# Application of Differential Evolution Algorithm Based on Mixed Penalty Function Screening Criterion in Imbalanced Data Integration Classification

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## Abstract

**:**

## 1. Introduction

**A: Source of The Idea**

**B: Technical Route**

## 2. Prerequisite Knowledge

#### 2.1. Basic Steps of the $DE$ Algorithm

#### 2.1.1. Population Initialization

#### 2.1.2. Individual Mutation

#### 2.1.3. Individual Crossover

#### 2.1.4. Individual Selection

#### 2.2. Internal Penalty Function

#### Normalization of the Internal Penalty Function

#### 2.3. External Penalty Function

#### Normalization of the External Penalty Function

## 3. Differential Evolution Integration Algorithm Based on Mixed Penalty Function Screening Criteria

#### 3.1. Mixed Penalty Function

#### 3.2. The Screening Criteria of the Mixed Penalty Function

#### 3.3. The Processing of the Constraint Conditions

#### 3.4. Implementation of Differential Evolution Integrated Algorithm Based on the Screening Criterion of the Mixed Penalty Function

**STEP 1**: Initializing variables: According to the number of the population and the number of individuals, let $t=0$, then generate the initial population within $\psi $ parents individuals and $\phi $ progeny individuals and each of the individuals corresponds with $\{({x}_{i}^{t},{\delta}_{i})|i=1,2,\cdots ,NP\}$.

**STEP 2**: Calculating the fitness function value of the initial variable: Calculate the fitness function values of $\psi $ parent individuals and record the maximum value and minimum value. At the same time, calculate the fitness function values at each iterative points inside and outside the feasible domain under the condition of the initial conditions and records the maximum value and minimum value.

**STEP 3**: According to Formulas $(5)$, $(17)$ and $(18)$, generate corresponding $\psi $ mutation individuals.

**STEP 4**: Calculating the fitness function value of corresponding $\psi $ mutation individuals and calculating the probability of the fitness function value of the iterative points of different regions based on screening criterion of the mixed penalty function.

**STEP 5**: Conducting selection and crossover operations for corresponding $\psi $ mutation individuals according to Formulas $(6)$ and $(7)$, generating $\phi $ progeny individuals and calculating their fitness function values and recording the iterative points that accord with the accuracy of the problem.

**STEP 6**: Consisting of the $\psi $ mutation individuals and the $\phi $ progeny individuals generated by the selection and crossover operations into a new $\psi +\phi $ individuals. According to the screening criteria, choose $\psi $ individuals of t generation as $t+1$ generation parents individuals and record the iterative points ${x}_{k+1}$ that accord with the accuracy of the problem.

**STEP 7**: Generating a series of iterative points $\{{x}_{{k}_{1}}\}$,$\{{x}_{{k}_{2}}\}$ in two parts inside and outside the feasible domain.

**STEP 8**: According to the iterative points sequence $\{{x}_{{k}_{1}}\}$,$\{{x}_{{k}_{2}}\}$ generated by step 7, the former are arranged in a monotonic decreasing sequence and the latter is arranged in a monotonic increasing sequence.

**STEP 9**: Calculating $P{r}_{1},P{r}_{2}$ according to the probability formula of the screening criteria.

**STEP 10**: Substituting the result of step 9 into Formulas $(12)$ or $(13)$, calculating the fitness function values and iterative points and judging whether the accuracy is satisfied.

**STEP 11**: If the iterative points satisfy the screening criteria and accuracy, then terminate the algorithm flows, otherwise let $t=t+1$, return step 3.

## 4. Theoretical Analysis of DE-MPFSC Algorithm

#### 4.1. Validity Analysis of DE-MPFSC Algorithm

**Definition**

**1**

**.**Let $\{{\widehat{X}}_{n};n\ge 0\}$ be a random variable with a discrete value. The whole discrete values are recorded as S, which is called the state space. If $\forall n\ge 1,{i}_{k}\in S(k\le n+1)$ and $P\{{\widehat{X}}_{n+1}=\frac{{i}_{n+1}}{{\widehat{X}}_{n}}={i}_{n},\cdots ,{\widehat{X}}_{0}={i}_{0}\}=P\{{\widehat{X}}_{n+1}=\frac{{i}_{n+1}}{{\widehat{X}}_{n}}={i}_{n}\}$, then $\{{\widehat{X}}_{n};n\ge 0\}$ is called the Markov chain.

