# Fuzzy Heuristics and Decision Tree for Classification of Statistical Feature-Based Control Chart Patterns

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Sample Patterns Generation

#### 2.2. Features Extraction

- Skewness: The symmetry of shape distribution. The estimate of the skewness in data points from ${X}_{1}$ to ${X}_{n}$ is;

- Mean Square Value:

- CUSUM: It is the cumulative sum of values. The last statistical value of CUSUM is taken as the feature in this study. The general formula for upper and lower CUSUM statistics are:

- Autocorrelation: Exists when later data is dependent on previous data.

- Kurtosis: Measures the peakness of a distribution

- SLOPE: The first order line fitting. The slope m is used as a feature in this study.

#### 2.3. Classifier Design and Development

#### 2.4. Performance Evaluation

## 3. Development of Heuristic Mamdani Fuzzy Classifier

## 4. Development of Decision Tree Classifier

## 5. Results and Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**An example of membership functions for feature MEAN with its respective indicative patterns.

**Figure 7.**Classification tree for control chart patterns based on partitions of statistical features.

Pattern Type | Parameters | Parameter Ranges | Standard Equations |
---|---|---|---|

Trend up (TU) or Trend down (TD) | Slope (${\gamma}_{1}$) | 0.005 to 0.025 | ${y}_{t}=\mathsf{\mu}+{N}_{t}\pm {\gamma}_{1}t$ |

Shift up (SU) or Shift down (SD) | Shift (${\gamma}_{2}$) | 0.005 to 2.5 | ${y}_{t}=\mathsf{\mu}+{N}_{t}\pm {\gamma}_{2}t$ |

Cyclic (CYC) | Amplitude (${\gamma}_{3}$) Frequency (${\gamma}_{4}=10$) | 0 to 1.8 | ${y}_{t}=\mathsf{\mu}+{N}_{t}\pm {\gamma}_{3}\mathrm{sin}\left(\frac{2\pi t}{{\gamma}_{4}}\right)$ |

Stratification (STRA) | Stratification (${\gamma}_{5}$) | 0.1 to 0.6 | ${y}_{t}=\mathsf{\mu}+{N}_{t}+{\gamma}_{5}{N}_{t}$ |

Systematic (SYS) | Departure (${\gamma}_{6}$) | 0.005 to 2.5 | ${y}_{t}$ = μ + ${N}_{t}$ ± ${\gamma}_{6}{\left(-1\right)}^{t}$ |

Normal (NOR) | NA | μ = 0, $\mathsf{\sigma}=1$ | ${y}_{t}=\mathsf{\mu}+{N}_{t}$ |

Rule | Description of IF-THEN Rules for the Fuzzy Classifier |
---|---|

1 | If (MEAN is HIGH) and (SD is MED) and (SLOPE is VHIGH) then (Pattern is Trend up) |

2 | If (MEAN is HIGH) and (SLOPE is HIGH) then (Pattern is Shift up) |

3 | If (MEAN is LOW) and (SD is MED) and (SLOPE is LOW) then (Pattern is Trend down) |

4 | If (MEAN is LOW) and (SLOPE is HIGH) then (Pattern is Shift down) |

5 | If (MEAN is MED) and (SD is HIGH) and (MSV is MED) and (CUSUM is HIGH) and (Autocorrelation is HIGH) and (Range is HIGH) and (Kurtosis is MED) and (SLOPE is VLOW) then (Pattern is Cyclic) |

6 | If (MEAN is MED) and (SD is HIGH) and (MSV is HIGH) and (CUSUM is MED) and (Autocorrelation is LOW) and (Range is HIGH) and (Kurtosis is LOW) and (SLOPE is VHIGH) then (Pattern is Systematic) |

7 | If (MEAN is MED) and (SD is LOW) and (Autocorrelation is HIGH) and (Range is LOW) and (Kurtosis is HIGH) and (SLOPE is HIGH) then (Pattern is Stratification) |

8 | If (MEAN is MED) and (MSV is LOW) and (CUSUM is LOW) and (Autocorrelation is LOW) and (Range is MED) and (Kurtosis is HIGH) and (SLOPE is HIGH) then (Pattern is Normal) |

9 | If (MEAN is MED) and (SD is LOW) and (Range is LOW) and (Kurtosis is HIGH) and (SLOPE is HIGH) then (Pattern is Stratification) |

