# An Improved Bacterial-Foraging Optimization-Based Machine Learning Framework for Predicting the Severity of Somatization Disorder

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

- (a)
- First, in order to fully explore the potential of the KELM classifier, we introduce an opposition-based, learning-strategy-enhanced BFO to adaptively determine the two key parameters of KELM, which aided the KELM classifier in more efficiently achieving the maximum classification performance.
- (b)
- The resulting model, IBFO-KELM, is applied to serve as a computer-aided decision-making tool for predicting the severity of somatization disorder.
- (c)
- The proposed IBFO-KELM method achieves superior results, and offers more stable and robust results when compared to the four other KELM models.

## 2. Background Information

#### 2.1. Kernel Extreme Learning Machine (KELM)

#### 2.2. Bacterial Foraging Optimization (BFO)

- (1)
- Chemotaxis: Chemotaxis operation is the core of the algorithm, which simulates the foraging behavior of E. coli moving and tumbling. In poorer areas, the bacteria tumble more frequently, while bacteria move in areas where food is more abundant. The chemotaxis operation of the ith bacterium can be represented as$$\begin{array}{l}{\theta}^{i}\left(j+1,k,l\right)={\theta}^{i}\left(j,k,l\right)+C\left(i\right)\ast dc{t}_{i}\\ dc{t}_{i}=\frac{\Delta \left(i\right)}{\sqrt{{\Delta}^{T}\left(i\right)\Delta \left(i\right)}}\end{array}$$
_{i}). Δ is a random vector between −1 and 1. - (2)
- Swarming: In the chemotactic of bacteria to the foraging process, in addition to searching for food in their own way, there is both gravitation and repulsion among the individual bacteria. Bacteria will generate attractive information to allow individual bacteria to travel to the center of the population, bringing them together; at the same time, individual bacteria are kept at a distance based on their respective repulsion information.
- (3)
- Reproduction: According to the natural mechanism of survival of the fittest, after some time, bacteria with weak ability to seek food will eventually be eliminated, and bacteria with strong feeding ability will breed offspring to maintain the size of the population. By simulating this phenomenon, a reproduction operation is proposed. In S-sized populations, S/2 bacteria with poor fitness were eliminated and S/2 individuals with higher fitness self-replicated after the bacteria performed the chemotaxis operator. After the execution of the reproduction operation, the offspring will inherit the fine characteristics of the parent completely, protect the good individuals, and greatly accelerate the speed towards the global optimal solution.
- (4)
- Elimination–Dispersal: In the process of bacterial foraging, do not rule out the occurrence of unexpected conditions leading to the death of bacteria or causing them to migrate to another new area. By simulating this phenomenon, an elimination–dispersal operation has been proposed. This operation occurs with a certain probability Ped. When the bacterial individual satisfies the probability Ped, then the individual of the bacterial dies and randomly generates a new individual anywhere in the solution space. This bacterium may be different from the original bacterial, which helps to jump out of the local optimal solution and promote the search for the global optimal solution.

#### 2.3. Improved Bacterial Foraging Optimization (IBFO)

## 3. Proposed IBFO-KELM Model

_{i}represents the average test error rates achieved by the KELM classifier via the five-fold CV during the inner parameter optimization procedure.

Algorithm 1. Pseudo-code of the improved bacterial foraging optimization (IBFO) strategy. |

BeginInitialize dimension p, population S, chemotactic steps Nc, swimming length Ns, reproduction steps Nre, elimination-dispersal steps Ned, elimination-dispersal probability Ped, step size C(i). Calculate the corresponding opposite solutions of bacterial populations based on opposition-based learning. From the original and its corresponding opposite solutions of bacterial populations, S superior individuals are selected as the initial solutions of bacterial populations. for ell = 1:Nedfor K = 1:Nrefor j = 1:NcIntertime = Intertime + 1; for i = 1:s J(i,j,K,ell) = fobj(P(:,i,j,K,ell)); Jlast = J(i,j,K,ell); Tumble according to Equation (5)m = 0; while m < Nsm = m + 1; if J(i,j + 1,K,ell) < JlastJlast = J(i,j + 1,K,ell); Tumble according to Equation (5)elsem = Ns; EndEndEnd End /*Reprodution*/Jhealth = sum(J(:,:,K,ell),2); [Jhealth,sortind] = sort(Jhealth); P(:,:,1,K + 1,ell) = P(:,sortind,Nc + 1,K,ell); for i = 1:SrP(:,i + Sr,1,K + 1,ell) = P(:,i,1,K + 1,ell); EndEnd /*Elimination-Dispersal*/for m = 1:sif Ped > rand Reinitialize bacteria mEnd End End End |