**Lemma**

**1**

**.**The joint distribution of homogeneous Markov chain ${p}_{ij}^{n}=P\{{\widehat{X}}_{n+m}=\frac{j}{{\widehat{X}}_{m}}=i\}$ is determined by the initial distribution $P\{{\widehat{X}}_{0}=i\}={p}_{i},i\in S$ and the individual transition probability ${p}_{ij}=P\{{\widehat{X}}_{n}=\frac{j}{{\widehat{X}}_{n-1}}=i\},(i,j\in S)$.

**Lemma**

**2**

**Definition**

**2**

**.**Let $\{{\widehat{X}}_{n};n\ge 0\}$ be the finite homogeneous Markov chain, ${p}_{ij}^{n}$ is the nth iterative individual transition probability. If $\exists n\ge 1$ makes ${p}_{ij}^{n}\ge 0$, which is called a case in which status i can be transferred to state j; otherwise, is called another case in which the status i cannot be transferred to state j.

**Definition**

**3**

**.**For Markov chain $\{{\widehat{X}}_{n};n\ge 0\}$, the greatest common divisor ${d}_{i}$ of the state set $\{i|n\ge 1,{p}_{ij}^{n}\ge 0\}$ is called the generalized period of the state set i. If ${d}_{i}>1$, then state set i is periodical. If ${d}_{i}=1$, then state set i is nonperiodic. If ${d}_{i}$ is a nonpositive real number, then i cannot be periodical.

**Lemma**

**3**

**.**If Markov chain $\{{\widehat{X}}_{n};n\ge 0\}$ is irreducible, that is, that all individual states are connected to each other, and $\exists j\le N$, and $\exists j\le N$ make ${p}_{ij}>0$, then the chain is nonperiodic. Its transfer matrix is the primitive random matrix and there is a stationary distribution on the transfer matrix. Where there is a stationary distribution and a limited distribution ${lim}_{n\to \infty}P\{{X}_{n}=j\}={o}_{j}$ on the transfer random matrix of the Markov chain being nonperiodic, irreducible and a finite state.

#### Markov Processing Model of DE-MPFSC Algorithm

**Theorem**

**1.**

**Proof.**

#### 4.2. Convergence Analysis of DE-MPFSC Algorithm

**Definition**

**4**

**Definition**

**5**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

## 5. Entanglement and Numerical Test of DE-MPFSC Algorithm Stability

#### 5.1. Entanglement Degree $\frac{1}{\xi}$ and Entanglement Degree Error $\zeta $

**Lemma**

**4**

**.**Let ${f}_{\epsilon}$ be a continuous differentiable function defined in a complete normed linear space and ${f}_{\epsilon}({v}_{1},\cdots ,{v}_{n})\in {L}_{2}({\mathbb{R}}_{n}),M\in Sp(2n+{P}_{t}.\mathbb{R})$, ${M}^{n}$ is a complete normed linear space. $\{{S}_{i}^{n}|i=1,2,\cdots ,n\}$ is a generalized n dimension complete normed linear subspace and ${M}^{n}={S}_{1}^{n}\cup {S}_{2}^{n}\cup \cdots \cup {S}_{n}^{n}$ forms an open cover of ${M}^{n}$, then $\forall {\lambda}_{i}^{t}\in \{{\lambda}_{i}^{t}|i=1,2,\cdots ,n\},\exists {S}_{i}^{n},{lim}_{t\to \infty}{\lambda}_{i}^{t}\sim {\lambda}_{i}$ or ${lim}_{t\to \infty}{\lambda}_{i}^{t}={({\lambda}_{\epsilon})}_{i},{\lambda}_{i}\in {S}_{i}^{n}$. If $det(B)\ne 0$, then the following formula,

