# Optimal Design of Interval Type-2 Fuzzy Heart Rate Level Classification Systems Using the Bird Swarm Algorithm

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## Abstract

**:**

## 1. Introduction

## 2. Basic Concepts

#### 2.1. Heart Rate

**Air temperature**: When temperatures and humidity soar, the heart pumps a little more blood, so the heart rate may increase, but usually no more than five to ten beats per minute.

**The position of the body**: At rest, sitting or standing, the heart rate is usually the same. Sometimes, when the person stands up for the first 15 to 20 s, the heart rate may go up a bit, but after a couple of minutes, it should be set.

**Emotions**: If the person is stressed, anxious or “extraordinarily happy or sad” the emotions can raise the heart rate.

**Body size**: Body size usually does not change pulse. If the person is very obese, it is possible to see a resting heart rate greater than normal, but usually no more than 100 beats per minute.

**The use of medications**: Drugs that block adrenaline (beta blockers) tend to slow the heart rate, while too much thyroid medicine or too high a dose tends to raise it [18].

#### 2.2. Bird Swarm Algorithm

_{i,j}is the best previous position in the i

_{th}bird and g

_{j}is the best previous position shared in the swarm.

_{i}corresponds to the best value fitness in the ith position and sumFit signifies the sum of the best fitness value of the swarms. $\epsilon $ is used to avoid the error in zero-division. mean

_{j}is the jth element of the average place of the entire swarm. a

_{1}and a

_{2}are positive constants in (0,2).

#### 2.3. Blood Pressure

#### 2.4. High Blood Pressure

#### 2.5. Type-1 Fuzzy Logic

#### 2.6. Interval Type-2 Fuzzy Logic

#### 2.7. Related Works

## 3. Problem Statement and Proposed Method

#### 3.1. Design of the Type-1 Fuzzy Inference System with Trapezoidal Membership Functions for the Classification of the Heart Rate Level

#### 3.2. Design of the Type-1 Fuzzy Inference System with Gaussian Membership Functions for the Classification of the Heart Rate Level

- If (Age is Child) and (Pulse is VeryLow) then (PLevels is Low) (1)
- If (Age is Child) and (Pulse is Low) then (PLevels is Low) (1)
- If (Age is Child) and (Pulse is Normal) then (PLevels is Excellent) (1)
- If (Age is Child) and (Pulse is High) then (PLevels is Excellent) (1)
- If (Age is Child) and (Pulse is VeryHigh) then (PLevels is AboveAV) (1)
- If (Age is Young) and (Pulse is VeryLow) then (PLevels is Low) (1)
- If (Age is Young) and (Pulse is Low) then (PLevels is BelowAV) (1)
- If (Age is Young) and (Pulse is Normal) then (PLevels is Excellent) (1)
- If (Age is Young) and (Pulse is High) then (PLevels is AboveAV) (1)
- If (Age is Young) and (Pulse is VeryHigh) then (PLevels is Very_High) (1)
- If (Age is Adult) and (Pulse is VeryLow) then (PLevels is Low) (1)
- If (Age is Adult) and (Pulse is Low) then (PLevels is BelowAV) (1)
- If (Age is Adult) and (Pulse is Normal) then (PLevels is Excellent) (1)
- If (Age is Adult) and (Pulse is High) then (PLevels is AboveAV) (1)
- If (Age is Adult) and (Pulse is VeryHigh) then (PLevels is Very_High) (1)
- If (Age is Elder) and (Pulse is High) then (PLevels is Very_High) (1)
- If (Age is Elder) and (Pulse is VeryHigh) then (PLevels is Very_High) (1)
- If (Age is Elder) and (Pulse is VeryLow) then (PLevels is Low) (1)
- If (Age is Elder) and (Pulse is Low) then (PLevels is Excellent) (1)
- If (Age is Elder) and (Pulse is Normal) then (PLevels is Excellent) (1)

