# Attribute Reduction Based on Lift and Random Sampling

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Basic Concept of Rough Set

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

#### 2.2. Attribute Reduction

**Definition**

**5.**

- (1)
- A meets the constraint ${\mathcal{C}}_{\rho}^{U}$;
- (2)
- $\forall {A}^{\prime}\subset A$, ${A}^{\prime}$ does not meet the constraint ${\mathcal{C}}_{\rho}^{U}$.

**Example**

**1.**

- (1)
- The greater the degree of the constrain obtained by $rho$, the less the level of approximation will be described. One example of such a metric is the definition of approximation quality.
- (2)
- The less the degree of the constrain we derived by utilizing $rho$, the less the amount of approximation in the dataset. One example of such a metric is the idea of decision error rate.

#### 2.3. Obtaining Reduct

Algorithm 1: Forward Greedy Searching for Attribute Reduction (FGSAR) |

#### 2.4. Strategy Based on Sample

- Bucket based searching for attribute reduction. Liu et al. [10] have considered the bucket method for fast obtaining reduct. A hash function is used in their technique, and every sample in the dataset will then be mapped into a number of separate buckets. Following the intrinsic properties of such a mapping, only samples from the same bucket must be compared rather than samples from the entire dataset. According to this viewpoint, the computing burden of information granulation may be minimized. As a result, the technique for obtaining reduct by the bucket-based strategy is identical to that of Algorithm 1, except for the device of information granulation.
- Positive approximation for attribute reduction (PAAR). Qian et al. [11] have presented the method based on positive approximation to calculate reduct rapidly. The essence to positive approximation is to compress gradually the sample space. The compressing process should be guided by the values associated with the constrain $rho$. The following are the precise steps of the attribute reduction based on positive approximation.
- (1)
- The hypothetical reduct A will be defined as ∅, as well as the sample compressed space ${U}^{\prime}$ will be defined as U.
- (2)
- By using the constrain of $\rho \left({U}^{\prime},A\cup \left\{a\right\}\right)$, analyse all hypothetical $a\in AT-A$ which in ${U}^{\prime}$.
- (3)
- Using the acquired constrain, choose one qualified attribute $b\in AT-A$ then merge in b to A.
- (4)
- Based on A, calculate $\rho \left({U}^{\prime},A\right)$ and then reanalysis construction of the sample compressed space ${U}^{\prime}$.
- (5)
- If the specified constraint is met, produce reduct A; otherwise, back to step(2).

- Random sampling accelerator for attribute reduction(RSAR). Chen et al. [12] examined the above-mentioned strategies, they found that: (1) Regardless of the searching technique, information granulation over the entire dataset is required; (2) Information granulation over the dataset always needs to be regenerated in each iteration throughout the entire searching process. In this aspect, sample distribution may influence the effectiveness of searching. Therefore, the two above-mentioned strategies have their own restrictions since they are directly tied to sample distribution. The restriction of BBSAR is that Bucket strategy will become inefficient when sample is too centralized, which is time-consuming. The restriction of PAAR is that the sample distribution strongly affect the construction of positive approximation, and then affect the effectiveness of attribute reduction. In view of this, Chen et al. [12] developed a new random sampling strategy. The following shows the exact structure of the random sampling being used to derive reduct.
- (1)
- The samples were randomly separated to n sample groups of equal size: ${U}_{1},\cdots ,{U}_{n}$.
- (2)
- Compute the reduct ${A}_{1}$ over ${U}_{1}$; reduct ${A}_{1}$ will then provide advice for computing the reduct ${A}_{2}$ over ${U}_{1}\cup {U}_{2}$, and so on.
- (3)
- Get the reduct ${A}_{n}$, use it as the ultimate reduct over the entire dataset.

## 3. Lift for Attribute Reduction

#### 3.1. Theoretical Foundations

- (1)
- Collect all cluster centers into ${U}_{1}$.
- (2)
- The other samples were randomly separated to n − 1 sample groups of equal size: ${U}_{2},\cdots ,{U}_{n}$.
- (3)
- Compute the first reduct ${A}_{1}$ over ${U}_{1}$; this reduct will then provide advice for computing the second reduct ${A}_{2}$ over ${U}_{1}\cup {U}_{2}$; ${A}_{2}$ will also provide advice for computing ${A}_{3}$ over ${U}_{1}\cup {U}_{2}\cup {U}_{3}$ and so on.
- (4)
- Get the n-th reduct ${A}_{n}$, use it as the ultimate reduct over the entire dataset.