#### 5.2. Entangled Image and Analysis of DE-MPFSC Algorithm

#### 5.3. Numerical Test and Analysis of DE-MPFSC Algorithm

#### 5.4. Test and Analysis of DE-MPFSC Algorithm about Several Classic DE Algorithms

#### 5.4.1. Entanglement Degree Test of DE-MPFSC Algorithm about Several Classic DE Algorithms

#### 5.4.2. Numerical Test of DE-MPFSC Algorithm about Several Classic DE Algorithms

## 6. Empirical Analysis

#### 6.1. Verification Data Sets

#### 6.2. Verification Indicator

#### 6.3. The Test Analysis

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The entangled image and error of the differential evolution (DE) algorithm running 200 times.

**Figure 2.**The entangled image and error of the Differential Evolution Integration Algorithm Based on Mixed Penalty Function Screening Criteria (DE-MPFSC) algorithm running 200 times.

**Figure 12.**The Entanglement Degree, Correction Rate and Loss Curve of the (Self-Adapting Parameter Setting in Differential Evolution) JDE algorithm.

**Figure 13.**The Entanglement Degree, Correction Rate and Loss Curve of the Opposition Based Differential Evolution (OBDE) algorithm.

**Figure 14.**The Entanglement Degree, Correction Rate and Loss Curve of the (Differential Evolution with Global and Local neighborhoods) DEGL algorithm.

**Figure 15.**The Entanglement Degree, Correction Rate and Loss Curve of the (Self-Adaptive Differential Evolution) SADE algorithm.

**Table 1.**Numerical Analysis of Entanglement Degree $\frac{1}{\xi}$ and Entanglement Degree Error $\zeta $ of the Differential Evolution (DE) algorithm and the Mixed Penalty Function Screening Criteria (DE-MPFSC) algorithm 200 times.

NP | $DE$ | $DE-MPFSC$ | ||||||||

F | $CR$ | $\xi $ | $\zeta $ | $\epsilon \%$ | ${F}^{*}$ | $C{R}^{*}$ | $\xi $ | $\zeta $ | $\epsilon \%$ | |

200 | 0.1 | 0.9 | 0.8 | 0.5 | $\pm 0.625$ | 0.1 | 0.9 | 0.12 | 0.05 | $\pm 0.416$ |

200 | / | / | 0.73 | 0.66 | $\pm 0.904$ | 0.5 | 0.6 | 0.20 | 0.06 | $\pm 0.3$ |

200 | / | / | 0.81 | 0.49 | $\pm 0.604$ | 0.3 | 0.2 | 0.19 | 0.11 | $\pm 0.578$ |

200 | / | / | 0.75 | 0.54 | $\pm 0.72$ | 0.2 | 0.3 | 0.15 | 0.03 | $\pm 0.2$ |

200 | / | / | 0.8 | 0.56 | $\pm 0.7$ | 0.2 | 0.3 | 0.15 | 0.02 | $\pm 0.13$ |

Average | $DE$ | $DE-MPFSC$ | ||||||||

0.1 | 0.9 | 0.778 | 0.55 | $\pm 0.7106$ | 0.26 | 0.46 | 0.162 | 0.054 | $\pm 0.3248$ | |

Variance | $DE$ | $DE-MPFSC$ | ||||||||

0 | 0 | 0.00127 | 0.0046 | $\pm 0.01407$ | 0.023 | 0.083 | 0.00107 | 0.00123 | $\pm 0.03164$ |

**Table 2.**Numerical Analysis of Entanglement Degree $\frac{1}{\xi}$ and Entanglement Degree Error $\zeta $ of the DE algorithm and the DE-MPFSC algorithm 400 times.