#### 3.3. Optimization of the Type-1 Fuzzy Inference System with the BSA

#### 3.4. Design and Optimization of the Interval Type-2 Fuzzy Systems

## 4. Knowledge Representation

#### 4.1. Knowledge Representation of the Optimized Type-1 Fuzzy System with Trapezoidal Membership Functions

#### 4.1.1. Input Variables

#### 4.1.1.1. Input Age

#### 4.1.1.2. Input Heart Rate

#### 4.1.2. Output Variable

#### 4.2. Knowledge Representation of the Optimized Type-1 Fuzzy System with Gaussian Membership Functions

#### 4.2.1. Input Variables

#### 4.2.1.1. Input Age

#### 4.2.1.2. Input Heart Rate

#### 4.2.2. Output Variable

#### 4.3. Knowledge Representation of the Optimized Interval Type-2 Fuzzy System with Trapezoidal Membership Functions

#### 4.3.1. Input Variables

#### 4.3.1.1. Input Age

#### 4.3.1.2. Input Heart Rate

#### 4.3.2. Output Variable

#### 4.4. Knowledge Representation of the Optimized Interval Type-2 Fuzzy System with Gaussian Membership Functions

#### 4.4.1. Input Variables

#### 4.4.1.1. Input Age

#### 4.4.1.2. Input Heart Rate

#### 4.4.2. Output Variable

## 5. Results

#### 5.1. Optimization of Type-1 Fuzzy Systems

#### 5.2. Optimization of the Interval Type-2 Fuzzy Systems

#### 5.3. Tests with Patients Using the Type-1 Fuzzy Systems

#### 5.4. Tests with Patients Using Interval Type-2 Fuzzy Systems

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Age | Men | Women | ||||||
---|---|---|---|---|---|---|---|---|

Ill | Normal | Good | Excellent | Ill | Normal | Good | Excellent | |

20–29 | 86 or more | 70–84 | 62–68 | 60 or less | 96 or more | 78–94 | 72–76 | 70 or less |

30–39 | 86 or more | 72–84 | 64–70 | 62 or less | 98 or more | 80–96 | 72–78 | 70 or less |

40–49 | 90 or more | 74–88 | 66–72 | 64 or less | 100 or more | 80–98 | 74–78 | 72 or less |

50 or more | 90 or more | 76–88 | 68–74 | 66 o or less | 104 or more | 84–102 | 76–86 | 74 or less |

No | M | pop | DimT | DimG | FQ | c1 | c2 | a1 | a2 |
---|---|---|---|---|---|---|---|---|---|