#### 3.2. Detailed Algorithm

**Remark**

**1.**

Algorithm 2: Attribute Reduction Based on Lift and Random Sampling |

## 4. Analysis of Experiment

#### 4.1. Datasets

#### 4.2. Basic Experiment Setting

#### 4.3. Time Consumption

#### 4.4. Classification Performance

#### 4.5. Discussion

- (1)
- The higher the dimension of the datasets is, the more information the selected sample can carry. Then, the more information can better help to guide the subsequent attribute reduction. Therefore, ARLRS algorithm performs well in handling big data.
- (2)
- Due to the low value density of big data, it is often necessary to preferentially extract relevant and useful information from a large amount of datasets. The above-mentioned needs are well solved by introducing Lift. The most representative samples found by LIFT make the unclear information under each label have available structure, from the perspective of samples, thus different samples form a certain association. Such preprocessing is very effective in the processing big data, so ARLRS will have certain advantages in processing big data.

## 5. Conclusions and Future Perspectives

- (1)
- In light of the uncertainty existed in constraints and classification performances, the resultant reduct may result in over-fitting. As a result, in future investigation, we will try balance the efficiency of attribute reduction with the classification performance.
- (2)
- The attribute reduction strategy presented in this research is only applied from the sample’s perspective. Therefore, we will try to investigate some novel algorithms taking into account both samples and attributes for improving more efficiency.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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ID | Datasets | # Samples | # Attributes | # Labels | Attributes Type |
---|---|---|---|---|---|