NP | $DE$ | $DE-MPFSC$ | ||||||||

F | $CR$ | $\xi $ | $\zeta $ | $\epsilon \%$ | ${F}^{*}$ | $C{R}^{*}$ | $\xi $ | $\zeta $ | $\epsilon \%$ | |

400 | 0.1 | 0.9 | 0.77 | 0.43 | $\pm 0.558$ | 0.1 | 0.9 | 0.15 | 0.015 | $\pm 0.1$ |

400 | / | / | 0.85 | 0.29 | $\pm 0.341$ | 0.4 | 0.5 | 0.25 | 0.02 | $\pm 0.08$ |

400 | / | / | 0.81 | 0.55 | $\pm 0.679$ | 0.2 | 0.2 | 0.20 | 0.05 | $\pm 0.25$ |

400 | / | / | 0.80 | 0.63 | $\pm 0.787$ | 0.2 | 0.3 | 0.30 | 0.042 | $\pm 0.14$ |

400 | / | / | 0.79 | 0.51 | $\pm 0.645$ | 0.2 | 0.3 | 0.22 | 0.02 | $\pm 0.09$ |

Average | $DE$ | $DE-MPFSC$ | ||||||||

0.1 | 0.9 | 0.804 | 0.482 | $\pm 0.602$ | 0.22 | 0.44 | 0.224 | 0.0294 | $\pm 0.132$ | |

Variance | $DE$ | $DE-MPFSC$ | ||||||||

0 | 0 | 0.00088 | 0.01672 | $\pm 0.02802$ | 0.012 | 0.078 | 0.00313 | 0.00024 | $\pm 0.00487$ |

**Table 3.**Numerical Analysis of Entanglement Degree $\frac{1}{\xi}$ and Entanglement Degree Error $\zeta $ of the DE algorithm and the DE-MPFSC algorithm 600 times.

NP | $DE$ | $DE-MPFSC$ | ||||||||

F | $CR$ | $\xi $ | $\zeta $ | $\epsilon \%$ | ${F}^{*}$ | $C{R}^{*}$ | $\xi $ | $\zeta $ | $\epsilon \%$ | |

600 | 0.1 | 0.9 | 0.69 | 0.38 | $\pm 0.550$ | 0.1 | 0.9 | 0.15 | 0.015 | $\pm 0.1$ |

600 | / | / | 0.70 | 0.43 | $\pm 0.614$ | 0.6 | 0.8 | 0.33 | 0.10 | $\pm 0.303$ |

600 | / | / | 0.81 | 0.59 | $\pm 0.728$ | 0.3 | 0.4 | 0.19 | 0.09 | $\pm 0.474$ |

600 | / | / | 0.75 | 0.41 | $\pm 0.547$ | 0.2 | 0.3 | 0.22 | 0.011 | $\pm 0.05$ |

600 | / | / | 0.79 | 0.49 | $\pm 0.620$ | 0.2 | 0.3 | 0.27 | 0.07 | $\pm 0.259$ |

Average | $DE$ | $DE-MPFSC$ | ||||||||

0.1 | 0.9 | 0.748 | 0.46 | $\pm 0.6118$ | 0.28 | 0.54 | 0.232 | 0.0572 | $\pm 0.2372$ | |

Variance | $DE$ | $DE-MPFSC$ | ||||||||

0 | 0 | 0.00282 | 0.0069 | $\pm 0.0054$ | 0.037 | 0.083 | 0.00492 | 0.00175 | $\pm 0.02869$ |

**Table 4.**Numerical Analysis of Entanglement Degree $\frac{1}{\xi}$ and Entanglement Degree Error $\zeta $ of the DE algorithm and the DE-MPFSC algorithm 800 times.

NP | $DE$ | $DE-MPFSC$ | ||||||||

F | $CR$ | $\xi $ | $\zeta $ | $\epsilon \%$ | ${F}^{*}$ | $C{R}^{*}$ | $\xi $ | $\zeta $ | $\epsilon \%$ | |

800 | 0.1 | 0.9 | 0.63 | 0.44 | $\pm 0.698$ | 0.1 | 0.9 | 0.15 | 0.015 | $\pm 0.1$ |

800 | / | / | 0.64 | 0.31 | $\pm 0.484$ | 0.5 | 0.4 | 0.27 | 0.03 | $\pm 0.111$ |

800 | / | / | 0.75 | 0.50 | $\pm 0.667$ | 0.3 | 0.3 | 0.33 | 0.06 | $\pm 0.182$ |

