | $2.76\times {10}^{-1}$ |

${f}_{7}$ | $1.28\times {10}^{1}$ | $9.33\times {10}^{-1}$ | $2.06\times {10}^{0}$ | $1.01\times {10}^{0}$ | $4.95\times {10}^{0}$ | $1.04\times {10}^{0}$ |

${f}_{8}$ | $1.22\times {10}^{3}$ | $6.35\times {10}^{1}$ | $3.30\times {10}^{0}$ | $3.64\times {10}^{0}$ | $1.22\times {10}^{3}$ | $6.37\times {10}^{1}$ |

${f}_{9}$ | $4.94\times {10}^{0}$ | $8.02\times {10}^{-1}$ | $7.74\times {10}^{-1}$ | $3.43\times {10}^{-1}$ | $3.73\times {10}^{0}$ | $9.46\times {10}^{-2}$ |

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**MDPI and ACS Style**

Mwaura, J.; Engelbrecht, A.P.; Nepomuceno, F.V.
Diversity Measures for Niching Algorithms. *Algorithms* **2021**, *14*, 36.
https://doi.org/10.3390/a14020036

**AMA Style**

Mwaura J, Engelbrecht AP, Nepomuceno FV.
Diversity Measures for Niching Algorithms. *Algorithms*. 2021; 14(2):36.
https://doi.org/10.3390/a14020036

**Chicago/Turabian Style**

Mwaura, Jonathan, Andries P. Engelbrecht, and Filipe V. Nepomuceno.
2021. "Diversity Measures for Niching Algorithms" *Algorithms* 14, no. 2: 36.
https://doi.org/10.3390/a14020036