1 | Australian Credit Approval [36] | 690 | 14 | 2 | Real |

2 | Breast Cancer Wisconsin (Diagnostic) [37] | 569 | 30 | 2 | Real |

3 | Cardiotocography [38] | 2126 | 21 | 10 | Real |

4 | Connectionist Bench (Sonar, Mines vs. Rocks) [39] | 208 | 60 | 2 | Real |

5 | Crowdsourced Mapping [40] | 10,845 | 28 | 6 | Real |

6 | Diabetic Retinopathy Debrecen [41] | 1151 | 19 | 2 | Integer & Real |

7 | DrivFace [42] | 606 | 6400 | 3 | Real |

8 | Forest Type Mapping [43] | 523 | 27 | 4 | Integer & Real |

9 | Glass Identification [44] | 214 | 9 | 6 | Real |

10 | Ionosphere [45] | 351 | 34 | 2 | Integer & Real |

11 | MAGIC Gamma Telescope [46] | 19,020 | 28 | 6 | Real |

12 | Parkinson Multiple Sound Recording [47] | 1208 | 26 | 2 | Real |

13 | QSAR Biodegradation [48] | 1055 | 41 | 2 | Real |

14 | Musk (Version 1) [49] | 476 | 166 | 2 | Integer |

15 | Page Blocks Classification [50] | 5473 | 10 | 5 | Integer & Real |

16 | Urban Land Cover [51] | 675 | 147 | 9 | Integer & Real |

17 | Quality Assessment of Digital Colposcopies [52] | 287 | 62 | 2 | Real |

ID | ARLRS | RSAR | FGSAR | PAAR | BBSAR | AGAR |
---|---|---|---|---|---|---|

1 | 1.3439 | 1.6046 | 1.9683 | 1.9533 | 3.0378 | 2.0001 |

2 | 2.8514 | 3.2616 | 3.7553 | 3.6474 | 4.9878 | 6.3496 |

3 | 1.7156 | 2.3740 | 3.6215 | 3.3452 | 4.0248 | 2.6923 |

4 | 3.114 | 3.8712 | 4.6534 | 4.4532 | 4.9458 | 6.3526 |

5 | 0.0331 | 0.0351 | 0.3561 | 0.3424 | 0.4138 | 0.0493 |

6 | 1.5043 | 1.6542 | 3.3462 | 3.3977 | 2.7745 | 3.4527 |

7 | 21.5071 | 21.6074 | 22.9543 | 22.6464 | 32.7441 | 24.2342 |

8 | 3.2707 | 4.9173 | 10.2732 | 10.2480 | 4.4253 | 4.5429 |

9 | 36.6825 | 39.9173 | 55.4524 | 54.2809 | 46.355 | 40.6939 |

10 | 2.2156 | 2.3740 | 3.5643 | 3.5754 | 4.0248 | 2.5551 |

11 | 32.6542 | 34.7645 | 35.3475 | 35.3342 | 35.4578 | 34.9578 |

12 | 2.5127 | 2.7545 | 3.1642 | 3.0008 | 2.8361 | 3.147 |

13 | 9.6467 | 11.9454 | 13.6468 | 13.4522 | 12.5647 | 12.3949 |

14 | 16.9655 | 18.3361 | 19.7468 | 19.2312 | 19.1485 | 18.5435 |

15 | 44.4554 | 46.1265 | 48.4641 | 48.8529 | 45.5659 | 48.4844 |

16 | 3.6125 | 4.0415 | 4.8353 | 4.1237 | 4.7264 | 4.9645 |

17 | 38.4124 | 41.5415 | 43.8515 | 43.4529 | 45.4554 | 44.5455 |

ID | RSAR | FGSAR | PAAR | BBSAR | AGAR |
---|---|---|---|---|---|

1 | 16.37 | 31.19 | 41.17 | 30.34 | 32.81 |

2 | 12.58 | 24.07 | 21.82 | 20.53 | 19.31 |

3 | 27.73 | 35.12 | 27.82 | 18.21 | 26.34 |

4 | 19.63 | 33.14 | 30.13 | 25.75 | 7.22 |

5 | 2.36 | 23.38 | 2.65 | 2.71 | 17.78 |

6 | 9.06 | 14.64 | 42.91 | 34.52 | 27.98 |

7 | 13.15 | 19.01 | 18.62 | 4.57 | 5.99 |

8 | 8.11 | 33.85 | 32.42 | 34.91 | 9.86 |

9 | 20.13 | 37.84 | 38.03 | 24.24 | 13.29 |

10 | 10.23 | 11.09 | 22.97 | 23.13 | 30.23 |

11 | 6.07 | 7.02 | 7.88 | 7.91 | 6.59 |

12 | 25.97 | 20.59 | 16.27 | 11.40 | 20.16 |

13 | 19.24 | 9.39 | 7.71 | 23.22 | 22.17 |

14 | 25.09 | 18.28 | 13.71 | 12.93 | 6.77 |

15 | 3.62 | 8.27 | 6.00 | 2.44 | 8.31 |

16 | 15.23 | 13.88 | 14.76 | 17.84 | 20.76 |

17 | 31.25 | 4.33 | 6.12 | 7.32 | 15.9 |

avg | 15.63 | 20.29 | 20.64 | 17.79 | 17.14 |

ID | ARLRS | RSAR | FGSAR | PAAR | BBSAR | AGAR |
---|---|---|---|---|---|---|

1 | 7.14 | 7.32 | 8.03 | 8.03 | 8.03 | 7.17 |

2 | 8.67 | 8.21 | 9.19 | 9.19 | 9.19 | 8.11 |

3 | 7.38 | 8.66 | 7.57 | 7.57 | 7.57 | 8.45 |

4 | 7.13 | 7.43 | 8.17 | 8.17 | 8.17 | 7.43 |

5 | 14.34 | 15.32 | 14.38 | 14.38 | 14.38 | 16.64 |

6 | 9.46 | 9.21 | 8.32 | 8.32 | 8.32 | 8.76 |

7 | 8.74 | 9.87 | 7.15 | 7.15 | 7.15 | 9.75 |

8 | 8.65 | 9.07 | 9.25 | 9.25 | 9.25 | 8.98 |

9 | 7.91 | 9.87 | 8.57 | 8.57 | 8.57 | 8.07 |

10 | 8.65 | 8.97 | 8.78 | 8.78 | 8.78 | 9.21 |

11 | 23.87 | 24.58 | 25.32 | 25.32 | 25.32 | 27.98 |

12 | 7.03 | 10.34 | 10.01 | 10.01 | 10.01 | 9.28 |

13 | 6.98 | 9.01 | 8.34 | 8.34 | 8.34 | 7.32 |

14 | 8.75 | 8.65 | 8.90 | 8.90 | 8.90 | 10.76 |

15 | 13.66 | 15.32 | 16.24 | 16.24 | 16.24 | 15.23 |

16 | 14.31 | 15.09 | 15.65 | 15.65 | 15.65 | 16.32 |

17 | 8.02 | 8.06 | 8.19 | 8.19 | 8.19 | 9.15 |

**Table 5.**Based on KNN classifier, the classification accuracies of reducts produced by six attribute reduction algorithms.