800 | / | / | 0.70 | 0.50 | $\pm 0.714$ | 0.2 | 0.3 | 0.30 | 0.017 | $\pm 0.057$ |

800 | / | / | 0.71 | 0.36 | $\pm 0.507$ | 0.2 | 0.3 | 0.27 | 0.02 | $\pm 0.074$ |

Average | $DE$ | $DE-MPFSC$ | ||||||||

0.1 | 0.9 | 0.686 | 0.422 | $\pm 0.614$ | 0.26 | 0.44 | 0.264 | 0.0284 | $\pm 0.1048$ | |

Variance | $DE$ | $DE-MPFSC$ | ||||||||

0 | 0 | 0.00253 | 0.00722 | $\pm 0.01205$ | 0.023 | 0.068 | 0.00468 | 0.00035 | $\pm 0.00231$ |

**Table 5.**Numerical Analysis of Entanglement Degree $\frac{1}{\xi}$ and Entanglement Degree Error $\zeta $ of the DE algorithm and the DE-MPFSC algorithm 1000 times.

NP | $DE$ | $DE-MPFSC$ | ||||||||

F | $CR$ | $\xi $ | $\zeta $ | $\epsilon \%$ | ${F}^{*}$ | $C{R}^{*}$ | $\xi $ | $\zeta $ | $\epsilon \%$ | |

1000 | 0.1 | 0.9 | 0.69 | 0.40 | $\pm 0.579$ | 0.1 | 0.9 | 0.15 | 0.015 | $\pm 0.1$ |

1000 | / | / | 0.82 | 0.66 | $\pm 0.805$ | 0.3 | 0.4 | 0.30 | 0.04 | $\pm 0.133$ |

1000 | / | / | 0.67 | 0.49 | $\pm 0.731$ | 0.3 | 0.3 | 0.29 | 0.02 | $\pm 0.068$ |

1000 | / | / | 0.77 | 0.58 | $\pm 0.753$ | 0.2 | 0.3 | 0.27 | 0.019 | $\pm 0.070$ |

1000 | / | / | 0.79 | 0.44 | $\pm 0.557$ | 0.2 | 0.3 | 0.32 | 0.018 | $\pm 0.056$ |

Average | $DE$ | $DE-MPFSC$ | ||||||||

0.1 | 0.9 | 0.748 | 0.514 | $\pm 0.685$ | 0.22 | 0.44 | 0.266 | 0.0224 | $\pm 0.0854$ | |

Variance | $DE$ | $DE-MPFSC$ | ||||||||

0 | 0 | 0.00422 | 0.01118 | $\pm 0.01219$ | 0.007 | 0.068 | 0.00453 | 0.0001 | $\pm 0.00097$ |

**Table 6.**Numerical Analysis of Entanglement Degree $\frac{1}{\xi}$ and Entanglement Degree Error $\zeta $ of the DE algorithm and the DE-MPFSC algorithm Several Times.

NP | $JDE$ | $OBDE$ | $DEGL$ | $SADE$ | ||||||||||||||||

F | $CR$ | $\xi $ | $\zeta $ | $\epsilon \%$ | F | $CR$ | $\xi $ | $\zeta $ | $\epsilon \%$ | F | $CR$ | $\xi $ | $\zeta $ | $\epsilon \%$ | F | $CR$ | $\xi $ | $\zeta $ | $\epsilon \%$ | |

200 | 0.2 | 0.3 | ∞ | ∞ | $\pm 0.40$ | 0.2 | 0.3 | − | − | $\pm 1$ | 0.2 | 0.3 | ∞ | ∞ | $\pm 1$ | 0.2 | 0.3 | $1.04$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $1.04$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $\pm 0.30$ |

200 | / | / | ∞ | ∞ | $\pm 0.45$ | / | / | − | − | $\pm 1$ | / | / | ∞ | ∞ | $\pm 1$ | / | / | $1.06$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $1.06$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $\pm 0.40$ |

200 | / | / | ∞ | ∞ | $\pm 0.40$ | / | / | ∞ | ∞ | $\pm 1$ | / | / | − | − | $\pm 1$ | / | / | $0.81$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $0.81$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $\pm 0.35$ |