## 4. Experimental Design

#### 4.1. Somatization Disorder Data Description

#### 4.2. Experimental Setup

^{−5}, 2

^{−3}, …, 2

^{15}} and γ ∈ {2

^{−15}, 2

^{−13}, …, 2

^{5}}. A population swarm size of eight, chemotactic step number of five, swimming length of four, reproduction step number of five, elimination–dispersal event number of two, and elimination–dispersal probability of 0.25 were selected for BFO-KELM and IBFO-KELM. The chemotaxis step value was established through trial and error, as shown in the experimental results section. For PSO, the maximum velocity was set to about 60% of the dynamic range of the variable on each dimension for the continuous type of dimensions. The two acceleration coefficients c

_{1}and c

_{2}were set as 2.05, the inertia weight was set to one.In order to determine the validity and accuracy of the results, the k-fold CV [38] was used to evaluate the classification performance of the model. A nested stratified 10-fold CV, which has been widely used in previous research, was used for the purposes of this study [39]. The classification performance evaluation was conducted in the outer loop. Since a 10-fold CV was used in the outer loop, the classifiers were evaluated in one independent fold of data, and the other nine folds of data were left for training. The parameter optimization process was performed in the inner loop. Since a five-fold CV was used in the inner loop, the IBFO-KELM searched for the optimal values of C and γ in the remaining nine folds of data. The nine folds of data were further split into one fold of data for the performance evaluation, and four folds of data were left for training. To evaluate the proposed method, commonly used evaluation criteria such as classification accuracy (ACC), sensitivity, specificity and Matthews correlation coefficients (MCC) were analyzed.

## 5. Experimental Results and Discussion

#### 5.1. Benchmark Function Validation

_{1}–f

_{5}) benchmark functions, one multimodal (f

_{6}) benchmark functions and three fixed-dimension multimodal (f

_{7}–f

_{9}) benchmark functions. Moreover, the performance of the IBFO is also compared with the PSO, Bat algorithm (BA) and conventional BFO. As mentioned in the literature, the proposed IBFO has superior results for these benchmark functions compared to the PSO, BA and original BFO. A total of 30 independent runs of the four algorithms were performed on each benchmark function. The maximum number of iterations and population size for all algorithms were set to 600 and 50, respectively. We recorded the average (Ave) and standard deviation (Std) for each benchmark problem.

_{1}–f

_{9}, it can be seen that IBFO has better searching ability and is more stable, which means that it also has better robustness. According to the performance of f

_{1}–f

_{9}in Figure 2, it is obvious that although the proposed IBFO can converge slowly in the early stages in some functions, the final result is optimal as the number of iterations increases.

#### 5.2. Results of the Somatization Disorder Diagnosis

## 6. Conclusions and Future Work

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**Convergence curves of the nine multidimensional benchmark functions for the four algorithms when dimension =2.

**Figure 3.**Training accuracy surfaces of KELM and KELM with parameters obtained by the grid-search based KELM (Grid-KELM) for several folds. (

**a**) Training accuracy surface of fold 2 for KELM; (

**b**) Training accuracy surface of fold 4 for KELM; (

**c**) Training accuracy surface of fold 6 for KELM; (

**d**) Training accuracy surface of fold 8 for KELM.

**Figure 4.**Relationship between the iteration and training accuracy of improved bacterial foraging optimization based KELM (IBFO-KELM), bacterial foraging optimization (BFO-KELM), particle swarm optimization based KELM (PSO-KELM) and genetic algorithm (GA-KELM).