10 | If (Range is LOW) then (Pattern is Stratification) |

Leaf No. | IF Decision Rule | THEN (Pattern Type) |
---|---|---|

1 | IF MEAN < 1.677 AND SD < −9.213 | Trend up |

2 | IF MEAN < 1.677 AND SD ≥ −9.213 | Shift up |

3 | IF MEAN ≥ 1.677 AND MSV ≥ −9.946 AND SD < −9.512 | Normal |

4 | IF MEAN ≥ 1.677 AND MSV ≥ −9.946 AND SD ≥ −9.512 | Stratification |

5 | IF MEAN ≥ 1.677 AND MSV ≥ −9.946 AND MEAN ≥ −1.259 AND SLOPE ≥ 4.106 | Trend down |

6 | IF MEAN ≥ 1.677 AND MSV ≥ −9.946 AND MEAN ≥ −1.259 AND SLOPE < 4.106 | Shift down |

7 | IF MEAN ≥ 1.677 AND MSV < −9.946 AND MEAN < −1.259 AND MEAN ≥ −0.1789 | Cyclic |

8 | IF MEAN ≥ 1.677 AND MSV < −9.946 AND MEAN < −1.259 AND MEAN < −0.1789 | Systematic |

Pattern | Fuzzy Classifier | DT Classifier |
---|---|---|

Normal | 100.0% | 96.1% |

Trend up | 85.4% | 100.0% |

Trend down | 96.3% | 96.5% |

Shift up | 100.0% | 96.0% |

Shift down | 100.0% | 100.0% |

Cyclic | 98.4% | 100.0% |

Systematic | 86.0% | 100.0% |

Stratification | 100.0% | 100.0% |

Overall Recognition accuracy ($\mathsf{\mu}$) | 95.76% | 98.58% |

Standard Deviation ($\mathsf{\sigma}$) | 1.09 | 0.48 |

Classified Patterns by Fuzzy Classifier | |||||||||
---|---|---|---|---|---|---|---|---|---|

Normal | Trend Up | Trend Down | Shift Up | Shift Down | Cyclic | Systematic | Stratification | ||

True Pattern Class | Normal | 100% | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

Trend up | 2.4% | 85.4% | 0 | 12.2% | 0 | 0 | 0 | 0 | |

Trend down | 0 | 0 | 96.3% | 0 | 3.7% | 0 | 0 | 0 | |

Shift up | 0 | 0 | 0 | 100% | 0 | 0 | 0 | 0 | |

Shift down | 0 | 0 | 0 | 0 | 100% | 0 | 0 | 0 | |

Cyclic | 1.6% | 0 | 0 | 0 | 0 | 98.4% | 0 | 0 | |

Systematic | 7.8% | 0 | 0 | 0 | 0 | 6.2% | 86% | 0 | |

Stratification | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 100% |

Classified Patterns by DT Classifier | |||||||||
---|---|---|---|---|---|---|---|---|---|

Normal | Trend Up | Trend Down | Shift Up | Shift Down | Cyclic | Systematic | Stratification | ||

True Pattern Class | Normal | 96.1% | 0 | 0 | 0 | 0 | 0 | 3.9% | 0 |

Trend up | 0 | 100% | 0 | 0 | 0 | 0 | 0 | 0 | |

Trend down | 0 | 0 | 96.5% | 0 | 3.5% | 0 | 0 | 0 | |

Shift up | 0 | 4% | 0 | 96% | 0 | 0 | 0 | 0 | |

Shift down | 0 | 0 | 0 | 0 | 100% | 0 | 0 | 0 | |

Cyclic | 0 | 0 | 0 | 0 | 0 | 100% | 0 | 0 | |

Systematic | 0 | 0 | 0 | 0 | 0 | 0 | 100% | 0 | |

Stratification | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 100% |

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**MDPI and ACS Style**

Zaman, M.; Hassan, A.
Fuzzy Heuristics and Decision Tree for Classification of Statistical Feature-Based Control Chart Patterns. *Symmetry* **2021**, *13*, 110.
https://doi.org/10.3390/sym13010110

**AMA Style**

Zaman M, Hassan A.
Fuzzy Heuristics and Decision Tree for Classification of Statistical Feature-Based Control Chart Patterns. *Symmetry*. 2021; 13(1):110.
https://doi.org/10.3390/sym13010110

**Chicago/Turabian Style**

Zaman, Munawar, and Adnan Hassan.
2021. "Fuzzy Heuristics and Decision Tree for Classification of Statistical Feature-Based Control Chart Patterns" *Symmetry* 13, no. 1: 110.
https://doi.org/10.3390/sym13010110