200 | / | / | − | − | $\pm 0.40$ | / | / | − | − | $\pm 1$ | / | / | ∞ | ∞ | $\pm 1$ | / | / | $2.43$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | $2.43$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | $\pm 0.35$ |

200 | / | / | − | − | $\pm 0.45$ | / | / | ∞ | ∞ | $\pm 1$ | / | / | − | ∞ | $\pm 1$ | / | / | $2.77$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | $2.77$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | $\pm 0.40$ |

NP | $JDE$ | $OBDE$ | $DEGL$ | $SADE$ | ||||||||||||||||

F | $CR$ | $\xi $ | $\zeta $ | $\epsilon \%$ | F | $CR$ | $\xi $ | $\zeta $ | $\epsilon \%$ | F | $CR$ | $\xi $ | $\zeta $ | $\epsilon \%$ | F | $CR$ | $\xi $ | $\zeta $ | $\epsilon \%$ | |

400 | 0.2 | 0.3 | $0.88$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{0}$ | $0.49$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{0}$ | $\pm 0.45$ | 0.2 | 0.3 | $0.53$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{0}$ | $0.60$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{0}$ | $\pm 1$ | 0.2 | 0.3 | $2.04$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $1.34$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $\pm 0.25$ | 0.2 | 0.3 | $2.42$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $1.06$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $\pm 0.15$ |

400 | / | / | $5.44$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | $1.49$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $\pm 0.40$ | / | / | $0.58$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{0}$ | $0.59$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{0}$ | $\pm 1$ | / | / | $1.74$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $1.41$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $\pm 0.25$ | / | / | $2.41$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $1.05$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $\pm 0.15$ |

400 | / | / | $6.05$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | $0.52$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{0}$ | $\pm 0.45$ | / | / | $1.61$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $4.72$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | $\pm 1$ | / | / | $1.72$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $3.72$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | $\pm 0.25$ | / | / | $6.23$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | $2.95$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | $\pm 0.15$ |

400 | / | / | $2.15$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $0.47$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{0}$ | $\pm 0.40$ | / | / | $4.22$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | $4.51$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | $\pm 1$ | / | / | $1.73$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $1.38$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $\pm 0.25$ | / | / | $6.49$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | $3.02$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $\pm 0.15$ |

400 | / | / | $2.33$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $1.13$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | $\pm 0.40$ | / | / | $1.67$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $1.69$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $\pm 1$ | / | / | $5.16$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | $3.70$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | $\pm 0.25$ | / | / | $2.43$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $1.08$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $\pm 0.15$ |

NP | $JDE$ | $OBDE$ | $DEGL$ | $SADE$ | ||||||||||||||||

F | $CR$ | $\xi $ | $\zeta $ | $\epsilon \%$ | F | $CR$ | $\xi $ | $\zeta $ | $\epsilon \%$ | F | $CR$ | $\xi $ | $\zeta $ | $\epsilon \%$ | F | $CR$ | $\xi $ | $\zeta $ | $\epsilon \%$ | |

600 | 0.2 | 0.3 | − | − | $\pm 1$ | 0.2 | 0.3 | $4.44$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | $0.49$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{0}$ | $\pm 0.45$ | 0.2 | 0.3 | $1.62$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $3.71$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | $\pm 0.15$ | 0.2 | 0.3 | $2.20$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $1.11$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $\pm 0.1$ |

600 | / | / | − | − | $\pm 1$ | / | / | $1.62$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $0.51$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{0}$ | $\pm 0.46$ | / | / | $4.30$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | $1.38$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $\pm 0.1$ | / | / | $2.18$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $1.10$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $\pm 0.1$ |

600 | / | / | ∞ | ∞ | $\pm 1$ | / | / | $1.62$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $0.47$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{0}$ | $\pm 0.40$ | / | / | $1.56$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $3.70$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | $\pm 0.1$ | / | / | $2.17$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $2.67$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | $\pm 0.15$ |