1 | 1000 | 20 | 56 | 28 | 19 | 0.5 | 0.5 | 2 | 2 |

2 | 870 | 24 | 56 | 28 | 28 | 0.8 | 0.8 | 1.5 | 1.5 |

3 | 714 | 28 | 56 | 28 | 18 | 1.2 | 1.2 | 0.4 | 0.4 |

4 | 625 | 32 | 56 | 28 | 15 | 1.5 | 1.5 | 0.1 | 0.1 |

5 | 571 | 36 | 56 | 28 | 6 | 1.8 | 1.8 | 0.8 | 0.8 |

6 | 574 | 38 | 56 | 28 | 21 | 2 | 2 | 1 | 1 |

7 | 454 | 44 | 56 | 28 | 25 | 2.33 | 2.33 | 1.3 | 1.3 |

8 | 416 | 48 | 56 | 28 | 6 | 2.48 | 2.48 | 0.6 | 0.6 |

9 | 400 | 50 | 56 | 28 | 28 | 2.76 | 2.76 | 0.9 | 0.9 |

10 | 357 | 56 | 56 | 28 | 20 | 3 | 3 | 1.1 | 1.1 |

11 | 338 | 60 | 56 | 28 | 10 | 3.18 | 3.18 | 1.9 | 1.9 |

12 | 322 | 62 | 56 | 28 | 21 | 3.22 | 3.22 | 0.5 | 0.5 |

13 | 307 | 66 | 56 | 28 | 1 | 3.45 | 3.45 | 1.5 | 1.5 |

14 | 285 | 70 | 56 | 28 | 13 | 3.56 | 3.56 | 0.7 | 0.7 |

15 | 278 | 72 | 56 | 28 | 2 | 4 | 4 | 1.3 | 1.3 |

16 | 256 | 78 | 56 | 28 | 24 | 0.4 | 0.4 | 1.8 | 1.8 |

17 | 250 | 80 | 56 | 28 | 19 | 0.7 | 0.7 | 0.3 | 0.3 |

18 | 235 | 86 | 56 | 28 | 1 | 1.15 | 1.15 | 0.9 | 0.9 |

19 | 227 | 88 | 56 | 28 | 24 | 1.34 | 1.34 | 1 | 1 |

20 | 208 | 96 | 56 | 28 | 22 | 1.45 | 1.45 | 2 | 2 |

21 | 202 | 100 | 56 | 28 | 15 | 1.67 | 1.67 | 0.6 | 0.6 |

22 | 166 | 120 | 56 | 28 | 4 | 1.78 | 1.78 | 0.3 | 0.3 |

23 | 133 | 150 | 56 | 28 | 16 | 1.92 | 1.92 | 1.5 | 1.5 |

24 | 111 | 180 | 56 | 28 | 2 | 2.18 | 2.18 | 1.2 | 1.2 |

25 | 100 | 200 | 56 | 28 | 2 | 2.39 | 2.39 | 1.8 | 1.8 |

26 | 95 | 210 | 56 | 28 | 21 | 2.56 | 2.56 | 0.7 | 0.7 |

27 | 90 | 220 | 56 | 28 | 22 | 2.83 | 2.83 | 0.9 | 0.9 |

28 | 87 | 230 | 56 | 28 | 15 | 3.4 | 3.4 | 1.5 | 1.5 |

29 | 83 | 240 | 56 | 28 | 19 | 3.7 | 3.7 | 1.7 | 1.7 |

30 | 80 | 250 | 56 | 28 | 20 | 4 | 4 | 2 | 2 |

Parameter | Value |
---|---|

M | 285 |

pop | 70 |

DimT | 56 |

DimG | 28 |

FQ | 13 |

c1 | 3.56 |

c2 | 3.56 |

a1 | 0.7 |

a2 | 0.7 |

No. | MFTra | MFGauss |
---|---|---|

1 | 97.50% | 87.50% |

2 | 95% | 87.50% |

3 | 97.50% | 92.50% |

4 | 90% | 87.50% |

5 | 97.50% | 87.50% |

6 | 90% | 92.50% |

7 | 92.50% | 90% |

8 | 87.50% | 92.50% |

9 | 92.50% | 92.50% |

10 | 95% | 90% |

11 | 95% | 92.50% |

12 | 90% | 95% |

13 | 92.50% | 90% |

14 | 100% | 95% |

15 | 97.50% | 90% |

16 | 95% | 90% |

17 | 95% | 87.50% |

18 | 97.50% | 90% |

19 | 92.50% | 92.50% |

20 | 95% | 92.50% |

21 | 95% | 92.50% |

22 | 92.50% | 85% |

23 | 95% | 90% |

24 | 95% | 87.50% |

25 | 100% | 90% |

26 | 95% | 87.50% |

27 | 97.50% | 92.50% |

28 | 100% | 87.50% |

29 | 92.50% | 90% |

30 | 90% | 92.50% |

No. | MFTra | MFGauss |
---|---|---|

1 | 87.5% | 87.5% |

2 | 87.5% | 92.5% |

3 | 87.5% | 92.5% |

4 | 85% | 87.5% |

5 | 87.5% | 90% |

6 | 85% | 90% |

7 | 87.5% | 90% |

8 | 87.5% | 90% |

9 | 87.5% | 90% |

10 | 85% | 90% |

11 | 90% | 87.5% |

12 | 90% | 90% |

13 | 90% | 90% |

14 | 90% | 90% |

15 | 87.5% | 90% |

16 | 85% | 90% |

17 | 87.5% | 90% |

18 | 90% | 90% |

19 | 87.5% | 87.5% |

20 | 90% | 87.5% |

21 | 90% | 92.5% |

22 | 97.5% | 90% |

23 | 90% | 90% |

24 | 90% | 90% |

25 | 87.5% | 87.5% |

26 | 87.5% | 87.5% |

27 | 90% | 90% |

28 | 87.5% | 90% |

29 | 87.5% | 90% |

30 | 87.5% | 90% |

Trapezoidal Membership Functions | Gaussians Membership Functions | ||
---|---|---|---|