ID | ARLRS | RSAR | FGSAR | PAAR | BBSAR | AGAR |
---|---|---|---|---|---|---|

1 | 0.8555 | 0.8742 | 0.8168 | 0.8168 | 0.8557 | 0.9175 |

2 | 0.9785 | 0.9000 | 0.8534 | 0.8534 | 0.9347 | 0.9456 |

3 | 0.9399 | 0.9320 | 0.8824 | 0.8824 | 0.8356 | 0.8943 |

4 | 0.9569 | 0.9642 | 0.8074 | 0.8074 | 0.9741 | 0.9456 |

5 | 0.8848 | 0.8351 | 0.8022 | 0.8022 | 0.8149 | 0.8624 |

6 | 0.8449 | 0.8272 | 0.9074 | 0.9074 | 0.8736 | 0.8488 |

7 | 0.9089 | 0.9003 | 0.8075 | 0.8075 | 0.8572 | 0.8197 |

8 | 0.8437 | 0.8326 | 0.8731 | 0.8731 | 0.8186 | 0.8279 |

9 | 0.8441 | 0.8741 | 0.8507 | 0.8507 | 0.9117 | 0.7986 |

10 | 0.9575 | 0.9320 | 0.9269 | 0.9269 | 0.8356 | 0.8943 |

11 | 0.9145 | 0.8487 | 0.8366 | 0.8366 | 0.8869 | 0.8747 |

12 | 0.9035 | 0.9055 | 0.9266 | 0.9266 | 0.8845 | 0.8647 |

13 | 0.8364 | 0.8641 | 0.8644 | 0.8644 | 0.9153 | 0.8279 |

14 | 0.9634 | 0.9534 | 0.9128 | 0.9128 | 0.8934 | 0.8634 |

15 | 0.9314 | 0.9169 | 0.8674 | 0.8674 | 0.8467 | 0.8469 |

16 | 0.9299 | 0.9064 | 0.8796 | 0.8796 | 0.9074 | 0.8534 |

17 | 0.8036 | 0.8712 | 0.8642 | 0.8642 | 0.8541 | 0.8779 |

**Table 6.**Based on SVM classifier, the classification accuracies of reducts produced by six attribute reduction algorithms.

ID | ARLRS | RSAR | FGSAR | PAAR | BBSAR | AGAR |
---|---|---|---|---|---|---|

1 | 0.8396 | 0.8614 | 0.8318 | 0.8318 | 0.8496 | 0.8779 |

2 | 0.8984 | 0.8401 | 0.8302 | 0.8302 | 0.7121 | 0.7326 |

3 | 0.8351 | 0.8741 | 0.9164 | 0.9164 | 0.8813 | 0.8801 |

4 | 0.8433 | 0.8381 | 0.8041 | 0.8041 | 0.8701 | 0.8912 |

5 | 0.9103 | 0.8701 | 0.9328 | 0.9328 | 0.9488 | 0.8468 |

6 | 0.9434 | 0.9362 | 0.9723 | 0.9723 | 0.9356 | 0.9147 |

7 | 0.8841 | 0.8766 | 0.8452 | 0.8452 | 0.8141 | 0.7998 |

8 | 0.8321 | 0.8311 | 0.9723 | 0.9723 | 0.7318 | 0.7323 |

9 | 0.9194 | 0.9248 | 0.9049 | 0.9049 | 0.8722 | 0.8413 |

10 | 0.8954 | 0.8888 | 0.7763 | 0.7763 | 0.8052 | 0.7113 |

11 | 0.9165 | 0.8971 | 0.9460 | 0.9460 | 0.8741 | 0.8492 |

12 | 0.9365 | 0.9207 | 0.9194 | 0.9194 | 0.8279 | 0.7812 |

13 | 0.9153 | 0.8803 | 0.7436 | 0.7436 | 0.7799 | 0.8940 |

14 | 0.8915 | 0.8766 | 0.8622 | 0.8622 | 0.8348 | 0.8786 |

15 | 0.9255 | 0.8903 | 0.8802 | 0.8802 | 0.8584 | 0.9205 |

16 | 0.9044 | 0.8909 | 0.8633 | 0.8633 | 0.8261 | 0.8759 |

17 | 0.8759 | 0.9204 | 0.8816 | 0.8816 | 0.9362 | 0.8294 |

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**MDPI and ACS Style**

Chen, Q.; Xu, T.; Chen, J.
Attribute Reduction Based on Lift and Random Sampling. *Symmetry* **2022**, *14*, 1828.
https://doi.org/10.3390/sym14091828

**AMA Style**

Chen Q, Xu T, Chen J.
Attribute Reduction Based on Lift and Random Sampling. *Symmetry*. 2022; 14(9):1828.
https://doi.org/10.3390/sym14091828

**Chicago/Turabian Style**

Chen, Qing, Taihua Xu, and Jianjun Chen.
2022. "Attribute Reduction Based on Lift and Random Sampling" *Symmetry* 14, no. 9: 1828.
https://doi.org/10.3390/sym14091828