600 | / | / | − | − | $\pm 1$ | / | / | $4.41$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | $1.38$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $\pm 0.40$ | / | / | $4.28$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | $1.36$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $\pm 0.15$ | / | / | $5.91$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | $1.02$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $\pm 0.15$ |

600 | / | / | ∞ | ∞ | $\pm 1$ | / | / | $4.49$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | $1.16$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $\pm 0.38$ | / | / | $1.59$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $3.69$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | $\pm 0.1$ | / | / | $2.17$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $1.00$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $\pm 0.1$ |

NP | $JDE$ | $OBDE$ | $DEGL$ | $SADE$ | ||||||||||||||||

F | $CR$ | $\xi $ | $\zeta $ | $\epsilon \%$ | F | $CR$ | $\xi $ | $\zeta $ | $\epsilon \%$ | F | $CR$ | $\xi $ | $\zeta $ | $\epsilon \%$ | F | $CR$ | $\xi $ | $\zeta $ | $\epsilon \%$ | |

800 | 0.2 | 0.3 | $5.49$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | $1.41$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $\pm 0.6$ | 0.2 | 0.3 | $2.28$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $0.30$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{0}$ | $\pm 0.40$ | 0.2 | 0.3 | $2.01$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | $1.31$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $\pm 0.30$ | 0.2 | 0.3 | $2.14$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $1.01$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $\pm 0.15$ |

800 | / | / | $5.12$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | $1.44$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $\pm 0.6$ | / | / | $2.25$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $0.35$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{0}$ | $\pm 0.30$ | / | / | $2.03$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $1.38$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $\pm 0.40$ | / | / | $2.15$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $1.06$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $\pm 0.15$ |

800 | / | / | $15.70$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | $3.84$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | $\pm 0.6$ | / | / | $2.36$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $2.54$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | $\pm 0.40$ | / | / | $2.03$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $3.72$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | $\pm 0.50$ | / | / | $2.08$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $2.82$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | $\pm 0.2$ |

800 | / | / | $2.06$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $3.99$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | $\pm 0.6$ | / | / | $2.17$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $0.37$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{0}$ | $\pm 0.30$ | / | / | $2.05$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $1.34$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $\pm 0.60$ | / | / | $5.57$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | $0.95$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $\pm 0.2$ |

800 | / | / | $15.86$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | $1.54$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $\pm 0.6$ | / | / | $2.22$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $1.01$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $\pm 0.40$ | / | / | $5.55$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | $3.62$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | $\pm 0.75$ | / | / | $1.52$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $0.96$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $\pm 0.2$ |

NP | $JDE$ | $OBDE$ | $DEGL$ | $SADE$ | ||||||||||||||||

F | $CR$ | $\xi $ | $\zeta $ | $\epsilon \%$ | F | $CR$ | $\xi $ | $\zeta $ | $\epsilon \%$ | F | $CR$ | $\xi $ | $\zeta $ | $\epsilon \%$ | F | $CR$ | $\xi $ | $\zeta $ | $\epsilon \%$ | |

1000 | 0.2 | 0.3 | − | − | $\pm 1$ | 0.2 | 0.3 | ∞ | ∞ | $\pm 1$ | 0.2 | 0.3 | ∞ | ∞ | $\pm 1$ | 0.2 | 0.3 | $1.93$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $1.03$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $\pm 0.1$ |

1000 | / | / | − | − | $\pm 0.1$ | / | / | ∞ | ∞ | $\pm 1$ | / | / | ∞ | ∞ | $\pm 1$ | / | / | $1.88$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $0.96$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $\pm 0.1$ |

1000 | / | / | ∞ | ∞ | $\pm 1$ | / | / | − | − | $\pm 1$ | / | / | ∞ | ∞ | $\pm 1$ | / | / | $1.90$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $2.60$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | $\pm 0.1$ |

1000 | / | / | ∞ | ∞ | $\pm 1$ | / | / | − | − | $\pm 1$ | / | / | − | − | $\pm 1$ | / | / | $5.09$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | $0.87$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $\pm 0.1$ |