Feature | Description |
---|---|

F1 | Headache |

F2 | Dizzy or fainted |

F3 | Chest pain |

F4 | Low back pain |

F5 | Nausea or upset stomach |

F6 | Muscle soreness |

F7 | Having breathe difficulty |

F8 | A series of chills or fever |

F9 | Body tingling or prickling |

F10 | The throat is infarcted |

F11 | Feeling that part of the body is weak |

F12 | Feeling the weight of your hands or feet |

F13 | Somatization severity |

Function | Range | Minimum |
---|---|---|

${f}_{1}(x)={\displaystyle {\sum}_{i=1}^{n}\left|{x}_{i}\right|}+{\displaystyle {\prod}_{i=1}^{n}\left|{x}_{i}\right|}$ | [−10, 10] | 0 |

${f}_{2}(x)={\displaystyle {\sum}_{i=1}^{n}{({\displaystyle {\sum}_{j-1}^{i}}{x}_{j})}^{2}}$ | [−100, 100] | 0 |

${f}_{3}(x)={\mathrm{max}}_{i}\left\{\right|{x}_{i}|,1\le i\le n\}$ | [−100, 100] | 0 |

${f}_{4}(x)={\displaystyle {\sum}_{i=1}^{n}{([{x}_{i}+0.5])}^{2}}$ | [−100, 100] | 0 |

${f}_{5}(x)={\displaystyle {\sum}_{i=1}^{n}i{x}_{i}{}^{4}+\mathrm{random}}$[0,1] | [−1.28, 1.28] | 0 |

${f}_{6}(x)={\displaystyle {\sum}_{i=1}^{n}-{x}_{i}\mathrm{sin}(\sqrt{\left|{x}_{i}\right|})}$ | [−500, 500] | −418.9829 × 5 |

${f}_{7}(x)={(\frac{1}{500}+{\displaystyle {\sum}_{j=1}^{25}\frac{1}{j+{\displaystyle {\sum}_{i=1}^{2}{({x}_{i}-{\mathrm{a}}_{ij})}^{6}}}})}^{-1}$ | [−65, 65] | 1 |

${f}_{8}(x)={\displaystyle {\sum}_{i=1}^{11}{[{\mathrm{a}}_{i}-\frac{{x}_{1}({\mathrm{b}}_{i}^{2}+{\mathrm{b}}_{i}{x}_{2})}{{\mathrm{b}}_{i}^{2}+{\mathrm{b}}_{i}{x}_{3}+{x}_{4}}]}^{2}}$ | [−5, 5] | 0.00030 |

${f}_{9}(x)=-{\displaystyle {\sum}_{i=1}^{10}}{[(x-{\mathrm{a}}_{i}){(x-{\mathrm{a}}_{i})}^{T}+{\mathrm{c}}_{i}]}^{-1}$ | [0, 10] | −10.5363 |

Methods | ||||||||
---|---|---|---|---|---|---|---|---|

PSO | BA | BFO | IBFO | |||||

Ave | Std | Ave | Std | Ave | Std | Ave | Std | |

f_{1} | 0.0185 | 0.0097 | 0.0113 | 0.0053 | 0.0075 | 0.0042 | 0.0057 | 0.0031 |

f_{2} | 0.0006 | 0.0005 | 0.0001 | 0.0001 | 4.07 × 10^{−5} | 4.76 × 10^{−5} | 2.83 × 10^{−5} | 4.83 × 10^{−5} |

f_{3} | 0.0135 | 0.0079 | 0.0095 | 0.0045 | 0.2072 | 1.1059 | 0.0046 | 0.0029 |

f_{4} | 0.0003 | 0.0003 | 9.88 × 10^{−5} | 0.0001 | 5.11 × 10^{−5} | 0.0001 | 3.13 × 10^{−5} | 3.76 × 10^{−5} |

f_{5} | 0.0070 | 0.0045 | 0.0020 | 0.0016 | 0.00045 | 0.0003 | 0.0004 | 0.0003 |

f_{6} | −793.55 | 65.4056 | −763.56 | 77.9863 | −720.08 | 76.2664 | −797.41 | 56.1899 |

f_{7} | 1.9873 | 1.5529 | 2.8100 | 1.9357 | 1.7915 | 1.0204 | 1.3947 | 0.8472 |

f_{8} | 0.0012 | 0.0002 | 0.0081 | 0.0095 | 0.0007 | 0.0002 | 0.0007 | 0.0002 |

f_{9} | −5.1904 | 1.9864 | −5.8112 | 3.1206 | −10.3323 | 0.9752 | −10.5104 | 0.0135 |

**Table 4.**The classification results of IBFO-KELM (kernel extreme learning machine) with different chemotaxis step size.