BSA | CSA | BSA | CSA |

94.58% | 88.33% | 90.33% | 89.66% |

Parameter | Value |
---|---|

M | 202 |

pop | 100 |

DimT | 56 |

FQ | 15 |

c1 | 1.67 |

c2 | 1.67 |

a1 | 0.6 |

a2 | 0.6 |

Parameter | Value |
---|---|

M | 87 |

pop | 230 |

DimG | 28 |

FQ | 15 |

c1 | 3.4 |

c2 | 3.4 |

a1 | 1.5 |

a2 | 1.5 |

No | MFTra | MFGauss |
---|---|---|

1 | 92.50% | 92.50% |

2 | 92.50% | 90% |

3 | 92.50% | 87.50% |

4 | 95% | 90% |

5 | 92.50% | 87.50% |

6 | 92.50% | 92.50% |

7 | 92.50% | 95% |

8 | 92.50% | 90% |

9 | 92.50% | 92.50% |

10 | 92.50% | 95% |

11 | 92.50% | 97.50% |

12 | 92.50% | 87.50% |

13 | 92.50% | 95% |

14 | 92.50% | 92.50% |

15 | 95% | 92.50% |

16 | 92.50% | 85% |

17 | 95% | 92.50% |

18 | 92.50% | 92.50% |

19 | 92.50% | 92.50% |

20 | 95% | 97.50% |

21 | 97.50% | 90% |

22 | 87.50% | 97.50% |

23 | 92.50% | 100% |

24 | 92.50% | 92.50% |

25 | 92.50% | 87.50% |

26 | 92.50% | 90% |

27 | 92.50% | 97.50% |

28 | 92.50% | 100% |

29 | 92.50% | 97.50% |

30 | 95% | 92.50% |

No | MFTra | MFGauss |
---|---|---|

1 | 92.50% | 92.50% |

2 | 90% | 92.50% |

3 | 90% | 92.50% |

4 | 90% | 95% |

5 | 90% | 92.50% |

6 | 90% | 95% |

7 | 90% | 92.50% |

8 | 90% | 92.50% |

9 | 90% | 92.50% |

10 | 90% | 87.50% |

11 | 90% | 95% |

12 | 90% | 92.50% |

13 | 90% | 95% |

14 | 90% | 92.50% |

15 | 90% | 95% |

16 | 90% | 92.50% |

17 | 90% | 95% |

18 | 90% | 95% |

19 | 90% | 92.50% |

20 | 90% | 95% |

21 | 90% | 95% |

22 | 90% | 95% |

23 | 90% | 92.50% |

24 | 90% | 92.50% |

25 | 90% | 87.50% |

26 | 90% | 92.50% |

27 | 90% | 92.50% |

28 | 90% | 92.50% |

29 | 90% | 87.50% |

30 | 90% | 92.50% |

Tapezoidal Membership Functions | Gaussians Membership Functions | ||
---|---|---|---|

BSA | CSA | BSA | CSA |

92.92% | 92.83% | 92.78% | 88.33% |

No. | Age | Pulse | Real | Trapezoidal | Gaussian | ||
---|---|---|---|---|---|---|---|