1000 | / | / | − | − | $\pm 1$ | / | / | ∞ | ∞ | $\pm 1$ | / | / | ∞ | ∞ | $\pm 1$ | / | / | $1.91$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $0.84$ $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $\pm 0.1$ |

**Table 7.**The situation description of various types of data about the UCI machine learning datasets.

Datasets | Sample Capacity | N.V. Datasets | C.V. Datasets | M.V. Datasets |
---|---|---|---|---|

Dermatology | 400 | 5 | 30 | 8 |

Credit Approval | 572 | 9 | 13 | 4 |

Automobile | 382 | 15 | 27 | 17 |

Sponge | 108 | 6 | 32 | 28 |

Contraceptive Method Choice (CMC) | 1458 | 4 | 21 | 16 |

Soybean | 453 | 2 | 39 | 14 |

Glass | 314 | 5 | 4 | 12 |

Data Sets | $CA$ | $ARI$ | $NMI$ | |||||||||

$SLCE$ | $CLCE$ | $KNN$ | $SKNN$ | $SLCE$ | $CLCE$ | $KNN$ | $SKNN$ | $SLCE$ | $CLCE$ | $KNN$ | $SKNN$ | |

$Der.$ | 0.430 | 0.772 | 0.674 | 0.710 | 0.152 | 0.667 | 0.666 | 0.458 | 0.351 | 0.728 | 0.588 | 0.579 |

$C.A.$ | 0.576 | 0.753 | 0.726 | 0.756 | 0.261 | 0.275 | 0.215 | 0.274 | 0.206 | 0.217 | 0.167 | 0.213 |

$Aut.$ | 0.523 | 0.537 | 0.529 | 0.516 | 0.168 | 0.148 | 0.157 | 0.153 | 0.265 | 0.257 | 0.237 | 0.269 |

$Spo.$ | 0.780 | 0.790 | 0.738 | 0.726 | 0.424 | 0.438 | 0.523 | 0.447 | 0.705 | 0.701 | 0.707 | 0.747 |

■(CMC) | 0.428 | 0.429 | 0.427 | 0.410 | 0.016 | 0.501 | 0.516 | 0.424 | 0.032 | 0.039 | 0.020 | 0.015 |

$Soy.$ | 0.545 | 0.668 | 0.631 | 0.634 | 0.431 | 0.436 | 0.356 | 0.346 | 0.622 | 0.637 | 0.638 | 0.696 |

■Glass | 0.473 | 0.546 | 0.524 | 0.525 | 0.199 | 0.154 | 0.158 | 0.168 | 0.323 | 0.302 | 0.315 | 0.226 |

Average | $CA$ | $ARI$ | $NMI$ | |||||||||

0.322 | 0.262 | 0.267 | 0.183 | 0.783 | 0.203 | 0.358 | 0.283 | 0.437 | 0.392 | 0.438 | 0.395 | |

Variance | $CA$ | $ARI$ | $NMI$ | |||||||||

0.002 | 0.0012 | 0.120 | 0.050 | 0.030 | 0.070 | 0.090 | 0.030 | 0.0013 | 0.0042 | 0.0912 | 0.095 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Gao, Y.; Wang, K.; Gao, C.; Shen, Y.; Li, T. Application of Differential Evolution Algorithm Based on Mixed Penalty Function Screening Criterion in Imbalanced Data Integration Classification. *Mathematics* **2019**, *7*, 1237.
https://doi.org/10.3390/math7121237

**AMA Style**

Gao Y, Wang K, Gao C, Shen Y, Li T. Application of Differential Evolution Algorithm Based on Mixed Penalty Function Screening Criterion in Imbalanced Data Integration Classification. *Mathematics*. 2019; 7(12):1237.
https://doi.org/10.3390/math7121237

**Chicago/Turabian Style**

Gao, Yuelin, Kaiguang Wang, Chenyang Gao, Yulong Shen, and Teng Li. 2019. "Application of Differential Evolution Algorithm Based on Mixed Penalty Function Screening Criterion in Imbalanced Data Integration Classification" *Mathematics* 7, no. 12: 1237.
https://doi.org/10.3390/math7121237