Step Size | IBFO-KELM | |||
---|---|---|---|---|

ACC | MCC | Sensitivity | Specificity | |

0.05 | 0.9213 (0.0389) | 0.8227 (0.0903) | 0.9679 (0.035) | 0.8286 (0.0768) |

0.1 | 0.9402 (0.0362) | 0.8653 (0.0824) | 0.9713 (0.0282) | 0.8786 (0.0678) |

0.15 | 0.9697 (0.0351) | 0.9243 (0.0907) | 0.9729 (0.0351) | 0.9600 (0.0843) |

0.2 | 0.9476 (0.0512) | 0.8850 (0.1089) | 0.9679 (0.0544) | 0.9071 (0.0828) |

0.25 | 0.9378 (0.0427) | 0.8614 (0.0965) | 0.9786 (0.0301) | 0.8571 (0.1117) |

0.3 | 0.9211 (0.0377) | 0.8241 (0.0836) | 0.9675 (0.0365) | 0.8286 (0.1075) |

**Table 5.**Classification performance obtained by the five methods in terms of ACC (classification accuracy), MCC (Matthew’s correlation coefficient), sensitivity, and specificity.

Method | Metrics | |||
---|---|---|---|---|

ACC | MCC | Sensitivity | Specificity | |

IBFO-KELM | 0.9697 ± 0.0351 | 0.9243 ± 0.0907 | 0.9729 ± 0.0351 | 0.9600 ± 0.0843 |

BFO-KELM | 0.9329 ± 0.0362 | 0.8280 ± 0.0850 | 0.9586 ± 0.0491 | 0.8550 ± 0.1012 |

PSO-KELM | 0.9282 ± 0.0266 | 0.8056 ± 0.0813 | 0.9657 ± 0.0362 | 0.8050 ± 0.1383 |

GA-KELM | 0.9176 ± 0.0662 | 0.7775 ± 0.1879 | 0.9595 ± 0.0349 | 0.8000 ± 0.2494 |

Grid-KELM | 0.9076 ± 0.0679 | 0.7592 ± 0.1637 | 0.9448 ± 0.0724 | 0.7900 ± 0.1647 |

**Table 6.**p-values of the Wilcoxon test of IBFO-KELM results versus other four methods (p > 0.05 are shown in bold).

Method | p-Value | |||
---|---|---|---|---|

ACC | MCC | Sensitivity | Specificity | |

BFO-KELM | 0.03 | 0.03 | 0.24 | 0.06 |

PSO-KELM | 0.02 | 0.02 | 0.41 | 0.02 |

GA-KELM | 0.03 | 0.04 | 0.61 | 0.08 |

Grid-KELM | 0.02 | 0.02 | 0.23 | 0.02 |

Method | Metrics | |||
---|---|---|---|---|

ACC | MCC | Sensitivity | Specificity | |

NB | 0.9182 ± 0.0543 | 0.7766 ± 0.1495 | 0.9800 ± 0.0322 | 0.7350 ± 0.1634 |

SVM | 0.8971 ± 0.0735 | 0.7122 ± 0.2148 | 0.9595 ± 0.0469 | 0.7100 ± 0.2601 |

RF | 0.9382 ± 0.0405 | 0.8337 ± 0.1133 | 0.9652 ± 0.0367 | 0.8500 ± 0.1414 |

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**MDPI and ACS Style**

Lv, X.; Chen, H.; Zhang, Q.; Li, X.; Huang, H.; Wang, G.
An Improved Bacterial-Foraging Optimization-Based Machine Learning Framework for Predicting the Severity of Somatization Disorder. *Algorithms* **2018**, *11*, 17.
https://doi.org/10.3390/a11020017

**AMA Style**

Lv X, Chen H, Zhang Q, Li X, Huang H, Wang G.
An Improved Bacterial-Foraging Optimization-Based Machine Learning Framework for Predicting the Severity of Somatization Disorder. *Algorithms*. 2018; 11(2):17.
https://doi.org/10.3390/a11020017

**Chicago/Turabian Style**

Lv, Xinen, Huiling Chen, Qian Zhang, Xujie Li, Hui Huang, and Gang Wang.
2018. "An Improved Bacterial-Foraging Optimization-Based Machine Learning Framework for Predicting the Severity of Somatization Disorder" *Algorithms* 11, no. 2: 17.
https://doi.org/10.3390/a11020017