No Optimized FIS | Optimized FIS | No Optimized FIS | Optimized FIS | ||||

1 | 25 | 84 | Excnt | Excnt | Excnt | Excnt | Excnt |

2 | 83 | 95 | AboveAV | Excnt | AboveAV | AboveAV | AboveAV |

3 | 15 | 114 | AboveAV | AboveAV | AboveAV | AboveAV | AboveAV |

4 | 34 | 72 | Excnt | Excnt | Excnt | Excnt | Excnt |

5 | 42 | 135 | AboveAV | AboveAV | AboveAV | AboveAV | AboveAV |

6 | 91 | 97 | VHigh | AboveAV | VHigh | AboveAV | VHigh |

7 | 45 | 60 | BelowAV | BelowAV | BelowAV | Excnt | Excnt |

8 | 56 | 87 | Excnt | Excnt | Excnt | AboveAV | Exc |

9 | 75 | 102 | VHigh | VHigh | VHigh | VHigh | VHigh |

10 | 9 | 120 | Excnt | Excnt | Excnt | Excnt | Excnt |

11 | 14 | 92 | Excnt | Excnt | Excnt | Excnt | Excnt |

12 | 38 | 78 | Excnt | Excnt | Excnt | Excnt | Excnt |

13 | 29 | 80 | Excnt | Excnt | Excnt | Excnt | Excnt |

14 | 21 | 62 | Excnt | Excnt | Excnt | Excnt | Excnt |

15 | 6 | 115 | Excnt | Excnt | Excnt | Excnt | Excnt |

No. | Age | Pulse | Real | Trapezoidal | Gaussian | ||
---|---|---|---|---|---|---|---|

No Optimized FIS | Optimized FIS | No Optimized FIS | Optimized FIS | ||||

1 | 46 | 75 | Excnt | Excnt | Excnt | Excnt | Excnt |

2 | 28 | 88 | Excnt | Excnt | Excnt | Excnt | Excnt |

3 | 30 | 69 | Excnt | Excnt | Excnt | Excnt | Excnt |

4 | 33 | 59 | BelowAV | BelowAV | BelowAV | Excnt | BelowAV |

5 | 31 | 68 | Excnt | Excnt | Excnt | Excnt | Excnt |

6 | 32 | 71 | Excnt | Excnt | Excnt | Excnt | Excnt |

7 | 32 | 66 | Excnt | Excnt | Excnt | Excnt | Excnt |

8 | 27 | 66 | Excnt | Excnt | Excnt | Excnt | Excnt |

9 | 31 | 72 | Excnt | Excnt | Excnt | Excnt | Excnt |

10 | 30 | 76 | Excnt | Excnt | Excnt | Excnt | Excnt |

11 | 32 | 81 | Excnt | Excnt | Excnt | Excnt | Excnt |

12 | 28 | 76 | Excnt | Excnt | Excnt | Excnt | Excnt |

13 | 31 | 85 | Excnt | Excnt | Excnt | Excnt | Excnt |

14 | 26 | 85 | Excnt | Excnt | Excnt | Excnt | Excnt |

15 | 31 | 77 | Excnt | Excnt | Excnt | Excnt | Excnt |

16 | 29 | 77 | Excnt | Excnt | Excnt | Excnt | Excnt |

17 | 45 | 69 | Excnt | Excnt | Excnt | Excnt | Excnt |

18 | 27 | 63 | Excnt | Excnt | Excnt | Excnt | Excnt |

19 | 25 | 107 | AvobeAV | AvobeAV | AvobeAV | AvobeAV | AvobeAV |

20 | 25 | 95 | AvobeAV | Excnt | Excnt | Excnt | AvobeAV |

No. | Age | Pulse | Real | Trapezoidal | Gaussian | ||
---|---|---|---|---|---|---|---|

No Optimized FIS | Optimized FIS | No Optimized FIS | Optimized FIS | ||||

1 | 25 | 84 | Excnt | Excnt | Excnt | Excnt | Excnt |

2 | 83 | 95 | AvobeAV | AvobeAV | AvobeAV | AvobeAV | AvobeAV |

3 | 15 | 114 | AvobeAV | AvobeAV | AvobeAV | AvobeAV | AvobeAV |

4 | 34 | 72 | V | Exc | Exc | Exc | Exc |

5 | 42 | 135 | AvobeAV | AvobeAV | AvobeAV | AvobeAV | AvobeAV |

6 | 91 | 97 | VHigh | AboveAV | VHigh | AboveAV | VHigh |

7 | 45 | 60 | BelowAV | Excnt | Excnt | Excnt | BelowAV |

8 | 56 | 87 | Excnt | Excnt | Excnt | AboveAV | Excnt |

9 | 75 | 102 | VHigh | VHigh | VHigh | VHigh | VHigh |

10 | 9 | 120 | Excnt | Excnt | Excnt | Excnt | Excnt |

11 | 14 | 92 | Excnt | Excnt | AboveAV | Excnt | Excnt |

12 | 38 | 78 | Excnt | Excnt | Excnt | Excnt | Excnt |

13 | 29 | 80 | Excnt | Excnt | Excnt | Excnt | Excnt |

14 | 21 | 62 | Excnt | Excnt | Excnt | Excnt | Excnt |

15 | 6 | 115 | Excnt | Excnt | Excnt | Excnt | Excnt |

No. | Age | Pulse | Real | Trapezoidal | Gaussian | ||
---|---|---|---|---|---|---|---|

No Optimized FIS | Optimized FIS | No Optimized FIS | Optimized FIS | ||||

1 | 46 | 75 | Excnt | Excnt | Excnt | Excnt | Excnt |

2 | 28 | 88 | Excnt | Excnt | Excnt | Excnt | Excnt |

3 | 30 | 69 | Excnt | Excnt | Excnt | Excnt | Excnt |

4 | 33 | 59 | BelowAV | Excnt | BelowAV | Excnt | BelowAV |

5 | 31 | 68 | Excnt | Excnt | Excnt | Excnt | Excnt |

6 | 32 | 71 | Excnt | Excnt | Excnt | Excnt | Excnt |

7 | 32 | 66 | Excnt | Excnt | Excnt | Excnt | Excnt |

8 | 27 | 66 | Excnt | Excnt | Excnt | Excnt | Excnt |

9 | 31 | 72 | Excnt | Excnt | Excnt | Excnt | Excnt |

10 | 30 | 76 | Excnt | Excnt | Excnt | Excnt | Excnt |

11 | 32 | 81 | Excnt | Excnt | Excnt | Excnt | Excnt |

12 | 28 | 76 | Excnt | Excnt | Excnt | Excnt | Excnt |

13 | 31 | 85 | Excnt | Excnt | Excnt | Excnt | Excnt |

14 | 26 | 85 | Excnt | Excnt | Excnt | Excnt | Excnt |

15 | 31 | 77 | Excnt | Excnt | Excnt | Excnt | Excnt |

16 | 29 | 77 | Excnt | Excnt | Excnt | Excnt | Excnt |

17 | 45 | 69 | Excnt | Excnt | Excnt | Excnt | Excnt |

18 | 27 | 63 | Excnt | Excnt | Excnt | Excnt | Excnt |

19 | 25 | 107 | AboveAV | AboveAV | AboveAV | AboveAV | AboveAV |

20 | 25 | 95 | AboveAV | AboveAV | AboveAV | Excnt | AboveAV |

FST1 No Optimized | FST1 Optimized | FST2 No Optimized | FST2 Optimized | ||||
---|---|---|---|---|---|---|---|

Trap | Gauss | Trap | Gauss | Trap | Gauss | Trap | Gauss |

86.6% | 80% | 100% | 93.3% | 86.6% | 80% | 86.6% | 100% |

FST1 No Optimized | FST1 Optimized | FST2 No Optimized | FST2 Optimized | ||||
---|---|---|---|---|---|---|---|

Trap | Gauss | Trap | Gauss | Trap | Gauss | Trap | Gauss |

95% | 90% | 95% | 100% | 95% | 90% | 100% | 100% |

© 2018 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**

Miramontes, I.; Guzman, J.C.; Melin, P.; Prado-Arechiga, G.
Optimal Design of Interval Type-2 Fuzzy Heart Rate Level Classification Systems Using the Bird Swarm Algorithm. *Algorithms* **2018**, *11*, 206.
https://doi.org/10.3390/a11120206

**AMA Style**

Miramontes I, Guzman JC, Melin P, Prado-Arechiga G.
Optimal Design of Interval Type-2 Fuzzy Heart Rate Level Classification Systems Using the Bird Swarm Algorithm. *Algorithms*. 2018; 11(12):206.
https://doi.org/10.3390/a11120206

**Chicago/Turabian Style**

Miramontes, Ivette, Juan Carlos Guzman, Patricia Melin, and German Prado-Arechiga.
2018. "Optimal Design of Interval Type-2 Fuzzy Heart Rate Level Classification Systems Using the Bird Swarm Algorithm" *Algorithms* 11, no. 12: 206.
https://doi.org/10.3390/a11120206